xref: /freebsd/contrib/arm-optimized-routines/math/aarch64/experimental/asin_3u.c (revision f3087bef11543b42e0d69b708f367097a4118d24)
1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner  * Double-precision asin(x) function.
3*f3087befSAndrew Turner  *
4*f3087befSAndrew Turner  * Copyright (c) 2023-2024, Arm Limited.
5*f3087befSAndrew Turner  * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*f3087befSAndrew Turner  */
7*f3087befSAndrew Turner 
8*f3087befSAndrew Turner #include "poly_scalar_f64.h"
9*f3087befSAndrew Turner #include "math_config.h"
10*f3087befSAndrew Turner #include "test_sig.h"
11*f3087befSAndrew Turner #include "test_defs.h"
12*f3087befSAndrew Turner 
13*f3087befSAndrew Turner #define AbsMask 0x7fffffffffffffff
14*f3087befSAndrew Turner #define Half 0x3fe0000000000000
15*f3087befSAndrew Turner #define One 0x3ff0000000000000
16*f3087befSAndrew Turner #define PiOver2 0x1.921fb54442d18p+0
17*f3087befSAndrew Turner #define Small 0x3e50000000000000 /* 2^-26.  */
18*f3087befSAndrew Turner #define Small16 0x3e50
19*f3087befSAndrew Turner #define QNaN 0x7ff8
20*f3087befSAndrew Turner 
21*f3087befSAndrew Turner /* Fast implementation of double-precision asin(x) based on polynomial
22*f3087befSAndrew Turner    approximation.
23*f3087befSAndrew Turner 
24*f3087befSAndrew Turner    For x < Small, approximate asin(x) by x. Small = 2^-26 for correct rounding.
25*f3087befSAndrew Turner 
26*f3087befSAndrew Turner    For x in [Small, 0.5], use an order 11 polynomial P such that the final
27*f3087befSAndrew Turner    approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2).
28*f3087befSAndrew Turner 
29*f3087befSAndrew Turner    The largest observed error in this region is 1.01 ulps,
30*f3087befSAndrew Turner    asin(0x1.da9735b5a9277p-2) got 0x1.ed78525a927efp-2
31*f3087befSAndrew Turner 			     want 0x1.ed78525a927eep-2.
32*f3087befSAndrew Turner 
33*f3087befSAndrew Turner    No cheap approximation can be obtained near x = 1, since the function is not
34*f3087befSAndrew Turner    continuously differentiable on 1.
35*f3087befSAndrew Turner 
36*f3087befSAndrew Turner    For x in [0.5, 1.0], we use a method based on a trigonometric identity
37*f3087befSAndrew Turner 
38*f3087befSAndrew Turner      asin(x) = pi/2 - acos(x)
39*f3087befSAndrew Turner 
40*f3087befSAndrew Turner    and a generalized power series expansion of acos(y) near y=1, that reads as
41*f3087befSAndrew Turner 
42*f3087befSAndrew Turner      acos(y)/sqrt(2y) ~ 1 + 1/12 * y + 3/160 * y^2 + ... (1)
43*f3087befSAndrew Turner 
44*f3087befSAndrew Turner    The Taylor series of asin(z) near z = 0, reads as
45*f3087befSAndrew Turner 
46*f3087befSAndrew Turner      asin(z) ~ z + z^3 P(z^2) = z + z^3 * (1/6 + 3/40 z^2 + ...).
47*f3087befSAndrew Turner 
48*f3087befSAndrew Turner    Therefore, (1) can be written in terms of P(y/2) or even asin(y/2)
49*f3087befSAndrew Turner 
50*f3087befSAndrew Turner      acos(y) ~ sqrt(2y) (1 + y/2 * P(y/2)) = 2 * sqrt(y/2) (1 + y/2 * P(y/2)
51*f3087befSAndrew Turner 
52*f3087befSAndrew Turner    Hence, if we write z = (1-x)/2, z is near 0 when x approaches 1 and
53*f3087befSAndrew Turner 
54*f3087befSAndrew Turner      asin(x) ~ pi/2 - acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z)).
55*f3087befSAndrew Turner 
56*f3087befSAndrew Turner    The largest observed error in this region is 2.69 ulps,
57*f3087befSAndrew Turner    asin(0x1.044e8cefee301p-1) got 0x1.1111dd54ddf96p-1
58*f3087befSAndrew Turner 			     want 0x1.1111dd54ddf99p-1.  */
59*f3087befSAndrew Turner double
asin(double x)60*f3087befSAndrew Turner asin (double x)
61*f3087befSAndrew Turner {
62*f3087befSAndrew Turner   uint64_t ix = asuint64 (x);
63*f3087befSAndrew Turner   uint64_t ia = ix & AbsMask;
64*f3087befSAndrew Turner   uint64_t ia16 = ia >> 48;
65*f3087befSAndrew Turner   double ax = asdouble (ia);
66*f3087befSAndrew Turner   uint64_t sign = ix & ~AbsMask;
67*f3087befSAndrew Turner 
68*f3087befSAndrew Turner   /* Special values and invalid range.  */
69*f3087befSAndrew Turner   if (unlikely (ia16 == QNaN))
70*f3087befSAndrew Turner     return x;
71*f3087befSAndrew Turner   if (ia > One)
72*f3087befSAndrew Turner     return __math_invalid (x);
73*f3087befSAndrew Turner   if (ia16 < Small16)
74*f3087befSAndrew Turner     return x;
75*f3087befSAndrew Turner 
76*f3087befSAndrew Turner   /* Evaluate polynomial Q(x) = y + y * z * P(z) with
77*f3087befSAndrew Turner      z2 = x ^ 2         and z = |x|     , if |x| < 0.5
78*f3087befSAndrew Turner      z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5.  */
79*f3087befSAndrew Turner   double z2 = ax < 0.5 ? x * x : fma (-0.5, ax, 0.5);
80*f3087befSAndrew Turner   double z = ax < 0.5 ? ax : sqrt (z2);
81*f3087befSAndrew Turner 
82*f3087befSAndrew Turner   /* Use a single polynomial approximation P for both intervals.  */
83*f3087befSAndrew Turner   double z4 = z2 * z2;
84*f3087befSAndrew Turner   double z8 = z4 * z4;
85*f3087befSAndrew Turner   double z16 = z8 * z8;
86*f3087befSAndrew Turner   double p = estrin_11_f64 (z2, z4, z8, z16, __asin_poly);
87*f3087befSAndrew Turner 
88*f3087befSAndrew Turner   /* Finalize polynomial: z + z * z2 * P(z2).  */
89*f3087befSAndrew Turner   p = fma (z * z2, p, z);
90*f3087befSAndrew Turner 
91*f3087befSAndrew Turner   /* asin(|x|) = Q(|x|)         , for |x| < 0.5
92*f3087befSAndrew Turner 	       = pi/2 - 2 Q(|x|), for |x| >= 0.5.  */
93*f3087befSAndrew Turner   double y = ax < 0.5 ? p : fma (-2.0, p, PiOver2);
94*f3087befSAndrew Turner 
95*f3087befSAndrew Turner   /* Copy sign.  */
96*f3087befSAndrew Turner   return asdouble (asuint64 (y) | sign);
97*f3087befSAndrew Turner }
98*f3087befSAndrew Turner 
99*f3087befSAndrew Turner TEST_SIG (S, D, 1, asin, -1.0, 1.0)
100*f3087befSAndrew Turner TEST_ULP (asin, 2.20)
101*f3087befSAndrew Turner TEST_INTERVAL (asin, 0, Small, 5000)
102*f3087befSAndrew Turner TEST_INTERVAL (asin, Small, 0.5, 50000)
103*f3087befSAndrew Turner TEST_INTERVAL (asin, 0.5, 1.0, 50000)
104*f3087befSAndrew Turner TEST_INTERVAL (asin, 1.0, 0x1p11, 50000)
105*f3087befSAndrew Turner TEST_INTERVAL (asin, 0x1p11, inf, 20000)
106*f3087befSAndrew Turner TEST_INTERVAL (asin, -0, -inf, 20000)
107