Diffusion FreeBSD src repository 33c82f11c267

Improve accuracy of asinf(3) and acosf(3)
33c82f11c267
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Description

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)