The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
README for libm-test math test suite
====================================
The libm-test math test suite tests a number of function points of
math functions in the GNU C library. The following sections contain a
brief overview. Please note that the test drivers and the Perl script
"gen-libm-test.pl" have some options. A full list of options is
available with --help (for the test drivers) and -h for
"gen-libm-test.pl".
What is tested?
===============
The tests just evaluate the functions at specified points and compare
the results with precomputed values and the requirements of the ISO
C99 standard.
Besides testing the special values mandated by IEEE 754 (infinities,
NaNs and minus zero), some more or less random values are tested.
Files that are part of libm-test
================================
The main file is "libm-test.inc". It is independent of the target
platform and the specific real floating type and format and contains
placeholder test "templates" for math functions defined in libm.
The file, along with a generated file named "auto-libm-test-out",
is preprocessed by the Perl script "gen-libm-test.pl" to expand
the templates and produce a set of test cases for each math function
that are specific to the target platform but still independent of
the real floating type. The results of the processing are
"libm-test.c" and a file "libm-test-ulps.h" with platform specific
deltas by which the actual math function results may deviate from
the expected results and still be considered correct.
The test drivers "test-double.c", "test-float.c", and "test-ldouble.c"
test the normal double, float and long double implementation of libm.
The test drivers with an 'i' in their name ("test-idouble.c",
"test-ifloat.c", and "test-ildoubl.c") test the corresponding inline
functions (where available - otherwise they also test the real
functions in libm). Each driver selects the desired real floating
type to exercise the math functions to test with (float, double, or
long double) by defining a small set of macros just before including
the generic "libm-test.c" file. Each driver also either defines or
undefines the __NO_MATH_INLINES macro just before including
"libm-test.c" to select either the real or inline functions,
respectively. Each driver is compiled into a single executable test
program with the corresponding name.
As mentioned above, the "gen-libm-test.pl" script looks for a file
named "libm-test-ulps" in the platform specific sysdep directory (or
its fpu or nofpu subdirectory) and for each variant (real floating
type and rounding mode) of every tested function reads from it the
maximum difference expressed as Units of Least Precision (ULP) the
actual result of the function may deviate from the expected result
before it's considered incorrect.
The "auto-libm-test-out" file contains sets of test cases to exercise,
the conditions under which to exercise each, and the expected results.
The file is generated by the "gen-auto-libm-tests" program from the
"auto-libm-test-in" file. See the comments in gen-auto-libm-tests.c
for details about the content and format of the -in and -out files.
How can I generate "libm-test-ulps"?
====================================
To automatically generate a new "libm-test-ulps" run "make regen-ulps".
This generates the file "math/NewUlps" in the build directory. The file
contains the sorted results of all the tests. You can use the "NewUlps"
file as the machine's updated "libm-test-ulps" file. Copy "NewUlps" to
"libm-test-ulps" in the appropriate machine sysdep directory. Verify
the changes, post your patch, and check it in after review.
To manually generate a new "libm-test-ulps" file, first remove "ULPs"
file in the current directory, then you can execute for example:
./testrun.sh math/test-double -u --ignore-max-ulp=yes
This generates a file "ULPs" with all double ULPs in it, ignoring any
previously calculated ULPs, and running with the newly built dynamic
loader and math library (assumes you didn't install your build). Now
generate the ULPs for all other formats, the tests will be appending the
data to the "ULPs" file. As final step run "gen-libm-test.pl" with the
file as input and ask to generate a pretty printed output in the file
"NewUlps":
gen-libm-test.pl -u ULPs -n
Copy "NewUlps" to "libm-test-ulps" in the appropriate machine sysdep
directory.
Note that the test drivers have an option "-u" to output an unsorted
list of all epsilons that the functions have. The output can be read
in directly but it's better to pretty print it first.
"gen-libm-test.pl" has an option to generate a pretty-printed and
sorted new ULPs file from the output of the test drivers.
Contents of libm-test-ulps
==========================
Since libm-test-ulps can be generated automatically, just a few notes.
The file contains lines for maximal errors of single functions, like:
Function "yn":
idouble: 6
The keywords are float, ifloat, double, idouble, ldouble and ildouble
(the prefix i stands for inline).
Adding tests to libm-test.inc
=============================
The tests are evaluated by a set of special test macros. The macros
start with "TEST_" followed by a specification the input values, an
underscore and a specification of the output values. As an example,
the test macro for a function with input of type FLOAT (FLOAT is
either float, double, long double) and output of type FLOAT is
"TEST_f_f". The macro's parameter are the name of the function, the
input parameter, output parameter and optionally one exception
parameter.
The accepted parameter types are:
- "f" for FLOAT
- "b" for boolean - just tests if the output parameter evaluates to 0
or 1 (only for output).
- "c" for complex. This parameter needs two values, first the real,
then the imaginary part.
- "i" for int.
- "l" for long int.
- "L" for long long int.
- "F" for the address of a FLOAT (only as input parameter)
- "I" for the address of an int (only as input parameter)
How to read the test output
===========================
Running each test on its own at the default level of verbosity will
print on stdout a line describing the implementation of math functions
exercised by the test (float, double, or long double), along with
whether the inline set has been selected, regardless of whether or
not any inline functions actually exist. This is then followed by
the details of test failures (if any). The output concludes by
a summary listing the number of test cases exercised and the number
of test failures uncovered.
For each test failure (and for each test case at higher levels of
verbosity), the output contains the name of the function under test
and its arguments or conditions that triggered the failure. Note
that the name of the function in the output need not correspond
exactly to the name of the math function actually invoked. For example,
the output will refer to the "acos" function even if the actual function
under test is acosf (for the float version) or acosl (for the long
double version). Also note that the function arguments may be shown
in either the decimal or the hexadecimal floating point format which
may or may not correspond to the format used in the auto-libm-test-in
file. Besides the name of the function, for each test failure the
output contains the actual and expected results and the difference
between the two, printed in both the decimal and hexadecimal
floating point format, and the ULP and maximum ULP for the test
case.
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