| Commit message (Collapse) | Author | Age | Files | Lines |
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This patch fixes the remaining part of bug 16560, spurious underflows
from exp2 of arguments close to 0 (when the result is close to 1, so
should not underflow), by just using 1+x instead of a more complicated
calculation when the argument is sufficiently small.
Tested for x86_64, x86 and mips64.
[BZ #16560]
* math/e_exp2l.c [LDBL_MANT_DIG == 106] (LDBL_EPSILON): Undefine
and redefine.
(__ieee754_exp2l): Do not multiply small fractional parts by
M_LN2l.
* sysdeps/i386/fpu/e_exp2l.S (__ieee754_exp2l): Just add 1 to
small argument.
* sysdeps/ieee754/dbl-64/e_exp2.c (__ieee754_exp2): Likewise.
* sysdeps/ieee754/flt-32/e_exp2f.c (__ieee754_exp2f): Likewise.
* sysdeps/x86_64/fpu/e_exp2l.S (__ieee754_exp2l): Likewise.
* math/auto-libm-test-in: Add more tests of exp2.
* math/auto-libm-test-out: Regenerated.
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This patch makes sincos set errno to EDOM when passed an infinity,
similarly to sin and cos.
Tested for x86_64, x86, powerpc and mips64. I don't know if the
architecture-specific implementations for ia64 and m68k might need
corresponding fixes.
2015-02-11 Joseph Myers <joseph@codesourcery.com>
[BZ #15467]
* sysdeps/ieee754/dbl-64/s_sincos.c: Include <errno.h>.
(__sincos): Set errno to EDOM for infinite argument.
* sysdeps/ieee754/flt-32/s_sincosf.c: Include <errno.h>.
(SINCOSF_FUNC): Set errno to EDOM for infinite argument.
* sysdeps/ieee754/ldbl-128/s_sincosl.c: Include <errno.h>.
(__sincosl): Set errno to EDOM for infinite argument.
* sysdeps/ieee754/ldbl-128ibm/s_sincosl.c: Include <errno.h>.
(__sincosl): Set errno to EDOM for infinite argument.
* sysdeps/ieee754/ldbl-96/s_sincosl.c: Include <errno.h>.
(__sincosl): Set errno to EDOM for infinite argument.
* math/libm-test.inc (sincos_test_data): Test errno setting.
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Platforms with 64-bit registers where 32-bit values need to have the
high 32 bits set in a particular way need to have an explicit cast
when using the 64-bit sysdeps/ieee754/dbl-64/wordsize-64 version
of llround() as lround(). This includes tilegx32, and likely MIPS.
x32 does not need this, and AArch64 ILP32 will not either. Require
it to be specified in sysdep.h to be explicit.
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Continuing the fixes for C90 libm functions calling C99 fe* functions,
this patch fixes the case of fegetround by making it a weak alias of
__fegetround and making the affected code call __fegetround.
Tested for x86_64 (testsuite, and that disassembly of installed shared
libraries is unchanged by the patch). Also tested for ARM
(soft-float) that fegetround failures disappear from the linknamespace
test failures (feholdexcept, fesetenv, fesetround and feupdateenv
remain to be addressed before bug 17748 is fully fixed, although this
patch may suffice to fix the failures in some cases, when the libc_fe*
functions are implemented but there is no architecture-specific sqrt
implementation in use so there were failures from fegetround used by
sqrt but no other such failures).
[BZ #17748]
* include/fenv.h (__fegetround): Declare. Use libm_hidden_proto.
* math/fegetround.c (fegetround): Rename to __fegetround and
define as weak alias of __fegetround. Use libm_hidden_weak.
* sysdeps/aarch64/fpu/fegetround.c (fegetround): Likewise.
* sysdeps/alpha/fpu/fegetround.c (fegetround): Likewise.
* sysdeps/arm/fegetround.c (fegetround): Likewise.
* sysdeps/hppa/fpu/fegetround.c (fegetround): Likewise.
* sysdeps/i386/fpu/fegetround.c (fegetround): Likewise.
* sysdeps/ia64/fpu/fegetround.c (fegetround): Likewise.
* sysdeps/m68k/fpu/fegetround.c (fegetround): Likewise.
* sysdeps/mips/fpu/fegetround.c (fegetround): Likewise.
* sysdeps/powerpc/fpu/fegetround.c (fegetround): Likewise.
Undefine after rather than before function definition; use
parentheses around function name in definition.
(__fegetround): Also undefine macro after function definition.
* sysdeps/powerpc/nofpu/fegetround.c (fegetround): Rename to
__fegetround and define as weak alias of __fegetround. Use
libm_hidden_weak. Do not undefine as macro.
* sysdeps/powerpc/powerpc32/e500/nofpu/fegetround.c (fegetround):
Likewise.
* sysdeps/s390/fpu/fegetround.c (fegetround): Rename to
__fegetround and define as weak alias of __fegetround. Use
libm_hidden_weak.
* sysdeps/sh/sh4/fpu/fegetround.c (fegetround): Likewise.
* sysdeps/sparc/fpu/fegetround.c (fegetround): Likewise.
* sysdeps/tile/math_private.h (__fegetround): New inline function.
* sysdeps/x86_64/fpu/fegetround.c (fegetround): Rename to
__fegetround and define as weak alias of __fegetround. Use
libm_hidden_weak.
* sysdeps/ieee754/dbl-64/e_sqrt.c (__ieee754_sqrt): Use
__fegetround instead of fegetround.
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libm uses symbols mpone and mptwo for internal purposes. This patch
moves them to the implementation namespace (__mpone and __mptwo).
Tested for x86_64 (testsuite, and that installed stripped shared
libraries are unchanged by the patch).
[BZ #17616]
* sysdeps/ieee754/dbl-64/mpa.c (mpone): Rename to __mpone.
(mptwo): Rename to __mptwo.
(__inv): Use __mptwo instead of mptwo.
* sysdeps/ieee754/dbl-64/mpa.h (mpone): Rename to __mpone.
(mptwo): Rename to __mptwo.
* sysdeps/ieee754/dbl-64/mpatan.c (__mpatan): Use __mpone instead
of mpone and __mptwo instead of mptwo.
* sysdeps/ieee754/dbl-64/mpatan2.c (__mpatan2): Use __mpone
instead of mpone.
* sysdeps/ieee754/dbl-64/mpexp.c (__mpexp): Likewise.
* sysdeps/ieee754/dbl-64/mplog.c (__mplog): Likewise.
* sysdeps/ieee754/dbl-64/sincos32.c (__c32): Use __mpone instead
of mpone and __mptwo instead of mptwo.
(__mpranred): Use __mpone instead of mpone.
* conform/Makefile (test-xfail-ISO/math.h/linknamespace): Remove
variable.
(test-xfail-ISO99/complex.h/linknamespace): Likewise.
(test-xfail-ISO99/math.h/linknamespace): Likewise.
(test-xfail-ISO99/tgmath.h/linknamespace): Likewise.
(test-xfail-ISO11/complex.h/linknamespace): Likewise.
(test-xfail-ISO11/math.h/linknamespace): Likewise.
(test-xfail-ISO11/tgmath.h/linknamespace): Likewise.
(test-xfail-XPG3/math.h/linknamespace): Likewise.
(test-xfail-XPG4/math.h/linknamespace): Likewise.
(test-xfail-POSIX/math.h/linknamespace): Likewise.
(test-xfail-UNIX98/math.h/linknamespace): Likewise.
(test-xfail-XOPEN2K/complex.h/linknamespace): Likewise.
(test-xfail-XOPEN2K/math.h/linknamespace): Likewise.
(test-xfail-XOPEN2K/tgmath.h/linknamespace): Likewise.
(test-xfail-POSIX2008/complex.h/linknamespace): Likewise.
(test-xfail-POSIX2008/math.h/linknamespace): Likewise.
(test-xfail-POSIX2008/tgmath.h/linknamespace): Likewise.
(test-xfail-XOPEN2K8/complex.h/linknamespace): Likewise.
(test-xfail-XOPEN2K8/math.h/linknamespace): Likewise.
(test-xfail-XOPEN2K8/tgmath.h/linknamespace): Likewise.
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This patch fixes bugs 16561 and 16562, bad results of yn in overflow
cases in non-default rounding modes, both because an intermediate
overflow in the recurrence does not get detected if the result is not
an infinity and because an overflowing result may occur in the wrong
sign. The fix is to set FE_TONEAREST mode internally for the parts of
the function where such overflows can occur (which includes the call
to y1 - where yn is used to compute a Bessel function of order -1,
negating the result of y1 isn't correct for overflowing results in
directed rounding modes) and then compute an overflowing value in the
original rounding mode if the to-nearest result was an infinity.
Tested x86_64 and x86 and ulps updated accordingly. Also tested for
mips64 and powerpc32 to test the ldbl-128 and ldbl-128ibm changes.
(The tests for these bugs were added in my previous y1 patch, so the
only thing this patch has to do with the testsuite is enable yn
testing in all rounding modes.)
[BZ #16561]
[BZ #16562]
* sysdeps/ieee754/dbl-64/e_jn.c: Include <float.h>.
(__ieee754_yn): Set FE_TONEAREST mode internally and then
recompute overflowing results in original rounding mode.
* sysdeps/ieee754/flt-32/e_jnf.c: Include <float.h>.
(__ieee754_ynf): Set FE_TONEAREST mode internally and then
recompute overflowing results in original rounding mode.
* sysdeps/ieee754/ldbl-128/e_jnl.c: Include <float.h>.
(__ieee754_ynl): Set FE_TONEAREST mode internally and then
recompute overflowing results in original rounding mode.
* sysdeps/ieee754/ldbl-128ibm/e_jnl.c: Include <float.h>.
(__ieee754_ynl): Set FE_TONEAREST mode internally and then
recompute overflowing results in original rounding mode.
* sysdeps/ieee754/ldbl-96/e_jnl.c: Include <float.h>.
(__ieee754_ynl): Set FE_TONEAREST mode internally and then
recompute overflowing results in original rounding mode.
* sysdeps/i386/fpu/fenv_private.h [!__SSE2_MATH__]
(libc_feholdsetround_ctx): New macro.
* math/libm-test.inc (yn_test): Use ALL_RM_TEST.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps : Likewise.
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This patch fixes spurious underflows from exp10 for arguments near 0
(part of bug 16560; that bug also includes spurious underflows from
exp2, which are not fixed by this patch). The problem is underflows
in the internal computation converting the exp10 argument to arguments
for exp (with extra precision), and the fix is simply to return 1
early for arguments near enough to 0 (just as arguments with large
enough magnitude have their own overflow / underflow logic at the
start of the function).
Tested x86_64 and x86 and ulps updated accordingly; also tested for
powerpc32 and mips64 to validate the ldbl-128ibm and ldbl-128 changes.
[BZ #16560]
* sysdeps/ieee754/dbl-64/e_exp10.c (__ieee754_exp10): Return 1 for
arguments close to 0.
* sysdeps/ieee754/ldbl-128/e_exp10l.c (__ieee754_exp10l):
Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_exp10l.c (__ieee754_exp10l):
Likewise.
* math/auto-libm-test-in: Add more tests of exp10.
* math/auto-libm-test-out: Regenerated.
* sysdeps/x86_64/fpu/libm-test-ulps: Update.
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0 (bug 17061).
This patch fixes bug 16354, spurious underflows from cosh when a tiny
argument is passed to expm1 and expm1 correctly underflows although
the final result of cosh should be 1. As noted in that bug, some
cases are latent because of expm1 implementations not raising
underflow (bug 16353), but all the implementations are fixed
similarly. They already contained checks for tiny arguments, but the
checks were too late to avoid underflow from expm1 (although they
would avoid underflow from subsequent squaring of the result of
expm1); they are moved before the expm1 calls.
The thresholds used for considering arguments tiny are not
particularly consistent in how they relate to the precision of the
floating-point format in question. They are, however, all sufficient
to ensure that the round-to-nearest result of cosh is indeed 1 below
the threshold (although sometimes they are smaller than necessary).
But the previous logic did not return 1, but the previously computed 1
+ expm1(abs(x)) value. And the thresholds in the ldbl-128 and
ldbl-128ibm code (0x1p-71L - I suspect 0x3f8b was intended in the code
instead of 0x3fb8 - and (roughly) 0x1p-55L) are not sufficient for
that value to be 1. So by moving the test for tiny arguments, and
consequently returning 1 directly now the expm1 value hasn't been
computed by that point, this patch also fixes bug 17061, the (large
number of ulps) inaccuracy for small arguments in those
implementations. Tests for that bug are duly added.
Tested x86_64 and x86 and ulps updated accordingly. Also tested for
mips64 and powerpc32 to validate the ldbl-128 and ldbl-128ibm changes.
[BZ #16354]
[BZ #17061]
* sysdeps/ieee754/dbl-64/e_cosh.c (__ieee754_cosh): Check for
small arguments before calling __expm1.
* sysdeps/ieee754/flt-32/e_coshf.c (__ieee754_coshf): Check for
small arguments before calling __expm1f.
* sysdeps/ieee754/ldbl-128/e_coshl.c (__ieee754_coshl): Check for
small arguments before calling __expm1l.
* sysdeps/ieee754/ldbl-128ibm/e_coshl.c (__ieee754_coshl):
Likewise.
* sysdeps/ieee754/ldbl-96/e_coshl.c (__ieee754_coshl): Likewise.
* math/auto-libm-test-in: Add more cosh tests. Do not allow
spurious underflow for some cosh tests.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
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This patch fixes bug 17050, missing errno setting for y1 overflow (for
small positive arguments). An appropriate check is added for overflow
directly in the __ieee754_y1 implementation, similar to the check
present for yn (doing it there rather than in the wrapper also avoids
yn needing to repeat the check when called for order 1 or -1 and it
uses __ieee754_y1).
Tested x86_64 and x86; no ulps update needed. Also tested for mips64
to verify the ldbl-128 fix (the ldbl-128ibm code just #includes the
ldbl-128 file).
[BZ #17050]
* sysdeps/ieee754/dbl-64/e_j1.c: Include <errno.h>.
(__ieee754_y1): Set errno if return value overflows.
* sysdeps/ieee754/flt-32/e_j1f.c: Include <errno.h>.
(__ieee754_y1f): Set errno if return value overflows.
* sysdeps/ieee754/ldbl-128/e_j1l.c: Include <errno.h>.
(__ieee754_y1l): Set errno if return value overflows.
* sysdeps/ieee754/ldbl-96/e_j1l.c: Include <errno.h>.
(__ieee754_y1l): Set errno if return value overflows.
* math/auto-libm-test-in: Add more tests of y0, y1 and yn.
* math/auto-libm-test-out: Regenerated.
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This patch fixes bug 16315, bad pow handling of overflow/underflow in
non-default rounding modes. Tests of pow are duly converted to
ALL_RM_TEST to run all tests in all rounding modes.
There are two main issues here. First, various implementations
compute a negative result by negating a positive result, but this
yields inappropriate overflow / underflow values for directed
rounding, so either overflow / underflow results need recomputing in
the correct sign, or the relevant overflowing / underflowing operation
needs to be made to have a result of the correct sign. Second, the
dbl-64 implementation sets FE_TONEAREST internally; in the overflow /
underflow case, the result needs recomputing in the original rounding
mode.
Tested x86_64 and x86 and ulps updated accordingly.
[BZ #16315]
* sysdeps/i386/fpu/e_pow.S (__ieee754_pow): Ensure possibly
overflowing or underflowing operations take place with sign of
result.
* sysdeps/i386/fpu/e_powf.S (__ieee754_powf): Likewise.
* sysdeps/i386/fpu/e_powl.S (__ieee754_powl): Likewise.
* sysdeps/ieee754/dbl-64/e_pow.c: Include <math.h>.
(__ieee754_pow): Recompute overflowing and underflowing results in
original rounding mode.
* sysdeps/x86/fpu/powl_helper.c: Include <stdbool.h>.
(__powl_helper): Allow negative argument X and scale negated value
as needed. Avoid passing value outside [-1, 1] to f2xm1.
* sysdeps/x86_64/fpu/e_powl.S (__ieee754_powl): Ensure possibly
overflowing or underflowing operations take place with sign of
result.
* sysdeps/x86_64/fpu/multiarch/e_pow.c [HAVE_FMA4_SUPPORT]:
Include <math.h>.
* math/auto-libm-test-in: Add more tests of pow.
* math/auto-libm-test-out: Regenerated.
* math/libm-test.inc (pow_test): Use ALL_RM_TEST.
(pow_tonearest_test_data): Remove.
(pow_test_tonearest): Likewise.
(pow_towardzero_test_data): Likewise.
(pow_test_towardzero): Likewise.
(pow_downward_test_data): Likewise.
(pow_test_downward): Likewise.
(pow_upward_test_data): Likewise.
(pow_test_upward): Likewise.
(main): Don't call removed functions.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
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Errno is not set and the testcases will fail.
Now the scalbln-aliases are removed in i386/m68
and the wrappers are used when calling the scalbln-functions.
On ia64 only scalblnf has its own implementation.
For scalbln and scalblnl the ieee754/dbl-64 and ieee754/ldbl-96 are used, thus
the wrappers are needed, too.
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Bug 16516 reports spurious underflows from erf (for all floating-point
types), when the result is close to underflowing but does not actually
underflow.
erf (x) is about (2/sqrt(pi))*x for x close to 0, so there are
subnormal arguments for which it does not underflow. The various
implementations do (x + efx*x) (for efx = 2/sqrt(pi) - 1), for greater
accuracy than if just using a single multiplication by an
approximation to 2/sqrt(pi) (effectively, this way there are a few
more bits in the approximation to 2/sqrt(pi)). This can introduce
underflows when efx*x underflows even though the final result does
not, so a scaled calculation with 8*efx is done in these cases - but 8
is not a big enough scale factor to avoid all such underflows. 16 is
(any underflows with a scale factor of 16 would only occur when the
final result underflows), so this patch changes the code to use that
factor. Rather than recomputing all the values of the efx8 variable,
it is removed, leaving it to the compiler's constant folding to
compute 16*efx. As such scaling can also lose underflows when the
final scaling down happens to be exact, appropriate checks are added
to ensure underflow exceptions occur when required in such cases.
Tested x86_64 and x86; no ulps updates needed. Also spot-checked for
powerpc32 and mips64 to verify the changes to the ldbl-128ibm and
ldbl-128 implementations.
[BZ #16516]
* sysdeps/ieee754/dbl-64/s_erf.c (efx8): Remove variable.
(__erf): Scale by 16 instead of 8 in potentially underflowing
case. Ensure exception if result actually underflows.
* sysdeps/ieee754/flt-32/s_erff.c (efx8): Remove variable.
(__erff): Scale by 16 instead of 8 in potentially underflowing
case. Ensure exception if result actually underflows.
* sysdeps/ieee754/ldbl-128/s_erfl.c: Include <float.h>.
(efx8): Remove variable.
(__erfl): Scale by 16 instead of 8 in potentially underflowing
case. Ensure exception if result actually underflows.
* sysdeps/ieee754/ldbl-128ibm/s_erfl.c: Include <float.h>.
(efx8): Remove variable.
(__erfl): Scale by 16 instead of 8 in potentially underflowing
case. Ensure exception if result actually underflows.
* sysdeps/ieee754/ldbl-96/s_erfl.c: Include <float.h>.
(efx8): Remove variable.
(__erfl): Scale by 16 instead of 8 in potentially underflowing
case. Ensure exception if result actually underflows.
* math/auto-libm-test-in: Add more tests of erf.
* math/auto-libm-test-out: Regenerated.
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My recent exp patch introduced warnings about implicit __isinf
declarations in exp because e_exp.c didn't include <math.h>. This
patch fixes this. Because <math.h> can't be included after
<math_private.h> (because of macro definitions of __nan*), it was
necessary to put an include in sysdeps/x86_64/fpu/multiarch/e_exp.c as
well.
Tested x86_64.
* sysdeps/ieee754/dbl-64/e_exp.c: Include <math.h>.
* sysdeps/x86_64/fpu/multiarch/e_exp.c
[HAVE_FMA4_SUPPORT || HAVE_AVX_SUPPORT]: Likewise.
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The dbl-64 version of exp needs round-to-nearest mode for its internal
computations, but that has the consequence of inappropriate
overflowing and underflowing results in other rounding modes. This
patch fixes this by recomputing the relevant results in cases where
the round-to-nearest result overflows to infinity or underflows to
zero (most of the diffs are actually just consequent reindentation).
Tests are enabled in all rounding modes for complex functions using
exp - but not for cexp because it turns out there are bugs causing
spurious underflows for cexp for some tests, which will need to be
fixed separately (I suspect ccos ccosh csin csinh ctan ctanh have
similar bugs, just not shown by the present set of test inputs).
Tested x86_64 and x86 and ulps updated accordingly.
[BZ #16284]
* sysdeps/ieee754/dbl-64/e_exp.c (__ieee754_exp): Use original
rounding mode to recompute results that overflow to infinity or
underflow to zero.
* math/auto-libm-test-in: Don't mark tests as expected to fail for
bug 16284.
* math/auto-libm-test-out: Regenerated.
* math/libm-test.inc (ccos_test): Use ALL_RM_TEST.
(ccosh_test): Likewise.
(csin_test_data): Use plus_oflow.
(csin_test): Use ALL_RM_TEST.
(csinh_test_data): Use plus_oflow.
(csinh_test): Use ALL_RM_TEST.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
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According to ISO C Annex F, log (1) should be +0 in all rounding
modes, but some implementations in glibc wrongly return -0 in
round-downward mode (mapping to log1p (x - 1) is problematic because 1
- 1 is -0 in round-downward mode, and log1p (-0) is -0). This patch
fixes this. (It helps with some implementations of other functions
such as acosh, log2 and log10 that call out to log, but not enough to
enable all-rounding-modes testing for those functions without further
fixes to other implementations of them.)
Tested x86_64 and x86 and ulps updated accordingly, and did spot tests
for mips64 for the ldbl-128 fix, and i586 for the sysdeps/i386/fpu
implementations shadowed by those in sysdeps/i386/i686/fpu.
[BZ #16731]
* sysdeps/i386/fpu/e_log.S (__ieee754_log): Take absolute value
when x - 1 is zero.
* sysdeps/i386/fpu/e_logf.S (__ieee754_logf): Likewise.
* sysdeps/i386/fpu/e_logl.S (__ieee754_logl): Likewise.
* sysdeps/i386/i686/fpu/e_logl.S (__ieee754_logl): Likewise.
* sysdeps/ieee754/dbl-64/e_log.c (__ieee754_log): Return +0 when
argument is 1.
* sysdeps/ieee754/ldbl-128/e_logl.c (__ieee754_logl): Likewise.
* sysdeps/x86_64/fpu/e_logl.S: Take absolute value when x - 1 is
zero.
* math/libm-test.inc (log_test): Use ALL_RM_TEST.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
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In 84ba214c, I removed some redundant sign computations and in the
process, I incorrectly got rid of a temporary variable, thus passing
the absolute value of the input to bsloww1. This caused #16623.
This fix undoes the incorrect change.
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An incorrect variable name was used during the refactoring done in
commit 4aafb73c.
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This patch consolidates the multiple copies of code that looks up sin
and cos of a number from the lookup table and computes the final
value, into static functions. This does not have a noticeable
performance impact since the functions are inlined by gcc.
There is further scope for consolidation in the functions but they
cause a more noticable impact on performance (>5%) due to which I have
held back on them.
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Removed more redundant computations in the slow paths of the sin and
cos functions. The notable change is the passing of the most
significant bits of X to the slow functions to check if X is positive
so that just the absolute value of x can be passed and the repeated
ABS() operation is avoided.
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The A and DA arguments in reduce_and_compute are useless and hence
have been removed.
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There are multiple points in the code where the absolute value of a
number is computed multiple times or is computed even though the value
can only be positive. This change removes those redundant
computations. Tested on x86_64 to verify that there were no
regressions in the testsuite.
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- Remove redundant mynumber union definitions
- Clean up a clumsy ternary operator
- Rename TAYLOR_SINCOS to TAYLOR_SIN since we're only expanding the
sin Taylor series in it.
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Consolidate conditionals in multiple precision sin and cos functions
to prepare the code for addition of probe points.
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Add systemtap probes to various slow paths in libm so that application
developers may use systemtap to find out if their applications are
hitting these slow paths. We have added probes for pow, exp, log,
tan, atan and atan2.
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Consolidated common Taylor series polynomials into macros in s_sin.c
to make it a bit cleaner.
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