[Bug 281001] Improve accuracy of asinf(x) and acosf(x)

https://bugs.freebsd.org/bugzilla/show_bug.cgi?id=281001

--- Comment #6 from commit-hook@FreeBSD.org ---
A commit in branch stable/13 references this bug:

URL:
https://cgit.FreeBSD.org/src/commit/?id=75bca9e7553c70d97b36df30ab05bbbbacb8a499

commit 75bca9e7553c70d97b36df30ab05bbbbacb8a499
Author:     Steve Kargl <kargl@FreeBSD.org>
AuthorDate: 2024-08-29 19:44:48 +0000
Commit:     Dimitry Andric <dim@FreeBSD.org>
CommitDate: 2024-09-08 07:37:57 +0000

    Improve accuracy of asinf(3) and acosf(3)

    This uses a better rational approximation to improve the accuracy of
    both functions. For exhaustive testing of asinf(3) in the interval, the
    current libm gives:

        % ./tlibm asin -fPED -x 0x1p-12f -X 1
        Interval tested for asinf: [0.000244141,1]
               ulp <= 0.5:  97.916% 98564994 |  97.916% 98564994
        0.5 <  ulp <  0.6:  2.038% 2051023 |  99.953% 100616017
        0.6 <  ulp <  0.7:  0.047%   47254 | 100.000% 100663271
        0.7 <  ulp <  0.8:  0.000%      25 | 100.000% 100663296
        Max ulp: 0.729891 at 5.00732839e-01

    which isn't too bad given that much of the computation is actually done
    in double floating point.

    With the new rational approximation, exhaustive testing yields:

        % ./tlibm asin -fPED -x 0x1p-12f -X 1
        Interval tested for asinf: [0.000244141,1]
               ulp <= 0.5:  99.711% 100372643 |  99.711% 100372643
        0.5 <  ulp <  0.6:  0.288%  290357 | 100.000% 100663000
        0.6 <  ulp <  0.7:  0.000%     296 | 100.000% 100663296
        Max ulp: 0.636344 at 5.09706438e-01

    Similarly, for exhaustive testing of asinf(3) in the interval, the
    current libm gives:

        % ./tlibm acos -fPED -x -1 -X -0x1p-12f
        Interval tested for acosf: [-1,-0.000244141]
               ulp <= 0.5:  97.008% 97651921 |  97.008% 97651921
        0.5 <  ulp <  0.6:   2.441%  2457242 |  99.450% 100109163
        0.6 <  ulp <  0.7:   0.472%   475503 |  99.922% 100584666
        0.7 <  ulp <  0.8:   0.071%    71309 |  99.993% 100655975
        0.8 <  ulp <  0.9:   0.007%     7319 | 100.000% 100663294
        0.9 <  ulp <  1.0:   0.000%        2 | 100.000% 100663296
        Max ulp: 0.914007 at -5.01484931e-01

        % ./tlibm acos -fPED -x 0x1p-12f -X 1
        Interval tested for acosf: [0.000244141,1]
               ulp <= 0.5:  97.317% 97962530 |  97.317% 97962530
        0.5 <  ulp <  0.6:   2.340%  2355182 |  99.657% 100317712
        0.6 <  ulp <  0.7:   0.314%   316134 |  99.971% 100633846
        0.7 <  ulp <  0.8:   0.029%    29450 | 100.000% 100663296
        Max ulp: 0.796035 at 4.99814630e-01

    With the new rational approximation, exhaustive testing yields:

        % ./tlibm acos -fPED -x -1 -X -0x1p-12f
        Interval tested for acosf: [-1,-0.000244141]
               ulp <= 0.5:  97.010% 97653245 |  97.010% 97653245
        0.5 <  ulp <  0.6:   2.442%  2458373 |  99.452% 100111618
        0.6 <  ulp <  0.7:   0.473%   476012 |  99.925% 100587630
        0.7 <  ulp <  0.8:   0.068%    68603 |  99.993% 100656233
        0.8 <  ulp <  0.9:   0.007%     7063 | 100.000% 100663296
        Max ulp: 0.896189 at -5.04511118e-01

        % ./tlibm acos -fPED -x 0x1p-12f -X 1
        Interval tested for acosf: [0.000244141,1]
               ulp <= 0.5:  97.650% 98298175 |  97.650% 98298175
        0.5 <  ulp <  0.6:   2.028%  2041709 |  99.679% 100339884
        0.6 <  ulp <  0.7:   0.292%   293555 |  99.970% 100633439
        0.7 <  ulp <  0.8:   0.030%    29857 | 100.000% 100663296
        Max ulp: 0.775875 at 4.91849005e-01

    PR:             281001
    MFC after:      1 week

    (cherry picked from commit 41e016289f77deb88b0ef1ec3f7b2ab3515ac7c8)

 lib/msun/src/e_acosf.c | 20 +++++++++++++-------
 lib/msun/src/e_asinf.c | 22 ++++++++++++++--------
 2 files changed, 27 insertions(+), 15 deletions(-)

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