1*412f47f9SXin Li /*
2*412f47f9SXin Li * Double-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 "math_config.h"
9*412f47f9SXin Li #include "poly_scalar_f64.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 Pi 0x1.921fb54442d18p+1
18*412f47f9SXin Li #define Small 0x3c90000000000000 /* 2^-53. */
19*412f47f9SXin Li #define Small16 0x3c90
20*412f47f9SXin Li #define QNaN 0x7ff8
21*412f47f9SXin Li
22*412f47f9SXin Li /* Fast implementation of double-precision acos(x) based on polynomial
23*412f47f9SXin Li approximation of double-precision asin(x).
24*412f47f9SXin Li
25*412f47f9SXin Li For x < Small, approximate acos(x) by pi/2 - x. Small = 2^-53 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 11 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 1.18 ulps,
36*412f47f9SXin Li acos(0x1.fbab0a7c460f6p-2) got 0x1.0d54d1985c068p+0
37*412f47f9SXin Li want 0x1.0d54d1985c069p+0.
38*412f47f9SXin Li
39*412f47f9SXin Li For |x| in [0.5, 1.0], use the following development of acos(x) near x = 1
40*412f47f9SXin Li
41*412f47f9SXin Li acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z))
42*412f47f9SXin Li
43*412f47f9SXin Li where z = (1-x)/2, z is near 0 when x approaches 1, and P contributes to the
44*412f47f9SXin Li approximation of asin near 0.
45*412f47f9SXin Li
46*412f47f9SXin Li The largest observed error in this region is 1.52 ulps,
47*412f47f9SXin Li acos(0x1.23d362722f591p-1) got 0x1.edbbedf8a7d6ep-1
48*412f47f9SXin Li want 0x1.edbbedf8a7d6cp-1.
49*412f47f9SXin Li
50*412f47f9SXin Li For x in [-1.0, -0.5], use this other identity to deduce the negative inputs
51*412f47f9SXin Li from their absolute value: acos(x) = pi - acos(-x). */
52*412f47f9SXin Li double
acos(double x)53*412f47f9SXin Li acos (double x)
54*412f47f9SXin Li {
55*412f47f9SXin Li uint64_t ix = asuint64 (x);
56*412f47f9SXin Li uint64_t ia = ix & AbsMask;
57*412f47f9SXin Li uint64_t ia16 = ia >> 48;
58*412f47f9SXin Li double ax = asdouble (ia);
59*412f47f9SXin Li uint64_t sign = ix & ~AbsMask;
60*412f47f9SXin Li
61*412f47f9SXin Li /* Special values and invalid range. */
62*412f47f9SXin Li if (unlikely (ia16 == QNaN))
63*412f47f9SXin Li return x;
64*412f47f9SXin Li if (ia > One)
65*412f47f9SXin Li return __math_invalid (x);
66*412f47f9SXin Li if (ia16 < Small16)
67*412f47f9SXin Li return PiOver2 - x;
68*412f47f9SXin Li
69*412f47f9SXin Li /* Evaluate polynomial Q(|x|) = z + z * z2 * P(z2) with
70*412f47f9SXin Li z2 = x ^ 2 and z = |x| , if |x| < 0.5
71*412f47f9SXin Li z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
72*412f47f9SXin Li double z2 = ax < 0.5 ? x * x : fma (-0.5, ax, 0.5);
73*412f47f9SXin Li double z = ax < 0.5 ? ax : sqrt (z2);
74*412f47f9SXin Li
75*412f47f9SXin Li /* Use a single polynomial approximation P for both intervals. */
76*412f47f9SXin Li double z4 = z2 * z2;
77*412f47f9SXin Li double z8 = z4 * z4;
78*412f47f9SXin Li double z16 = z8 * z8;
79*412f47f9SXin Li double p = estrin_11_f64 (z2, z4, z8, z16, __asin_poly);
80*412f47f9SXin Li
81*412f47f9SXin Li /* Finalize polynomial: z + z * z2 * P(z2). */
82*412f47f9SXin Li p = fma (z * z2, p, z);
83*412f47f9SXin Li
84*412f47f9SXin Li /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
85*412f47f9SXin Li = pi - 2 Q(|x|), for -1.0 < x <= -0.5
86*412f47f9SXin Li = 2 Q(|x|) , for -0.5 < x < 0.0. */
87*412f47f9SXin Li if (ax < 0.5)
88*412f47f9SXin Li return PiOver2 - asdouble (asuint64 (p) | sign);
89*412f47f9SXin Li
90*412f47f9SXin Li return (x <= -0.5) ? fma (-2.0, p, Pi) : 2.0 * p;
91*412f47f9SXin Li }
92*412f47f9SXin Li
93*412f47f9SXin Li PL_SIG (S, D, 1, acos, -1.0, 1.0)
94*412f47f9SXin Li PL_TEST_ULP (acos, 1.02)
95*412f47f9SXin Li PL_TEST_INTERVAL (acos, 0, Small, 5000)
96*412f47f9SXin Li PL_TEST_INTERVAL (acos, Small, 0.5, 50000)
97*412f47f9SXin Li PL_TEST_INTERVAL (acos, 0.5, 1.0, 50000)
98*412f47f9SXin Li PL_TEST_INTERVAL (acos, 1.0, 0x1p11, 50000)
99*412f47f9SXin Li PL_TEST_INTERVAL (acos, 0x1p11, inf, 20000)
100*412f47f9SXin Li PL_TEST_INTERVAL (acos, -0, -inf, 20000)
101