[Bug 295884] [libm] Implementation for acospi[fl], asinpi[fl], and atanpi[fl]

https://bugs.freebsd.org/bugzilla/show_bug.cgi?id=295884

--- Comment #2 from Steve Kargl <kargl@FreeBSD.org> ---
The attached patch implements the inverse half-cycle
trigonometric functions:

   asinpi(x) = asin(x) / pi                        Eq. (1)
   acospi(x) = acos(x) / pi
   atanpi(x) = atan(x) / pi

Implemention details are contained in src/s_asinpi.c and
src/a_atanpi.c, where the details for acospi(x) appear in
the former.

*************

CAVEAT EMPTOR: The ld128 code has been only compiled.  It has
not been tested for correctness due to lack of hardware.

*************

Code compiled on AMD Ryzen 7 7700X system run FreeBSD 16.0-CURRENT
main-n284956-de9fe28ab847.


Exhaustive testing of acospif(x), asinpif(x), and atanpif(x)
on the indicated intervals yields

% ./tlibm acospi -fPE -x 0x1p-120 -X 1
Interval tested for acospif: [7.52316e-37,1]
      ulp <= 0.5: 99.627% 1002878299 |  99.627% 1002878299
0.5 < ulp <= 0.6:  0.277%    2789599 |  99.904% 1005667898
0.6 < ulp <= 0.7:  0.096%     965062 | 100.000% 1006632960
Max ulp: 0.63661975 at 5.96046412e-08 0x1.fffffep-25

% ./tlibm asinpi -fPED -x 0x1p-120f -X 1.f
Interval tested for asinpif: [7.52316e-37,1]
      ulp <= 0.5: 99.851% 1005129353 |  99.851% 1005129353
0.5 < ulp <= 0.6:  0.149%    1501097 | 100.000% 1006630450
0.6 < ulp <= 0.7:  0.000%       2510 | 100.000% 1006632960
Max ulp: 0.68957579 at 5.04878759e-01 0x1.027f78p-1

% ./tlibm atanpi -fPE -x 0x1p-120 -X max > zatanpif.txt &
Interval tested for atanpif: [7.52316e-37,3.40282e+38]
      ulp <= 0.5: 99.865% 2077574602 |  99.865% 2077574602
0.5 < ulp <= 0.6:  0.131%    2735011 |  99.997% 2080309613
0.6 < ulp <= 0.7:  0.003%      65170 | 100.000% 2080374783
Max ulp: 0.68433094 at 5.01186252e-01 0x1.009b7cp-1


Testing the double and long double version cannot be done
in an exhaustive manner.  For 300 M values, uniformily 
distributed in the indicated interals, one finds the max ULP:


Interval tested for acospi: [9.31323e-10,0.25]
  xm =  2.4423788416892520e-01, /* 0x3fcf432f, 0xde79920f */
libm =  4.2146222480005391e-01, /* 0x3fdaf93c, 0xb201001c */
mpfr =  4.2146222480005396e-01, /* 0x3fdaf93c, 0xb201001d */
 ULP = 0.50499351466286857

Interval tested for acospi: [0.25,0.5]
  xm =  4.9689430915631438e-01, /* 0x3fdfcd1d, 0xc9d945c6 */
libm =  3.3447366122373884e-01, /* 0x3fd56804, 0x371513ef */
mpfr =  3.3447366122373889e-01, /* 0x3fd56804, 0x371513f0 */
 ULP = 0.57195275455053829

Interval tested for acospi: [0.5,0.75]
  xm =  5.0238623667462079e-01, /* 0x3fe0138c, 0x4d0f4be0 */
libm =  3.3245556599062825e-01, /* 0x3fd546f3, 0xb5d36303 */
mpfr =  3.3245556599062820e-01, /* 0x3fd546f3, 0xb5d36302 */
 ULP = 0.63427929243758807

Interval tested for acospi: [0.75,1]
  xm =  7.5853651919512177e-01, /* 0x3fe845ee, 0x60d8789f */
libm =  2.2591472240382732e-01, /* 0x3fcceac6, 0x0c3465ce */
mpfr =  2.2591472240382729e-01, /* 0x3fcceac6, 0x0c3465cd */
 ULP = 0.56915750216472161


Interval tested for asinpi: [9.31323e-10,0.25]
  xm =  1.9502362835488171e-01, /* 0x3fc8f688, 0xc4dda0fb */
libm =  6.2478354989018887e-02, /* 0x3faffd29, 0xb6c57c61 */
mpfr =  6.2478354989018881e-02, /* 0x3faffd29, 0xb6c57c60 */
 ULP = 0.52347765415885006

Interval tested for asinpi: [0.25,0.5]
  xm =  4.9937103583123676e-01, /* 0x3fdff5b1, 0xeeddbf62 */
libm =  1.6643553767987129e-01, /* 0x3fc54dc2, 0x7b9d15a4 */
mpfr =  1.6643553767987126e-01, /* 0x3fc54dc2, 0x7b9d15a3 */
 ULP = 0.66214688371031072

Interval tested for asinpi: [0.5,0.75]
  xm =  5.0228515250761718e-01, /* 0x3fe012b8, 0x4fe92bbb */
libm =  1.6750722213679006e-01, /* 0x3fc570e0, 0x6c75edd5 */
mpfr =  1.6750722213679009e-01, /* 0x3fc570e0, 0x6c75edd6 */
 ULP = 0.78223048105528226

Interval tested for asinpi: [0.75,1]
  xm =  7.5425933001419776e-01, /* 0x3fe822e4, 0x7663a4aa */
libm =  2.7200385380185182e-01, /* 0x3fd16882, 0xda1dc13b */
mpfr =  2.7200385380185188e-01, /* 0x3fd16882, 0xda1dc13c */
 ULP = 0.53747973176773822

Interval tested for atanpi: [9.31323e-10,0.25]
  xm =  1.9666113418757322e-01, /* 0x3fc92c31, 0x29dd6d2f */
libm =  6.1810387818117797e-02, /* 0x3fafa59c, 0x7476baa5 */
mpfr =  6.1810387818117804e-02, /* 0x3fafa59c, 0x7476baa6 */
 ULP = 0.54674297446584263

Interval tested for atanpi: [0.25,0.5]
  xm =  4.1312119637707068e-01, /* 0x3fda7093, 0xe2ee5494 */
libm =  1.2470309560460152e-01, /* 0x3fbfec8a, 0xc554ebec */
mpfr =  1.2470309560460154e-01, /* 0x3fbfec8a, 0xc554ebed */
 ULP = 0.73116638175113347

Interval tested for atanpi: [0.5,0.75]
  xm =  5.0018949583396499e-01, /* 0x3fe0018d, 0x66cd1b82 */
libm =  1.4763186871058706e-01, /* 0x3fc2e599, 0xdffacb8f */
mpfr =  1.4763186871058709e-01, /* 0x3fc2e599, 0xdffacb90 */
 ULP = 0.69192753950764663

Interval tested for atanpi: [0.75,1]
  xm =  7.5007880583359599e-01, /* 0x3fe800a5, 0x448f4c03 */
libm =  2.0484881828445453e-01, /* 0x3fca387c, 0x6f93f71f */
mpfr =  2.0484881828445450e-01, /* 0x3fca387c, 0x6f93f71e */
 ULP = 0.65765471872064396

Interval tested for atanpi: [1,2]
  xm =  1.0103228000344093e+00, /* 0x3ff02a48, 0x3d88d0a2 */
libm =  2.5163447403817019e-01, /* 0x3fd01ac7, 0x7b229108 */
mpfr =  2.5163447403817013e-01, /* 0x3fd01ac7, 0x7b229107 */
 ULP = 0.67409519689166042

Interval tested for atanpi: [2,4]
  xm =  2.0231383267437946e+00, /* 0x40002f63, 0x25a530a9 */
libm =  3.5387589538123299e-01, /* 0x3fd6a5e7, 0x156053c6 */
mpfr =  3.5387589538123293e-01, /* 0x3fd6a5e7, 0x156053c5 */
 ULP = 0.69695587476021503

Interval tested for atanpi: [4,1.79769e+308]
  xm =  4.0000000000000000e+00, /* 0x40100000, 0x00000000 */
libm =  4.2202086962263069e-01, /* 0x3fdb0263, 0xd2508e31 */
mpfr =  4.2202086962263069e-01, /* 0x3fdb0263, 0xd2508e31 */
 ULP = 0.27709400511686716

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