1/*
2 * Double-precision acos(x) function.
3 *
4 * Copyright (c) 2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8#include "math_config.h"
9#include "poly_scalar_f64.h"
10#include "pl_sig.h"
11#include "pl_test.h"
12
13#define AbsMask (0x7fffffffffffffff)
14#define Half (0x3fe0000000000000)
15#define One (0x3ff0000000000000)
16#define PiOver2 (0x1.921fb54442d18p+0)
17#define Pi (0x1.921fb54442d18p+1)
18#define Small (0x3c90000000000000) /* 2^-53.  */
19#define Small16 (0x3c90)
20#define QNaN (0x7ff8)
21
22/* Fast implementation of double-precision acos(x) based on polynomial
23   approximation of double-precision asin(x).
24
25   For x < Small, approximate acos(x) by pi/2 - x. Small = 2^-53 for correct
26   rounding.
27
28   For |x| in [Small, 0.5], use the trigonometric identity
29
30     acos(x) = pi/2 - asin(x)
31
32   and use an order 11 polynomial P such that the final approximation of asin is
33   an odd polynomial: asin(x) ~ x + x^3 * P(x^2).
34
35   The largest observed error in this region is 1.18 ulps,
36   acos(0x1.fbab0a7c460f6p-2) got 0x1.0d54d1985c068p+0
37			     want 0x1.0d54d1985c069p+0.
38
39   For |x| in [0.5, 1.0], use the following development of acos(x) near x = 1
40
41     acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z))
42
43   where z = (1-x)/2, z is near 0 when x approaches 1, and P contributes to the
44   approximation of asin near 0.
45
46   The largest observed error in this region is 1.52 ulps,
47   acos(0x1.23d362722f591p-1) got 0x1.edbbedf8a7d6ep-1
48			     want 0x1.edbbedf8a7d6cp-1.
49
50   For x in [-1.0, -0.5], use this other identity to deduce the negative inputs
51   from their absolute value: acos(x) = pi - acos(-x).  */
52double
53acos (double x)
54{
55  uint64_t ix = asuint64 (x);
56  uint64_t ia = ix & AbsMask;
57  uint64_t ia16 = ia >> 48;
58  double ax = asdouble (ia);
59  uint64_t sign = ix & ~AbsMask;
60
61  /* Special values and invalid range.  */
62  if (unlikely (ia16 == QNaN))
63    return x;
64  if (ia > One)
65    return __math_invalid (x);
66  if (ia16 < Small16)
67    return PiOver2 - x;
68
69  /* Evaluate polynomial Q(|x|) = z + z * z2 * P(z2) with
70     z2 = x ^ 2         and z = |x|     , if |x| < 0.5
71     z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5.  */
72  double z2 = ax < 0.5 ? x * x : fma (-0.5, ax, 0.5);
73  double z = ax < 0.5 ? ax : sqrt (z2);
74
75  /* Use a single polynomial approximation P for both intervals.  */
76  double z4 = z2 * z2;
77  double z8 = z4 * z4;
78  double z16 = z8 * z8;
79  double p = estrin_11_f64 (z2, z4, z8, z16, __asin_poly);
80
81  /* Finalize polynomial: z + z * z2 * P(z2).  */
82  p = fma (z * z2, p, z);
83
84  /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
85	       = pi - 2 Q(|x|), for -1.0 < x <= -0.5
86	       = 2 Q(|x|)     , for -0.5 < x < 0.0.  */
87  if (ax < 0.5)
88    return PiOver2 - asdouble (asuint64 (p) | sign);
89
90  return (x <= -0.5) ? fma (-2.0, p, Pi) : 2.0 * p;
91}
92
93PL_SIG (S, D, 1, acos, -1.0, 1.0)
94PL_TEST_ULP (acos, 1.02)
95PL_TEST_INTERVAL (acos, 0, Small, 5000)
96PL_TEST_INTERVAL (acos, Small, 0.5, 50000)
97PL_TEST_INTERVAL (acos, 0.5, 1.0, 50000)
98PL_TEST_INTERVAL (acos, 1.0, 0x1p11, 50000)
99PL_TEST_INTERVAL (acos, 0x1p11, inf, 20000)
100PL_TEST_INTERVAL (acos, -0, -inf, 20000)
101