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diff --git a/manual/math.texi b/manual/math.texi index f2736c2c45..c9a4c3a611 100644 --- a/manual/math.texi +++ b/manual/math.texi @@ -61,6 +61,7 @@ same as @code{double}. * Exponents and Logarithms:: Also pow and sqrt. * Hyperbolic Functions:: sinh, cosh, tanh, etc. * Special Functions:: Bessel, gamma, erf. +* Errors in Math Functions:: Known Maximum Errors in Math Functions. * Pseudo-Random Numbers:: Functions for generating pseudo-random numbers. * FP Function Optimizations:: Fast code or small code. @@ -1203,6 +1204,47 @@ is negative, @code{yn} signals a domain error; if it is zero, @code{yn} signals overflow and returns @math{-@infinity}. @end deftypefun +@node Errors in Math Functions +@section Known Maximum Errors in Math Functions +@cindex math errors +@cindex ulps + +This section lists the known errors of the functions in the math +library. Errors are measured in ``units of the last place''. This is a +measure for the relative error. For a number @math{z} with the +representation @math{d.d@dots{}d@mul{}2^e} (we assume IEEE +floating-point numbers with base 2) the ULP is represented by + +@tex +$$\frac{|{\mathrm d.d\dots d - (z/2^e)|}{2^{p-1}}$$ +@end tex +@ifnottex +@smallexample +|d.d...d - (z / 2^e)| / 2^(p - 1) +@end smallexample +@end ifnottex + +@noindent +where @math{p} is the number of bits in the mantissa of the +floating-point number representation. Ideally the error for all +functions is always less than 0.5ulps. Using rounding bits this is also +possible and normally implemented for the basic operations. To achieve +the same for the complex math functions requires a lot more work and +this was not spend so far. + +Therefore many of the functions in the math library have errors. The +table lists the maximum error for each function which is exposed by one +of the existing tests in the test suite. It is tried to cover as much +as possible and really list the maximum error (or at least a ballpark +figure) but this is often not achieved due to the large search space. + +The table lists the ULP values for different architectures. Different +architectures have different results since their hardware support for +floating-point operations varies and also the existing hardware support +is different. + +@include libm-err.texi + @node Pseudo-Random Numbers @section Pseudo-Random Numbers @cindex random numbers |