IEEE-754 单精度表示的最大绝对和相对误差？

Question

I'm looking to find the maximum overall absolute and relative error of IEEE-754 single precision representation. Sign: 1 bit, Exponent: 8 bits, Significand: 23 bits.

I understood that when normalised, the maximum number of digits in the significand would be 23 (and we assume a sign bit and exponent of 8 obviously). Hence if any extra digits turned up, then the error would propagate from 2^-24 onwards ie 2^-24, 2^-25, 2^-26... Hence I completed a geometric infinite sum of this to find an error: so i got 2^-23. However, I'm unsure whether this is correct for the relative error. Relative error would be the ((true value-given value)/true value)*100. I'm not sure if this is a wrong approach.

Additionally, I'm confused on how to find an absolute error. Could anyone assist please. Thanks in advance.

Answer 1

All finite IEEE-754 single precision are exact. There is no error in the value itself.

A calculation/conversion may incur an error as there are only about 2 ³² different IEEE-754 single precision values and there are infinite possible calculations results. Typically a nearby single precision value is selected when the true result is not encodable.

If we limit the discussion to calculation results that are within a pair of finite single precision values, then the error could be at most 1.0 ULP ^*1 .

Note: finite range +/-3.4028235... × 10 ³⁸ or FLT_MAX

Within that range, the absolute difference between the true result and the encoded single precision is then at most FLT_MAX - next_smallest_float(FLOAT_MAX) . This is close to FLOAT_MAX * pow(2,-24) (about 2.03 * 10 ³¹ ). Single precision has a 24-bit significand (23-bits explicitly encoded, 1 implied).

Outside that range the absolute error can be infinite.

For many calculations, when the results are in the normal single precision range, the relative error is within 1.0 * ULP of the correct answer ^*1 . For transcendental calculations like sine , the error is within 2.0 * ULP of the correct answer. That can be much worse for weak implementations.

When the true result is small and the single precision value is a non-zero sub-normal , the relative error grows as the true value nears 0.0 until 0.5 * pow(2,0) or 1/2. Note this is considering the relative error as:

relative_error_IEEE = |true value - IEEE value|/IEEE value

When the IEEE value is zero or the relative error is determined as below, the relative error approaches infinity.

relative_error_true = |true value - IEEE value|/true value

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^*1 Common calculations like +,-,*,/ should be within 0.5 ULP when the rounding mode is round-to-nearest .

Answer 2

The largest error is 10141204801825835211973625643008, and the largest relative error is 0.5:

>>> (2**(0xfe-150)* 0xffffff - 2**(0xfe-150)* 0xfffffe)/2 
10141204801825835211973625643008L
>>> 2**(0xfe-150)* 0xffffff
340282346638528859811704183484516925440L
>>> print ("%100.100f\n" % (10141204801825835211973625643007.0/340282346638528859811704183484516925440.0))
0.0000000298023241640522577793688714653530524856250849552452564239501953125000000000000000000000000000

>>> print("%151.151f\n" % ( ( 2**(0x0-150)* 0x000002 - 2**(0x0-150)* 0x000001 )/2 ))
0.0000000000000000000000000000000000000000000003503246160812042677309323958224790328200654854691289429392670709724477706714651503716595470905303955078125

>>> print("%151.151f\n" % (2**(0x0-150)* 0x00001))
0.0000000000000000000000000000000000000000000007006492321624085354618647916449580656401309709382578858785341419448955413429303007433190941810607910156250

>>> 3503246160812042677309323958224790328200654854691289429392670709724477706714651503716595470905303955078125.0/7006492321624085354618647916449580656401309709382578858785341419448955413429303007433190941810607910156250
0.5