1*412f47f9SXin Li /*
2*412f47f9SXin Li * Double-precision asin(x) function.
3*412f47f9SXin Li *
4*412f47f9SXin Li * Copyright (c) 2023-2024, Arm Limited.
5*412f47f9SXin Li * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*412f47f9SXin Li */
7*412f47f9SXin Li
8*412f47f9SXin Li #include "poly_scalar_f64.h"
9*412f47f9SXin Li #include "math_config.h"
10*412f47f9SXin Li #include "pl_sig.h"
11*412f47f9SXin Li #include "pl_test.h"
12*412f47f9SXin Li
13*412f47f9SXin Li #define AbsMask 0x7fffffffffffffff
14*412f47f9SXin Li #define Half 0x3fe0000000000000
15*412f47f9SXin Li #define One 0x3ff0000000000000
16*412f47f9SXin Li #define PiOver2 0x1.921fb54442d18p+0
17*412f47f9SXin Li #define Small 0x3e50000000000000 /* 2^-26. */
18*412f47f9SXin Li #define Small16 0x3e50
19*412f47f9SXin Li #define QNaN 0x7ff8
20*412f47f9SXin Li
21*412f47f9SXin Li /* Fast implementation of double-precision asin(x) based on polynomial
22*412f47f9SXin Li approximation.
23*412f47f9SXin Li
24*412f47f9SXin Li For x < Small, approximate asin(x) by x. Small = 2^-26 for correct rounding.
25*412f47f9SXin Li
26*412f47f9SXin Li For x in [Small, 0.5], use an order 11 polynomial P such that the final
27*412f47f9SXin Li approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2).
28*412f47f9SXin Li
29*412f47f9SXin Li The largest observed error in this region is 1.01 ulps,
30*412f47f9SXin Li asin(0x1.da9735b5a9277p-2) got 0x1.ed78525a927efp-2
31*412f47f9SXin Li want 0x1.ed78525a927eep-2.
32*412f47f9SXin Li
33*412f47f9SXin Li No cheap approximation can be obtained near x = 1, since the function is not
34*412f47f9SXin Li continuously differentiable on 1.
35*412f47f9SXin Li
36*412f47f9SXin Li For x in [0.5, 1.0], we use a method based on a trigonometric identity
37*412f47f9SXin Li
38*412f47f9SXin Li asin(x) = pi/2 - acos(x)
39*412f47f9SXin Li
40*412f47f9SXin Li and a generalized power series expansion of acos(y) near y=1, that reads as
41*412f47f9SXin Li
42*412f47f9SXin Li acos(y)/sqrt(2y) ~ 1 + 1/12 * y + 3/160 * y^2 + ... (1)
43*412f47f9SXin Li
44*412f47f9SXin Li The Taylor series of asin(z) near z = 0, reads as
45*412f47f9SXin Li
46*412f47f9SXin Li asin(z) ~ z + z^3 P(z^2) = z + z^3 * (1/6 + 3/40 z^2 + ...).
47*412f47f9SXin Li
48*412f47f9SXin Li Therefore, (1) can be written in terms of P(y/2) or even asin(y/2)
49*412f47f9SXin Li
50*412f47f9SXin Li acos(y) ~ sqrt(2y) (1 + y/2 * P(y/2)) = 2 * sqrt(y/2) (1 + y/2 * P(y/2)
51*412f47f9SXin Li
52*412f47f9SXin Li Hence, if we write z = (1-x)/2, z is near 0 when x approaches 1 and
53*412f47f9SXin Li
54*412f47f9SXin Li asin(x) ~ pi/2 - acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z)).
55*412f47f9SXin Li
56*412f47f9SXin Li The largest observed error in this region is 2.69 ulps,
57*412f47f9SXin Li asin(0x1.044e8cefee301p-1) got 0x1.1111dd54ddf96p-1
58*412f47f9SXin Li want 0x1.1111dd54ddf99p-1. */
59*412f47f9SXin Li double
asin(double x)60*412f47f9SXin Li asin (double x)
61*412f47f9SXin Li {
62*412f47f9SXin Li uint64_t ix = asuint64 (x);
63*412f47f9SXin Li uint64_t ia = ix & AbsMask;
64*412f47f9SXin Li uint64_t ia16 = ia >> 48;
65*412f47f9SXin Li double ax = asdouble (ia);
66*412f47f9SXin Li uint64_t sign = ix & ~AbsMask;
67*412f47f9SXin Li
68*412f47f9SXin Li /* Special values and invalid range. */
69*412f47f9SXin Li if (unlikely (ia16 == QNaN))
70*412f47f9SXin Li return x;
71*412f47f9SXin Li if (ia > One)
72*412f47f9SXin Li return __math_invalid (x);
73*412f47f9SXin Li if (ia16 < Small16)
74*412f47f9SXin Li return x;
75*412f47f9SXin Li
76*412f47f9SXin Li /* Evaluate polynomial Q(x) = y + y * z * P(z) with
77*412f47f9SXin Li z2 = x ^ 2 and z = |x| , if |x| < 0.5
78*412f47f9SXin Li z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
79*412f47f9SXin Li double z2 = ax < 0.5 ? x * x : fma (-0.5, ax, 0.5);
80*412f47f9SXin Li double z = ax < 0.5 ? ax : sqrt (z2);
81*412f47f9SXin Li
82*412f47f9SXin Li /* Use a single polynomial approximation P for both intervals. */
83*412f47f9SXin Li double z4 = z2 * z2;
84*412f47f9SXin Li double z8 = z4 * z4;
85*412f47f9SXin Li double z16 = z8 * z8;
86*412f47f9SXin Li double p = estrin_11_f64 (z2, z4, z8, z16, __asin_poly);
87*412f47f9SXin Li
88*412f47f9SXin Li /* Finalize polynomial: z + z * z2 * P(z2). */
89*412f47f9SXin Li p = fma (z * z2, p, z);
90*412f47f9SXin Li
91*412f47f9SXin Li /* asin(|x|) = Q(|x|) , for |x| < 0.5
92*412f47f9SXin Li = pi/2 - 2 Q(|x|), for |x| >= 0.5. */
93*412f47f9SXin Li double y = ax < 0.5 ? p : fma (-2.0, p, PiOver2);
94*412f47f9SXin Li
95*412f47f9SXin Li /* Copy sign. */
96*412f47f9SXin Li return asdouble (asuint64 (y) | sign);
97*412f47f9SXin Li }
98*412f47f9SXin Li
99*412f47f9SXin Li PL_SIG (S, D, 1, asin, -1.0, 1.0)
100*412f47f9SXin Li PL_TEST_ULP (asin, 2.20)
101*412f47f9SXin Li PL_TEST_INTERVAL (asin, 0, Small, 5000)
102*412f47f9SXin Li PL_TEST_INTERVAL (asin, Small, 0.5, 50000)
103*412f47f9SXin Li PL_TEST_INTERVAL (asin, 0.5, 1.0, 50000)
104*412f47f9SXin Li PL_TEST_INTERVAL (asin, 1.0, 0x1p11, 50000)
105*412f47f9SXin Li PL_TEST_INTERVAL (asin, 0x1p11, inf, 20000)
106*412f47f9SXin Li PL_TEST_INTERVAL (asin, -0, -inf, 20000)
107