Re: issue with tgammaf

[Paul Zimmermann <Paul.Zimmermann@inria.fr>]

       Dear Keith,

> Do you know that newlib has two different sets of 32-bit math functions?
> Arm supplied a new set of many common math functions that use 64-bit
> doubles for the intermediate computations? I'd be very interested in
> measuring the effect of that code on accuracy. The new paths are
> selected by compiling newlib with -D__OBSOLETE_MATH_DEFAULT=0 while the
> old can be selected with -D__OBSOLETE_MATH_DEFAULT=1

here are the differences. For each function, the first line is newlib-4.1.0
with default configuration, the second line is newlib-4.1.0 with
-D__OBSOLETE_MATH_DEFAULT=1. Functions that do not appear (like acos) give
the same results in both cases (thus probably there is no new code).

For each line, "errors" is the number of binary32 inputs that are incorrectly
rounded, "errors2" is the number of inputs with an error >= 2 ulps, and "maxerr"
is the maximal error.

In summary, the old code (__OBSOLETE_MATH_DEFAULT=1) is always better in terms
of accuracy, with huge differences for some functions (cos, exp, exp10, exp2,
log, log2, sin, pow).

For bivariate functions (atan2, hypot, pow), the tests are not exhaustive.

Best regards,
Paul

acosh:
Total: errors=244658623 (5.72%) errors2=2698 maxerr=2.01e+00 ulp(s)
Total: errors=243413455 (5.69%) errors2=2698 maxerr=2.01e+00 ulp(s)

asinh:
Total: errors=542122908 (12.67%) errors2=2748 maxerr=1.78e+00 ulp(s)
Total: errors=539621602 (12.61%) errors2=2748 maxerr=1.78e+00 ulp(s)

cos:
Total: errors=209833072 (4.90%) errors2=6 maxerr=2.91e+00 ulp(s)
Total: errors=28209648 (0.66%) errors2=0 maxerr=7.76e-01 ulp(s)

cosh:
Total: errors=23905668 (0.56%) errors2=7706 maxerr=2.51e+00 ulp(s)
Total: errors=17868390 (0.42%) errors2=3558 maxerr=1.89e+00 ulp(s)

erf:
Total: errors=126741900 (2.96%) errors2=0 maxerr=9.68e-01 ulp(s)
Total: errors=126734730 (2.96%) errors2=0 maxerr=9.68e-01 ulp(s)

erfc:
Total: errors=21247299 (0.50%) errors2=1131209 maxerr=6.39e+01 ulp(s)
Total: errors=20982747 (0.49%) errors2=1067774 maxerr=6.39e+01 ulp(s)

exp:
Total: errors=17982847 (0.42%) errors2=0 maxerr=9.11e-01 ulp(s)
Total: errors=170646 (0.00%) errors2=0 maxerr=5.02e-01 ulp(s)

exp10:
Total: errors=18423203 (0.43%) errors2=0 maxerr=1.06e+00 ulp(s)
Total: errors=172143 (0.00%) errors2=0 maxerr=5.03e-01 ulp(s)

exp2:
Total: errors=18401203 (0.43%) errors2=0 maxerr=1.02e+00 ulp(s)
Total: errors=168362 (0.00%) errors2=0 maxerr=5.02e-01 ulp(s)

j0:
Total: errors=1338235574 (31.28%) errors2=279528826 maxerr=6.18e+06 ulp(s)
Total: errors=1334176546 (31.19%) errors2=269351612 maxerr=6.18e+06 ulp(s)

j1:
Total: errors=1818091384 (42.50%) errors2=1376362116 maxerr=1.68e+07 ulp(s)
Total: errors=1816660580 (42.46%) errors2=1373004084 maxerr=1.68e+07 ulp(s)

lgamma:
Total: errors=510903809 (11.94%) errors2=13277834 maxerr=7.50e+06 ulp(s)
Total: errors=506865167 (11.85%) errors2=13400241 maxerr=7.50e+06 ulp(s)

log:
Total: errors=13363494 (0.31%) errors2=0 maxerr=8.88e-01 ulp(s)
Total: errors=416908 (0.01%) errors2=0 maxerr=8.18e-01 ulp(s)

log10:
Total: errors=30061115 (0.70%) errors2=91958 maxerr=2.10e+00 ulp(s)
Total: errors=29787060 (0.70%) errors2=62225 maxerr=2.07e+00 ulp(s)

log2:
Total: errors=602745869 (14.09%) errors2=258 maxerr=1.65e+00 ulp(s)
Total: errors=313550 (0.01%) errors2=0 maxerr=7.52e-01 ulp(s)

sin:
Total: errors=206155238 (4.82%) errors2=0 maxerr=1.37e+00 ulp(s)
Total: errors=29362804 (0.69%) errors2=0 maxerr=5.61e-01 ulp(s)

sinh:
Total: errors=74587762 (1.74%) errors2=38924 maxerr=2.51e+00 ulp(s)
Total: errors=71328304 (1.67%) errors2=34776 maxerr=1.89e+00 ulp(s)

tgamma:
Total: errors=2028164922 (47.41%) errors2=1833526367 maxerr=2.39e+02 ulp(s)
Total: errors=2026865970 (47.38%) errors2=1832940352 maxerr=2.39e+02 ulp(s)

y0:
Total: errors=1306144386 (30.53%) errors2=191859954 maxerr=4.84e+06 ulp(s)
Total: errors=1304302538 (30.49%) errors2=187031173 maxerr=4.84e+06 ulp(s)

y1:
Total: errors=1201178797 (28.08%) errors2=153321647 maxerr=6.18e+06 ulp(s)
Total: errors=1199229206 (28.03%) errors2=148430893 maxerr=6.18e+06 ulp(s)

pow:
pow 0 -1 0x1.d55902p-1,-0x1.fe037ep+9 [1.69e+02] 168.527 168.5268728137016
pow 0 -1 0x1.025736p+0,0x1.309f94p+13 [0.817] 0.816736 0.816736102104187