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.

ChangeLog

@@ -1,5 +1,10 @@
 1999-10-29  Ulrich Drepper  <drepper@cygnus.com>
 
+	* 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.
+
 	* stdlib/tst-strtod.c: Add more hexadecimal tests.
 
 	* stdlib/strtod.c: Handle hexadecimal numbers with leading digit

manual/Makefile

@@ -69,7 +69,7 @@ examples = $(filter-out $(foreach d, $(add-ons), ../$d/%.c.texi), \
 chapters.% top-menu.%: libc-texinfo.sh $(texis)
 	AWK=$(AWK) $(SHELL) $< '$(chapters)' '$(add-chapters)' '$(appendices)'
 
-libc.dvi libc.info: chapters.texi top-menu.texi
+libc.dvi libc.info: chapters.texi top-menu.texi libm-err.texi
 libc.dvi: texinfo.tex
 
 # Generate the summary from the Texinfo source files for each chapter.
@@ -89,6 +89,15 @@ dir-add.info: xtract-typefun.awk $(texis)
 	 echo "END-INFO-DIR-ENTRY") > $@.new
 	mv -f $@.new $@
 
+# The table with the math errors is generated.
+libm-err.texi: stamp-libm-err
+stamp-libm-err: libm-err-tab.pl $(wildcard $(foreach dir,$(sysdirs),\
+						     $(dir)/libm-test-ulps))
+	pwd=`pwd`; \
+	$(PERL) $< $$pwd/.. > libm-err-tmp
+	$(move-if-change) libm-err-tmp libm-err.texi
+	touch $@
+
 # Generate Texinfo files from the C source for the example programs.
 %.c.texi: examples/%.c
 	sed -e 's,[{}],@&,g'				\
@@ -106,13 +115,14 @@ dir-add.info: xtract-typefun.awk $(texis)
 
 # Distribution.
 minimal-dist = summary.awk texis.awk tsort.awk libc-texinfo.sh libc.texinfo \
+	       libm-err.texi stamp-libm-err				    \
 	       $(filter-out summary.texi, $(nonexamples))		    \
 	       $(patsubst %.c.texi,examples/%.c, $(examples))
 
 doc-only-dist = Makefile COPYING.LIB
 distribute = $(minimal-dist) $(examples) stdio-fp.c	\
 	     libc.info* libc.?? libc.??s texinfo.tex	\
-	     xtract-typefun.awk dir-add.info dir
+	     xtract-typefun.awk dir-add.info dir libm-err-tab.pl
 export distribute := $(distribute)
 
 tar-it = tar chovf $@ $^

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