summaryrefslogtreecommitdiff
path: root/lib/Analysis/ScalarEvolution.cpp
blob: 0512429ad8ad383efa030cf290e7b778f948cd0d (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
3042
3043
3044
3045
3046
3047
3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116
3117
3118
3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
3164
3165
3166
3167
3168
3169
3170
3171
3172
3173
3174
3175
3176
3177
3178
3179
3180
3181
3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
3206
3207
3208
3209
3210
3211
3212
3213
3214
3215
3216
3217
3218
3219
3220
3221
3222
3223
3224
3225
3226
3227
3228
3229
3230
3231
3232
3233
3234
3235
3236
3237
3238
3239
3240
3241
3242
3243
3244
3245
3246
3247
3248
3249
3250
3251
3252
3253
3254
3255
3256
3257
3258
3259
3260
3261
3262
3263
3264
3265
3266
3267
3268
3269
3270
3271
3272
3273
3274
3275
3276
3277
3278
3279
3280
3281
3282
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
3326
3327
3328
3329
3330
3331
3332
3333
3334
3335
3336
3337
3338
3339
3340
3341
3342
3343
3344
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
3360
3361
3362
3363
3364
3365
3366
3367
3368
3369
3370
3371
3372
3373
3374
3375
3376
3377
3378
3379
3380
3381
3382
3383
3384
3385
3386
3387
3388
3389
3390
3391
3392
3393
3394
3395
3396
3397
3398
3399
3400
3401
3402
3403
3404
3405
3406
3407
3408
3409
3410
3411
3412
3413
3414
3415
3416
3417
3418
3419
3420
3421
3422
3423
3424
3425
3426
3427
3428
3429
3430
3431
3432
3433
3434
3435
3436
3437
3438
3439
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
3450
3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
3466
3467
3468
3469
3470
3471
3472
3473
3474
3475
3476
3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
3489
3490
3491
3492
3493
3494
3495
3496
3497
3498
3499
3500
3501
3502
3503
3504
3505
3506
3507
3508
3509
3510
3511
3512
3513
3514
3515
3516
3517
3518
3519
3520
3521
3522
3523
3524
3525
3526
3527
3528
3529
3530
3531
3532
3533
3534
3535
3536
3537
3538
3539
3540
3541
3542
3543
3544
3545
3546
3547
3548
3549
3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
3568
3569
3570
3571
3572
3573
3574
3575
3576
3577
3578
3579
3580
3581
3582
3583
3584
3585
3586
3587
3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
3601
3602
3603
3604
3605
3606
3607
3608
3609
3610
3611
3612
3613
3614
3615
3616
3617
3618
3619
3620
3621
3622
3623
3624
3625
3626
3627
3628
3629
3630
3631
3632
3633
3634
3635
3636
3637
3638
3639
3640
3641
3642
3643
3644
3645
3646
3647
3648
3649
3650
3651
3652
3653
3654
3655
3656
3657
3658
3659
3660
3661
3662
3663
3664
3665
3666
3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684
3685
3686
3687
3688
3689
3690
3691
3692
3693
3694
3695
3696
3697
3698
3699
3700
3701
3702
3703
3704
3705
3706
3707
3708
3709
3710
3711
3712
3713
3714
3715
3716
3717
3718
3719
3720
3721
3722
3723
3724
3725
3726
3727
3728
3729
3730
3731
3732
3733
3734
3735
3736
3737
3738
3739
3740
3741
3742
3743
3744
3745
3746
3747
3748
3749
3750
3751
3752
3753
3754
3755
3756
3757
3758
3759
3760
3761
3762
3763
3764
3765
3766
3767
3768
3769
3770
3771
3772
3773
3774
3775
3776
3777
3778
3779
3780
3781
3782
3783
3784
3785
3786
3787
3788
3789
3790
3791
3792
3793
3794
3795
3796
3797
3798
3799
3800
3801
3802
3803
3804
3805
3806
3807
3808
3809
3810
3811
3812
3813
3814
3815
3816
3817
3818
3819
3820
3821
3822
3823
3824
3825
3826
3827
3828
3829
3830
3831
3832
3833
3834
3835
3836
3837
3838
3839
3840
3841
3842
3843
3844
3845
3846
3847
3848
3849
3850
3851
3852
3853
3854
3855
3856
3857
3858
3859
3860
3861
3862
3863
3864
3865
3866
3867
3868
3869
3870
3871
3872
3873
3874
3875
3876
3877
3878
3879
3880
3881
3882
3883
3884
3885
3886
3887
3888
3889
3890
3891
3892
3893
3894
3895
3896
3897
3898
3899
3900
3901
3902
3903
3904
3905
3906
3907
3908
3909
3910
3911
3912
3913
3914
3915
3916
3917
3918
3919
3920
3921
3922
3923
3924
3925
3926
3927
3928
3929
3930
3931
3932
3933
3934
3935
3936
3937
3938
3939
3940
3941
3942
3943
3944
3945
3946
3947
3948
3949
3950
3951
3952
3953
3954
3955
3956
3957
3958
3959
3960
3961
3962
3963
3964
3965
3966
3967
3968
3969
3970
3971
3972
3973
3974
3975
3976
3977
3978
3979
3980
3981
3982
3983
3984
3985
3986
3987
3988
3989
3990
3991
3992
3993
3994
3995
3996
3997
3998
3999
4000
4001
4002
4003
4004
4005
4006
4007
4008
4009
4010
4011
4012
4013
4014
4015
4016
4017
4018
4019
4020
4021
4022
4023
4024
4025
4026
4027
4028
4029
4030
4031
4032
4033
4034
4035
4036
4037
4038
4039
4040
4041
4042
4043
4044
4045
4046
4047
4048
4049
4050
4051
4052
4053
4054
4055
4056
4057
4058
4059
4060
4061
4062
4063
4064
4065
4066
4067
4068
4069
4070
4071
4072
4073
4074
4075
4076
4077
4078
4079
4080
4081
4082
4083
4084
4085
4086
4087
4088
4089
4090
4091
4092
4093
4094
4095
4096
4097
4098
4099
4100
4101
4102
4103
4104
4105
4106
4107
4108
4109
4110
4111
4112
4113
4114
4115
4116
4117
4118
4119
4120
4121
4122
4123
4124
4125
4126
4127
4128
4129
4130
4131
4132
4133
4134
4135
4136
4137
4138
4139
4140
4141
4142
4143
4144
4145
4146
4147
4148
4149
4150
4151
4152
4153
4154
4155
4156
4157
4158
4159
4160
4161
4162
4163
4164
4165
4166
4167
4168
4169
4170
4171
4172
4173
4174
4175
4176
4177
4178
4179
4180
4181
4182
4183
4184
4185
4186
4187
4188
4189
4190
4191
4192
4193
4194
4195
4196
4197
4198
4199
4200
4201
4202
4203
4204
4205
4206
4207
4208
4209
4210
4211
4212
4213
4214
4215
4216
4217
4218
4219
4220
4221
4222
4223
4224
4225
4226
4227
4228
4229
4230
4231
4232
4233
4234
4235
4236
4237
4238
4239
4240
4241
4242
4243
4244
4245
4246
4247
4248
4249
4250
4251
4252
4253
4254
4255
4256
4257
4258
4259
4260
4261
4262
4263
4264
4265
4266
4267
4268
4269
4270
4271
4272
4273
4274
4275
4276
4277
4278
4279
4280
4281
4282
4283
4284
4285
4286
4287
4288
4289
4290
4291
4292
4293
4294
4295
4296
4297
4298
4299
4300
4301
4302
4303
4304
4305
4306
4307
4308
4309
4310
4311
4312
4313
4314
4315
4316
4317
4318
4319
4320
4321
4322
4323
4324
4325
4326
4327
4328
4329
4330
4331
4332
4333
4334
4335
4336
4337
4338
4339
4340
4341
4342
4343
4344
4345
4346
4347
4348
4349
4350
4351
4352
4353
4354
4355
4356
4357
4358
4359
4360
4361
4362
4363
4364
4365
4366
4367
4368
4369
4370
4371
4372
4373
4374
4375
4376
4377
4378
4379
4380
4381
4382
4383
4384
4385
4386
4387
4388
4389
4390
4391
4392
4393
4394
4395
4396
4397
4398
4399
4400
4401
4402
4403
4404
4405
4406
4407
4408
4409
4410
4411
4412
4413
4414
4415
4416
4417
4418
4419
4420
4421
4422
4423
4424
4425
4426
4427
4428
4429
4430
4431
4432
4433
4434
4435
4436
4437
4438
4439
4440
4441
4442
4443
4444
4445
4446
4447
4448
4449
4450
4451
4452
4453
4454
4455
4456
4457
4458
4459
4460
4461
4462
4463
4464
4465
4466
4467
4468
4469
4470
4471
4472
4473
4474
4475
4476
4477
4478
4479
4480
4481
4482
4483
4484
4485
4486
4487
4488
4489
4490
4491
4492
4493
4494
4495
4496
4497
4498
4499
4500
4501
4502
4503
4504
4505
4506
4507
4508
4509
4510
4511
4512
4513
4514
4515
4516
4517
4518
4519
4520
4521
4522
4523
4524
4525
4526
4527
4528
4529
4530
4531
4532
4533
4534
4535
4536
4537
4538
4539
4540
4541
4542
4543
4544
4545
4546
4547
4548
4549
4550
4551
4552
4553
4554
4555
4556
4557
4558
4559
4560
4561
4562
4563
4564
4565
4566
4567
4568
4569
4570
4571
4572
4573
4574
4575
4576
4577
4578
4579
4580
4581
4582
4583
4584
4585
4586
4587
4588
4589
4590
4591
4592
4593
4594
4595
4596
4597
4598
4599
4600
4601
4602
4603
4604
4605
4606
4607
4608
4609
4610
4611
4612
4613
4614
4615
4616
4617
4618
4619
4620
4621
4622
4623
4624
4625
4626
4627
4628
4629
4630
4631
4632
4633
4634
4635
4636
4637
4638
4639
4640
4641
4642
4643
4644
4645
4646
4647
4648
4649
4650
4651
4652
4653
4654
4655
4656
4657
4658
4659
4660
4661
4662
4663
4664
4665
4666
4667
4668
4669
4670
4671
4672
4673
4674
4675
4676
4677
4678
4679
4680
4681
4682
4683
4684
4685
4686
4687
4688
4689
4690
4691
4692
4693
4694
4695
4696
4697
4698
4699
4700
4701
4702
4703
4704
4705
4706
4707
4708
4709
4710
4711
4712
4713
4714
4715
4716
4717
4718
4719
4720
4721
4722
4723
4724
4725
4726
4727
4728
4729
4730
4731
4732
4733
4734
4735
4736
4737
4738
4739
4740
4741
4742
4743
4744
4745
4746
4747
4748
4749
4750
4751
4752
4753
4754
4755
4756
4757
4758
4759
4760
4761
4762
4763
4764
4765
4766
4767
4768
4769
4770
4771
4772
4773
4774
4775
4776
4777
4778
4779
4780
4781
4782
4783
4784
4785
4786
4787
4788
4789
4790
4791
4792
4793
4794
4795
4796
4797
4798
4799
4800
4801
4802
4803
4804
4805
4806
4807
4808
4809
4810
4811
4812
4813
4814
4815
4816
4817
4818
4819
4820
4821
4822
4823
4824
4825
4826
4827
4828
4829
4830
4831
4832
4833
4834
4835
4836
4837
4838
4839
4840
4841
4842
4843
4844
4845
4846
4847
4848
4849
4850
4851
4852
4853
4854
4855
4856
4857
4858
4859
4860
4861
4862
4863
4864
4865
4866
4867
4868
4869
4870
4871
4872
4873
4874
4875
4876
4877
4878
4879
4880
4881
4882
4883
4884
4885
4886
4887
4888
4889
4890
4891
4892
4893
4894
4895
4896
4897
4898
4899
4900
4901
4902
4903
4904
4905
4906
4907
4908
4909
4910
4911
4912
4913
4914
4915
4916
4917
4918
4919
4920
4921
4922
4923
4924
4925
4926
4927
4928
4929
4930
4931
4932
4933
4934
4935
4936
4937
4938
4939
4940
4941
4942
4943
4944
4945
4946
4947
4948
4949
4950
4951
4952
4953
4954
4955
4956
4957
4958
4959
4960
4961
4962
4963
4964
4965
4966
4967
4968
4969
4970
4971
4972
4973
4974
4975
4976
4977
4978
4979
4980
4981
4982
4983
4984
4985
4986
4987
4988
4989
4990
4991
4992
4993
4994
4995
4996
4997
4998
4999
5000
5001
5002
5003
5004
5005
5006
5007
5008
5009
5010
5011
5012
5013
5014
5015
5016
5017
5018
5019
5020
5021
5022
5023
5024
5025
5026
5027
5028
5029
5030
5031
5032
5033
5034
5035
5036
5037
5038
5039
5040
5041
5042
5043
5044
5045
5046
5047
5048
5049
5050
5051
5052
5053
5054
5055
5056
5057
5058
5059
5060
5061
5062
5063
5064
5065
5066
5067
5068
5069
5070
5071
5072
5073
5074
5075
5076
5077
5078
5079
5080
5081
5082
5083
5084
5085
5086
5087
5088
5089
5090
5091
5092
5093
5094
5095
5096
5097
5098
5099
5100
5101
5102
5103
5104
5105
5106
5107
5108
5109
5110
5111
5112
5113
5114
5115
5116
5117
5118
5119
5120
5121
5122
5123
5124
5125
5126
5127
5128
5129
5130
5131
5132
5133
5134
5135
5136
5137
5138
5139
5140
5141
5142
5143
5144
5145
5146
5147
5148
5149
5150
5151
5152
5153
5154
5155
5156
5157
5158
5159
5160
5161
5162
5163
5164
5165
5166
5167
5168
5169
5170
5171
5172
5173
5174
5175
5176
5177
5178
5179
5180
5181
5182
5183
5184
5185
5186
5187
5188
5189
5190
5191
5192
5193
5194
5195
5196
5197
5198
5199
5200
5201
5202
5203
5204
5205
5206
5207
5208
5209
5210
5211
5212
5213
5214
5215
5216
5217
5218
5219
5220
5221
5222
5223
5224
5225
5226
5227
5228
5229
5230
5231
5232
5233
5234
5235
5236
5237
5238
5239
5240
5241
5242
5243
5244
5245
5246
5247
5248
5249
5250
5251
5252
5253
5254
5255
5256
5257
5258
5259
5260
5261
5262
5263
5264
5265
5266
5267
5268
5269
5270
5271
5272
5273
5274
5275
5276
5277
5278
5279
5280
5281
5282
5283
5284
5285
5286
5287
5288
5289
5290
5291
5292
5293
5294
5295
5296
5297
5298
5299
5300
5301
5302
5303
5304
5305
5306
5307
5308
5309
5310
5311
5312
5313
5314
5315
5316
5317
5318
5319
5320
5321
5322
5323
5324
5325
5326
5327
5328
5329
5330
5331
5332
5333
5334
5335
5336
5337
5338
5339
5340
5341
5342
5343
5344
5345
5346
5347
5348
5349
5350
5351
5352
5353
5354
5355
5356
5357
5358
5359
5360
5361
5362
5363
5364
5365
5366
5367
5368
5369
5370
5371
5372
5373
5374
5375
5376
5377
5378
5379
5380
5381
5382
5383
5384
5385
5386
5387
5388
5389
5390
5391
5392
5393
5394
5395
5396
5397
5398
5399
5400
5401
5402
5403
5404
5405
5406
5407
5408
5409
5410
5411
5412
5413
5414
5415
5416
5417
5418
5419
5420
5421
5422
5423
5424
5425
5426
5427
5428
5429
5430
5431
5432
5433
5434
5435
5436
5437
5438
5439
5440
5441
5442
5443
5444
5445
5446
5447
5448
5449
5450
5451
5452
5453
5454
5455
5456
5457
5458
5459
5460
5461
5462
5463
5464
5465
5466
5467
5468
5469
5470
5471
5472
5473
5474
5475
5476
5477
5478
5479
5480
5481
5482
5483
5484
5485
5486
5487
5488
5489
5490
5491
5492
5493
5494
5495
5496
5497
5498
5499
5500
5501
5502
5503
5504
5505
5506
5507
5508
5509
5510
5511
5512
5513
5514
5515
5516
5517
5518
5519
5520
5521
5522
5523
5524
5525
5526
5527
5528
5529
5530
5531
5532
5533
5534
5535
5536
5537
5538
5539
5540
5541
5542
5543
5544
5545
5546
5547
5548
5549
5550
5551
5552
5553
5554
5555
5556
5557
5558
5559
5560
5561
5562
5563
5564
5565
5566
5567
5568
5569
5570
5571
5572
5573
5574
5575
5576
5577
5578
5579
5580
5581
5582
5583
5584
5585
5586
5587
5588
5589
5590
5591
5592
5593
5594
5595
5596
5597
5598
5599
5600
5601
5602
5603
5604
5605
5606
5607
5608
5609
5610
5611
5612
5613
5614
5615
5616
5617
5618
5619
5620
5621
5622
5623
5624
5625
5626
5627
5628
5629
5630
5631
5632
5633
5634
5635
5636
5637
5638
5639
5640
5641
5642
5643
5644
5645
5646
5647
5648
5649
5650
5651
5652
5653
5654
5655
5656
5657
5658
5659
5660
5661
5662
5663
5664
5665
5666
5667
5668
5669
5670
5671
5672
5673
5674
5675
5676
5677
5678
5679
5680
5681
5682
5683
5684
5685
5686
5687
5688
5689
5690
5691
5692
5693
5694
5695
5696
5697
5698
5699
5700
5701
5702
5703
5704
5705
5706
5707
5708
5709
5710
5711
5712
5713
5714
5715
5716
5717
5718
5719
5720
5721
5722
5723
5724
5725
5726
5727
5728
5729
5730
5731
5732
5733
5734
5735
5736
5737
5738
5739
5740
5741
5742
5743
5744
5745
5746
5747
5748
5749
5750
5751
5752
5753
5754
5755
5756
5757
5758
5759
5760
5761
5762
5763
5764
5765
5766
5767
5768
5769
5770
5771
5772
5773
5774
5775
5776
5777
5778
5779
5780
5781
5782
5783
5784
5785
5786
5787
5788
5789
5790
5791
5792
5793
5794
5795
5796
5797
5798
5799
5800
5801
5802
5803
5804
5805
5806
5807
5808
5809
5810
5811
5812
5813
5814
5815
5816
5817
5818
5819
5820
5821
5822
5823
5824
5825
5826
5827
5828
5829
5830
5831
5832
5833
5834
5835
5836
5837
5838
5839
5840
5841
5842
5843
5844
5845
5846
5847
5848
5849
5850
5851
5852
5853
5854
5855
5856
5857
5858
5859
5860
5861
5862
5863
5864
5865
5866
5867
5868
5869
5870
5871
5872
5873
5874
5875
5876
5877
5878
5879
5880
5881
5882
5883
5884
5885
5886
5887
5888
5889
5890
5891
5892
5893
5894
5895
5896
5897
5898
5899
5900
5901
5902
5903
5904
5905
5906
5907
5908
5909
5910
5911
5912
5913
5914
5915
5916
5917
5918
5919
5920
5921
5922
5923
5924
5925
5926
5927
5928
5929
5930
5931
5932
5933
5934
5935
5936
5937
5938
5939
5940
5941
5942
5943
5944
5945
5946
5947
5948
5949
5950
5951
5952
5953
5954
5955
5956
5957
5958
5959
5960
5961
5962
5963
5964
5965
5966
5967
5968
5969
5970
5971
5972
5973
5974
5975
5976
5977
5978
5979
5980
5981
5982
5983
5984
5985
5986
5987
5988
5989
5990
5991
5992
5993
5994
5995
5996
5997
5998
5999
6000
6001
6002
6003
6004
6005
6006
6007
6008
6009
6010
6011
6012
6013
6014
6015
6016
6017
6018
6019
6020
6021
6022
6023
6024
6025
6026
6027
6028
6029
6030
6031
6032
6033
6034
6035
6036
6037
6038
6039
6040
6041
6042
6043
6044
6045
6046
6047
6048
6049
6050
6051
6052
6053
6054
6055
6056
6057
6058
6059
6060
6061
6062
6063
6064
6065
6066
6067
6068
6069
6070
6071
6072
6073
6074
6075
6076
6077
6078
6079
6080
6081
6082
6083
6084
6085
6086
6087
6088
6089
6090
6091
6092
6093
6094
6095
6096
6097
6098
6099
6100
6101
6102
6103
6104
6105
6106
6107
6108
6109
6110
6111
6112
6113
6114
6115
6116
6117
6118
6119
6120
6121
6122
6123
6124
6125
6126
6127
6128
6129
6130
6131
6132
6133
6134
6135
6136
6137
6138
6139
6140
6141
6142
6143
6144
6145
6146
6147
6148
6149
6150
6151
6152
6153
6154
6155
6156
6157
6158
6159
6160
6161
6162
6163
6164
6165
6166
6167
6168
6169
6170
6171
6172
6173
6174
6175
6176
6177
6178
6179
6180
6181
6182
6183
6184
6185
6186
6187
6188
6189
6190
6191
6192
6193
6194
6195
6196
6197
6198
6199
6200
6201
6202
6203
6204
6205
6206
6207
6208
6209
6210
6211
6212
6213
6214
6215
6216
6217
6218
6219
6220
6221
6222
6223
6224
6225
6226
6227
6228
6229
6230
6231
6232
6233
6234
6235
6236
6237
6238
6239
6240
6241
6242
6243
6244
6245
6246
6247
6248
6249
6250
6251
6252
6253
6254
6255
6256
6257
6258
6259
6260
6261
6262
6263
6264
6265
6266
6267
6268
6269
6270
6271
6272
6273
6274
6275
6276
6277
6278
6279
6280
6281
6282
6283
6284
6285
6286
6287
6288
6289
6290
6291
6292
6293
6294
6295
6296
6297
6298
6299
6300
6301
6302
6303
6304
6305
6306
6307
6308
6309
6310
6311
6312
6313
6314
6315
6316
6317
6318
6319
6320
6321
6322
6323
6324
6325
6326
6327
6328
6329
6330
6331
6332
6333
6334
6335
6336
6337
6338
6339
6340
6341
6342
6343
6344
6345
6346
6347
6348
6349
6350
6351
6352
6353
6354
6355
6356
6357
6358
6359
6360
6361
6362
6363
6364
6365
6366
6367
6368
6369
6370
6371
6372
6373
6374
6375
6376
6377
6378
6379
6380
6381
6382
6383
6384
6385
6386
6387
6388
6389
6390
6391
6392
6393
6394
6395
6396
6397
6398
6399
6400
6401
6402
6403
6404
6405
6406
6407
6408
6409
6410
6411
6412
6413
6414
6415
6416
6417
6418
6419
6420
6421
6422
6423
6424
6425
6426
6427
6428
6429
6430
6431
6432
6433
6434
6435
6436
6437
6438
6439
6440
6441
6442
6443
6444
6445
6446
6447
6448
6449
6450
6451
6452
6453
6454
6455
6456
6457
6458
6459
6460
6461
6462
6463
6464
6465
6466
6467
6468
6469
6470
6471
6472
6473
6474
6475
6476
6477
6478
6479
6480
6481
6482
6483
6484
6485
6486
6487
6488
6489
6490
6491
6492
6493
6494
6495
6496
6497
6498
6499
6500
6501
6502
6503
6504
6505
6506
6507
6508
6509
6510
6511
6512
6513
6514
6515
6516
6517
6518
6519
6520
6521
6522
6523
6524
6525
6526
6527
6528
6529
6530
6531
6532
6533
6534
6535
6536
6537
6538
6539
6540
6541
6542
6543
6544
6545
6546
6547
6548
6549
6550
6551
6552
6553
6554
6555
6556
6557
6558
6559
6560
6561
6562
6563
6564
6565
6566
6567
6568
6569
6570
6571
6572
6573
6574
6575
6576
6577
6578
6579
6580
6581
6582
6583
6584
6585
6586
6587
6588
6589
6590
6591
6592
6593
6594
6595
6596
6597
6598
6599
6600
6601
6602
6603
6604
6605
6606
6607
6608
6609
6610
6611
6612
6613
6614
6615
6616
6617
6618
6619
6620
6621
6622
6623
6624
6625
6626
6627
6628
6629
6630
6631
6632
6633
6634
6635
6636
6637
6638
6639
6640
6641
6642
6643
6644
6645
6646
6647
6648
6649
6650
6651
6652
6653
6654
6655
6656
6657
6658
6659
6660
6661
6662
6663
6664
6665
6666
6667
6668
6669
6670
6671
6672
6673
6674
6675
6676
6677
6678
6679
6680
6681
6682
6683
6684
6685
6686
6687
6688
6689
6690
6691
6692
6693
6694
6695
6696
6697
6698
6699
6700
6701
6702
6703
6704
6705
6706
6707
6708
6709
6710
6711
6712
6713
6714
6715
6716
6717
6718
6719
6720
6721
6722
6723
6724
6725
6726
6727
6728
6729
6730
6731
6732
6733
6734
6735
6736
6737
6738
6739
6740
6741
6742
6743
6744
6745
6746
6747
6748
6749
6750
6751
6752
6753
6754
6755
6756
6757
6758
6759
6760
6761
6762
6763
6764
6765
6766
6767
6768
6769
6770
6771
6772
6773
6774
6775
6776
6777
6778
6779
6780
6781
6782
6783
6784
6785
6786
6787
6788
6789
6790
6791
6792
6793
6794
6795
6796
6797
6798
6799
6800
6801
6802
6803
6804
6805
6806
6807
6808
6809
6810
6811
6812
6813
6814
6815
6816
6817
6818
6819
6820
6821
6822
6823
6824
6825
6826
6827
6828
6829
6830
6831
6832
6833
6834
6835
6836
6837
6838
6839
6840
6841
6842
6843
6844
6845
6846
6847
6848
6849
6850
6851
6852
6853
6854
6855
6856
6857
6858
6859
6860
6861
6862
6863
6864
6865
6866
6867
6868
6869
6870
6871
6872
6873
6874
6875
6876
6877
6878
6879
6880
6881
6882
6883
6884
6885
6886
6887
6888
6889
6890
6891
6892
6893
6894
6895
6896
6897
6898
6899
6900
6901
6902
6903
6904
6905
6906
6907
6908
6909
6910
6911
6912
6913
6914
6915
6916
6917
6918
6919
6920
6921
6922
6923
6924
6925
6926
6927
6928
6929
6930
6931
6932
6933
6934
6935
6936
6937
6938
6939
6940
6941
6942
6943
6944
6945
6946
6947
6948
6949
6950
6951
6952
6953
6954
6955
6956
6957
6958
6959
6960
6961
6962
6963
6964
6965
6966
6967
6968
6969
6970
6971
6972
6973
6974
6975
6976
6977
6978
6979
6980
6981
6982
6983
6984
6985
6986
6987
6988
6989
6990
6991
6992
6993
6994
6995
6996
6997
6998
6999
7000
7001
7002
7003
7004
7005
7006
7007
7008
7009
7010
7011
7012
7013
7014
7015
7016
7017
7018
7019
7020
7021
7022
7023
7024
7025
7026
7027
7028
7029
7030
7031
7032
7033
7034
7035
7036
7037
7038
7039
7040
7041
7042
7043
7044
7045
7046
7047
7048
7049
7050
7051
7052
7053
7054
7055
7056
7057
7058
7059
7060
7061
7062
7063
7064
7065
7066
7067
7068
7069
7070
7071
7072
7073
7074
7075
7076
7077
7078
7079
7080
7081
7082
7083
7084
7085
7086
7087
7088
7089
7090
7091
7092
7093
7094
7095
7096
7097
7098
7099
7100
7101
7102
7103
7104
7105
7106
7107
7108
7109
7110
7111
7112
7113
7114
7115
7116
7117
7118
7119
7120
7121
//===- ScalarEvolution.cpp - Scalar Evolution Analysis ----------*- C++ -*-===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// This file contains the implementation of the scalar evolution analysis
// engine, which is used primarily to analyze expressions involving induction
// variables in loops.
//
// There are several aspects to this library.  First is the representation of
// scalar expressions, which are represented as subclasses of the SCEV class.
// These classes are used to represent certain types of subexpressions that we
// can handle. We only create one SCEV of a particular shape, so
// pointer-comparisons for equality are legal.
//
// One important aspect of the SCEV objects is that they are never cyclic, even
// if there is a cycle in the dataflow for an expression (ie, a PHI node).  If
// the PHI node is one of the idioms that we can represent (e.g., a polynomial
// recurrence) then we represent it directly as a recurrence node, otherwise we
// represent it as a SCEVUnknown node.
//
// In addition to being able to represent expressions of various types, we also
// have folders that are used to build the *canonical* representation for a
// particular expression.  These folders are capable of using a variety of
// rewrite rules to simplify the expressions.
//
// Once the folders are defined, we can implement the more interesting
// higher-level code, such as the code that recognizes PHI nodes of various
// types, computes the execution count of a loop, etc.
//
// TODO: We should use these routines and value representations to implement
// dependence analysis!
//
//===----------------------------------------------------------------------===//
//
// There are several good references for the techniques used in this analysis.
//
//  Chains of recurrences -- a method to expedite the evaluation
//  of closed-form functions
//  Olaf Bachmann, Paul S. Wang, Eugene V. Zima
//
//  On computational properties of chains of recurrences
//  Eugene V. Zima
//
//  Symbolic Evaluation of Chains of Recurrences for Loop Optimization
//  Robert A. van Engelen
//
//  Efficient Symbolic Analysis for Optimizing Compilers
//  Robert A. van Engelen
//
//  Using the chains of recurrences algebra for data dependence testing and
//  induction variable substitution
//  MS Thesis, Johnie Birch
//
//===----------------------------------------------------------------------===//

#define DEBUG_TYPE "scalar-evolution"
#include "llvm/Analysis/ScalarEvolution.h"
#include "llvm/ADT/STLExtras.h"
#include "llvm/ADT/SmallPtrSet.h"
#include "llvm/ADT/Statistic.h"
#include "llvm/Analysis/ConstantFolding.h"
#include "llvm/Analysis/Dominators.h"
#include "llvm/Analysis/InstructionSimplify.h"
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/Analysis/ScalarEvolutionExpressions.h"
#include "llvm/Analysis/ValueTracking.h"
#include "llvm/Assembly/Writer.h"
#include "llvm/IR/Constants.h"
#include "llvm/IR/DataLayout.h"
#include "llvm/IR/DerivedTypes.h"
#include "llvm/IR/GlobalAlias.h"
#include "llvm/IR/GlobalVariable.h"
#include "llvm/IR/Instructions.h"
#include "llvm/IR/LLVMContext.h"
#include "llvm/IR/Operator.h"
#include "llvm/Support/CommandLine.h"
#include "llvm/Support/ConstantRange.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/ErrorHandling.h"
#include "llvm/Support/GetElementPtrTypeIterator.h"
#include "llvm/Support/InstIterator.h"
#include "llvm/Support/MathExtras.h"
#include "llvm/Support/raw_ostream.h"
#include "llvm/Target/TargetLibraryInfo.h"
#include <algorithm>
using namespace llvm;

STATISTIC(NumArrayLenItCounts,
          "Number of trip counts computed with array length");
STATISTIC(NumTripCountsComputed,
          "Number of loops with predictable loop counts");
STATISTIC(NumTripCountsNotComputed,
          "Number of loops without predictable loop counts");
STATISTIC(NumBruteForceTripCountsComputed,
          "Number of loops with trip counts computed by force");

static cl::opt<unsigned>
MaxBruteForceIterations("scalar-evolution-max-iterations", cl::ReallyHidden,
                        cl::desc("Maximum number of iterations SCEV will "
                                 "symbolically execute a constant "
                                 "derived loop"),
                        cl::init(100));

// FIXME: Enable this with XDEBUG when the test suite is clean.
static cl::opt<bool>
VerifySCEV("verify-scev",
           cl::desc("Verify ScalarEvolution's backedge taken counts (slow)"));

INITIALIZE_PASS_BEGIN(ScalarEvolution, "scalar-evolution",
                "Scalar Evolution Analysis", false, true)
INITIALIZE_PASS_DEPENDENCY(LoopInfo)
INITIALIZE_PASS_DEPENDENCY(DominatorTree)
INITIALIZE_PASS_DEPENDENCY(TargetLibraryInfo)
INITIALIZE_PASS_END(ScalarEvolution, "scalar-evolution",
                "Scalar Evolution Analysis", false, true)
char ScalarEvolution::ID = 0;

//===----------------------------------------------------------------------===//
//                           SCEV class definitions
//===----------------------------------------------------------------------===//

//===----------------------------------------------------------------------===//
// Implementation of the SCEV class.
//

#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
void SCEV::dump() const {
  print(dbgs());
  dbgs() << '\n';
}
#endif

void SCEV::print(raw_ostream &OS) const {
  switch (getSCEVType()) {
  case scConstant:
    WriteAsOperand(OS, cast<SCEVConstant>(this)->getValue(), false);
    return;
  case scTruncate: {
    const SCEVTruncateExpr *Trunc = cast<SCEVTruncateExpr>(this);
    const SCEV *Op = Trunc->getOperand();
    OS << "(trunc " << *Op->getType() << " " << *Op << " to "
       << *Trunc->getType() << ")";
    return;
  }
  case scZeroExtend: {
    const SCEVZeroExtendExpr *ZExt = cast<SCEVZeroExtendExpr>(this);
    const SCEV *Op = ZExt->getOperand();
    OS << "(zext " << *Op->getType() << " " << *Op << " to "
       << *ZExt->getType() << ")";
    return;
  }
  case scSignExtend: {
    const SCEVSignExtendExpr *SExt = cast<SCEVSignExtendExpr>(this);
    const SCEV *Op = SExt->getOperand();
    OS << "(sext " << *Op->getType() << " " << *Op << " to "
       << *SExt->getType() << ")";
    return;
  }
  case scAddRecExpr: {
    const SCEVAddRecExpr *AR = cast<SCEVAddRecExpr>(this);
    OS << "{" << *AR->getOperand(0);
    for (unsigned i = 1, e = AR->getNumOperands(); i != e; ++i)
      OS << ",+," << *AR->getOperand(i);
    OS << "}<";
    if (AR->getNoWrapFlags(FlagNUW))
      OS << "nuw><";
    if (AR->getNoWrapFlags(FlagNSW))
      OS << "nsw><";
    if (AR->getNoWrapFlags(FlagNW) &&
        !AR->getNoWrapFlags((NoWrapFlags)(FlagNUW | FlagNSW)))
      OS << "nw><";
    WriteAsOperand(OS, AR->getLoop()->getHeader(), /*PrintType=*/false);
    OS << ">";
    return;
  }
  case scAddExpr:
  case scMulExpr:
  case scUMaxExpr:
  case scSMaxExpr: {
    const SCEVNAryExpr *NAry = cast<SCEVNAryExpr>(this);
    const char *OpStr = 0;
    switch (NAry->getSCEVType()) {
    case scAddExpr: OpStr = " + "; break;
    case scMulExpr: OpStr = " * "; break;
    case scUMaxExpr: OpStr = " umax "; break;
    case scSMaxExpr: OpStr = " smax "; break;
    }
    OS << "(";
    for (SCEVNAryExpr::op_iterator I = NAry->op_begin(), E = NAry->op_end();
         I != E; ++I) {
      OS << **I;
      if (llvm::next(I) != E)
        OS << OpStr;
    }
    OS << ")";
    switch (NAry->getSCEVType()) {
    case scAddExpr:
    case scMulExpr:
      if (NAry->getNoWrapFlags(FlagNUW))
        OS << "<nuw>";
      if (NAry->getNoWrapFlags(FlagNSW))
        OS << "<nsw>";
    }
    return;
  }
  case scUDivExpr: {
    const SCEVUDivExpr *UDiv = cast<SCEVUDivExpr>(this);
    OS << "(" << *UDiv->getLHS() << " /u " << *UDiv->getRHS() << ")";
    return;
  }
  case scUnknown: {
    const SCEVUnknown *U = cast<SCEVUnknown>(this);
    Type *AllocTy;
    if (U->isSizeOf(AllocTy)) {
      OS << "sizeof(" << *AllocTy << ")";
      return;
    }
    if (U->isAlignOf(AllocTy)) {
      OS << "alignof(" << *AllocTy << ")";
      return;
    }

    Type *CTy;
    Constant *FieldNo;
    if (U->isOffsetOf(CTy, FieldNo)) {
      OS << "offsetof(" << *CTy << ", ";
      WriteAsOperand(OS, FieldNo, false);
      OS << ")";
      return;
    }

    // Otherwise just print it normally.
    WriteAsOperand(OS, U->getValue(), false);
    return;
  }
  case scCouldNotCompute:
    OS << "***COULDNOTCOMPUTE***";
    return;
  default: break;
  }
  llvm_unreachable("Unknown SCEV kind!");
}

Type *SCEV::getType() const {
  switch (getSCEVType()) {
  case scConstant:
    return cast<SCEVConstant>(this)->getType();
  case scTruncate:
  case scZeroExtend:
  case scSignExtend:
    return cast<SCEVCastExpr>(this)->getType();
  case scAddRecExpr:
  case scMulExpr:
  case scUMaxExpr:
  case scSMaxExpr:
    return cast<SCEVNAryExpr>(this)->getType();
  case scAddExpr:
    return cast<SCEVAddExpr>(this)->getType();
  case scUDivExpr:
    return cast<SCEVUDivExpr>(this)->getType();
  case scUnknown:
    return cast<SCEVUnknown>(this)->getType();
  case scCouldNotCompute:
    llvm_unreachable("Attempt to use a SCEVCouldNotCompute object!");
  default:
    llvm_unreachable("Unknown SCEV kind!");
  }
}

bool SCEV::isZero() const {
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(this))
    return SC->getValue()->isZero();
  return false;
}

bool SCEV::isOne() const {
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(this))
    return SC->getValue()->isOne();
  return false;
}

bool SCEV::isAllOnesValue() const {
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(this))
    return SC->getValue()->isAllOnesValue();
  return false;
}

/// isNonConstantNegative - Return true if the specified scev is negated, but
/// not a constant.
bool SCEV::isNonConstantNegative() const {
  const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(this);
  if (!Mul) return false;

  // If there is a constant factor, it will be first.
  const SCEVConstant *SC = dyn_cast<SCEVConstant>(Mul->getOperand(0));
  if (!SC) return false;

  // Return true if the value is negative, this matches things like (-42 * V).
  return SC->getValue()->getValue().isNegative();
}

SCEVCouldNotCompute::SCEVCouldNotCompute() :
  SCEV(FoldingSetNodeIDRef(), scCouldNotCompute) {}

bool SCEVCouldNotCompute::classof(const SCEV *S) {
  return S->getSCEVType() == scCouldNotCompute;
}

const SCEV *ScalarEvolution::getConstant(ConstantInt *V) {
  FoldingSetNodeID ID;
  ID.AddInteger(scConstant);
  ID.AddPointer(V);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  SCEV *S = new (SCEVAllocator) SCEVConstant(ID.Intern(SCEVAllocator), V);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

const SCEV *ScalarEvolution::getConstant(const APInt& Val) {
  return getConstant(ConstantInt::get(getContext(), Val));
}

const SCEV *
ScalarEvolution::getConstant(Type *Ty, uint64_t V, bool isSigned) {
  IntegerType *ITy = cast<IntegerType>(getEffectiveSCEVType(Ty));
  return getConstant(ConstantInt::get(ITy, V, isSigned));
}

SCEVCastExpr::SCEVCastExpr(const FoldingSetNodeIDRef ID,
                           unsigned SCEVTy, const SCEV *op, Type *ty)
  : SCEV(ID, SCEVTy), Op(op), Ty(ty) {}

SCEVTruncateExpr::SCEVTruncateExpr(const FoldingSetNodeIDRef ID,
                                   const SCEV *op, Type *ty)
  : SCEVCastExpr(ID, scTruncate, op, ty) {
  assert((Op->getType()->isIntegerTy() || Op->getType()->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot truncate non-integer value!");
}

SCEVZeroExtendExpr::SCEVZeroExtendExpr(const FoldingSetNodeIDRef ID,
                                       const SCEV *op, Type *ty)
  : SCEVCastExpr(ID, scZeroExtend, op, ty) {
  assert((Op->getType()->isIntegerTy() || Op->getType()->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot zero extend non-integer value!");
}

SCEVSignExtendExpr::SCEVSignExtendExpr(const FoldingSetNodeIDRef ID,
                                       const SCEV *op, Type *ty)
  : SCEVCastExpr(ID, scSignExtend, op, ty) {
  assert((Op->getType()->isIntegerTy() || Op->getType()->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot sign extend non-integer value!");
}

void SCEVUnknown::deleted() {
  // Clear this SCEVUnknown from various maps.
  SE->forgetMemoizedResults(this);

  // Remove this SCEVUnknown from the uniquing map.
  SE->UniqueSCEVs.RemoveNode(this);

  // Release the value.
  setValPtr(0);
}

void SCEVUnknown::allUsesReplacedWith(Value *New) {
  // Clear this SCEVUnknown from various maps.
  SE->forgetMemoizedResults(this);

  // Remove this SCEVUnknown from the uniquing map.
  SE->UniqueSCEVs.RemoveNode(this);

  // Update this SCEVUnknown to point to the new value. This is needed
  // because there may still be outstanding SCEVs which still point to
  // this SCEVUnknown.
  setValPtr(New);
}

bool SCEVUnknown::isSizeOf(Type *&AllocTy) const {
  if (ConstantExpr *VCE = dyn_cast<ConstantExpr>(getValue()))
    if (VCE->getOpcode() == Instruction::PtrToInt)
      if (ConstantExpr *CE = dyn_cast<ConstantExpr>(VCE->getOperand(0)))
        if (CE->getOpcode() == Instruction::GetElementPtr &&
            CE->getOperand(0)->isNullValue() &&
            CE->getNumOperands() == 2)
          if (ConstantInt *CI = dyn_cast<ConstantInt>(CE->getOperand(1)))
            if (CI->isOne()) {
              AllocTy = cast<PointerType>(CE->getOperand(0)->getType())
                                 ->getElementType();
              return true;
            }

  return false;
}

bool SCEVUnknown::isAlignOf(Type *&AllocTy) const {
  if (ConstantExpr *VCE = dyn_cast<ConstantExpr>(getValue()))
    if (VCE->getOpcode() == Instruction::PtrToInt)
      if (ConstantExpr *CE = dyn_cast<ConstantExpr>(VCE->getOperand(0)))
        if (CE->getOpcode() == Instruction::GetElementPtr &&
            CE->getOperand(0)->isNullValue()) {
          Type *Ty =
            cast<PointerType>(CE->getOperand(0)->getType())->getElementType();
          if (StructType *STy = dyn_cast<StructType>(Ty))
            if (!STy->isPacked() &&
                CE->getNumOperands() == 3 &&
                CE->getOperand(1)->isNullValue()) {
              if (ConstantInt *CI = dyn_cast<ConstantInt>(CE->getOperand(2)))
                if (CI->isOne() &&
                    STy->getNumElements() == 2 &&
                    STy->getElementType(0)->isIntegerTy(1)) {
                  AllocTy = STy->getElementType(1);
                  return true;
                }
            }
        }

  return false;
}

bool SCEVUnknown::isOffsetOf(Type *&CTy, Constant *&FieldNo) const {
  if (ConstantExpr *VCE = dyn_cast<ConstantExpr>(getValue()))
    if (VCE->getOpcode() == Instruction::PtrToInt)
      if (ConstantExpr *CE = dyn_cast<ConstantExpr>(VCE->getOperand(0)))
        if (CE->getOpcode() == Instruction::GetElementPtr &&
            CE->getNumOperands() == 3 &&
            CE->getOperand(0)->isNullValue() &&
            CE->getOperand(1)->isNullValue()) {
          Type *Ty =
            cast<PointerType>(CE->getOperand(0)->getType())->getElementType();
          // Ignore vector types here so that ScalarEvolutionExpander doesn't
          // emit getelementptrs that index into vectors.
          if (Ty->isStructTy() || Ty->isArrayTy()) {
            CTy = Ty;
            FieldNo = CE->getOperand(2);
            return true;
          }
        }

  return false;
}

//===----------------------------------------------------------------------===//
//                               SCEV Utilities
//===----------------------------------------------------------------------===//

namespace {
  /// SCEVComplexityCompare - Return true if the complexity of the LHS is less
  /// than the complexity of the RHS.  This comparator is used to canonicalize
  /// expressions.
  class SCEVComplexityCompare {
    const LoopInfo *const LI;
  public:
    explicit SCEVComplexityCompare(const LoopInfo *li) : LI(li) {}

    // Return true or false if LHS is less than, or at least RHS, respectively.
    bool operator()(const SCEV *LHS, const SCEV *RHS) const {
      return compare(LHS, RHS) < 0;
    }

    // Return negative, zero, or positive, if LHS is less than, equal to, or
    // greater than RHS, respectively. A three-way result allows recursive
    // comparisons to be more efficient.
    int compare(const SCEV *LHS, const SCEV *RHS) const {
      // Fast-path: SCEVs are uniqued so we can do a quick equality check.
      if (LHS == RHS)
        return 0;

      // Primarily, sort the SCEVs by their getSCEVType().
      unsigned LType = LHS->getSCEVType(), RType = RHS->getSCEVType();
      if (LType != RType)
        return (int)LType - (int)RType;

      // Aside from the getSCEVType() ordering, the particular ordering
      // isn't very important except that it's beneficial to be consistent,
      // so that (a + b) and (b + a) don't end up as different expressions.
      switch (LType) {
      case scUnknown: {
        const SCEVUnknown *LU = cast<SCEVUnknown>(LHS);
        const SCEVUnknown *RU = cast<SCEVUnknown>(RHS);

        // Sort SCEVUnknown values with some loose heuristics. TODO: This is
        // not as complete as it could be.
        const Value *LV = LU->getValue(), *RV = RU->getValue();

        // Order pointer values after integer values. This helps SCEVExpander
        // form GEPs.
        bool LIsPointer = LV->getType()->isPointerTy(),
             RIsPointer = RV->getType()->isPointerTy();
        if (LIsPointer != RIsPointer)
          return (int)LIsPointer - (int)RIsPointer;

        // Compare getValueID values.
        unsigned LID = LV->getValueID(),
                 RID = RV->getValueID();
        if (LID != RID)
          return (int)LID - (int)RID;

        // Sort arguments by their position.
        if (const Argument *LA = dyn_cast<Argument>(LV)) {
          const Argument *RA = cast<Argument>(RV);
          unsigned LArgNo = LA->getArgNo(), RArgNo = RA->getArgNo();
          return (int)LArgNo - (int)RArgNo;
        }

        // For instructions, compare their loop depth, and their operand
        // count.  This is pretty loose.
        if (const Instruction *LInst = dyn_cast<Instruction>(LV)) {
          const Instruction *RInst = cast<Instruction>(RV);

          // Compare loop depths.
          const BasicBlock *LParent = LInst->getParent(),
                           *RParent = RInst->getParent();
          if (LParent != RParent) {
            unsigned LDepth = LI->getLoopDepth(LParent),
                     RDepth = LI->getLoopDepth(RParent);
            if (LDepth != RDepth)
              return (int)LDepth - (int)RDepth;
          }

          // Compare the number of operands.
          unsigned LNumOps = LInst->getNumOperands(),
                   RNumOps = RInst->getNumOperands();
          return (int)LNumOps - (int)RNumOps;
        }

        return 0;
      }

      case scConstant: {
        const SCEVConstant *LC = cast<SCEVConstant>(LHS);
        const SCEVConstant *RC = cast<SCEVConstant>(RHS);

        // Compare constant values.
        const APInt &LA = LC->getValue()->getValue();
        const APInt &RA = RC->getValue()->getValue();
        unsigned LBitWidth = LA.getBitWidth(), RBitWidth = RA.getBitWidth();
        if (LBitWidth != RBitWidth)
          return (int)LBitWidth - (int)RBitWidth;
        return LA.ult(RA) ? -1 : 1;
      }

      case scAddRecExpr: {
        const SCEVAddRecExpr *LA = cast<SCEVAddRecExpr>(LHS);
        const SCEVAddRecExpr *RA = cast<SCEVAddRecExpr>(RHS);

        // Compare addrec loop depths.
        const Loop *LLoop = LA->getLoop(), *RLoop = RA->getLoop();
        if (LLoop != RLoop) {
          unsigned LDepth = LLoop->getLoopDepth(),
                   RDepth = RLoop->getLoopDepth();
          if (LDepth != RDepth)
            return (int)LDepth - (int)RDepth;
        }

        // Addrec complexity grows with operand count.
        unsigned LNumOps = LA->getNumOperands(), RNumOps = RA->getNumOperands();
        if (LNumOps != RNumOps)
          return (int)LNumOps - (int)RNumOps;

        // Lexicographically compare.
        for (unsigned i = 0; i != LNumOps; ++i) {
          long X = compare(LA->getOperand(i), RA->getOperand(i));
          if (X != 0)
            return X;
        }

        return 0;
      }

      case scAddExpr:
      case scMulExpr:
      case scSMaxExpr:
      case scUMaxExpr: {
        const SCEVNAryExpr *LC = cast<SCEVNAryExpr>(LHS);
        const SCEVNAryExpr *RC = cast<SCEVNAryExpr>(RHS);

        // Lexicographically compare n-ary expressions.
        unsigned LNumOps = LC->getNumOperands(), RNumOps = RC->getNumOperands();
        if (LNumOps != RNumOps)
          return (int)LNumOps - (int)RNumOps;

        for (unsigned i = 0; i != LNumOps; ++i) {
          if (i >= RNumOps)
            return 1;
          long X = compare(LC->getOperand(i), RC->getOperand(i));
          if (X != 0)
            return X;
        }
        return (int)LNumOps - (int)RNumOps;
      }

      case scUDivExpr: {
        const SCEVUDivExpr *LC = cast<SCEVUDivExpr>(LHS);
        const SCEVUDivExpr *RC = cast<SCEVUDivExpr>(RHS);

        // Lexicographically compare udiv expressions.
        long X = compare(LC->getLHS(), RC->getLHS());
        if (X != 0)
          return X;
        return compare(LC->getRHS(), RC->getRHS());
      }

      case scTruncate:
      case scZeroExtend:
      case scSignExtend: {
        const SCEVCastExpr *LC = cast<SCEVCastExpr>(LHS);
        const SCEVCastExpr *RC = cast<SCEVCastExpr>(RHS);

        // Compare cast expressions by operand.
        return compare(LC->getOperand(), RC->getOperand());
      }

      default:
        llvm_unreachable("Unknown SCEV kind!");
      }
    }
  };
}

/// GroupByComplexity - Given a list of SCEV objects, order them by their
/// complexity, and group objects of the same complexity together by value.
/// When this routine is finished, we know that any duplicates in the vector are
/// consecutive and that complexity is monotonically increasing.
///
/// Note that we go take special precautions to ensure that we get deterministic
/// results from this routine.  In other words, we don't want the results of
/// this to depend on where the addresses of various SCEV objects happened to
/// land in memory.
///
static void GroupByComplexity(SmallVectorImpl<const SCEV *> &Ops,
                              LoopInfo *LI) {
  if (Ops.size() < 2) return;  // Noop
  if (Ops.size() == 2) {
    // This is the common case, which also happens to be trivially simple.
    // Special case it.
    const SCEV *&LHS = Ops[0], *&RHS = Ops[1];
    if (SCEVComplexityCompare(LI)(RHS, LHS))
      std::swap(LHS, RHS);
    return;
  }

  // Do the rough sort by complexity.
  std::stable_sort(Ops.begin(), Ops.end(), SCEVComplexityCompare(LI));

  // Now that we are sorted by complexity, group elements of the same
  // complexity.  Note that this is, at worst, N^2, but the vector is likely to
  // be extremely short in practice.  Note that we take this approach because we
  // do not want to depend on the addresses of the objects we are grouping.
  for (unsigned i = 0, e = Ops.size(); i != e-2; ++i) {
    const SCEV *S = Ops[i];
    unsigned Complexity = S->getSCEVType();

    // If there are any objects of the same complexity and same value as this
    // one, group them.
    for (unsigned j = i+1; j != e && Ops[j]->getSCEVType() == Complexity; ++j) {
      if (Ops[j] == S) { // Found a duplicate.
        // Move it to immediately after i'th element.
        std::swap(Ops[i+1], Ops[j]);
        ++i;   // no need to rescan it.
        if (i == e-2) return;  // Done!
      }
    }
  }
}



//===----------------------------------------------------------------------===//
//                      Simple SCEV method implementations
//===----------------------------------------------------------------------===//

/// BinomialCoefficient - Compute BC(It, K).  The result has width W.
/// Assume, K > 0.
static const SCEV *BinomialCoefficient(const SCEV *It, unsigned K,
                                       ScalarEvolution &SE,
                                       Type *ResultTy) {
  // Handle the simplest case efficiently.
  if (K == 1)
    return SE.getTruncateOrZeroExtend(It, ResultTy);

  // We are using the following formula for BC(It, K):
  //
  //   BC(It, K) = (It * (It - 1) * ... * (It - K + 1)) / K!
  //
  // Suppose, W is the bitwidth of the return value.  We must be prepared for
  // overflow.  Hence, we must assure that the result of our computation is
  // equal to the accurate one modulo 2^W.  Unfortunately, division isn't
  // safe in modular arithmetic.
  //
  // However, this code doesn't use exactly that formula; the formula it uses
  // is something like the following, where T is the number of factors of 2 in
  // K! (i.e. trailing zeros in the binary representation of K!), and ^ is
  // exponentiation:
  //
  //   BC(It, K) = (It * (It - 1) * ... * (It - K + 1)) / 2^T / (K! / 2^T)
  //
  // This formula is trivially equivalent to the previous formula.  However,
  // this formula can be implemented much more efficiently.  The trick is that
  // K! / 2^T is odd, and exact division by an odd number *is* safe in modular
  // arithmetic.  To do exact division in modular arithmetic, all we have
  // to do is multiply by the inverse.  Therefore, this step can be done at
  // width W.
  //
  // The next issue is how to safely do the division by 2^T.  The way this
  // is done is by doing the multiplication step at a width of at least W + T
  // bits.  This way, the bottom W+T bits of the product are accurate. Then,
  // when we perform the division by 2^T (which is equivalent to a right shift
  // by T), the bottom W bits are accurate.  Extra bits are okay; they'll get
  // truncated out after the division by 2^T.
  //
  // In comparison to just directly using the first formula, this technique
  // is much more efficient; using the first formula requires W * K bits,
  // but this formula less than W + K bits. Also, the first formula requires
  // a division step, whereas this formula only requires multiplies and shifts.
  //
  // It doesn't matter whether the subtraction step is done in the calculation
  // width or the input iteration count's width; if the subtraction overflows,
  // the result must be zero anyway.  We prefer here to do it in the width of
  // the induction variable because it helps a lot for certain cases; CodeGen
  // isn't smart enough to ignore the overflow, which leads to much less
  // efficient code if the width of the subtraction is wider than the native
  // register width.
  //
  // (It's possible to not widen at all by pulling out factors of 2 before
  // the multiplication; for example, K=2 can be calculated as
  // It/2*(It+(It*INT_MIN/INT_MIN)+-1). However, it requires
  // extra arithmetic, so it's not an obvious win, and it gets
  // much more complicated for K > 3.)

  // Protection from insane SCEVs; this bound is conservative,
  // but it probably doesn't matter.
  if (K > 1000)
    return SE.getCouldNotCompute();

  unsigned W = SE.getTypeSizeInBits(ResultTy);

  // Calculate K! / 2^T and T; we divide out the factors of two before
  // multiplying for calculating K! / 2^T to avoid overflow.
  // Other overflow doesn't matter because we only care about the bottom
  // W bits of the result.
  APInt OddFactorial(W, 1);
  unsigned T = 1;
  for (unsigned i = 3; i <= K; ++i) {
    APInt Mult(W, i);
    unsigned TwoFactors = Mult.countTrailingZeros();
    T += TwoFactors;
    Mult = Mult.lshr(TwoFactors);
    OddFactorial *= Mult;
  }

  // We need at least W + T bits for the multiplication step
  unsigned CalculationBits = W + T;

  // Calculate 2^T, at width T+W.
  APInt DivFactor = APInt::getOneBitSet(CalculationBits, T);

  // Calculate the multiplicative inverse of K! / 2^T;
  // this multiplication factor will perform the exact division by
  // K! / 2^T.
  APInt Mod = APInt::getSignedMinValue(W+1);
  APInt MultiplyFactor = OddFactorial.zext(W+1);
  MultiplyFactor = MultiplyFactor.multiplicativeInverse(Mod);
  MultiplyFactor = MultiplyFactor.trunc(W);

  // Calculate the product, at width T+W
  IntegerType *CalculationTy = IntegerType::get(SE.getContext(),
                                                      CalculationBits);
  const SCEV *Dividend = SE.getTruncateOrZeroExtend(It, CalculationTy);
  for (unsigned i = 1; i != K; ++i) {
    const SCEV *S = SE.getMinusSCEV(It, SE.getConstant(It->getType(), i));
    Dividend = SE.getMulExpr(Dividend,
                             SE.getTruncateOrZeroExtend(S, CalculationTy));
  }

  // Divide by 2^T
  const SCEV *DivResult = SE.getUDivExpr(Dividend, SE.getConstant(DivFactor));

  // Truncate the result, and divide by K! / 2^T.

  return SE.getMulExpr(SE.getConstant(MultiplyFactor),
                       SE.getTruncateOrZeroExtend(DivResult, ResultTy));
}

/// evaluateAtIteration - Return the value of this chain of recurrences at
/// the specified iteration number.  We can evaluate this recurrence by
/// multiplying each element in the chain by the binomial coefficient
/// corresponding to it.  In other words, we can evaluate {A,+,B,+,C,+,D} as:
///
///   A*BC(It, 0) + B*BC(It, 1) + C*BC(It, 2) + D*BC(It, 3)
///
/// where BC(It, k) stands for binomial coefficient.
///
const SCEV *SCEVAddRecExpr::evaluateAtIteration(const SCEV *It,
                                                ScalarEvolution &SE) const {
  const SCEV *Result = getStart();
  for (unsigned i = 1, e = getNumOperands(); i != e; ++i) {
    // The computation is correct in the face of overflow provided that the
    // multiplication is performed _after_ the evaluation of the binomial
    // coefficient.
    const SCEV *Coeff = BinomialCoefficient(It, i, SE, getType());
    if (isa<SCEVCouldNotCompute>(Coeff))
      return Coeff;

    Result = SE.getAddExpr(Result, SE.getMulExpr(getOperand(i), Coeff));
  }
  return Result;
}

//===----------------------------------------------------------------------===//
//                    SCEV Expression folder implementations
//===----------------------------------------------------------------------===//

const SCEV *ScalarEvolution::getTruncateExpr(const SCEV *Op,
                                             Type *Ty) {
  assert(getTypeSizeInBits(Op->getType()) > getTypeSizeInBits(Ty) &&
         "This is not a truncating conversion!");
  assert(isSCEVable(Ty) &&
         "This is not a conversion to a SCEVable type!");
  Ty = getEffectiveSCEVType(Ty);

  FoldingSetNodeID ID;
  ID.AddInteger(scTruncate);
  ID.AddPointer(Op);
  ID.AddPointer(Ty);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;

  // Fold if the operand is constant.
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
    return getConstant(
      cast<ConstantInt>(ConstantExpr::getTrunc(SC->getValue(), Ty)));

  // trunc(trunc(x)) --> trunc(x)
  if (const SCEVTruncateExpr *ST = dyn_cast<SCEVTruncateExpr>(Op))
    return getTruncateExpr(ST->getOperand(), Ty);

  // trunc(sext(x)) --> sext(x) if widening or trunc(x) if narrowing
  if (const SCEVSignExtendExpr *SS = dyn_cast<SCEVSignExtendExpr>(Op))
    return getTruncateOrSignExtend(SS->getOperand(), Ty);

  // trunc(zext(x)) --> zext(x) if widening or trunc(x) if narrowing
  if (const SCEVZeroExtendExpr *SZ = dyn_cast<SCEVZeroExtendExpr>(Op))
    return getTruncateOrZeroExtend(SZ->getOperand(), Ty);

  // trunc(x1+x2+...+xN) --> trunc(x1)+trunc(x2)+...+trunc(xN) if we can
  // eliminate all the truncates.
  if (const SCEVAddExpr *SA = dyn_cast<SCEVAddExpr>(Op)) {
    SmallVector<const SCEV *, 4> Operands;
    bool hasTrunc = false;
    for (unsigned i = 0, e = SA->getNumOperands(); i != e && !hasTrunc; ++i) {
      const SCEV *S = getTruncateExpr(SA->getOperand(i), Ty);
      hasTrunc = isa<SCEVTruncateExpr>(S);
      Operands.push_back(S);
    }
    if (!hasTrunc)
      return getAddExpr(Operands);
    UniqueSCEVs.FindNodeOrInsertPos(ID, IP);  // Mutates IP, returns NULL.
  }

  // trunc(x1*x2*...*xN) --> trunc(x1)*trunc(x2)*...*trunc(xN) if we can
  // eliminate all the truncates.
  if (const SCEVMulExpr *SM = dyn_cast<SCEVMulExpr>(Op)) {
    SmallVector<const SCEV *, 4> Operands;
    bool hasTrunc = false;
    for (unsigned i = 0, e = SM->getNumOperands(); i != e && !hasTrunc; ++i) {
      const SCEV *S = getTruncateExpr(SM->getOperand(i), Ty);
      hasTrunc = isa<SCEVTruncateExpr>(S);
      Operands.push_back(S);
    }
    if (!hasTrunc)
      return getMulExpr(Operands);
    UniqueSCEVs.FindNodeOrInsertPos(ID, IP);  // Mutates IP, returns NULL.
  }

  // If the input value is a chrec scev, truncate the chrec's operands.
  if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Op)) {
    SmallVector<const SCEV *, 4> Operands;
    for (unsigned i = 0, e = AddRec->getNumOperands(); i != e; ++i)
      Operands.push_back(getTruncateExpr(AddRec->getOperand(i), Ty));
    return getAddRecExpr(Operands, AddRec->getLoop(), SCEV::FlagAnyWrap);
  }

  // The cast wasn't folded; create an explicit cast node. We can reuse
  // the existing insert position since if we get here, we won't have
  // made any changes which would invalidate it.
  SCEV *S = new (SCEVAllocator) SCEVTruncateExpr(ID.Intern(SCEVAllocator),
                                                 Op, Ty);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

const SCEV *ScalarEvolution::getZeroExtendExpr(const SCEV *Op,
                                               Type *Ty) {
  assert(getTypeSizeInBits(Op->getType()) < getTypeSizeInBits(Ty) &&
         "This is not an extending conversion!");
  assert(isSCEVable(Ty) &&
         "This is not a conversion to a SCEVable type!");
  Ty = getEffectiveSCEVType(Ty);

  // Fold if the operand is constant.
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
    return getConstant(
      cast<ConstantInt>(ConstantExpr::getZExt(SC->getValue(), Ty)));

  // zext(zext(x)) --> zext(x)
  if (const SCEVZeroExtendExpr *SZ = dyn_cast<SCEVZeroExtendExpr>(Op))
    return getZeroExtendExpr(SZ->getOperand(), Ty);

  // Before doing any expensive analysis, check to see if we've already
  // computed a SCEV for this Op and Ty.
  FoldingSetNodeID ID;
  ID.AddInteger(scZeroExtend);
  ID.AddPointer(Op);
  ID.AddPointer(Ty);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;

  // zext(trunc(x)) --> zext(x) or x or trunc(x)
  if (const SCEVTruncateExpr *ST = dyn_cast<SCEVTruncateExpr>(Op)) {
    // It's possible the bits taken off by the truncate were all zero bits. If
    // so, we should be able to simplify this further.
    const SCEV *X = ST->getOperand();
    ConstantRange CR = getUnsignedRange(X);
    unsigned TruncBits = getTypeSizeInBits(ST->getType());
    unsigned NewBits = getTypeSizeInBits(Ty);
    if (CR.truncate(TruncBits).zeroExtend(NewBits).contains(
            CR.zextOrTrunc(NewBits)))
      return getTruncateOrZeroExtend(X, Ty);
  }

  // If the input value is a chrec scev, and we can prove that the value
  // did not overflow the old, smaller, value, we can zero extend all of the
  // operands (often constants).  This allows analysis of something like
  // this:  for (unsigned char X = 0; X < 100; ++X) { int Y = X; }
  if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Op))
    if (AR->isAffine()) {
      const SCEV *Start = AR->getStart();
      const SCEV *Step = AR->getStepRecurrence(*this);
      unsigned BitWidth = getTypeSizeInBits(AR->getType());
      const Loop *L = AR->getLoop();

      // If we have special knowledge that this addrec won't overflow,
      // we don't need to do any further analysis.
      if (AR->getNoWrapFlags(SCEV::FlagNUW))
        return getAddRecExpr(getZeroExtendExpr(Start, Ty),
                             getZeroExtendExpr(Step, Ty),
                             L, AR->getNoWrapFlags());

      // Check whether the backedge-taken count is SCEVCouldNotCompute.
      // Note that this serves two purposes: It filters out loops that are
      // simply not analyzable, and it covers the case where this code is
      // being called from within backedge-taken count analysis, such that
      // attempting to ask for the backedge-taken count would likely result
      // in infinite recursion. In the later case, the analysis code will
      // cope with a conservative value, and it will take care to purge
      // that value once it has finished.
      const SCEV *MaxBECount = getMaxBackedgeTakenCount(L);
      if (!isa<SCEVCouldNotCompute>(MaxBECount)) {
        // Manually compute the final value for AR, checking for
        // overflow.

        // Check whether the backedge-taken count can be losslessly casted to
        // the addrec's type. The count is always unsigned.
        const SCEV *CastedMaxBECount =
          getTruncateOrZeroExtend(MaxBECount, Start->getType());
        const SCEV *RecastedMaxBECount =
          getTruncateOrZeroExtend(CastedMaxBECount, MaxBECount->getType());
        if (MaxBECount == RecastedMaxBECount) {
          Type *WideTy = IntegerType::get(getContext(), BitWidth * 2);
          // Check whether Start+Step*MaxBECount has no unsigned overflow.
          const SCEV *ZMul = getMulExpr(CastedMaxBECount, Step);
          const SCEV *ZAdd = getZeroExtendExpr(getAddExpr(Start, ZMul), WideTy);
          const SCEV *WideStart = getZeroExtendExpr(Start, WideTy);
          const SCEV *WideMaxBECount =
            getZeroExtendExpr(CastedMaxBECount, WideTy);
          const SCEV *OperandExtendedAdd =
            getAddExpr(WideStart,
                       getMulExpr(WideMaxBECount,
                                  getZeroExtendExpr(Step, WideTy)));
          if (ZAdd == OperandExtendedAdd) {
            // Cache knowledge of AR NUW, which is propagated to this AddRec.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNUW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getZeroExtendExpr(Start, Ty),
                                 getZeroExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
          // Similar to above, only this time treat the step value as signed.
          // This covers loops that count down.
          OperandExtendedAdd =
            getAddExpr(WideStart,
                       getMulExpr(WideMaxBECount,
                                  getSignExtendExpr(Step, WideTy)));
          if (ZAdd == OperandExtendedAdd) {
            // Cache knowledge of AR NW, which is propagated to this AddRec.
            // Negative step causes unsigned wrap, but it still can't self-wrap.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getZeroExtendExpr(Start, Ty),
                                 getSignExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
        }

        // If the backedge is guarded by a comparison with the pre-inc value
        // the addrec is safe. Also, if the entry is guarded by a comparison
        // with the start value and the backedge is guarded by a comparison
        // with the post-inc value, the addrec is safe.
        if (isKnownPositive(Step)) {
          const SCEV *N = getConstant(APInt::getMinValue(BitWidth) -
                                      getUnsignedRange(Step).getUnsignedMax());
          if (isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_ULT, AR, N) ||
              (isLoopEntryGuardedByCond(L, ICmpInst::ICMP_ULT, Start, N) &&
               isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_ULT,
                                           AR->getPostIncExpr(*this), N))) {
            // Cache knowledge of AR NUW, which is propagated to this AddRec.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNUW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getZeroExtendExpr(Start, Ty),
                                 getZeroExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
        } else if (isKnownNegative(Step)) {
          const SCEV *N = getConstant(APInt::getMaxValue(BitWidth) -
                                      getSignedRange(Step).getSignedMin());
          if (isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_UGT, AR, N) ||
              (isLoopEntryGuardedByCond(L, ICmpInst::ICMP_UGT, Start, N) &&
               isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_UGT,
                                           AR->getPostIncExpr(*this), N))) {
            // Cache knowledge of AR NW, which is propagated to this AddRec.
            // Negative step causes unsigned wrap, but it still can't self-wrap.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getZeroExtendExpr(Start, Ty),
                                 getSignExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
        }
      }
    }

  // The cast wasn't folded; create an explicit cast node.
  // Recompute the insert position, as it may have been invalidated.
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  SCEV *S = new (SCEVAllocator) SCEVZeroExtendExpr(ID.Intern(SCEVAllocator),
                                                   Op, Ty);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

// Get the limit of a recurrence such that incrementing by Step cannot cause
// signed overflow as long as the value of the recurrence within the loop does
// not exceed this limit before incrementing.
static const SCEV *getOverflowLimitForStep(const SCEV *Step,
                                           ICmpInst::Predicate *Pred,
                                           ScalarEvolution *SE) {
  unsigned BitWidth = SE->getTypeSizeInBits(Step->getType());
  if (SE->isKnownPositive(Step)) {
    *Pred = ICmpInst::ICMP_SLT;
    return SE->getConstant(APInt::getSignedMinValue(BitWidth) -
                           SE->getSignedRange(Step).getSignedMax());
  }
  if (SE->isKnownNegative(Step)) {
    *Pred = ICmpInst::ICMP_SGT;
    return SE->getConstant(APInt::getSignedMaxValue(BitWidth) -
                       SE->getSignedRange(Step).getSignedMin());
  }
  return 0;
}

// The recurrence AR has been shown to have no signed wrap. Typically, if we can
// prove NSW for AR, then we can just as easily prove NSW for its preincrement
// or postincrement sibling. This allows normalizing a sign extended AddRec as
// such: {sext(Step + Start),+,Step} => {(Step + sext(Start),+,Step} As a
// result, the expression "Step + sext(PreIncAR)" is congruent with
// "sext(PostIncAR)"
static const SCEV *getPreStartForSignExtend(const SCEVAddRecExpr *AR,
                                            Type *Ty,
                                            ScalarEvolution *SE) {
  const Loop *L = AR->getLoop();
  const SCEV *Start = AR->getStart();
  const SCEV *Step = AR->getStepRecurrence(*SE);

  // Check for a simple looking step prior to loop entry.
  const SCEVAddExpr *SA = dyn_cast<SCEVAddExpr>(Start);
  if (!SA)
    return 0;

  // Create an AddExpr for "PreStart" after subtracting Step. Full SCEV
  // subtraction is expensive. For this purpose, perform a quick and dirty
  // difference, by checking for Step in the operand list.
  SmallVector<const SCEV *, 4> DiffOps;
  for (SCEVAddExpr::op_iterator I = SA->op_begin(), E = SA->op_end();
       I != E; ++I) {
    if (*I != Step)
      DiffOps.push_back(*I);
  }
  if (DiffOps.size() == SA->getNumOperands())
    return 0;

  // This is a postinc AR. Check for overflow on the preinc recurrence using the
  // same three conditions that getSignExtendedExpr checks.

  // 1. NSW flags on the step increment.
  const SCEV *PreStart = SE->getAddExpr(DiffOps, SA->getNoWrapFlags());
  const SCEVAddRecExpr *PreAR = dyn_cast<SCEVAddRecExpr>(
    SE->getAddRecExpr(PreStart, Step, L, SCEV::FlagAnyWrap));

  if (PreAR && PreAR->getNoWrapFlags(SCEV::FlagNSW))
    return PreStart;

  // 2. Direct overflow check on the step operation's expression.
  unsigned BitWidth = SE->getTypeSizeInBits(AR->getType());
  Type *WideTy = IntegerType::get(SE->getContext(), BitWidth * 2);
  const SCEV *OperandExtendedStart =
    SE->getAddExpr(SE->getSignExtendExpr(PreStart, WideTy),
                   SE->getSignExtendExpr(Step, WideTy));
  if (SE->getSignExtendExpr(Start, WideTy) == OperandExtendedStart) {
    // Cache knowledge of PreAR NSW.
    if (PreAR)
      const_cast<SCEVAddRecExpr *>(PreAR)->setNoWrapFlags(SCEV::FlagNSW);
    // FIXME: this optimization needs a unit test
    DEBUG(dbgs() << "SCEV: untested prestart overflow check\n");
    return PreStart;
  }

  // 3. Loop precondition.
  ICmpInst::Predicate Pred;
  const SCEV *OverflowLimit = getOverflowLimitForStep(Step, &Pred, SE);

  if (OverflowLimit &&
      SE->isLoopEntryGuardedByCond(L, Pred, PreStart, OverflowLimit)) {
    return PreStart;
  }
  return 0;
}

// Get the normalized sign-extended expression for this AddRec's Start.
static const SCEV *getSignExtendAddRecStart(const SCEVAddRecExpr *AR,
                                            Type *Ty,
                                            ScalarEvolution *SE) {
  const SCEV *PreStart = getPreStartForSignExtend(AR, Ty, SE);
  if (!PreStart)
    return SE->getSignExtendExpr(AR->getStart(), Ty);

  return SE->getAddExpr(SE->getSignExtendExpr(AR->getStepRecurrence(*SE), Ty),
                        SE->getSignExtendExpr(PreStart, Ty));
}

const SCEV *ScalarEvolution::getSignExtendExpr(const SCEV *Op,
                                               Type *Ty) {
  assert(getTypeSizeInBits(Op->getType()) < getTypeSizeInBits(Ty) &&
         "This is not an extending conversion!");
  assert(isSCEVable(Ty) &&
         "This is not a conversion to a SCEVable type!");
  Ty = getEffectiveSCEVType(Ty);

  // Fold if the operand is constant.
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
    return getConstant(
      cast<ConstantInt>(ConstantExpr::getSExt(SC->getValue(), Ty)));

  // sext(sext(x)) --> sext(x)
  if (const SCEVSignExtendExpr *SS = dyn_cast<SCEVSignExtendExpr>(Op))
    return getSignExtendExpr(SS->getOperand(), Ty);

  // sext(zext(x)) --> zext(x)
  if (const SCEVZeroExtendExpr *SZ = dyn_cast<SCEVZeroExtendExpr>(Op))
    return getZeroExtendExpr(SZ->getOperand(), Ty);

  // Before doing any expensive analysis, check to see if we've already
  // computed a SCEV for this Op and Ty.
  FoldingSetNodeID ID;
  ID.AddInteger(scSignExtend);
  ID.AddPointer(Op);
  ID.AddPointer(Ty);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;

  // If the input value is provably positive, build a zext instead.
  if (isKnownNonNegative(Op))
    return getZeroExtendExpr(Op, Ty);

  // sext(trunc(x)) --> sext(x) or x or trunc(x)
  if (const SCEVTruncateExpr *ST = dyn_cast<SCEVTruncateExpr>(Op)) {
    // It's possible the bits taken off by the truncate were all sign bits. If
    // so, we should be able to simplify this further.
    const SCEV *X = ST->getOperand();
    ConstantRange CR = getSignedRange(X);
    unsigned TruncBits = getTypeSizeInBits(ST->getType());
    unsigned NewBits = getTypeSizeInBits(Ty);
    if (CR.truncate(TruncBits).signExtend(NewBits).contains(
            CR.sextOrTrunc(NewBits)))
      return getTruncateOrSignExtend(X, Ty);
  }

  // If the input value is a chrec scev, and we can prove that the value
  // did not overflow the old, smaller, value, we can sign extend all of the
  // operands (often constants).  This allows analysis of something like
  // this:  for (signed char X = 0; X < 100; ++X) { int Y = X; }
  if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Op))
    if (AR->isAffine()) {
      const SCEV *Start = AR->getStart();
      const SCEV *Step = AR->getStepRecurrence(*this);
      unsigned BitWidth = getTypeSizeInBits(AR->getType());
      const Loop *L = AR->getLoop();

      // If we have special knowledge that this addrec won't overflow,
      // we don't need to do any further analysis.
      if (AR->getNoWrapFlags(SCEV::FlagNSW))
        return getAddRecExpr(getSignExtendAddRecStart(AR, Ty, this),
                             getSignExtendExpr(Step, Ty),
                             L, SCEV::FlagNSW);

      // Check whether the backedge-taken count is SCEVCouldNotCompute.
      // Note that this serves two purposes: It filters out loops that are
      // simply not analyzable, and it covers the case where this code is
      // being called from within backedge-taken count analysis, such that
      // attempting to ask for the backedge-taken count would likely result
      // in infinite recursion. In the later case, the analysis code will
      // cope with a conservative value, and it will take care to purge
      // that value once it has finished.
      const SCEV *MaxBECount = getMaxBackedgeTakenCount(L);
      if (!isa<SCEVCouldNotCompute>(MaxBECount)) {
        // Manually compute the final value for AR, checking for
        // overflow.

        // Check whether the backedge-taken count can be losslessly casted to
        // the addrec's type. The count is always unsigned.
        const SCEV *CastedMaxBECount =
          getTruncateOrZeroExtend(MaxBECount, Start->getType());
        const SCEV *RecastedMaxBECount =
          getTruncateOrZeroExtend(CastedMaxBECount, MaxBECount->getType());
        if (MaxBECount == RecastedMaxBECount) {
          Type *WideTy = IntegerType::get(getContext(), BitWidth * 2);
          // Check whether Start+Step*MaxBECount has no signed overflow.
          const SCEV *SMul = getMulExpr(CastedMaxBECount, Step);
          const SCEV *SAdd = getSignExtendExpr(getAddExpr(Start, SMul), WideTy);
          const SCEV *WideStart = getSignExtendExpr(Start, WideTy);
          const SCEV *WideMaxBECount =
            getZeroExtendExpr(CastedMaxBECount, WideTy);
          const SCEV *OperandExtendedAdd =
            getAddExpr(WideStart,
                       getMulExpr(WideMaxBECount,
                                  getSignExtendExpr(Step, WideTy)));
          if (SAdd == OperandExtendedAdd) {
            // Cache knowledge of AR NSW, which is propagated to this AddRec.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNSW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getSignExtendAddRecStart(AR, Ty, this),
                                 getSignExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
          // Similar to above, only this time treat the step value as unsigned.
          // This covers loops that count up with an unsigned step.
          OperandExtendedAdd =
            getAddExpr(WideStart,
                       getMulExpr(WideMaxBECount,
                                  getZeroExtendExpr(Step, WideTy)));
          if (SAdd == OperandExtendedAdd) {
            // Cache knowledge of AR NSW, which is propagated to this AddRec.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNSW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getSignExtendAddRecStart(AR, Ty, this),
                                 getZeroExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
        }

        // If the backedge is guarded by a comparison with the pre-inc value
        // the addrec is safe. Also, if the entry is guarded by a comparison
        // with the start value and the backedge is guarded by a comparison
        // with the post-inc value, the addrec is safe.
        ICmpInst::Predicate Pred;
        const SCEV *OverflowLimit = getOverflowLimitForStep(Step, &Pred, this);
        if (OverflowLimit &&
            (isLoopBackedgeGuardedByCond(L, Pred, AR, OverflowLimit) ||
             (isLoopEntryGuardedByCond(L, Pred, Start, OverflowLimit) &&
              isLoopBackedgeGuardedByCond(L, Pred, AR->getPostIncExpr(*this),
                                          OverflowLimit)))) {
          // Cache knowledge of AR NSW, then propagate NSW to the wide AddRec.
          const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNSW);
          return getAddRecExpr(getSignExtendAddRecStart(AR, Ty, this),
                               getSignExtendExpr(Step, Ty),
                               L, AR->getNoWrapFlags());
        }
      }
    }

  // The cast wasn't folded; create an explicit cast node.
  // Recompute the insert position, as it may have been invalidated.
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  SCEV *S = new (SCEVAllocator) SCEVSignExtendExpr(ID.Intern(SCEVAllocator),
                                                   Op, Ty);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

/// getAnyExtendExpr - Return a SCEV for the given operand extended with
/// unspecified bits out to the given type.
///
const SCEV *ScalarEvolution::getAnyExtendExpr(const SCEV *Op,
                                              Type *Ty) {
  assert(getTypeSizeInBits(Op->getType()) < getTypeSizeInBits(Ty) &&
         "This is not an extending conversion!");
  assert(isSCEVable(Ty) &&
         "This is not a conversion to a SCEVable type!");
  Ty = getEffectiveSCEVType(Ty);

  // Sign-extend negative constants.
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
    if (SC->getValue()->getValue().isNegative())
      return getSignExtendExpr(Op, Ty);

  // Peel off a truncate cast.
  if (const SCEVTruncateExpr *T = dyn_cast<SCEVTruncateExpr>(Op)) {
    const SCEV *NewOp = T->getOperand();
    if (getTypeSizeInBits(NewOp->getType()) < getTypeSizeInBits(Ty))
      return getAnyExtendExpr(NewOp, Ty);
    return getTruncateOrNoop(NewOp, Ty);
  }

  // Next try a zext cast. If the cast is folded, use it.
  const SCEV *ZExt = getZeroExtendExpr(Op, Ty);
  if (!isa<SCEVZeroExtendExpr>(ZExt))
    return ZExt;

  // Next try a sext cast. If the cast is folded, use it.
  const SCEV *SExt = getSignExtendExpr(Op, Ty);
  if (!isa<SCEVSignExtendExpr>(SExt))
    return SExt;

  // Force the cast to be folded into the operands of an addrec.
  if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Op)) {
    SmallVector<const SCEV *, 4> Ops;
    for (SCEVAddRecExpr::op_iterator I = AR->op_begin(), E = AR->op_end();
         I != E; ++I)
      Ops.push_back(getAnyExtendExpr(*I, Ty));
    return getAddRecExpr(Ops, AR->getLoop(), SCEV::FlagNW);
  }

  // If the expression is obviously signed, use the sext cast value.
  if (isa<SCEVSMaxExpr>(Op))
    return SExt;

  // Absent any other information, use the zext cast value.
  return ZExt;
}

/// CollectAddOperandsWithScales - Process the given Ops list, which is
/// a list of operands to be added under the given scale, update the given
/// map. This is a helper function for getAddRecExpr. As an example of
/// what it does, given a sequence of operands that would form an add
/// expression like this:
///
///    m + n + 13 + (A * (o + p + (B * q + m + 29))) + r + (-1 * r)
///
/// where A and B are constants, update the map with these values:
///
///    (m, 1+A*B), (n, 1), (o, A), (p, A), (q, A*B), (r, 0)
///
/// and add 13 + A*B*29 to AccumulatedConstant.
/// This will allow getAddRecExpr to produce this:
///
///    13+A*B*29 + n + (m * (1+A*B)) + ((o + p) * A) + (q * A*B)
///
/// This form often exposes folding opportunities that are hidden in
/// the original operand list.
///
/// Return true iff it appears that any interesting folding opportunities
/// may be exposed. This helps getAddRecExpr short-circuit extra work in
/// the common case where no interesting opportunities are present, and
/// is also used as a check to avoid infinite recursion.
///
static bool
CollectAddOperandsWithScales(DenseMap<const SCEV *, APInt> &M,
                             SmallVectorImpl<const SCEV *> &NewOps,
                             APInt &AccumulatedConstant,
                             const SCEV *const *Ops, size_t NumOperands,
                             const APInt &Scale,
                             ScalarEvolution &SE) {
  bool Interesting = false;

  // Iterate over the add operands. They are sorted, with constants first.
  unsigned i = 0;
  while (const SCEVConstant *C = dyn_cast<SCEVConstant>(Ops[i])) {
    ++i;
    // Pull a buried constant out to the outside.
    if (Scale != 1 || AccumulatedConstant != 0 || C->getValue()->isZero())
      Interesting = true;
    AccumulatedConstant += Scale * C->getValue()->getValue();
  }

  // Next comes everything else. We're especially interested in multiplies
  // here, but they're in the middle, so just visit the rest with one loop.
  for (; i != NumOperands; ++i) {
    const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(Ops[i]);
    if (Mul && isa<SCEVConstant>(Mul->getOperand(0))) {
      APInt NewScale =
        Scale * cast<SCEVConstant>(Mul->getOperand(0))->getValue()->getValue();
      if (Mul->getNumOperands() == 2 && isa<SCEVAddExpr>(Mul->getOperand(1))) {
        // A multiplication of a constant with another add; recurse.
        const SCEVAddExpr *Add = cast<SCEVAddExpr>(Mul->getOperand(1));
        Interesting |=
          CollectAddOperandsWithScales(M, NewOps, AccumulatedConstant,
                                       Add->op_begin(), Add->getNumOperands(),
                                       NewScale, SE);
      } else {
        // A multiplication of a constant with some other value. Update
        // the map.
        SmallVector<const SCEV *, 4> MulOps(Mul->op_begin()+1, Mul->op_end());
        const SCEV *Key = SE.getMulExpr(MulOps);
        std::pair<DenseMap<const SCEV *, APInt>::iterator, bool> Pair =
          M.insert(std::make_pair(Key, NewScale));
        if (Pair.second) {
          NewOps.push_back(Pair.first->first);
        } else {
          Pair.first->second += NewScale;
          // The map already had an entry for this value, which may indicate
          // a folding opportunity.
          Interesting = true;
        }
      }
    } else {
      // An ordinary operand. Update the map.
      std::pair<DenseMap<const SCEV *, APInt>::iterator, bool> Pair =
        M.insert(std::make_pair(Ops[i], Scale));
      if (Pair.second) {
        NewOps.push_back(Pair.first->first);
      } else {
        Pair.first->second += Scale;
        // The map already had an entry for this value, which may indicate
        // a folding opportunity.
        Interesting = true;
      }
    }
  }

  return Interesting;
}

namespace {
  struct APIntCompare {
    bool operator()(const APInt &LHS, const APInt &RHS) const {
      return LHS.ult(RHS);
    }
  };
}

/// getAddExpr - Get a canonical add expression, or something simpler if
/// possible.
const SCEV *ScalarEvolution::getAddExpr(SmallVectorImpl<const SCEV *> &Ops,
                                        SCEV::NoWrapFlags Flags) {
  assert(!(Flags & ~(SCEV::FlagNUW | SCEV::FlagNSW)) &&
         "only nuw or nsw allowed");
  assert(!Ops.empty() && "Cannot get empty add!");
  if (Ops.size() == 1) return Ops[0];
#ifndef NDEBUG
  Type *ETy = getEffectiveSCEVType(Ops[0]->getType());
  for (unsigned i = 1, e = Ops.size(); i != e; ++i)
    assert(getEffectiveSCEVType(Ops[i]->getType()) == ETy &&
           "SCEVAddExpr operand types don't match!");
#endif

  // If FlagNSW is true and all the operands are non-negative, infer FlagNUW.
  // And vice-versa.
  int SignOrUnsignMask = SCEV::FlagNUW | SCEV::FlagNSW;
  SCEV::NoWrapFlags SignOrUnsignWrap = maskFlags(Flags, SignOrUnsignMask);
  if (SignOrUnsignWrap && (SignOrUnsignWrap != SignOrUnsignMask)) {
    bool All = true;
    for (SmallVectorImpl<const SCEV *>::const_iterator I = Ops.begin(),
         E = Ops.end(); I != E; ++I)
      if (!isKnownNonNegative(*I)) {
        All = false;
        break;
      }
    if (All) Flags = setFlags(Flags, (SCEV::NoWrapFlags)SignOrUnsignMask);
  }

  // Sort by complexity, this groups all similar expression types together.
  GroupByComplexity(Ops, LI);

  // If there are any constants, fold them together.
  unsigned Idx = 0;
  if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {
    ++Idx;
    assert(Idx < Ops.size());
    while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
      // We found two constants, fold them together!
      Ops[0] = getConstant(LHSC->getValue()->getValue() +
                           RHSC->getValue()->getValue());
      if (Ops.size() == 2) return Ops[0];
      Ops.erase(Ops.begin()+1);  // Erase the folded element
      LHSC = cast<SCEVConstant>(Ops[0]);
    }

    // If we are left with a constant zero being added, strip it off.
    if (LHSC->getValue()->isZero()) {
      Ops.erase(Ops.begin());
      --Idx;
    }

    if (Ops.size() == 1) return Ops[0];
  }

  // Okay, check to see if the same value occurs in the operand list more than
  // once.  If so, merge them together into an multiply expression.  Since we
  // sorted the list, these values are required to be adjacent.
  Type *Ty = Ops[0]->getType();
  bool FoundMatch = false;
  for (unsigned i = 0, e = Ops.size(); i != e-1; ++i)
    if (Ops[i] == Ops[i+1]) {      //  X + Y + Y  -->  X + Y*2
      // Scan ahead to count how many equal operands there are.
      unsigned Count = 2;
      while (i+Count != e && Ops[i+Count] == Ops[i])
        ++Count;
      // Merge the values into a multiply.
      const SCEV *Scale = getConstant(Ty, Count);
      const SCEV *Mul = getMulExpr(Scale, Ops[i]);
      if (Ops.size() == Count)
        return Mul;
      Ops[i] = Mul;
      Ops.erase(Ops.begin()+i+1, Ops.begin()+i+Count);
      --i; e -= Count - 1;
      FoundMatch = true;
    }
  if (FoundMatch)
    return getAddExpr(Ops, Flags);

  // Check for truncates. If all the operands are truncated from the same
  // type, see if factoring out the truncate would permit the result to be
  // folded. eg., trunc(x) + m*trunc(n) --> trunc(x + trunc(m)*n)
  // if the contents of the resulting outer trunc fold to something simple.
  for (; Idx < Ops.size() && isa<SCEVTruncateExpr>(Ops[Idx]); ++Idx) {
    const SCEVTruncateExpr *Trunc = cast<SCEVTruncateExpr>(Ops[Idx]);
    Type *DstType = Trunc->getType();
    Type *SrcType = Trunc->getOperand()->getType();
    SmallVector<const SCEV *, 8> LargeOps;
    bool Ok = true;
    // Check all the operands to see if they can be represented in the
    // source type of the truncate.
    for (unsigned i = 0, e = Ops.size(); i != e; ++i) {
      if (const SCEVTruncateExpr *T = dyn_cast<SCEVTruncateExpr>(Ops[i])) {
        if (T->getOperand()->getType() != SrcType) {
          Ok = false;
          break;
        }
        LargeOps.push_back(T->getOperand());
      } else if (const SCEVConstant *C = dyn_cast<SCEVConstant>(Ops[i])) {
        LargeOps.push_back(getAnyExtendExpr(C, SrcType));
      } else if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(Ops[i])) {
        SmallVector<const SCEV *, 8> LargeMulOps;
        for (unsigned j = 0, f = M->getNumOperands(); j != f && Ok; ++j) {
          if (const SCEVTruncateExpr *T =
                dyn_cast<SCEVTruncateExpr>(M->getOperand(j))) {
            if (T->getOperand()->getType() != SrcType) {
              Ok = false;
              break;
            }
            LargeMulOps.push_back(T->getOperand());
          } else if (const SCEVConstant *C =
                       dyn_cast<SCEVConstant>(M->getOperand(j))) {
            LargeMulOps.push_back(getAnyExtendExpr(C, SrcType));
          } else {
            Ok = false;
            break;
          }
        }
        if (Ok)
          LargeOps.push_back(getMulExpr(LargeMulOps));
      } else {
        Ok = false;
        break;
      }
    }
    if (Ok) {
      // Evaluate the expression in the larger type.
      const SCEV *Fold = getAddExpr(LargeOps, Flags);
      // If it folds to something simple, use it. Otherwise, don't.
      if (isa<SCEVConstant>(Fold) || isa<SCEVUnknown>(Fold))
        return getTruncateExpr(Fold, DstType);
    }
  }

  // Skip past any other cast SCEVs.
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scAddExpr)
    ++Idx;

  // If there are add operands they would be next.
  if (Idx < Ops.size()) {
    bool DeletedAdd = false;
    while (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Ops[Idx])) {
      // If we have an add, expand the add operands onto the end of the operands
      // list.
      Ops.erase(Ops.begin()+Idx);
      Ops.append(Add->op_begin(), Add->op_end());
      DeletedAdd = true;
    }

    // If we deleted at least one add, we added operands to the end of the list,
    // and they are not necessarily sorted.  Recurse to resort and resimplify
    // any operands we just acquired.
    if (DeletedAdd)
      return getAddExpr(Ops);
  }

  // Skip over the add expression until we get to a multiply.
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scMulExpr)
    ++Idx;

  // Check to see if there are any folding opportunities present with
  // operands multiplied by constant values.
  if (Idx < Ops.size() && isa<SCEVMulExpr>(Ops[Idx])) {
    uint64_t BitWidth = getTypeSizeInBits(Ty);
    DenseMap<const SCEV *, APInt> M;
    SmallVector<const SCEV *, 8> NewOps;
    APInt AccumulatedConstant(BitWidth, 0);
    if (CollectAddOperandsWithScales(M, NewOps, AccumulatedConstant,
                                     Ops.data(), Ops.size(),
                                     APInt(BitWidth, 1), *this)) {
      // Some interesting folding opportunity is present, so its worthwhile to
      // re-generate the operands list. Group the operands by constant scale,
      // to avoid multiplying by the same constant scale multiple times.
      std::map<APInt, SmallVector<const SCEV *, 4>, APIntCompare> MulOpLists;
      for (SmallVectorImpl<const SCEV *>::const_iterator I = NewOps.begin(),
           E = NewOps.end(); I != E; ++I)
        MulOpLists[M.find(*I)->second].push_back(*I);
      // Re-generate the operands list.
      Ops.clear();
      if (AccumulatedConstant != 0)
        Ops.push_back(getConstant(AccumulatedConstant));
      for (std::map<APInt, SmallVector<const SCEV *, 4>, APIntCompare>::iterator
           I = MulOpLists.begin(), E = MulOpLists.end(); I != E; ++I)
        if (I->first != 0)
          Ops.push_back(getMulExpr(getConstant(I->first),
                                   getAddExpr(I->second)));
      if (Ops.empty())
        return getConstant(Ty, 0);
      if (Ops.size() == 1)
        return Ops[0];
      return getAddExpr(Ops);
    }
  }

  // If we are adding something to a multiply expression, make sure the
  // something is not already an operand of the multiply.  If so, merge it into
  // the multiply.
  for (; Idx < Ops.size() && isa<SCEVMulExpr>(Ops[Idx]); ++Idx) {
    const SCEVMulExpr *Mul = cast<SCEVMulExpr>(Ops[Idx]);
    for (unsigned MulOp = 0, e = Mul->getNumOperands(); MulOp != e; ++MulOp) {
      const SCEV *MulOpSCEV = Mul->getOperand(MulOp);
      if (isa<SCEVConstant>(MulOpSCEV))
        continue;
      for (unsigned AddOp = 0, e = Ops.size(); AddOp != e; ++AddOp)
        if (MulOpSCEV == Ops[AddOp]) {
          // Fold W + X + (X * Y * Z)  -->  W + (X * ((Y*Z)+1))
          const SCEV *InnerMul = Mul->getOperand(MulOp == 0);
          if (Mul->getNumOperands() != 2) {
            // If the multiply has more than two operands, we must get the
            // Y*Z term.
            SmallVector<const SCEV *, 4> MulOps(Mul->op_begin(),
                                                Mul->op_begin()+MulOp);
            MulOps.append(Mul->op_begin()+MulOp+1, Mul->op_end());
            InnerMul = getMulExpr(MulOps);
          }
          const SCEV *One = getConstant(Ty, 1);
          const SCEV *AddOne = getAddExpr(One, InnerMul);
          const SCEV *OuterMul = getMulExpr(AddOne, MulOpSCEV);
          if (Ops.size() == 2) return OuterMul;
          if (AddOp < Idx) {
            Ops.erase(Ops.begin()+AddOp);
            Ops.erase(Ops.begin()+Idx-1);
          } else {
            Ops.erase(Ops.begin()+Idx);
            Ops.erase(Ops.begin()+AddOp-1);
          }
          Ops.push_back(OuterMul);
          return getAddExpr(Ops);
        }

      // Check this multiply against other multiplies being added together.
      for (unsigned OtherMulIdx = Idx+1;
           OtherMulIdx < Ops.size() && isa<SCEVMulExpr>(Ops[OtherMulIdx]);
           ++OtherMulIdx) {
        const SCEVMulExpr *OtherMul = cast<SCEVMulExpr>(Ops[OtherMulIdx]);
        // If MulOp occurs in OtherMul, we can fold the two multiplies
        // together.
        for (unsigned OMulOp = 0, e = OtherMul->getNumOperands();
             OMulOp != e; ++OMulOp)
          if (OtherMul->getOperand(OMulOp) == MulOpSCEV) {
            // Fold X + (A*B*C) + (A*D*E) --> X + (A*(B*C+D*E))
            const SCEV *InnerMul1 = Mul->getOperand(MulOp == 0);
            if (Mul->getNumOperands() != 2) {
              SmallVector<const SCEV *, 4> MulOps(Mul->op_begin(),
                                                  Mul->op_begin()+MulOp);
              MulOps.append(Mul->op_begin()+MulOp+1, Mul->op_end());
              InnerMul1 = getMulExpr(MulOps);
            }
            const SCEV *InnerMul2 = OtherMul->getOperand(OMulOp == 0);
            if (OtherMul->getNumOperands() != 2) {
              SmallVector<const SCEV *, 4> MulOps(OtherMul->op_begin(),
                                                  OtherMul->op_begin()+OMulOp);
              MulOps.append(OtherMul->op_begin()+OMulOp+1, OtherMul->op_end());
              InnerMul2 = getMulExpr(MulOps);
            }
            const SCEV *InnerMulSum = getAddExpr(InnerMul1,InnerMul2);
            const SCEV *OuterMul = getMulExpr(MulOpSCEV, InnerMulSum);
            if (Ops.size() == 2) return OuterMul;
            Ops.erase(Ops.begin()+Idx);
            Ops.erase(Ops.begin()+OtherMulIdx-1);
            Ops.push_back(OuterMul);
            return getAddExpr(Ops);
          }
      }
    }
  }

  // If there are any add recurrences in the operands list, see if any other
  // added values are loop invariant.  If so, we can fold them into the
  // recurrence.
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scAddRecExpr)
    ++Idx;

  // Scan over all recurrences, trying to fold loop invariants into them.
  for (; Idx < Ops.size() && isa<SCEVAddRecExpr>(Ops[Idx]); ++Idx) {
    // Scan all of the other operands to this add and add them to the vector if
    // they are loop invariant w.r.t. the recurrence.
    SmallVector<const SCEV *, 8> LIOps;
    const SCEVAddRecExpr *AddRec = cast<SCEVAddRecExpr>(Ops[Idx]);
    const Loop *AddRecLoop = AddRec->getLoop();
    for (unsigned i = 0, e = Ops.size(); i != e; ++i)
      if (isLoopInvariant(Ops[i], AddRecLoop)) {
        LIOps.push_back(Ops[i]);
        Ops.erase(Ops.begin()+i);
        --i; --e;
      }

    // If we found some loop invariants, fold them into the recurrence.
    if (!LIOps.empty()) {
      //  NLI + LI + {Start,+,Step}  -->  NLI + {LI+Start,+,Step}
      LIOps.push_back(AddRec->getStart());

      SmallVector<const SCEV *, 4> AddRecOps(AddRec->op_begin(),
                                             AddRec->op_end());
      AddRecOps[0] = getAddExpr(LIOps);

      // Build the new addrec. Propagate the NUW and NSW flags if both the
      // outer add and the inner addrec are guaranteed to have no overflow.
      // Always propagate NW.
      Flags = AddRec->getNoWrapFlags(setFlags(Flags, SCEV::FlagNW));
      const SCEV *NewRec = getAddRecExpr(AddRecOps, AddRecLoop, Flags);

      // If all of the other operands were loop invariant, we are done.
      if (Ops.size() == 1) return NewRec;

      // Otherwise, add the folded AddRec by the non-invariant parts.
      for (unsigned i = 0;; ++i)
        if (Ops[i] == AddRec) {
          Ops[i] = NewRec;
          break;
        }
      return getAddExpr(Ops);
    }

    // Okay, if there weren't any loop invariants to be folded, check to see if
    // there are multiple AddRec's with the same loop induction variable being
    // added together.  If so, we can fold them.
    for (unsigned OtherIdx = Idx+1;
         OtherIdx < Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
         ++OtherIdx)
      if (AddRecLoop == cast<SCEVAddRecExpr>(Ops[OtherIdx])->getLoop()) {
        // Other + {A,+,B}<L> + {C,+,D}<L>  -->  Other + {A+C,+,B+D}<L>
        SmallVector<const SCEV *, 4> AddRecOps(AddRec->op_begin(),
                                               AddRec->op_end());
        for (; OtherIdx != Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
             ++OtherIdx)
          if (const SCEVAddRecExpr *OtherAddRec =
                dyn_cast<SCEVAddRecExpr>(Ops[OtherIdx]))
            if (OtherAddRec->getLoop() == AddRecLoop) {
              for (unsigned i = 0, e = OtherAddRec->getNumOperands();
                   i != e; ++i) {
                if (i >= AddRecOps.size()) {
                  AddRecOps.append(OtherAddRec->op_begin()+i,
                                   OtherAddRec->op_end());
                  break;
                }
                AddRecOps[i] = getAddExpr(AddRecOps[i],
                                          OtherAddRec->getOperand(i));
              }
              Ops.erase(Ops.begin() + OtherIdx); --OtherIdx;
            }
        // Step size has changed, so we cannot guarantee no self-wraparound.
        Ops[Idx] = getAddRecExpr(AddRecOps, AddRecLoop, SCEV::FlagAnyWrap);
        return getAddExpr(Ops);
      }

    // Otherwise couldn't fold anything into this recurrence.  Move onto the
    // next one.
  }

  // Okay, it looks like we really DO need an add expr.  Check to see if we
  // already have one, otherwise create a new one.
  FoldingSetNodeID ID;
  ID.AddInteger(scAddExpr);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  void *IP = 0;
  SCEVAddExpr *S =
    static_cast<SCEVAddExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));
  if (!S) {
    const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
    std::uninitialized_copy(Ops.begin(), Ops.end(), O);
    S = new (SCEVAllocator) SCEVAddExpr(ID.Intern(SCEVAllocator),
                                        O, Ops.size());
    UniqueSCEVs.InsertNode(S, IP);
  }
  S->setNoWrapFlags(Flags);
  return S;
}

static uint64_t umul_ov(uint64_t i, uint64_t j, bool &Overflow) {
  uint64_t k = i*j;
  if (j > 1 && k / j != i) Overflow = true;
  return k;
}

/// Compute the result of "n choose k", the binomial coefficient.  If an
/// intermediate computation overflows, Overflow will be set and the return will
/// be garbage. Overflow is not cleared on absence of overflow.
static uint64_t Choose(uint64_t n, uint64_t k, bool &Overflow) {
  // We use the multiplicative formula:
  //     n(n-1)(n-2)...(n-(k-1)) / k(k-1)(k-2)...1 .
  // At each iteration, we take the n-th term of the numeral and divide by the
  // (k-n)th term of the denominator.  This division will always produce an
  // integral result, and helps reduce the chance of overflow in the
  // intermediate computations. However, we can still overflow even when the
  // final result would fit.

  if (n == 0 || n == k) return 1;
  if (k > n) return 0;

  if (k > n/2)
    k = n-k;

  uint64_t r = 1;
  for (uint64_t i = 1; i <= k; ++i) {
    r = umul_ov(r, n-(i-1), Overflow);
    r /= i;
  }
  return r;
}

/// getMulExpr - Get a canonical multiply expression, or something simpler if
/// possible.
const SCEV *ScalarEvolution::getMulExpr(SmallVectorImpl<const SCEV *> &Ops,
                                        SCEV::NoWrapFlags Flags) {
  assert(Flags == maskFlags(Flags, SCEV::FlagNUW | SCEV::FlagNSW) &&
         "only nuw or nsw allowed");
  assert(!Ops.empty() && "Cannot get empty mul!");
  if (Ops.size() == 1) return Ops[0];
#ifndef NDEBUG
  Type *ETy = getEffectiveSCEVType(Ops[0]->getType());
  for (unsigned i = 1, e = Ops.size(); i != e; ++i)
    assert(getEffectiveSCEVType(Ops[i]->getType()) == ETy &&
           "SCEVMulExpr operand types don't match!");
#endif

  // If FlagNSW is true and all the operands are non-negative, infer FlagNUW.
  // And vice-versa.
  int SignOrUnsignMask = SCEV::FlagNUW | SCEV::FlagNSW;
  SCEV::NoWrapFlags SignOrUnsignWrap = maskFlags(Flags, SignOrUnsignMask);
  if (SignOrUnsignWrap && (SignOrUnsignWrap != SignOrUnsignMask)) {
    bool All = true;
    for (SmallVectorImpl<const SCEV *>::const_iterator I = Ops.begin(),
         E = Ops.end(); I != E; ++I)
      if (!isKnownNonNegative(*I)) {
        All = false;
        break;
      }
    if (All) Flags = setFlags(Flags, (SCEV::NoWrapFlags)SignOrUnsignMask);
  }

  // Sort by complexity, this groups all similar expression types together.
  GroupByComplexity(Ops, LI);

  // If there are any constants, fold them together.
  unsigned Idx = 0;
  if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {

    // C1*(C2+V) -> C1*C2 + C1*V
    if (Ops.size() == 2)
      if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Ops[1]))
        if (Add->getNumOperands() == 2 &&
            isa<SCEVConstant>(Add->getOperand(0)))
          return getAddExpr(getMulExpr(LHSC, Add->getOperand(0)),
                            getMulExpr(LHSC, Add->getOperand(1)));

    ++Idx;
    while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
      // We found two constants, fold them together!
      ConstantInt *Fold = ConstantInt::get(getContext(),
                                           LHSC->getValue()->getValue() *
                                           RHSC->getValue()->getValue());
      Ops[0] = getConstant(Fold);
      Ops.erase(Ops.begin()+1);  // Erase the folded element
      if (Ops.size() == 1) return Ops[0];
      LHSC = cast<SCEVConstant>(Ops[0]);
    }

    // If we are left with a constant one being multiplied, strip it off.
    if (cast<SCEVConstant>(Ops[0])->getValue()->equalsInt(1)) {
      Ops.erase(Ops.begin());
      --Idx;
    } else if (cast<SCEVConstant>(Ops[0])->getValue()->isZero()) {
      // If we have a multiply of zero, it will always be zero.
      return Ops[0];
    } else if (Ops[0]->isAllOnesValue()) {
      // If we have a mul by -1 of an add, try distributing the -1 among the
      // add operands.
      if (Ops.size() == 2) {
        if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Ops[1])) {
          SmallVector<const SCEV *, 4> NewOps;
          bool AnyFolded = false;
          for (SCEVAddRecExpr::op_iterator I = Add->op_begin(),
                 E = Add->op_end(); I != E; ++I) {
            const SCEV *Mul = getMulExpr(Ops[0], *I);
            if (!isa<SCEVMulExpr>(Mul)) AnyFolded = true;
            NewOps.push_back(Mul);
          }
          if (AnyFolded)
            return getAddExpr(NewOps);
        }
        else if (const SCEVAddRecExpr *
                 AddRec = dyn_cast<SCEVAddRecExpr>(Ops[1])) {
          // Negation preserves a recurrence's no self-wrap property.
          SmallVector<const SCEV *, 4> Operands;
          for (SCEVAddRecExpr::op_iterator I = AddRec->op_begin(),
                 E = AddRec->op_end(); I != E; ++I) {
            Operands.push_back(getMulExpr(Ops[0], *I));
          }
          return getAddRecExpr(Operands, AddRec->getLoop(),
                               AddRec->getNoWrapFlags(SCEV::FlagNW));
        }
      }
    }

    if (Ops.size() == 1)
      return Ops[0];
  }

  // Skip over the add expression until we get to a multiply.
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scMulExpr)
    ++Idx;

  // If there are mul operands inline them all into this expression.
  if (Idx < Ops.size()) {
    bool DeletedMul = false;
    while (const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(Ops[Idx])) {
      // If we have an mul, expand the mul operands onto the end of the operands
      // list.
      Ops.erase(Ops.begin()+Idx);
      Ops.append(Mul->op_begin(), Mul->op_end());
      DeletedMul = true;
    }

    // If we deleted at least one mul, we added operands to the end of the list,
    // and they are not necessarily sorted.  Recurse to resort and resimplify
    // any operands we just acquired.
    if (DeletedMul)
      return getMulExpr(Ops);
  }

  // If there are any add recurrences in the operands list, see if any other
  // added values are loop invariant.  If so, we can fold them into the
  // recurrence.
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scAddRecExpr)
    ++Idx;

  // Scan over all recurrences, trying to fold loop invariants into them.
  for (; Idx < Ops.size() && isa<SCEVAddRecExpr>(Ops[Idx]); ++Idx) {
    // Scan all of the other operands to this mul and add them to the vector if
    // they are loop invariant w.r.t. the recurrence.
    SmallVector<const SCEV *, 8> LIOps;
    const SCEVAddRecExpr *AddRec = cast<SCEVAddRecExpr>(Ops[Idx]);
    const Loop *AddRecLoop = AddRec->getLoop();
    for (unsigned i = 0, e = Ops.size(); i != e; ++i)
      if (isLoopInvariant(Ops[i], AddRecLoop)) {
        LIOps.push_back(Ops[i]);
        Ops.erase(Ops.begin()+i);
        --i; --e;
      }

    // If we found some loop invariants, fold them into the recurrence.
    if (!LIOps.empty()) {
      //  NLI * LI * {Start,+,Step}  -->  NLI * {LI*Start,+,LI*Step}
      SmallVector<const SCEV *, 4> NewOps;
      NewOps.reserve(AddRec->getNumOperands());
      const SCEV *Scale = getMulExpr(LIOps);
      for (unsigned i = 0, e = AddRec->getNumOperands(); i != e; ++i)
        NewOps.push_back(getMulExpr(Scale, AddRec->getOperand(i)));

      // Build the new addrec. Propagate the NUW and NSW flags if both the
      // outer mul and the inner addrec are guaranteed to have no overflow.
      //
      // No self-wrap cannot be guaranteed after changing the step size, but
      // will be inferred if either NUW or NSW is true.
      Flags = AddRec->getNoWrapFlags(clearFlags(Flags, SCEV::FlagNW));
      const SCEV *NewRec = getAddRecExpr(NewOps, AddRecLoop, Flags);

      // If all of the other operands were loop invariant, we are done.
      if (Ops.size() == 1) return NewRec;

      // Otherwise, multiply the folded AddRec by the non-invariant parts.
      for (unsigned i = 0;; ++i)
        if (Ops[i] == AddRec) {
          Ops[i] = NewRec;
          break;
        }
      return getMulExpr(Ops);
    }

    // Okay, if there weren't any loop invariants to be folded, check to see if
    // there are multiple AddRec's with the same loop induction variable being
    // multiplied together.  If so, we can fold them.
    for (unsigned OtherIdx = Idx+1;
         OtherIdx < Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
         ++OtherIdx) {
      if (AddRecLoop != cast<SCEVAddRecExpr>(Ops[OtherIdx])->getLoop())
        continue;

      // {A1,+,A2,+,...,+,An}<L> * {B1,+,B2,+,...,+,Bn}<L>
      // = {x=1 in [ sum y=x..2x [ sum z=max(y-x, y-n)..min(x,n) [
      //       choose(x, 2x)*choose(2x-y, x-z)*A_{y-z}*B_z
      //   ]]],+,...up to x=2n}.
      // Note that the arguments to choose() are always integers with values
      // known at compile time, never SCEV objects.
      //
      // The implementation avoids pointless extra computations when the two
      // addrec's are of different length (mathematically, it's equivalent to
      // an infinite stream of zeros on the right).
      bool OpsModified = false;
      for (; OtherIdx != Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
           ++OtherIdx) {
        const SCEVAddRecExpr *OtherAddRec =
          dyn_cast<SCEVAddRecExpr>(Ops[OtherIdx]);
        if (!OtherAddRec || OtherAddRec->getLoop() != AddRecLoop)
          continue;

        bool Overflow = false;
        Type *Ty = AddRec->getType();
        bool LargerThan64Bits = getTypeSizeInBits(Ty) > 64;
        SmallVector<const SCEV*, 7> AddRecOps;
        for (int x = 0, xe = AddRec->getNumOperands() +
               OtherAddRec->getNumOperands() - 1; x != xe && !Overflow; ++x) {
          const SCEV *Term = getConstant(Ty, 0);
          for (int y = x, ye = 2*x+1; y != ye && !Overflow; ++y) {
            uint64_t Coeff1 = Choose(x, 2*x - y, Overflow);
            for (int z = std::max(y-x, y-(int)AddRec->getNumOperands()+1),
                   ze = std::min(x+1, (int)OtherAddRec->getNumOperands());
                 z < ze && !Overflow; ++z) {
              uint64_t Coeff2 = Choose(2*x - y, x-z, Overflow);
              uint64_t Coeff;
              if (LargerThan64Bits)
                Coeff = umul_ov(Coeff1, Coeff2, Overflow);
              else
                Coeff = Coeff1*Coeff2;
              const SCEV *CoeffTerm = getConstant(Ty, Coeff);
              const SCEV *Term1 = AddRec->getOperand(y-z);
              const SCEV *Term2 = OtherAddRec->getOperand(z);
              Term = getAddExpr(Term, getMulExpr(CoeffTerm, Term1,Term2));
            }
          }
          AddRecOps.push_back(Term);
        }
        if (!Overflow) {
          const SCEV *NewAddRec = getAddRecExpr(AddRecOps, AddRec->getLoop(),
                                                SCEV::FlagAnyWrap);
          if (Ops.size() == 2) return NewAddRec;
          Ops[Idx] = NewAddRec;
          Ops.erase(Ops.begin() + OtherIdx); --OtherIdx;
          OpsModified = true;
          AddRec = dyn_cast<SCEVAddRecExpr>(NewAddRec);
          if (!AddRec)
            break;
        }
      }
      if (OpsModified)
        return getMulExpr(Ops);
    }

    // Otherwise couldn't fold anything into this recurrence.  Move onto the
    // next one.
  }

  // Okay, it looks like we really DO need an mul expr.  Check to see if we
  // already have one, otherwise create a new one.
  FoldingSetNodeID ID;
  ID.AddInteger(scMulExpr);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  void *IP = 0;
  SCEVMulExpr *S =
    static_cast<SCEVMulExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));
  if (!S) {
    const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
    std::uninitialized_copy(Ops.begin(), Ops.end(), O);
    S = new (SCEVAllocator) SCEVMulExpr(ID.Intern(SCEVAllocator),
                                        O, Ops.size());
    UniqueSCEVs.InsertNode(S, IP);
  }
  S->setNoWrapFlags(Flags);
  return S;
}

/// getUDivExpr - Get a canonical unsigned division expression, or something
/// simpler if possible.
const SCEV *ScalarEvolution::getUDivExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  assert(getEffectiveSCEVType(LHS->getType()) ==
         getEffectiveSCEVType(RHS->getType()) &&
         "SCEVUDivExpr operand types don't match!");

  if (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS)) {
    if (RHSC->getValue()->equalsInt(1))
      return LHS;                               // X udiv 1 --> x
    // If the denominator is zero, the result of the udiv is undefined. Don't
    // try to analyze it, because the resolution chosen here may differ from
    // the resolution chosen in other parts of the compiler.
    if (!RHSC->getValue()->isZero()) {
      // Determine if the division can be folded into the operands of
      // its operands.
      // TODO: Generalize this to non-constants by using known-bits information.
      Type *Ty = LHS->getType();
      unsigned LZ = RHSC->getValue()->getValue().countLeadingZeros();
      unsigned MaxShiftAmt = getTypeSizeInBits(Ty) - LZ - 1;
      // For non-power-of-two values, effectively round the value up to the
      // nearest power of two.
      if (!RHSC->getValue()->getValue().isPowerOf2())
        ++MaxShiftAmt;
      IntegerType *ExtTy =
        IntegerType::get(getContext(), getTypeSizeInBits(Ty) + MaxShiftAmt);
      if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(LHS))
        if (const SCEVConstant *Step =
            dyn_cast<SCEVConstant>(AR->getStepRecurrence(*this))) {
          // {X,+,N}/C --> {X/C,+,N/C} if safe and N/C can be folded.
          const APInt &StepInt = Step->getValue()->getValue();
          const APInt &DivInt = RHSC->getValue()->getValue();
          if (!StepInt.urem(DivInt) &&
              getZeroExtendExpr(AR, ExtTy) ==
              getAddRecExpr(getZeroExtendExpr(AR->getStart(), ExtTy),
                            getZeroExtendExpr(Step, ExtTy),
                            AR->getLoop(), SCEV::FlagAnyWrap)) {
            SmallVector<const SCEV *, 4> Operands;
            for (unsigned i = 0, e = AR->getNumOperands(); i != e; ++i)
              Operands.push_back(getUDivExpr(AR->getOperand(i), RHS));
            return getAddRecExpr(Operands, AR->getLoop(),
                                 SCEV::FlagNW);
          }
          /// Get a canonical UDivExpr for a recurrence.
          /// {X,+,N}/C => {Y,+,N}/C where Y=X-(X%N). Safe when C%N=0.
          // We can currently only fold X%N if X is constant.
          const SCEVConstant *StartC = dyn_cast<SCEVConstant>(AR->getStart());
          if (StartC && !DivInt.urem(StepInt) &&
              getZeroExtendExpr(AR, ExtTy) ==
              getAddRecExpr(getZeroExtendExpr(AR->getStart(), ExtTy),
                            getZeroExtendExpr(Step, ExtTy),
                            AR->getLoop(), SCEV::FlagAnyWrap)) {
            const APInt &StartInt = StartC->getValue()->getValue();
            const APInt &StartRem = StartInt.urem(StepInt);
            if (StartRem != 0)
              LHS = getAddRecExpr(getConstant(StartInt - StartRem), Step,
                                  AR->getLoop(), SCEV::FlagNW);
          }
        }
      // (A*B)/C --> A*(B/C) if safe and B/C can be folded.
      if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(LHS)) {
        SmallVector<const SCEV *, 4> Operands;
        for (unsigned i = 0, e = M->getNumOperands(); i != e; ++i)
          Operands.push_back(getZeroExtendExpr(M->getOperand(i), ExtTy));
        if (getZeroExtendExpr(M, ExtTy) == getMulExpr(Operands))
          // Find an operand that's safely divisible.
          for (unsigned i = 0, e = M->getNumOperands(); i != e; ++i) {
            const SCEV *Op = M->getOperand(i);
            const SCEV *Div = getUDivExpr(Op, RHSC);
            if (!isa<SCEVUDivExpr>(Div) && getMulExpr(Div, RHSC) == Op) {
              Operands = SmallVector<const SCEV *, 4>(M->op_begin(),
                                                      M->op_end());
              Operands[i] = Div;
              return getMulExpr(Operands);
            }
          }
      }
      // (A+B)/C --> (A/C + B/C) if safe and A/C and B/C can be folded.
      if (const SCEVAddExpr *A = dyn_cast<SCEVAddExpr>(LHS)) {
        SmallVector<const SCEV *, 4> Operands;
        for (unsigned i = 0, e = A->getNumOperands(); i != e; ++i)
          Operands.push_back(getZeroExtendExpr(A->getOperand(i), ExtTy));
        if (getZeroExtendExpr(A, ExtTy) == getAddExpr(Operands)) {
          Operands.clear();
          for (unsigned i = 0, e = A->getNumOperands(); i != e; ++i) {
            const SCEV *Op = getUDivExpr(A->getOperand(i), RHS);
            if (isa<SCEVUDivExpr>(Op) ||
                getMulExpr(Op, RHS) != A->getOperand(i))
              break;
            Operands.push_back(Op);
          }
          if (Operands.size() == A->getNumOperands())
            return getAddExpr(Operands);
        }
      }

      // Fold if both operands are constant.
      if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(LHS)) {
        Constant *LHSCV = LHSC->getValue();
        Constant *RHSCV = RHSC->getValue();
        return getConstant(cast<ConstantInt>(ConstantExpr::getUDiv(LHSCV,
                                                                   RHSCV)));
      }
    }
  }

  FoldingSetNodeID ID;
  ID.AddInteger(scUDivExpr);
  ID.AddPointer(LHS);
  ID.AddPointer(RHS);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  SCEV *S = new (SCEVAllocator) SCEVUDivExpr(ID.Intern(SCEVAllocator),
                                             LHS, RHS);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}


/// getAddRecExpr - Get an add recurrence expression for the specified loop.
/// Simplify the expression as much as possible.
const SCEV *ScalarEvolution::getAddRecExpr(const SCEV *Start, const SCEV *Step,
                                           const Loop *L,
                                           SCEV::NoWrapFlags Flags) {
  SmallVector<const SCEV *, 4> Operands;
  Operands.push_back(Start);
  if (const SCEVAddRecExpr *StepChrec = dyn_cast<SCEVAddRecExpr>(Step))
    if (StepChrec->getLoop() == L) {
      Operands.append(StepChrec->op_begin(), StepChrec->op_end());
      return getAddRecExpr(Operands, L, maskFlags(Flags, SCEV::FlagNW));
    }

  Operands.push_back(Step);
  return getAddRecExpr(Operands, L, Flags);
}

/// getAddRecExpr - Get an add recurrence expression for the specified loop.
/// Simplify the expression as much as possible.
const SCEV *
ScalarEvolution::getAddRecExpr(SmallVectorImpl<const SCEV *> &Operands,
                               const Loop *L, SCEV::NoWrapFlags Flags) {
  if (Operands.size() == 1) return Operands[0];
#ifndef NDEBUG
  Type *ETy = getEffectiveSCEVType(Operands[0]->getType());
  for (unsigned i = 1, e = Operands.size(); i != e; ++i)
    assert(getEffectiveSCEVType(Operands[i]->getType()) == ETy &&
           "SCEVAddRecExpr operand types don't match!");
  for (unsigned i = 0, e = Operands.size(); i != e; ++i)
    assert(isLoopInvariant(Operands[i], L) &&
           "SCEVAddRecExpr operand is not loop-invariant!");
#endif

  if (Operands.back()->isZero()) {
    Operands.pop_back();
    return getAddRecExpr(Operands, L, SCEV::FlagAnyWrap); // {X,+,0}  -->  X
  }

  // It's tempting to want to call getMaxBackedgeTakenCount count here and
  // use that information to infer NUW and NSW flags. However, computing a
  // BE count requires calling getAddRecExpr, so we may not yet have a
  // meaningful BE count at this point (and if we don't, we'd be stuck
  // with a SCEVCouldNotCompute as the cached BE count).

  // If FlagNSW is true and all the operands are non-negative, infer FlagNUW.
  // And vice-versa.
  int SignOrUnsignMask = SCEV::FlagNUW | SCEV::FlagNSW;
  SCEV::NoWrapFlags SignOrUnsignWrap = maskFlags(Flags, SignOrUnsignMask);
  if (SignOrUnsignWrap && (SignOrUnsignWrap != SignOrUnsignMask)) {
    bool All = true;
    for (SmallVectorImpl<const SCEV *>::const_iterator I = Operands.begin(),
         E = Operands.end(); I != E; ++I)
      if (!isKnownNonNegative(*I)) {
        All = false;
        break;
      }
    if (All) Flags = setFlags(Flags, (SCEV::NoWrapFlags)SignOrUnsignMask);
  }

  // Canonicalize nested AddRecs in by nesting them in order of loop depth.
  if (const SCEVAddRecExpr *NestedAR = dyn_cast<SCEVAddRecExpr>(Operands[0])) {
    const Loop *NestedLoop = NestedAR->getLoop();
    if (L->contains(NestedLoop) ?
        (L->getLoopDepth() < NestedLoop->getLoopDepth()) :
        (!NestedLoop->contains(L) &&
         DT->dominates(L->getHeader(), NestedLoop->getHeader()))) {
      SmallVector<const SCEV *, 4> NestedOperands(NestedAR->op_begin(),
                                                  NestedAR->op_end());
      Operands[0] = NestedAR->getStart();
      // AddRecs require their operands be loop-invariant with respect to their
      // loops. Don't perform this transformation if it would break this
      // requirement.
      bool AllInvariant = true;
      for (unsigned i = 0, e = Operands.size(); i != e; ++i)
        if (!isLoopInvariant(Operands[i], L)) {
          AllInvariant = false;
          break;
        }
      if (AllInvariant) {
        // Create a recurrence for the outer loop with the same step size.
        //
        // The outer recurrence keeps its NW flag but only keeps NUW/NSW if the
        // inner recurrence has the same property.
        SCEV::NoWrapFlags OuterFlags =
          maskFlags(Flags, SCEV::FlagNW | NestedAR->getNoWrapFlags());

        NestedOperands[0] = getAddRecExpr(Operands, L, OuterFlags);
        AllInvariant = true;
        for (unsigned i = 0, e = NestedOperands.size(); i != e; ++i)
          if (!isLoopInvariant(NestedOperands[i], NestedLoop)) {
            AllInvariant = false;
            break;
          }
        if (AllInvariant) {
          // Ok, both add recurrences are valid after the transformation.
          //
          // The inner recurrence keeps its NW flag but only keeps NUW/NSW if
          // the outer recurrence has the same property.
          SCEV::NoWrapFlags InnerFlags =
            maskFlags(NestedAR->getNoWrapFlags(), SCEV::FlagNW | Flags);
          return getAddRecExpr(NestedOperands, NestedLoop, InnerFlags);
        }
      }
      // Reset Operands to its original state.
      Operands[0] = NestedAR;
    }
  }

  // Okay, it looks like we really DO need an addrec expr.  Check to see if we
  // already have one, otherwise create a new one.
  FoldingSetNodeID ID;
  ID.AddInteger(scAddRecExpr);
  for (unsigned i = 0, e = Operands.size(); i != e; ++i)
    ID.AddPointer(Operands[i]);
  ID.AddPointer(L);
  void *IP = 0;
  SCEVAddRecExpr *S =
    static_cast<SCEVAddRecExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));
  if (!S) {
    const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Operands.size());
    std::uninitialized_copy(Operands.begin(), Operands.end(), O);
    S = new (SCEVAllocator) SCEVAddRecExpr(ID.Intern(SCEVAllocator),
                                           O, Operands.size(), L);
    UniqueSCEVs.InsertNode(S, IP);
  }
  S->setNoWrapFlags(Flags);
  return S;
}

const SCEV *ScalarEvolution::getSMaxExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  SmallVector<const SCEV *, 2> Ops;
  Ops.push_back(LHS);
  Ops.push_back(RHS);
  return getSMaxExpr(Ops);
}

const SCEV *
ScalarEvolution::getSMaxExpr(SmallVectorImpl<const SCEV *> &Ops) {
  assert(!Ops.empty() && "Cannot get empty smax!");
  if (Ops.size() == 1) return Ops[0];
#ifndef NDEBUG
  Type *ETy = getEffectiveSCEVType(Ops[0]->getType());
  for (unsigned i = 1, e = Ops.size(); i != e; ++i)
    assert(getEffectiveSCEVType(Ops[i]->getType()) == ETy &&
           "SCEVSMaxExpr operand types don't match!");
#endif

  // Sort by complexity, this groups all similar expression types together.
  GroupByComplexity(Ops, LI);

  // If there are any constants, fold them together.
  unsigned Idx = 0;
  if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {
    ++Idx;
    assert(Idx < Ops.size());
    while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
      // We found two constants, fold them together!
      ConstantInt *Fold = ConstantInt::get(getContext(),
                              APIntOps::smax(LHSC->getValue()->getValue(),
                                             RHSC->getValue()->getValue()));
      Ops[0] = getConstant(Fold);
      Ops.erase(Ops.begin()+1);  // Erase the folded element
      if (Ops.size() == 1) return Ops[0];
      LHSC = cast<SCEVConstant>(Ops[0]);
    }

    // If we are left with a constant minimum-int, strip it off.
    if (cast<SCEVConstant>(Ops[0])->getValue()->isMinValue(true)) {
      Ops.erase(Ops.begin());
      --Idx;
    } else if (cast<SCEVConstant>(Ops[0])->getValue()->isMaxValue(true)) {
      // If we have an smax with a constant maximum-int, it will always be
      // maximum-int.
      return Ops[0];
    }

    if (Ops.size() == 1) return Ops[0];
  }

  // Find the first SMax
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scSMaxExpr)
    ++Idx;

  // Check to see if one of the operands is an SMax. If so, expand its operands
  // onto our operand list, and recurse to simplify.
  if (Idx < Ops.size()) {
    bool DeletedSMax = false;
    while (const SCEVSMaxExpr *SMax = dyn_cast<SCEVSMaxExpr>(Ops[Idx])) {
      Ops.erase(Ops.begin()+Idx);
      Ops.append(SMax->op_begin(), SMax->op_end());
      DeletedSMax = true;
    }

    if (DeletedSMax)
      return getSMaxExpr(Ops);
  }

  // Okay, check to see if the same value occurs in the operand list twice.  If
  // so, delete one.  Since we sorted the list, these values are required to
  // be adjacent.
  for (unsigned i = 0, e = Ops.size()-1; i != e; ++i)
    //  X smax Y smax Y  -->  X smax Y
    //  X smax Y         -->  X, if X is always greater than Y
    if (Ops[i] == Ops[i+1] ||
        isKnownPredicate(ICmpInst::ICMP_SGE, Ops[i], Ops[i+1])) {
      Ops.erase(Ops.begin()+i+1, Ops.begin()+i+2);
      --i; --e;
    } else if (isKnownPredicate(ICmpInst::ICMP_SLE, Ops[i], Ops[i+1])) {
      Ops.erase(Ops.begin()+i, Ops.begin()+i+1);
      --i; --e;
    }

  if (Ops.size() == 1) return Ops[0];

  assert(!Ops.empty() && "Reduced smax down to nothing!");

  // Okay, it looks like we really DO need an smax expr.  Check to see if we
  // already have one, otherwise create a new one.
  FoldingSetNodeID ID;
  ID.AddInteger(scSMaxExpr);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
  std::uninitialized_copy(Ops.begin(), Ops.end(), O);
  SCEV *S = new (SCEVAllocator) SCEVSMaxExpr(ID.Intern(SCEVAllocator),
                                             O, Ops.size());
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

const SCEV *ScalarEvolution::getUMaxExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  SmallVector<const SCEV *, 2> Ops;
  Ops.push_back(LHS);
  Ops.push_back(RHS);
  return getUMaxExpr(Ops);
}

const SCEV *
ScalarEvolution::getUMaxExpr(SmallVectorImpl<const SCEV *> &Ops) {
  assert(!Ops.empty() && "Cannot get empty umax!");
  if (Ops.size() == 1) return Ops[0];
#ifndef NDEBUG
  Type *ETy = getEffectiveSCEVType(Ops[0]->getType());
  for (unsigned i = 1, e = Ops.size(); i != e; ++i)
    assert(getEffectiveSCEVType(Ops[i]->getType()) == ETy &&
           "SCEVUMaxExpr operand types don't match!");
#endif

  // Sort by complexity, this groups all similar expression types together.
  GroupByComplexity(Ops, LI);

  // If there are any constants, fold them together.
  unsigned Idx = 0;
  if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {
    ++Idx;
    assert(Idx < Ops.size());
    while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
      // We found two constants, fold them together!
      ConstantInt *Fold = ConstantInt::get(getContext(),
                              APIntOps::umax(LHSC->getValue()->getValue(),
                                             RHSC->getValue()->getValue()));
      Ops[0] = getConstant(Fold);
      Ops.erase(Ops.begin()+1);  // Erase the folded element
      if (Ops.size() == 1) return Ops[0];
      LHSC = cast<SCEVConstant>(Ops[0]);
    }

    // If we are left with a constant minimum-int, strip it off.
    if (cast<SCEVConstant>(Ops[0])->getValue()->isMinValue(false)) {
      Ops.erase(Ops.begin());
      --Idx;
    } else if (cast<SCEVConstant>(Ops[0])->getValue()->isMaxValue(false)) {
      // If we have an umax with a constant maximum-int, it will always be
      // maximum-int.
      return Ops[0];
    }

    if (Ops.size() == 1) return Ops[0];
  }

  // Find the first UMax
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scUMaxExpr)
    ++Idx;

  // Check to see if one of the operands is a UMax. If so, expand its operands
  // onto our operand list, and recurse to simplify.
  if (Idx < Ops.size()) {
    bool DeletedUMax = false;
    while (const SCEVUMaxExpr *UMax = dyn_cast<SCEVUMaxExpr>(Ops[Idx])) {
      Ops.erase(Ops.begin()+Idx);
      Ops.append(UMax->op_begin(), UMax->op_end());
      DeletedUMax = true;
    }

    if (DeletedUMax)
      return getUMaxExpr(Ops);
  }

  // Okay, check to see if the same value occurs in the operand list twice.  If
  // so, delete one.  Since we sorted the list, these values are required to
  // be adjacent.
  for (unsigned i = 0, e = Ops.size()-1; i != e; ++i)
    //  X umax Y umax Y  -->  X umax Y
    //  X umax Y         -->  X, if X is always greater than Y
    if (Ops[i] == Ops[i+1] ||
        isKnownPredicate(ICmpInst::ICMP_UGE, Ops[i], Ops[i+1])) {
      Ops.erase(Ops.begin()+i+1, Ops.begin()+i+2);
      --i; --e;
    } else if (isKnownPredicate(ICmpInst::ICMP_ULE, Ops[i], Ops[i+1])) {
      Ops.erase(Ops.begin()+i, Ops.begin()+i+1);
      --i; --e;
    }

  if (Ops.size() == 1) return Ops[0];

  assert(!Ops.empty() && "Reduced umax down to nothing!");

  // Okay, it looks like we really DO need a umax expr.  Check to see if we
  // already have one, otherwise create a new one.
  FoldingSetNodeID ID;
  ID.AddInteger(scUMaxExpr);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
  std::uninitialized_copy(Ops.begin(), Ops.end(), O);
  SCEV *S = new (SCEVAllocator) SCEVUMaxExpr(ID.Intern(SCEVAllocator),
                                             O, Ops.size());
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

const SCEV *ScalarEvolution::getSMinExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  // ~smax(~x, ~y) == smin(x, y).
  return getNotSCEV(getSMaxExpr(getNotSCEV(LHS), getNotSCEV(RHS)));
}

const SCEV *ScalarEvolution::getUMinExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  // ~umax(~x, ~y) == umin(x, y)
  return getNotSCEV(getUMaxExpr(getNotSCEV(LHS), getNotSCEV(RHS)));
}

const SCEV *ScalarEvolution::getSizeOfExpr(Type *IntTy, Type *AllocTy) {
  // If we have DataLayout, we can bypass creating a target-independent
  // constant expression and then folding it back into a ConstantInt.
  // This is just a compile-time optimization.
  if (TD)
    return getConstant(IntTy, TD->getTypeAllocSize(AllocTy));

  Constant *C = ConstantExpr::getSizeOf(AllocTy);
  if (ConstantExpr *CE = dyn_cast<ConstantExpr>(C))
    if (Constant *Folded = ConstantFoldConstantExpression(CE, TD, TLI))
      C = Folded;
  Type *Ty = getEffectiveSCEVType(PointerType::getUnqual(AllocTy));
  assert(Ty == IntTy && "Effective SCEV type doesn't match");
  return getTruncateOrZeroExtend(getSCEV(C), Ty);
}

const SCEV *ScalarEvolution::getOffsetOfExpr(Type *IntTy,
                                             StructType *STy,
                                             unsigned FieldNo) {
  // If we have DataLayout, we can bypass creating a target-independent
  // constant expression and then folding it back into a ConstantInt.
  // This is just a compile-time optimization.
  if (TD) {
    return getConstant(IntTy,
                       TD->getStructLayout(STy)->getElementOffset(FieldNo));
  }

  Constant *C = ConstantExpr::getOffsetOf(STy, FieldNo);
  if (ConstantExpr *CE = dyn_cast<ConstantExpr>(C))
    if (Constant *Folded = ConstantFoldConstantExpression(CE, TD, TLI))
      C = Folded;

  Type *Ty = getEffectiveSCEVType(PointerType::getUnqual(STy));
  return getTruncateOrZeroExtend(getSCEV(C), Ty);
}

const SCEV *ScalarEvolution::getUnknown(Value *V) {
  // Don't attempt to do anything other than create a SCEVUnknown object
  // here.  createSCEV only calls getUnknown after checking for all other
  // interesting possibilities, and any other code that calls getUnknown
  // is doing so in order to hide a value from SCEV canonicalization.

  FoldingSetNodeID ID;
  ID.AddInteger(scUnknown);
  ID.AddPointer(V);
  void *IP = 0;
  if (SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) {
    assert(cast<SCEVUnknown>(S)->getValue() == V &&
           "Stale SCEVUnknown in uniquing map!");
    return S;
  }
  SCEV *S = new (SCEVAllocator) SCEVUnknown(ID.Intern(SCEVAllocator), V, this,
                                            FirstUnknown);
  FirstUnknown = cast<SCEVUnknown>(S);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

//===----------------------------------------------------------------------===//
//            Basic SCEV Analysis and PHI Idiom Recognition Code
//

/// isSCEVable - Test if values of the given type are analyzable within
/// the SCEV framework. This primarily includes integer types, and it
/// can optionally include pointer types if the ScalarEvolution class
/// has access to target-specific information.
bool ScalarEvolution::isSCEVable(Type *Ty) const {
  // Integers and pointers are always SCEVable.
  return Ty->isIntegerTy() || Ty->isPointerTy();
}

/// getTypeSizeInBits - Return the size in bits of the specified type,
/// for which isSCEVable must return true.
uint64_t ScalarEvolution::getTypeSizeInBits(Type *Ty) const {
  assert(isSCEVable(Ty) && "Type is not SCEVable!");

  // If we have a DataLayout, use it!
  if (TD)
    return TD->getTypeSizeInBits(Ty);

  // Integer types have fixed sizes.
  if (Ty->isIntegerTy())
    return Ty->getPrimitiveSizeInBits();

  // The only other support type is pointer. Without DataLayout, conservatively
  // assume pointers are 64-bit.
  assert(Ty->isPointerTy() && "isSCEVable permitted a non-SCEVable type!");
  return 64;
}

/// getEffectiveSCEVType - Return a type with the same bitwidth as
/// the given type and which represents how SCEV will treat the given
/// type, for which isSCEVable must return true. For pointer types,
/// this is the pointer-sized integer type.
Type *ScalarEvolution::getEffectiveSCEVType(Type *Ty) const {
  assert(isSCEVable(Ty) && "Type is not SCEVable!");

  if (Ty->isIntegerTy()) {
    return Ty;
  }

  // The only other support type is pointer.
  assert(Ty->isPointerTy() && "Unexpected non-pointer non-integer type!");

  if (TD)
    return TD->getIntPtrType(Ty);

  // Without DataLayout, conservatively assume pointers are 64-bit.
  return Type::getInt64Ty(getContext());
}

const SCEV *ScalarEvolution::getCouldNotCompute() {
  return &CouldNotCompute;
}

namespace {
  // Helper class working with SCEVTraversal to figure out if a SCEV contains
  // a SCEVUnknown with null value-pointer. FindInvalidSCEVUnknown::FindOne
  // is set iff if find such SCEVUnknown.
  //
  struct FindInvalidSCEVUnknown {
    bool FindOne;
    FindInvalidSCEVUnknown() { FindOne = false; }
    bool follow(const SCEV *S) {
      switch (S->getSCEVType()) {
      case scConstant:
        return false;
      case scUnknown:
        if (!cast<SCEVUnknown>(S)->getValue())
          FindOne = true;
        return false;
      default:
        return true;
      }
    }
    bool isDone() const { return FindOne; }
  };
}

bool ScalarEvolution::checkValidity(const SCEV *S) const {
  FindInvalidSCEVUnknown F;
  SCEVTraversal<FindInvalidSCEVUnknown> ST(F);
  ST.visitAll(S);

  return !F.FindOne;
}

/// getSCEV - Return an existing SCEV if it exists, otherwise analyze the
/// expression and create a new one.
const SCEV *ScalarEvolution::getSCEV(Value *V) {
  assert(isSCEVable(V->getType()) && "Value is not SCEVable!");

  ValueExprMapType::iterator I = ValueExprMap.find_as(V);
  if (I != ValueExprMap.end()) {
    const SCEV *S = I->second;
    if (checkValidity(S))
      return S;
    else
      ValueExprMap.erase(I);
  }
  const SCEV *S = createSCEV(V);

  // The process of creating a SCEV for V may have caused other SCEVs
  // to have been created, so it's necessary to insert the new entry
  // from scratch, rather than trying to remember the insert position
  // above.
  ValueExprMap.insert(std::make_pair(SCEVCallbackVH(V, this), S));
  return S;
}

/// getNegativeSCEV - Return a SCEV corresponding to -V = -1*V
///
const SCEV *ScalarEvolution::getNegativeSCEV(const SCEV *V) {
  if (const SCEVConstant *VC = dyn_cast<SCEVConstant>(V))
    return getConstant(
               cast<ConstantInt>(ConstantExpr::getNeg(VC->getValue())));

  Type *Ty = V->getType();
  Ty = getEffectiveSCEVType(Ty);
  return getMulExpr(V,
                  getConstant(cast<ConstantInt>(Constant::getAllOnesValue(Ty))));
}

/// getNotSCEV - Return a SCEV corresponding to ~V = -1-V
const SCEV *ScalarEvolution::getNotSCEV(const SCEV *V) {
  if (const SCEVConstant *VC = dyn_cast<SCEVConstant>(V))
    return getConstant(
                cast<ConstantInt>(ConstantExpr::getNot(VC->getValue())));

  Type *Ty = V->getType();
  Ty = getEffectiveSCEVType(Ty);
  const SCEV *AllOnes =
                   getConstant(cast<ConstantInt>(Constant::getAllOnesValue(Ty)));
  return getMinusSCEV(AllOnes, V);
}

/// getMinusSCEV - Return LHS-RHS.  Minus is represented in SCEV as A+B*-1.
const SCEV *ScalarEvolution::getMinusSCEV(const SCEV *LHS, const SCEV *RHS,
                                          SCEV::NoWrapFlags Flags) {
  assert(!maskFlags(Flags, SCEV::FlagNUW) && "subtraction does not have NUW");

  // Fast path: X - X --> 0.
  if (LHS == RHS)
    return getConstant(LHS->getType(), 0);

  // X - Y --> X + -Y
  return getAddExpr(LHS, getNegativeSCEV(RHS), Flags);
}

/// getTruncateOrZeroExtend - Return a SCEV corresponding to a conversion of the
/// input value to the specified type.  If the type must be extended, it is zero
/// extended.
const SCEV *
ScalarEvolution::getTruncateOrZeroExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot truncate or zero extend with non-integer arguments!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  if (getTypeSizeInBits(SrcTy) > getTypeSizeInBits(Ty))
    return getTruncateExpr(V, Ty);
  return getZeroExtendExpr(V, Ty);
}

/// getTruncateOrSignExtend - Return a SCEV corresponding to a conversion of the
/// input value to the specified type.  If the type must be extended, it is sign
/// extended.
const SCEV *
ScalarEvolution::getTruncateOrSignExtend(const SCEV *V,
                                         Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot truncate or zero extend with non-integer arguments!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  if (getTypeSizeInBits(SrcTy) > getTypeSizeInBits(Ty))
    return getTruncateExpr(V, Ty);
  return getSignExtendExpr(V, Ty);
}

/// getNoopOrZeroExtend - Return a SCEV corresponding to a conversion of the
/// input value to the specified type.  If the type must be extended, it is zero
/// extended.  The conversion must not be narrowing.
const SCEV *
ScalarEvolution::getNoopOrZeroExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot noop or zero extend with non-integer arguments!");
  assert(getTypeSizeInBits(SrcTy) <= getTypeSizeInBits(Ty) &&
         "getNoopOrZeroExtend cannot truncate!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  return getZeroExtendExpr(V, Ty);
}

/// getNoopOrSignExtend - Return a SCEV corresponding to a conversion of the
/// input value to the specified type.  If the type must be extended, it is sign
/// extended.  The conversion must not be narrowing.
const SCEV *
ScalarEvolution::getNoopOrSignExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot noop or sign extend with non-integer arguments!");
  assert(getTypeSizeInBits(SrcTy) <= getTypeSizeInBits(Ty) &&
         "getNoopOrSignExtend cannot truncate!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  return getSignExtendExpr(V, Ty);
}

/// getNoopOrAnyExtend - Return a SCEV corresponding to a conversion of
/// the input value to the specified type. If the type must be extended,
/// it is extended with unspecified bits. The conversion must not be
/// narrowing.
const SCEV *
ScalarEvolution::getNoopOrAnyExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot noop or any extend with non-integer arguments!");
  assert(getTypeSizeInBits(SrcTy) <= getTypeSizeInBits(Ty) &&
         "getNoopOrAnyExtend cannot truncate!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  return getAnyExtendExpr(V, Ty);
}

/// getTruncateOrNoop - Return a SCEV corresponding to a conversion of the
/// input value to the specified type.  The conversion must not be widening.
const SCEV *
ScalarEvolution::getTruncateOrNoop(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot truncate or noop with non-integer arguments!");
  assert(getTypeSizeInBits(SrcTy) >= getTypeSizeInBits(Ty) &&
         "getTruncateOrNoop cannot extend!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  return getTruncateExpr(V, Ty);
}

/// getUMaxFromMismatchedTypes - Promote the operands to the wider of
/// the types using zero-extension, and then perform a umax operation
/// with them.
const SCEV *ScalarEvolution::getUMaxFromMismatchedTypes(const SCEV *LHS,
                                                        const SCEV *RHS) {
  const SCEV *PromotedLHS = LHS;
  const SCEV *PromotedRHS = RHS;

  if (getTypeSizeInBits(LHS->getType()) > getTypeSizeInBits(RHS->getType()))
    PromotedRHS = getZeroExtendExpr(RHS, LHS->getType());
  else
    PromotedLHS = getNoopOrZeroExtend(LHS, RHS->getType());

  return getUMaxExpr(PromotedLHS, PromotedRHS);
}

/// getUMinFromMismatchedTypes - Promote the operands to the wider of
/// the types using zero-extension, and then perform a umin operation
/// with them.
const SCEV *ScalarEvolution::getUMinFromMismatchedTypes(const SCEV *LHS,
                                                        const SCEV *RHS) {
  const SCEV *PromotedLHS = LHS;
  const SCEV *PromotedRHS = RHS;

  if (getTypeSizeInBits(LHS->getType()) > getTypeSizeInBits(RHS->getType()))
    PromotedRHS = getZeroExtendExpr(RHS, LHS->getType());
  else
    PromotedLHS = getNoopOrZeroExtend(LHS, RHS->getType());

  return getUMinExpr(PromotedLHS, PromotedRHS);
}

/// getPointerBase - Transitively follow the chain of pointer-type operands
/// until reaching a SCEV that does not have a single pointer operand. This
/// returns a SCEVUnknown pointer for well-formed pointer-type expressions,
/// but corner cases do exist.
const SCEV *ScalarEvolution::getPointerBase(const SCEV *V) {
  // A pointer operand may evaluate to a nonpointer expression, such as null.
  if (!V->getType()->isPointerTy())
    return V;

  if (const SCEVCastExpr *Cast = dyn_cast<SCEVCastExpr>(V)) {
    return getPointerBase(Cast->getOperand());
  }
  else if (const SCEVNAryExpr *NAry = dyn_cast<SCEVNAryExpr>(V)) {
    const SCEV *PtrOp = 0;
    for (SCEVNAryExpr::op_iterator I = NAry->op_begin(), E = NAry->op_end();
         I != E; ++I) {
      if ((*I)->getType()->isPointerTy()) {
        // Cannot find the base of an expression with multiple pointer operands.
        if (PtrOp)
          return V;
        PtrOp = *I;
      }
    }
    if (!PtrOp)
      return V;
    return getPointerBase(PtrOp);
  }
  return V;
}

/// PushDefUseChildren - Push users of the given Instruction
/// onto the given Worklist.
static void
PushDefUseChildren(Instruction *I,
                   SmallVectorImpl<Instruction *> &Worklist) {
  // Push the def-use children onto the Worklist stack.
  for (Value::use_iterator UI = I->use_begin(), UE = I->use_end();
       UI != UE; ++UI)
    Worklist.push_back(cast<Instruction>(*UI));
}

/// ForgetSymbolicValue - This looks up computed SCEV values for all
/// instructions that depend on the given instruction and removes them from
/// the ValueExprMapType map if they reference SymName. This is used during PHI
/// resolution.
void
ScalarEvolution::ForgetSymbolicName(Instruction *PN, const SCEV *SymName) {
  SmallVector<Instruction *, 16> Worklist;
  PushDefUseChildren(PN, Worklist);

  SmallPtrSet<Instruction *, 8> Visited;
  Visited.insert(PN);
  while (!Worklist.empty()) {
    Instruction *I = Worklist.pop_back_val();
    if (!Visited.insert(I)) continue;

    ValueExprMapType::iterator It =
      ValueExprMap.find_as(static_cast<Value *>(I));
    if (It != ValueExprMap.end()) {
      const SCEV *Old = It->second;

      // Short-circuit the def-use traversal if the symbolic name
      // ceases to appear in expressions.
      if (Old != SymName && !hasOperand(Old, SymName))
        continue;

      // SCEVUnknown for a PHI either means that it has an unrecognized
      // structure, it's a PHI that's in the progress of being computed
      // by createNodeForPHI, or it's a single-value PHI. In the first case,
      // additional loop trip count information isn't going to change anything.
      // In the second case, createNodeForPHI will perform the necessary
      // updates on its own when it gets to that point. In the third, we do
      // want to forget the SCEVUnknown.
      if (!isa<PHINode>(I) ||
          !isa<SCEVUnknown>(Old) ||
          (I != PN && Old == SymName)) {
        forgetMemoizedResults(Old);
        ValueExprMap.erase(It);
      }
    }

    PushDefUseChildren(I, Worklist);
  }
}

/// createNodeForPHI - PHI nodes have two cases.  Either the PHI node exists in
/// a loop header, making it a potential recurrence, or it doesn't.
///
const SCEV *ScalarEvolution::createNodeForPHI(PHINode *PN) {
  if (const Loop *L = LI->getLoopFor(PN->getParent()))
    if (L->getHeader() == PN->getParent()) {
      // The loop may have multiple entrances or multiple exits; we can analyze
      // this phi as an addrec if it has a unique entry value and a unique
      // backedge value.
      Value *BEValueV = 0, *StartValueV = 0;
      for (unsigned i = 0, e = PN->getNumIncomingValues(); i != e; ++i) {
        Value *V = PN->getIncomingValue(i);
        if (L->contains(PN->getIncomingBlock(i))) {
          if (!BEValueV) {
            BEValueV = V;
          } else if (BEValueV != V) {
            BEValueV = 0;
            break;
          }
        } else if (!StartValueV) {
          StartValueV = V;
        } else if (StartValueV != V) {
          StartValueV = 0;
          break;
        }
      }
      if (BEValueV && StartValueV) {
        // While we are analyzing this PHI node, handle its value symbolically.
        const SCEV *SymbolicName = getUnknown(PN);
        assert(ValueExprMap.find_as(PN) == ValueExprMap.end() &&
               "PHI node already processed?");
        ValueExprMap.insert(std::make_pair(SCEVCallbackVH(PN, this), SymbolicName));

        // Using this symbolic name for the PHI, analyze the value coming around
        // the back-edge.
        const SCEV *BEValue = getSCEV(BEValueV);

        // NOTE: If BEValue is loop invariant, we know that the PHI node just
        // has a special value for the first iteration of the loop.

        // If the value coming around the backedge is an add with the symbolic
        // value we just inserted, then we found a simple induction variable!
        if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(BEValue)) {
          // If there is a single occurrence of the symbolic value, replace it
          // with a recurrence.
          unsigned FoundIndex = Add->getNumOperands();
          for (unsigned i = 0, e = Add->getNumOperands(); i != e; ++i)
            if (Add->getOperand(i) == SymbolicName)
              if (FoundIndex == e) {
                FoundIndex = i;
                break;
              }

          if (FoundIndex != Add->getNumOperands()) {
            // Create an add with everything but the specified operand.
            SmallVector<const SCEV *, 8> Ops;
            for (unsigned i = 0, e = Add->getNumOperands(); i != e; ++i)
              if (i != FoundIndex)
                Ops.push_back(Add->getOperand(i));
            const SCEV *Accum = getAddExpr(Ops);

            // This is not a valid addrec if the step amount is varying each
            // loop iteration, but is not itself an addrec in this loop.
            if (isLoopInvariant(Accum, L) ||
                (isa<SCEVAddRecExpr>(Accum) &&
                 cast<SCEVAddRecExpr>(Accum)->getLoop() == L)) {
              SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap;

              // If the increment doesn't overflow, then neither the addrec nor
              // the post-increment will overflow.
              if (const AddOperator *OBO = dyn_cast<AddOperator>(BEValueV)) {
                if (OBO->hasNoUnsignedWrap())
                  Flags = setFlags(Flags, SCEV::FlagNUW);
                if (OBO->hasNoSignedWrap())
                  Flags = setFlags(Flags, SCEV::FlagNSW);
              } else if (const GEPOperator *GEP =
                         dyn_cast<GEPOperator>(BEValueV)) {
                // If the increment is an inbounds GEP, then we know the address
                // space cannot be wrapped around. We cannot make any guarantee
                // about signed or unsigned overflow because pointers are
                // unsigned but we may have a negative index from the base
                // pointer.
                if (GEP->isInBounds())
                  Flags = setFlags(Flags, SCEV::FlagNW);
              }

              const SCEV *StartVal = getSCEV(StartValueV);
              const SCEV *PHISCEV = getAddRecExpr(StartVal, Accum, L, Flags);

              // Since the no-wrap flags are on the increment, they apply to the
              // post-incremented value as well.
              if (isLoopInvariant(Accum, L))
                (void)getAddRecExpr(getAddExpr(StartVal, Accum),
                                    Accum, L, Flags);

              // Okay, for the entire analysis of this edge we assumed the PHI
              // to be symbolic.  We now need to go back and purge all of the
              // entries for the scalars that use the symbolic expression.
              ForgetSymbolicName(PN, SymbolicName);
              ValueExprMap[SCEVCallbackVH(PN, this)] = PHISCEV;
              return PHISCEV;
            }
          }
        } else if (const SCEVAddRecExpr *AddRec =
                     dyn_cast<SCEVAddRecExpr>(BEValue)) {
          // Otherwise, this could be a loop like this:
          //     i = 0;  for (j = 1; ..; ++j) { ....  i = j; }
          // In this case, j = {1,+,1}  and BEValue is j.
          // Because the other in-value of i (0) fits the evolution of BEValue
          // i really is an addrec evolution.
          if (AddRec->getLoop() == L && AddRec->isAffine()) {
            const SCEV *StartVal = getSCEV(StartValueV);

            // If StartVal = j.start - j.stride, we can use StartVal as the
            // initial step of the addrec evolution.
            if (StartVal == getMinusSCEV(AddRec->getOperand(0),
                                         AddRec->getOperand(1))) {
              // FIXME: For constant StartVal, we should be able to infer
              // no-wrap flags.
              const SCEV *PHISCEV =
                getAddRecExpr(StartVal, AddRec->getOperand(1), L,
                              SCEV::FlagAnyWrap);

              // Okay, for the entire analysis of this edge we assumed the PHI
              // to be symbolic.  We now need to go back and purge all of the
              // entries for the scalars that use the symbolic expression.
              ForgetSymbolicName(PN, SymbolicName);
              ValueExprMap[SCEVCallbackVH(PN, this)] = PHISCEV;
              return PHISCEV;
            }
          }
        }
      }
    }

  // If the PHI has a single incoming value, follow that value, unless the
  // PHI's incoming blocks are in a different loop, in which case doing so
  // risks breaking LCSSA form. Instcombine would normally zap these, but
  // it doesn't have DominatorTree information, so it may miss cases.
  if (Value *V = SimplifyInstruction(PN, TD, TLI, DT))
    if (LI->replacementPreservesLCSSAForm(PN, V))
      return getSCEV(V);

  // If it's not a loop phi, we can't handle it yet.
  return getUnknown(PN);
}

/// createNodeForGEP - Expand GEP instructions into add and multiply
/// operations. This allows them to be analyzed by regular SCEV code.
///
const SCEV *ScalarEvolution::createNodeForGEP(GEPOperator *GEP) {
  Type *IntPtrTy = getEffectiveSCEVType(GEP->getType());
  Value *Base = GEP->getOperand(0);
  // Don't attempt to analyze GEPs over unsized objects.
  if (!Base->getType()->getPointerElementType()->isSized())
    return getUnknown(GEP);

  // Don't blindly transfer the inbounds flag from the GEP instruction to the
  // Add expression, because the Instruction may be guarded by control flow
  // and the no-overflow bits may not be valid for the expression in any
  // context.
  SCEV::NoWrapFlags Wrap = GEP->isInBounds() ? SCEV::FlagNSW : SCEV::FlagAnyWrap;

  const SCEV *TotalOffset = getConstant(IntPtrTy, 0);
  gep_type_iterator GTI = gep_type_begin(GEP);
  for (GetElementPtrInst::op_iterator I = llvm::next(GEP->op_begin()),
                                      E = GEP->op_end();
       I != E; ++I) {
    Value *Index = *I;
    // Compute the (potentially symbolic) offset in bytes for this index.
    if (StructType *STy = dyn_cast<StructType>(*GTI++)) {
      // For a struct, add the member offset.
      unsigned FieldNo = cast<ConstantInt>(Index)->getZExtValue();
      const SCEV *FieldOffset = getOffsetOfExpr(IntPtrTy, STy, FieldNo);

      // Add the field offset to the running total offset.
      TotalOffset = getAddExpr(TotalOffset, FieldOffset);
    } else {
      // For an array, add the element offset, explicitly scaled.
      const SCEV *ElementSize = getSizeOfExpr(IntPtrTy, *GTI);
      const SCEV *IndexS = getSCEV(Index);
      // Getelementptr indices are signed.
      IndexS = getTruncateOrSignExtend(IndexS, IntPtrTy);

      // Multiply the index by the element size to compute the element offset.
      const SCEV *LocalOffset = getMulExpr(IndexS, ElementSize, Wrap);

      // Add the element offset to the running total offset.
      TotalOffset = getAddExpr(TotalOffset, LocalOffset);
    }
  }

  // Get the SCEV for the GEP base.
  const SCEV *BaseS = getSCEV(Base);

  // Add the total offset from all the GEP indices to the base.
  return getAddExpr(BaseS, TotalOffset, Wrap);
}

/// GetMinTrailingZeros - Determine the minimum number of zero bits that S is
/// guaranteed to end in (at every loop iteration).  It is, at the same time,
/// the minimum number of times S is divisible by 2.  For example, given {4,+,8}
/// it returns 2.  If S is guaranteed to be 0, it returns the bitwidth of S.
uint32_t
ScalarEvolution::GetMinTrailingZeros(const SCEV *S) {
  if (const SCEVConstant *C = dyn_cast<SCEVConstant>(S))
    return C->getValue()->getValue().countTrailingZeros();

  if (const SCEVTruncateExpr *T = dyn_cast<SCEVTruncateExpr>(S))
    return std::min(GetMinTrailingZeros(T->getOperand()),
                    (uint32_t)getTypeSizeInBits(T->getType()));

  if (const SCEVZeroExtendExpr *E = dyn_cast<SCEVZeroExtendExpr>(S)) {
    uint32_t OpRes = GetMinTrailingZeros(E->getOperand());
    return OpRes == getTypeSizeInBits(E->getOperand()->getType()) ?
             getTypeSizeInBits(E->getType()) : OpRes;
  }

  if (const SCEVSignExtendExpr *E = dyn_cast<SCEVSignExtendExpr>(S)) {
    uint32_t OpRes = GetMinTrailingZeros(E->getOperand());
    return OpRes == getTypeSizeInBits(E->getOperand()->getType()) ?
             getTypeSizeInBits(E->getType()) : OpRes;
  }

  if (const SCEVAddExpr *A = dyn_cast<SCEVAddExpr>(S)) {
    // The result is the min of all operands results.
    uint32_t MinOpRes = GetMinTrailingZeros(A->getOperand(0));
    for (unsigned i = 1, e = A->getNumOperands(); MinOpRes && i != e; ++i)
      MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(A->getOperand(i)));
    return MinOpRes;
  }

  if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(S)) {
    // The result is the sum of all operands results.
    uint32_t SumOpRes = GetMinTrailingZeros(M->getOperand(0));
    uint32_t BitWidth = getTypeSizeInBits(M->getType());
    for (unsigned i = 1, e = M->getNumOperands();
         SumOpRes != BitWidth && i != e; ++i)
      SumOpRes = std::min(SumOpRes + GetMinTrailingZeros(M->getOperand(i)),
                          BitWidth);
    return SumOpRes;
  }

  if (const SCEVAddRecExpr *A = dyn_cast<SCEVAddRecExpr>(S)) {
    // The result is the min of all operands results.
    uint32_t MinOpRes = GetMinTrailingZeros(A->getOperand(0));
    for (unsigned i = 1, e = A->getNumOperands(); MinOpRes && i != e; ++i)
      MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(A->getOperand(i)));
    return MinOpRes;
  }

  if (const SCEVSMaxExpr *M = dyn_cast<SCEVSMaxExpr>(S)) {
    // The result is the min of all operands results.
    uint32_t MinOpRes = GetMinTrailingZeros(M->getOperand(0));
    for (unsigned i = 1, e = M->getNumOperands(); MinOpRes && i != e; ++i)
      MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(M->getOperand(i)));
    return MinOpRes;
  }

  if (const SCEVUMaxExpr *M = dyn_cast<SCEVUMaxExpr>(S)) {
    // The result is the min of all operands results.
    uint32_t MinOpRes = GetMinTrailingZeros(M->getOperand(0));
    for (unsigned i = 1, e = M->getNumOperands(); MinOpRes && i != e; ++i)
      MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(M->getOperand(i)));
    return MinOpRes;
  }

  if (const SCEVUnknown *U = dyn_cast<SCEVUnknown>(S)) {
    // For a SCEVUnknown, ask ValueTracking.
    unsigned BitWidth = getTypeSizeInBits(U->getType());
    APInt Zeros(BitWidth, 0), Ones(BitWidth, 0);
    ComputeMaskedBits(U->getValue(), Zeros, Ones);
    return Zeros.countTrailingOnes();
  }

  // SCEVUDivExpr
  return 0;
}

/// getUnsignedRange - Determine the unsigned range for a particular SCEV.
///
ConstantRange
ScalarEvolution::getUnsignedRange(const SCEV *S) {
  // See if we've computed this range already.
  DenseMap<const SCEV *, ConstantRange>::iterator I = UnsignedRanges.find(S);
  if (I != UnsignedRanges.end())
    return I->second;

  if (const SCEVConstant *C = dyn_cast<SCEVConstant>(S))
    return setUnsignedRange(C, ConstantRange(C->getValue()->getValue()));

  unsigned BitWidth = getTypeSizeInBits(S->getType());
  ConstantRange ConservativeResult(BitWidth, /*isFullSet=*/true);

  // If the value has known zeros, the maximum unsigned value will have those
  // known zeros as well.
  uint32_t TZ = GetMinTrailingZeros(S);
  if (TZ != 0)
    ConservativeResult =
      ConstantRange(APInt::getMinValue(BitWidth),
                    APInt::getMaxValue(BitWidth).lshr(TZ).shl(TZ) + 1);

  if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(S)) {
    ConstantRange X = getUnsignedRange(Add->getOperand(0));
    for (unsigned i = 1, e = Add->getNumOperands(); i != e; ++i)
      X = X.add(getUnsignedRange(Add->getOperand(i)));
    return setUnsignedRange(Add, ConservativeResult.intersectWith(X));
  }

  if (const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(S)) {
    ConstantRange X = getUnsignedRange(Mul->getOperand(0));
    for (unsigned i = 1, e = Mul->getNumOperands(); i != e; ++i)
      X = X.multiply(getUnsignedRange(Mul->getOperand(i)));
    return setUnsignedRange(Mul, ConservativeResult.intersectWith(X));
  }

  if (const SCEVSMaxExpr *SMax = dyn_cast<SCEVSMaxExpr>(S)) {
    ConstantRange X = getUnsignedRange(SMax->getOperand(0));
    for (unsigned i = 1, e = SMax->getNumOperands(); i != e; ++i)
      X = X.smax(getUnsignedRange(SMax->getOperand(i)));
    return setUnsignedRange(SMax, ConservativeResult.intersectWith(X));
  }

  if (const SCEVUMaxExpr *UMax = dyn_cast<SCEVUMaxExpr>(S)) {
    ConstantRange X = getUnsignedRange(UMax->getOperand(0));
    for (unsigned i = 1, e = UMax->getNumOperands(); i != e; ++i)
      X = X.umax(getUnsignedRange(UMax->getOperand(i)));
    return setUnsignedRange(UMax, ConservativeResult.intersectWith(X));
  }

  if (const SCEVUDivExpr *UDiv = dyn_cast<SCEVUDivExpr>(S)) {
    ConstantRange X = getUnsignedRange(UDiv->getLHS());
    ConstantRange Y = getUnsignedRange(UDiv->getRHS());
    return setUnsignedRange(UDiv, ConservativeResult.intersectWith(X.udiv(Y)));
  }

  if (const SCEVZeroExtendExpr *ZExt = dyn_cast<SCEVZeroExtendExpr>(S)) {
    ConstantRange X = getUnsignedRange(ZExt->getOperand());
    return setUnsignedRange(ZExt,
      ConservativeResult.intersectWith(X.zeroExtend(BitWidth)));
  }

  if (const SCEVSignExtendExpr *SExt = dyn_cast<SCEVSignExtendExpr>(S)) {
    ConstantRange X = getUnsignedRange(SExt->getOperand());
    return setUnsignedRange(SExt,
      ConservativeResult.intersectWith(X.signExtend(BitWidth)));
  }

  if (const SCEVTruncateExpr *Trunc = dyn_cast<SCEVTruncateExpr>(S)) {
    ConstantRange X = getUnsignedRange(Trunc->getOperand());
    return setUnsignedRange(Trunc,
      ConservativeResult.intersectWith(X.truncate(BitWidth)));
  }

  if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(S)) {
    // If there's no unsigned wrap, the value will never be less than its
    // initial value.
    if (AddRec->getNoWrapFlags(SCEV::FlagNUW))
      if (const SCEVConstant *C = dyn_cast<SCEVConstant>(AddRec->getStart()))
        if (!C->getValue()->isZero())
          ConservativeResult =
            ConservativeResult.intersectWith(
              ConstantRange(C->getValue()->getValue(), APInt(BitWidth, 0)));

    // TODO: non-affine addrec
    if (AddRec->isAffine()) {
      Type *Ty = AddRec->getType();
      const SCEV *MaxBECount = getMaxBackedgeTakenCount(AddRec->getLoop());
      if (!isa<SCEVCouldNotCompute>(MaxBECount) &&
          getTypeSizeInBits(MaxBECount->getType()) <= BitWidth) {
        MaxBECount = getNoopOrZeroExtend(MaxBECount, Ty);

        const SCEV *Start = AddRec->getStart();
        const SCEV *Step = AddRec->getStepRecurrence(*this);

        ConstantRange StartRange = getUnsignedRange(Start);
        ConstantRange StepRange = getSignedRange(Step);
        ConstantRange MaxBECountRange = getUnsignedRange(MaxBECount);
        ConstantRange EndRange =
          StartRange.add(MaxBECountRange.multiply(StepRange));

        // Check for overflow. This must be done with ConstantRange arithmetic
        // because we could be called from within the ScalarEvolution overflow
        // checking code.
        ConstantRange ExtStartRange = StartRange.zextOrTrunc(BitWidth*2+1);
        ConstantRange ExtStepRange = StepRange.sextOrTrunc(BitWidth*2+1);
        ConstantRange ExtMaxBECountRange =
          MaxBECountRange.zextOrTrunc(BitWidth*2+1);
        ConstantRange ExtEndRange = EndRange.zextOrTrunc(BitWidth*2+1);
        if (ExtStartRange.add(ExtMaxBECountRange.multiply(ExtStepRange)) !=
            ExtEndRange)
          return setUnsignedRange(AddRec, ConservativeResult);

        APInt Min = APIntOps::umin(StartRange.getUnsignedMin(),
                                   EndRange.getUnsignedMin());
        APInt Max = APIntOps::umax(StartRange.getUnsignedMax(),
                                   EndRange.getUnsignedMax());
        if (Min.isMinValue() && Max.isMaxValue())
          return setUnsignedRange(AddRec, ConservativeResult);
        return setUnsignedRange(AddRec,
          ConservativeResult.intersectWith(ConstantRange(Min, Max+1)));
      }
    }

    return setUnsignedRange(AddRec, ConservativeResult);
  }

  if (const SCEVUnknown *U = dyn_cast<SCEVUnknown>(S)) {
    // For a SCEVUnknown, ask ValueTracking.
    APInt Zeros(BitWidth, 0), Ones(BitWidth, 0);
    ComputeMaskedBits(U->getValue(), Zeros, Ones, TD);
    if (Ones == ~Zeros + 1)
      return setUnsignedRange(U, ConservativeResult);
    return setUnsignedRange(U,
      ConservativeResult.intersectWith(ConstantRange(Ones, ~Zeros + 1)));
  }

  return setUnsignedRange(S, ConservativeResult);
}

/// getSignedRange - Determine the signed range for a particular SCEV.
///
ConstantRange
ScalarEvolution::getSignedRange(const SCEV *S) {
  // See if we've computed this range already.
  DenseMap<const SCEV *, ConstantRange>::iterator I = SignedRanges.find(S);
  if (I != SignedRanges.end())
    return I->second;

  if (const SCEVConstant *C = dyn_cast<SCEVConstant>(S))
    return setSignedRange(C, ConstantRange(C->getValue()->getValue()));

  unsigned BitWidth = getTypeSizeInBits(S->getType());
  ConstantRange ConservativeResult(BitWidth, /*isFullSet=*/true);

  // If the value has known zeros, the maximum signed value will have those
  // known zeros as well.
  uint32_t TZ = GetMinTrailingZeros(S);
  if (TZ != 0)
    ConservativeResult =
      ConstantRange(APInt::getSignedMinValue(BitWidth),
                    APInt::getSignedMaxValue(BitWidth).ashr(TZ).shl(TZ) + 1);

  if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(S)) {
    ConstantRange X = getSignedRange(Add->getOperand(0));
    for (unsigned i = 1, e = Add->getNumOperands(); i != e; ++i)
      X = X.add(getSignedRange(Add->getOperand(i)));
    return setSignedRange(Add, ConservativeResult.intersectWith(X));
  }

  if (const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(S)) {
    ConstantRange X = getSignedRange(Mul->getOperand(0));
    for (unsigned i = 1, e = Mul->getNumOperands(); i != e; ++i)
      X = X.multiply(getSignedRange(Mul->getOperand(i)));
    return setSignedRange(Mul, ConservativeResult.intersectWith(X));
  }

  if (const SCEVSMaxExpr *SMax = dyn_cast<SCEVSMaxExpr>(S)) {
    ConstantRange X = getSignedRange(SMax->getOperand(0));
    for (unsigned i = 1, e = SMax->getNumOperands(); i != e; ++i)
      X = X.smax(getSignedRange(SMax->getOperand(i)));
    return setSignedRange(SMax, ConservativeResult.intersectWith(X));
  }

  if (const SCEVUMaxExpr *UMax = dyn_cast<SCEVUMaxExpr>(S)) {
    ConstantRange X = getSignedRange(UMax->getOperand(0));
    for (unsigned i = 1, e = UMax->getNumOperands(); i != e; ++i)
      X = X.umax(getSignedRange(UMax->getOperand(i)));
    return setSignedRange(UMax, ConservativeResult.intersectWith(X));
  }

  if (const SCEVUDivExpr *UDiv = dyn_cast<SCEVUDivExpr>(S)) {
    ConstantRange X = getSignedRange(UDiv->getLHS());
    ConstantRange Y = getSignedRange(UDiv->getRHS());
    return setSignedRange(UDiv, ConservativeResult.intersectWith(X.udiv(Y)));
  }

  if (const SCEVZeroExtendExpr *ZExt = dyn_cast<SCEVZeroExtendExpr>(S)) {
    ConstantRange X = getSignedRange(ZExt->getOperand());
    return setSignedRange(ZExt,
      ConservativeResult.intersectWith(X.zeroExtend(BitWidth)));
  }

  if (const SCEVSignExtendExpr *SExt = dyn_cast<SCEVSignExtendExpr>(S)) {
    ConstantRange X = getSignedRange(SExt->getOperand());
    return setSignedRange(SExt,
      ConservativeResult.intersectWith(X.signExtend(BitWidth)));
  }

  if (const SCEVTruncateExpr *Trunc = dyn_cast<SCEVTruncateExpr>(S)) {
    ConstantRange X = getSignedRange(Trunc->getOperand());
    return setSignedRange(Trunc,
      ConservativeResult.intersectWith(X.truncate(BitWidth)));
  }

  if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(S)) {
    // If there's no signed wrap, and all the operands have the same sign or
    // zero, the value won't ever change sign.
    if (AddRec->getNoWrapFlags(SCEV::FlagNSW)) {
      bool AllNonNeg = true;
      bool AllNonPos = true;
      for (unsigned i = 0, e = AddRec->getNumOperands(); i != e; ++i) {
        if (!isKnownNonNegative(AddRec->getOperand(i))) AllNonNeg = false;
        if (!isKnownNonPositive(AddRec->getOperand(i))) AllNonPos = false;
      }
      if (AllNonNeg)
        ConservativeResult = ConservativeResult.intersectWith(
          ConstantRange(APInt(BitWidth, 0),
                        APInt::getSignedMinValue(BitWidth)));
      else if (AllNonPos)
        ConservativeResult = ConservativeResult.intersectWith(
          ConstantRange(APInt::getSignedMinValue(BitWidth),
                        APInt(BitWidth, 1)));
    }

    // TODO: non-affine addrec
    if (AddRec->isAffine()) {
      Type *Ty = AddRec->getType();
      const SCEV *MaxBECount = getMaxBackedgeTakenCount(AddRec->getLoop());
      if (!isa<SCEVCouldNotCompute>(MaxBECount) &&
          getTypeSizeInBits(MaxBECount->getType()) <= BitWidth) {
        MaxBECount = getNoopOrZeroExtend(MaxBECount, Ty);

        const SCEV *Start = AddRec->getStart();
        const SCEV *Step = AddRec->getStepRecurrence(*this);

        ConstantRange StartRange = getSignedRange(Start);
        ConstantRange StepRange = getSignedRange(Step);
        ConstantRange MaxBECountRange = getUnsignedRange(MaxBECount);
        ConstantRange EndRange =
          StartRange.add(MaxBECountRange.multiply(StepRange));

        // Check for overflow. This must be done with ConstantRange arithmetic
        // because we could be called from within the ScalarEvolution overflow
        // checking code.
        ConstantRange ExtStartRange = StartRange.sextOrTrunc(BitWidth*2+1);
        ConstantRange ExtStepRange = StepRange.sextOrTrunc(BitWidth*2+1);
        ConstantRange ExtMaxBECountRange =
          MaxBECountRange.zextOrTrunc(BitWidth*2+1);
        ConstantRange ExtEndRange = EndRange.sextOrTrunc(BitWidth*2+1);
        if (ExtStartRange.add(ExtMaxBECountRange.multiply(ExtStepRange)) !=
            ExtEndRange)
          return setSignedRange(AddRec, ConservativeResult);

        APInt Min = APIntOps::smin(StartRange.getSignedMin(),
                                   EndRange.getSignedMin());
        APInt Max = APIntOps::smax(StartRange.getSignedMax(),
                                   EndRange.getSignedMax());
        if (Min.isMinSignedValue() && Max.isMaxSignedValue())
          return setSignedRange(AddRec, ConservativeResult);
        return setSignedRange(AddRec,
          ConservativeResult.intersectWith(ConstantRange(Min, Max+1)));
      }
    }

    return setSignedRange(AddRec, ConservativeResult);
  }

  if (const SCEVUnknown *U = dyn_cast<SCEVUnknown>(S)) {
    // For a SCEVUnknown, ask ValueTracking.
    if (!U->getValue()->getType()->isIntegerTy() && !TD)
      return setSignedRange(U, ConservativeResult);
    unsigned NS = ComputeNumSignBits(U->getValue(), TD);
    if (NS <= 1)
      return setSignedRange(U, ConservativeResult);
    return setSignedRange(U, ConservativeResult.intersectWith(
      ConstantRange(APInt::getSignedMinValue(BitWidth).ashr(NS - 1),
                    APInt::getSignedMaxValue(BitWidth).ashr(NS - 1)+1)));
  }

  return setSignedRange(S, ConservativeResult);
}

/// createSCEV - We know that there is no SCEV for the specified value.
/// Analyze the expression.
///
const SCEV *ScalarEvolution::createSCEV(Value *V) {
  if (!isSCEVable(V->getType()))
    return getUnknown(V);

  unsigned Opcode = Instruction::UserOp1;
  if (Instruction *I = dyn_cast<Instruction>(V)) {
    Opcode = I->getOpcode();

    // Don't attempt to analyze instructions in blocks that aren't
    // reachable. Such instructions don't matter, and they aren't required
    // to obey basic rules for definitions dominating uses which this
    // analysis depends on.
    if (!DT->isReachableFromEntry(I->getParent()))
      return getUnknown(V);
  } else if (ConstantExpr *CE = dyn_cast<ConstantExpr>(V))
    Opcode = CE->getOpcode();
  else if (ConstantInt *CI = dyn_cast<ConstantInt>(V))
    return getConstant(CI);
  else if (isa<ConstantPointerNull>(V))
    return getConstant(V->getType(), 0);
  else if (GlobalAlias *GA = dyn_cast<GlobalAlias>(V))
    return GA->mayBeOverridden() ? getUnknown(V) : getSCEV(GA->getAliasee());
  else
    return getUnknown(V);

  Operator *U = cast<Operator>(V);
  switch (Opcode) {
  case Instruction::Add: {
    // The simple thing to do would be to just call getSCEV on both operands
    // and call getAddExpr with the result. However if we're looking at a
    // bunch of things all added together, this can be quite inefficient,
    // because it leads to N-1 getAddExpr calls for N ultimate operands.
    // Instead, gather up all the operands and make a single getAddExpr call.
    // LLVM IR canonical form means we need only traverse the left operands.
    //
    // Don't apply this instruction's NSW or NUW flags to the new
    // expression. The instruction may be guarded by control flow that the
    // no-wrap behavior depends on. Non-control-equivalent instructions can be
    // mapped to the same SCEV expression, and it would be incorrect to transfer
    // NSW/NUW semantics to those operations.
    SmallVector<const SCEV *, 4> AddOps;
    AddOps.push_back(getSCEV(U->getOperand(1)));
    for (Value *Op = U->getOperand(0); ; Op = U->getOperand(0)) {
      unsigned Opcode = Op->getValueID() - Value::InstructionVal;
      if (Opcode != Instruction::Add && Opcode != Instruction::Sub)
        break;
      U = cast<Operator>(Op);
      const SCEV *Op1 = getSCEV(U->getOperand(1));
      if (Opcode == Instruction::Sub)
        AddOps.push_back(getNegativeSCEV(Op1));
      else
        AddOps.push_back(Op1);
    }
    AddOps.push_back(getSCEV(U->getOperand(0)));
    return getAddExpr(AddOps);
  }
  case Instruction::Mul: {
    // Don't transfer NSW/NUW for the same reason as AddExpr.
    SmallVector<const SCEV *, 4> MulOps;
    MulOps.push_back(getSCEV(U->getOperand(1)));
    for (Value *Op = U->getOperand(0);
         Op->getValueID() == Instruction::Mul + Value::InstructionVal;
         Op = U->getOperand(0)) {
      U = cast<Operator>(Op);
      MulOps.push_back(getSCEV(U->getOperand(1)));
    }
    MulOps.push_back(getSCEV(U->getOperand(0)));
    return getMulExpr(MulOps);
  }
  case Instruction::UDiv:
    return getUDivExpr(getSCEV(U->getOperand(0)),
                       getSCEV(U->getOperand(1)));
  case Instruction::Sub:
    return getMinusSCEV(getSCEV(U->getOperand(0)),
                        getSCEV(U->getOperand(1)));
  case Instruction::And:
    // For an expression like x&255 that merely masks off the high bits,
    // use zext(trunc(x)) as the SCEV expression.
    if (ConstantInt *CI = dyn_cast<ConstantInt>(U->getOperand(1))) {
      if (CI->isNullValue())
        return getSCEV(U->getOperand(1));
      if (CI->isAllOnesValue())
        return getSCEV(U->getOperand(0));
      const APInt &A = CI->getValue();

      // Instcombine's ShrinkDemandedConstant may strip bits out of
      // constants, obscuring what would otherwise be a low-bits mask.
      // Use ComputeMaskedBits to compute what ShrinkDemandedConstant
      // knew about to reconstruct a low-bits mask value.
      unsigned LZ = A.countLeadingZeros();
      unsigned BitWidth = A.getBitWidth();
      APInt KnownZero(BitWidth, 0), KnownOne(BitWidth, 0);
      ComputeMaskedBits(U->getOperand(0), KnownZero, KnownOne, TD);

      APInt EffectiveMask = APInt::getLowBitsSet(BitWidth, BitWidth - LZ);

      if (LZ != 0 && !((~A & ~KnownZero) & EffectiveMask))
        return
          getZeroExtendExpr(getTruncateExpr(getSCEV(U->getOperand(0)),
                                IntegerType::get(getContext(), BitWidth - LZ)),
                            U->getType());
    }
    break;

  case Instruction::Or:
    // If the RHS of the Or is a constant, we may have something like:
    // X*4+1 which got turned into X*4|1.  Handle this as an Add so loop
    // optimizations will transparently handle this case.
    //
    // In order for this transformation to be safe, the LHS must be of the
    // form X*(2^n) and the Or constant must be less than 2^n.
    if (ConstantInt *CI = dyn_cast<ConstantInt>(U->getOperand(1))) {
      const SCEV *LHS = getSCEV(U->getOperand(0));
      const APInt &CIVal = CI->getValue();
      if (GetMinTrailingZeros(LHS) >=
          (CIVal.getBitWidth() - CIVal.countLeadingZeros())) {
        // Build a plain add SCEV.
        const SCEV *S = getAddExpr(LHS, getSCEV(CI));
        // If the LHS of the add was an addrec and it has no-wrap flags,
        // transfer the no-wrap flags, since an or won't introduce a wrap.
        if (const SCEVAddRecExpr *NewAR = dyn_cast<SCEVAddRecExpr>(S)) {
          const SCEVAddRecExpr *OldAR = cast<SCEVAddRecExpr>(LHS);
          const_cast<SCEVAddRecExpr *>(NewAR)->setNoWrapFlags(
            OldAR->getNoWrapFlags());
        }
        return S;
      }
    }
    break;
  case Instruction::Xor:
    if (ConstantInt *CI = dyn_cast<ConstantInt>(U->getOperand(1))) {
      // If the RHS of the xor is a signbit, then this is just an add.
      // Instcombine turns add of signbit into xor as a strength reduction step.
      if (CI->getValue().isSignBit())
        return getAddExpr(getSCEV(U->getOperand(0)),
                          getSCEV(U->getOperand(1)));

      // If the RHS of xor is -1, then this is a not operation.
      if (CI->isAllOnesValue())
        return getNotSCEV(getSCEV(U->getOperand(0)));

      // Model xor(and(x, C), C) as and(~x, C), if C is a low-bits mask.
      // This is a variant of the check for xor with -1, and it handles
      // the case where instcombine has trimmed non-demanded bits out
      // of an xor with -1.
      if (BinaryOperator *BO = dyn_cast<BinaryOperator>(U->getOperand(0)))
        if (ConstantInt *LCI = dyn_cast<ConstantInt>(BO->getOperand(1)))
          if (BO->getOpcode() == Instruction::And &&
              LCI->getValue() == CI->getValue())
            if (const SCEVZeroExtendExpr *Z =
                  dyn_cast<SCEVZeroExtendExpr>(getSCEV(U->getOperand(0)))) {
              Type *UTy = U->getType();
              const SCEV *Z0 = Z->getOperand();
              Type *Z0Ty = Z0->getType();
              unsigned Z0TySize = getTypeSizeInBits(Z0Ty);

              // If C is a low-bits mask, the zero extend is serving to
              // mask off the high bits. Complement the operand and
              // re-apply the zext.
              if (APIntOps::isMask(Z0TySize, CI->getValue()))
                return getZeroExtendExpr(getNotSCEV(Z0), UTy);

              // If C is a single bit, it may be in the sign-bit position
              // before the zero-extend. In this case, represent the xor
              // using an add, which is equivalent, and re-apply the zext.
              APInt Trunc = CI->getValue().trunc(Z0TySize);
              if (Trunc.zext(getTypeSizeInBits(UTy)) == CI->getValue() &&
                  Trunc.isSignBit())
                return getZeroExtendExpr(getAddExpr(Z0, getConstant(Trunc)),
                                         UTy);
            }
    }
    break;

  case Instruction::Shl:
    // Turn shift left of a constant amount into a multiply.
    if (ConstantInt *SA = dyn_cast<ConstantInt>(U->getOperand(1))) {
      uint32_t BitWidth = cast<IntegerType>(U->getType())->getBitWidth();

      // If the shift count is not less than the bitwidth, the result of
      // the shift is undefined. Don't try to analyze it, because the
      // resolution chosen here may differ from the resolution chosen in
      // other parts of the compiler.
      if (SA->getValue().uge(BitWidth))
        break;

      Constant *X = ConstantInt::get(getContext(),
        APInt::getOneBitSet(BitWidth, SA->getZExtValue()));
      return getMulExpr(getSCEV(U->getOperand(0)), getSCEV(X));
    }
    break;

  case Instruction::LShr:
    // Turn logical shift right of a constant into a unsigned divide.
    if (ConstantInt *SA = dyn_cast<ConstantInt>(U->getOperand(1))) {
      uint32_t BitWidth = cast<IntegerType>(U->getType())->getBitWidth();

      // If the shift count is not less than the bitwidth, the result of
      // the shift is undefined. Don't try to analyze it, because the
      // resolution chosen here may differ from the resolution chosen in
      // other parts of the compiler.
      if (SA->getValue().uge(BitWidth))
        break;

      Constant *X = ConstantInt::get(getContext(),
        APInt::getOneBitSet(BitWidth, SA->getZExtValue()));
      return getUDivExpr(getSCEV(U->getOperand(0)), getSCEV(X));
    }
    break;

  case Instruction::AShr:
    // For a two-shift sext-inreg, use sext(trunc(x)) as the SCEV expression.
    if (ConstantInt *CI = dyn_cast<ConstantInt>(U->getOperand(1)))
      if (Operator *L = dyn_cast<Operator>(U->getOperand(0)))
        if (L->getOpcode() == Instruction::Shl &&
            L->getOperand(1) == U->getOperand(1)) {
          uint64_t BitWidth = getTypeSizeInBits(U->getType());

          // If the shift count is not less than the bitwidth, the result of
          // the shift is undefined. Don't try to analyze it, because the
          // resolution chosen here may differ from the resolution chosen in
          // other parts of the compiler.
          if (CI->getValue().uge(BitWidth))
            break;

          uint64_t Amt = BitWidth - CI->getZExtValue();
          if (Amt == BitWidth)
            return getSCEV(L->getOperand(0));       // shift by zero --> noop
          return
            getSignExtendExpr(getTruncateExpr(getSCEV(L->getOperand(0)),
                                              IntegerType::get(getContext(),
                                                               Amt)),
                              U->getType());
        }
    break;

  case Instruction::Trunc:
    return getTruncateExpr(getSCEV(U->getOperand(0)), U->getType());

  case Instruction::ZExt:
    return getZeroExtendExpr(getSCEV(U->getOperand(0)), U->getType());

  case Instruction::SExt:
    return getSignExtendExpr(getSCEV(U->getOperand(0)), U->getType());

  case Instruction::BitCast:
    // BitCasts are no-op casts so we just eliminate the cast.
    if (isSCEVable(U->getType()) && isSCEVable(U->getOperand(0)->getType()))
      return getSCEV(U->getOperand(0));
    break;

  // It's tempting to handle inttoptr and ptrtoint as no-ops, however this can
  // lead to pointer expressions which cannot safely be expanded to GEPs,
  // because ScalarEvolution doesn't respect the GEP aliasing rules when
  // simplifying integer expressions.

  case Instruction::GetElementPtr:
    return createNodeForGEP(cast<GEPOperator>(U));

  case Instruction::PHI:
    return createNodeForPHI(cast<PHINode>(U));

  case Instruction::Select:
    // This could be a smax or umax that was lowered earlier.
    // Try to recover it.
    if (ICmpInst *ICI = dyn_cast<ICmpInst>(U->getOperand(0))) {
      Value *LHS = ICI->getOperand(0);
      Value *RHS = ICI->getOperand(1);
      switch (ICI->getPredicate()) {
      case ICmpInst::ICMP_SLT:
      case ICmpInst::ICMP_SLE:
        std::swap(LHS, RHS);
        // fall through
      case ICmpInst::ICMP_SGT:
      case ICmpInst::ICMP_SGE:
        // a >s b ? a+x : b+x  ->  smax(a, b)+x
        // a >s b ? b+x : a+x  ->  smin(a, b)+x
        if (LHS->getType() == U->getType()) {
          const SCEV *LS = getSCEV(LHS);
          const SCEV *RS = getSCEV(RHS);
          const SCEV *LA = getSCEV(U->getOperand(1));
          const SCEV *RA = getSCEV(U->getOperand(2));
          const SCEV *LDiff = getMinusSCEV(LA, LS);
          const SCEV *RDiff = getMinusSCEV(RA, RS);
          if (LDiff == RDiff)
            return getAddExpr(getSMaxExpr(LS, RS), LDiff);
          LDiff = getMinusSCEV(LA, RS);
          RDiff = getMinusSCEV(RA, LS);
          if (LDiff == RDiff)
            return getAddExpr(getSMinExpr(LS, RS), LDiff);
        }
        break;
      case ICmpInst::ICMP_ULT:
      case ICmpInst::ICMP_ULE:
        std::swap(LHS, RHS);
        // fall through
      case ICmpInst::ICMP_UGT:
      case ICmpInst::ICMP_UGE:
        // a >u b ? a+x : b+x  ->  umax(a, b)+x
        // a >u b ? b+x : a+x  ->  umin(a, b)+x
        if (LHS->getType() == U->getType()) {
          const SCEV *LS = getSCEV(LHS);
          const SCEV *RS = getSCEV(RHS);
          const SCEV *LA = getSCEV(U->getOperand(1));
          const SCEV *RA = getSCEV(U->getOperand(2));
          const SCEV *LDiff = getMinusSCEV(LA, LS);
          const SCEV *RDiff = getMinusSCEV(RA, RS);
          if (LDiff == RDiff)
            return getAddExpr(getUMaxExpr(LS, RS), LDiff);
          LDiff = getMinusSCEV(LA, RS);
          RDiff = getMinusSCEV(RA, LS);
          if (LDiff == RDiff)
            return getAddExpr(getUMinExpr(LS, RS), LDiff);
        }
        break;
      case ICmpInst::ICMP_NE:
        // n != 0 ? n+x : 1+x  ->  umax(n, 1)+x
        if (LHS->getType() == U->getType() &&
            isa<ConstantInt>(RHS) &&
            cast<ConstantInt>(RHS)->isZero()) {
          const SCEV *One = getConstant(LHS->getType(), 1);
          const SCEV *LS = getSCEV(LHS);
          const SCEV *LA = getSCEV(U->getOperand(1));
          const SCEV *RA = getSCEV(U->getOperand(2));
          const SCEV *LDiff = getMinusSCEV(LA, LS);
          const SCEV *RDiff = getMinusSCEV(RA, One);
          if (LDiff == RDiff)
            return getAddExpr(getUMaxExpr(One, LS), LDiff);
        }
        break;
      case ICmpInst::ICMP_EQ:
        // n == 0 ? 1+x : n+x  ->  umax(n, 1)+x
        if (LHS->getType() == U->getType() &&
            isa<ConstantInt>(RHS) &&
            cast<ConstantInt>(RHS)->isZero()) {
          const SCEV *One = getConstant(LHS->getType(), 1);
          const SCEV *LS = getSCEV(LHS);
          const SCEV *LA = getSCEV(U->getOperand(1));
          const SCEV *RA = getSCEV(U->getOperand(2));
          const SCEV *LDiff = getMinusSCEV(LA, One);
          const SCEV *RDiff = getMinusSCEV(RA, LS);
          if (LDiff == RDiff)
            return getAddExpr(getUMaxExpr(One, LS), LDiff);
        }
        break;
      default:
        break;
      }
    }

  default: // We cannot analyze this expression.
    break;
  }

  return getUnknown(V);
}



//===----------------------------------------------------------------------===//
//                   Iteration Count Computation Code
//

/// getSmallConstantTripCount - Returns the maximum trip count of this loop as a
/// normal unsigned value. Returns 0 if the trip count is unknown or not
/// constant. Will also return 0 if the maximum trip count is very large (>=
/// 2^32).
///
/// This "trip count" assumes that control exits via ExitingBlock. More
/// precisely, it is the number of times that control may reach ExitingBlock
/// before taking the branch. For loops with multiple exits, it may not be the
/// number times that the loop header executes because the loop may exit
/// prematurely via another branch.
///
/// FIXME: We conservatively call getBackedgeTakenCount(L) instead of
/// getExitCount(L, ExitingBlock) to compute a safe trip count considering all
/// loop exits. getExitCount() may return an exact count for this branch
/// assuming no-signed-wrap. The number of well-defined iterations may actually
/// be higher than this trip count if this exit test is skipped and the loop
/// exits via a different branch. Ideally, getExitCount() would know whether it
/// depends on a NSW assumption, and we would only fall back to a conservative
/// trip count in that case.
unsigned ScalarEvolution::
getSmallConstantTripCount(Loop *L, BasicBlock * /*ExitingBlock*/) {
  const SCEVConstant *ExitCount =
    dyn_cast<SCEVConstant>(getBackedgeTakenCount(L));
  if (!ExitCount)
    return 0;

  ConstantInt *ExitConst = ExitCount->getValue();

  // Guard against huge trip counts.
  if (ExitConst->getValue().getActiveBits() > 32)
    return 0;

  // In case of integer overflow, this returns 0, which is correct.
  return ((unsigned)ExitConst->getZExtValue()) + 1;
}

/// getSmallConstantTripMultiple - Returns the largest constant divisor of the
/// trip count of this loop as a normal unsigned value, if possible. This
/// means that the actual trip count is always a multiple of the returned
/// value (don't forget the trip count could very well be zero as well!).
///
/// Returns 1 if the trip count is unknown or not guaranteed to be the
/// multiple of a constant (which is also the case if the trip count is simply
/// constant, use getSmallConstantTripCount for that case), Will also return 1
/// if the trip count is very large (>= 2^32).
///
/// As explained in the comments for getSmallConstantTripCount, this assumes
/// that control exits the loop via ExitingBlock.
unsigned ScalarEvolution::
getSmallConstantTripMultiple(Loop *L, BasicBlock * /*ExitingBlock*/) {
  const SCEV *ExitCount = getBackedgeTakenCount(L);
  if (ExitCount == getCouldNotCompute())
    return 1;

  // Get the trip count from the BE count by adding 1.
  const SCEV *TCMul = getAddExpr(ExitCount,
                                 getConstant(ExitCount->getType(), 1));
  // FIXME: SCEV distributes multiplication as V1*C1 + V2*C1. We could attempt
  // to factor simple cases.
  if (const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(TCMul))
    TCMul = Mul->getOperand(0);

  const SCEVConstant *MulC = dyn_cast<SCEVConstant>(TCMul);
  if (!MulC)
    return 1;

  ConstantInt *Result = MulC->getValue();

  // Guard against huge trip counts (this requires checking
  // for zero to handle the case where the trip count == -1 and the
  // addition wraps).
  if (!Result || Result->getValue().getActiveBits() > 32 ||
      Result->getValue().getActiveBits() == 0)
    return 1;

  return (unsigned)Result->getZExtValue();
}

// getExitCount - Get the expression for the number of loop iterations for which
// this loop is guaranteed not to exit via ExitingBlock. Otherwise return
// SCEVCouldNotCompute.
const SCEV *ScalarEvolution::getExitCount(Loop *L, BasicBlock *ExitingBlock) {
  return getBackedgeTakenInfo(L).getExact(ExitingBlock, this);
}

/// getBackedgeTakenCount - If the specified loop has a predictable
/// backedge-taken count, return it, otherwise return a SCEVCouldNotCompute
/// object. The backedge-taken count is the number of times the loop header
/// will be branched to from within the loop. This is one less than the
/// trip count of the loop, since it doesn't count the first iteration,
/// when the header is branched to from outside the loop.
///
/// Note that it is not valid to call this method on a loop without a
/// loop-invariant backedge-taken count (see
/// hasLoopInvariantBackedgeTakenCount).
///
const SCEV *ScalarEvolution::getBackedgeTakenCount(const Loop *L) {
  return getBackedgeTakenInfo(L).getExact(this);
}

/// getMaxBackedgeTakenCount - Similar to getBackedgeTakenCount, except
/// return the least SCEV value that is known never to be less than the
/// actual backedge taken count.
const SCEV *ScalarEvolution::getMaxBackedgeTakenCount(const Loop *L) {
  return getBackedgeTakenInfo(L).getMax(this);
}

/// PushLoopPHIs - Push PHI nodes in the header of the given loop
/// onto the given Worklist.
static void
PushLoopPHIs(const Loop *L, SmallVectorImpl<Instruction *> &Worklist) {
  BasicBlock *Header = L->getHeader();

  // Push all Loop-header PHIs onto the Worklist stack.
  for (BasicBlock::iterator I = Header->begin();
       PHINode *PN = dyn_cast<PHINode>(I); ++I)
    Worklist.push_back(PN);
}

const ScalarEvolution::BackedgeTakenInfo &
ScalarEvolution::getBackedgeTakenInfo(const Loop *L) {
  // Initially insert an invalid entry for this loop. If the insertion
  // succeeds, proceed to actually compute a backedge-taken count and
  // update the value. The temporary CouldNotCompute value tells SCEV
  // code elsewhere that it shouldn't attempt to request a new
  // backedge-taken count, which could result in infinite recursion.
  std::pair<DenseMap<const Loop *, BackedgeTakenInfo>::iterator, bool> Pair =
    BackedgeTakenCounts.insert(std::make_pair(L, BackedgeTakenInfo()));
  if (!Pair.second)
    return Pair.first->second;

  // ComputeBackedgeTakenCount may allocate memory for its result. Inserting it
  // into the BackedgeTakenCounts map transfers ownership. Otherwise, the result
  // must be cleared in this scope.
  BackedgeTakenInfo Result = ComputeBackedgeTakenCount(L);

  if (Result.getExact(this) != getCouldNotCompute()) {
    assert(isLoopInvariant(Result.getExact(this), L) &&
           isLoopInvariant(Result.getMax(this), L) &&
           "Computed backedge-taken count isn't loop invariant for loop!");
    ++NumTripCountsComputed;
  }
  else if (Result.getMax(this) == getCouldNotCompute() &&
           isa<PHINode>(L->getHeader()->begin())) {
    // Only count loops that have phi nodes as not being computable.
    ++NumTripCountsNotComputed;
  }

  // Now that we know more about the trip count for this loop, forget any
  // existing SCEV values for PHI nodes in this loop since they are only
  // conservative estimates made without the benefit of trip count
  // information. This is similar to the code in forgetLoop, except that
  // it handles SCEVUnknown PHI nodes specially.
  if (Result.hasAnyInfo()) {
    SmallVector<Instruction *, 16> Worklist;
    PushLoopPHIs(L, Worklist);

    SmallPtrSet<Instruction *, 8> Visited;
    while (!Worklist.empty()) {
      Instruction *I = Worklist.pop_back_val();
      if (!Visited.insert(I)) continue;

      ValueExprMapType::iterator It =
        ValueExprMap.find_as(static_cast<Value *>(I));
      if (It != ValueExprMap.end()) {
        const SCEV *Old = It->second;

        // SCEVUnknown for a PHI either means that it has an unrecognized
        // structure, or it's a PHI that's in the progress of being computed
        // by createNodeForPHI.  In the former case, additional loop trip
        // count information isn't going to change anything. In the later
        // case, createNodeForPHI will perform the necessary updates on its
        // own when it gets to that point.
        if (!isa<PHINode>(I) || !isa<SCEVUnknown>(Old)) {
          forgetMemoizedResults(Old);
          ValueExprMap.erase(It);
        }
        if (PHINode *PN = dyn_cast<PHINode>(I))
          ConstantEvolutionLoopExitValue.erase(PN);
      }

      PushDefUseChildren(I, Worklist);
    }
  }

  // Re-lookup the insert position, since the call to
  // ComputeBackedgeTakenCount above could result in a
  // recusive call to getBackedgeTakenInfo (on a different
  // loop), which would invalidate the iterator computed
  // earlier.
  return BackedgeTakenCounts.find(L)->second = Result;
}

/// forgetLoop - This method should be called by the client when it has
/// changed a loop in a way that may effect ScalarEvolution's ability to
/// compute a trip count, or if the loop is deleted.
void ScalarEvolution::forgetLoop(const Loop *L) {
  // Drop any stored trip count value.
  DenseMap<const Loop*, BackedgeTakenInfo>::iterator BTCPos =
    BackedgeTakenCounts.find(L);
  if (BTCPos != BackedgeTakenCounts.end()) {
    BTCPos->second.clear();
    BackedgeTakenCounts.erase(BTCPos);
  }

  // Drop information about expressions based on loop-header PHIs.
  SmallVector<Instruction *, 16> Worklist;
  PushLoopPHIs(L, Worklist);

  SmallPtrSet<Instruction *, 8> Visited;
  while (!Worklist.empty()) {
    Instruction *I = Worklist.pop_back_val();
    if (!Visited.insert(I)) continue;

    ValueExprMapType::iterator It =
      ValueExprMap.find_as(static_cast<Value *>(I));
    if (It != ValueExprMap.end()) {
      forgetMemoizedResults(It->second);
      ValueExprMap.erase(It);
      if (PHINode *PN = dyn_cast<PHINode>(I))
        ConstantEvolutionLoopExitValue.erase(PN);
    }

    PushDefUseChildren(I, Worklist);
  }

  // Forget all contained loops too, to avoid dangling entries in the
  // ValuesAtScopes map.
  for (Loop::iterator I = L->begin(), E = L->end(); I != E; ++I)
    forgetLoop(*I);
}

/// forgetValue - This method should be called by the client when it has
/// changed a value in a way that may effect its value, or which may
/// disconnect it from a def-use chain linking it to a loop.
void ScalarEvolution::forgetValue(Value *V) {
  Instruction *I = dyn_cast<Instruction>(V);
  if (!I) return;

  // Drop information about expressions based on loop-header PHIs.
  SmallVector<Instruction *, 16> Worklist;
  Worklist.push_back(I);

  SmallPtrSet<Instruction *, 8> Visited;
  while (!Worklist.empty()) {
    I = Worklist.pop_back_val();
    if (!Visited.insert(I)) continue;

    ValueExprMapType::iterator It =
      ValueExprMap.find_as(static_cast<Value *>(I));
    if (It != ValueExprMap.end()) {
      forgetMemoizedResults(It->second);
      ValueExprMap.erase(It);
      if (PHINode *PN = dyn_cast<PHINode>(I))
        ConstantEvolutionLoopExitValue.erase(PN);
    }

    PushDefUseChildren(I, Worklist);
  }
}

/// getExact - Get the exact loop backedge taken count considering all loop
/// exits. A computable result can only be return for loops with a single exit.
/// Returning the minimum taken count among all exits is incorrect because one
/// of the loop's exit limit's may have been skipped. HowFarToZero assumes that
/// the limit of each loop test is never skipped. This is a valid assumption as
/// long as the loop exits via that test. For precise results, it is the
/// caller's responsibility to specify the relevant loop exit using
/// getExact(ExitingBlock, SE).
const SCEV *
ScalarEvolution::BackedgeTakenInfo::getExact(ScalarEvolution *SE) const {
  // If any exits were not computable, the loop is not computable.
  if (!ExitNotTaken.isCompleteList()) return SE->getCouldNotCompute();

  // We need exactly one computable exit.
  if (!ExitNotTaken.ExitingBlock) return SE->getCouldNotCompute();
  assert(ExitNotTaken.ExactNotTaken && "uninitialized not-taken info");

  const SCEV *BECount = 0;
  for (const ExitNotTakenInfo *ENT = &ExitNotTaken;
       ENT != 0; ENT = ENT->getNextExit()) {

    assert(ENT->ExactNotTaken != SE->getCouldNotCompute() && "bad exit SCEV");

    if (!BECount)
      BECount = ENT->ExactNotTaken;
    else if (BECount != ENT->ExactNotTaken)
      return SE->getCouldNotCompute();
  }
  assert(BECount && "Invalid not taken count for loop exit");
  return BECount;
}

/// getExact - Get the exact not taken count for this loop exit.
const SCEV *
ScalarEvolution::BackedgeTakenInfo::getExact(BasicBlock *ExitingBlock,
                                             ScalarEvolution *SE) const {
  for (const ExitNotTakenInfo *ENT = &ExitNotTaken;
       ENT != 0; ENT = ENT->getNextExit()) {

    if (ENT->ExitingBlock == ExitingBlock)
      return ENT->ExactNotTaken;
  }
  return SE->getCouldNotCompute();
}

/// getMax - Get the max backedge taken count for the loop.
const SCEV *
ScalarEvolution::BackedgeTakenInfo::getMax(ScalarEvolution *SE) const {
  return Max ? Max : SE->getCouldNotCompute();
}

bool ScalarEvolution::BackedgeTakenInfo::hasOperand(const SCEV *S,
                                                    ScalarEvolution *SE) const {
  if (Max && Max != SE->getCouldNotCompute() && SE->hasOperand(Max, S))
    return true;

  if (!ExitNotTaken.ExitingBlock)
    return false;

  for (const ExitNotTakenInfo *ENT = &ExitNotTaken;
       ENT != 0; ENT = ENT->getNextExit()) {

    if (ENT->ExactNotTaken != SE->getCouldNotCompute()
        && SE->hasOperand(ENT->ExactNotTaken, S)) {
      return true;
    }
  }
  return false;
}

/// Allocate memory for BackedgeTakenInfo and copy the not-taken count of each
/// computable exit into a persistent ExitNotTakenInfo array.
ScalarEvolution::BackedgeTakenInfo::BackedgeTakenInfo(
  SmallVectorImpl< std::pair<BasicBlock *, const SCEV *> > &ExitCounts,
  bool Complete, const SCEV *MaxCount) : Max(MaxCount) {

  if (!Complete)
    ExitNotTaken.setIncomplete();

  unsigned NumExits = ExitCounts.size();
  if (NumExits == 0) return;

  ExitNotTaken.ExitingBlock = ExitCounts[0].first;
  ExitNotTaken.ExactNotTaken = ExitCounts[0].second;
  if (NumExits == 1) return;

  // Handle the rare case of multiple computable exits.
  ExitNotTakenInfo *ENT = new ExitNotTakenInfo[NumExits-1];

  ExitNotTakenInfo *PrevENT = &ExitNotTaken;
  for (unsigned i = 1; i < NumExits; ++i, PrevENT = ENT, ++ENT) {
    PrevENT->setNextExit(ENT);
    ENT->ExitingBlock = ExitCounts[i].first;
    ENT->ExactNotTaken = ExitCounts[i].second;
  }
}

/// clear - Invalidate this result and free the ExitNotTakenInfo array.
void ScalarEvolution::BackedgeTakenInfo::clear() {
  ExitNotTaken.ExitingBlock = 0;
  ExitNotTaken.ExactNotTaken = 0;
  delete[] ExitNotTaken.getNextExit();
}

/// ComputeBackedgeTakenCount - Compute the number of times the backedge
/// of the specified loop will execute.
ScalarEvolution::BackedgeTakenInfo
ScalarEvolution::ComputeBackedgeTakenCount(const Loop *L) {
  SmallVector<BasicBlock *, 8> ExitingBlocks;
  L->getExitingBlocks(ExitingBlocks);

  // Examine all exits and pick the most conservative values.
  const SCEV *MaxBECount = getCouldNotCompute();
  bool CouldComputeBECount = true;
  SmallVector<std::pair<BasicBlock *, const SCEV *>, 4> ExitCounts;
  for (unsigned i = 0, e = ExitingBlocks.size(); i != e; ++i) {
    ExitLimit EL = ComputeExitLimit(L, ExitingBlocks[i]);
    if (EL.Exact == getCouldNotCompute())
      // We couldn't compute an exact value for this exit, so
      // we won't be able to compute an exact value for the loop.
      CouldComputeBECount = false;
    else
      ExitCounts.push_back(std::make_pair(ExitingBlocks[i], EL.Exact));

    if (MaxBECount == getCouldNotCompute())
      MaxBECount = EL.Max;
    else if (EL.Max != getCouldNotCompute()) {
      // We cannot take the "min" MaxBECount, because non-unit stride loops may
      // skip some loop tests. Taking the max over the exits is sufficiently
      // conservative.  TODO: We could do better taking into consideration
      // that (1) the loop has unit stride (2) the last loop test is
      // less-than/greater-than (3) any loop test is less-than/greater-than AND
      // falls-through some constant times less then the other tests.
      MaxBECount = getUMaxFromMismatchedTypes(MaxBECount, EL.Max);
    }
  }

  return BackedgeTakenInfo(ExitCounts, CouldComputeBECount, MaxBECount);
}

/// ComputeExitLimit - Compute the number of times the backedge of the specified
/// loop will execute if it exits via the specified block.
ScalarEvolution::ExitLimit
ScalarEvolution::ComputeExitLimit(const Loop *L, BasicBlock *ExitingBlock) {

  // Okay, we've chosen an exiting block.  See what condition causes us to
  // exit at this block.
  //
  // FIXME: we should be able to handle switch instructions (with a single exit)
  BranchInst *ExitBr = dyn_cast<BranchInst>(ExitingBlock->getTerminator());
  if (ExitBr == 0) return getCouldNotCompute();
  assert(ExitBr->isConditional() && "If unconditional, it can't be in loop!");

  // At this point, we know we have a conditional branch that determines whether
  // the loop is exited.  However, we don't know if the branch is executed each
  // time through the loop.  If not, then the execution count of the branch will
  // not be equal to the trip count of the loop.
  //
  // Currently we check for this by checking to see if the Exit branch goes to
  // the loop header.  If so, we know it will always execute the same number of
  // times as the loop.  We also handle the case where the exit block *is* the
  // loop header.  This is common for un-rotated loops.
  //
  // If both of those tests fail, walk up the unique predecessor chain to the
  // header, stopping if there is an edge that doesn't exit the loop. If the
  // header is reached, the execution count of the branch will be equal to the
  // trip count of the loop.
  //
  //  More extensive analysis could be done to handle more cases here.
  //
  if (ExitBr->getSuccessor(0) != L->getHeader() &&
      ExitBr->getSuccessor(1) != L->getHeader() &&
      ExitBr->getParent() != L->getHeader()) {
    // The simple checks failed, try climbing the unique predecessor chain
    // up to the header.
    bool Ok = false;
    for (BasicBlock *BB = ExitBr->getParent(); BB; ) {
      BasicBlock *Pred = BB->getUniquePredecessor();
      if (!Pred)
        return getCouldNotCompute();
      TerminatorInst *PredTerm = Pred->getTerminator();
      for (unsigned i = 0, e = PredTerm->getNumSuccessors(); i != e; ++i) {
        BasicBlock *PredSucc = PredTerm->getSuccessor(i);
        if (PredSucc == BB)
          continue;
        // If the predecessor has a successor that isn't BB and isn't
        // outside the loop, assume the worst.
        if (L->contains(PredSucc))
          return getCouldNotCompute();
      }
      if (Pred == L->getHeader()) {
        Ok = true;
        break;
      }
      BB = Pred;
    }
    if (!Ok)
      return getCouldNotCompute();
  }

  // Proceed to the next level to examine the exit condition expression.
  return ComputeExitLimitFromCond(L, ExitBr->getCondition(),
                                  ExitBr->getSuccessor(0),
                                  ExitBr->getSuccessor(1),
                                  /*IsSubExpr=*/false);
}

/// ComputeExitLimitFromCond - Compute the number of times the
/// backedge of the specified loop will execute if its exit condition
/// were a conditional branch of ExitCond, TBB, and FBB.
///
/// @param IsSubExpr is true if ExitCond does not directly control the exit
/// branch. In this case, we cannot assume that the loop only exits when the
/// condition is true and cannot infer that failing to meet the condition prior
/// to integer wraparound results in undefined behavior.
ScalarEvolution::ExitLimit
ScalarEvolution::ComputeExitLimitFromCond(const Loop *L,
                                          Value *ExitCond,
                                          BasicBlock *TBB,
                                          BasicBlock *FBB,
                                          bool IsSubExpr) {
  // Check if the controlling expression for this loop is an And or Or.
  if (BinaryOperator *BO = dyn_cast<BinaryOperator>(ExitCond)) {
    if (BO->getOpcode() == Instruction::And) {
      // Recurse on the operands of the and.
      bool EitherMayExit = L->contains(TBB);
      ExitLimit EL0 = ComputeExitLimitFromCond(L, BO->getOperand(0), TBB, FBB,
                                               IsSubExpr || EitherMayExit);
      ExitLimit EL1 = ComputeExitLimitFromCond(L, BO->getOperand(1), TBB, FBB,
                                               IsSubExpr || EitherMayExit);
      const SCEV *BECount = getCouldNotCompute();
      const SCEV *MaxBECount = getCouldNotCompute();
      if (EitherMayExit) {
        // Both conditions must be true for the loop to continue executing.
        // Choose the less conservative count.
        if (EL0.Exact == getCouldNotCompute() ||
            EL1.Exact == getCouldNotCompute())
          BECount = getCouldNotCompute();
        else
          BECount = getUMinFromMismatchedTypes(EL0.Exact, EL1.Exact);
        if (EL0.Max == getCouldNotCompute())
          MaxBECount = EL1.Max;
        else if (EL1.Max == getCouldNotCompute())
          MaxBECount = EL0.Max;
        else
          MaxBECount = getUMinFromMismatchedTypes(EL0.Max, EL1.Max);
      } else {
        // Both conditions must be true at the same time for the loop to exit.
        // For now, be conservative.
        assert(L->contains(FBB) && "Loop block has no successor in loop!");
        if (EL0.Max == EL1.Max)
          MaxBECount = EL0.Max;
        if (EL0.Exact == EL1.Exact)
          BECount = EL0.Exact;
      }

      return ExitLimit(BECount, MaxBECount);
    }
    if (BO->getOpcode() == Instruction::Or) {
      // Recurse on the operands of the or.
      bool EitherMayExit = L->contains(FBB);
      ExitLimit EL0 = ComputeExitLimitFromCond(L, BO->getOperand(0), TBB, FBB,
                                               IsSubExpr || EitherMayExit);
      ExitLimit EL1 = ComputeExitLimitFromCond(L, BO->getOperand(1), TBB, FBB,
                                               IsSubExpr || EitherMayExit);
      const SCEV *BECount = getCouldNotCompute();
      const SCEV *MaxBECount = getCouldNotCompute();
      if (EitherMayExit) {
        // Both conditions must be false for the loop to continue executing.
        // Choose the less conservative count.
        if (EL0.Exact == getCouldNotCompute() ||
            EL1.Exact == getCouldNotCompute())
          BECount = getCouldNotCompute();
        else
          BECount = getUMinFromMismatchedTypes(EL0.Exact, EL1.Exact);
        if (EL0.Max == getCouldNotCompute())
          MaxBECount = EL1.Max;
        else if (EL1.Max == getCouldNotCompute())
          MaxBECount = EL0.Max;
        else
          MaxBECount = getUMinFromMismatchedTypes(EL0.Max, EL1.Max);
      } else {
        // Both conditions must be false at the same time for the loop to exit.
        // For now, be conservative.
        assert(L->contains(TBB) && "Loop block has no successor in loop!");
        if (EL0.Max == EL1.Max)
          MaxBECount = EL0.Max;
        if (EL0.Exact == EL1.Exact)
          BECount = EL0.Exact;
      }

      return ExitLimit(BECount, MaxBECount);
    }
  }

  // With an icmp, it may be feasible to compute an exact backedge-taken count.
  // Proceed to the next level to examine the icmp.
  if (ICmpInst *ExitCondICmp = dyn_cast<ICmpInst>(ExitCond))
    return ComputeExitLimitFromICmp(L, ExitCondICmp, TBB, FBB, IsSubExpr);

  // Check for a constant condition. These are normally stripped out by
  // SimplifyCFG, but ScalarEvolution may be used by a pass which wishes to
  // preserve the CFG and is temporarily leaving constant conditions
  // in place.
  if (ConstantInt *CI = dyn_cast<ConstantInt>(ExitCond)) {
    if (L->contains(FBB) == !CI->getZExtValue())
      // The backedge is always taken.
      return getCouldNotCompute();
    else
      // The backedge is never taken.
      return getConstant(CI->getType(), 0);
  }

  // If it's not an integer or pointer comparison then compute it the hard way.
  return ComputeExitCountExhaustively(L, ExitCond, !L->contains(TBB));
}

/// ComputeExitLimitFromICmp - Compute the number of times the
/// backedge of the specified loop will execute if its exit condition
/// were a conditional branch of the ICmpInst ExitCond, TBB, and FBB.
ScalarEvolution::ExitLimit
ScalarEvolution::ComputeExitLimitFromICmp(const Loop *L,
                                          ICmpInst *ExitCond,
                                          BasicBlock *TBB,
                                          BasicBlock *FBB,
                                          bool IsSubExpr) {

  // If the condition was exit on true, convert the condition to exit on false
  ICmpInst::Predicate Cond;
  if (!L->contains(FBB))
    Cond = ExitCond->getPredicate();
  else
    Cond = ExitCond->getInversePredicate();

  // Handle common loops like: for (X = "string"; *X; ++X)
  if (LoadInst *LI = dyn_cast<LoadInst>(ExitCond->getOperand(0)))
    if (Constant *RHS = dyn_cast<Constant>(ExitCond->getOperand(1))) {
      ExitLimit ItCnt =
        ComputeLoadConstantCompareExitLimit(LI, RHS, L, Cond);
      if (ItCnt.hasAnyInfo())
        return ItCnt;
    }

  const SCEV *LHS = getSCEV(ExitCond->getOperand(0));
  const SCEV *RHS = getSCEV(ExitCond->getOperand(1));

  // Try to evaluate any dependencies out of the loop.
  LHS = getSCEVAtScope(LHS, L);
  RHS = getSCEVAtScope(RHS, L);

  // At this point, we would like to compute how many iterations of the
  // loop the predicate will return true for these inputs.
  if (isLoopInvariant(LHS, L) && !isLoopInvariant(RHS, L)) {
    // If there is a loop-invariant, force it into the RHS.
    std::swap(LHS, RHS);
    Cond = ICmpInst::getSwappedPredicate(Cond);
  }

  // Simplify the operands before analyzing them.
  (void)SimplifyICmpOperands(Cond, LHS, RHS);

  // If we have a comparison of a chrec against a constant, try to use value
  // ranges to answer this query.
  if (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS))
    if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(LHS))
      if (AddRec->getLoop() == L) {
        // Form the constant range.
        ConstantRange CompRange(
            ICmpInst::makeConstantRange(Cond, RHSC->getValue()->getValue()));

        const SCEV *Ret = AddRec->getNumIterationsInRange(CompRange, *this);
        if (!isa<SCEVCouldNotCompute>(Ret)) return Ret;
      }

  switch (Cond) {
  case ICmpInst::ICMP_NE: {                     // while (X != Y)
    // Convert to: while (X-Y != 0)
    ExitLimit EL = HowFarToZero(getMinusSCEV(LHS, RHS), L, IsSubExpr);
    if (EL.hasAnyInfo()) return EL;
    break;
  }
  case ICmpInst::ICMP_EQ: {                     // while (X == Y)
    // Convert to: while (X-Y == 0)
    ExitLimit EL = HowFarToNonZero(getMinusSCEV(LHS, RHS), L);
    if (EL.hasAnyInfo()) return EL;
    break;
  }
  case ICmpInst::ICMP_SLT: {
    ExitLimit EL = HowManyLessThans(LHS, RHS, L, true, IsSubExpr);
    if (EL.hasAnyInfo()) return EL;
    break;
  }
  case ICmpInst::ICMP_SGT: {
    ExitLimit EL = HowManyLessThans(getNotSCEV(LHS),
                                    getNotSCEV(RHS), L, true, IsSubExpr);
    if (EL.hasAnyInfo()) return EL;
    break;
  }
  case ICmpInst::ICMP_ULT: {
    ExitLimit EL = HowManyLessThans(LHS, RHS, L, false, IsSubExpr);
    if (EL.hasAnyInfo()) return EL;
    break;
  }
  case ICmpInst::ICMP_UGT: {
    ExitLimit EL = HowManyLessThans(getNotSCEV(LHS),
                                    getNotSCEV(RHS), L, false, IsSubExpr);
    if (EL.hasAnyInfo()) return EL;
    break;
  }
  default:
#if 0
    dbgs() << "ComputeBackedgeTakenCount ";
    if (ExitCond->getOperand(0)->getType()->isUnsigned())
      dbgs() << "[unsigned] ";
    dbgs() << *LHS << "   "
         << Instruction::getOpcodeName(Instruction::ICmp)
         << "   " << *RHS << "\n";
#endif
    break;
  }
  return ComputeExitCountExhaustively(L, ExitCond, !L->contains(TBB));
}

static ConstantInt *
EvaluateConstantChrecAtConstant(const SCEVAddRecExpr *AddRec, ConstantInt *C,
                                ScalarEvolution &SE) {
  const SCEV *InVal = SE.getConstant(C);
  const SCEV *Val = AddRec->evaluateAtIteration(InVal, SE);
  assert(isa<SCEVConstant>(Val) &&
         "Evaluation of SCEV at constant didn't fold correctly?");
  return cast<SCEVConstant>(Val)->getValue();
}

/// ComputeLoadConstantCompareExitLimit - Given an exit condition of
/// 'icmp op load X, cst', try to see if we can compute the backedge
/// execution count.
ScalarEvolution::ExitLimit
ScalarEvolution::ComputeLoadConstantCompareExitLimit(
  LoadInst *LI,
  Constant *RHS,
  const Loop *L,
  ICmpInst::Predicate predicate) {

  if (LI->isVolatile()) return getCouldNotCompute();

  // Check to see if the loaded pointer is a getelementptr of a global.
  // TODO: Use SCEV instead of manually grubbing with GEPs.
  GetElementPtrInst *GEP = dyn_cast<GetElementPtrInst>(LI->getOperand(0));
  if (!GEP) return getCouldNotCompute();

  // Make sure that it is really a constant global we are gepping, with an
  // initializer, and make sure the first IDX is really 0.
  GlobalVariable *GV = dyn_cast<GlobalVariable>(GEP->getOperand(0));
  if (!GV || !GV->isConstant() || !GV->hasDefinitiveInitializer() ||
      GEP->getNumOperands() < 3 || !isa<Constant>(GEP->getOperand(1)) ||
      !cast<Constant>(GEP->getOperand(1))->isNullValue())
    return getCouldNotCompute();

  // Okay, we allow one non-constant index into the GEP instruction.
  Value *VarIdx = 0;
  std::vector<Constant*> Indexes;
  unsigned VarIdxNum = 0;
  for (unsigned i = 2, e = GEP->getNumOperands(); i != e; ++i)
    if (ConstantInt *CI = dyn_cast<ConstantInt>(GEP->getOperand(i))) {
      Indexes.push_back(CI);
    } else if (!isa<ConstantInt>(GEP->getOperand(i))) {
      if (VarIdx) return getCouldNotCompute();  // Multiple non-constant idx's.
      VarIdx = GEP->getOperand(i);
      VarIdxNum = i-2;
      Indexes.push_back(0);
    }

  // Loop-invariant loads may be a byproduct of loop optimization. Skip them.
  if (!VarIdx)
    return getCouldNotCompute();

  // Okay, we know we have a (load (gep GV, 0, X)) comparison with a constant.
  // Check to see if X is a loop variant variable value now.
  const SCEV *Idx = getSCEV(VarIdx);
  Idx = getSCEVAtScope(Idx, L);

  // We can only recognize very limited forms of loop index expressions, in
  // particular, only affine AddRec's like {C1,+,C2}.
  const SCEVAddRecExpr *IdxExpr = dyn_cast<SCEVAddRecExpr>(Idx);
  if (!IdxExpr || !IdxExpr->isAffine() || isLoopInvariant(IdxExpr, L) ||
      !isa<SCEVConstant>(IdxExpr->getOperand(0)) ||
      !isa<SCEVConstant>(IdxExpr->getOperand(1)))
    return getCouldNotCompute();

  unsigned MaxSteps = MaxBruteForceIterations;
  for (unsigned IterationNum = 0; IterationNum != MaxSteps; ++IterationNum) {
    ConstantInt *ItCst = ConstantInt::get(
                           cast<IntegerType>(IdxExpr->getType()), IterationNum);
    ConstantInt *Val = EvaluateConstantChrecAtConstant(IdxExpr, ItCst, *this);

    // Form the GEP offset.
    Indexes[VarIdxNum] = Val;

    Constant *Result = ConstantFoldLoadThroughGEPIndices(GV->getInitializer(),
                                                         Indexes);
    if (Result == 0) break;  // Cannot compute!

    // Evaluate the condition for this iteration.
    Result = ConstantExpr::getICmp(predicate, Result, RHS);
    if (!isa<ConstantInt>(Result)) break;  // Couldn't decide for sure
    if (cast<ConstantInt>(Result)->getValue().isMinValue()) {
#if 0
      dbgs() << "\n***\n*** Computed loop count " << *ItCst
             << "\n*** From global " << *GV << "*** BB: " << *L->getHeader()
             << "***\n";
#endif
      ++NumArrayLenItCounts;
      return getConstant(ItCst);   // Found terminating iteration!
    }
  }
  return getCouldNotCompute();
}


/// CanConstantFold - Return true if we can constant fold an instruction of the
/// specified type, assuming that all operands were constants.
static bool CanConstantFold(const Instruction *I) {
  if (isa<BinaryOperator>(I) || isa<CmpInst>(I) ||
      isa<SelectInst>(I) || isa<CastInst>(I) || isa<GetElementPtrInst>(I) ||
      isa<LoadInst>(I))
    return true;

  if (const CallInst *CI = dyn_cast<CallInst>(I))
    if (const Function *F = CI->getCalledFunction())
      return canConstantFoldCallTo(F);
  return false;
}

/// Determine whether this instruction can constant evolve within this loop
/// assuming its operands can all constant evolve.
static bool canConstantEvolve(Instruction *I, const Loop *L) {
  // An instruction outside of the loop can't be derived from a loop PHI.
  if (!L->contains(I)) return false;

  if (isa<PHINode>(I)) {
    if (L->getHeader() == I->getParent())
      return true;
    else
      // We don't currently keep track of the control flow needed to evaluate
      // PHIs, so we cannot handle PHIs inside of loops.
      return false;
  }

  // If we won't be able to constant fold this expression even if the operands
  // are constants, bail early.
  return CanConstantFold(I);
}

/// getConstantEvolvingPHIOperands - Implement getConstantEvolvingPHI by
/// recursing through each instruction operand until reaching a loop header phi.
static PHINode *
getConstantEvolvingPHIOperands(Instruction *UseInst, const Loop *L,
                               DenseMap<Instruction *, PHINode *> &PHIMap) {

  // Otherwise, we can evaluate this instruction if all of its operands are
  // constant or derived from a PHI node themselves.
  PHINode *PHI = 0;
  for (Instruction::op_iterator OpI = UseInst->op_begin(),
         OpE = UseInst->op_end(); OpI != OpE; ++OpI) {

    if (isa<Constant>(*OpI)) continue;

    Instruction *OpInst = dyn_cast<Instruction>(*OpI);
    if (!OpInst || !canConstantEvolve(OpInst, L)) return 0;

    PHINode *P = dyn_cast<PHINode>(OpInst);
    if (!P)
      // If this operand is already visited, reuse the prior result.
      // We may have P != PHI if this is the deepest point at which the
      // inconsistent paths meet.
      P = PHIMap.lookup(OpInst);
    if (!P) {
      // Recurse and memoize the results, whether a phi is found or not.
      // This recursive call invalidates pointers into PHIMap.
      P = getConstantEvolvingPHIOperands(OpInst, L, PHIMap);
      PHIMap[OpInst] = P;
    }
    if (P == 0) return 0;        // Not evolving from PHI
    if (PHI && PHI != P) return 0;  // Evolving from multiple different PHIs.
    PHI = P;
  }
  // This is a expression evolving from a constant PHI!
  return PHI;
}

/// getConstantEvolvingPHI - Given an LLVM value and a loop, return a PHI node
/// in the loop that V is derived from.  We allow arbitrary operations along the
/// way, but the operands of an operation must either be constants or a value
/// derived from a constant PHI.  If this expression does not fit with these
/// constraints, return null.
static PHINode *getConstantEvolvingPHI(Value *V, const Loop *L) {
  Instruction *I = dyn_cast<Instruction>(V);
  if (I == 0 || !canConstantEvolve(I, L)) return 0;

  if (PHINode *PN = dyn_cast<PHINode>(I)) {
    return PN;
  }

  // Record non-constant instructions contained by the loop.
  DenseMap<Instruction *, PHINode *> PHIMap;
  return getConstantEvolvingPHIOperands(I, L, PHIMap);
}

/// EvaluateExpression - Given an expression that passes the
/// getConstantEvolvingPHI predicate, evaluate its value assuming the PHI node
/// in the loop has the value PHIVal.  If we can't fold this expression for some
/// reason, return null.
static Constant *EvaluateExpression(Value *V, const Loop *L,
                                    DenseMap<Instruction *, Constant *> &Vals,
                                    const DataLayout *TD,
                                    const TargetLibraryInfo *TLI) {
  // Convenient constant check, but redundant for recursive calls.
  if (Constant *C = dyn_cast<Constant>(V)) return C;
  Instruction *I = dyn_cast<Instruction>(V);
  if (!I) return 0;

  if (Constant *C = Vals.lookup(I)) return C;

  // An instruction inside the loop depends on a value outside the loop that we
  // weren't given a mapping for, or a value such as a call inside the loop.
  if (!canConstantEvolve(I, L)) return 0;

  // An unmapped PHI can be due to a branch or another loop inside this loop,
  // or due to this not being the initial iteration through a loop where we
  // couldn't compute the evolution of this particular PHI last time.
  if (isa<PHINode>(I)) return 0;

  std::vector<Constant*> Operands(I->getNumOperands());

  for (unsigned i = 0, e = I->getNumOperands(); i != e; ++i) {
    Instruction *Operand = dyn_cast<Instruction>(I->getOperand(i));
    if (!Operand) {
      Operands[i] = dyn_cast<Constant>(I->getOperand(i));
      if (!Operands[i]) return 0;
      continue;
    }
    Constant *C = EvaluateExpression(Operand, L, Vals, TD, TLI);
    Vals[Operand] = C;
    if (!C) return 0;
    Operands[i] = C;
  }

  if (CmpInst *CI = dyn_cast<CmpInst>(I))
    return ConstantFoldCompareInstOperands(CI->getPredicate(), Operands[0],
                                           Operands[1], TD, TLI);
  if (LoadInst *LI = dyn_cast<LoadInst>(I)) {
    if (!LI->isVolatile())
      return ConstantFoldLoadFromConstPtr(Operands[0], TD);
  }
  return ConstantFoldInstOperands(I->getOpcode(), I->getType(), Operands, TD,
                                  TLI);
}

/// getConstantEvolutionLoopExitValue - If we know that the specified Phi is
/// in the header of its containing loop, we know the loop executes a
/// constant number of times, and the PHI node is just a recurrence
/// involving constants, fold it.
Constant *
ScalarEvolution::getConstantEvolutionLoopExitValue(PHINode *PN,
                                                   const APInt &BEs,
                                                   const Loop *L) {
  DenseMap<PHINode*, Constant*>::const_iterator I =
    ConstantEvolutionLoopExitValue.find(PN);
  if (I != ConstantEvolutionLoopExitValue.end())
    return I->second;

  if (BEs.ugt(MaxBruteForceIterations))
    return ConstantEvolutionLoopExitValue[PN] = 0;  // Not going to evaluate it.

  Constant *&RetVal = ConstantEvolutionLoopExitValue[PN];

  DenseMap<Instruction *, Constant *> CurrentIterVals;
  BasicBlock *Header = L->getHeader();
  assert(PN->getParent() == Header && "Can't evaluate PHI not in loop header!");

  // Since the loop is canonicalized, the PHI node must have two entries.  One
  // entry must be a constant (coming in from outside of the loop), and the
  // second must be derived from the same PHI.
  bool SecondIsBackedge = L->contains(PN->getIncomingBlock(1));
  PHINode *PHI = 0;
  for (BasicBlock::iterator I = Header->begin();
       (PHI = dyn_cast<PHINode>(I)); ++I) {
    Constant *StartCST =
      dyn_cast<Constant>(PHI->getIncomingValue(!SecondIsBackedge));
    if (StartCST == 0) continue;
    CurrentIterVals[PHI] = StartCST;
  }
  if (!CurrentIterVals.count(PN))
    return RetVal = 0;

  Value *BEValue = PN->getIncomingValue(SecondIsBackedge);

  // Execute the loop symbolically to determine the exit value.
  if (BEs.getActiveBits() >= 32)
    return RetVal = 0; // More than 2^32-1 iterations?? Not doing it!

  unsigned NumIterations = BEs.getZExtValue(); // must be in range
  unsigned IterationNum = 0;
  for (; ; ++IterationNum) {
    if (IterationNum == NumIterations)
      return RetVal = CurrentIterVals[PN];  // Got exit value!

    // Compute the value of the PHIs for the next iteration.
    // EvaluateExpression adds non-phi values to the CurrentIterVals map.
    DenseMap<Instruction *, Constant *> NextIterVals;
    Constant *NextPHI = EvaluateExpression(BEValue, L, CurrentIterVals, TD,
                                           TLI);
    if (NextPHI == 0)
      return 0;        // Couldn't evaluate!
    NextIterVals[PN] = NextPHI;

    bool StoppedEvolving = NextPHI == CurrentIterVals[PN];

    // Also evaluate the other PHI nodes.  However, we don't get to stop if we
    // cease to be able to evaluate one of them or if they stop evolving,
    // because that doesn't necessarily prevent us from computing PN.
    SmallVector<std::pair<PHINode *, Constant *>, 8> PHIsToCompute;
    for (DenseMap<Instruction *, Constant *>::const_iterator
           I = CurrentIterVals.begin(), E = CurrentIterVals.end(); I != E; ++I){
      PHINode *PHI = dyn_cast<PHINode>(I->first);
      if (!PHI || PHI == PN || PHI->getParent() != Header) continue;
      PHIsToCompute.push_back(std::make_pair(PHI, I->second));
    }
    // We use two distinct loops because EvaluateExpression may invalidate any
    // iterators into CurrentIterVals.
    for (SmallVectorImpl<std::pair<PHINode *, Constant*> >::const_iterator
             I = PHIsToCompute.begin(), E = PHIsToCompute.end(); I != E; ++I) {
      PHINode *PHI = I->first;
      Constant *&NextPHI = NextIterVals[PHI];
      if (!NextPHI) {   // Not already computed.
        Value *BEValue = PHI->getIncomingValue(SecondIsBackedge);
        NextPHI = EvaluateExpression(BEValue, L, CurrentIterVals, TD, TLI);
      }
      if (NextPHI != I->second)
        StoppedEvolving = false;
    }

    // If all entries in CurrentIterVals == NextIterVals then we can stop
    // iterating, the loop can't continue to change.
    if (StoppedEvolving)
      return RetVal = CurrentIterVals[PN];

    CurrentIterVals.swap(NextIterVals);
  }
}

/// ComputeExitCountExhaustively - If the loop is known to execute a
/// constant number of times (the condition evolves only from constants),
/// try to evaluate a few iterations of the loop until we get the exit
/// condition gets a value of ExitWhen (true or false).  If we cannot
/// evaluate the trip count of the loop, return getCouldNotCompute().
const SCEV *ScalarEvolution::ComputeExitCountExhaustively(const Loop *L,
                                                          Value *Cond,
                                                          bool ExitWhen) {
  PHINode *PN = getConstantEvolvingPHI(Cond, L);
  if (PN == 0) return getCouldNotCompute();

  // If the loop is canonicalized, the PHI will have exactly two entries.
  // That's the only form we support here.
  if (PN->getNumIncomingValues() != 2) return getCouldNotCompute();

  DenseMap<Instruction *, Constant *> CurrentIterVals;
  BasicBlock *Header = L->getHeader();
  assert(PN->getParent() == Header && "Can't evaluate PHI not in loop header!");

  // One entry must be a constant (coming in from outside of the loop), and the
  // second must be derived from the same PHI.
  bool SecondIsBackedge = L->contains(PN->getIncomingBlock(1));
  PHINode *PHI = 0;
  for (BasicBlock::iterator I = Header->begin();
       (PHI = dyn_cast<PHINode>(I)); ++I) {
    Constant *StartCST =
      dyn_cast<Constant>(PHI->getIncomingValue(!SecondIsBackedge));
    if (StartCST == 0) continue;
    CurrentIterVals[PHI] = StartCST;
  }
  if (!CurrentIterVals.count(PN))
    return getCouldNotCompute();

  // Okay, we find a PHI node that defines the trip count of this loop.  Execute
  // the loop symbolically to determine when the condition gets a value of
  // "ExitWhen".

  unsigned MaxIterations = MaxBruteForceIterations;   // Limit analysis.
  for (unsigned IterationNum = 0; IterationNum != MaxIterations;++IterationNum){
    ConstantInt *CondVal =
      dyn_cast_or_null<ConstantInt>(EvaluateExpression(Cond, L, CurrentIterVals,
                                                       TD, TLI));

    // Couldn't symbolically evaluate.
    if (!CondVal) return getCouldNotCompute();

    if (CondVal->getValue() == uint64_t(ExitWhen)) {
      ++NumBruteForceTripCountsComputed;
      return getConstant(Type::getInt32Ty(getContext()), IterationNum);
    }

    // Update all the PHI nodes for the next iteration.
    DenseMap<Instruction *, Constant *> NextIterVals;

    // Create a list of which PHIs we need to compute. We want to do this before
    // calling EvaluateExpression on them because that may invalidate iterators
    // into CurrentIterVals.
    SmallVector<PHINode *, 8> PHIsToCompute;
    for (DenseMap<Instruction *, Constant *>::const_iterator
           I = CurrentIterVals.begin(), E = CurrentIterVals.end(); I != E; ++I){
      PHINode *PHI = dyn_cast<PHINode>(I->first);
      if (!PHI || PHI->getParent() != Header) continue;
      PHIsToCompute.push_back(PHI);
    }
    for (SmallVectorImpl<PHINode *>::const_iterator I = PHIsToCompute.begin(),
             E = PHIsToCompute.end(); I != E; ++I) {
      PHINode *PHI = *I;
      Constant *&NextPHI = NextIterVals[PHI];
      if (NextPHI) continue;    // Already computed!

      Value *BEValue = PHI->getIncomingValue(SecondIsBackedge);
      NextPHI = EvaluateExpression(BEValue, L, CurrentIterVals, TD, TLI);
    }
    CurrentIterVals.swap(NextIterVals);
  }

  // Too many iterations were needed to evaluate.
  return getCouldNotCompute();
}

/// getSCEVAtScope - Return a SCEV expression for the specified value
/// at the specified scope in the program.  The L value specifies a loop
/// nest to evaluate the expression at, where null is the top-level or a
/// specified loop is immediately inside of the loop.
///
/// This method can be used to compute the exit value for a variable defined
/// in a loop by querying what the value will hold in the parent loop.
///
/// In the case that a relevant loop exit value cannot be computed, the
/// original value V is returned.
const SCEV *ScalarEvolution::getSCEVAtScope(const SCEV *V, const Loop *L) {
  // Check to see if we've folded this expression at this loop before.
  std::map<const Loop *, const SCEV *> &Values = ValuesAtScopes[V];
  std::pair<std::map<const Loop *, const SCEV *>::iterator, bool> Pair =
    Values.insert(std::make_pair(L, static_cast<const SCEV *>(0)));
  if (!Pair.second)
    return Pair.first->second ? Pair.first->second : V;

  // Otherwise compute it.
  const SCEV *C = computeSCEVAtScope(V, L);
  ValuesAtScopes[V][L] = C;
  return C;
}

/// This builds up a Constant using the ConstantExpr interface.  That way, we
/// will return Constants for objects which aren't represented by a
/// SCEVConstant, because SCEVConstant is restricted to ConstantInt.
/// Returns NULL if the SCEV isn't representable as a Constant.
static Constant *BuildConstantFromSCEV(const SCEV *V) {
  switch (V->getSCEVType()) {
    default:  // TODO: smax, umax.
    case scCouldNotCompute:
    case scAddRecExpr:
      break;
    case scConstant:
      return cast<SCEVConstant>(V)->getValue();
    case scUnknown:
      return dyn_cast<Constant>(cast<SCEVUnknown>(V)->getValue());
    case scSignExtend: {
      const SCEVSignExtendExpr *SS = cast<SCEVSignExtendExpr>(V);
      if (Constant *CastOp = BuildConstantFromSCEV(SS->getOperand()))
        return ConstantExpr::getSExt(CastOp, SS->getType());
      break;
    }
    case scZeroExtend: {
      const SCEVZeroExtendExpr *SZ = cast<SCEVZeroExtendExpr>(V);
      if (Constant *CastOp = BuildConstantFromSCEV(SZ->getOperand()))
        return ConstantExpr::getZExt(CastOp, SZ->getType());
      break;
    }
    case scTruncate: {
      const SCEVTruncateExpr *ST = cast<SCEVTruncateExpr>(V);
      if (Constant *CastOp = BuildConstantFromSCEV(ST->getOperand()))
        return ConstantExpr::getTrunc(CastOp, ST->getType());
      break;
    }
    case scAddExpr: {
      const SCEVAddExpr *SA = cast<SCEVAddExpr>(V);
      if (Constant *C = BuildConstantFromSCEV(SA->getOperand(0))) {
        if (PointerType *PTy = dyn_cast<PointerType>(C->getType())) {
          unsigned AS = PTy->getAddressSpace();
          Type *DestPtrTy = Type::getInt8PtrTy(C->getContext(), AS);
          C = ConstantExpr::getBitCast(C, DestPtrTy);
        }
        for (unsigned i = 1, e = SA->getNumOperands(); i != e; ++i) {
          Constant *C2 = BuildConstantFromSCEV(SA->getOperand(i));
          if (!C2) return 0;

          // First pointer!
          if (!C->getType()->isPointerTy() && C2->getType()->isPointerTy()) {
            unsigned AS = C2->getType()->getPointerAddressSpace();
            std::swap(C, C2);
            Type *DestPtrTy = Type::getInt8PtrTy(C->getContext(), AS);
            // The offsets have been converted to bytes.  We can add bytes to an
            // i8* by GEP with the byte count in the first index.
            C = ConstantExpr::getBitCast(C, DestPtrTy);
          }

          // Don't bother trying to sum two pointers. We probably can't
          // statically compute a load that results from it anyway.
          if (C2->getType()->isPointerTy())
            return 0;

          if (PointerType *PTy = dyn_cast<PointerType>(C->getType())) {
            if (PTy->getElementType()->isStructTy())
              C2 = ConstantExpr::getIntegerCast(
                  C2, Type::getInt32Ty(C->getContext()), true);
            C = ConstantExpr::getGetElementPtr(C, C2);
          } else
            C = ConstantExpr::getAdd(C, C2);
        }
        return C;
      }
      break;
    }
    case scMulExpr: {
      const SCEVMulExpr *SM = cast<SCEVMulExpr>(V);
      if (Constant *C = BuildConstantFromSCEV(SM->getOperand(0))) {
        // Don't bother with pointers at all.
        if (C->getType()->isPointerTy()) return 0;
        for (unsigned i = 1, e = SM->getNumOperands(); i != e; ++i) {
          Constant *C2 = BuildConstantFromSCEV(SM->getOperand(i));
          if (!C2 || C2->getType()->isPointerTy()) return 0;
          C = ConstantExpr::getMul(C, C2);
        }
        return C;
      }
      break;
    }
    case scUDivExpr: {
      const SCEVUDivExpr *SU = cast<SCEVUDivExpr>(V);
      if (Constant *LHS = BuildConstantFromSCEV(SU->getLHS()))
        if (Constant *RHS = BuildConstantFromSCEV(SU->getRHS()))
          if (LHS->getType() == RHS->getType())
            return ConstantExpr::getUDiv(LHS, RHS);
      break;
    }
  }
  return 0;
}

const SCEV *ScalarEvolution::computeSCEVAtScope(const SCEV *V, const Loop *L) {
  if (isa<SCEVConstant>(V)) return V;

  // If this instruction is evolved from a constant-evolving PHI, compute the
  // exit value from the loop without using SCEVs.
  if (const SCEVUnknown *SU = dyn_cast<SCEVUnknown>(V)) {
    if (Instruction *I = dyn_cast<Instruction>(SU->getValue())) {
      const Loop *LI = (*this->LI)[I->getParent()];
      if (LI && LI->getParentLoop() == L)  // Looking for loop exit value.
        if (PHINode *PN = dyn_cast<PHINode>(I))
          if (PN->getParent() == LI->getHeader()) {
            // Okay, there is no closed form solution for the PHI node.  Check
            // to see if the loop that contains it has a known backedge-taken
            // count.  If so, we may be able to force computation of the exit
            // value.
            const SCEV *BackedgeTakenCount = getBackedgeTakenCount(LI);
            if (const SCEVConstant *BTCC =
                  dyn_cast<SCEVConstant>(BackedgeTakenCount)) {
              // Okay, we know how many times the containing loop executes.  If
              // this is a constant evolving PHI node, get the final value at
              // the specified iteration number.
              Constant *RV = getConstantEvolutionLoopExitValue(PN,
                                                   BTCC->getValue()->getValue(),
                                                               LI);
              if (RV) return getSCEV(RV);
            }
          }

      // Okay, this is an expression that we cannot symbolically evaluate
      // into a SCEV.  Check to see if it's possible to symbolically evaluate
      // the arguments into constants, and if so, try to constant propagate the
      // result.  This is particularly useful for computing loop exit values.
      if (CanConstantFold(I)) {
        SmallVector<Constant *, 4> Operands;
        bool MadeImprovement = false;
        for (unsigned i = 0, e = I->getNumOperands(); i != e; ++i) {
          Value *Op = I->getOperand(i);
          if (Constant *C = dyn_cast<Constant>(Op)) {
            Operands.push_back(C);
            continue;
          }

          // If any of the operands is non-constant and if they are
          // non-integer and non-pointer, don't even try to analyze them
          // with scev techniques.
          if (!isSCEVable(Op->getType()))
            return V;

          const SCEV *OrigV = getSCEV(Op);
          const SCEV *OpV = getSCEVAtScope(OrigV, L);
          MadeImprovement |= OrigV != OpV;

          Constant *C = BuildConstantFromSCEV(OpV);
          if (!C) return V;
          if (C->getType() != Op->getType())
            C = ConstantExpr::getCast(CastInst::getCastOpcode(C, false,
                                                              Op->getType(),
                                                              false),
                                      C, Op->getType());
          Operands.push_back(C);
        }

        // Check to see if getSCEVAtScope actually made an improvement.
        if (MadeImprovement) {
          Constant *C = 0;
          if (const CmpInst *CI = dyn_cast<CmpInst>(I))
            C = ConstantFoldCompareInstOperands(CI->getPredicate(),
                                                Operands[0], Operands[1], TD,
                                                TLI);
          else if (const LoadInst *LI = dyn_cast<LoadInst>(I)) {
            if (!LI->isVolatile())
              C = ConstantFoldLoadFromConstPtr(Operands[0], TD);
          } else
            C = ConstantFoldInstOperands(I->getOpcode(), I->getType(),
                                         Operands, TD, TLI);
          if (!C) return V;
          return getSCEV(C);
        }
      }
    }

    // This is some other type of SCEVUnknown, just return it.
    return V;
  }

  if (const SCEVCommutativeExpr *Comm = dyn_cast<SCEVCommutativeExpr>(V)) {
    // Avoid performing the look-up in the common case where the specified
    // expression has no loop-variant portions.
    for (unsigned i = 0, e = Comm->getNumOperands(); i != e; ++i) {
      const SCEV *OpAtScope = getSCEVAtScope(Comm->getOperand(i), L);
      if (OpAtScope != Comm->getOperand(i)) {
        // Okay, at least one of these operands is loop variant but might be
        // foldable.  Build a new instance of the folded commutative expression.
        SmallVector<const SCEV *, 8> NewOps(Comm->op_begin(),
                                            Comm->op_begin()+i);
        NewOps.push_back(OpAtScope);

        for (++i; i != e; ++i) {
          OpAtScope = getSCEVAtScope(Comm->getOperand(i), L);
          NewOps.push_back(OpAtScope);
        }
        if (isa<SCEVAddExpr>(Comm))
          return getAddExpr(NewOps);
        if (isa<SCEVMulExpr>(Comm))
          return getMulExpr(NewOps);
        if (isa<SCEVSMaxExpr>(Comm))
          return getSMaxExpr(NewOps);
        if (isa<SCEVUMaxExpr>(Comm))
          return getUMaxExpr(NewOps);
        llvm_unreachable("Unknown commutative SCEV type!");
      }
    }
    // If we got here, all operands are loop invariant.
    return Comm;
  }

  if (const SCEVUDivExpr *Div = dyn_cast<SCEVUDivExpr>(V)) {
    const SCEV *LHS = getSCEVAtScope(Div->getLHS(), L);
    const SCEV *RHS = getSCEVAtScope(Div->getRHS(), L);
    if (LHS == Div->getLHS() && RHS == Div->getRHS())
      return Div;   // must be loop invariant
    return getUDivExpr(LHS, RHS);
  }

  // If this is a loop recurrence for a loop that does not contain L, then we
  // are dealing with the final value computed by the loop.
  if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(V)) {
    // First, attempt to evaluate each operand.
    // Avoid performing the look-up in the common case where the specified
    // expression has no loop-variant portions.
    for (unsigned i = 0, e = AddRec->getNumOperands(); i != e; ++i) {
      const SCEV *OpAtScope = getSCEVAtScope(AddRec->getOperand(i), L);
      if (OpAtScope == AddRec->getOperand(i))
        continue;

      // Okay, at least one of these operands is loop variant but might be
      // foldable.  Build a new instance of the folded commutative expression.
      SmallVector<const SCEV *, 8> NewOps(AddRec->op_begin(),
                                          AddRec->op_begin()+i);
      NewOps.push_back(OpAtScope);
      for (++i; i != e; ++i)
        NewOps.push_back(getSCEVAtScope(AddRec->getOperand(i), L));

      const SCEV *FoldedRec =
        getAddRecExpr(NewOps, AddRec->getLoop(),
                      AddRec->getNoWrapFlags(SCEV::FlagNW));
      AddRec = dyn_cast<SCEVAddRecExpr>(FoldedRec);
      // The addrec may be folded to a nonrecurrence, for example, if the
      // induction variable is multiplied by zero after constant folding. Go
      // ahead and return the folded value.
      if (!AddRec)
        return FoldedRec;
      break;
    }

    // If the scope is outside the addrec's loop, evaluate it by using the
    // loop exit value of the addrec.
    if (!AddRec->getLoop()->contains(L)) {
      // To evaluate this recurrence, we need to know how many times the AddRec
      // loop iterates.  Compute this now.
      const SCEV *BackedgeTakenCount = getBackedgeTakenCount(AddRec->getLoop());
      if (BackedgeTakenCount == getCouldNotCompute()) return AddRec;

      // Then, evaluate the AddRec.
      return AddRec->evaluateAtIteration(BackedgeTakenCount, *this);
    }

    return AddRec;
  }

  if (const SCEVZeroExtendExpr *Cast = dyn_cast<SCEVZeroExtendExpr>(V)) {
    const SCEV *Op = getSCEVAtScope(Cast->getOperand(), L);
    if (Op == Cast->getOperand())
      return Cast;  // must be loop invariant
    return getZeroExtendExpr(Op, Cast->getType());
  }

  if (const SCEVSignExtendExpr *Cast = dyn_cast<SCEVSignExtendExpr>(V)) {
    const SCEV *Op = getSCEVAtScope(Cast->getOperand(), L);
    if (Op == Cast->getOperand())
      return Cast;  // must be loop invariant
    return getSignExtendExpr(Op, Cast->getType());
  }

  if (const SCEVTruncateExpr *Cast = dyn_cast<SCEVTruncateExpr>(V)) {
    const SCEV *Op = getSCEVAtScope(Cast->getOperand(), L);
    if (Op == Cast->getOperand())
      return Cast;  // must be loop invariant
    return getTruncateExpr(Op, Cast->getType());
  }

  llvm_unreachable("Unknown SCEV type!");
}

/// getSCEVAtScope - This is a convenience function which does
/// getSCEVAtScope(getSCEV(V), L).
const SCEV *ScalarEvolution::getSCEVAtScope(Value *V, const Loop *L) {
  return getSCEVAtScope(getSCEV(V), L);
}

/// SolveLinEquationWithOverflow - Finds the minimum unsigned root of the
/// following equation:
///
///     A * X = B (mod N)
///
/// where N = 2^BW and BW is the common bit width of A and B. The signedness of
/// A and B isn't important.
///
/// If the equation does not have a solution, SCEVCouldNotCompute is returned.
static const SCEV *SolveLinEquationWithOverflow(const APInt &A, const APInt &B,
                                               ScalarEvolution &SE) {
  uint32_t BW = A.getBitWidth();
  assert(BW == B.getBitWidth() && "Bit widths must be the same.");
  assert(A != 0 && "A must be non-zero.");

  // 1. D = gcd(A, N)
  //
  // The gcd of A and N may have only one prime factor: 2. The number of
  // trailing zeros in A is its multiplicity
  uint32_t Mult2 = A.countTrailingZeros();
  // D = 2^Mult2

  // 2. Check if B is divisible by D.
  //
  // B is divisible by D if and only if the multiplicity of prime factor 2 for B
  // is not less than multiplicity of this prime factor for D.
  if (B.countTrailingZeros() < Mult2)
    return SE.getCouldNotCompute();

  // 3. Compute I: the multiplicative inverse of (A / D) in arithmetic
  // modulo (N / D).
  //
  // (N / D) may need BW+1 bits in its representation.  Hence, we'll use this
  // bit width during computations.
  APInt AD = A.lshr(Mult2).zext(BW + 1);  // AD = A / D
  APInt Mod(BW + 1, 0);
  Mod.setBit(BW - Mult2);  // Mod = N / D
  APInt I = AD.multiplicativeInverse(Mod);

  // 4. Compute the minimum unsigned root of the equation:
  // I * (B / D) mod (N / D)
  APInt Result = (I * B.lshr(Mult2).zext(BW + 1)).urem(Mod);

  // The result is guaranteed to be less than 2^BW so we may truncate it to BW
  // bits.
  return SE.getConstant(Result.trunc(BW));
}

/// SolveQuadraticEquation - Find the roots of the quadratic equation for the
/// given quadratic chrec {L,+,M,+,N}.  This returns either the two roots (which
/// might be the same) or two SCEVCouldNotCompute objects.
///
static std::pair<const SCEV *,const SCEV *>
SolveQuadraticEquation(const SCEVAddRecExpr *AddRec, ScalarEvolution &SE) {
  assert(AddRec->getNumOperands() == 3 && "This is not a quadratic chrec!");
  const SCEVConstant *LC = dyn_cast<SCEVConstant>(AddRec->getOperand(0));
  const SCEVConstant *MC = dyn_cast<SCEVConstant>(AddRec->getOperand(1));
  const SCEVConstant *NC = dyn_cast<SCEVConstant>(AddRec->getOperand(2));

  // We currently can only solve this if the coefficients are constants.
  if (!LC || !MC || !NC) {
    const SCEV *CNC = SE.getCouldNotCompute();
    return std::make_pair(CNC, CNC);
  }

  uint32_t BitWidth = LC->getValue()->getValue().getBitWidth();
  const APInt &L = LC->getValue()->getValue();
  const APInt &M = MC->getValue()->getValue();
  const APInt &N = NC->getValue()->getValue();
  APInt Two(BitWidth, 2);
  APInt Four(BitWidth, 4);

  {
    using namespace APIntOps;
    const APInt& C = L;
    // Convert from chrec coefficients to polynomial coefficients AX^2+BX+C
    // The B coefficient is M-N/2
    APInt B(M);
    B -= sdiv(N,Two);

    // The A coefficient is N/2
    APInt A(N.sdiv(Two));

    // Compute the B^2-4ac term.
    APInt SqrtTerm(B);
    SqrtTerm *= B;
    SqrtTerm -= Four * (A * C);

    if (SqrtTerm.isNegative()) {
      // The loop is provably infinite.
      const SCEV *CNC = SE.getCouldNotCompute();
      return std::make_pair(CNC, CNC);
    }

    // Compute sqrt(B^2-4ac). This is guaranteed to be the nearest
    // integer value or else APInt::sqrt() will assert.
    APInt SqrtVal(SqrtTerm.sqrt());

    // Compute the two solutions for the quadratic formula.
    // The divisions must be performed as signed divisions.
    APInt NegB(-B);
    APInt TwoA(A << 1);
    if (TwoA.isMinValue()) {
      const SCEV *CNC = SE.getCouldNotCompute();
      return std::make_pair(CNC, CNC);
    }

    LLVMContext &Context = SE.getContext();

    ConstantInt *Solution1 =
      ConstantInt::get(Context, (NegB + SqrtVal).sdiv(TwoA));
    ConstantInt *Solution2 =
      ConstantInt::get(Context, (NegB - SqrtVal).sdiv(TwoA));

    return std::make_pair(SE.getConstant(Solution1),
                          SE.getConstant(Solution2));
  } // end APIntOps namespace
}

/// HowFarToZero - Return the number of times a backedge comparing the specified
/// value to zero will execute.  If not computable, return CouldNotCompute.
///
/// This is only used for loops with a "x != y" exit test. The exit condition is
/// now expressed as a single expression, V = x-y. So the exit test is
/// effectively V != 0.  We know and take advantage of the fact that this
/// expression only being used in a comparison by zero context.
ScalarEvolution::ExitLimit
ScalarEvolution::HowFarToZero(const SCEV *V, const Loop *L, bool IsSubExpr) {
  // If the value is a constant
  if (const SCEVConstant *C = dyn_cast<SCEVConstant>(V)) {
    // If the value is already zero, the branch will execute zero times.
    if (C->getValue()->isZero()) return C;
    return getCouldNotCompute();  // Otherwise it will loop infinitely.
  }

  const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(V);
  if (!AddRec || AddRec->getLoop() != L)
    return getCouldNotCompute();

  // If this is a quadratic (3-term) AddRec {L,+,M,+,N}, find the roots of
  // the quadratic equation to solve it.
  if (AddRec->isQuadratic() && AddRec->getType()->isIntegerTy()) {
    std::pair<const SCEV *,const SCEV *> Roots =
      SolveQuadraticEquation(AddRec, *this);
    const SCEVConstant *R1 = dyn_cast<SCEVConstant>(Roots.first);
    const SCEVConstant *R2 = dyn_cast<SCEVConstant>(Roots.second);
    if (R1 && R2) {
#if 0
      dbgs() << "HFTZ: " << *V << " - sol#1: " << *R1
             << "  sol#2: " << *R2 << "\n";
#endif
      // Pick the smallest positive root value.
      if (ConstantInt *CB =
          dyn_cast<ConstantInt>(ConstantExpr::getICmp(CmpInst::ICMP_ULT,
                                                      R1->getValue(),
                                                      R2->getValue()))) {
        if (CB->getZExtValue() == false)
          std::swap(R1, R2);   // R1 is the minimum root now.

        // We can only use this value if the chrec ends up with an exact zero
        // value at this index.  When solving for "X*X != 5", for example, we
        // should not accept a root of 2.
        const SCEV *Val = AddRec->evaluateAtIteration(R1, *this);
        if (Val->isZero())
          return R1;  // We found a quadratic root!
      }
    }
    return getCouldNotCompute();
  }

  // Otherwise we can only handle this if it is affine.
  if (!AddRec->isAffine())
    return getCouldNotCompute();

  // If this is an affine expression, the execution count of this branch is
  // the minimum unsigned root of the following equation:
  //
  //     Start + Step*N = 0 (mod 2^BW)
  //
  // equivalent to:
  //
  //             Step*N = -Start (mod 2^BW)
  //
  // where BW is the common bit width of Start and Step.

  // Get the initial value for the loop.
  const SCEV *Start = getSCEVAtScope(AddRec->getStart(), L->getParentLoop());
  const SCEV *Step = getSCEVAtScope(AddRec->getOperand(1), L->getParentLoop());

  // For now we handle only constant steps.
  //
  // TODO: Handle a nonconstant Step given AddRec<NUW>. If the
  // AddRec is NUW, then (in an unsigned sense) it cannot be counting up to wrap
  // to 0, it must be counting down to equal 0. Consequently, N = Start / -Step.
  // We have not yet seen any such cases.
  const SCEVConstant *StepC = dyn_cast<SCEVConstant>(Step);
  if (StepC == 0 || StepC->getValue()->equalsInt(0))
    return getCouldNotCompute();

  // For positive steps (counting up until unsigned overflow):
  //   N = -Start/Step (as unsigned)
  // For negative steps (counting down to zero):
  //   N = Start/-Step
  // First compute the unsigned distance from zero in the direction of Step.
  bool CountDown = StepC->getValue()->getValue().isNegative();
  const SCEV *Distance = CountDown ? Start : getNegativeSCEV(Start);

  // Handle unitary steps, which cannot wraparound.
  // 1*N = -Start; -1*N = Start (mod 2^BW), so:
  //   N = Distance (as unsigned)
  if (StepC->getValue()->equalsInt(1) || StepC->getValue()->isAllOnesValue()) {
    ConstantRange CR = getUnsignedRange(Start);
    const SCEV *MaxBECount;
    if (!CountDown && CR.getUnsignedMin().isMinValue())
      // When counting up, the worst starting value is 1, not 0.
      MaxBECount = CR.getUnsignedMax().isMinValue()
        ? getConstant(APInt::getMinValue(CR.getBitWidth()))
        : getConstant(APInt::getMaxValue(CR.getBitWidth()));
    else
      MaxBECount = getConstant(CountDown ? CR.getUnsignedMax()
                                         : -CR.getUnsignedMin());
    return ExitLimit(Distance, MaxBECount);
  }

  // If the recurrence is known not to wraparound, unsigned divide computes the
  // back edge count. (Ideally we would have an "isexact" bit for udiv). We know
  // that the value will either become zero (and thus the loop terminates), that
  // the loop will terminate through some other exit condition first, or that
  // the loop has undefined behavior.  This means we can't "miss" the exit
  // value, even with nonunit stride.
  //
  // This is only valid for expressions that directly compute the loop exit. It
  // is invalid for subexpressions in which the loop may exit through this
  // branch even if this subexpression is false. In that case, the trip count
  // computed by this udiv could be smaller than the number of well-defined
  // iterations.
  if (!IsSubExpr && AddRec->getNoWrapFlags(SCEV::FlagNW))
    return getUDivExpr(Distance, CountDown ? getNegativeSCEV(Step) : Step);

  // Then, try to solve the above equation provided that Start is constant.
  if (const SCEVConstant *StartC = dyn_cast<SCEVConstant>(Start))
    return SolveLinEquationWithOverflow(StepC->getValue()->getValue(),
                                        -StartC->getValue()->getValue(),
                                        *this);
  return getCouldNotCompute();
}

/// HowFarToNonZero - Return the number of times a backedge checking the
/// specified value for nonzero will execute.  If not computable, return
/// CouldNotCompute
ScalarEvolution::ExitLimit
ScalarEvolution::HowFarToNonZero(const SCEV *V, const Loop *L) {
  // Loops that look like: while (X == 0) are very strange indeed.  We don't
  // handle them yet except for the trivial case.  This could be expanded in the
  // future as needed.

  // If the value is a constant, check to see if it is known to be non-zero
  // already.  If so, the backedge will execute zero times.
  if (const SCEVConstant *C = dyn_cast<SCEVConstant>(V)) {
    if (!C->getValue()->isNullValue())
      return getConstant(C->getType(), 0);
    return getCouldNotCompute();  // Otherwise it will loop infinitely.
  }

  // We could implement others, but I really doubt anyone writes loops like
  // this, and if they did, they would already be constant folded.
  return getCouldNotCompute();
}

/// getPredecessorWithUniqueSuccessorForBB - Return a predecessor of BB
/// (which may not be an immediate predecessor) which has exactly one
/// successor from which BB is reachable, or null if no such block is
/// found.
///
std::pair<BasicBlock *, BasicBlock *>
ScalarEvolution::getPredecessorWithUniqueSuccessorForBB(BasicBlock *BB) {
  // If the block has a unique predecessor, then there is no path from the
  // predecessor to the block that does not go through the direct edge
  // from the predecessor to the block.
  if (BasicBlock *Pred = BB->getSinglePredecessor())
    return std::make_pair(Pred, BB);

  // A loop's header is defined to be a block that dominates the loop.
  // If the header has a unique predecessor outside the loop, it must be
  // a block that has exactly one successor that can reach the loop.
  if (Loop *L = LI->getLoopFor(BB))
    return std::make_pair(L->getLoopPredecessor(), L->getHeader());

  return std::pair<BasicBlock *, BasicBlock *>();
}

/// HasSameValue - SCEV structural equivalence is usually sufficient for
/// testing whether two expressions are equal, however for the purposes of
/// looking for a condition guarding a loop, it can be useful to be a little
/// more general, since a front-end may have replicated the controlling
/// expression.
///
static bool HasSameValue(const SCEV *A, const SCEV *B) {
  // Quick check to see if they are the same SCEV.
  if (A == B) return true;

  // Otherwise, if they're both SCEVUnknown, it's possible that they hold
  // two different instructions with the same value. Check for this case.
  if (const SCEVUnknown *AU = dyn_cast<SCEVUnknown>(A))
    if (const SCEVUnknown *BU = dyn_cast<SCEVUnknown>(B))
      if (const Instruction *AI = dyn_cast<Instruction>(AU->getValue()))
        if (const Instruction *BI = dyn_cast<Instruction>(BU->getValue()))
          if (AI->isIdenticalTo(BI) && !AI->mayReadFromMemory())
            return true;

  // Otherwise assume they may have a different value.
  return false;
}

/// SimplifyICmpOperands - Simplify LHS and RHS in a comparison with
/// predicate Pred. Return true iff any changes were made.
///
bool ScalarEvolution::SimplifyICmpOperands(ICmpInst::Predicate &Pred,
                                           const SCEV *&LHS, const SCEV *&RHS,
                                           unsigned Depth) {
  bool Changed = false;

  // If we hit the max recursion limit bail out.
  if (Depth >= 3)
    return false;

  // Canonicalize a constant to the right side.
  if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(LHS)) {
    // Check for both operands constant.
    if (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS)) {
      if (ConstantExpr::getICmp(Pred,
                                LHSC->getValue(),
                                RHSC->getValue())->isNullValue())
        goto trivially_false;
      else
        goto trivially_true;
    }
    // Otherwise swap the operands to put the constant on the right.
    std::swap(LHS, RHS);
    Pred = ICmpInst::getSwappedPredicate(Pred);
    Changed = true;
  }

  // If we're comparing an addrec with a value which is loop-invariant in the
  // addrec's loop, put the addrec on the left. Also make a dominance check,
  // as both operands could be addrecs loop-invariant in each other's loop.
  if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(RHS)) {
    const Loop *L = AR->getLoop();
    if (isLoopInvariant(LHS, L) && properlyDominates(LHS, L->getHeader())) {
      std::swap(LHS, RHS);
      Pred = ICmpInst::getSwappedPredicate(Pred);
      Changed = true;
    }
  }

  // If there's a constant operand, canonicalize comparisons with boundary
  // cases, and canonicalize *-or-equal comparisons to regular comparisons.
  if (const SCEVConstant *RC = dyn_cast<SCEVConstant>(RHS)) {
    const APInt &RA = RC->getValue()->getValue();
    switch (Pred) {
    default: llvm_unreachable("Unexpected ICmpInst::Predicate value!");
    case ICmpInst::ICMP_EQ:
    case ICmpInst::ICMP_NE:
      // Fold ((-1) * %a) + %b == 0 (equivalent to %b-%a == 0) into %a == %b.
      if (!RA)
        if (const SCEVAddExpr *AE = dyn_cast<SCEVAddExpr>(LHS))
          if (const SCEVMulExpr *ME = dyn_cast<SCEVMulExpr>(AE->getOperand(0)))
            if (AE->getNumOperands() == 2 && ME->getNumOperands() == 2 &&
                ME->getOperand(0)->isAllOnesValue()) {
              RHS = AE->getOperand(1);
              LHS = ME->getOperand(1);
              Changed = true;
            }
      break;
    case ICmpInst::ICMP_UGE:
      if ((RA - 1).isMinValue()) {
        Pred = ICmpInst::ICMP_NE;
        RHS = getConstant(RA - 1);
        Changed = true;
        break;
      }
      if (RA.isMaxValue()) {
        Pred = ICmpInst::ICMP_EQ;
        Changed = true;
        break;
      }
      if (RA.isMinValue()) goto trivially_true;

      Pred = ICmpInst::ICMP_UGT;
      RHS = getConstant(RA - 1);
      Changed = true;
      break;
    case ICmpInst::ICMP_ULE:
      if ((RA + 1).isMaxValue()) {
        Pred = ICmpInst::ICMP_NE;
        RHS = getConstant(RA + 1);
        Changed = true;
        break;
      }
      if (RA.isMinValue()) {
        Pred = ICmpInst::ICMP_EQ;
        Changed = true;
        break;
      }
      if (RA.isMaxValue()) goto trivially_true;

      Pred = ICmpInst::ICMP_ULT;
      RHS = getConstant(RA + 1);
      Changed = true;
      break;
    case ICmpInst::ICMP_SGE:
      if ((RA - 1).isMinSignedValue()) {
        Pred = ICmpInst::ICMP_NE;
        RHS = getConstant(RA - 1);
        Changed = true;
        break;
      }
      if (RA.isMaxSignedValue()) {
        Pred = ICmpInst::ICMP_EQ;
        Changed = true;
        break;
      }
      if (RA.isMinSignedValue()) goto trivially_true;

      Pred = ICmpInst::ICMP_SGT;
      RHS = getConstant(RA - 1);
      Changed = true;
      break;
    case ICmpInst::ICMP_SLE:
      if ((RA + 1).isMaxSignedValue()) {
        Pred = ICmpInst::ICMP_NE;
        RHS = getConstant(RA + 1);
        Changed = true;
        break;
      }
      if (RA.isMinSignedValue()) {
        Pred = ICmpInst::ICMP_EQ;
        Changed = true;
        break;
      }
      if (RA.isMaxSignedValue()) goto trivially_true;

      Pred = ICmpInst::ICMP_SLT;
      RHS = getConstant(RA + 1);
      Changed = true;
      break;
    case ICmpInst::ICMP_UGT:
      if (RA.isMinValue()) {
        Pred = ICmpInst::ICMP_NE;
        Changed = true;
        break;
      }
      if ((RA + 1).isMaxValue()) {
        Pred = ICmpInst::ICMP_EQ;
        RHS = getConstant(RA + 1);
        Changed = true;
        break;
      }
      if (RA.isMaxValue()) goto trivially_false;
      break;
    case ICmpInst::ICMP_ULT:
      if (RA.isMaxValue()) {
        Pred = ICmpInst::ICMP_NE;
        Changed = true;
        break;
      }
      if ((RA - 1).isMinValue()) {
        Pred = ICmpInst::ICMP_EQ;
        RHS = getConstant(RA - 1);
        Changed = true;
        break;
      }
      if (RA.isMinValue()) goto trivially_false;
      break;
    case ICmpInst::ICMP_SGT:
      if (RA.isMinSignedValue()) {
        Pred = ICmpInst::ICMP_NE;
        Changed = true;
        break;
      }
      if ((RA + 1).isMaxSignedValue()) {
        Pred = ICmpInst::ICMP_EQ;
        RHS = getConstant(RA + 1);
        Changed = true;
        break;
      }
      if (RA.isMaxSignedValue()) goto trivially_false;
      break;
    case ICmpInst::ICMP_SLT:
      if (RA.isMaxSignedValue()) {
        Pred = ICmpInst::ICMP_NE;
        Changed = true;
        break;
      }
      if ((RA - 1).isMinSignedValue()) {
       Pred = ICmpInst::ICMP_EQ;
       RHS = getConstant(RA - 1);
        Changed = true;
       break;
      }
      if (RA.isMinSignedValue()) goto trivially_false;
      break;
    }
  }

  // Check for obvious equality.
  if (HasSameValue(LHS, RHS)) {
    if (ICmpInst::isTrueWhenEqual(Pred))
      goto trivially_true;
    if (ICmpInst::isFalseWhenEqual(Pred))
      goto trivially_false;
  }

  // If possible, canonicalize GE/LE comparisons to GT/LT comparisons, by
  // adding or subtracting 1 from one of the operands.
  switch (Pred) {
  case ICmpInst::ICMP_SLE:
    if (!getSignedRange(RHS).getSignedMax().isMaxSignedValue()) {
      RHS = getAddExpr(getConstant(RHS->getType(), 1, true), RHS,
                       SCEV::FlagNSW);
      Pred = ICmpInst::ICMP_SLT;
      Changed = true;
    } else if (!getSignedRange(LHS).getSignedMin().isMinSignedValue()) {
      LHS = getAddExpr(getConstant(RHS->getType(), (uint64_t)-1, true), LHS,
                       SCEV::FlagNSW);
      Pred = ICmpInst::ICMP_SLT;
      Changed = true;
    }
    break;
  case ICmpInst::ICMP_SGE:
    if (!getSignedRange(RHS).getSignedMin().isMinSignedValue()) {
      RHS = getAddExpr(getConstant(RHS->getType(), (uint64_t)-1, true), RHS,
                       SCEV::FlagNSW);
      Pred = ICmpInst::ICMP_SGT;
      Changed = true;
    } else if (!getSignedRange(LHS).getSignedMax().isMaxSignedValue()) {
      LHS = getAddExpr(getConstant(RHS->getType(), 1, true), LHS,
                       SCEV::FlagNSW);
      Pred = ICmpInst::ICMP_SGT;
      Changed = true;
    }
    break;
  case ICmpInst::ICMP_ULE:
    if (!getUnsignedRange(RHS).getUnsignedMax().isMaxValue()) {
      RHS = getAddExpr(getConstant(RHS->getType(), 1, true), RHS,
                       SCEV::FlagNUW);
      Pred = ICmpInst::ICMP_ULT;
      Changed = true;
    } else if (!getUnsignedRange(LHS).getUnsignedMin().isMinValue()) {
      LHS = getAddExpr(getConstant(RHS->getType(), (uint64_t)-1, true), LHS,
                       SCEV::FlagNUW);
      Pred = ICmpInst::ICMP_ULT;
      Changed = true;
    }
    break;
  case ICmpInst::ICMP_UGE:
    if (!getUnsignedRange(RHS).getUnsignedMin().isMinValue()) {
      RHS = getAddExpr(getConstant(RHS->getType(), (uint64_t)-1, true), RHS,
                       SCEV::FlagNUW);
      Pred = ICmpInst::ICMP_UGT;
      Changed = true;
    } else if (!getUnsignedRange(LHS).getUnsignedMax().isMaxValue()) {
      LHS = getAddExpr(getConstant(RHS->getType(), 1, true), LHS,
                       SCEV::FlagNUW);
      Pred = ICmpInst::ICMP_UGT;
      Changed = true;
    }
    break;
  default:
    break;
  }

  // TODO: More simplifications are possible here.

  // Recursively simplify until we either hit a recursion limit or nothing
  // changes.
  if (Changed)
    return SimplifyICmpOperands(Pred, LHS, RHS, Depth+1);

  return Changed;

trivially_true:
  // Return 0 == 0.
  LHS = RHS = getConstant(ConstantInt::getFalse(getContext()));
  Pred = ICmpInst::ICMP_EQ;
  return true;

trivially_false:
  // Return 0 != 0.
  LHS = RHS = getConstant(ConstantInt::getFalse(getContext()));
  Pred = ICmpInst::ICMP_NE;
  return true;
}

bool ScalarEvolution::isKnownNegative(const SCEV *S) {
  return getSignedRange(S).getSignedMax().isNegative();
}

bool ScalarEvolution::isKnownPositive(const SCEV *S) {
  return getSignedRange(S).getSignedMin().isStrictlyPositive();
}

bool ScalarEvolution::isKnownNonNegative(const SCEV *S) {
  return !getSignedRange(S).getSignedMin().isNegative();
}

bool ScalarEvolution::isKnownNonPositive(const SCEV *S) {
  return !getSignedRange(S).getSignedMax().isStrictlyPositive();
}

bool ScalarEvolution::isKnownNonZero(const SCEV *S) {
  return isKnownNegative(S) || isKnownPositive(S);
}

bool ScalarEvolution::isKnownPredicate(ICmpInst::Predicate Pred,
                                       const SCEV *LHS, const SCEV *RHS) {
  // Canonicalize the inputs first.
  (void)SimplifyICmpOperands(Pred, LHS, RHS);

  // If LHS or RHS is an addrec, check to see if the condition is true in
  // every iteration of the loop.
  if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(LHS))
    if (isLoopEntryGuardedByCond(
          AR->getLoop(), Pred, AR->getStart(), RHS) &&
        isLoopBackedgeGuardedByCond(
          AR->getLoop(), Pred, AR->getPostIncExpr(*this), RHS))
      return true;
  if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(RHS))
    if (isLoopEntryGuardedByCond(
          AR->getLoop(), Pred, LHS, AR->getStart()) &&
        isLoopBackedgeGuardedByCond(
          AR->getLoop(), Pred, LHS, AR->getPostIncExpr(*this)))
      return true;

  // Otherwise see what can be done with known constant ranges.
  return isKnownPredicateWithRanges(Pred, LHS, RHS);
}

bool
ScalarEvolution::isKnownPredicateWithRanges(ICmpInst::Predicate Pred,
                                            const SCEV *LHS, const SCEV *RHS) {
  if (HasSameValue(LHS, RHS))
    return ICmpInst::isTrueWhenEqual(Pred);

  // This code is split out from isKnownPredicate because it is called from
  // within isLoopEntryGuardedByCond.
  switch (Pred) {
  default:
    llvm_unreachable("Unexpected ICmpInst::Predicate value!");
  case ICmpInst::ICMP_SGT:
    Pred = ICmpInst::ICMP_SLT;
    std::swap(LHS, RHS);
  case ICmpInst::ICMP_SLT: {
    ConstantRange LHSRange = getSignedRange(LHS);
    ConstantRange RHSRange = getSignedRange(RHS);
    if (LHSRange.getSignedMax().slt(RHSRange.getSignedMin()))
      return true;
    if (LHSRange.getSignedMin().sge(RHSRange.getSignedMax()))
      return false;
    break;
  }
  case ICmpInst::ICMP_SGE:
    Pred = ICmpInst::ICMP_SLE;
    std::swap(LHS, RHS);
  case ICmpInst::ICMP_SLE: {
    ConstantRange LHSRange = getSignedRange(LHS);
    ConstantRange RHSRange = getSignedRange(RHS);
    if (LHSRange.getSignedMax().sle(RHSRange.getSignedMin()))
      return true;
    if (LHSRange.getSignedMin().sgt(RHSRange.getSignedMax()))
      return false;
    break;
  }
  case ICmpInst::ICMP_UGT:
    Pred = ICmpInst::ICMP_ULT;
    std::swap(LHS, RHS);
  case ICmpInst::ICMP_ULT: {
    ConstantRange LHSRange = getUnsignedRange(LHS);
    ConstantRange RHSRange = getUnsignedRange(RHS);
    if (LHSRange.getUnsignedMax().ult(RHSRange.getUnsignedMin()))
      return true;
    if (LHSRange.getUnsignedMin().uge(RHSRange.getUnsignedMax()))
      return false;
    break;
  }
  case ICmpInst::ICMP_UGE:
    Pred = ICmpInst::ICMP_ULE;
    std::swap(LHS, RHS);
  case ICmpInst::ICMP_ULE: {
    ConstantRange LHSRange = getUnsignedRange(LHS);
    ConstantRange RHSRange = getUnsignedRange(RHS);
    if (LHSRange.getUnsignedMax().ule(RHSRange.getUnsignedMin()))
      return true;
    if (LHSRange.getUnsignedMin().ugt(RHSRange.getUnsignedMax()))
      return false;
    break;
  }
  case ICmpInst::ICMP_NE: {
    if (getUnsignedRange(LHS).intersectWith(getUnsignedRange(RHS)).isEmptySet())
      return true;
    if (getSignedRange(LHS).intersectWith(getSignedRange(RHS)).isEmptySet())
      return true;

    const SCEV *Diff = getMinusSCEV(LHS, RHS);
    if (isKnownNonZero(Diff))
      return true;
    break;
  }
  case ICmpInst::ICMP_EQ:
    // The check at the top of the function catches the case where
    // the values are known to be equal.
    break;
  }
  return false;
}

/// isLoopBackedgeGuardedByCond - Test whether the backedge of the loop is
/// protected by a conditional between LHS and RHS.  This is used to
/// to eliminate casts.
bool
ScalarEvolution::isLoopBackedgeGuardedByCond(const Loop *L,
                                             ICmpInst::Predicate Pred,
                                             const SCEV *LHS, const SCEV *RHS) {
  // Interpret a null as meaning no loop, where there is obviously no guard
  // (interprocedural conditions notwithstanding).
  if (!L) return true;

  BasicBlock *Latch = L->getLoopLatch();
  if (!Latch)
    return false;

  BranchInst *LoopContinuePredicate =
    dyn_cast<BranchInst>(Latch->getTerminator());
  if (!LoopContinuePredicate ||
      LoopContinuePredicate->isUnconditional())
    return false;

  return isImpliedCond(Pred, LHS, RHS,
                       LoopContinuePredicate->getCondition(),
                       LoopContinuePredicate->getSuccessor(0) != L->getHeader());
}

/// isLoopEntryGuardedByCond - Test whether entry to the loop is protected
/// by a conditional between LHS and RHS.  This is used to help avoid max
/// expressions in loop trip counts, and to eliminate casts.
bool
ScalarEvolution::isLoopEntryGuardedByCond(const Loop *L,
                                          ICmpInst::Predicate Pred,
                                          const SCEV *LHS, const SCEV *RHS) {
  // Interpret a null as meaning no loop, where there is obviously no guard
  // (interprocedural conditions notwithstanding).
  if (!L) return false;

  // Starting at the loop predecessor, climb up the predecessor chain, as long
  // as there are predecessors that can be found that have unique successors
  // leading to the original header.
  for (std::pair<BasicBlock *, BasicBlock *>
         Pair(L->getLoopPredecessor(), L->getHeader());
       Pair.first;
       Pair = getPredecessorWithUniqueSuccessorForBB(Pair.first)) {

    BranchInst *LoopEntryPredicate =
      dyn_cast<BranchInst>(Pair.first->getTerminator());
    if (!LoopEntryPredicate ||
        LoopEntryPredicate->isUnconditional())
      continue;

    if (isImpliedCond(Pred, LHS, RHS,
                      LoopEntryPredicate->getCondition(),
                      LoopEntryPredicate->getSuccessor(0) != Pair.second))
      return true;
  }

  return false;
}

/// RAII wrapper to prevent recursive application of isImpliedCond.
/// ScalarEvolution's PendingLoopPredicates set must be empty unless we are
/// currently evaluating isImpliedCond.
struct MarkPendingLoopPredicate {
  Value *Cond;
  DenseSet<Value*> &LoopPreds;
  bool Pending;

  MarkPendingLoopPredicate(Value *C, DenseSet<Value*> &LP)
    : Cond(C), LoopPreds(LP) {
    Pending = !LoopPreds.insert(Cond).second;
  }
  ~MarkPendingLoopPredicate() {
    if (!Pending)
      LoopPreds.erase(Cond);
  }
};

/// isImpliedCond - Test whether the condition described by Pred, LHS,
/// and RHS is true whenever the given Cond value evaluates to true.
bool ScalarEvolution::isImpliedCond(ICmpInst::Predicate Pred,
                                    const SCEV *LHS, const SCEV *RHS,
                                    Value *FoundCondValue,
                                    bool Inverse) {
  MarkPendingLoopPredicate Mark(FoundCondValue, PendingLoopPredicates);
  if (Mark.Pending)
    return false;

  // Recursively handle And and Or conditions.
  if (BinaryOperator *BO = dyn_cast<BinaryOperator>(FoundCondValue)) {
    if (BO->getOpcode() == Instruction::And) {
      if (!Inverse)
        return isImpliedCond(Pred, LHS, RHS, BO->getOperand(0), Inverse) ||
               isImpliedCond(Pred, LHS, RHS, BO->getOperand(1), Inverse);
    } else if (BO->getOpcode() == Instruction::Or) {
      if (Inverse)
        return isImpliedCond(Pred, LHS, RHS, BO->getOperand(0), Inverse) ||
               isImpliedCond(Pred, LHS, RHS, BO->getOperand(1), Inverse);
    }
  }

  ICmpInst *ICI = dyn_cast<ICmpInst>(FoundCondValue);
  if (!ICI) return false;

  // Bail if the ICmp's operands' types are wider than the needed type
  // before attempting to call getSCEV on them. This avoids infinite
  // recursion, since the analysis of widening casts can require loop
  // exit condition information for overflow checking, which would
  // lead back here.
  if (getTypeSizeInBits(LHS->getType()) <
      getTypeSizeInBits(ICI->getOperand(0)->getType()))
    return false;

  // Now that we found a conditional branch that dominates the loop or controls
  // the loop latch. Check to see if it is the comparison we are looking for.
  ICmpInst::Predicate FoundPred;
  if (Inverse)
    FoundPred = ICI->getInversePredicate();
  else
    FoundPred = ICI->getPredicate();

  const SCEV *FoundLHS = getSCEV(ICI->getOperand(0));
  const SCEV *FoundRHS = getSCEV(ICI->getOperand(1));

  // Balance the types. The case where FoundLHS' type is wider than
  // LHS' type is checked for above.
  if (getTypeSizeInBits(LHS->getType()) >
      getTypeSizeInBits(FoundLHS->getType())) {
    if (CmpInst::isSigned(Pred)) {
      FoundLHS = getSignExtendExpr(FoundLHS, LHS->getType());
      FoundRHS = getSignExtendExpr(FoundRHS, LHS->getType());
    } else {
      FoundLHS = getZeroExtendExpr(FoundLHS, LHS->getType());
      FoundRHS = getZeroExtendExpr(FoundRHS, LHS->getType());
    }
  }

  // Canonicalize the query to match the way instcombine will have
  // canonicalized the comparison.
  if (SimplifyICmpOperands(Pred, LHS, RHS))
    if (LHS == RHS)
      return CmpInst::isTrueWhenEqual(Pred);
  if (SimplifyICmpOperands(FoundPred, FoundLHS, FoundRHS))
    if (FoundLHS == FoundRHS)
      return CmpInst::isFalseWhenEqual(FoundPred);

  // Check to see if we can make the LHS or RHS match.
  if (LHS == FoundRHS || RHS == FoundLHS) {
    if (isa<SCEVConstant>(RHS)) {
      std::swap(FoundLHS, FoundRHS);
      FoundPred = ICmpInst::getSwappedPredicate(FoundPred);
    } else {
      std::swap(LHS, RHS);
      Pred = ICmpInst::getSwappedPredicate(Pred);
    }
  }

  // Check whether the found predicate is the same as the desired predicate.
  if (FoundPred == Pred)
    return isImpliedCondOperands(Pred, LHS, RHS, FoundLHS, FoundRHS);

  // Check whether swapping the found predicate makes it the same as the
  // desired predicate.
  if (ICmpInst::getSwappedPredicate(FoundPred) == Pred) {
    if (isa<SCEVConstant>(RHS))
      return isImpliedCondOperands(Pred, LHS, RHS, FoundRHS, FoundLHS);
    else
      return isImpliedCondOperands(ICmpInst::getSwappedPredicate(Pred),
                                   RHS, LHS, FoundLHS, FoundRHS);
  }

  // Check whether the actual condition is beyond sufficient.
  if (FoundPred == ICmpInst::ICMP_EQ)
    if (ICmpInst::isTrueWhenEqual(Pred))
      if (isImpliedCondOperands(Pred, LHS, RHS, FoundLHS, FoundRHS))
        return true;
  if (Pred == ICmpInst::ICMP_NE)
    if (!ICmpInst::isTrueWhenEqual(FoundPred))
      if (isImpliedCondOperands(FoundPred, LHS, RHS, FoundLHS, FoundRHS))
        return true;

  // Otherwise assume the worst.
  return false;
}

/// isImpliedCondOperands - Test whether the condition described by Pred,
/// LHS, and RHS is true whenever the condition described by Pred, FoundLHS,
/// and FoundRHS is true.
bool ScalarEvolution::isImpliedCondOperands(ICmpInst::Predicate Pred,
                                            const SCEV *LHS, const SCEV *RHS,
                                            const SCEV *FoundLHS,
                                            const SCEV *FoundRHS) {
  return isImpliedCondOperandsHelper(Pred, LHS, RHS,
                                     FoundLHS, FoundRHS) ||
         // ~x < ~y --> x > y
         isImpliedCondOperandsHelper(Pred, LHS, RHS,
                                     getNotSCEV(FoundRHS),
                                     getNotSCEV(FoundLHS));
}

/// isImpliedCondOperandsHelper - Test whether the condition described by
/// Pred, LHS, and RHS is true whenever the condition described by Pred,
/// FoundLHS, and FoundRHS is true.
bool
ScalarEvolution::isImpliedCondOperandsHelper(ICmpInst::Predicate Pred,
                                             const SCEV *LHS, const SCEV *RHS,
                                             const SCEV *FoundLHS,
                                             const SCEV *FoundRHS) {
  switch (Pred) {
  default: llvm_unreachable("Unexpected ICmpInst::Predicate value!");
  case ICmpInst::ICMP_EQ:
  case ICmpInst::ICMP_NE:
    if (HasSameValue(LHS, FoundLHS) && HasSameValue(RHS, FoundRHS))
      return true;
    break;
  case ICmpInst::ICMP_SLT:
  case ICmpInst::ICMP_SLE:
    if (isKnownPredicateWithRanges(ICmpInst::ICMP_SLE, LHS, FoundLHS) &&
        isKnownPredicateWithRanges(ICmpInst::ICMP_SGE, RHS, FoundRHS))
      return true;
    break;
  case ICmpInst::ICMP_SGT:
  case ICmpInst::ICMP_SGE:
    if (isKnownPredicateWithRanges(ICmpInst::ICMP_SGE, LHS, FoundLHS) &&
        isKnownPredicateWithRanges(ICmpInst::ICMP_SLE, RHS, FoundRHS))
      return true;
    break;
  case ICmpInst::ICMP_ULT:
  case ICmpInst::ICMP_ULE:
    if (isKnownPredicateWithRanges(ICmpInst::ICMP_ULE, LHS, FoundLHS) &&
        isKnownPredicateWithRanges(ICmpInst::ICMP_UGE, RHS, FoundRHS))
      return true;
    break;
  case ICmpInst::ICMP_UGT:
  case ICmpInst::ICMP_UGE:
    if (isKnownPredicateWithRanges(ICmpInst::ICMP_UGE, LHS, FoundLHS) &&
        isKnownPredicateWithRanges(ICmpInst::ICMP_ULE, RHS, FoundRHS))
      return true;
    break;
  }

  return false;
}

/// getBECount - Subtract the end and start values and divide by the step,
/// rounding up, to get the number of times the backedge is executed. Return
/// CouldNotCompute if an intermediate computation overflows.
const SCEV *ScalarEvolution::getBECount(const SCEV *Start,
                                        const SCEV *End,
                                        const SCEV *Step,
                                        bool NoWrap) {
  assert(!isKnownNegative(Step) &&
         "This code doesn't handle negative strides yet!");

  Type *Ty = Start->getType();

  // When Start == End, we have an exact BECount == 0. Short-circuit this case
  // here because SCEV may not be able to determine that the unsigned division
  // after rounding is zero.
  if (Start == End)
    return getConstant(Ty, 0);

  const SCEV *NegOne = getConstant(Ty, (uint64_t)-1);
  const SCEV *Diff = getMinusSCEV(End, Start);
  const SCEV *RoundUp = getAddExpr(Step, NegOne);

  // Add an adjustment to the difference between End and Start so that
  // the division will effectively round up.
  const SCEV *Add = getAddExpr(Diff, RoundUp);

  if (!NoWrap) {
    // Check Add for unsigned overflow.
    // TODO: More sophisticated things could be done here.
    Type *WideTy = IntegerType::get(getContext(),
                                          getTypeSizeInBits(Ty) + 1);
    const SCEV *EDiff = getZeroExtendExpr(Diff, WideTy);
    const SCEV *ERoundUp = getZeroExtendExpr(RoundUp, WideTy);
    const SCEV *OperandExtendedAdd = getAddExpr(EDiff, ERoundUp);
    if (getZeroExtendExpr(Add, WideTy) != OperandExtendedAdd)
      return getCouldNotCompute();
  }

  return getUDivExpr(Add, Step);
}

/// HowManyLessThans - Return the number of times a backedge containing the
/// specified less-than comparison will execute.  If not computable, return
/// CouldNotCompute.
///
/// @param IsSubExpr is true when the LHS < RHS condition does not directly
/// control the branch. In this case, we can only compute an iteration count for
/// a subexpression that cannot overflow before evaluating true.
ScalarEvolution::ExitLimit
ScalarEvolution::HowManyLessThans(const SCEV *LHS, const SCEV *RHS,
                                  const Loop *L, bool isSigned,
                                  bool IsSubExpr) {
  // Only handle:  "ADDREC < LoopInvariant".
  if (!isLoopInvariant(RHS, L)) return getCouldNotCompute();

  const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(LHS);
  if (!AddRec || AddRec->getLoop() != L)
    return getCouldNotCompute();

  if (AddRec->isAffine()) {
    unsigned BitWidth = getTypeSizeInBits(AddRec->getType());
    const SCEV *Step = AddRec->getStepRecurrence(*this);

    if (Step->isZero())
      return getCouldNotCompute();
    if (Step->isOne()) {
      // With unit stride, the iteration never steps past the limit value.
    } else if (isKnownPositive(Step)) {
      // Test whether a positive iteration can step past the limit value and
      // past the maximum value for its type in a single step. The NSW/NUW flags
      // can imply that stepping past RHS would immediately result in undefined
      // behavior. No self-wrap is not useful here because the loop counter may
      // signed or unsigned wrap but continue iterating and terminate with
      // defined behavior without ever self-wrapping.
      const SCEV *One = getConstant(Step->getType(), 1);
      if (isSigned) {
        if (!AddRec->getNoWrapFlags(SCEV::FlagNSW)) {
          APInt Max = APInt::getSignedMaxValue(BitWidth);
          if ((Max - getSignedRange(getMinusSCEV(Step, One)).getSignedMax())
              .slt(getSignedRange(RHS).getSignedMax()))
            return getCouldNotCompute();
        }
      } else if (!AddRec->getNoWrapFlags(SCEV::FlagNUW)){
        APInt Max = APInt::getMaxValue(BitWidth);
        if ((Max - getUnsignedRange(getMinusSCEV(Step, One)).getUnsignedMax())
              .ult(getUnsignedRange(RHS).getUnsignedMax()))
          return getCouldNotCompute();
      }
    } else
      // TODO: Handle negative strides here and below.
      return getCouldNotCompute();

    // We know the LHS is of the form {n,+,s} and the RHS is some loop-invariant
    // m.  So, we count the number of iterations in which {n,+,s} < m is true.
    // Note that we cannot simply return max(m-n,0)/s because it's not safe to
    // treat m-n as signed nor unsigned due to overflow possibility.

    // First, we get the value of the LHS in the first iteration: n
    const SCEV *Start = AddRec->getOperand(0);

    // Determine the minimum constant start value.
    const SCEV *MinStart = getConstant(isSigned ?
      getSignedRange(Start).getSignedMin() :
      getUnsignedRange(Start).getUnsignedMin());

    // If we know that the condition is true in order to enter the loop,
    // then we know that it will run exactly (m-n)/s times. Otherwise, we
    // only know that it will execute (max(m,n)-n)/s times. In both cases,
    // the division must round up.
    const SCEV *End = RHS;
    if (!isLoopEntryGuardedByCond(L,
                                  isSigned ? ICmpInst::ICMP_SLT :
                                             ICmpInst::ICMP_ULT,
                                  getMinusSCEV(Start, Step), RHS))
      End = isSigned ? getSMaxExpr(RHS, Start)
                     : getUMaxExpr(RHS, Start);

    // Determine the maximum constant end value.
    const SCEV *MaxEnd = getConstant(isSigned ?
      getSignedRange(End).getSignedMax() :
      getUnsignedRange(End).getUnsignedMax());

    // If MaxEnd is within a step of the maximum integer value in its type,
    // adjust it down to the minimum value which would produce the same effect.
    // This allows the subsequent ceiling division of (N+(step-1))/step to
    // compute the correct value.
    const SCEV *StepMinusOne = getMinusSCEV(Step,
                                            getConstant(Step->getType(), 1));
    MaxEnd = isSigned ?
      getSMinExpr(MaxEnd,
                  getMinusSCEV(getConstant(APInt::getSignedMaxValue(BitWidth)),
                               StepMinusOne)) :
      getUMinExpr(MaxEnd,
                  getMinusSCEV(getConstant(APInt::getMaxValue(BitWidth)),
                               StepMinusOne));

    // If the loop counter does not self-wrap, then the trip count may be
    // computed by dividing the distance by the step. This is independent of
    // signed or unsigned wrap.
    bool NoWrap = false;
    if (!IsSubExpr) {
      NoWrap = AddRec->getNoWrapFlags(
        (SCEV::NoWrapFlags)(((isSigned ? SCEV::FlagNSW : SCEV::FlagNUW))
                            | SCEV::FlagNW));
    }
    // Finally, we subtract these two values and divide, rounding up, to get
    // the number of times the backedge is executed.
    const SCEV *BECount = getBECount(Start, End, Step, NoWrap);

    // The maximum backedge count is similar, except using the minimum start
    // value and the maximum end value.
    // If we already have an exact constant BECount, use it instead.
    const SCEV *MaxBECount = isa<SCEVConstant>(BECount) ? BECount
      : getBECount(MinStart, MaxEnd, Step, NoWrap);

    // If the stride is nonconstant, and NoWrap == true, then
    // getBECount(MinStart, MaxEnd) may not compute. This would result in an
    // exact BECount and invalid MaxBECount, which should be avoided to catch
    // more optimization opportunities.
    if (isa<SCEVCouldNotCompute>(MaxBECount))
      MaxBECount = BECount;

    return ExitLimit(BECount, MaxBECount);
  }

  return getCouldNotCompute();
}

/// getNumIterationsInRange - Return the number of iterations of this loop that
/// produce values in the specified constant range.  Another way of looking at
/// this is that it returns the first iteration number where the value is not in
/// the condition, thus computing the exit count. If the iteration count can't
/// be computed, an instance of SCEVCouldNotCompute is returned.
const SCEV *SCEVAddRecExpr::getNumIterationsInRange(ConstantRange Range,
                                                    ScalarEvolution &SE) const {
  if (Range.isFullSet())  // Infinite loop.
    return SE.getCouldNotCompute();

  // If the start is a non-zero constant, shift the range to simplify things.
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(getStart()))
    if (!SC->getValue()->isZero()) {
      SmallVector<const SCEV *, 4> Operands(op_begin(), op_end());
      Operands[0] = SE.getConstant(SC->getType(), 0);
      const SCEV *Shifted = SE.getAddRecExpr(Operands, getLoop(),
                                             getNoWrapFlags(FlagNW));
      if (const SCEVAddRecExpr *ShiftedAddRec =
            dyn_cast<SCEVAddRecExpr>(Shifted))
        return ShiftedAddRec->getNumIterationsInRange(
                           Range.subtract(SC->getValue()->getValue()), SE);
      // This is strange and shouldn't happen.
      return SE.getCouldNotCompute();
    }

  // The only time we can solve this is when we have all constant indices.
  // Otherwise, we cannot determine the overflow conditions.
  for (unsigned i = 0, e = getNumOperands(); i != e; ++i)
    if (!isa<SCEVConstant>(getOperand(i)))
      return SE.getCouldNotCompute();


  // Okay at this point we know that all elements of the chrec are constants and
  // that the start element is zero.

  // First check to see if the range contains zero.  If not, the first
  // iteration exits.
  unsigned BitWidth = SE.getTypeSizeInBits(getType());
  if (!Range.contains(APInt(BitWidth, 0)))
    return SE.getConstant(getType(), 0);

  if (isAffine()) {
    // If this is an affine expression then we have this situation:
    //   Solve {0,+,A} in Range  ===  Ax in Range

    // We know that zero is in the range.  If A is positive then we know that
    // the upper value of the range must be the first possible exit value.
    // If A is negative then the lower of the range is the last possible loop
    // value.  Also note that we already checked for a full range.
    APInt One(BitWidth,1);
    APInt A     = cast<SCEVConstant>(getOperand(1))->getValue()->getValue();
    APInt End = A.sge(One) ? (Range.getUpper() - One) : Range.getLower();

    // The exit value should be (End+A)/A.
    APInt ExitVal = (End + A).udiv(A);
    ConstantInt *ExitValue = ConstantInt::get(SE.getContext(), ExitVal);

    // Evaluate at the exit value.  If we really did fall out of the valid
    // range, then we computed our trip count, otherwise wrap around or other
    // things must have happened.
    ConstantInt *Val = EvaluateConstantChrecAtConstant(this, ExitValue, SE);
    if (Range.contains(Val->getValue()))
      return SE.getCouldNotCompute();  // Something strange happened

    // Ensure that the previous value is in the range.  This is a sanity check.
    assert(Range.contains(
           EvaluateConstantChrecAtConstant(this,
           ConstantInt::get(SE.getContext(), ExitVal - One), SE)->getValue()) &&
           "Linear scev computation is off in a bad way!");
    return SE.getConstant(ExitValue);
  } else if (isQuadratic()) {
    // If this is a quadratic (3-term) AddRec {L,+,M,+,N}, find the roots of the
    // quadratic equation to solve it.  To do this, we must frame our problem in
    // terms of figuring out when zero is crossed, instead of when
    // Range.getUpper() is crossed.
    SmallVector<const SCEV *, 4> NewOps(op_begin(), op_end());
    NewOps[0] = SE.getNegativeSCEV(SE.getConstant(Range.getUpper()));
    const SCEV *NewAddRec = SE.getAddRecExpr(NewOps, getLoop(),
                                             // getNoWrapFlags(FlagNW)
                                             FlagAnyWrap);

    // Next, solve the constructed addrec
    std::pair<const SCEV *,const SCEV *> Roots =
      SolveQuadraticEquation(cast<SCEVAddRecExpr>(NewAddRec), SE);
    const SCEVConstant *R1 = dyn_cast<SCEVConstant>(Roots.first);
    const SCEVConstant *R2 = dyn_cast<SCEVConstant>(Roots.second);
    if (R1) {
      // Pick the smallest positive root value.
      if (ConstantInt *CB =
          dyn_cast<ConstantInt>(ConstantExpr::getICmp(ICmpInst::ICMP_ULT,
                         R1->getValue(), R2->getValue()))) {
        if (CB->getZExtValue() == false)
          std::swap(R1, R2);   // R1 is the minimum root now.

        // Make sure the root is not off by one.  The returned iteration should
        // not be in the range, but the previous one should be.  When solving
        // for "X*X < 5", for example, we should not return a root of 2.
        ConstantInt *R1Val = EvaluateConstantChrecAtConstant(this,
                                                             R1->getValue(),
                                                             SE);
        if (Range.contains(R1Val->getValue())) {
          // The next iteration must be out of the range...
          ConstantInt *NextVal =
                ConstantInt::get(SE.getContext(), R1->getValue()->getValue()+1);

          R1Val = EvaluateConstantChrecAtConstant(this, NextVal, SE);
          if (!Range.contains(R1Val->getValue()))
            return SE.getConstant(NextVal);
          return SE.getCouldNotCompute();  // Something strange happened
        }

        // If R1 was not in the range, then it is a good return value.  Make
        // sure that R1-1 WAS in the range though, just in case.
        ConstantInt *NextVal =
               ConstantInt::get(SE.getContext(), R1->getValue()->getValue()-1);
        R1Val = EvaluateConstantChrecAtConstant(this, NextVal, SE);
        if (Range.contains(R1Val->getValue()))
          return R1;
        return SE.getCouldNotCompute();  // Something strange happened
      }
    }
  }

  return SE.getCouldNotCompute();
}



//===----------------------------------------------------------------------===//
//                   SCEVCallbackVH Class Implementation
//===----------------------------------------------------------------------===//

void ScalarEvolution::SCEVCallbackVH::deleted() {
  assert(SE && "SCEVCallbackVH called with a null ScalarEvolution!");
  if (PHINode *PN = dyn_cast<PHINode>(getValPtr()))
    SE->ConstantEvolutionLoopExitValue.erase(PN);
  SE->ValueExprMap.erase(getValPtr());
  // this now dangles!
}

void ScalarEvolution::SCEVCallbackVH::allUsesReplacedWith(Value *V) {
  assert(SE && "SCEVCallbackVH called with a null ScalarEvolution!");

  // Forget all the expressions associated with users of the old value,
  // so that future queries will recompute the expressions using the new
  // value.
  Value *Old = getValPtr();
  SmallVector<User *, 16> Worklist;
  SmallPtrSet<User *, 8> Visited;
  for (Value::use_iterator UI = Old->use_begin(), UE = Old->use_end();
       UI != UE; ++UI)
    Worklist.push_back(*UI);
  while (!Worklist.empty()) {
    User *U = Worklist.pop_back_val();
    // Deleting the Old value will cause this to dangle. Postpone
    // that until everything else is done.
    if (U == Old)
      continue;
    if (!Visited.insert(U))
      continue;
    if (PHINode *PN = dyn_cast<PHINode>(U))
      SE->ConstantEvolutionLoopExitValue.erase(PN);
    SE->ValueExprMap.erase(U);
    for (Value::use_iterator UI = U->use_begin(), UE = U->use_end();
         UI != UE; ++UI)
      Worklist.push_back(*UI);
  }
  // Delete the Old value.
  if (PHINode *PN = dyn_cast<PHINode>(Old))
    SE->ConstantEvolutionLoopExitValue.erase(PN);
  SE->ValueExprMap.erase(Old);
  // this now dangles!
}

ScalarEvolution::SCEVCallbackVH::SCEVCallbackVH(Value *V, ScalarEvolution *se)
  : CallbackVH(V), SE(se) {}

//===----------------------------------------------------------------------===//
//                   ScalarEvolution Class Implementation
//===----------------------------------------------------------------------===//

ScalarEvolution::ScalarEvolution()
  : FunctionPass(ID), FirstUnknown(0) {
  initializeScalarEvolutionPass(*PassRegistry::getPassRegistry());
}

bool ScalarEvolution::runOnFunction(Function &F) {
  this->F = &F;
  LI = &getAnalysis<LoopInfo>();
  TD = getAnalysisIfAvailable<DataLayout>();
  TLI = &getAnalysis<TargetLibraryInfo>();
  DT = &getAnalysis<DominatorTree>();
  return false;
}

void ScalarEvolution::releaseMemory() {
  // Iterate through all the SCEVUnknown instances and call their
  // destructors, so that they release their references to their values.
  for (SCEVUnknown *U = FirstUnknown; U; U = U->Next)
    U->~SCEVUnknown();
  FirstUnknown = 0;

  ValueExprMap.clear();

  // Free any extra memory created for ExitNotTakenInfo in the unlikely event
  // that a loop had multiple computable exits.
  for (DenseMap<const Loop*, BackedgeTakenInfo>::iterator I =
         BackedgeTakenCounts.begin(), E = BackedgeTakenCounts.end();
       I != E; ++I) {
    I->second.clear();
  }

  assert(PendingLoopPredicates.empty() && "isImpliedCond garbage");

  BackedgeTakenCounts.clear();
  ConstantEvolutionLoopExitValue.clear();
  ValuesAtScopes.clear();
  LoopDispositions.clear();
  BlockDispositions.clear();
  UnsignedRanges.clear();
  SignedRanges.clear();
  UniqueSCEVs.clear();
  SCEVAllocator.Reset();
}

void ScalarEvolution::getAnalysisUsage(AnalysisUsage &AU) const {
  AU.setPreservesAll();
  AU.addRequiredTransitive<LoopInfo>();
  AU.addRequiredTransitive<DominatorTree>();
  AU.addRequired<TargetLibraryInfo>();
}

bool ScalarEvolution::hasLoopInvariantBackedgeTakenCount(const Loop *L) {
  return !isa<SCEVCouldNotCompute>(getBackedgeTakenCount(L));
}

static void PrintLoopInfo(raw_ostream &OS, ScalarEvolution *SE,
                          const Loop *L) {
  // Print all inner loops first
  for (Loop::iterator I = L->begin(), E = L->end(); I != E; ++I)
    PrintLoopInfo(OS, SE, *I);

  OS << "Loop ";
  WriteAsOperand(OS, L->getHeader(), /*PrintType=*/false);
  OS << ": ";

  SmallVector<BasicBlock *, 8> ExitBlocks;
  L->getExitBlocks(ExitBlocks);
  if (ExitBlocks.size() != 1)
    OS << "<multiple exits> ";

  if (SE->hasLoopInvariantBackedgeTakenCount(L)) {
    OS << "backedge-taken count is " << *SE->getBackedgeTakenCount(L);
  } else {
    OS << "Unpredictable backedge-taken count. ";
  }

  OS << "\n"
        "Loop ";
  WriteAsOperand(OS, L->getHeader(), /*PrintType=*/false);
  OS << ": ";

  if (!isa<SCEVCouldNotCompute>(SE->getMaxBackedgeTakenCount(L))) {
    OS << "max backedge-taken count is " << *SE->getMaxBackedgeTakenCount(L);
  } else {
    OS << "Unpredictable max backedge-taken count. ";
  }

  OS << "\n";
}

void ScalarEvolution::print(raw_ostream &OS, const Module *) const {
  // ScalarEvolution's implementation of the print method is to print
  // out SCEV values of all instructions that are interesting. Doing
  // this potentially causes it to create new SCEV objects though,
  // which technically conflicts with the const qualifier. This isn't
  // observable from outside the class though, so casting away the
  // const isn't dangerous.
  ScalarEvolution &SE = *const_cast<ScalarEvolution *>(this);

  OS << "Classifying expressions for: ";
  WriteAsOperand(OS, F, /*PrintType=*/false);
  OS << "\n";
  for (inst_iterator I = inst_begin(F), E = inst_end(F); I != E; ++I)
    if (isSCEVable(I->getType()) && !isa<CmpInst>(*I)) {
      OS << *I << '\n';
      OS << "  -->  ";
      const SCEV *SV = SE.getSCEV(&*I);
      SV->print(OS);

      const Loop *L = LI->getLoopFor((*I).getParent());

      const SCEV *AtUse = SE.getSCEVAtScope(SV, L);
      if (AtUse != SV) {
        OS << "  -->  ";
        AtUse->print(OS);
      }

      if (L) {
        OS << "\t\t" "Exits: ";
        const SCEV *ExitValue = SE.getSCEVAtScope(SV, L->getParentLoop());
        if (!SE.isLoopInvariant(ExitValue, L)) {
          OS << "<<Unknown>>";
        } else {
          OS << *ExitValue;
        }
      }

      OS << "\n";
    }

  OS << "Determining loop execution counts for: ";
  WriteAsOperand(OS, F, /*PrintType=*/false);
  OS << "\n";
  for (LoopInfo::iterator I = LI->begin(), E = LI->end(); I != E; ++I)
    PrintLoopInfo(OS, &SE, *I);
}

ScalarEvolution::LoopDisposition
ScalarEvolution::getLoopDisposition(const SCEV *S, const Loop *L) {
  std::map<const Loop *, LoopDisposition> &Values = LoopDispositions[S];
  std::pair<std::map<const Loop *, LoopDisposition>::iterator, bool> Pair =
    Values.insert(std::make_pair(L, LoopVariant));
  if (!Pair.second)
    return Pair.first->second;

  LoopDisposition D = computeLoopDisposition(S, L);
  return LoopDispositions[S][L] = D;
}

ScalarEvolution::LoopDisposition
ScalarEvolution::computeLoopDisposition(const SCEV *S, const Loop *L) {
  switch (S->getSCEVType()) {
  case scConstant:
    return LoopInvariant;
  case scTruncate:
  case scZeroExtend:
  case scSignExtend:
    return getLoopDisposition(cast<SCEVCastExpr>(S)->getOperand(), L);
  case scAddRecExpr: {
    const SCEVAddRecExpr *AR = cast<SCEVAddRecExpr>(S);

    // If L is the addrec's loop, it's computable.
    if (AR->getLoop() == L)
      return LoopComputable;

    // Add recurrences are never invariant in the function-body (null loop).
    if (!L)
      return LoopVariant;

    // This recurrence is variant w.r.t. L if L contains AR's loop.
    if (L->contains(AR->getLoop()))
      return LoopVariant;

    // This recurrence is invariant w.r.t. L if AR's loop contains L.
    if (AR->getLoop()->contains(L))
      return LoopInvariant;

    // This recurrence is variant w.r.t. L if any of its operands
    // are variant.
    for (SCEVAddRecExpr::op_iterator I = AR->op_begin(), E = AR->op_end();
         I != E; ++I)
      if (!isLoopInvariant(*I, L))
        return LoopVariant;

    // Otherwise it's loop-invariant.
    return LoopInvariant;
  }
  case scAddExpr:
  case scMulExpr:
  case scUMaxExpr:
  case scSMaxExpr: {
    const SCEVNAryExpr *NAry = cast<SCEVNAryExpr>(S);
    bool HasVarying = false;
    for (SCEVNAryExpr::op_iterator I = NAry->op_begin(), E = NAry->op_end();
         I != E; ++I) {
      LoopDisposition D = getLoopDisposition(*I, L);
      if (D == LoopVariant)
        return LoopVariant;
      if (D == LoopComputable)
        HasVarying = true;
    }
    return HasVarying ? LoopComputable : LoopInvariant;
  }
  case scUDivExpr: {
    const SCEVUDivExpr *UDiv = cast<SCEVUDivExpr>(S);
    LoopDisposition LD = getLoopDisposition(UDiv->getLHS(), L);
    if (LD == LoopVariant)
      return LoopVariant;
    LoopDisposition RD = getLoopDisposition(UDiv->getRHS(), L);
    if (RD == LoopVariant)
      return LoopVariant;
    return (LD == LoopInvariant && RD == LoopInvariant) ?
           LoopInvariant : LoopComputable;
  }
  case scUnknown:
    // All non-instruction values are loop invariant.  All instructions are loop
    // invariant if they are not contained in the specified loop.
    // Instructions are never considered invariant in the function body
    // (null loop) because they are defined within the "loop".
    if (Instruction *I = dyn_cast<Instruction>(cast<SCEVUnknown>(S)->getValue()))
      return (L && !L->contains(I)) ? LoopInvariant : LoopVariant;
    return LoopInvariant;
  case scCouldNotCompute:
    llvm_unreachable("Attempt to use a SCEVCouldNotCompute object!");
  default: llvm_unreachable("Unknown SCEV kind!");
  }
}

bool ScalarEvolution::isLoopInvariant(const SCEV *S, const Loop *L) {
  return getLoopDisposition(S, L) == LoopInvariant;
}

bool ScalarEvolution::hasComputableLoopEvolution(const SCEV *S, const Loop *L) {
  return getLoopDisposition(S, L) == LoopComputable;
}

ScalarEvolution::BlockDisposition
ScalarEvolution::getBlockDisposition(const SCEV *S, const BasicBlock *BB) {
  std::map<const BasicBlock *, BlockDisposition> &Values = BlockDispositions[S];
  std::pair<std::map<const BasicBlock *, BlockDisposition>::iterator, bool>
    Pair = Values.insert(std::make_pair(BB, DoesNotDominateBlock));
  if (!Pair.second)
    return Pair.first->second;

  BlockDisposition D = computeBlockDisposition(S, BB);
  return BlockDispositions[S][BB] = D;
}

ScalarEvolution::BlockDisposition
ScalarEvolution::computeBlockDisposition(const SCEV *S, const BasicBlock *BB) {
  switch (S->getSCEVType()) {
  case scConstant:
    return ProperlyDominatesBlock;
  case scTruncate:
  case scZeroExtend:
  case scSignExtend:
    return getBlockDisposition(cast<SCEVCastExpr>(S)->getOperand(), BB);
  case scAddRecExpr: {
    // This uses a "dominates" query instead of "properly dominates" query
    // to test for proper dominance too, because the instruction which
    // produces the addrec's value is a PHI, and a PHI effectively properly
    // dominates its entire containing block.
    const SCEVAddRecExpr *AR = cast<SCEVAddRecExpr>(S);
    if (!DT->dominates(AR->getLoop()->getHeader(), BB))
      return DoesNotDominateBlock;
  }
  // FALL THROUGH into SCEVNAryExpr handling.
  case scAddExpr:
  case scMulExpr:
  case scUMaxExpr:
  case scSMaxExpr: {
    const SCEVNAryExpr *NAry = cast<SCEVNAryExpr>(S);
    bool Proper = true;
    for (SCEVNAryExpr::op_iterator I = NAry->op_begin(), E = NAry->op_end();
         I != E; ++I) {
      BlockDisposition D = getBlockDisposition(*I, BB);
      if (D == DoesNotDominateBlock)
        return DoesNotDominateBlock;
      if (D == DominatesBlock)
        Proper = false;
    }
    return Proper ? ProperlyDominatesBlock : DominatesBlock;
  }
  case scUDivExpr: {
    const SCEVUDivExpr *UDiv = cast<SCEVUDivExpr>(S);
    const SCEV *LHS = UDiv->getLHS(), *RHS = UDiv->getRHS();
    BlockDisposition LD = getBlockDisposition(LHS, BB);
    if (LD == DoesNotDominateBlock)
      return DoesNotDominateBlock;
    BlockDisposition RD = getBlockDisposition(RHS, BB);
    if (RD == DoesNotDominateBlock)
      return DoesNotDominateBlock;
    return (LD == ProperlyDominatesBlock && RD == ProperlyDominatesBlock) ?
      ProperlyDominatesBlock : DominatesBlock;
  }
  case scUnknown:
    if (Instruction *I =
          dyn_cast<Instruction>(cast<SCEVUnknown>(S)->getValue())) {
      if (I->getParent() == BB)
        return DominatesBlock;
      if (DT->properlyDominates(I->getParent(), BB))
        return ProperlyDominatesBlock;
      return DoesNotDominateBlock;
    }
    return ProperlyDominatesBlock;
  case scCouldNotCompute:
    llvm_unreachable("Attempt to use a SCEVCouldNotCompute object!");
  default:
    llvm_unreachable("Unknown SCEV kind!");
  }
}

bool ScalarEvolution::dominates(const SCEV *S, const BasicBlock *BB) {
  return getBlockDisposition(S, BB) >= DominatesBlock;
}

bool ScalarEvolution::properlyDominates(const SCEV *S, const BasicBlock *BB) {
  return getBlockDisposition(S, BB) == ProperlyDominatesBlock;
}

namespace {
// Search for a SCEV expression node within an expression tree.
// Implements SCEVTraversal::Visitor.
struct SCEVSearch {
  const SCEV *Node;
  bool IsFound;

  SCEVSearch(const SCEV *N): Node(N), IsFound(false) {}

  bool follow(const SCEV *S) {
    IsFound |= (S == Node);
    return !IsFound;
  }
  bool isDone() const { return IsFound; }
};
}

bool ScalarEvolution::hasOperand(const SCEV *S, const SCEV *Op) const {
  SCEVSearch Search(Op);
  visitAll(S, Search);
  return Search.IsFound;
}

void ScalarEvolution::forgetMemoizedResults(const SCEV *S) {
  ValuesAtScopes.erase(S);
  LoopDispositions.erase(S);
  BlockDispositions.erase(S);
  UnsignedRanges.erase(S);
  SignedRanges.erase(S);

  for (DenseMap<const Loop*, BackedgeTakenInfo>::iterator I =
         BackedgeTakenCounts.begin(), E = BackedgeTakenCounts.end(); I != E; ) {
    BackedgeTakenInfo &BEInfo = I->second;
    if (BEInfo.hasOperand(S, this)) {
      BEInfo.clear();
      BackedgeTakenCounts.erase(I++);
    }
    else
      ++I;
  }
}

typedef DenseMap<const Loop *, std::string> VerifyMap;

/// replaceSubString - Replaces all occurences of From in Str with To.
static void replaceSubString(std::string &Str, StringRef From, StringRef To) {
  size_t Pos = 0;
  while ((Pos = Str.find(From, Pos)) != std::string::npos) {
    Str.replace(Pos, From.size(), To.data(), To.size());
    Pos += To.size();
  }
}

/// getLoopBackedgeTakenCounts - Helper method for verifyAnalysis.
static void
getLoopBackedgeTakenCounts(Loop *L, VerifyMap &Map, ScalarEvolution &SE) {
  for (Loop::reverse_iterator I = L->rbegin(), E = L->rend(); I != E; ++I) {
    getLoopBackedgeTakenCounts(*I, Map, SE); // recurse.

    std::string &S = Map[L];
    if (S.empty()) {
      raw_string_ostream OS(S);
      SE.getBackedgeTakenCount(L)->print(OS);

      // false and 0 are semantically equivalent. This can happen in dead loops.
      replaceSubString(OS.str(), "false", "0");
      // Remove wrap flags, their use in SCEV is highly fragile.
      // FIXME: Remove this when SCEV gets smarter about them.
      replaceSubString(OS.str(), "<nw>", "");
      replaceSubString(OS.str(), "<nsw>", "");
      replaceSubString(OS.str(), "<nuw>", "");
    }
  }
}

void ScalarEvolution::verifyAnalysis() const {
  if (!VerifySCEV)
    return;

  ScalarEvolution &SE = *const_cast<ScalarEvolution *>(this);

  // Gather stringified backedge taken counts for all loops using SCEV's caches.
  // FIXME: It would be much better to store actual values instead of strings,
  //        but SCEV pointers will change if we drop the caches.
  VerifyMap BackedgeDumpsOld, BackedgeDumpsNew;
  for (LoopInfo::reverse_iterator I = LI->rbegin(), E = LI->rend(); I != E; ++I)
    getLoopBackedgeTakenCounts(*I, BackedgeDumpsOld, SE);

  // Gather stringified backedge taken counts for all loops without using
  // SCEV's caches.
  SE.releaseMemory();
  for (LoopInfo::reverse_iterator I = LI->rbegin(), E = LI->rend(); I != E; ++I)
    getLoopBackedgeTakenCounts(*I, BackedgeDumpsNew, SE);

  // Now compare whether they're the same with and without caches. This allows
  // verifying that no pass changed the cache.
  assert(BackedgeDumpsOld.size() == BackedgeDumpsNew.size() &&
         "New loops suddenly appeared!");

  for (VerifyMap::iterator OldI = BackedgeDumpsOld.begin(),
                           OldE = BackedgeDumpsOld.end(),
                           NewI = BackedgeDumpsNew.begin();
       OldI != OldE; ++OldI, ++NewI) {
    assert(OldI->first == NewI->first && "Loop order changed!");

    // Compare the stringified SCEVs. We don't care if undef backedgetaken count
    // changes.
    // FIXME: We currently ignore SCEV changes from/to CouldNotCompute. This
    // means that a pass is buggy or SCEV has to learn a new pattern but is
    // usually not harmful.
    if (OldI->second != NewI->second &&
        OldI->second.find("undef") == std::string::npos &&
        NewI->second.find("undef") == std::string::npos &&
        OldI->second != "***COULDNOTCOMPUTE***" &&
        NewI->second != "***COULDNOTCOMPUTE***") {
      dbgs() << "SCEVValidator: SCEV for loop '"
             << OldI->first->getHeader()->getName()
             << "' changed from '" << OldI->second
             << "' to '" << NewI->second << "'!\n";
      std::abort();
    }
  }

  // TODO: Verify more things.
}