1*412f47f9SXin Li /*
2*412f47f9SXin Li * Single-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_f32.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 0x7fffffff
14*412f47f9SXin Li #define Half 0x3f000000
15*412f47f9SXin Li #define One 0x3f800000
16*412f47f9SXin Li #define PiOver2f 0x1.921fb6p+0f
17*412f47f9SXin Li #define Small 0x39800000 /* 2^-12. */
18*412f47f9SXin Li #define Small12 0x398
19*412f47f9SXin Li #define QNaN 0x7fc
20*412f47f9SXin Li
21*412f47f9SXin Li /* Fast implementation of single-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^-12 for correct rounding.
25*412f47f9SXin Li
26*412f47f9SXin Li For x in [Small, 0.5], use order 4 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 0.83 ulps,
30*412f47f9SXin Li asinf(0x1.ea00f4p-2) got 0x1.fef15ep-2 want 0x1.fef15cp-2.
31*412f47f9SXin Li
32*412f47f9SXin Li No cheap approximation can be obtained near x = 1, since the function is not
33*412f47f9SXin Li continuously differentiable on 1.
34*412f47f9SXin Li
35*412f47f9SXin Li For x in [0.5, 1.0], we use a method based on a trigonometric identity
36*412f47f9SXin Li
37*412f47f9SXin Li asin(x) = pi/2 - acos(x)
38*412f47f9SXin Li
39*412f47f9SXin Li and a generalized power series expansion of acos(y) near y=1, that reads as
40*412f47f9SXin Li
41*412f47f9SXin Li acos(y)/sqrt(2y) ~ 1 + 1/12 * y + 3/160 * y^2 + ... (1)
42*412f47f9SXin Li
43*412f47f9SXin Li The Taylor series of asin(z) near z = 0, reads as
44*412f47f9SXin Li
45*412f47f9SXin Li asin(z) ~ z + z^3 P(z^2) = z + z^3 * (1/6 + 3/40 z^2 + ...).
46*412f47f9SXin Li
47*412f47f9SXin Li Therefore, (1) can be written in terms of P(y/2) or even asin(y/2)
48*412f47f9SXin Li
49*412f47f9SXin Li acos(y) ~ sqrt(2y) (1 + y/2 * P(y/2)) = 2 * sqrt(y/2) (1 + y/2 * P(y/2)
50*412f47f9SXin Li
51*412f47f9SXin Li Hence, if we write z = (1-x)/2, z is near 0 when x approaches 1 and
52*412f47f9SXin Li
53*412f47f9SXin Li asin(x) ~ pi/2 - acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z)).
54*412f47f9SXin Li
55*412f47f9SXin Li The largest observed error in this region is 2.41 ulps,
56*412f47f9SXin Li asinf(0x1.00203ep-1) got 0x1.0c3a64p-1 want 0x1.0c3a6p-1. */
57*412f47f9SXin Li float
asinf(float x)58*412f47f9SXin Li asinf (float x)
59*412f47f9SXin Li {
60*412f47f9SXin Li uint32_t ix = asuint (x);
61*412f47f9SXin Li uint32_t ia = ix & AbsMask;
62*412f47f9SXin Li uint32_t ia12 = ia >> 20;
63*412f47f9SXin Li float ax = asfloat (ia);
64*412f47f9SXin Li uint32_t sign = ix & ~AbsMask;
65*412f47f9SXin Li
66*412f47f9SXin Li /* Special values and invalid range. */
67*412f47f9SXin Li if (unlikely (ia12 == QNaN))
68*412f47f9SXin Li return x;
69*412f47f9SXin Li if (ia > One)
70*412f47f9SXin Li return __math_invalidf (x);
71*412f47f9SXin Li if (ia12 < Small12)
72*412f47f9SXin Li return x;
73*412f47f9SXin Li
74*412f47f9SXin Li /* Evaluate polynomial Q(x) = y + y * z * P(z) with
75*412f47f9SXin Li z2 = x ^ 2 and z = |x| , if |x| < 0.5
76*412f47f9SXin Li z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
77*412f47f9SXin Li float z2 = ax < 0.5 ? x * x : fmaf (-0.5f, ax, 0.5f);
78*412f47f9SXin Li float z = ax < 0.5 ? ax : sqrtf (z2);
79*412f47f9SXin Li
80*412f47f9SXin Li /* Use a single polynomial approximation P for both intervals. */
81*412f47f9SXin Li float p = horner_4_f32 (z2, __asinf_poly);
82*412f47f9SXin Li /* Finalize polynomial: z + z * z2 * P(z2). */
83*412f47f9SXin Li p = fmaf (z * z2, p, z);
84*412f47f9SXin Li
85*412f47f9SXin Li /* asin(|x|) = Q(|x|) , for |x| < 0.5
86*412f47f9SXin Li = pi/2 - 2 Q(|x|), for |x| >= 0.5. */
87*412f47f9SXin Li float y = ax < 0.5 ? p : fmaf (-2.0f, p, PiOver2f);
88*412f47f9SXin Li
89*412f47f9SXin Li /* Copy sign. */
90*412f47f9SXin Li return asfloat (asuint (y) | sign);
91*412f47f9SXin Li }
92*412f47f9SXin Li
93*412f47f9SXin Li PL_SIG (S, F, 1, asin, -1.0, 1.0)
94*412f47f9SXin Li PL_TEST_ULP (asinf, 1.91)
95*412f47f9SXin Li PL_TEST_INTERVAL (asinf, 0, Small, 5000)
96*412f47f9SXin Li PL_TEST_INTERVAL (asinf, Small, 0.5, 50000)
97*412f47f9SXin Li PL_TEST_INTERVAL (asinf, 0.5, 1.0, 50000)
98*412f47f9SXin Li PL_TEST_INTERVAL (asinf, 1.0, 0x1p11, 50000)
99*412f47f9SXin Li PL_TEST_INTERVAL (asinf, 0x1p11, inf, 20000)
100*412f47f9SXin Li PL_TEST_INTERVAL (asinf, -0, -inf, 20000)
101