1*412f47f9SXin Li /*
2*412f47f9SXin Li * Single-precision acos(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 Pif 0x1.921fb6p+1f
18*412f47f9SXin Li #define Small 0x32800000 /* 2^-26. */
19*412f47f9SXin Li #define Small12 0x328
20*412f47f9SXin Li #define QNaN 0x7fc
21*412f47f9SXin Li
22*412f47f9SXin Li /* Fast implementation of single-precision acos(x) based on polynomial
23*412f47f9SXin Li approximation of single-precision asin(x).
24*412f47f9SXin Li
25*412f47f9SXin Li For x < Small, approximate acos(x) by pi/2 - x. Small = 2^-26 for correct
26*412f47f9SXin Li rounding.
27*412f47f9SXin Li
28*412f47f9SXin Li For |x| in [Small, 0.5], use the trigonometric identity
29*412f47f9SXin Li
30*412f47f9SXin Li acos(x) = pi/2 - asin(x)
31*412f47f9SXin Li
32*412f47f9SXin Li and use an order 4 polynomial P such that the final approximation of asin is
33*412f47f9SXin Li an odd polynomial: asin(x) ~ x + x^3 * P(x^2).
34*412f47f9SXin Li
35*412f47f9SXin Li The largest observed error in this region is 1.16 ulps,
36*412f47f9SXin Li acosf(0x1.ffbeccp-2) got 0x1.0c27f8p+0 want 0x1.0c27f6p+0.
37*412f47f9SXin Li
38*412f47f9SXin Li For |x| in [0.5, 1.0], use the following development of acos(x) near x = 1
39*412f47f9SXin Li
40*412f47f9SXin Li acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z))
41*412f47f9SXin Li
42*412f47f9SXin Li where z = (1-x)/2, z is near 0 when x approaches 1, and P contributes to the
43*412f47f9SXin Li approximation of asin near 0.
44*412f47f9SXin Li
45*412f47f9SXin Li The largest observed error in this region is 1.32 ulps,
46*412f47f9SXin Li acosf(0x1.15ba56p-1) got 0x1.feb33p-1 want 0x1.feb32ep-1.
47*412f47f9SXin Li
48*412f47f9SXin Li For x in [-1.0, -0.5], use this other identity to deduce the negative inputs
49*412f47f9SXin Li from their absolute value.
50*412f47f9SXin Li
51*412f47f9SXin Li acos(x) = pi - acos(-x)
52*412f47f9SXin Li
53*412f47f9SXin Li The largest observed error in this region is 1.28 ulps,
54*412f47f9SXin Li acosf(-0x1.002072p-1) got 0x1.0c1e84p+1 want 0x1.0c1e82p+1. */
55*412f47f9SXin Li float
acosf(float x)56*412f47f9SXin Li acosf (float x)
57*412f47f9SXin Li {
58*412f47f9SXin Li uint32_t ix = asuint (x);
59*412f47f9SXin Li uint32_t ia = ix & AbsMask;
60*412f47f9SXin Li uint32_t ia12 = ia >> 20;
61*412f47f9SXin Li float ax = asfloat (ia);
62*412f47f9SXin Li uint32_t sign = ix & ~AbsMask;
63*412f47f9SXin Li
64*412f47f9SXin Li /* Special values and invalid range. */
65*412f47f9SXin Li if (unlikely (ia12 == QNaN))
66*412f47f9SXin Li return x;
67*412f47f9SXin Li if (ia > One)
68*412f47f9SXin Li return __math_invalidf (x);
69*412f47f9SXin Li if (ia12 < Small12)
70*412f47f9SXin Li return PiOver2f - x;
71*412f47f9SXin Li
72*412f47f9SXin Li /* Evaluate polynomial Q(|x|) = z + z * z2 * P(z2) with
73*412f47f9SXin Li z2 = x ^ 2 and z = |x| , if |x| < 0.5
74*412f47f9SXin Li z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
75*412f47f9SXin Li float z2 = ax < 0.5 ? x * x : fmaf (-0.5f, ax, 0.5f);
76*412f47f9SXin Li float z = ax < 0.5 ? ax : sqrtf (z2);
77*412f47f9SXin Li
78*412f47f9SXin Li /* Use a single polynomial approximation P for both intervals. */
79*412f47f9SXin Li float p = horner_4_f32 (z2, __asinf_poly);
80*412f47f9SXin Li /* Finalize polynomial: z + z * z2 * P(z2). */
81*412f47f9SXin Li p = fmaf (z * z2, p, z);
82*412f47f9SXin Li
83*412f47f9SXin Li /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
84*412f47f9SXin Li = pi - 2 Q(|x|), for -1.0 < x <= -0.5
85*412f47f9SXin Li = 2 Q(|x|) , for -0.5 < x < 0.0. */
86*412f47f9SXin Li if (ax < 0.5)
87*412f47f9SXin Li return PiOver2f - asfloat (asuint (p) | sign);
88*412f47f9SXin Li
89*412f47f9SXin Li return (x <= -0.5) ? fmaf (-2.0f, p, Pif) : 2.0f * p;
90*412f47f9SXin Li }
91*412f47f9SXin Li
92*412f47f9SXin Li PL_SIG (S, F, 1, acos, -1.0, 1.0)
93*412f47f9SXin Li PL_TEST_ULP (acosf, 0.82)
94*412f47f9SXin Li PL_TEST_INTERVAL (acosf, 0, Small, 5000)
95*412f47f9SXin Li PL_TEST_INTERVAL (acosf, Small, 0.5, 50000)
96*412f47f9SXin Li PL_TEST_INTERVAL (acosf, 0.5, 1.0, 50000)
97*412f47f9SXin Li PL_TEST_INTERVAL (acosf, 1.0, 0x1p11, 50000)
98*412f47f9SXin Li PL_TEST_INTERVAL (acosf, 0x1p11, inf, 20000)
99*412f47f9SXin Li PL_TEST_INTERVAL (acosf, -0, -inf, 20000)
100