25:
667:, which implies that in rounding to nearest, the rounded result is within 0.5 ulp of the mathematically exact result, using John Harrison's definition; conversely, this property implies that the distance between the rounded result and the mathematically exact result is minimized (but for the halfway cases, it is satisfied by two consecutive floating-point numbers). Reputable
685:
Since the 2010s, advances in floating-point mathematics have allowed correctly rounded functions to be almost as fast in average as these earlier, less accurate functions. A correctly rounded function would also be fully reproducible.
347:
933:
1058:
for a single-precision number contains 24 bits, the first integer that is not exactly representable is 2+1, and this value rounds to 2 in round to nearest, ties to even. Thus the result is equal to 2.
501:
1324:
433:
1877:
Muller, Jean-Michel (2005–11). "On the definition of ulp(x)". INRIA Technical Report 5504. ACM Transactions on
Mathematical Software, Vol. V, No. N, November 2005. Retrieved in 2012-03 from
827:
783:
1266:
871:
266:
540:
1140:
745:
2042:
Goldberg, David (1991–03). "Rounding Error" in "What Every
Computer Scientist Should Know About Floating-Point Arithmetic". Computing Surveys, ACM, March 1991. Retrieved from
616:
376:
642:
1094:
953:
721:
584:
564:
204:
184:
1665:, which represent the positive difference between 1.0 and the next greater representable number in the corresponding type (i.e. the ulp of one).
1815:
Muller, Jean-Michel; Brunie, Nicolas; de
Dinechin, Florent; Jeannerod, Claude-Pierre; Joldes, Mioara; Lefèvre, Vincent; Melquiond, Guillaume;
272:
1944:
1797:
876:
1928:
89:
61:
42:
1674:
1051:
442:
1680:
68:
2059:
1840:
1271:
678:
to between 0.5 and about 1 ulp. Only a few libraries compute them within 0.5 ulp, this problem being complex due to the
75:
1691:
standard library provides access to the next floating-point number in some given direction via the instance properties
644:), assuming that the exponent range is not upper-bounded. These definitions differ only at signed powers of the radix.
381:
108:
788:
750:
57:
1239:
832:
1987:
956:
46:
724:
209:
1426:
513:
1099:
2078:
1688:
1878:
1067:
730:
690:
which theoretically would only produce one incorrect rounding out of 1000 random floating-point inputs.
2043:
436:
82:
1610:
679:
589:
1800:
2083:
675:
147:
35:
2013:
1613:
library provides functions to calculate the next floating-point number in some given direction:
660:
354:
152:
1960:
1768:
1731:
1583:
652:
621:
651:
specification—followed by all modern floating-point hardware—requires that the result of an
664:
1913:
Brisebarre, Nicolas; Hanrot, Guillaume; Muller, Jean-Michel; Zimmermann, Paul (May 2024).
1079:
8:
1054:(binary32) and repeatedly add 1 until the operation does not change the value. Since the
671:
1892:
1787:
938:
706:
668:
569:
549:
189:
169:
126:
1857:
2055:
1836:
1791:
1828:
1777:
1571:
504:
122:
1742:
1737:
1669:
723:
be a positive floating-point number and assume that the active rounding mode is
1816:
142:
1832:
2072:
2014:"FloatingPoint - Swift Standard Library | Apple Developer Documentation"
1725:
1824:
935:, depending on the value of the least significant digit and the exponent of
1764:"What Every Computer Scientist Should Know About Floating-Point Arithmetic"
1782:
1763:
1915:"Correctly-rounded evaluation of a function: why, how, and at what cost?"
1055:
656:
510:
Another definition, suggested by John
Harrison, is slightly different:
1879:
http://ljk.imag.fr/membres/Carine.Lucas/TPScilab/JMMuller/ulp-toms.pdf
1076:
as a floating point value by finding the two double values bracketing
342:{\displaystyle \operatorname {ulp} (x)=b^{\max\{e,\,e_{\min }\}-p+1}}
1201:// p0 is smaller than π, so find next number representable as double
151:(rightmost digit) represents if it is 1. It is used as a measure of
24:
1914:
1720:
648:
2044:
http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html#689
928:{\displaystyle \operatorname {RN} (x+1)=x+\operatorname {ulp} (x)}
1796:(With the addendum "Differences Among IEEE 754 Implementations":
1912:
1814:
655:
operation (addition, subtraction, multiplication, division, and
1988:"ulpOfOne - FloatingPoint | Apple Developer Documentation"
688:
An earlier, intermediate milestone was the 0.501 ulp functions,
1606:
to calculate the floating-point distance between two doubles.
378:
is the minimal exponent of the normal numbers. In particular,
164:
1570:. Similarly to Example 1, the result is 2 because the
1602:
to obtain nearby (and distant) floating-point values, and
496:{\displaystyle \operatorname {ulp} (x)=b^{e_{\min }-p+1}}
1858:"A Machine-Checked Theory of Floating Point Arithmetic"
1385:// same result when using the standard library function
1071:
1319:{\displaystyle \operatorname {ulp} (\pi )=p_{1}-p_{0}}
1231:// -> 3.1415926535897936 (hex: 0x1.921fb54442d19p1)
1198:// -> 3.141592653589793 (hex: 0x1.921fb54442d18p1)
1274:
1242:
1102:
1082:
941:
879:
835:
791:
753:
733:
709:
624:
592:
572:
552:
516:
445:
384:
357:
275:
212:
192:
172:
1937:
49:. Unsourced material may be challenged and removed.
1379:// -> 4.44089209850062616169452667236328125E-16
1318:
1260:
1134:
1088:
947:
927:
865:
821:
777:
739:
715:
636:
610:
578:
558:
534:
495:
427:
370:
341:
260:
198:
178:
1574:floating-point format uses a 53-bit significand.
428:{\displaystyle \operatorname {ulp} (x)=b^{e-p+1}}
2070:
474:
363:
317:
299:
1415:// -> 4.440892098500626E-16 (hex: 0x1.0p-51)
1331:// ulp(Ď€) is the difference between p1 and p0
822:{\displaystyle \operatorname {RN} (x+1)>x}
778:{\displaystyle \operatorname {ulp} (x)\leq 1}
1921:
322:
302:
1429:, also typed at an interactive prompt, is:
1810:
1808:
1261:{\displaystyle \operatorname {ulp} (\pi )}
866:{\displaystyle \operatorname {RN} (x+1)=x}
1781:
1699:. It also provides the instance property
311:
141:) is the spacing between two consecutive
109:Learn how and when to remove this message
1761:
1672:standard library provides the functions
955:. This is demonstrated in the following
542:is the distance between the two closest
261:{\displaystyle b^{e}\leq |x|<b^{e+1}}
1805:
535:{\displaystyle \operatorname {ulp} (x)}
2071:
2054:. Boston: Birkhäuser. pp. 32–37.
2049:
1684:. They were introduced with Java 1.5.
1174:// truncate to a double floating point
1135:{\displaystyle p_{0}<\pi <p_{1}}
2052:Handbook of floating-point arithmetic
1821:Handbook of Floating-Point Arithmetic
959:code typed at an interactive prompt:
1855:
1711:) for Swift's floating-point types.
1707:(which corresponds to C macros like
47:adding citations to reliable sources
18:
1577:
740:{\displaystyle \operatorname {RN} }
13:
1953:
1168:"3.14159265358979323846"
163:The most common definition is: In
14:
2095:
1890:
1728:, part 1 requires an ulp function
1604:boost::math::float_distance(a, b)
1893:"A Logarithm Too Clever by Half"
23:
2036:
2006:
1980:
1962:ISO/IEC 9899:1999 specification
1566:and repeatedly double it until
1382:// (this is precisely 2**(-51))
34:needs additional citations for
1906:
1884:
1871:
1849:
1762:Goldberg, David (March 1991).
1755:
1653:. It also provides the macros
1287:
1281:
1255:
1249:
922:
916:
898:
886:
854:
842:
810:
798:
766:
760:
725:round to nearest, ties to even
529:
523:
458:
452:
397:
391:
288:
282:
235:
227:
16:Floating-point accuracy metric
1:
1748:
611:{\displaystyle a\leq x\leq b}
158:
145:numbers, i.e., the value the
2050:Muller, Jean-Michel (2010).
1562:In this case, we start with
1420:
1061:
698:
7:
1714:
1147:// π with 20 decimal digits
693:
10:
2100:
1600:boost::math::float_advance
1833:10.1007/978-3-319-76526-6
1819:; Torres, Serge (2018) .
1066:The following example in
663:since 2008) be correctly
371:{\displaystyle e_{\min }}
155:in numeric calculations.
1974:The nexttoward functions
1592:boost::math::float_prior
1431:
1328:
1144:
1050:Here we start with 0 in
961:
676:transcendental functions
58:"Unit in the last place"
1970:The nextafter functions
1588:boost::math::float_next
1586:provides the functions
637:{\displaystyle a\neq b}
546:floating-point numbers
148:least significant digit
135:unit of least precision
1703:and the type property
1596:boost::math::nextafter
1320:
1262:
1136:
1090:
949:
929:
867:
823:
779:
741:
717:
638:
612:
580:
560:
536:
497:
429:
372:
343:
262:
200:
180:
131:unit in the last place
1968:. p. 237, §7.12.11.3
1783:10.1145/103162.103163
1769:ACM Computing Surveys
1732:Least significant bit
1321:
1263:
1137:
1091:
950:
930:
868:
824:
780:
742:
718:
680:Table-maker's dilemma
653:elementary arithmetic
639:
613:
581:
561:
537:
498:
430:
373:
344:
263:
201:
181:
1946:Boost float_distance
1425:Another example, in
1272:
1240:
1100:
1089:{\displaystyle \pi }
1080:
939:
877:
833:
789:
751:
731:
707:
622:
590:
570:
550:
514:
443:
382:
355:
273:
210:
190:
170:
43:improve this article
2079:Computer arithmetic
1930:Boost float_advance
1584:Boost C++ libraries
1557:9007199254740996.0
1527:9007199254740992.0
1316:
1258:
1132:
1086:
945:
925:
863:
819:
775:
737:
713:
674:compute the basic
634:
608:
586:(i.e., satisfying
576:
556:
532:
493:
425:
368:
339:
258:
196:
176:
127:numerical analysis
2061:978-0-8176-4704-9
1842:978-3-319-76525-9
1268:is determined as
948:{\displaystyle x}
716:{\displaystyle x}
579:{\displaystyle b}
559:{\displaystyle a}
199:{\displaystyle p}
179:{\displaystyle b}
119:
118:
111:
93:
2091:
2065:
2030:
2029:
2027:
2025:
2010:
2004:
2003:
2001:
1999:
1984:
1978:
1977:
1967:
1957:
1951:
1950:
1941:
1935:
1934:
1925:
1919:
1918:
1910:
1904:
1903:
1901:
1899:
1891:Kahan, William.
1888:
1882:
1875:
1869:
1868:
1866:
1864:
1856:Harrison, John.
1853:
1847:
1846:
1812:
1803:
1795:
1785:
1759:
1710:
1706:
1702:
1698:
1694:
1683:
1677:
1675:Math.ulp(double)
1664:
1660:
1656:
1652:
1648:
1644:
1640:
1636:
1632:
1628:
1624:
1620:
1616:
1605:
1601:
1597:
1593:
1589:
1578:Language support
1572:double-precision
1569:
1565:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1325:
1323:
1322:
1317:
1315:
1314:
1302:
1301:
1267:
1265:
1264:
1259:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1141:
1139:
1138:
1133:
1131:
1130:
1112:
1111:
1095:
1093:
1092:
1087:
1074:
1052:single precision
1045:
1042:
1039:
1036:
1032:
1029:
1026:
1023:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
954:
952:
951:
946:
934:
932:
931:
926:
872:
870:
869:
864:
828:
826:
825:
820:
784:
782:
781:
776:
746:
744:
743:
738:
722:
720:
719:
714:
689:
659:since 1985, and
643:
641:
640:
635:
617:
615:
614:
609:
585:
583:
582:
577:
565:
563:
562:
557:
541:
539:
538:
533:
502:
500:
499:
494:
492:
491:
478:
477:
434:
432:
431:
426:
424:
423:
377:
375:
374:
369:
367:
366:
350:
348:
346:
345:
340:
338:
337:
321:
320:
267:
265:
264:
259:
257:
256:
238:
230:
222:
221:
205:
203:
202:
197:
185:
183:
182:
177:
123:computer science
114:
107:
103:
100:
94:
92:
51:
27:
19:
2099:
2098:
2094:
2093:
2092:
2090:
2089:
2088:
2069:
2068:
2062:
2039:
2034:
2033:
2023:
2021:
2012:
2011:
2007:
1997:
1995:
1986:
1985:
1981:
1972:and §7.12.11.4
1965:
1959:
1958:
1954:
1943:
1942:
1938:
1927:
1926:
1922:
1911:
1907:
1897:
1895:
1889:
1885:
1876:
1872:
1862:
1860:
1854:
1850:
1843:
1817:Revol, Nathalie
1813:
1806:
1760:
1756:
1751:
1743:Round-off error
1738:Machine epsilon
1717:
1708:
1704:
1700:
1696:
1692:
1681:Math.ulp(float)
1679:
1673:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1634:
1630:
1626:
1622:
1618:
1614:
1603:
1599:
1595:
1591:
1587:
1580:
1567:
1563:
1560:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1423:
1418:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1310:
1306:
1297:
1293:
1273:
1270:
1269:
1241:
1238:
1237:
1234:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1126:
1122:
1107:
1103:
1101:
1098:
1097:
1081:
1078:
1077:
1072:
1064:
1048:
1047:
1043:
1040:
1037:
1034:
1030:
1027:
1024:
1021:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
940:
937:
936:
878:
875:
874:
834:
831:
830:
790:
787:
786:
752:
749:
748:
732:
729:
728:
708:
705:
704:
701:
696:
687:
623:
620:
619:
591:
588:
587:
571:
568:
567:
551:
548:
547:
515:
512:
511:
473:
469:
468:
464:
444:
441:
440:
407:
403:
383:
380:
379:
362:
358:
356:
353:
352:
316:
312:
298:
294:
274:
271:
270:
269:
246:
242:
234:
226:
217:
213:
211:
208:
207:
191:
188:
187:
186:with precision
171:
168:
167:
161:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
2097:
2087:
2086:
2084:Floating point
2081:
2067:
2066:
2060:
2047:
2038:
2035:
2032:
2031:
2005:
1979:
1952:
1936:
1920:
1905:
1883:
1870:
1848:
1841:
1823:(2 ed.).
1804:
1753:
1752:
1750:
1747:
1746:
1745:
1740:
1735:
1729:
1723:
1716:
1713:
1651:<math.h>
1649:, declared in
1579:
1576:
1432:
1422:
1419:
1329:
1313:
1309:
1305:
1300:
1296:
1292:
1289:
1286:
1283:
1280:
1277:
1257:
1254:
1251:
1248:
1245:
1145:
1129:
1125:
1121:
1118:
1115:
1110:
1106:
1085:
1063:
1060:
962:
944:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
862:
859:
856:
853:
850:
847:
844:
841:
838:
818:
815:
812:
809:
806:
803:
800:
797:
794:
774:
771:
768:
765:
762:
759:
756:
736:
712:
700:
697:
695:
692:
633:
630:
627:
607:
604:
601:
598:
595:
575:
555:
531:
528:
525:
522:
519:
490:
487:
484:
481:
476:
472:
467:
463:
460:
457:
454:
451:
448:
437:normal numbers
422:
419:
416:
413:
410:
406:
402:
399:
396:
393:
390:
387:
365:
361:
336:
333:
330:
327:
324:
319:
315:
310:
307:
304:
301:
297:
293:
290:
287:
284:
281:
278:
255:
252:
249:
245:
241:
237:
233:
229:
225:
220:
216:
195:
175:
160:
157:
143:floating-point
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
2096:
2085:
2082:
2080:
2077:
2076:
2074:
2063:
2057:
2053:
2048:
2045:
2041:
2040:
2019:
2015:
2009:
1993:
1989:
1983:
1975:
1971:
1964:
1963:
1956:
1948:
1947:
1940:
1932:
1931:
1924:
1916:
1909:
1894:
1887:
1880:
1874:
1859:
1852:
1844:
1838:
1834:
1830:
1826:
1822:
1818:
1811:
1809:
1801:
1798:
1793:
1789:
1784:
1779:
1775:
1771:
1770:
1765:
1758:
1754:
1744:
1741:
1739:
1736:
1733:
1730:
1727:
1726:ISO/IEC 10967
1724:
1722:
1719:
1718:
1712:
1690:
1685:
1682:
1676:
1671:
1666:
1612:
1607:
1585:
1575:
1573:
1539:>>>
1530:>>>
1521:>>>
1458:>>>
1446:>>>
1434:>>>
1430:
1428:
1327:
1311:
1307:
1303:
1298:
1294:
1290:
1284:
1278:
1275:
1252:
1246:
1243:
1143:
1127:
1123:
1119:
1116:
1113:
1108:
1104:
1083:
1075:
1070:approximates
1069:
1059:
1057:
1053:
960:
958:
942:
919:
913:
910:
907:
904:
901:
895:
892:
889:
883:
880:
860:
857:
851:
848:
845:
839:
836:
829:. Otherwise,
816:
813:
807:
804:
801:
795:
792:
772:
769:
763:
757:
754:
734:
726:
710:
691:
683:
681:
677:
673:
670:
666:
662:
658:
654:
650:
645:
631:
628:
625:
605:
602:
599:
596:
593:
573:
553:
545:
526:
520:
517:
508:
506:
488:
485:
482:
479:
470:
465:
461:
455:
449:
446:
438:
420:
417:
414:
411:
408:
404:
400:
394:
388:
385:
359:
334:
331:
328:
325:
313:
308:
305:
295:
291:
285:
279:
276:
253:
250:
247:
243:
239:
231:
223:
218:
214:
193:
173:
166:
156:
154:
150:
149:
144:
140:
136:
132:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
2051:
2037:Bibliography
2022:. Retrieved
2017:
2008:
1996:. Retrieved
1991:
1982:
1973:
1969:
1961:
1955:
1945:
1939:
1929:
1923:
1908:
1896:. Retrieved
1886:
1873:
1861:. Retrieved
1851:
1820:
1773:
1767:
1757:
1686:
1667:
1663:LDBL_EPSILON
1608:
1581:
1561:
1424:
1235:
1065:
1049:
1046:1.6777216e7
1033:1.6777215e7
1020:1.6777216e7
702:
684:
646:
543:
509:
162:
146:
138:
134:
130:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2020:. Apple Inc
1994:. Apple Inc
1898:14 November
1776:(1): 5–48.
1709:FLT_EPSILON
1659:DBL_EPSILON
1655:FLT_EPSILON
1647:long double
1643:nexttowardl
1619:nexttowardf
1192:doubleValue
1056:significand
657:square root
2073:Categories
1825:Birkhäuser
1749:References
1639:nextafterl
1631:nexttoward
1615:nextafterf
1611:C language
1367:BigDecimal
1346:BigDecimal
1334:BigDecimal
1162:BigDecimal
1150:BigDecimal
727:, denoted
544:straddling
505:subnormals
159:Definition
99:March 2015
69:newspapers
2024:18 August
2018:Apple Inc
1998:18 August
1992:Apple Inc
1792:222008826
1627:nextafter
1568:x = x + 1
1421:Example 3
1304:−
1285:π
1279:
1253:π
1247:
1117:π
1084:π
1062:Example 2
914:
884:
840:
796:
770:≤
758:
699:Example 1
672:libraries
629:≠
603:≤
597:≤
521:
480:−
450:
412:−
389:
326:−
280:
224:≤
1721:IEEE 754
1715:See also
1705:ulpOfOne
1693:nextDown
1358:subtract
694:Examples
649:IEEE 754
153:accuracy
1863:17 July
1391:ulpMath
957:Haskell
785:, then
669:numeric
665:rounded
268:, then
83:scholar
2058:
1839:
1790:
1697:nextUp
1635:double
1427:Python
1388:double
1219:nextUp
1204:double
1177:double
439:, and
351:where
85:
78:
71:
64:
56:
1966:(PDF)
1788:S2CID
1734:(LSB)
1689:Swift
1623:float
1564:x = 1
1461:while
1236:Then
1035:>
1022:>
1018:Float
979:->
967:until
964:>
747:. If
206:, if
165:radix
90:JSTOR
76:books
2056:ISBN
2026:2019
2000:2019
1900:2008
1865:2013
1837:ISBN
1695:and
1687:The
1678:and
1670:Java
1668:The
1645:for
1641:and
1633:for
1629:and
1621:for
1617:and
1609:The
1598:and
1582:The
1518:...
1500:...
1482:...
1397:Math
1213:Math
1120:<
1114:<
1068:Java
814:>
703:Let
647:The
618:and
566:and
503:for
435:for
240:<
125:and
62:news
1829:doi
1778:doi
1701:ulp
1443:1.0
1403:ulp
1376:));
1364:new
1343:new
1337:ulp
1276:ulp
1244:ulp
1195:();
1159:new
911:ulp
873:or
755:ulp
661:FMA
518:ulp
475:min
447:ulp
386:ulp
364:min
318:min
300:max
277:ulp
139:ulp
133:or
121:In
45:by
2075::
2016:.
1990:.
1835:.
1827:.
1807:^
1802:).
1799:,
1786:.
1774:23
1772:.
1766:.
1661:,
1657:,
1637:,
1625:,
1594:,
1590:,
1536:53
1467:!=
1412:);
1409:p0
1373:p0
1355:).
1352:p1
1326:.
1228:);
1225:p0
1207:p1
1180:p0
1171:);
1142:.
1096::
1038:it
1025:it
1015:::
985:==
881:RN
837:RN
793:RN
735:RN
682:.
507:.
129:,
2064:.
2046:.
2028:.
2002:.
1976:.
1949:.
1933:.
1917:.
1902:.
1881:.
1867:.
1845:.
1831::
1794:.
1780::
1554:1
1551:+
1548:2
1545:+
1542:x
1533:p
1524:x
1515:1
1512:+
1509:p
1506:=
1503:p
1497:2
1494:*
1491:x
1488:=
1485:x
1479::
1476:1
1473:+
1470:x
1464:x
1455:0
1452:=
1449:p
1440:=
1437:x
1406:(
1400:.
1394:=
1370:(
1361:(
1349:(
1340:=
1312:0
1308:p
1299:1
1295:p
1291:=
1288:)
1282:(
1256:)
1250:(
1222:(
1216:.
1210:=
1189:.
1186:Ď€
1183:=
1165:(
1156:=
1153:Ď€
1128:1
1124:p
1109:0
1105:p
1073:Ď€
1044:1
1041:+
1031:1
1028:-
1012:0
1009:)
1006:1
1003:+
1000:(
997:)
994:1
991:+
988:x
982:x
976:x
973:\
970:(
943:x
923:)
920:x
917:(
908:+
905:x
902:=
899:)
896:1
893:+
890:x
887:(
861:x
858:=
855:)
852:1
849:+
846:x
843:(
817:x
811:)
808:1
805:+
802:x
799:(
773:1
767:)
764:x
761:(
711:x
632:b
626:a
606:b
600:x
594:a
574:b
554:a
530:)
527:x
524:(
489:1
486:+
483:p
471:e
466:b
462:=
459:)
456:x
453:(
421:1
418:+
415:p
409:e
405:b
401:=
398:)
395:x
392:(
360:e
349:,
335:1
332:+
329:p
323:}
314:e
309:,
306:e
303:{
296:b
292:=
289:)
286:x
283:(
254:1
251:+
248:e
244:b
236:|
232:x
228:|
219:e
215:b
194:p
174:b
137:(
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.