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authorUlrich Drepper <drepper@redhat.com>1999-10-30 06:56:10 +0000
committerUlrich Drepper <drepper@redhat.com>1999-10-30 06:56:10 +0000
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Update.
* manual/math.texi (Errors in Math Functions): New section. * math/libm-err-tab.pl: Moved to... * manual/libm-err-tab.pl: ...here. * manual/Makefile: Add rules for math function error table generation.
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@@ -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