366:
637:
Between 2=4,503,599,627,370,496 and 2=9,007,199,254,740,992 the representable numbers are exactly the integers. For the next range, from 2 to 2, everything is multiplied by 2, so the representable numbers are the even ones, etc. Conversely, for the previous range from 2 to 2, the spacing is 0.5, etc.
1913:
data encoding format supports numeric values, and the grammar to which numeric expressions must conform has no limits on the precision or range of the numbers so encoded. However, RFC 8259 advises that, since IEEE 754 binary64 numbers are widely implemented, good interoperability can be achieved by
348:
precision (2 ≈ 1.11 × 10). If a decimal string with at most 15 significant digits is converted to the IEEE 754 double-precision format, giving a normal number, and then converted back to a decimal string with the same number of digits, the final result should match the original
1298:
floating-point standard does not specify endianness. Theoretically, this means that even standard IEEE floating-point data written by one machine might not be readable by another. However, on modern standard computers (i.e., implementing IEEE 754), one may safely assume that the endianness is the
1864:
provides the types SHORT-FLOAT, SINGLE-FLOAT, DOUBLE-FLOAT and LONG-FLOAT. Most implementations provide SINGLE-FLOATs and DOUBLE-FLOATs with the other types appropriate synonyms. Common Lisp provides exceptions for catching floating-point underflows and overflows, and the inexact floating-point
1260:
356:
having an implicit integer bit of value 1 (except for special data, see the exponent encoding below). With the 52 bits of the fraction (F) significand appearing in the memory format, the total precision is therefore 53 bits (approximately 16 decimal digits, 53
648:
The 11 bit width of the exponent allows the representation of numbers between 10 and 10, with full 15–17 decimal digits precision. By compromising precision, the subnormal representation allows even smaller values up to about 5 × 10.
632:
1759:
Using double-precision floating-point variables is usually slower than working with their single precision counterparts. One area of computing where this is a particular issue is parallel code running on GPUs. For example, when using
349:
string. If an IEEE 754 double-precision number is converted to a decimal string with at least 17 significant digits, and then converted back to double-precision representation, the final result must match the original number.
505:
1131:
295:
Double-precision binary floating-point is a commonly used format on PCs, due to its wider range over single-precision floating point, in spite of its performance and bandwidth cost. It is commonly known simply as
1150:
1286:
processors that have mixed-endian floating-point representation for double-precision numbers: each of the two 32-bit words is stored as little-endian, but the most significant word is stored first.
2118:
960:
341:: an exponent value of 1023 represents the actual zero. Exponents range from −1022 to +1023 because exponents of −1023 (all 0s) and +1024 (all 1s) are reserved for special numbers.
1877:
before version 1.2, every implementation had to be IEEE 754 compliant. Version 1.2 allowed implementations to bring extra precision in intermediate computations for platforms like
1830:
type corresponds to double precision. However, on 32-bit x86 with extended precision by default, some compilers may not conform to the C standard or the arithmetic may suffer from
735:
887:
813:
1775:
Additionally, many mathematical functions (e.g., sin, cos, atan2, log, exp and sqrt) need more computations to give accurate double-precision results, and are therefore slower.
1290:
floating point stores little-endian 16-bit words in big-endian order. Because there have been many floating-point formats with no network standard representation for them, the
641:
The spacing as a fraction of the numbers in the range from 2 to 2 is 2. The maximum relative rounding error when rounding a number to the nearest representable one (the
516:
1810:(or when SSE2 is not used, for compatibility purpose) and with extended precision used by default, software may have difficulties to fulfill some requirements.
392:
1865:
exception, as per IEEE 754. No infinities and NaNs are described in the ANSI standard, however, several implementations do provide these as extensions.
1278:
Although many processors use little-endian storage for all types of data (integer, floating point), there are a number of hardware architectures where
2345:
173:
400:
277:
2105:
1070:
2716:
2350:
1802:
Doubles are implemented in many programming languages in different ways such as the following. On processors with only dynamic precision, such as
1768:
platform, calculations with double precision can take, depending on hardware, from 2 to 32 times as long to complete compared to those done using
661:
representation, with the zero offset being 1023; also known as exponent bias in the IEEE 754 standard. Examples of such representations would be:
183:
2335:
1769:
1710:
family processors, use the most significant bit of the significand field to indicate a quiet NaN; this is what is recommended by IEEE 754. The
153:
64:
2323:
2224:
146:
1958:
1255:{\displaystyle (-1)^{\text{sign}}\times 2^{1-1023}\times 0.{\text{fraction}}=(-1)^{\text{sign}}\times 2^{-1022}\times 0.{\text{fraction}}}
2008:
1898:
108:. Before the widespread adoption of IEEE 754-1985, the representation and properties of floating-point data types depended on the
177:
2474:
167:
157:
2062:
2591:
2396:
2328:
2290:
251:
225:
210:
2777:
2772:
2491:
2421:
2269:
1291:
270:
2746:
1984:
1922:
1874:
1299:
same for floating-point numbers as for integers, making the conversion straightforward regardless of data type. Small
2767:
2501:
2369:
2156:
1926:
1743:
Value = 2 × 1.Fraction – Note that
Fraction must not be converted to decimal here = 2 × (15 5555 5555 5555
2679:
2631:
2543:
2521:
2516:
2444:
2310:
1885:
was introduced to enforce strict IEEE 754 computations. Strict floating point has been restored in Java 17.
740:
126:
42:
2553:
2217:
1784:
Integers from −2 to 2 (−9,007,199,254,740,992 to 9,007,199,254,740,992) can be exactly represented.
1294:
standard uses big-endian IEEE 754 as its representation. It may therefore appear strange that the widespread
2706:
2621:
2029:
919:
263:
220:
1282:
numbers are represented in big-endian form while integers are represented in little-endian form. There are
365:
2449:
2305:
2264:
2259:
691:
323:
129:
94:
45:
846:
772:
2439:
2414:
109:
2241:
230:
101:
2711:
2689:
2616:
2469:
2461:
2381:
2210:
1699:
345:
2694:
2674:
2626:
2601:
2386:
2355:
189:
1914:
implementations processing JSON if they expect no more precision or range than binary64 offers.
627:{\displaystyle (-1)^{\text{sign}}\left(1+\sum _{i=1}^{52}b_{52-i}2^{-i}\right)\times 2^{e-1023}}
1951:
2581:
2511:
2486:
2300:
2295:
2004:
330:
The sign bit determines the sign of the number (including when this number is zero, which is
2726:
2611:
2409:
1722:
1707:
1283:
1137:
1010:
2731:
2596:
2548:
2481:
1300:
642:
74:
1787:
Integers between 2 and 2 = 18,014,398,509,481,984 round to a multiple of 2 (even number).
112:
and computer model, and upon decisions made by programming-language implementers. E.g.,
2684:
2506:
2496:
2404:
1952:"Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic"
1822:. Double precision is not required by the standards (except by the optional annex F of
1739:= 1021 Exponent Bias = 1023 (constant value; see above) Fraction = 5 5555 5555 5555
1279:
377:
215:
2761:
2606:
1064:
Except for the above exceptions, the entire double-precision number is described by:
658:
372:
338:
86:
53:
2563:
2538:
2181:
2054:
1819:
2741:
2736:
2586:
2533:
2360:
2135:
1861:
1058:
1002:
353:
331:
320:
89:. IEEE 754 specifies additional floating-point formats, including 32-bit base-2
57:
17:
500:{\displaystyle (-1)^{\text{sign}}(1.b_{51}b_{50}...b_{0})_{2}\times 2^{e-1023}}
2646:
2641:
2558:
2526:
2431:
2374:
1894:
1272:
117:
1126:{\displaystyle (-1)^{\text{sign}}\times 2^{e-1023}\times 1.{\text{fraction}}}
371:
The real value assumed by a given 64-bit double-precision datum with a given
2721:
2699:
2656:
2651:
2318:
2274:
2233:
205:
1976:
1790:
Integers between 2 and 2 = 36,028,797,018,963,968 round to a multiple of 4.
2636:
1882:
1831:
1751:= 0.333333333333333314829616256247390992939472198486328125 ≈ 1/3
1295:
1034:
314:
308:
136:
113:
71:
1518:≙ +2 × 2 = 2 ≈ 4.9406564584124654 × 10 (Min. subnormal positive double)
657:
The double-precision binary floating-point exponent is encoded using an
1842:
1711:
105:
1761:
1303:
using special floating-point formats may be another matter, however.
337:
The exponent field is an 11-bit unsigned integer from 0 to 2047, in
2085:
1681:
0 10000000000 1001001000011111101101010100010001000010110100011000
1661:
0 01111111101 0101010101010101010101010101010101010101010101010101
1644:
0 11111111111 1111111111111111111111111111111111111111111111111111
1631:
0 11111111111 1000000000000000000000000000000000000000000000000001
1618:
0 11111111111 0000000000000000000000000000000000000000000000000001
1605:
1 11111111111 0000000000000000000000000000000000000000000000000000
1592:
0 11111111111 0000000000000000000000000000000000000000000000000000
1579:
1 00000000000 0000000000000000000000000000000000000000000000000000
1566:
0 00000000000 0000000000000000000000000000000000000000000000000000
1549:
0 11111111110 1111111111111111111111111111111111111111111111111111
1536:
0 00000000001 0000000000000000000000000000000000000000000000000000
1523:
0 00000000000 1111111111111111111111111111111111111111111111111111
1510:
0 00000000000 0000000000000000000000000000000000000000000000000001
1485:
0 01111111000 1000000000000000000000000000000000000000000000000000
1464:
0 10000000011 0111000000000000000000000000000000000000000000000000
1443:
0 10000000001 1000000000000000000000000000000000000000000000000000
1422:
0 10000000001 0100000000000000000000000000000000000000000000000000
1405:
0 10000000001 0000000000000000000000000000000000000000000000000000
1384:
0 10000000000 1000000000000000000000000000000000000000000000000000
1367:
1 10000000000 0000000000000000000000000000000000000000000000000000
1354:
0 10000000000 0000000000000000000000000000000000000000000000000000
1341:
0 01111111111 0000000000000000000000000000000000000000000000000010
1328:
0 01111111111 0000000000000000000000000000000000000000000000000001
1315:
0 01111111111 0000000000000000000000000000000000000000000000000000
235:
1544:≙ +2 × 1 ≈ 2.2250738585072014 × 10 (Min. normal positive double)
1531:≙ +2 × (1 − 2) ≈ 2.2250738585072009 × 10 (Max. subnormal double)
2254:
1910:
1901:
shall be done using double-precision floating-point arithmetic.
1807:
1765:
1336:≙ +2 × (1 + 2) ≈ 1.0000000000000002, the smallest number > 1
291:
IEEE 754 double-precision binary floating-point format: binary64
2206:
2136:"The JavaScript Object Notation (JSON) Data Interchange Format"
2055:"Bug 323 – optimized code gives strange floating point results"
2249:
2030:"Nvidia's New Titan V Pushes 110 Teraflops From A Single Chip"
1878:
1823:
1803:
1703:
1695:
1287:
1042:
63:
Double precision may be chosen when the range or precision of
49:
2202:
1845:
provides several integer and real types, and the 64-bit type
1557:≙ +2 × (1 + (1 − 2)) ≈ 1.7976931348623157 × 10 (Max. double)
1826:, covering IEEE 754 arithmetic), but on most systems, the
1702:
and depend on the processor. Most processors, such as the
2086:"JEP 306: Restore Always-Strict Floating-Point Semantics"
1731:
Given the hexadecimal representation 3FD5 5555 5555 5555
1725:, because of the odd number of bits in the significand.
77:, the 64-bit base-2 format is officially referred to as
1639:≙ NaN (qNaN on most processors, such as x86 and ARM)
1626:≙ NaN (sNaN on most processors, such as x86 and ARM)
1153:
1073:
922:
849:
775:
694:
519:
403:
380:
344:
The 53-bit significand precision gives from 15 to 17
2005:"pack – convert a list into a binary representation"
1714:
processors use the bit to indicate a signaling NaN.
2665:
2574:
2460:
2430:
2395:
2283:
2240:
1793:Integers between 2 and 2 round to a multiple of 2.
1254:
1144:= 0) the double-precision number is described by:
1125:
954:
881:
807:
729:
626:
499:
386:
361:(2) ≈ 15.955). The bits are laid out as follows:
2138:. Internet Engineering Task Force. December 2017
2113:(5th ed.). Ecma International. p. 29, §8.5
1755:Execution speed with double-precision arithmetic
2182:"Documentation - The Zig Programming Language"
2218:
271:
93:and, more recently, base-10 representations (
8:
2157:"Data Types - The Rust Programming Language"
1849:, accessible via Fortran's intrinsic module
2225:
2211:
2203:
2107:ECMA-262 ECMAScript Language Specification
278:
264:
122:
1247:
1232:
1219:
1198:
1180:
1167:
1152:
1118:
1100:
1087:
1072:
946:
927:
921:
873:
854:
848:
799:
780:
774:
718:
699:
693:
612:
591:
575:
565:
554:
533:
518:
485:
472:
462:
443:
433:
417:
402:
379:
104:to provide floating-point data types was
52:in computer memory; it represents a wide
1942:
1779:Precision limitations on integer values
1652:≙ NaN (an alternative encoding of NaN)
1061:. All bit patterns are valid encoding.
243:
197:
135:
125:
955:{\displaystyle 2^{2046-1023}=2^{1023}}
116:'s double-precision data type was the
56:of numeric values by using a floating
31:Double-precision floating-point format
1964:from the original on 8 February 2012.
7:
1747:× 2) = 2 × 15 5555 5555 5555
1349:≙ +2 × (1 + 2) ≈ 1.0000000000000004
730:{\displaystyle 2^{1-1023}=2^{-1022}}
300:. The IEEE 754 standard specifies a
1853:, corresponds to double precision.
882:{\displaystyle 2^{1029-1023}=2^{6}}
808:{\displaystyle 2^{1023-1023}=2^{0}}
2065:from the original on 30 April 2018
1818:C and C++ offer a wide variety of
25:
2124:from the original on 2012-03-13.
1950:William Kahan (1 October 1997).
1721:rounds down, instead of up like
1698:are not completely specified in
1669:≙ +2 × (1 + 2 + 2 + ... + 2) ≈ /
1271:This section is an excerpt from
364:
326:: 53 bits (52 explicitly stored)
2011:from the original on 2009-02-18
1987:from the original on 2018-07-03
352:The format is written with the
226:IBM floating-point architecture
1216:
1206:
1164:
1154:
1084:
1074:
1057:is the fractional part of the
530:
520:
469:
423:
414:
404:
1:
2291:Arbitrary-precision or bignum
1735:, Sign = 0 Exponent = 3FD
27:64-bit computer number format
1975:Savard, John J. G. (2018) ,
1897:standard, all arithmetic in
1273:Endianness § Floating point
2794:
1696:Encodings of qNaN and sNaN
1270:
346:significant decimal digits
2632:Strongly typed identifier
1613:≙ −∞ (negative infinity)
1600:≙ +∞ (positive infinity)
1308:Double-precision examples
394:and a 52-bit fraction is
1977:"Floating-Point Formats"
984:have a special meaning:
2707:Parametric polymorphism
1001:is used to represent a
739:(smallest exponent for
221:Microsoft Binary Format
120:floating-point format.
67:would be insufficient.
48:, usually occupying 64
1256:
1127:
956:
883:
809:
731:
628:
570:
501:
388:
95:decimal floating point
1685:= 4009 21FB 5444 2D18
1665:= 3FD5 5555 5555 5555
1648:≙ 7FFF FFFF FFFF FFFF
1635:≙ 7FF8 0000 0000 0001
1622:≙ 7FF0 0000 0000 0001
1609:≙ FFF0 0000 0000 0000
1596:≙ 7FF0 0000 0000 0000
1583:≙ 8000 0000 0000 0000
1570:≙ 0000 0000 0000 0000
1553:≙ 7FEF FFFF FFFF FFFF
1540:≙ 0010 0000 0000 0000
1527:≙ 000F FFFF FFFF FFFF
1514:≙ 0000 0000 0000 0001
1501:= 0.01171875 (3/256)
1489:≙ 3F88 0000 0000 0000
1468:≙ 4037 0000 0000 0000
1447:≙ 4018 0000 0000 0000
1426:≙ 4014 0000 0000 0000
1409:≙ 4010 0000 0000 0000
1388:≙ 4008 0000 0000 0000
1371:≙ C000 0000 0000 0000
1358:≙ 4000 0000 0000 0000
1345:≙ 3FF0 0000 0000 0002
1332:≙ 3FF0 0000 0000 0001
1319:≙ 3FF0 0000 0000 0000
1257:
1128:
1033:is used to represent
957:
884:
810:
732:
629:
550:
502:
389:
110:computer manufacturer
102:programming languages
2778:Floating point types
1937:Notes and references
1893:As specified by the
1151:
1071:
920:
847:
773:
692:
517:
401:
378:
2773:Computer arithmetic
2712:Primitive data type
2617:Recursive data type
2470:Algebraic data type
2346:Quadruple precision
964:(highest exponent)
252:Arbitrary precision
2675:Abstract data type
2356:Extended precision
2315:Reduced precision
1881:. Thus a modifier
1252:
1123:
952:
879:
805:
727:
645:) is therefore 2.
624:
497:
384:
236:G.711 8-bit floats
190:Extended precision
33:(sometimes called
2768:Binary arithmetic
2755:
2754:
2487:Associative array
2351:Octuple precision
2161:doc.rust-lang.org
2084:Darcy, Joseph D.
1693:
1692:
1676:
1675:
1656:
1655:
1561:
1560:
1505:
1504:
1379:
1378:
1250:
1222:
1201:
1170:
1138:subnormal numbers
1121:
1090:
1011:subnormal numbers
968:
967:
653:Exponent encoding
536:
420:
387:{\displaystyle e}
288:
287:
100:One of the first
16:(Redirected from
2785:
2727:Type constructor
2612:Opaque data type
2544:Record or Struct
2341:Double precision
2336:Single precision
2227:
2220:
2213:
2204:
2197:
2196:
2194:
2192:
2178:
2172:
2171:
2169:
2167:
2153:
2147:
2146:
2144:
2143:
2132:
2126:
2125:
2123:
2112:
2102:
2096:
2095:
2093:
2092:
2081:
2075:
2074:
2072:
2070:
2051:
2045:
2044:
2042:
2041:
2026:
2020:
2019:
2017:
2016:
2001:
1995:
1994:
1993:
1992:
1972:
1966:
1965:
1963:
1956:
1947:
1932:
1852:
1848:
1829:
1820:arithmetic types
1770:single precision
1728:In more detail:
1723:single precision
1678:
1677:
1658:
1657:
1563:
1562:
1507:
1506:
1381:
1380:
1312:
1311:
1301:embedded systems
1261:
1259:
1258:
1253:
1251:
1248:
1240:
1239:
1224:
1223:
1220:
1202:
1199:
1191:
1190:
1172:
1171:
1168:
1132:
1130:
1129:
1124:
1122:
1119:
1111:
1110:
1092:
1091:
1088:
1032:
1025:
1000:
993:
983:
976:
961:
959:
958:
953:
951:
950:
938:
937:
911:
904:
888:
886:
885:
880:
878:
877:
865:
864:
838:
831:
814:
812:
811:
806:
804:
803:
791:
790:
764:
757:
736:
734:
733:
728:
726:
725:
710:
709:
683:
676:
664:
663:
633:
631:
630:
625:
623:
622:
604:
600:
599:
598:
586:
585:
569:
564:
538:
537:
534:
506:
504:
503:
498:
496:
495:
477:
476:
467:
466:
448:
447:
438:
437:
422:
421:
418:
393:
391:
390:
385:
368:
280:
273:
266:
123:
91:single precision
81:; it was called
65:single precision
21:
18:Double precision
2793:
2792:
2788:
2787:
2786:
2784:
2783:
2782:
2758:
2757:
2756:
2751:
2732:Type conversion
2667:
2661:
2597:Enumerated type
2570:
2456:
2450:null-terminated
2426:
2391:
2279:
2236:
2231:
2201:
2200:
2190:
2188:
2180:
2179:
2175:
2165:
2163:
2155:
2154:
2150:
2141:
2139:
2134:
2133:
2129:
2121:
2115:The Number Type
2110:
2104:
2103:
2099:
2090:
2088:
2083:
2082:
2078:
2068:
2066:
2053:
2052:
2048:
2039:
2037:
2028:
2027:
2023:
2014:
2012:
2003:
2002:
1998:
1990:
1988:
1974:
1973:
1969:
1961:
1954:
1949:
1948:
1944:
1939:
1930:
1920:
1907:
1891:
1871:
1859:
1851:iso_fortran_env
1850:
1846:
1840:
1832:double rounding
1827:
1816:
1800:
1798:Implementations
1781:
1757:
1752:
1750:
1746:
1742:
1738:
1734:
1720:
1706:family and the
1688:
1684:
1672:
1668:
1664:
1651:
1647:
1638:
1634:
1625:
1621:
1612:
1608:
1599:
1595:
1586:
1582:
1573:
1569:
1556:
1552:
1543:
1539:
1530:
1526:
1517:
1513:
1500:
1496:
1492:
1488:
1479:
1475:
1471:
1467:
1458:
1454:
1450:
1446:
1437:
1433:
1429:
1425:
1416:
1412:
1408:
1399:
1395:
1391:
1387:
1374:
1370:
1361:
1357:
1348:
1344:
1335:
1331:
1322:
1318:
1310:
1305:
1304:
1276:
1268:
1228:
1215:
1176:
1163:
1149:
1148:
1136:In the case of
1096:
1083:
1069:
1068:
1031:
1027:
1024:
1020:
999:
995:
992:
988:
982:
978:
975:
971:
942:
923:
918:
917:
910:
906:
903:
899:
869:
850:
845:
844:
837:
833:
830:
826:
795:
776:
771:
770:
763:
759:
756:
752:
714:
695:
690:
689:
682:
678:
675:
671:
655:
643:machine epsilon
608:
587:
571:
543:
539:
529:
515:
514:
481:
468:
458:
439:
429:
413:
399:
398:
376:
375:
373:biased exponent
360:
293:
284:
231:PMBus Linear-11
28:
23:
22:
15:
12:
11:
5:
2791:
2789:
2781:
2780:
2775:
2770:
2760:
2759:
2753:
2752:
2750:
2749:
2744:
2739:
2734:
2729:
2724:
2719:
2714:
2709:
2704:
2703:
2702:
2692:
2687:
2685:Data structure
2682:
2677:
2671:
2669:
2663:
2662:
2660:
2659:
2654:
2649:
2644:
2639:
2634:
2629:
2624:
2619:
2614:
2609:
2604:
2599:
2594:
2589:
2584:
2578:
2576:
2572:
2571:
2569:
2568:
2567:
2566:
2556:
2551:
2546:
2541:
2536:
2531:
2530:
2529:
2519:
2514:
2509:
2504:
2499:
2494:
2489:
2484:
2479:
2478:
2477:
2466:
2464:
2458:
2457:
2455:
2454:
2453:
2452:
2442:
2436:
2434:
2428:
2427:
2425:
2424:
2419:
2418:
2417:
2412:
2401:
2399:
2393:
2392:
2390:
2389:
2384:
2379:
2378:
2377:
2367:
2366:
2365:
2364:
2363:
2353:
2348:
2343:
2338:
2333:
2332:
2331:
2326:
2324:Half precision
2321:
2311:Floating point
2308:
2303:
2298:
2293:
2287:
2285:
2281:
2280:
2278:
2277:
2272:
2267:
2262:
2257:
2252:
2246:
2244:
2238:
2237:
2232:
2230:
2229:
2222:
2215:
2207:
2199:
2198:
2173:
2148:
2127:
2097:
2076:
2046:
2034:Tom's Hardware
2021:
1996:
1967:
1941:
1940:
1938:
1935:
1919:
1916:
1906:
1903:
1890:
1887:
1870:
1867:
1858:
1855:
1839:
1836:
1815:
1812:
1799:
1796:
1795:
1794:
1791:
1788:
1785:
1780:
1777:
1756:
1753:
1748:
1744:
1740:
1736:
1732:
1730:
1718:
1691:
1690:
1686:
1682:
1674:
1673:
1670:
1666:
1662:
1654:
1653:
1649:
1645:
1641:
1640:
1636:
1632:
1628:
1627:
1623:
1619:
1615:
1614:
1610:
1606:
1602:
1601:
1597:
1593:
1589:
1588:
1584:
1580:
1576:
1575:
1571:
1567:
1559:
1558:
1554:
1550:
1546:
1545:
1541:
1537:
1533:
1532:
1528:
1524:
1520:
1519:
1515:
1511:
1503:
1502:
1498:
1494:
1490:
1486:
1482:
1481:
1477:
1473:
1469:
1465:
1461:
1460:
1456:
1452:
1448:
1444:
1440:
1439:
1435:
1431:
1427:
1423:
1419:
1418:
1414:
1413:≙ +2 × 1 = 100
1410:
1406:
1402:
1401:
1397:
1393:
1389:
1385:
1377:
1376:
1375:≙ −2 × 1 = −2
1372:
1368:
1364:
1363:
1359:
1355:
1351:
1350:
1346:
1342:
1338:
1337:
1333:
1329:
1325:
1324:
1320:
1316:
1309:
1306:
1280:floating-point
1277:
1269:
1267:
1264:
1263:
1262:
1246:
1243:
1238:
1235:
1231:
1227:
1218:
1214:
1211:
1208:
1205:
1197:
1194:
1189:
1186:
1183:
1179:
1175:
1166:
1162:
1159:
1156:
1134:
1133:
1117:
1114:
1109:
1106:
1103:
1099:
1095:
1086:
1082:
1079:
1076:
1051:
1050:
1029:
1022:
1018:
997:
990:
980:
973:
970:The exponents
966:
965:
962:
949:
945:
941:
936:
933:
930:
926:
915:
913:
908:
901:
892:
891:
889:
876:
872:
868:
863:
860:
857:
853:
842:
840:
835:
828:
819:
818:
817:(zero offset)
815:
802:
798:
794:
789:
786:
783:
779:
768:
766:
761:
754:
745:
744:
741:normal numbers
737:
724:
721:
717:
713:
708:
705:
702:
698:
687:
685:
680:
673:
654:
651:
635:
634:
621:
618:
615:
611:
607:
603:
597:
594:
590:
584:
581:
578:
574:
568:
563:
560:
557:
553:
549:
546:
542:
532:
528:
525:
522:
508:
507:
494:
491:
488:
484:
480:
475:
471:
465:
461:
457:
454:
451:
446:
442:
436:
432:
428:
425:
416:
412:
409:
406:
383:
358:
328:
327:
318:
312:
292:
289:
286:
285:
283:
282:
275:
268:
260:
257:
256:
255:
254:
246:
245:
241:
240:
239:
238:
233:
228:
223:
218:
216:TensorFloat-32
213:
208:
200:
199:
195:
194:
193:
192:
187:
180:
170:
160:
150:
140:
139:
133:
132:
127:Floating-point
43:floating-point
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2790:
2779:
2776:
2774:
2771:
2769:
2766:
2765:
2763:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2701:
2698:
2697:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2672:
2670:
2664:
2658:
2655:
2653:
2650:
2648:
2645:
2643:
2640:
2638:
2635:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2610:
2608:
2607:Function type
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2588:
2585:
2583:
2580:
2579:
2577:
2573:
2565:
2562:
2561:
2560:
2557:
2555:
2552:
2550:
2547:
2545:
2542:
2540:
2537:
2535:
2532:
2528:
2525:
2524:
2523:
2520:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
2480:
2476:
2473:
2472:
2471:
2468:
2467:
2465:
2463:
2459:
2451:
2448:
2447:
2446:
2443:
2441:
2438:
2437:
2435:
2433:
2429:
2423:
2420:
2416:
2413:
2411:
2408:
2407:
2406:
2403:
2402:
2400:
2398:
2394:
2388:
2385:
2383:
2380:
2376:
2373:
2372:
2371:
2368:
2362:
2359:
2358:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2330:
2327:
2325:
2322:
2320:
2317:
2316:
2314:
2313:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2288:
2286:
2282:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2256:
2253:
2251:
2248:
2247:
2245:
2243:
2242:Uninterpreted
2239:
2235:
2228:
2223:
2221:
2216:
2214:
2209:
2208:
2205:
2187:
2183:
2177:
2174:
2162:
2158:
2152:
2149:
2137:
2131:
2128:
2120:
2116:
2109:
2108:
2101:
2098:
2087:
2080:
2077:
2064:
2060:
2056:
2050:
2047:
2035:
2031:
2025:
2022:
2010:
2006:
2000:
1997:
1986:
1982:
1978:
1971:
1968:
1960:
1957:. p. 4.
1953:
1946:
1943:
1936:
1934:
1928:
1924:
1917:
1915:
1912:
1904:
1902:
1900:
1896:
1888:
1886:
1884:
1880:
1876:
1868:
1866:
1863:
1856:
1854:
1844:
1837:
1835:
1833:
1825:
1821:
1813:
1811:
1809:
1805:
1797:
1792:
1789:
1786:
1783:
1782:
1778:
1776:
1773:
1771:
1767:
1763:
1754:
1729:
1726:
1724:
1717:By default, /
1715:
1713:
1709:
1705:
1701:
1697:
1680:
1679:
1660:
1659:
1643:
1642:
1630:
1629:
1617:
1616:
1604:
1603:
1591:
1590:
1578:
1577:
1565:
1564:
1548:
1547:
1535:
1534:
1522:
1521:
1509:
1508:
1484:
1483:
1472:≙ +2 × 1.0111
1463:
1462:
1442:
1441:
1421:
1420:
1404:
1403:
1383:
1382:
1366:
1365:
1362:≙ +2 × 1 = 2
1353:
1352:
1340:
1339:
1327:
1326:
1323:≙ +2 × 1 = 1
1314:
1313:
1307:
1302:
1297:
1293:
1289:
1285:
1281:
1274:
1265:
1244:
1241:
1236:
1233:
1229:
1225:
1212:
1209:
1203:
1195:
1192:
1187:
1184:
1181:
1177:
1173:
1160:
1157:
1147:
1146:
1145:
1143:
1139:
1115:
1112:
1107:
1104:
1101:
1097:
1093:
1080:
1077:
1067:
1066:
1065:
1062:
1060:
1056:
1048:
1044:
1040:
1036:
1019:
1016:
1012:
1008:
1004:
987:
986:
985:
963:
947:
943:
939:
934:
931:
928:
924:
916:
914:
897:
894:
893:
890:
874:
870:
866:
861:
858:
855:
851:
843:
841:
824:
821:
820:
816:
800:
796:
792:
787:
784:
781:
777:
769:
767:
750:
747:
746:
742:
738:
722:
719:
715:
711:
706:
703:
700:
696:
688:
686:
669:
666:
665:
662:
660:
659:offset-binary
652:
650:
646:
644:
639:
619:
616:
613:
609:
605:
601:
595:
592:
588:
582:
579:
576:
572:
566:
561:
558:
555:
551:
547:
544:
540:
526:
523:
513:
512:
511:
492:
489:
486:
482:
478:
473:
463:
459:
455:
452:
449:
444:
440:
434:
430:
426:
410:
407:
397:
396:
395:
381:
374:
369:
367:
362:
355:
350:
347:
342:
340:
335:
333:
325:
322:
319:
316:
313:
310:
307:
306:
305:
303:
299:
290:
281:
276:
274:
269:
267:
262:
261:
259:
258:
253:
250:
249:
248:
247:
242:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
203:
202:
201:
196:
191:
188:
185:
181:
179:
176:(binary128),
175:
171:
169:
165:
161:
159:
155:
151:
148:
144:
143:
142:
141:
138:
134:
131:
128:
124:
121:
119:
115:
111:
107:
103:
98:
96:
92:
88:
87:IEEE 754-1985
84:
80:
76:
73:
68:
66:
61:
59:
55:
54:dynamic range
51:
47:
46:number format
44:
40:
36:
32:
19:
2512:Intersection
2340:
2189:. Retrieved
2185:
2176:
2164:. Retrieved
2160:
2151:
2140:. Retrieved
2130:
2114:
2106:
2100:
2089:. Retrieved
2079:
2067:. Retrieved
2058:
2049:
2038:. Retrieved
2036:. 2017-12-08
2033:
2024:
2013:. Retrieved
1999:
1989:, retrieved
1980:
1970:
1945:
1921:
1918:Rust and Zig
1908:
1892:
1872:
1860:
1841:
1817:
1801:
1774:
1758:
1727:
1716:
1694:
1497:= 0.00000011
1141:
1135:
1063:
1054:
1052:
1046:
1038:
1014:
1006:
969:
895:
822:
748:
667:
656:
647:
640:
636:
509:
370:
363:
351:
343:
336:
329:
301:
297:
294:
244:Alternatives
166:(binary64),
163:
156:(binary32),
99:
90:
82:
78:
69:
62:
38:
34:
30:
29:
2742:Type theory
2737:Type system
2587:Bottom type
2534:Option type
2475:generalized
2361:Long double
2306:Fixed point
2186:ziglang.org
2059:gcc.gnu.org
1933:data type.
1862:Common Lisp
1857:Common Lisp
1430:≙ +2 × 1.01
1059:significand
1021:11111111111
1003:signed zero
989:00000000000
900:11111111110
827:10000000101
753:01111111111
672:00000000001
354:significand
339:biased form
321:Significand
304:as having:
186:(binary256)
58:radix point
2762:Categories
2647:Empty type
2642:Type class
2592:Collection
2549:Refinement
2527:metaobject
2375:signedness
2234:Data types
2142:2022-02-01
2091:2021-09-12
2040:2018-11-05
2015:2009-02-04
1991:2018-07-16
1899:JavaScript
1895:ECMAScript
1889:JavaScript
1493:≙ +2 × 1.1
1451:≙ +2 × 1.1
1392:≙ +2 × 1.1
1266:Endianness
178:decimal128
149:(binary16)
118:64-bit MBF
2722:Subtyping
2717:Interface
2700:metaclass
2652:Unit type
2622:Semaphore
2602:Exception
2507:Inductive
2497:Dependent
2462:Composite
2440:Character
2422:Reference
2319:Minifloat
2275:Bit array
2191:10 August
2166:10 August
1981:quadibloc
1929:have the
1814:C and C++
1242:×
1234:−
1226:×
1210:−
1193:×
1185:−
1174:×
1158:−
1113:×
1105:−
1094:×
1078:−
1041:= 0) and
1017:≠ 0); and
1009:= 0) and
932:−
859:−
785:−
720:−
704:−
617:−
606:×
593:−
580:−
552:∑
524:−
490:−
479:×
408:−
324:precision
317:: 11 bits
206:Minifloat
182:256-bit:
174:Quadruple
172:128-bit:
168:decimal64
158:decimal32
2747:Variable
2637:Top type
2502:Equality
2410:physical
2387:Rational
2382:Interval
2329:bfloat16
2119:Archived
2069:30 April
2063:Archived
2009:Archived
1985:archived
1959:Archived
1883:strictfp
1806:without
1700:IEEE 754
1296:IEEE 754
1249:fraction
1200:fraction
1120:fraction
315:Exponent
309:Sign bit
302:binary64
211:bfloat16
162:64-bit:
152:32-bit:
145:16-bit:
137:IEEE 754
114:GW-BASIC
79:binary64
75:standard
72:IEEE 754
2690:Generic
2666:Related
2582:Boolean
2539:Product
2415:virtual
2405:Address
2397:Pointer
2370:Integer
2301:Decimal
2296:Complex
2284:Numeric
1843:Fortran
1838:Fortran
1712:PA-RISC
1476:= 10111
912:=2046:
839:=1029:
765:=1023:
311:: 1 bit
184:Octuple
130:formats
106:Fortran
70:In the
41:) is a
39:float64
2680:Boxing
2668:topics
2627:Stream
2564:tagged
2522:Object
2445:String
1847:real64
1828:double
1762:NVIDIA
1053:where
332:signed
298:double
164:Double
154:Single
83:double
2575:Other
2559:Union
2492:Class
2482:Array
2265:Tryte
2122:(PDF)
2111:(PDF)
1962:(PDF)
1955:(PDF)
1689:≈ pi
1587:≙ −0
1574:≙ +0
1480:= 23
1455:= 110
1434:= 101
1049:≠ 0),
198:Other
2695:Kind
2657:Void
2517:List
2432:Text
2270:Word
2260:Trit
2255:Byte
2193:2024
2168:2024
2071:2018
1925:and
1923:Rust
1911:JSON
1909:The
1905:JSON
1875:Java
1869:Java
1808:SSE2
1766:CUDA
1459:= 6
1438:= 5
1417:= 4
1400:= 3
1396:= 11
1237:1022
1221:sign
1188:1023
1169:sign
1108:1023
1089:sign
1045:(if
1043:NaNs
1037:(if
1013:(if
1005:(if
977:and
948:1023
935:1023
929:2046
862:1023
856:1029
788:1023
782:1023
723:1022
707:1023
684:=1:
620:1023
535:sign
493:1023
419:sign
147:Half
50:bits
35:FP64
2554:Set
2250:Bit
1931:f64
1927:Zig
1879:x87
1873:On
1824:C99
1804:x86
1764:'s
1708:ARM
1704:x86
1292:XDR
1288:VAX
1284:ARM
1028:7ff
996:000
979:7ff
972:000
907:7fe
834:405
760:3ff
679:001
510:or
357:log
334:).
97:).
85:in
37:or
2764::
2184:.
2159:.
2117:.
2061:.
2057:.
2032:.
2007:.
1983:,
1979:,
1834:.
1772:.
1749:16
1745:16
1741:16
1737:16
1733:16
1687:16
1667:16
1650:16
1637:16
1624:16
1611:16
1598:16
1585:16
1572:16
1555:16
1542:16
1529:16
1516:16
1491:16
1470:16
1449:16
1428:16
1411:16
1390:16
1373:16
1360:16
1347:16
1334:16
1321:16
1245:0.
1196:0.
1116:1.
1030:16
998:16
981:16
974:16
909:16
836:16
762:16
743:)
681:16
577:52
567:52
445:50
435:51
427:1.
359:10
60:.
2226:e
2219:t
2212:v
2195:.
2170:.
2145:.
2094:.
2073:.
2043:.
2018:.
1719:3
1683:2
1671:3
1663:2
1646:2
1633:2
1620:2
1607:2
1594:2
1581:2
1568:2
1551:2
1538:2
1525:2
1512:2
1499:2
1495:2
1487:2
1478:2
1474:2
1466:2
1457:2
1453:2
1445:2
1436:2
1432:2
1424:2
1415:2
1407:2
1398:2
1394:2
1386:2
1369:2
1356:2
1343:2
1330:2
1317:2
1275:.
1230:2
1217:)
1213:1
1207:(
1204:=
1182:1
1178:2
1165:)
1161:1
1155:(
1142:e
1140:(
1102:e
1098:2
1085:)
1081:1
1075:(
1055:F
1047:F
1039:F
1035:∞
1026:=
1023:2
1015:F
1007:F
994:=
991:2
944:2
940:=
925:2
905:=
902:2
898:=
896:e
875:6
871:2
867:=
852:2
832:=
829:2
825:=
823:e
801:0
797:2
793:=
778:2
758:=
755:2
751:=
749:e
716:2
712:=
701:1
697:2
677:=
674:2
670:=
668:e
614:e
610:2
602:)
596:i
589:2
583:i
573:b
562:1
559:=
556:i
548:+
545:1
541:(
531:)
527:1
521:(
487:e
483:2
474:2
470:)
464:0
460:b
456:.
453:.
450:.
441:b
431:b
424:(
415:)
411:1
405:(
382:e
279:e
272:t
265:v
20:)
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