1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Single-precision acos(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_f32.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 0x7fffffff
14*f3087befSAndrew Turner #define Half 0x3f000000
15*f3087befSAndrew Turner #define One 0x3f800000
16*f3087befSAndrew Turner #define PiOver2f 0x1.921fb6p+0f
17*f3087befSAndrew Turner #define Pif 0x1.921fb6p+1f
18*f3087befSAndrew Turner #define Small 0x32800000 /* 2^-26. */
19*f3087befSAndrew Turner #define Small12 0x328
20*f3087befSAndrew Turner #define QNaN 0x7fc
21*f3087befSAndrew Turner
22*f3087befSAndrew Turner /* Fast implementation of single-precision acos(x) based on polynomial
23*f3087befSAndrew Turner approximation of single-precision asin(x).
24*f3087befSAndrew Turner
25*f3087befSAndrew Turner For x < Small, approximate acos(x) by pi/2 - x. Small = 2^-26 for correct
26*f3087befSAndrew Turner rounding.
27*f3087befSAndrew Turner
28*f3087befSAndrew Turner For |x| in [Small, 0.5], use the trigonometric identity
29*f3087befSAndrew Turner
30*f3087befSAndrew Turner acos(x) = pi/2 - asin(x)
31*f3087befSAndrew Turner
32*f3087befSAndrew Turner and use an order 4 polynomial P such that the final approximation of asin is
33*f3087befSAndrew Turner an odd polynomial: asin(x) ~ x + x^3 * P(x^2).
34*f3087befSAndrew Turner
35*f3087befSAndrew Turner The largest observed error in this region is 1.16 ulps,
36*f3087befSAndrew Turner acosf(0x1.ffbeccp-2) got 0x1.0c27f8p+0 want 0x1.0c27f6p+0.
37*f3087befSAndrew Turner
38*f3087befSAndrew Turner For |x| in [0.5, 1.0], use the following development of acos(x) near x = 1
39*f3087befSAndrew Turner
40*f3087befSAndrew Turner acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z))
41*f3087befSAndrew Turner
42*f3087befSAndrew Turner where z = (1-x)/2, z is near 0 when x approaches 1, and P contributes to the
43*f3087befSAndrew Turner approximation of asin near 0.
44*f3087befSAndrew Turner
45*f3087befSAndrew Turner The largest observed error in this region is 1.32 ulps,
46*f3087befSAndrew Turner acosf(0x1.15ba56p-1) got 0x1.feb33p-1 want 0x1.feb32ep-1.
47*f3087befSAndrew Turner
48*f3087befSAndrew Turner For x in [-1.0, -0.5], use this other identity to deduce the negative inputs
49*f3087befSAndrew Turner from their absolute value.
50*f3087befSAndrew Turner
51*f3087befSAndrew Turner acos(x) = pi - acos(-x)
52*f3087befSAndrew Turner
53*f3087befSAndrew Turner The largest observed error in this region is 1.28 ulps,
54*f3087befSAndrew Turner acosf(-0x1.002072p-1) got 0x1.0c1e84p+1 want 0x1.0c1e82p+1. */
55*f3087befSAndrew Turner float
acosf(float x)56*f3087befSAndrew Turner acosf (float x)
57*f3087befSAndrew Turner {
58*f3087befSAndrew Turner uint32_t ix = asuint (x);
59*f3087befSAndrew Turner uint32_t ia = ix & AbsMask;
60*f3087befSAndrew Turner uint32_t ia12 = ia >> 20;
61*f3087befSAndrew Turner float ax = asfloat (ia);
62*f3087befSAndrew Turner uint32_t sign = ix & ~AbsMask;
63*f3087befSAndrew Turner
64*f3087befSAndrew Turner /* Special values and invalid range. */
65*f3087befSAndrew Turner if (unlikely (ia12 == QNaN))
66*f3087befSAndrew Turner return x;
67*f3087befSAndrew Turner if (ia > One)
68*f3087befSAndrew Turner return __math_invalidf (x);
69*f3087befSAndrew Turner if (ia12 < Small12)
70*f3087befSAndrew Turner return PiOver2f - x;
71*f3087befSAndrew Turner
72*f3087befSAndrew Turner /* Evaluate polynomial Q(|x|) = z + z * z2 * P(z2) with
73*f3087befSAndrew Turner z2 = x ^ 2 and z = |x| , if |x| < 0.5
74*f3087befSAndrew Turner z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
75*f3087befSAndrew Turner float z2 = ax < 0.5 ? x * x : fmaf (-0.5f, ax, 0.5f);
76*f3087befSAndrew Turner float z = ax < 0.5 ? ax : sqrtf (z2);
77*f3087befSAndrew Turner
78*f3087befSAndrew Turner /* Use a single polynomial approximation P for both intervals. */
79*f3087befSAndrew Turner float p = horner_4_f32 (z2, __asinf_poly);
80*f3087befSAndrew Turner /* Finalize polynomial: z + z * z2 * P(z2). */
81*f3087befSAndrew Turner p = fmaf (z * z2, p, z);
82*f3087befSAndrew Turner
83*f3087befSAndrew Turner /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
84*f3087befSAndrew Turner = pi - 2 Q(|x|), for -1.0 < x <= -0.5
85*f3087befSAndrew Turner = 2 Q(|x|) , for -0.5 < x < 0.0. */
86*f3087befSAndrew Turner if (ax < 0.5)
87*f3087befSAndrew Turner return PiOver2f - asfloat (asuint (p) | sign);
88*f3087befSAndrew Turner
89*f3087befSAndrew Turner return (x <= -0.5) ? fmaf (-2.0f, p, Pif) : 2.0f * p;
90*f3087befSAndrew Turner }
91*f3087befSAndrew Turner
92*f3087befSAndrew Turner TEST_SIG (S, F, 1, acos, -1.0, 1.0)
93*f3087befSAndrew Turner TEST_ULP (acosf, 0.82)
94*f3087befSAndrew Turner TEST_INTERVAL (acosf, 0, Small, 5000)
95*f3087befSAndrew Turner TEST_INTERVAL (acosf, Small, 0.5, 50000)
96*f3087befSAndrew Turner TEST_INTERVAL (acosf, 0.5, 1.0, 50000)
97*f3087befSAndrew Turner TEST_INTERVAL (acosf, 1.0, 0x1p11, 50000)
98*f3087befSAndrew Turner TEST_INTERVAL (acosf, 0x1p11, inf, 20000)
99*f3087befSAndrew Turner TEST_INTERVAL (acosf, -0, -inf, 20000)
100