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The performance can be improved, i.e. all carries propagated more quickly by varying the block sizes. Accordingly the initial blocks of the adder are made smaller so as to quickly detect carry generates that must be propagated the furthers, the middle blocks are made larger because they are not the
2582:
The problem of determining the block sizes and number of levels required to make the physically fastest carry-skip adder is known as the 'carry-skip adder optimization problem'. This problem is made complex by the fact that a carry-skip adders are implemented with physical devices whose size and
1009:
propagate signal set to logic 1 (as opposed to a long ripple-carry chain, which would require the carry to ripple through each bit in the adder). The number of inputs of the AND-gate is equal to the width of the adder. For a large width, this becomes impractical and leads to additional delays,
2057:
2840:
V. G. Oklobdzija and E. R. Barnes, "Some
Optimal Schemes For ALU Implementation In VLSI Technology", Proceedings of the 7th Symposium on Computer Arithmetic ARITH-7, pp. 2-8. Reprinted in Computer Arithmetic, E. E. Swartzlander, (editor), Vol. II, pp. 137-142,
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blocks. The critical path consists of the ripple path and the skip element of the first block, the skip paths that are enclosed between the first and the last block, and finally the ripple-path of the last block.
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As the propagate signals are computed in parallel and are early available, the critical path for the skip logic in a carry-skip adder consists only of the delay imposed by the multiplexer (conditional skip).
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The carry-skip optimization problem for variable block sizes and multiple levels for an arbitrary device process node was solved by
Oklobdzija and Barnes at IBM and published in 1985.
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with little effort compared to other adders. The improvement of the worst-case delay is achieved by using several carry-skip adders to form a block-carry-skip adder.
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Unlike other fast adders, carry-skip adder performance is increased with only some of the combinations of input bits. This means, speed improvement is only
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This greatly reduces the latency of the adder through its critical path, since the carry bit for each block can now "skip" over blocks with a
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problem case, and then the most significant blocks are again made smaller so that the late arriving carry inputs can be processed quickly.
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Block-carry-skip adders are composed of a number of carry-skip adders. There are two types of block-carry-skip adders The two operands
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2213:
2052:{\displaystyle T_{FCSA}(n)=T_{CRA_{}}(m)+T_{CSK}+(k-2)\cdot T_{CSK}+T_{CRA}(m)=3D+m\cdot 2D+(k-1)\cdot 2D+(m+2)2D=(2m+k)\cdot 2D+5D}
2644:)would drive the select signal for three 2 to 1 multiplexers. The second set of 2 full adders would add the last two bits assuming
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The critical path of a carry-skip-adder begins at the first full-adder, passes through all adders and ends at the sum-bit
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The worst case for a simple one level ripple-carry adder occurs, when the propagate-condition is true for each digit pair
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because the AND-gate has to be built as a tree. A good width is achieved, when the sum-logic has the same depth like the
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791:-input AND-gate. The resulting bit is used as the select bit of a multiplexer that switches either the last carry-bit
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The first two full adders would add the first two bits together. The carry-out signal from the second full adder (
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1063:. Carry-skip-adders are chained (see block-carry-skip-adders) to reduce the overall critical path, since a single
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Breaking this down into more specific terms, in order to build a 4-bit carry-bypass adder, 6
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995:{\displaystyle s=p_{n-1}\wedge p_{n-2}\wedge \dots \wedge p_{1}\wedge p_{0}=p_{}}
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2610:) signal. The output would be a 4-bit bus X and a carry-out signal (
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are determined using an XOR-gate. When all propagate-conditions are
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is a logical 0. And the final set of full adders would assume that
2202:{\displaystyle 2D\cdot \left(2-n\cdot {\frac {1}{m^{2}}}\right)=0}
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787:, that is provided by the carry-ripple-chain is connected to the
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is often abbreviated as CSA, however, this can be confused with
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16-bit fixed-block-carry-skip adder with a block size of 4 bit.
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The multiplexers then control which output signal is used for
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Variable size block-carry-skip adders (VBA, Oklobdzija-Barnes)
2257:{\displaystyle \Rightarrow m_{1,2}=\pm {\sqrt {\frac {n}{2}}}}
1083:-bit carry-skip-adder has no real speed benefit compared to a
2564:{\displaystyle p_{}=p_{}\wedge p_{}\wedge p_{}\wedge p_{}}
611:
Full adder with additional generate and propagate signals.
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are further summarized and used to perform larger skips:
2856:
Explanation for critical path of the variable-skip adder
760:-input AND-gate and one multiplexer. Each propagate bit
597:{\displaystyle \tau _{CRA}(n)\approx n\cdot \tau _{VA}}
1549:{\displaystyle B=(b_{n-1},b_{n-2},\dots ,b_{1},b_{0})}
1453:{\displaystyle A=(a_{n-1},a_{n-2},\dots ,a_{1},a_{0})}
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Computer arithmetic: Algorithms and
Hardware Designs
2301:{\displaystyle \Rightarrow m={\sqrt {\frac {n}{2}}}}
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2062:The optimal block size for a given adder width
455:implementation that improves on the delay of a
1667:Should the block-size be constant or variable?
540:-bit adder and appears as the carry-out after
2583:other parameters also affects addition time.
1679:Fixed size block-carry-skip adders split the
1169:{\displaystyle \tau _{CSA}(n)=\tau _{CRA}(n)}
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2130:{\displaystyle {\frac {dT_{FCSA}(n)}{dm}}=0}
1670:Fixed block width vs. variable block width
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1272:{\displaystyle T_{SK}=T_{AND}(m)+T_{MUX}}
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1699:bits of the input bits into blocks of
1664:Why are block-carry-skip-adders used?
1199:-input AND-gate and one multiplexer.
1014:-input AND-gate and the multiplexer.
7:
752:-bit-carry-skip adder consists of a
2574:Thus making the adder even faster.
1675:Fixed size block-carry-skip adders
1340:{\displaystyle T_{CSK}=T_{MUX}=2D}
25:
1745:{\displaystyle k={\frac {n}{m}}}
615:For each operand input bit pair
236:Booth's multiplication algorithm
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1179:The skip-logic consists of a
654:{\displaystyle (a_{i},b_{i})}
513:{\displaystyle (a_{i},b_{i})}
341:Multiplyâaccumulate operation
82:Signed number representations
2332:Multilevel carry-skip adders
2066:is derived by equating to 0
371:Category:Computer arithmetic
756:-bit-carry-ripple-chain, a
2887:
366:Category:Binary arithmetic
2803:Parhami, Behrooz (2000).
1103:-bit ripple-carry adder.
661:the propagate-conditions
38:Arithmetic logic circuits
1719:bits each, resulting in
845:to the carry-out signal
718:, then the carry-in bit
335:Kochanski multiplication
231:Multiplication algorithm
27:Arithmetic logic circuit
2813:Oxford University Press
2590:Implementation overview
2578:Carry-skip optimization
1352:Block carry-skip adders
1056:{\displaystyle s_{n-1}}
1021:4 bit carry-skip adder.
871:{\displaystyle c_{out}}
470:Single carry-skip adder
77:Two's complement number
72:Ones' complement number
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2752:{\displaystyle X_{3}}
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2725:{\displaystyle X_{2}}
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2691:{\displaystyle C_{1}}
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2664:{\displaystyle C_{1}}
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2637:{\displaystyle C_{1}}
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838:{\displaystyle c_{0}}
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811:{\displaystyle c_{n}}
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780:{\displaystyle p_{i}}
740:
738:{\displaystyle c_{0}}
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404:Mechanical calculator
132:Carry-lookahead adder
2871:Adders (electronics)
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2374:{\displaystyle p_{}}
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174:Adderâsubtractor (±)
2606:, with a carry-in (
33:Part of a series on
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457:ripple-carry adder
449:carry-bypass adder
337:(exponentiation)
269:Division algorithm
157:Carry-select adder
127:Ripple-carry adder
18:Carry-bypass adder
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1712:{\displaystyle m}
1692:{\displaystyle n}
1569:{\displaystyle k}
1192:{\displaystyle m}
1096:{\displaystyle n}
1076:{\displaystyle n}
533:{\displaystyle n}
447:(also known as a
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287:Bitwise operation
226:Binary multiplier
142:KoggeâStone adder
16:(Redirected from
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2780:carry-save adder
2776:Carry-skip adder
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445:carry-skip adder
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317:Bit manipulation
246:Dadda multiplier
180:Adderâsubtractor
162:Carry-skip adder
152:Carry-save adder
137:BrentâKung adder
100:
43:Quick navigation
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203:Full subtractor
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2421:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2394:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2346:
2333:
2330:
2313:
2310:
2309:
2308:
2294:
2291:
2285:
2282:
2279:
2265:
2264:
2250:
2247:
2241:
2238:
2233:
2230:
2227:
2223:
2219:
2209:
2198:
2195:
2191:
2183:
2179:
2175:
2170:
2167:
2164:
2161:
2157:
2153:
2150:
2147:
2137:
2126:
2123:
2117:
2114:
2109:
2106:
2103:
2098:
2095:
2092:
2089:
2085:
2081:
2060:
2059:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1923:
1920:
1917:
1913:
1909:
1904:
1901:
1898:
1894:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1867:
1864:
1861:
1857:
1853:
1850:
1847:
1844:
1837:
1832:
1829:
1826:
1822:
1818:
1815:
1812:
1808:
1804:
1801:
1797:
1793:
1790:
1787:
1784:
1779:
1776:
1773:
1770:
1766:
1739:
1736:
1731:
1728:
1708:
1688:
1676:
1673:
1672:
1671:
1668:
1665:
1649:
1644:
1640:
1636:
1631:
1627:
1623:
1620:
1617:
1612:
1609:
1606:
1602:
1598:
1593:
1589:
1585:
1565:
1545:
1540:
1536:
1532:
1527:
1523:
1519:
1516:
1513:
1508:
1505:
1502:
1498:
1494:
1489:
1486:
1483:
1479:
1475:
1472:
1469:
1449:
1444:
1440:
1436:
1431:
1427:
1423:
1420:
1417:
1412:
1409:
1406:
1402:
1398:
1393:
1390:
1387:
1383:
1379:
1376:
1373:
1353:
1350:
1349:
1348:
1336:
1333:
1330:
1325:
1322:
1319:
1315:
1311:
1306:
1303:
1300:
1296:
1280:
1279:
1266:
1263:
1260:
1256:
1252:
1249:
1246:
1243:
1238:
1235:
1232:
1228:
1224:
1219:
1216:
1212:
1188:
1177:
1176:
1165:
1162:
1159:
1154:
1151:
1148:
1144:
1140:
1137:
1134:
1131:
1126:
1123:
1120:
1116:
1092:
1072:
1050:
1047:
1044:
1040:
1027:
1024:
1003:
1002:
989:
986:
983:
980:
977:
974:
971:
967:
963:
958:
954:
950:
945:
941:
937:
934:
931:
926:
923:
920:
916:
912:
907:
904:
901:
897:
893:
890:
865:
862:
859:
855:
832:
828:
805:
801:
774:
770:
732:
728:
701:
697:
693:
688:
684:
680:
675:
671:
650:
645:
641:
637:
632:
628:
624:
591:
588:
584:
580:
577:
574:
571:
568:
565:
560:
557:
554:
550:
529:
509:
504:
500:
496:
491:
487:
483:
471:
468:
439:
438:
436:
435:
428:
421:
413:
410:
409:
407:
406:
401:
396:
391:
382:
381:
380:
377:
376:
374:
373:
368:
359:
358:
357:
354:
353:
351:
350:
347:
346:
344:
343:
338:
328:
327:
326:
323:
322:
320:
319:
314:
309:
304:
299:
294:
289:
280:
279:
278:
275:
274:
272:
271:
266:
264:Binary Divider
257:
256:
255:
252:
251:
249:
248:
243:
238:
233:
228:
220:Multiplier (Ă)
219:
218:
217:
214:
213:
211:
210:
205:
200:
192:Subtractor (â)
191:
190:
189:
186:
185:
183:
182:
173:
172:
171:
168:
167:
165:
164:
159:
154:
149:
144:
139:
134:
129:
124:
119:
114:
105:
104:
103:
93:
92:
91:
88:
87:
85:
84:
79:
74:
69:
64:
59:
50:
49:
48:
45:
44:
40:
39:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2883:
2872:
2869:
2868:
2866:
2857:
2854:
2853:
2849:
2842:
2836:
2833:
2828:
2826:0-19-512583-5
2822:
2818:
2814:
2809:
2808:
2799:
2796:
2789:
2781:
2777:
2772:
2769:
2762:
2760:
2744:
2740:
2717:
2713:
2704:
2699:
2683:
2679:
2656:
2652:
2629:
2625:
2615:
2613:
2609:
2605:
2601:
2597:
2589:
2587:
2584:
2577:
2575:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2528:
2524:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2491:
2487:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2454:
2450:
2442:
2439:
2436:
2433:
2430:
2423:
2419:
2411:
2408:
2405:
2402:
2399:
2392:
2384:
2383:
2382:
2363:
2360:
2357:
2354:
2351:
2344:
2331:
2329:
2320:
2311:
2292:
2289:
2283:
2280:
2270:
2269:
2268:
2248:
2245:
2239:
2236:
2231:
2228:
2225:
2221:
2210:
2196:
2193:
2189:
2181:
2177:
2173:
2168:
2165:
2162:
2159:
2155:
2151:
2148:
2145:
2138:
2124:
2121:
2115:
2112:
2104:
2096:
2093:
2090:
2087:
2083:
2079:
2069:
2068:
2067:
2065:
2046:
2043:
2040:
2037:
2034:
2031:
2025:
2022:
2019:
2016:
2010:
2007:
2004:
1998:
1995:
1992:
1986:
1983:
1980:
1977:
1971:
1968:
1965:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1929:
1921:
1918:
1915:
1911:
1907:
1902:
1899:
1896:
1892:
1888:
1882:
1879:
1876:
1870:
1865:
1862:
1859:
1855:
1851:
1845:
1830:
1827:
1824:
1820:
1816:
1813:
1806:
1802:
1799:
1795:
1791:
1785:
1777:
1774:
1771:
1768:
1764:
1756:
1755:
1754:
1737:
1734:
1729:
1726:
1706:
1686:
1674:
1669:
1666:
1663:
1662:
1661:
1642:
1638:
1634:
1629:
1625:
1621:
1618:
1615:
1610:
1607:
1604:
1600:
1596:
1591:
1587:
1563:
1556:are split in
1538:
1534:
1530:
1525:
1521:
1517:
1514:
1511:
1506:
1503:
1500:
1496:
1492:
1487:
1484:
1481:
1477:
1470:
1467:
1442:
1438:
1434:
1429:
1425:
1421:
1418:
1415:
1410:
1407:
1404:
1400:
1396:
1391:
1388:
1385:
1381:
1374:
1371:
1358:
1351:
1334:
1331:
1328:
1323:
1320:
1317:
1313:
1309:
1304:
1301:
1298:
1294:
1286:
1285:
1284:
1264:
1261:
1258:
1254:
1250:
1244:
1236:
1233:
1230:
1226:
1222:
1217:
1214:
1210:
1202:
1201:
1200:
1186:
1160:
1152:
1149:
1146:
1142:
1138:
1132:
1124:
1121:
1118:
1114:
1106:
1105:
1104:
1090:
1070:
1048:
1045:
1042:
1038:
1025:
1019:
1015:
1013:
1008:
984:
981:
978:
975:
972:
965:
961:
956:
952:
948:
943:
939:
935:
932:
929:
924:
921:
918:
914:
910:
905:
902:
899:
895:
891:
888:
881:
880:
879:
863:
860:
857:
853:
830:
826:
803:
799:
790:
772:
768:
759:
755:
751:
746:
730:
726:
717:
699:
695:
691:
686:
682:
678:
673:
669:
643:
639:
635:
630:
626:
609:
605:
589:
586:
582:
578:
575:
572:
566:
558:
555:
552:
548:
527:
502:
498:
494:
489:
485:
469:
467:
465:
464:probabilistic
460:
458:
454:
450:
446:
434:
429:
427:
422:
420:
415:
414:
412:
411:
405:
402:
400:
397:
395:
392:
390:
387:
379:
378:
372:
369:
367:
364:
356:
355:
342:
339:
336:
333:
325:
324:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
277:
276:
270:
267:
265:
262:
254:
253:
247:
244:
242:
239:
237:
234:
232:
229:
227:
224:
216:
215:
209:
206:
204:
201:
199:
196:
188:
187:
181:
178:
170:
169:
163:
160:
158:
155:
153:
150:
148:
145:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
118:
115:
113:
110:
102:
101:
98:
90:
89:
83:
80:
78:
75:
73:
70:
68:
65:
63:
60:
58:
57:Binary number
55:
47:
46:
42:
41:
36:
32:
31:
19:
2835:
2806:
2798:
2775:
2771:
2702:
2700:
2616:
2611:
2607:
2603:
2602:and a 4-bit
2599:
2593:
2585:
2581:
2573:
2335:
2326:
2266:
2063:
2061:
1678:
1363:
1281:
1178:
1029:
1011:
1006:
1004:
788:
757:
753:
749:
747:
715:
614:
473:
463:
461:
448:
444:
442:
241:Wallace tree
161:
2596:full adders
1026:Performance
281:Bitwise ops
258:Divider (Ă·)
2815:. p.
2790:References
1576:blocks of
360:Categories
312:Bit shifts
198:Subtractor
147:Ling adder
122:Full adder
117:Half adder
94:Components
67:Logic gate
2525:∧
2488:∧
2451:∧
2278:⇒
2240:±
2218:⇒
2169:⋅
2163:−
2152:⋅
2032:⋅
1978:⋅
1969:−
1951:⋅
1889:⋅
1880:−
1619:…
1608:−
1515:…
1504:−
1485:−
1419:…
1408:−
1389:−
1143:τ
1115:τ
1046:−
982:−
949:∧
936:∧
933:⋯
930:∧
922:−
911:∧
903:−
692:⊕
583:τ
579:⋅
573:≈
549:τ
106:Adder (+)
2865:Category
451:) is an
383:See also
329:See also
2823:
2316:": -->
1660:bits.
51:Theory
2841:1985.
2763:Notes
1007:group
453:adder
112:Adder
2821:ISBN
2732:and
2703:COUT
2612:COUT
2318:edit
1460:and
748:The
716:true
2817:108
2614:).
2608:CIN
878:.
399:AGU
394:GPU
389:FPU
307:XOR
297:AND
292:NOT
2867::
2819:.
2811:.
2759:.
2705:,
2554:15
2542:12
2517:11
2412:15
604:.
466:.
443:A
302:OR
2829:.
2782:.
2745:3
2741:X
2718:2
2714:X
2684:1
2680:C
2657:1
2653:C
2630:1
2626:C
2604:B
2600:A
2557:]
2551:+
2548:i
2545::
2539:+
2536:i
2533:[
2529:p
2520:]
2514:+
2511:i
2508::
2505:8
2502:+
2499:i
2496:[
2492:p
2483:]
2480:7
2477:+
2474:i
2471::
2468:4
2465:+
2462:i
2459:[
2455:p
2446:]
2443:3
2440:+
2437:i
2434::
2431:i
2428:[
2424:p
2420:=
2415:]
2409:+
2406:i
2403::
2400:i
2397:[
2393:p
2367:]
2364:3
2361:+
2358:i
2355::
2352:i
2349:[
2345:p
2322:]
2293:2
2290:n
2284:=
2281:m
2249:2
2246:n
2237:=
2232:2
2229:,
2226:1
2222:m
2197:0
2194:=
2190:)
2182:2
2178:m
2174:1
2166:n
2160:2
2156:(
2149:D
2146:2
2125:0
2122:=
2116:m
2113:d
2108:)
2105:n
2102:(
2097:A
2094:S
2091:C
2088:F
2084:T
2080:d
2064:n
2047:D
2044:5
2041:+
2038:D
2035:2
2029:)
2026:k
2023:+
2020:m
2017:2
2014:(
2011:=
2008:D
2005:2
2002:)
1999:2
1996:+
1993:m
1990:(
1987:+
1984:D
1981:2
1975:)
1972:1
1966:k
1963:(
1960:+
1957:D
1954:2
1948:m
1945:+
1942:D
1939:3
1936:=
1933:)
1930:m
1927:(
1922:A
1919:R
1916:C
1912:T
1908:+
1903:K
1900:S
1897:C
1893:T
1886:)
1883:2
1877:k
1874:(
1871:+
1866:K
1863:S
1860:C
1856:T
1852:+
1849:)
1846:m
1843:(
1836:]
1831:t
1828:u
1825:o
1821:c
1817::
1814:0
1811:[
1807:A
1803:R
1800:C
1796:T
1792:=
1789:)
1786:n
1783:(
1778:A
1775:S
1772:C
1769:F
1765:T
1738:m
1735:n
1730:=
1727:k
1707:m
1687:n
1648:)
1643:1
1639:m
1635:,
1630:2
1626:m
1622:,
1616:,
1611:1
1605:k
1601:m
1597:,
1592:k
1588:m
1584:(
1564:k
1544:)
1539:0
1535:b
1531:,
1526:1
1522:b
1518:,
1512:,
1507:2
1501:n
1497:b
1493:,
1488:1
1482:n
1478:b
1474:(
1471:=
1468:B
1448:)
1443:0
1439:a
1435:,
1430:1
1426:a
1422:,
1416:,
1411:2
1405:n
1401:a
1397:,
1392:1
1386:n
1382:a
1378:(
1375:=
1372:A
1347:.
1335:D
1332:2
1329:=
1324:X
1321:U
1318:M
1314:T
1310:=
1305:K
1302:S
1299:C
1295:T
1265:X
1262:U
1259:M
1255:T
1251:+
1248:)
1245:m
1242:(
1237:D
1234:N
1231:A
1227:T
1223:=
1218:K
1215:S
1211:T
1187:m
1164:)
1161:n
1158:(
1153:A
1150:R
1147:C
1139:=
1136:)
1133:n
1130:(
1125:A
1122:S
1119:C
1091:n
1071:n
1049:1
1043:n
1039:s
1012:n
988:]
985:1
979:n
976::
973:0
970:[
966:p
962:=
957:0
953:p
944:1
940:p
925:2
919:n
915:p
906:1
900:n
896:p
892:=
889:s
864:t
861:u
858:o
854:c
831:0
827:c
804:n
800:c
789:n
773:i
769:p
758:n
754:n
750:n
731:0
727:c
700:i
696:b
687:i
683:a
679:=
674:i
670:p
649:)
644:i
640:b
636:,
631:i
627:a
623:(
590:A
587:V
576:n
570:)
567:n
564:(
559:A
556:R
553:C
528:n
508:)
503:i
499:b
495:,
490:i
486:a
482:(
432:e
425:t
418:v
20:)
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