1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Double-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 "math_config.h"
9*f3087befSAndrew Turner #include "poly_scalar_f64.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 Pi 0x1.921fb54442d18p+1
18*f3087befSAndrew Turner #define Small 0x3c90000000000000 /* 2^-53. */
19*f3087befSAndrew Turner #define Small16 0x3c90
20*f3087befSAndrew Turner #define QNaN 0x7ff8
21*f3087befSAndrew Turner
22*f3087befSAndrew Turner /* Fast implementation of double-precision acos(x) based on polynomial
23*f3087befSAndrew Turner approximation of double-precision asin(x).
24*f3087befSAndrew Turner
25*f3087befSAndrew Turner For x < Small, approximate acos(x) by pi/2 - x. Small = 2^-53 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 11 polynomial P such that the final approximation of asin
33*f3087befSAndrew Turner is 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 1.18 ulps,
36*f3087befSAndrew Turner acos(0x1.fbab0a7c460f6p-2) got 0x1.0d54d1985c068p+0
37*f3087befSAndrew Turner want 0x1.0d54d1985c069p+0.
38*f3087befSAndrew Turner
39*f3087befSAndrew Turner For |x| in [0.5, 1.0], use the following development of acos(x) near x = 1
40*f3087befSAndrew Turner
41*f3087befSAndrew Turner acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z))
42*f3087befSAndrew Turner
43*f3087befSAndrew Turner where z = (1-x)/2, z is near 0 when x approaches 1, and P contributes to the
44*f3087befSAndrew Turner approximation of asin near 0.
45*f3087befSAndrew Turner
46*f3087befSAndrew Turner The largest observed error in this region is 1.52 ulps,
47*f3087befSAndrew Turner acos(0x1.23d362722f591p-1) got 0x1.edbbedf8a7d6ep-1
48*f3087befSAndrew Turner want 0x1.edbbedf8a7d6cp-1.
49*f3087befSAndrew Turner
50*f3087befSAndrew Turner For x in [-1.0, -0.5], use this other identity to deduce the negative inputs
51*f3087befSAndrew Turner from their absolute value: acos(x) = pi - acos(-x). */
52*f3087befSAndrew Turner double
acos(double x)53*f3087befSAndrew Turner acos (double x)
54*f3087befSAndrew Turner {
55*f3087befSAndrew Turner uint64_t ix = asuint64 (x);
56*f3087befSAndrew Turner uint64_t ia = ix & AbsMask;
57*f3087befSAndrew Turner uint64_t ia16 = ia >> 48;
58*f3087befSAndrew Turner double ax = asdouble (ia);
59*f3087befSAndrew Turner uint64_t sign = ix & ~AbsMask;
60*f3087befSAndrew Turner
61*f3087befSAndrew Turner /* Special values and invalid range. */
62*f3087befSAndrew Turner if (unlikely (ia16 == QNaN))
63*f3087befSAndrew Turner return x;
64*f3087befSAndrew Turner if (ia > One)
65*f3087befSAndrew Turner return __math_invalid (x);
66*f3087befSAndrew Turner if (ia16 < Small16)
67*f3087befSAndrew Turner return PiOver2 - x;
68*f3087befSAndrew Turner
69*f3087befSAndrew Turner /* Evaluate polynomial Q(|x|) = z + z * z2 * P(z2) with
70*f3087befSAndrew Turner z2 = x ^ 2 and z = |x| , if |x| < 0.5
71*f3087befSAndrew Turner z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
72*f3087befSAndrew Turner double z2 = ax < 0.5 ? x * x : fma (-0.5, ax, 0.5);
73*f3087befSAndrew Turner double z = ax < 0.5 ? ax : sqrt (z2);
74*f3087befSAndrew Turner
75*f3087befSAndrew Turner /* Use a single polynomial approximation P for both intervals. */
76*f3087befSAndrew Turner double z4 = z2 * z2;
77*f3087befSAndrew Turner double z8 = z4 * z4;
78*f3087befSAndrew Turner double z16 = z8 * z8;
79*f3087befSAndrew Turner double p = estrin_11_f64 (z2, z4, z8, z16, __asin_poly);
80*f3087befSAndrew Turner
81*f3087befSAndrew Turner /* Finalize polynomial: z + z * z2 * P(z2). */
82*f3087befSAndrew Turner p = fma (z * z2, p, z);
83*f3087befSAndrew Turner
84*f3087befSAndrew Turner /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
85*f3087befSAndrew Turner = pi - 2 Q(|x|), for -1.0 < x <= -0.5
86*f3087befSAndrew Turner = 2 Q(|x|) , for -0.5 < x < 0.0. */
87*f3087befSAndrew Turner if (ax < 0.5)
88*f3087befSAndrew Turner return PiOver2 - asdouble (asuint64 (p) | sign);
89*f3087befSAndrew Turner
90*f3087befSAndrew Turner return (x <= -0.5) ? fma (-2.0, p, Pi) : 2.0 * p;
91*f3087befSAndrew Turner }
92*f3087befSAndrew Turner
93*f3087befSAndrew Turner TEST_SIG (S, D, 1, acos, -1.0, 1.0)
94*f3087befSAndrew Turner TEST_ULP (acos, 1.02)
95*f3087befSAndrew Turner TEST_INTERVAL (acos, 0, Small, 5000)
96*f3087befSAndrew Turner TEST_INTERVAL (acos, Small, 0.5, 50000)
97*f3087befSAndrew Turner TEST_INTERVAL (acos, 0.5, 1.0, 50000)
98*f3087befSAndrew Turner TEST_INTERVAL (acos, 1.0, 0x1p11, 50000)
99*f3087befSAndrew Turner TEST_INTERVAL (acos, 0x1p11, inf, 20000)
100*f3087befSAndrew Turner TEST_INTERVAL (acos, -0, -inf, 20000)
101