188:
180:
510:
1733:
1393:
Rounding errors inherent to floating point calculations may limit the use of comparisons for checking the exact equality of results. Choosing an acceptable range is a complex topic. A common technique is to use a comparison epsilon value to perform approximate comparisons. Depending on how lenient
1759:
to assess it sided against the dissenters. DEC had the study done in order to demonstrate that gradual underflow was a bad idea, but the study concluded the opposite, and DEC gave in. In 1985, the standard was ratified, but it had already become the de facto standard a year earlier, implemented by
1701:
The work within Intel worried other vendors, who set up a standardization effort to ensure a "level playing field". Kahan attended the second IEEE 754 standards working group meeting, held in
November 1977. He subsequently received permission from Intel to put forward a draft proposal based on his
1697:
had just come out in late 1977, and its floating point was highly regarded. However, seeking to market their chip to the broadest possible market, Intel wanted the best floating point possible, and Kahan went on to draw up specifications. Kahan initially recommended that the floating point base be
1389:
integers: If the sign bits differ, the negative number precedes the positive number, so 2's complement gives the correct result (except that negative zero and positive zero should be considered equal). If both values are positive, the 2's complement comparison again gives the correct result.
346:
integers. If two floating-point numbers have different signs, the sign-and-magnitude comparison also works with biased exponents. However, if both biased-exponent floating-point numbers are negative, then the ordering must be reversed. If the exponent were represented as, say, a 2's-complement
222:, where numbers are written to have a single non-zero digit to the left of the decimal point, we rewrite this number so it has a single 1 bit to the left of the "binary point". We simply multiply by the appropriate power of 2 to compensate for shifting the bits left by three positions:
1010:
The standard also recommends extended format(s) to be used to perform internal computations at a higher precision than that required for the final result, to minimise round-off errors: the standard only specifies minimum precision and exponent requirements for such formats. The
417:
A denormal number is represented with a biased exponent of all 0 bits, which represents an exponent of −126 in single precision (not −127), or −1022 in double precision (not −1023). In contrast, the smallest biased exponent representing a normal number is 1 (see
1709:
As an 8-bit exponent was not wide enough for some operations desired for double-precision numbers, e.g. to store the product of two 32-bit numbers, both Kahan's proposal and a counter-proposal by DEC therefore used 11 bits, like the time-tested
775:
As an example, 16,777,217 cannot be encoded as a 32-bit float as it will be rounded to 16,777,216. However, all integers within the representable range that are a power of 2 can be stored in a 32-bit float without rounding.
545:
Precision is defined as the minimum difference between two successive mantissa representations; thus it is a function only in the mantissa; while the gap is defined as the difference between two successive numbers.
1702:
work for their coprocessor; he was allowed to explain details of the format and its rationale, but not anything related to Intel's implementation architecture. The draft was co-written with Jerome Coonen and
1747:
chip was already released, but DEC remained opposed, to denormal numbers in particular, because of performance concerns and since it would give DEC a competitive advantage to standardise on DEC's format.
1512:, with a single unsigned infinity, by providing programmers with a mode selection option. In the interest of reducing the complexity of the final standard, the projective mode was dropped, however. The
1718:
from 1965. Kahan's proposal also provided for infinities, which are useful when dealing with division-by-zero conditions; not-a-number values, which are useful when dealing with invalid operations;
350:
The leading 1 bit is omitted since all numbers except zero start with a leading 1; the leading 1 is implicit and doesn't actually need to be stored which gives an extra bit of precision for "free."
3092:
1729:
Even before it was approved, the draft standard had been implemented by a number of manufacturers. The Intel 8087, which was announced in 1980, was the first chip to implement the draft standard.
1608:−x returns x with the sign reversed. This is different from 0−x in some cases, notably when x is 0. So −(0) is −0, but the sign of 0−0 depends on the rounding mode.
402:
occurs, IEEE 754 includes the ability to represent fractions smaller than are possible in the normalized representation, by making the implicit leading digit a 0. Such numbers are called
279:
1469:– rounds to the nearest value; if the number falls midway it is rounded to the nearest value with an even (zero) least significant bit, which means it is rounded up 50% of the time (in
2190:
1406:
for double-precision. Another common technique is ULP, which checks what the difference is in the last place digits, effectively checking how many steps away the two values are.
2735:
561:
The positive and negative numbers closest to zero (represented by the denormalized value with all 0s in the exponent field and the binary value 1 in the fraction field) are
791:
The positive and negative numbers closest to zero (represented by the denormalized value with all 0s in the Exp field and the binary value 1 in the
Fraction field) are
2332:
1677:
John Palmer, who managed the project, believed the effort should be backed by a standard unifying floating point operations across disparate processors. He contacted
4177:
1373:
with its associated order, except for the two combinations of bits for negative zero and positive zero, which sometimes require special attention (see below). The
585:
The finite positive and finite negative numbers furthest from zero (represented by the value with 254 in the exponent field and all 1s in the fraction field) are
815:
The finite positive and finite negative numbers furthest from zero (represented by the value with 2046 in the Exp field and all 1s in the fraction field) are
342:
integers. Using a biased exponent, the lesser of two positive floating-point numbers will come out "less than" the greater following the same ordering as for
3299:
4067:
573:
The positive and negative normalized numbers closest to zero (represented with the binary value 1 in the exponent field and 0 in the fraction field) are
48:. During its 23 years, it was the most widely used format for floating-point computation. It was implemented in software, in the form of floating-point
2832:
1674:. Intel hoped to be able to sell a chip containing good implementations of all the operations found in the widely varying maths software libraries.
1509:
803:
The positive and negative normalized numbers closest to zero (represented with the binary value 1 in the Exp field and 0 in the fraction field) are
2599:
2552:
2534:
513:
Relative precision of single (binary32) and double precision (binary64) numbers, compared with decimal representations using a fixed number of
2789:
2779:: A compendium of non-intuitive behaviours of floating-point on popular architectures, with implications for program verification and testing.
2344:
1914:
2191:"Intel and Floating-Point - Updating One of the Industry's Most Successful Standards - The Technology Vision for the Floating-Point Standard"
2485:
1505:
1370:
2603:
2201:
4103:
4093:
2942:
3368:
2887:
1897:
1840:
473:
are invalid, such as taking the square root of a negative number. The act of reaching an invalid result is called a floating-point
2015:
2387:
3067:
2033:
3349:
2258:
1564:
Comparison operations. Besides the more obvious results, IEEE 754 defines that −∞ = −∞, +∞ = +∞ and
53:
2699:
2575:"Molecular Expressions: Science, Optics & You - Olympus MIC-D: Integrated Circuit Gallery - Intel 8087 Math Coprocessor"
1508:, with separate positive and negative infinities. During drafting, there was a proposal for the standard to incorporate the
3324:
1756:
1690:
4162:
2825:
92:
430:
The biased-exponent field is filled with all 1 bits to indicate either infinity or an invalid result of a computation.
228:
3733:
3294:
2997:
2912:
2892:
2375:
1417:
338:
to the exponent so that numbers can in many cases be compared conveniently by the same hardware that compares signed
4136:
2865:
2841:
2556:
1428:
distinguish them (officially starting with Java version 1.1 but actually with 1.1.1), as do the comparison methods
183:
The number 0.15625 represented as a single-precision IEEE 754-1985 floating-point number. See text for explanation.
3703:
2902:
470:
411:
2574:
3137:
1979:
1682:
2070:
1997:
4172:
4167:
4141:
3002:
2992:
2972:
2818:
1625:
187:
179:
3314:
2875:
2794:
2113:
David
Stevenson (March 1981). "IEEE Task P754: A proposed standard for binary floating-point arithmetic".
1780:
1390:
Otherwise (two negative numbers), the correct FP ordering is the opposite of the 2's complement ordering.
530:
1601:. This is one of the few operations which operates on a NaN in a way resembling arithmetic. The function
291:
As illustrated in the pictures, the three fields in the IEEE 754 representation of this number are:
4108:
3231:
2967:
2663:
1537:
76:
1409:
Although negative zero and positive zero are generally considered equal for comparison purposes, some
3374:
3304:
3122:
3052:
2443:
2412:
1456:
The IEEE standard has four different rounding modes; the first is the default; the others are called
1410:
399:
4017:
4012:
4007:
4002:
3997:
3992:
3987:
2538:
1923:
1889:
1558:
1413:
219:
61:
49:
2233:
3256:
3132:
2770:
2744:
2680:
2628:
2416:
2165:
2130:
2095:
1799:
Precision: The number of decimal digits precision is calculated via number_of_mantissa_bits * Log
1378:
1015:
514:
438:
407:
343:
65:
29:
79:, providing definitions for four levels of precision, of which the two most commonly used are:
1955:
398:
meaning that the implicit leading binary digit is a 1. To reduce the loss of precision when an
3805:
3800:
3780:
3764:
3758:
3753:
3748:
3743:
3738:
3728:
3723:
3713:
3708:
3344:
3174:
2762:
2710:
2636:
2507:
2481:
2475:
2379:
1893:
1836:
1752:
1458:
1386:
509:
339:
3933:
3674:
3669:
3659:
3654:
3649:
3644:
3639:
3634:
3624:
3619:
3614:
3609:
3599:
3594:
3589:
3584:
3569:
3564:
3559:
3554:
3549:
2365:
Thornton, James E. (1970). Written at
Advanced Design Laboratory, Control Data Corporation.
3698:
3364:
2754:
2672:
2620:
2305:
2157:
2122:
2085:
1881:
1828:
1548:
784:
554:
403:
164:
156:
3127:
1719:
1703:
1698:
decimal but the hardware design of the coprocessor was too far along to make that change.
1686:
1150:
410:
as a normalized number, but they enable a gradual loss of precision when the result of an
1882:
3982:
3112:
3107:
3027:
2977:
2285:
1561:. For undirected rounding when halfway between two integers the even integer is chosen.
414:
is not exactly zero but is too close to zero to be represented by a normalized number.
33:
1726:, which can help avoid overflow and underflow when taking the reciprocal of a number.
1369:
Every possible bit combination is either a NaN or a number with a unique value in the
347:
number, comparison to see which of two numbers is greater would not be as convenient.
4156:
4032:
4027:
4022:
3972:
3967:
3962:
3952:
3928:
3904:
3892:
3881:
3870:
3858:
3853:
3848:
3843:
3830:
3819:
3309:
3289:
3142:
3117:
3047:
2937:
2882:
2774:
2684:
2611:
2366:
2263:
1951:
1723:
1678:
1470:
1078:
335:
200:
152:
45:
41:
2632:
2169:
2134:
4072:
4062:
3810:
3795:
3790:
3785:
3775:
3718:
3359:
3354:
3339:
3334:
3329:
3279:
2730:
2447:
2099:
1420:
Language
Specification, comparison and equality operators treat them as equal, but
1333:
478:
2805:
1706:, and was initially known as the "Kahan-Coonen-Stone proposal" or "K-C-S format".
1377:
has the special property that, excluding NaNs, any two numbers can be compared as
2718:
3944:
3684:
3438:
3319:
3284:
3274:
3251:
3246:
3241:
3236:
3219:
3214:
3204:
1671:
1542:
1517:
1018:
is the most commonly implemented extended format that meets these requirements.
370:
366:
2148:
William Kahan and John Palmer (1979). "On a proposed floating-point standard".
1832:
4057:
4052:
3384:
3266:
3199:
3194:
3189:
3184:
3179:
3169:
2799:
1744:
1736:
1694:
1513:
1382:
69:
2766:
2714:
2655:
2477:
Numerical
Analysis and Parallel Processing: Lectures given at The Lancaster …
1551:, it can be negative for two positive numbers. It returns the exact value of
4118:
4098:
3956:
3209:
3102:
3097:
3082:
3072:
3062:
3042:
3037:
3022:
3012:
3007:
2987:
2982:
2962:
2957:
2952:
2947:
2932:
2897:
2795:
Coprocessor.info: x87 FPU pictures, development and manufacturer information
2758:
2340:
2161:
2126:
1774:
37:
2656:"What Every Computer Scientist Should Know About Floating-Point Arithmetic"
2624:
828:
Some example range and gap values for given exponents in double precision:
598:
Some example range and gap values for given exponents in single precision:
2676:
2090:
1722:, which help mitigate problems caused by underflow; and a better balanced
477:
An exceptional result is represented by a special code called a NaN, for "
195:
Floating-point numbers in IEEE 754 format consist of three fields: a
3389:
3161:
3152:
2870:
2860:
2855:
1769:
1715:
1711:
203:, and a fraction. The following example illustrates the meaning of each.
196:
168:
148:
20:
2560:
1732:
1640:, which turns out to have different behavior than NOT(x = y) due to NaN.
68:
to implement the draft of what was to become IEEE 754-1985 was the
4113:
3909:
3875:
3824:
3769:
3664:
3629:
3604:
3579:
3574:
3544:
3539:
3534:
3528:
3522:
3517:
3512:
3507:
3501:
3495:
3490:
3485:
3480:
3474:
3468:
3463:
3458:
3453:
3398:
3032:
3017:
2051:
1856:
1803:(2). Thus ~7.2 and ~15.9 for single and double precision respectively.
1657:
returns the next representable value from x in the direction towards y
284:
Now we can read off the fraction and the exponent: the fraction is .01
3976:
3448:
3443:
3433:
3428:
3423:
3418:
3413:
3408:
3403:
3077:
2907:
1689:'s calculators. Kahan suggested that Intel use the floating point of
1026:
Here are some examples of single-precision IEEE 754 representations:
1777:
for simple examples of properties of IEEE 754 floating point numbers
500:= anything except all 0 bits (since all 0 bits represents infinity).
147:
The standard also defines representations for positive and negative
2749:
1646:
is true when "x is unordered with y", i.e., either x or y is a NaN.
1628:
for "x is a finite value", equivalent to −Inf < x < Inf
3834:
3689:
3224:
3087:
2927:
2197:
1731:
1667:
508:
215:
186:
178:
2810:
3057:
2922:
2917:
2383:
1520:
floating point co-processors both support this projective mode.
1416:
and similar constructs treat them as distinct. According to the
2814:
2333:"IEEE vs. Microsoft Binary Format; Rounding Issues (Complete)"
1012:
160:
57:
298:= 0, because the number is positive. (1 indicates negative.)
2300:
2298:
2296:
2294:
2071:"Handling Floating-Point Exceptions in Numeric Programs"
167:
to represent numbers smaller than shown above, and four
2731:"The pitfalls of verifying floating-point computations"
214:(that is, 1/8 + 1/32). (Subscripts indicate the number
40:, officially adopted in 1985 and superseded in 2008 by
2806:
IEEE754 (Single and Double precision) Online
Converter
2417:"Why do we need a floating-point arithmetic standard?"
394:
The number representations described above are called
2736:
2448:"How Java's Floating-Point Hurts Everyone Everywhere"
2078:
1477:
to distinguish it from another round-to-nearest mode)
231:
16:
First edition of the IEEE 754 floating-point standard
1913:
Hossam A. H. Fahmy; Shlomo Waser; Michael J. Flynn,
1547:
Floating point remainder. This is not like a normal
1495:– directed rounding towards negative infinity.
481:". All NaNs in IEEE 754-1985 have this format:
312:, so in this example the biased exponent is 124; in
4081:
4045:
3943:
3683:
3383:
3265:
3160:
3151:
2848:
2252:
2250:
2228:
2226:
2224:
2222:
1634:
a predicate for "x is a NaN", equivalent to "x ≠ x"
1489:– directed rounding towards positive infinity
448:= 0 for positive infinity, 1 for negative infinity.
2185:
2183:
2181:
2179:
2052:"Java Language and Virtual Machine Specifications"
1825:IEEE Standard for Binary Floating-Point Arithmetic
273:
155:", five exceptions to handle invalid results like
2539:"An Interview with the Old Man of Floating-Point"
2234:"An Interview with the Old Man of Floating-Point"
1670:was starting the development of a floating-point
320:, so the biased exponent in this example is 1020.
274:{\displaystyle 0.00101_{2}=1.01_{2}\times 2^{-3}}
2034:"Comparing Floating Point Numbers, 2012 Edition"
2508:"Names for Standardized Floating-Point Formats"
1374:
2826:
787:numbers occupy 64 bits. In double precision:
557:numbers occupy 32 bits. In single precision:
8:
1504:The IEEE standard employs (and extends) the
517:. Relative precision is defined here as ulp(
2604:"IEEE 754: An Interview with William Kahan"
2368:Design of a Computer: The Control Data 6600
2306:"IEEE 754: An Interview with William Kahan"
1394:the comparisons are, common values include
44:, and then again in 2019 by minor revision
3157:
2833:
2819:
2811:
2343:. 2006-11-21. Article ID KB35826, Q35826.
1755:lasted until 1981 when an expert hired by
1533:The following functions must be provided:
358:The number zero is represented specially:
191:The three fields in a 64bit IEEE 754 float
2748:
2089:
1956:"Lecture Notes on the Status of IEEE 754"
1685:, who had helped improve the accuracy of
262:
249:
236:
230:
1510:projectively extended real number system
1358:* Sign bit can be either 0 or 1 .
1028:
830:
600:
81:
2557:"History of IEEE Floating-Point Format"
2374:(1 ed.). Glenview, Illinois, USA:
1946:
1944:
1942:
1816:
1792:
4178:Computer-related introductions in 1985
1483:– directed rounding towards zero
7:
2288:and others after an IEEE 754 meeting
1961:. University of California, Berkeley
1585:Recommended functions and predicates
1506:affinely extended real number system
1371:affinely extended real number system
75:IEEE 754-1985 represents numbers in
2700:"Let's Get To The (Floating) Point"
2259:"A Conversation with William Kahan"
163:for representing those exceptions,
1385:issues apply). When comparing as
541:and the next representable number.
14:
1693:'s (DEC) VAX. The first VAX, the
1593:returns x with the sign of y, so
1884:Computer Organization and Design
1365:Comparing floating-point numbers
210:represented in binary is 0.00101
141:Approximately 16 decimal digits
2474:Turner, Peter R. (2013-12-21).
2393:from the original on 2020-08-28
2347:from the original on 2020-08-28
2257:Woehr, Jack, ed. (1997-11-01).
1538:Add, subtract, multiply, divide
1452:Rounding floating-point numbers
119:Approximately 7 decimal digits
2698:Chris Hecker (February 1996).
2284:W. Kahan 2003, pers. comm. to
439:Positive and negative infinity
434:Positive and negative infinity
1:
2654:David Goldberg (March 1991).
2236:. cs.berkeley.edu. 1998-02-20
2069:John R. Hauser (March 1996).
1691:Digital Equipment Corporation
1325:000 0000 0000 0000 0000 0000
1302:000 0000 0000 0000 0000 0000
1275:111 1111 1111 1111 1111 1111
1248:000 0000 0000 0000 0000 0000
1221:111 1111 1111 1111 1111 1111
1194:100 0000 0000 0000 0000 0000
1179:"Middle" denormalized number
1167:000 0000 0000 0000 0000 0001
1141:000 0000 0000 0000 0000 0000
1118:000 0000 0000 0000 0000 0000
1095:000 0000 0000 0000 0000 0000
1070:000 0000 0000 0000 0000 0000
426:Representation of non-numbers
419:
406:. They don't include as many
2559:. Connexions. Archived from
1712:60-bit floating-point format
1278:±(2−2) × 2 ≈ ±3.4
1206:Largest denormalized number
2802:— History and minutes
2729:David Monniaux (May 2008).
2376:Scott, Foresman and Company
1888:. Morgan Kaufmann. p.
1605:is new in the C99 standard.
1233:Smallest normalized number
834:Actual Exponent (unbiased)
604:Actual Exponent (unbiased)
4196:
4137:IEEE Standards Association
1833:10.1109/IEEESTD.1985.82928
1739:floating-point coprocessor
1500:Extending the real numbers
1402:for single-precision, and
1260:Largest normalized number
945:≈ 4095.999999999999545253
928:≈ 2047.999999999999772626
911:≈ 7.999999999999999111822
894:≈ 3.999999999999999555911
877:≈ 1.999999999999999777955
860:≈ 0.999999999999999888978
818:±(2−2) × 2 ≈ ±1.79769
588:±(2−2) × 2 ≈ ±3.40282
206:The decimal number 0.15625
52:, and in hardware, in the
18:
4127:
2038:randomascii.wordpress.com
1357:
533:in the representation of
471:floating-point arithmetic
175:Representation of numbers
1857:"ANSI/IEEE Std 754-2019"
1683:University of California
1559:Round to nearest integer
1524:Functions and predicates
1197:±2 × 2 = ±2 ≈ ±5.88
288:and the exponent is −3.
159:, special values called
4142:Category:IEEE standards
2759:10.1145/1353445.1353446
2707:Game Developer Magazine
2162:10.1145/1057520.1057522
2127:10.1109/C-M.1981.220377
1224:±(1−2) × 2 ≈ ±1.18
1170:±2 × 2 = ±2 ≈ ±1.4
537:, i.e. the gap between
28:is a historic industry
2625:10.1109/MC.1998.660194
2577:. micro.magnet.fsu.edu
2200:. 2016. Archived from
1781:Fixed-point arithmetic
1740:
1016:80-bit extended format
806:±1 × 2 ≈ ±2.22507
794:±2 × 2 ≈ ±4.94066
576:±1 × 2 ≈ ±1.17549
564:±2 × 2 ≈ ±1.40130
542:
531:unit in the last place
441:are represented thus:
304:= −3 + the "bias". In
275:
192:
184:
2677:10.1145/103162.103163
2664:ACM Computing Surveys
2444:Kahan, William Morton
2413:Kahan, William Morton
2091:10.1145/227699.227701
1735:
1375:binary representation
512:
334:IEEE 754 adds a
276:
190:
182:
2537:(20 February 1998).
2402:(1+13+181+2+2 pages)
1980:"Godot math_funcs.h"
1760:many manufacturers.
1473:this mode is called
1414:relational operators
1411:programming language
390:Denormalized numbers
229:
4163:Computer arithmetic
2446:; Darcy, Joseph D.
2040:. 26 February 2012.
1998:"Godot math_defs.h"
1954:(October 1, 1997).
1916:Computer Arithmetic
1751:The arguments over
1529:Standard operations
1151:denormalized number
505:Range and precision
469:Some operations of
220:scientific notation
2156:(Special): 13–21.
2056:Java Documentation
2016:"Godot MathfEx.cs"
1741:
1459:directed roundings
1379:sign and magnitude
1310:Negative infinity
1287:Positive infinity
979:18014398509481982
543:
515:significant digits
408:significant digits
344:sign and magnitude
271:
193:
185:
66:integrated circuit
4150:
4149:
4041:
4040:
2600:Charles Severance
2553:Charles Severance
2535:Charles Severance
2337:Microsoft Support
2150:SIGNUM Newsletter
1880:Hennessy (2009).
1753:gradual underflow
1362:
1361:
1003:
1002:
976:9007199254740992
962:9007199254740991
959:4503599627370496
773:
772:
715:≈ 4095.999755859
698:≈ 2047.999877930
681:≈ 7.999999523163
664:≈ 3.999999761581
647:≈ 1.999999880791
630:≈ 0.999999940395
145:
144:
124:Double precision
102:Single precision
32:for representing
4185:
3158:
2835:
2828:
2821:
2812:
2790:Comparing floats
2778:
2752:
2725:
2723:
2717:. Archived from
2704:
2694:
2692:
2691:
2660:
2650:
2648:
2647:
2641:
2635:. Archived from
2608:
2586:
2585:
2583:
2582:
2571:
2565:
2564:
2549:
2543:
2542:
2531:
2525:
2524:
2522:
2521:
2512:
2504:
2498:
2497:
2495:
2494:
2487:978-3-66239812-8
2471:
2465:
2464:
2462:
2461:
2452:
2440:
2434:
2433:
2431:
2430:
2421:
2409:
2403:
2401:
2399:
2398:
2392:
2373:
2362:
2356:
2355:
2353:
2352:
2329:
2323:
2322:
2320:
2319:
2310:
2302:
2289:
2282:
2276:
2275:
2273:
2272:
2254:
2245:
2244:
2242:
2241:
2230:
2217:
2215:
2213:
2212:
2206:
2195:
2187:
2174:
2173:
2145:
2139:
2138:
2110:
2104:
2103:
2093:
2075:
2066:
2060:
2059:
2048:
2042:
2041:
2030:
2024:
2023:
2012:
2006:
2005:
1994:
1988:
1987:
1976:
1970:
1969:
1967:
1966:
1960:
1948:
1937:
1936:
1935:
1934:
1928:
1922:, archived from
1921:
1910:
1904:
1903:
1887:
1877:
1871:
1870:
1868:
1867:
1861:754r.ucbtest.org
1853:
1847:
1846:
1821:
1804:
1797:
1720:denormal numbers
1656:
1651:
1645:
1639:
1633:
1623:
1618:
1613:
1604:
1600:
1596:
1592:
1579:
1571:
1554:
1553:x–(round(x/y)·y)
1549:modulo operation
1467:Round to Nearest
1447:
1443:
1439:
1435:
1431:
1427:
1423:
1405:
1401:
1397:
1281:
1254:
1227:
1200:
1173:
1038:Actual Exponent
1029:
1006:Extended formats
831:
821:
809:
797:
785:Double-precision
780:Double precision
601:
591:
579:
567:
555:Single-precision
550:Single precision
488:= either 0 or 1.
314:double precision
306:single precision
280:
278:
277:
272:
270:
269:
254:
253:
241:
240:
218:.) Analogous to
165:denormal numbers
157:division by zero
137:
133:
115:
111:
82:
4195:
4194:
4188:
4187:
4186:
4184:
4183:
4182:
4153:
4152:
4151:
4146:
4123:
4077:
4037:
3939:
3687:
3679:
3387:
3379:
3261:
3147:
2844:
2839:
2786:
2728:
2721:
2702:
2697:
2689:
2687:
2658:
2653:
2645:
2643:
2639:
2606:
2598:
2595:
2593:Further reading
2590:
2589:
2580:
2578:
2573:
2572:
2568:
2551:
2550:
2546:
2533:
2532:
2528:
2519:
2517:
2515:cs.berkeley.edu
2510:
2506:
2505:
2501:
2492:
2490:
2488:
2473:
2472:
2468:
2459:
2457:
2455:cs.berkeley.edu
2450:
2442:
2441:
2437:
2428:
2426:
2424:cs.berkeley.edu
2419:
2411:
2410:
2406:
2396:
2394:
2390:
2371:
2364:
2363:
2359:
2350:
2348:
2331:
2330:
2326:
2317:
2315:
2308:
2304:
2303:
2292:
2283:
2279:
2270:
2268:
2256:
2255:
2248:
2239:
2237:
2232:
2231:
2220:
2210:
2208:
2204:
2193:
2189:
2188:
2177:
2147:
2146:
2142:
2112:
2111:
2107:
2073:
2068:
2067:
2063:
2050:
2049:
2045:
2032:
2031:
2027:
2014:
2013:
2009:
2004:. 30 July 2022.
1996:
1995:
1991:
1986:. 30 July 2022.
1978:
1977:
1973:
1964:
1962:
1958:
1950:
1949:
1940:
1932:
1930:
1926:
1919:
1912:
1911:
1907:
1900:
1879:
1878:
1874:
1865:
1863:
1855:
1854:
1850:
1843:
1823:
1822:
1818:
1813:
1808:
1807:
1802:
1798:
1794:
1789:
1766:
1687:Hewlett-Packard
1664:
1654:
1649:
1644:unordered(x, y)
1643:
1637:
1631:
1621:
1616:
1611:
1602:
1599:copysign(x,1.0)
1598:
1594:
1590:
1587:
1577:
1575:
1569:
1567:
1552:
1531:
1526:
1502:
1493:Round toward −∞
1487:Round toward +∞
1475:roundTiesToEven
1454:
1445:
1441:
1437:
1433:
1429:
1425:
1421:
1403:
1399:
1395:
1367:
1279:
1252:
1225:
1198:
1171:
1047:Fraction field
1044:Exponent field
1024:
1008:
819:
807:
795:
782:
589:
577:
565:
552:
507:
492:biased exponent
467:
452:biased exponent
436:
428:
392:
377:biased exponent
356:
329:
302:biased exponent
287:
258:
245:
232:
227:
226:
213:
209:
201:biased exponent
177:
135:
131:
113:
109:
23:
17:
12:
11:
5:
4193:
4192:
4189:
4181:
4180:
4175:
4173:Floating point
4170:
4168:IEEE standards
4165:
4155:
4154:
4148:
4147:
4145:
4144:
4139:
4134:
4128:
4125:
4124:
4122:
4121:
4116:
4111:
4106:
4101:
4096:
4091:
4085:
4083:
4079:
4078:
4076:
4075:
4070:
4065:
4060:
4055:
4049:
4047:
4043:
4042:
4039:
4038:
4036:
4035:
4030:
4025:
4020:
4015:
4010:
4005:
4000:
3995:
3990:
3985:
3980:
3970:
3965:
3960:
3949:
3947:
3941:
3940:
3938:
3937:
3925:
3922:
3919:
3916:
3913:
3901:
3898:
3895:
3890:
3887:
3884:
3879:
3867:
3864:
3861:
3856:
3851:
3846:
3841:
3838:
3828:
3816:
3813:
3808:
3803:
3798:
3793:
3788:
3783:
3778:
3773:
3761:
3756:
3751:
3746:
3741:
3736:
3731:
3726:
3721:
3716:
3711:
3706:
3701:
3695:
3693:
3681:
3680:
3678:
3677:
3672:
3667:
3662:
3657:
3652:
3647:
3642:
3637:
3632:
3627:
3622:
3617:
3612:
3607:
3602:
3597:
3592:
3587:
3582:
3577:
3572:
3567:
3562:
3557:
3552:
3547:
3542:
3537:
3532:
3525:
3520:
3515:
3510:
3505:
3498:
3493:
3488:
3483:
3478:
3471:
3466:
3461:
3456:
3451:
3446:
3441:
3436:
3431:
3426:
3421:
3416:
3411:
3406:
3401:
3395:
3393:
3381:
3380:
3378:
3377:
3372:
3362:
3357:
3352:
3347:
3342:
3337:
3332:
3327:
3322:
3317:
3312:
3307:
3302:
3297:
3292:
3287:
3282:
3277:
3271:
3269:
3263:
3262:
3260:
3259:
3254:
3249:
3244:
3239:
3234:
3229:
3228:
3227:
3217:
3212:
3207:
3202:
3197:
3192:
3187:
3182:
3177:
3172:
3166:
3164:
3155:
3149:
3148:
3146:
3145:
3140:
3135:
3130:
3125:
3120:
3115:
3110:
3105:
3100:
3095:
3090:
3085:
3080:
3075:
3070:
3065:
3060:
3055:
3050:
3045:
3040:
3035:
3030:
3025:
3020:
3015:
3010:
3005:
3000:
2995:
2990:
2985:
2980:
2975:
2970:
2965:
2960:
2955:
2950:
2945:
2940:
2935:
2930:
2925:
2920:
2915:
2910:
2905:
2900:
2895:
2890:
2885:
2880:
2879:
2878:
2868:
2863:
2858:
2852:
2850:
2846:
2845:
2842:IEEE standards
2840:
2838:
2837:
2830:
2823:
2815:
2809:
2808:
2803:
2797:
2792:
2785:
2784:External links
2782:
2781:
2780:
2726:
2724:on 2007-02-03.
2695:
2651:
2619:(3): 114–115.
2602:(March 1998).
2594:
2591:
2588:
2587:
2566:
2563:on 2009-11-20.
2544:
2526:
2499:
2486:
2466:
2435:
2404:
2357:
2324:
2290:
2286:Mike Cowlishaw
2277:
2246:
2218:
2175:
2140:
2105:
2084:(2): 139–174.
2061:
2043:
2025:
2007:
1989:
1971:
1938:
1905:
1898:
1872:
1848:
1841:
1815:
1814:
1812:
1809:
1806:
1805:
1800:
1791:
1790:
1788:
1785:
1784:
1783:
1778:
1772:
1765:
1762:
1663:
1660:
1659:
1658:
1655:nextafter(x,y)
1652:
1647:
1641:
1635:
1629:
1619:
1614:
1609:
1606:
1586:
1583:
1582:
1581:
1573:
1565:
1562:
1556:
1545:
1540:
1530:
1527:
1525:
1522:
1501:
1498:
1497:
1496:
1490:
1484:
1481:Round toward 0
1478:
1453:
1450:
1387:2's-complement
1366:
1363:
1360:
1359:
1355:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1330:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1307:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1284:
1283:
1276:
1273:
1270:
1267:
1264:
1261:
1257:
1256:
1249:
1246:
1243:
1240:
1237:
1234:
1230:
1229:
1222:
1219:
1216:
1213:
1210:
1207:
1203:
1202:
1195:
1192:
1189:
1186:
1183:
1180:
1176:
1175:
1168:
1165:
1162:
1159:
1156:
1153:
1146:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1123:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1100:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1075:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1052:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1023:
1020:
1007:
1004:
1001:
1000:
999:≈ 1.99584e292
997:
996:≈ 1.79769e308
994:
993:≈ 8.98847e307
991:
988:
984:
983:
980:
977:
974:
971:
967:
966:
963:
960:
957:
954:
950:
949:
948:≈ 4.54747e-13
946:
943:
940:
937:
933:
932:
931:≈ 2.27374e-13
929:
926:
923:
920:
916:
915:
914:≈ 8.88178e-16
912:
909:
906:
903:
899:
898:
897:≈ 4.44089e-16
895:
892:
889:
886:
882:
881:
880:≈ 2.22045e-16
878:
875:
872:
869:
865:
864:
863:≈ 1.11022e-16
861:
858:
855:
852:
848:
847:
844:
841:
838:
835:
826:
825:
824:
823:
813:
812:
811:
801:
800:
799:
781:
778:
771:
770:
767:
764:
761:
758:
754:
753:
750:
747:
744:
741:
737:
736:
733:
730:
727:
724:
720:
719:
716:
713:
710:
707:
703:
702:
699:
696:
693:
690:
686:
685:
682:
679:
676:
673:
669:
668:
665:
662:
659:
656:
652:
651:
648:
645:
642:
639:
635:
634:
631:
628:
625:
622:
618:
617:
614:
611:
608:
605:
596:
595:
594:
593:
583:
582:
581:
571:
570:
569:
551:
548:
506:
503:
502:
501:
495:
489:
466:
463:
462:
461:
455:
449:
435:
432:
427:
424:
391:
388:
387:
386:
380:
374:
355:
352:
340:2's-complement
332:
331:
327:
321:
316:, the bias is
308:, the bias is
299:
285:
282:
281:
268:
265:
261:
257:
252:
248:
244:
239:
235:
211:
207:
176:
173:
143:
142:
139:
128:
125:
121:
120:
117:
106:
103:
99:
98:
95:
91:Range at full
89:
86:
34:floating-point
15:
13:
10:
9:
6:
4:
3:
2:
4191:
4190:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4160:
4158:
4143:
4140:
4138:
4135:
4133:
4130:
4129:
4126:
4120:
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4086:
4084:
4080:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4054:
4051:
4050:
4048:
4044:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3978:
3974:
3971:
3969:
3966:
3964:
3961:
3958:
3954:
3951:
3950:
3948:
3946:
3942:
3935:
3931:
3930:
3926:
3923:
3920:
3917:
3914:
3911:
3907:
3906:
3902:
3899:
3896:
3894:
3891:
3888:
3885:
3883:
3880:
3877:
3873:
3872:
3868:
3865:
3862:
3860:
3857:
3855:
3852:
3850:
3847:
3845:
3842:
3839:
3836:
3832:
3829:
3826:
3822:
3821:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3771:
3767:
3766:
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3710:
3707:
3705:
3702:
3700:
3697:
3696:
3694:
3691:
3686:
3682:
3676:
3673:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3651:
3648:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3533:
3531:
3530:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3503:
3499:
3497:
3494:
3492:
3489:
3487:
3484:
3482:
3479:
3477:
3476:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3396:
3394:
3391:
3386:
3382:
3376:
3373:
3370:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3272:
3270:
3268:
3264:
3258:
3255:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3226:
3225:WiMAX · d · e
3223:
3222:
3221:
3218:
3216:
3213:
3211:
3208:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3167:
3165:
3163:
3159:
3156:
3154:
3150:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
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3079:
3076:
3074:
3071:
3069:
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3051:
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3044:
3041:
3039:
3036:
3034:
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3026:
3024:
3021:
3019:
3016:
3014:
3011:
3009:
3006:
3004:
3001:
2999:
2996:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
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2941:
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2936:
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2931:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
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2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2877:
2874:
2873:
2872:
2869:
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2864:
2862:
2859:
2857:
2854:
2853:
2851:
2847:
2843:
2836:
2831:
2829:
2824:
2822:
2817:
2816:
2813:
2807:
2804:
2801:
2800:IEEE 854-1987
2798:
2796:
2793:
2791:
2788:
2787:
2783:
2776:
2772:
2768:
2764:
2760:
2756:
2751:
2746:
2742:
2738:
2737:
2732:
2727:
2720:
2716:
2712:
2708:
2701:
2696:
2686:
2682:
2678:
2674:
2670:
2666:
2665:
2657:
2652:
2642:on 2009-08-23
2638:
2634:
2630:
2626:
2622:
2618:
2614:
2613:
2612:IEEE Computer
2605:
2601:
2597:
2596:
2592:
2576:
2570:
2567:
2562:
2558:
2554:
2548:
2545:
2540:
2536:
2530:
2527:
2516:
2509:
2503:
2500:
2489:
2483:
2479:
2478:
2470:
2467:
2456:
2449:
2445:
2439:
2436:
2425:
2418:
2414:
2408:
2405:
2389:
2385:
2381:
2377:
2370:
2369:
2361:
2358:
2346:
2342:
2338:
2334:
2328:
2325:
2314:
2307:
2301:
2299:
2297:
2295:
2291:
2287:
2281:
2278:
2267:. drdobbs.com
2266:
2265:
2260:
2253:
2251:
2247:
2235:
2229:
2227:
2225:
2223:
2219:
2207:on 2016-03-04
2203:
2199:
2192:
2186:
2184:
2182:
2180:
2176:
2171:
2167:
2163:
2159:
2155:
2151:
2144:
2141:
2136:
2132:
2128:
2124:
2120:
2116:
2115:IEEE Computer
2109:
2106:
2101:
2097:
2092:
2087:
2083:
2079:
2072:
2065:
2062:
2057:
2053:
2047:
2044:
2039:
2035:
2029:
2026:
2021:
2017:
2011:
2008:
2003:
1999:
1993:
1990:
1985:
1981:
1975:
1972:
1957:
1953:
1952:William Kahan
1947:
1945:
1943:
1939:
1929:on 2010-10-08
1925:
1918:
1917:
1909:
1906:
1901:
1899:9780123744937
1895:
1891:
1886:
1885:
1876:
1873:
1862:
1858:
1852:
1849:
1844:
1842:0-7381-1165-1
1838:
1834:
1830:
1826:
1820:
1817:
1810:
1796:
1793:
1786:
1782:
1779:
1776:
1773:
1771:
1768:
1767:
1763:
1761:
1758:
1754:
1749:
1746:
1743:In 1980, the
1738:
1734:
1730:
1727:
1725:
1724:exponent bias
1721:
1717:
1713:
1707:
1705:
1699:
1696:
1692:
1688:
1684:
1680:
1679:William Kahan
1675:
1673:
1669:
1661:
1653:
1648:
1642:
1636:
1630:
1627:
1620:
1615:
1610:
1607:
1591:copysign(x,y)
1589:
1588:
1584:
1568: ≠
1563:
1560:
1557:
1550:
1546:
1544:
1541:
1539:
1536:
1535:
1534:
1528:
1523:
1521:
1519:
1515:
1511:
1507:
1499:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1472:
1471:IEEE 754-2008
1468:
1465:
1464:
1463:
1461:
1460:
1451:
1449:
1419:
1415:
1412:
1407:
1391:
1388:
1384:
1380:
1376:
1372:
1364:
1356:
1352:
1349:
1346:
1343:
1340:
1337:
1335:
1332:
1331:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1308:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1285:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1258:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1231:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1204:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1177:
1169:
1166:
1163:
1160:
1157:
1154:
1152:
1148:
1147:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1124:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1101:
1097:
1094:
1091:
1088:
1085:
1082:
1080:
1079:Negative zero
1077:
1076:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1053:
1049:
1046:
1043:
1041:Exp (biased)
1040:
1037:
1034:
1031:
1030:
1027:
1021:
1019:
1017:
1014:
1005:
998:
995:
992:
989:
986:
985:
981:
978:
975:
972:
969:
968:
964:
961:
958:
955:
952:
951:
947:
944:
941:
938:
935:
934:
930:
927:
924:
921:
918:
917:
913:
910:
907:
904:
901:
900:
896:
893:
890:
887:
884:
883:
879:
876:
873:
870:
867:
866:
862:
859:
856:
853:
850:
849:
845:
842:
839:
837:Exp (biased)
836:
833:
832:
829:
817:
816:
814:
805:
804:
802:
793:
792:
790:
789:
788:
786:
779:
777:
769:≈ 2.02824e31
768:
766:≈ 3.40282e38
765:
763:≈ 1.70141e38
762:
759:
756:
755:
751:
748:
745:
742:
739:
738:
734:
731:
728:
725:
722:
721:
718:≈ 2.44141e-4
717:
714:
711:
708:
705:
704:
701:≈ 1.22070e-4
700:
697:
694:
691:
688:
687:
684:≈ 4.76837e-7
683:
680:
677:
674:
671:
670:
667:≈ 2.38419e-7
666:
663:
660:
657:
654:
653:
650:≈ 1.19209e-7
649:
646:
643:
640:
637:
636:
633:≈ 5.96046e-8
632:
629:
626:
623:
620:
619:
615:
612:
609:
607:Exp (biased)
606:
603:
602:
599:
587:
586:
584:
575:
574:
572:
563:
562:
560:
559:
558:
556:
549:
547:
540:
536:
532:
528:
524:
520:
516:
511:
504:
499:
496:
494:= all 1 bits.
493:
490:
487:
484:
483:
482:
480:
476:
472:
464:
460:= all 0 bits.
459:
456:
454:= all 1 bits.
453:
450:
447:
444:
443:
442:
440:
433:
431:
425:
423:
421:
415:
413:
409:
405:
401:
397:
389:
384:
381:
378:
375:
372:
371:negative zero
368:
367:positive zero
364:
361:
360:
359:
353:
351:
348:
345:
341:
337:
325:
322:
319:
315:
311:
307:
303:
300:
297:
294:
293:
292:
289:
266:
263:
259:
255:
250:
246:
242:
237:
233:
225:
224:
223:
221:
217:
204:
202:
198:
189:
181:
174:
172:
170:
166:
162:
158:
154:
153:negative zero
150:
140:
129:
126:
123:
122:
118:
107:
104:
101:
100:
96:
94:
90:
87:
84:
83:
80:
78:
73:
71:
67:
63:
59:
55:
51:
47:
46:IEEE 754-2019
43:
42:IEEE 754-2008
39:
35:
31:
27:
26:IEEE 754-1985
22:
4131:
4088:
3927:
3903:
3869:
3818:
3763:
3527:
3500:
3473:
2740:
2734:
2719:the original
2706:
2688:. Retrieved
2668:
2662:
2644:. Retrieved
2637:the original
2616:
2610:
2579:. Retrieved
2569:
2561:the original
2547:
2529:
2518:. Retrieved
2514:
2502:
2491:. Retrieved
2480:. Springer.
2476:
2469:
2458:. Retrieved
2454:
2438:
2427:. Retrieved
2423:
2407:
2395:. Retrieved
2367:
2360:
2349:. Retrieved
2336:
2327:
2316:. Retrieved
2313:dr-chuck.com
2312:
2280:
2269:. Retrieved
2262:
2238:. Retrieved
2209:. Retrieved
2202:the original
2153:
2149:
2143:
2121:(3): 51–62.
2118:
2114:
2108:
2081:
2077:
2064:
2055:
2046:
2037:
2028:
2019:
2010:
2001:
1992:
1983:
1974:
1963:. Retrieved
1931:, retrieved
1924:the original
1915:
1908:
1883:
1875:
1864:. Retrieved
1860:
1851:
1824:
1819:
1795:
1750:
1742:
1728:
1708:
1704:Harold Stone
1700:
1676:
1665:
1638:x <> y
1532:
1503:
1492:
1486:
1480:
1474:
1466:
1457:
1455:
1408:
1392:
1368:
1334:Not a number
1025:
1009:
827:
783:
774:
597:
553:
544:
538:
534:
526:
525:, where ulp(
522:
518:
497:
491:
485:
479:Not a Number
474:
468:
457:
451:
445:
437:
429:
416:
395:
393:
382:
376:
362:
357:
349:
333:
323:
317:
313:
309:
305:
301:
295:
290:
283:
205:
194:
146:
74:
64:. The first
54:instructions
25:
24:
3704:legacy mode
2743:(3): 1–41.
2671:(1): 5–48.
1672:coprocessor
1612:scalb(y, N)
1576:(including
1543:Square root
1518:Intel 80287
1440:of classes
1434:compareTo()
1144:−1.0
1098:−0.0
396:normalized,
134:10 to ±1.80
36:numbers in
4157:Categories
4082:Superseded
3153:802 series
2750:cs/0701192
2690:2008-04-28
2646:2008-04-28
2581:2016-05-30
2520:2016-06-02
2493:2016-05-30
2460:2016-06-02
2429:2016-06-02
2397:2016-06-02
2351:2010-02-24
2318:2016-06-02
2271:2016-05-30
2264:Dr. Dobb's
2240:2016-05-30
2216:(11 pages)
2211:2016-05-30
2020:GitHub.com
2002:GitHub.com
1984:GitHub.com
1965:2007-04-12
1933:2011-01-02
1866:2019-08-06
1811:References
1745:Intel 8087
1737:Intel 8087
1695:VAX-11/780
1514:Intel 8087
1426:Math.max()
1422:Math.min()
1383:endianness
1381:integers (
1347:1111 1111
1322:1111 1111
1299:1111 1111
1272:1111 1110
1251:±2 ≈ ±1.18
1245:0000 0001
1218:0000 0000
1191:0000 0000
1164:0000 0000
1138:0111 1111
1126:Minus One
1115:0111 1111
1092:0000 0000
1067:0000 0000
475:exception.
112:10 to ±3.4
97:Precision
70:Intel 8087
19:See also:
3957:Bluetooth
2775:218578808
2767:0164-0925
2715:1073-922X
2709:: 19–24.
2685:222008826
2341:Microsoft
1775:Minifloat
1666:In 1976,
1626:predicate
1622:finite(x)
1438:compare()
1436:and even
1350:non zero
1149:Smallest
749:33554430
746:16777216
732:16777215
529:) is the
412:operation
400:underflow
326:= .01000…
264:−
256:×
93:precision
50:libraries
38:computers
4132:See also
4089:754-1985
4046:Proposed
3390:Ethernet
2876:Revision
2633:33291145
2388:Archived
2384:74-96462
2345:Archived
2170:16981715
2135:15523399
1827:. 1985.
1770:IEEE 754
1764:See also
1716:CDC 6600
1650:class(x)
1632:isnan(x)
1603:copysign
1572:for any
1430:equals()
1022:Examples
843:Maximum
840:Minimum
729:8388608
613:Maximum
610:Minimum
498:fraction
458:fraction
422:below).
420:examples
404:denormal
383:fraction
369:, 1 for
365:= 0 for
324:fraction
197:sign bit
169:rounding
149:infinity
127:64 bits
105:32 bits
56:of many
30:standard
21:IEEE 754
4073:P1906.1
3934:Wi-Fi 8
3910:Wi-Fi 7
3876:Wi-Fi 6
3825:Wi-Fi 5
3770:Wi-Fi 4
2849:Current
2100:9820157
1714:of the
1681:of the
1662:History
1617:logb(x)
1597:equals
234:0.00101
171:modes.
3977:Zigbee
3945:802.15
3685:802.11
2923:1149.1
2773:
2765:
2713:
2683:
2631:
2484:
2382:
2168:
2133:
2098:
1896:
1839:
1595:abs(x)
1446:Double
1050:Value
88:Width
85:Level
77:binary
4068:P1823
4063:P1699
4058:P1619
4053:P1363
3835:WiGig
3699:-1997
3690:Wi-Fi
3399:-1983
3385:802.3
3267:802.1
3143:42010
3138:29148
3133:16326
3128:16085
3123:14764
3118:12207
3113:11073
2771:S2CID
2745:arXiv
2722:(PDF)
2703:(PDF)
2681:S2CID
2659:(PDF)
2640:(PDF)
2629:S2CID
2607:(PDF)
2511:(PDF)
2451:(PDF)
2420:(PDF)
2391:(PDF)
2372:(PDF)
2309:(PDF)
2205:(PDF)
2198:Intel
2194:(PDF)
2166:S2CID
2131:S2CID
2096:S2CID
2074:(PDF)
1959:(PDF)
1927:(PDF)
1920:(PDF)
1787:Notes
1668:Intel
1442:Float
1404:1e-14
1239:−126
1212:−126
1185:−126
1158:−126
1086:−126
1061:−126
1055:Zero
1035:Sign
1032:Type
990:2046
987:1023
973:1076
956:1075
942:2048
939:1034
925:1024
922:1033
905:1025
888:1024
871:1023
854:1022
712:2048
695:1024
151:, a "
130:±2.23
108:±1.18
4119:1471
4114:1364
4109:1362
4104:1233
4099:1219
3369:LACP
3108:2050
3103:2030
3098:1905
3093:1904
3088:1902
3083:1901
3078:1900
3073:1855
3068:1850
3063:1849
3058:1815
3053:1801
3048:1800
3043:1733
3038:1722
3033:1685
3028:1675
3023:1667
3018:1666
3013:1619
3008:1613
3003:1603
2998:1596
2993:1588
2988:1584
2983:1547
2978:1541
2973:1516
2968:1497
2963:1451
2958:1394
2953:1355
2948:1284
2943:1278
2938:1275
2933:1164
2928:1154
2918:1076
2913:1016
2908:1014
2903:1003
2763:ISSN
2711:ISSN
2482:ISBN
2380:LCCN
1894:ISBN
1837:ISBN
1516:and
1444:and
1424:and
1418:Java
1400:1e-5
1396:1e-6
1353:NaN
1344:255
1341:128
1319:255
1316:128
1296:255
1293:128
1269:254
1266:127
1135:127
1121:1.0
1112:127
1103:One
1073:0.0
857:0.5
846:Gap
760:254
757:127
743:151
726:150
709:138
692:137
675:129
658:128
641:127
627:0.5
624:126
616:Gap
486:sign
446:sign
385:= 0.
379:= 0.
363:sign
354:Zero
336:bias
318:1023
296:sign
247:1.01
216:base
199:, a
161:NaNs
62:FPUs
60:and
58:CPUs
4094:830
4018:.4z
4013:.4g
4008:.4f
4003:.4e
3998:.4d
3993:.4c
3988:.4b
3983:.4a
3310:Qbb
3305:Qaz
3300:Qay
3295:Qat
3290:Qav
3257:.24
3252:.22
3247:.21
3242:.20
3237:.18
3232:.17
3220:.16
3215:.14
3210:.12
3205:.10
3162:802
2898:896
2893:829
2888:828
2883:854
2871:754
2866:730
2861:693
2856:488
2755:doi
2673:doi
2621:doi
2158:doi
2123:doi
2086:doi
1890:270
1829:doi
1757:DEC
1578:NaN
1570:NaN
1398:or
1328:−∞
1305:+∞
1282:10
1255:10
1228:10
1201:10
1174:10
1013:x87
970:53
953:52
936:11
919:10
851:−1
740:24
723:23
706:11
689:10
621:−1
465:NaN
310:127
138:10
116:10
4159::
4033:.7
4028:.6
4023:.5
3973:.4
3968:.3
3963:.2
3953:.1
3929:bn
3924:bk
3921:bi
3918:bh
3915:bf
3905:be
3900:bd
3897:bc
3893:bb
3889:ba
3886:az
3882:ay
3871:ax
3866:aq
3863:ak
3859:aj
3854:ai
3849:ah
3844:af
3840:ae
3831:ad
3820:ac
3815:aa
3675:df
3670:de
3665:dd
3660:db
3655:da
3650:cz
3645:cy
3640:cx
3635:cw
3630:cv
3625:cu
3620:ct
3615:cs
3610:cr
3605:cq
3600:cp
3595:cn
3590:cm
3585:ck
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