25:
667:, which implies that in rounding to nearest, the rounded result is within 0.5 ulp of the mathematically exact result, using John Harrison's definition; conversely, this property implies that the distance between the rounded result and the mathematically exact result is minimized (but for the halfway cases, it is satisfied by two consecutive floating-point numbers). Reputable
685:
Since the 2010s, advances in floating-point mathematics have allowed correctly rounded functions to be almost as fast in average as these earlier, less accurate functions. A correctly rounded function would also be fully reproducible.
347:
933:
1058:
for a single-precision number contains 24 bits, the first integer that is not exactly representable is 2+1, and this value rounds to 2 in round to nearest, ties to even. Thus the result is equal to 2.
501:
1324:
433:
1877:
Muller, Jean-Michel (2005–11). "On the definition of ulp(x)". INRIA Technical Report 5504. ACM Transactions on
Mathematical Software, Vol. V, No. N, November 2005. Retrieved in 2012-03 from
827:
783:
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871:
266:
540:
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Goldberg, David (1991–03). "Rounding Error" in "What Every
Computer Scientist Should Know About Floating-Point Arithmetic". Computing Surveys, ACM, March 1991. Retrieved from
616:
376:
642:
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204:
184:
42:
1665:, which represent the positive difference between 1.0 and the next greater representable number in the corresponding type (i.e. the ulp of one).
1051:
1815:
Muller, Jean-Michel; Brunie, Nicolas; de
Dinechin, Florent; Jeannerod, Claude-Pierre; Joldes, Mioara; Lefèvre, Vincent; Melquiond, Guillaume;
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to between 0.5 and about 1 ulp. Only a few libraries compute them within 0.5 ulp, this problem being complex due to the
75:
1691:
standard library provides access to the next floating-point number in some given direction via the instance properties
644:), assuming that the exponent range is not upper-bounded. These definitions differ only at signed powers of the radix.
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which theoretically would only produce one incorrect rounding out of 1000 random floating-point inputs.
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2013:
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library provides functions to calculate the next floating-point number in some given direction:
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specification—followed by all modern floating-point hardware—requires that the result of an
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1913:
Brisebarre, Nicolas; Hanrot, Guillaume; Muller, Jean-Michel; Zimmermann, Paul (May 2024).
1742:
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1079:
1054:(binary32) and repeatedly add 1 until the operation does not change the value. Since the
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be a positive floating-point number and assume that the active rounding mode is
656:
24:
1832:
2014:"FloatingPoint - Swift Standard Library | Apple Developer Documentation"
935:, depending on the value of the least significant digit and the exponent of
1764:"What Every Computer Scientist Should Know About Floating-Point Arithmetic"
1782:
1763:
1915:"Correctly-rounded evaluation of a function: why, how, and at what cost?"
1720:
664:
648:
510:
Another definition, suggested by John
Harrison, is slightly different:
1879:
http://ljk.imag.fr/membres/Carine.Lucas/TPScilab/JMMuller/ulp-toms.pdf
1076:
as a floating point value by finding the two double values bracketing
342:{\displaystyle \operatorname {ulp} (x)=b^{\max\{e,\,e_{\min }\}-p+1}}
1201:// p0 is smaller than π, so find next number representable as double
151:(rightmost digit) represents if it is 1. It is used as a measure of
1914:
2044:
http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html#689
928:{\displaystyle \operatorname {RN} (x+1)=x+\operatorname {ulp} (x)}
164:
1796:(With the addendum "Differences Among IEEE 754 Implementations":
655:
operation (addition, subtraction, multiplication, division, and
1988:"ulpOfOne - FloatingPoint | Apple Developer Documentation"
688:
An earlier, intermediate milestone was the 0.501 ulp functions,
1606:
to calculate the floating-point distance between two doubles.
378:
is the minimal exponent of the normal numbers. In particular,
18:
1570:. Similarly to Example 1, the result is 2 because the
1071:
1602:
to obtain nearby (and distant) floating-point values, and
496:{\displaystyle \operatorname {ulp} (x)=b^{e_{\min }-p+1}}
1858:"A Machine-Checked Theory of Floating Point Arithmetic"
1385:// same result when using the standard library function
1319:{\displaystyle \operatorname {ulp} (\pi )=p_{1}-p_{0}}
1231:// -> 3.1415926535897936 (hex: 0x1.921fb54442d19p1)
1198:// -> 3.141592653589793 (hex: 0x1.921fb54442d18p1)
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49:. Unsourced material may be challenged and removed.
1379:// -> 4.44089209850062616169452667236328125E-16
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1574:floating-point format uses a 53-bit significand.
428:{\displaystyle \operatorname {ulp} (x)=b^{e-p+1}}
474:
363:
317:
299:
1415:// -> 4.440892098500626E-16 (hex: 0x1.0p-51)
1331:// ulp(Ď€) is the difference between p1 and p0
822:{\displaystyle \operatorname {RN} (x+1)>x}
778:{\displaystyle \operatorname {ulp} (x)\leq 1}
8:
322:
302:
1429:, also typed at an interactive prompt, is:
1810:
1808:
1261:{\displaystyle \operatorname {ulp} (\pi )}
866:{\displaystyle \operatorname {RN} (x+1)=x}
1781:
1699:. It also provides the instance property
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141:) is the spacing between two consecutive
109:Learn how and when to remove this message
1672:standard library provides the functions
955:. This is demonstrated in the following
542:is the distance between the two closest
261:{\displaystyle b^{e}\leq |x|<b^{e+1}}
1754:
535:{\displaystyle \operatorname {ulp} (x)}
2054:. Boston: Birkhäuser. pp. 32–37.
1684:. They were introduced with Java 1.5.
1174:// truncate to a double floating point
1135:{\displaystyle p_{0}<\pi <p_{1}}
2052:Handbook of floating-point arithmetic
1821:Handbook of Floating-Point Arithmetic
959:code typed at an interactive prompt:
7:
1711:) for Swift's floating-point types.
1707:(which corresponds to C macros like
47:adding citations to reliable sources
740:{\displaystyle \operatorname {RN} }
1168:"3.14159265358979323846"
163:The most common definition is: In
14:
1728:, part 1 requires an ulp function
1604:boost::math::float_distance(a, b)
1893:"A Logarithm Too Clever by Half"
23:
1962:ISO/IEC 9899:1999 specification
1566:and repeatedly double it until
1382:// (this is precisely 2**(-51))
34:needs additional citations for
1762:Goldberg, David (March 1991).
1653:. It also provides the macros
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725:round to nearest, ties to even
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235:
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16:Floating-point accuracy metric
1:
611:{\displaystyle a\leq x\leq b}
145:numbers, i.e., the value the
2050:Muller, Jean-Michel (2010).
1562:In this case, we start with
1147:// π with 20 decimal digits
2100:
1600:boost::math::float_advance
1833:10.1007/978-3-319-76526-6
1819:; Torres, Serge (2018) .
1066:The following example in
663:since 2008) be correctly
371:{\displaystyle e_{\min }}
155:in numeric calculations.
1974:The nexttoward functions
1592:boost::math::float_prior
1431:
1328:
1144:
1050:Here we start with 0 in
961:
676:transcendental functions
58:"Unit in the last place"
1970:The nextafter functions
1588:boost::math::float_next
1586:provides the functions
637:{\displaystyle a\neq b}
546:floating-point numbers
148:least significant digit
135:unit of least precision
1703:and the type property
1596:boost::math::nextafter
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131:unit in the last place
1968:. p. 237, §7.12.11.3
1783:10.1145/103162.103163
1769:ACM Computing Surveys
1732:Least significant bit
1321:
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680:Table-maker's dilemma
653:elementary arithmetic
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1946:Boost float_distance
1425:Another example, in
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1089:{\displaystyle \pi }
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43:improve this article
2079:Computer arithmetic
1930:Boost float_advance
1584:Boost C++ libraries
1557:9007199254740996.0
1527:9007199254740992.0
1316:
1258:
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674:compute the basic
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586:(i.e., satisfying
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127:numerical analysis
2061:978-0-8176-4704-9
1842:978-3-319-76525-9
1268:is determined as
948:{\displaystyle x}
716:{\displaystyle x}
579:{\displaystyle b}
559:{\displaystyle a}
199:{\displaystyle p}
179:{\displaystyle b}
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1891:Kahan, William.
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1675:Math.ulp(double)
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1052:single precision
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689:
659:since 1985, and
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123:computer science
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51:
27:
19:
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2068:
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2039:
2034:
2033:
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2012:
2011:
2007:
1997:
1995:
1986:
1985:
1981:
1972:and §7.12.11.4
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1850:
1843:
1817:Revol, Nathalie
1814:
1813:
1806:
1761:
1760:
1756:
1751:
1743:Round-off error
1738:Machine epsilon
1717:
1708:
1704:
1700:
1696:
1692:
1681:Math.ulp(float)
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186:with precision
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2084:Floating point
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2047:
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2035:
2032:
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2005:
1979:
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1883:
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1841:
1823:(2 ed.).
1804:
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1651:<math.h>
1649:, declared in
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437:normal numbers
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143:floating-point
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13:
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1539:>>>
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1070:approximates
1069:
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60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
2051:
2037:Bibliography
2022:. Retrieved
2017:
2008:
1996:. Retrieved
1991:
1982:
1973:
1969:
1961:
1955:
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1939:
1929:
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1851:
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1667:
1663:LDBL_EPSILON
1608:
1581:
1561:
1424:
1235:
1065:
1049:
1046:1.6777216e7
1033:1.6777215e7
1020:1.6777216e7
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543:
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162:
146:
138:
134:
130:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2020:. Apple Inc
1994:. Apple Inc
1898:14 November
1776:(1): 5–48.
1709:FLT_EPSILON
1659:DBL_EPSILON
1655:FLT_EPSILON
1647:long double
1643:nexttowardl
1619:nexttowardf
1192:doubleValue
1056:significand
657:square root
2073:Categories
1825:Birkhäuser
1749:References
1639:nextafterl
1631:nexttoward
1615:nextafterf
1611:C language
1367:BigDecimal
1346:BigDecimal
1334:BigDecimal
1162:BigDecimal
1150:BigDecimal
727:, denoted
544:straddling
505:subnormals
159:Definition
99:March 2015
69:newspapers
2024:18 August
2018:Apple Inc
1998:18 August
1992:Apple Inc
1792:222008826
1627:nextafter
1568:x = x + 1
1421:Example 3
1304:−
1285:π
1279:
1253:π
1247:
1117:π
1084:π
1062:Example 2
914:
884:
840:
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770:≤
758:
699:Example 1
672:libraries
629:≠
603:≤
597:≤
521:
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450:
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389:
326:−
280:
224:≤
1721:IEEE 754
1715:See also
1705:ulpOfOne
1693:nextDown
1358:subtract
694:Examples
649:IEEE 754
153:accuracy
1863:17 July
1391:ulpMath
957:Haskell
785:, then
669:numeric
665:rounded
268:, then
83:scholar
2058:
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1790:
1697:nextUp
1635:double
1427:Python
1388:double
1219:nextUp
1204:double
1177:double
439:, and
351:where
85:
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1966:(PDF)
1788:S2CID
1734:(LSB)
1689:Swift
1623:float
1564:x = 1
1461:while
1236:Then
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1018:Float
979:->
967:until
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747:. If
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165:radix
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