DLP

ゞ浴缸里的玫瑰 2022-08-08 16:54 75阅读 0赞

## Definition ##

In [ radix][radix] *b*, if *x* has exponent *E*, then ULP(*x*) =[machine epsilon][] · *b**E*,[\[1\]][1] but alternative definitions exist in the numerics and computing literature for *ULP*, *exponent* and*machine epsilon*, and they may give different equalities.\[*[citation needed][]*\]

Another definition, suggested by John Harrison, is slightly different: ULP(*x*) is the distance between the two closest*straddling* floating-point numbers *a* and *b* (i.e., those with*a* ≤ *x* ≤ *b* and *a* ≠ *b*), assuming that the exponent range is not upper-bounded.[\[2\]][2][\[3\]][3] These definitions differ only at signed powers of the radix.\[*[citation needed][]*\]

The [ IEEE 754][IEEE 754] specification—followed by all modern floating-point hardware—requires that the result of an[elementary arithmetic][] operation (addition, subtraction, multiplication, division, and[square root][] since 1985, and [ FMA][FMA] since 2008) be [ correctly rounded][correctly rounded], 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[numeric][] [ libraries][libraries] compute the basic [ transcendental functions][transcendental functions] 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[Table-Maker's Dilemma][Table-Maker_s Dilemma].[\[4\]][4]

##     ##

## **Example:**    ##

Letx be a nonnegative floating-point number and assume that the active roundingattribute is round to nearest, ties to even, denoted RN. If ULP(x) is less thanor equal to 1, then RN(x + 1) > x. Otherwise, RN(x + 1) = x or RN(x + 1) = x+  ULP(x), depending on the value of theleast significant digit and the exponent of x. This is demonstrated in thefollowing Haskell code typed at an interactive prompt:\[citation needed\]

>until (\\x -> x == x+1) (+1) 0 :: Float

1.6777216e7

>it-1

1.6777215e7

>it+1

1.6777216e7

Herewe start with 0 in 32-bit single-precision and repeatedly add 1 until theoperation is idempotent. The result is equal to 224 since the significand for asingle-precision number in this example contains 24 bits.

Anotherexample, in Python, also typed at an interactive prompt, is:

>>>x = 1.0

>>>p = 0

>>>while x != x + 1:

...   x = x \* 2

...   p = p + 1

...

>>>x

9007199254740992.0

>>>p

>>>x + 2 + 1

9007199254740996.0

Inthis case, we start with x = 1 and repeatedly double it until x = x + 1. Theresult is 253, because the double-precision floating-point format uses a 53-bitsignificand.

[点击打开链接][Link 1]

[radix]: https://en.wikipedia.org/wiki/Radix
[machine epsilon]: https://en.wikipedia.org/wiki/Machine_epsilon
[1]: https://en.wikipedia.org/wiki/Unit_in_the_last_place#cite_note-1
[citation needed]: https://en.wikipedia.org/wiki/Wikipedia:Citation_needed
[2]: https://en.wikipedia.org/wiki/Unit_in_the_last_place#cite_note-2
[3]: https://en.wikipedia.org/wiki/Unit_in_the_last_place#cite_note-3
[IEEE 754]: https://en.wikipedia.org/wiki/IEEE_754
[elementary arithmetic]: https://en.wikipedia.org/wiki/Elementary_arithmetic
[square root]: https://en.wikipedia.org/wiki/Square_root
[FMA]: https://en.wikipedia.org/wiki/Fused_multiply%E2%80%93add
[correctly rounded]: https://en.wikipedia.org/wiki/Rounding#Table-maker.27s_dilemma
[numeric]: https://en.wikipedia.org/wiki/Numerical_analysis
[libraries]: https://en.wikipedia.org/wiki/Library_%28computing%29
[transcendental functions]: https://en.wikipedia.org/wiki/Transcendental_function
[Table-Maker_s Dilemma]: https://en.wikipedia.org/wiki/Table-maker%27s_dilemma
[4]: https://en.wikipedia.org/wiki/Unit_in_the_last_place#cite_note-4
[Link 1]: https://en.wikipedia.org/wiki/Unit_in_the_last_place#External_links