1233:
761:
602:
914:
478:. Mathematically, this is equivalent to taking the inverse (additive inverse) of a negative number, and then finding the SLI image for the inverse. Using one bit for the sign enables the representation of negative numbers.
1466:
1019:
439:(multiplicative inverse) of a small magnitude number, and then finding the SLI image for the reciprocal. Using one bit for the reciprocal sign enables the representation of extremely small numbers.
649:
491:
368:
312:
153:
1523:
1389:
1317:
990:
791:
1743:
1758:
188:
637:
218:
68:) arithmetic, and a parallel implementation of it. There has been extensive work on developing the SLI arithmetic algorithms and extending them to
1739:
1786:
1817:
1798:
1228:{\displaystyle s_{X}=\operatorname {sgn}(X),\,r_{X}=\operatorname {sgn}(|X|-|X|^{-1}),\,x=\psi (\max(|X|,|X|^{-1}))=\psi (|X|^{r_{X}})}
1732:
1400:
1622:
1827:
73:
756:{\displaystyle \varphi (x)={\begin{cases}x&{\text{if }}0\leq x<1\\e^{\varphi (x-1)}&{\text{if }}x\geq 1\end{cases}}}
597:{\displaystyle \psi (X)={\begin{cases}X&{\text{if }}0\leq X<1\\1+\psi (\ln X)&{\text{if }}X\geq 1\end{cases}}}
1858:
1557:
57:
The symmetric form of the LI system and its arithmetic operations were presented by
Clenshaw and Peter Turner in 1987.
323:
1617:
264:
1687:(Conference proceedings / The Lancaster Numerical Analysis Summer School 1987). Lecture Notes in Mathematics (LNM).
428:
and we can do away without a third state and use only one bit for the two states −1 and +1) as the reciprocal sign
97:
1477:
1551:
43:
1323:
1248:
936:
909:{\displaystyle \left.{\frac {d\varphi (x)}{dx}}\right|_{x=1}=\left.{\frac {d\varphi (e^{x})}{dx}}\right|_{x=0}.}
1680:
1578:
51:
1631:
1545:
436:
1704:
Clenshaw, Charles
William; Turner, Peter R. (1989-06-23) . "Root Squaring Using Level-Index Arithmetic".
471:
and we can do away without a third state and use only one bit for the two states −1 and +1) as the sign
673:
515:
1706:
1764:
161:
1769:
1587:
920:
610:
197:
1823:
1813:
1719:
1651:
1643:
1778:
1692:
1635:
1596:
47:
1715:
1013:
is the reciprocal sign (multiplicative inversion or not) as in the following equations:
1539:
69:
60:
Michael Anuta, Daniel Lozier, Nicolas
Schabanel and Turner developed the algorithm for
17:
1852:
1809:
639:
onto itself monotonically and so it is invertible on this interval. The inverse, the
447:
379:
1803:
85:
1723:
1683:; Turner, Peter R. (1989). "Level-index arithmetic: An introductory survey".
1647:
1639:
404:
and stores it (after substituting +1 for 0 for the reciprocal sign since for
1655:
1534:
373:
The symmetric form is used to allow negative exponents, if the magnitude of
39:
1842:
443:
1782:
1696:
1601:
1582:
450:(X) and stores it (after substituting +1 for 0 for the sign since for
926:
Formally, we can define the SLI representation for an arbitrary real
84:
The idea of the level-index system is to represent a non-negative
1461:{\displaystyle X=-{\dfrac {1}{1234567}}=-e^{-e^{e^{0.9711308}}}}
919:
The generalized logarithm function is closely related to the
1616:
Clenshaw, Charles
William; Turner, Peter R. (1988-10-01) .
849:
797:
749:
590:
1634:, Institute of Mathematics and Its Applications: 517–526.
1843:"C++ Implementation of Symmetric Level-Index Arithmetic"
446:
may also be used to allow negative numbers. One takes
1802:
Hayes, Brian (2017). "Chapter 8: Higher
Arithmetic".
1480:
1414:
1403:
1326:
1251:
1022:
939:
794:
652:
613:
494:
326:
267:
200:
164:
100:
435:. Mathematically, this is equivalent to taking the
1517:
1460:
1383:
1311:
1227:
984:
908:
755:
631:
596:
362:
306:
212:
182:
147:
1733:"Rechnerarithmetik: Logarithmische Zahlensysteme"
923:used in computer science analysis of algorithms.
1135:
363:{\displaystyle x=\ell +f=3+0.9711308=3.9711308.}
307:{\displaystyle X=1234567=e^{e^{e^{0.9711308}}}}
190:and the process of exponentiation is performed
1805:Foolproof, and Other Mathematical Meditations
148:{\displaystyle X=e^{e^{e^{\cdots ^{e^{f}}}}}}
8:
1518:{\displaystyle x=-\varphi (3.9711308)^{-1}.}
1002:is the sign (additive inversion or not) of
774:has no discontinuities as we go from level
1685:Numerical Analysis and Parallel Processing
1384:{\displaystyle s_{1}=+1,r_{1}=+1,x=1.0.\,}
1312:{\displaystyle s_{0}=+1,r_{0}=+1,x=0.0,\,}
985:{\displaystyle X=s_{X}\varphi (x)^{r_{X}}}
1600:
1503:
1479:
1448:
1443:
1435:
1413:
1402:
1380:
1353:
1331:
1325:
1308:
1278:
1256:
1250:
1214:
1209:
1204:
1195:
1171:
1166:
1157:
1149:
1141:
1122:
1107:
1102:
1093:
1085:
1077:
1059:
1054:
1027:
1021:
974:
969:
950:
938:
891:
867:
851:
832:
799:
793:
732:
709:
681:
668:
651:
612:
573:
523:
510:
493:
325:
294:
289:
284:
266:
199:
163:
131:
126:
121:
116:
111:
99:
1841:sli-c-library (hosted by Google Code),
1763:Hayes, Brian (September–October 2009).
1569:
38:) representation of numbers, and its
7:
1740:Friedrich-Schiller-Universität Jena
1731:Zehendner, Eberhard (Summer 2008).
785:(a very desirable property) since:
481:The mapping function is called the
1618:"The Symmetric Level-Index System"
623:
25:
1623:IMA Journal of Numerical Analysis
641:generalized exponential function
377:is less than 1. One takes
1789:from the original on 2018-07-09
1749:from the original on 2018-07-09
46:operations, were introduced by
1738:(Lecture script) (in German).
1500:
1493:
1471:and its SLI representation is
1222:
1205:
1196:
1192:
1183:
1180:
1167:
1158:
1150:
1142:
1138:
1132:
1116:
1103:
1094:
1086:
1078:
1074:
1048:
1042:
966:
959:
873:
860:
814:
808:
725:
713:
662:
656:
626:
614:
568:
556:
504:
498:
483:generalized logarithm function
1:
1558:Level (logarithmic quantity)
183:{\displaystyle 0\leq f<1}
1679:Clenshaw, Charles William;
1577:Clenshaw, Charles William;
632:{\displaystyle [0,\infty )}
409: = 1 =
213:{\displaystyle \ell \geq 0}
27:Type of computer arithmetic
1875:
1681:Olver, Frank William John
1579:Olver, Frank William John
1552:Logarithmic number system
396:| − |
1765:"The Higher Arithmetic"
1632:Oxford University Press
1583:"Beyond floating point"
76:arithmetic operations.
1640:10.1093/imanum/8.4.517
1546:Tapered floating point
1519:
1462:
1385:
1313:
1229:
986:
910:
766:The density of values
757:
633:
598:
364:
308:
214:
184:
149:
18:Level-index arithmetic
1800:. Also reprinted in:
1520:
1463:
1386:
1314:
1230:
987:
911:
758:
634:
599:
464:and uniquely defines
421:and uniquely defines
365:
309:
215:
185:
150:
62:symmetric level-index
1812:. pp. 113–126.
1478:
1401:
1324:
1249:
1020:
937:
792:
650:
611:
492:
485:. It is defined as
324:
265:
198:
162:
98:
1859:Computer arithmetic
1783:10.1511/2009.80.364
1242:= 0 or 1, we have:
317:so its LI image is
254:is the LI image of
1770:American Scientist
1742:. pp. 21–22.
1697:10.1007/BFb0085718
1588:Journal of the ACM
1515:
1458:
1423:
1381:
1309:
1225:
982:
921:iterated logarithm
906:
753:
748:
629:
594:
589:
360:
304:
210:
180:
145:
1602:10.1145/62.322429
1422:
885:
826:
735:
684:
576:
526:
16:(Redirected from
1866:
1831:
1819:978-0-26203686-3
1797:
1795:
1794:
1757:
1755:
1754:
1748:
1737:
1727:
1700:
1666:
1665:
1663:
1662:
1613:
1607:
1606:
1604:
1574:
1524:
1522:
1521:
1516:
1511:
1510:
1467:
1465:
1464:
1459:
1457:
1456:
1455:
1454:
1453:
1452:
1424:
1415:
1390:
1388:
1387:
1382:
1358:
1357:
1336:
1335:
1318:
1316:
1315:
1310:
1283:
1282:
1261:
1260:
1241:
1234:
1232:
1231:
1226:
1221:
1220:
1219:
1218:
1208:
1199:
1179:
1178:
1170:
1161:
1153:
1145:
1115:
1114:
1106:
1097:
1089:
1081:
1064:
1063:
1032:
1031:
1012:
1005:
1001:
991:
989:
988:
983:
981:
980:
979:
978:
955:
954:
930:(not 0 or 1) as
929:
915:
913:
912:
907:
902:
901:
890:
886:
884:
876:
872:
871:
852:
843:
842:
831:
827:
825:
817:
800:
784:
777:
773:
769:
762:
760:
759:
754:
752:
751:
736:
733:
729:
728:
685:
682:
643:, is defined by
638:
636:
635:
630:
603:
601:
600:
595:
593:
592:
577:
574:
527:
524:
477:
470:
463:
462: = 0.0
457:the LI image is
456:
434:
427:
420:
419: = 1.0
414:the LI image is
413:
403:
401:
395:
387:
376:
369:
367:
366:
361:
313:
311:
310:
305:
303:
302:
301:
300:
299:
298:
258:. For example,
257:
253:
239:
227:
223:
219:
217:
216:
211:
193:
189:
187:
186:
181:
154:
152:
151:
146:
144:
143:
142:
141:
140:
139:
138:
137:
136:
135:
90:
48:Charles Clenshaw
21:
1874:
1873:
1869:
1868:
1867:
1865:
1864:
1863:
1849:
1848:
1838:
1820:
1801:
1792:
1790:
1762:
1752:
1750:
1746:
1735:
1730:
1716:Springer-Verlag
1703:
1678:
1675:
1673:Further reading
1670:
1669:
1660:
1658:
1615:
1614:
1610:
1576:
1575:
1571:
1566:
1531:
1499:
1476:
1475:
1444:
1439:
1431:
1399:
1398:
1349:
1327:
1322:
1321:
1274:
1252:
1247:
1246:
1239:
1210:
1203:
1165:
1101:
1055:
1023:
1018:
1017:
1011:
1007:
1003:
1000:
996:
970:
965:
946:
935:
934:
927:
877:
863:
853:
848:
847:
818:
801:
796:
795:
790:
789:
779:
775:
771:
770:represented by
767:
747:
746:
730:
705:
702:
701:
679:
669:
648:
647:
609:
608:
588:
587:
571:
544:
543:
521:
511:
490:
489:
476:
472:
465:
458:
451:
433:
429:
422:
415:
405:
397:
391:
389:
378:
374:
322:
321:
290:
285:
280:
263:
262:
255:
241:
240:respectively.
237:
225:
221:
196:
195:
191:
160:
159:
127:
122:
117:
112:
107:
96:
95:
88:
82:
28:
23:
22:
15:
12:
11:
5:
1872:
1870:
1862:
1861:
1851:
1850:
1847:
1846:
1837:
1836:External links
1834:
1833:
1832:
1818:
1808:(1 ed.).
1777:(5): 364–368.
1760:
1728:
1701:
1674:
1671:
1668:
1667:
1608:
1595:(2): 319–328.
1568:
1567:
1565:
1562:
1561:
1560:
1555:
1549:
1543:
1540:Floating point
1537:
1530:
1527:
1526:
1525:
1514:
1509:
1506:
1502:
1498:
1495:
1492:
1489:
1486:
1483:
1469:
1468:
1451:
1447:
1442:
1438:
1434:
1430:
1427:
1421:
1418:
1412:
1409:
1406:
1392:
1391:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1356:
1352:
1348:
1345:
1342:
1339:
1334:
1330:
1319:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1281:
1277:
1273:
1270:
1267:
1264:
1259:
1255:
1236:
1235:
1224:
1217:
1213:
1207:
1202:
1198:
1194:
1191:
1188:
1185:
1182:
1177:
1174:
1169:
1164:
1160:
1156:
1152:
1148:
1144:
1140:
1137:
1134:
1131:
1128:
1125:
1121:
1118:
1113:
1110:
1105:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1073:
1070:
1067:
1062:
1058:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1030:
1026:
1009:
998:
993:
992:
977:
973:
968:
964:
961:
958:
953:
949:
945:
942:
917:
916:
905:
900:
897:
894:
889:
883:
880:
875:
870:
866:
862:
859:
856:
850:
846:
841:
838:
835:
830:
824:
821:
816:
813:
810:
807:
804:
798:
783: + 1
764:
763:
750:
745:
742:
739:
731:
727:
724:
721:
718:
715:
712:
708:
704:
703:
700:
697:
694:
691:
688:
680:
678:
675:
674:
672:
667:
664:
661:
658:
655:
628:
625:
622:
619:
616:
605:
604:
591:
586:
583:
580:
572:
570:
567:
564:
561:
558:
555:
552:
549:
546:
545:
542:
539:
536:
533:
530:
522:
520:
517:
516:
514:
509:
506:
503:
500:
497:
474:
469: = 0
455: = 0
431:
371:
370:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
315:
314:
297:
293:
288:
283:
279:
276:
273:
270:
209:
206:
203:
179:
176:
173:
170:
167:
156:
155:
134:
130:
125:
120:
115:
110:
106:
103:
81:
78:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1871:
1860:
1857:
1856:
1854:
1844:
1840:
1839:
1835:
1829:
1825:
1821:
1815:
1811:
1810:The MIT Press
1807:
1806:
1799:
1788:
1784:
1780:
1776:
1772:
1771:
1766:
1761:
1759:
1745:
1741:
1734:
1729:
1725:
1721:
1717:
1713:
1709:
1708:
1702:
1698:
1694:
1690:
1686:
1682:
1677:
1676:
1672:
1657:
1653:
1649:
1645:
1641:
1637:
1633:
1629:
1625:
1624:
1619:
1612:
1609:
1603:
1598:
1594:
1590:
1589:
1584:
1580:
1573:
1570:
1563:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1536:
1533:
1532:
1528:
1512:
1507:
1504:
1496:
1490:
1487:
1484:
1481:
1474:
1473:
1472:
1449:
1445:
1440:
1436:
1432:
1428:
1425:
1419:
1416:
1410:
1407:
1404:
1397:
1396:
1395:
1394:For example,
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1354:
1350:
1346:
1343:
1340:
1337:
1332:
1328:
1320:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1279:
1275:
1271:
1268:
1265:
1262:
1257:
1253:
1245:
1244:
1243:
1215:
1211:
1200:
1189:
1186:
1175:
1172:
1162:
1154:
1146:
1129:
1126:
1123:
1119:
1111:
1108:
1098:
1090:
1082:
1071:
1068:
1065:
1060:
1056:
1051:
1045:
1039:
1036:
1033:
1028:
1024:
1016:
1015:
1014:
975:
971:
962:
956:
951:
947:
943:
940:
933:
932:
931:
924:
922:
903:
898:
895:
892:
887:
881:
878:
868:
864:
857:
854:
844:
839:
836:
833:
828:
822:
819:
811:
805:
802:
788:
787:
786:
782:
743:
740:
737:
722:
719:
716:
710:
706:
698:
695:
692:
689:
686:
676:
670:
665:
659:
653:
646:
645:
644:
642:
620:
617:
584:
581:
578:
565:
562:
559:
553:
550:
547:
540:
537:
534:
531:
528:
518:
512:
507:
501:
495:
488:
487:
486:
484:
479:
468:
461:
454:
449:
445:
440:
438:
425:
418:
412:
408:
400:
394:
385:
381:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
320:
319:
318:
295:
291:
286:
281:
277:
274:
271:
268:
261:
260:
259:
252:
248:
244:
235:
231:
207:
204:
201:
177:
174:
171:
168:
165:
132:
128:
123:
118:
113:
108:
104:
101:
94:
93:
92:
87:
79:
77:
75:
71:
67:
63:
58:
55:
53:
49:
45:
41:
37:
33:
19:
1828:0-26203686-X
1804:
1791:. Retrieved
1774:
1768:
1751:. Retrieved
1711:
1705:
1688:
1684:
1659:. Retrieved
1627:
1621:
1611:
1592:
1586:
1572:
1470:
1393:
1238:whereas for
1237:
994:
925:
918:
780:
765:
640:
607:and it maps
606:
482:
480:
466:
459:
452:
441:
423:
416:
410:
406:
398:
392:
383:
372:
316:
250:
246:
242:
233:
229:
194:times, with
157:
83:
65:
61:
59:
56:
35:
31:
29:
1718:: 171–185.
86:real number
52:Frank Olver
32:level-index
1793:2018-07-09
1753:2018-07-09
1691:: 95–168.
1661:2018-07-10
1564:References
437:reciprocal
390:sgn(|
358:3.9711308.
80:Definition
44:arithmetic
40:algorithms
1724:0010-485X
1707:Computing
1648:0272-4979
1535:Tetration
1505:−
1497:3.9711308
1491:φ
1488:−
1450:0.9711308
1437:−
1429:−
1411:−
1190:ψ
1173:−
1130:ψ
1109:−
1091:−
1072:
1040:
957:φ
858:φ
806:φ
741:≥
720:−
711:φ
690:≤
654:φ
624:∞
582:≥
563:
554:ψ
532:≤
496:ψ
352:0.9711308
334:ℓ
296:0.9711308
205:≥
202:ℓ
169:≤
124:⋯
54:in 1984.
1853:Category
1787:Archived
1744:Archived
1656:42026743
1581:(1984).
1529:See also
734:if
683:if
575:if
525:if
444:sign bit
228:are the
1420:1234567
402:|)
275:1234567
70:complex
1826:
1816:
1722:
1654:
1646:
995:where
158:where
74:vector
1747:(PDF)
1736:(PDF)
1714:(2).
1630:(4).
1554:(LNS)
1548:(TFP)
382:(log(
234:index
230:level
1824:ISBN
1814:ISBN
1720:ISSN
1689:1397
1652:OCLC
1644:ISSN
1542:(FP)
1378:1.0.
1006:and
696:<
538:<
232:and
224:and
175:<
72:and
50:and
42:for
30:The
1779:doi
1693:doi
1636:doi
1597:doi
1303:0.0
1136:max
1069:sgn
1037:sgn
778:to
448:sgn
388:or
380:sgn
236:of
220:.
91:as
66:SLI
1855::
1822:.
1785:.
1775:97
1773:.
1767:.
1712:43
1710:.
1650:.
1642:.
1626:.
1620:.
1593:31
1591:.
1585:.
560:ln
442:A
426:=1
386:))
249:+
245:=
36:LI
1845:.
1830:.
1796:.
1781::
1756:.
1726:.
1699:.
1695::
1664:.
1638::
1628:8
1605:.
1599::
1513:.
1508:1
1501:)
1494:(
1485:=
1482:x
1446:e
1441:e
1433:e
1426:=
1417:1
1408:=
1405:X
1375:=
1372:x
1369:,
1366:1
1363:+
1360:=
1355:1
1351:r
1347:,
1344:1
1341:+
1338:=
1333:1
1329:s
1306:,
1300:=
1297:x
1294:,
1291:1
1288:+
1285:=
1280:0
1276:r
1272:,
1269:1
1266:+
1263:=
1258:0
1254:s
1240:X
1223:)
1216:X
1212:r
1206:|
1201:X
1197:|
1193:(
1187:=
1184:)
1181:)
1176:1
1168:|
1163:X
1159:|
1155:,
1151:|
1147:X
1143:|
1139:(
1133:(
1127:=
1124:x
1120:,
1117:)
1112:1
1104:|
1099:X
1095:|
1087:|
1083:X
1079:|
1075:(
1066:=
1061:X
1057:r
1052:,
1049:)
1046:X
1043:(
1034:=
1029:X
1025:s
1010:X
1008:r
1004:X
999:X
997:s
976:X
972:r
967:)
963:x
960:(
952:X
948:s
944:=
941:X
928:X
904:.
899:0
896:=
893:x
888:|
882:x
879:d
874:)
869:x
865:e
861:(
855:d
845:=
840:1
837:=
834:x
829:|
823:x
820:d
815:)
812:x
809:(
803:d
781:ℓ
776:ℓ
772:x
768:X
744:1
738:x
726:)
723:1
717:x
714:(
707:e
699:1
693:x
687:0
677:x
671:{
666:=
663:)
660:x
657:(
627:)
621:,
618:0
615:[
585:1
579:X
569:)
566:X
557:(
551:+
548:1
541:1
535:X
529:0
519:X
513:{
508:=
505:)
502:X
499:(
475:X
473:s
467:X
460:x
453:X
432:X
430:r
424:X
417:x
411:e
407:X
399:X
393:X
384:X
375:X
355:=
349:+
346:3
343:=
340:f
337:+
331:=
328:x
292:e
287:e
282:e
278:=
272:=
269:X
256:X
251:f
247:ℓ
243:x
238:X
226:f
222:ℓ
208:0
192:ℓ
178:1
172:f
166:0
133:f
129:e
119:e
114:e
109:e
105:=
102:X
89:X
64:(
34:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.