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).  */
52 double
53 acos (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 
93 PL_SIG (S, D, 1, acos, -1.0, 1.0)
94 PL_TEST_ULP (acos, 1.02)
95 PL_TEST_INTERVAL (acos, 0, Small, 5000)
96 PL_TEST_INTERVAL (acos, Small, 0.5, 50000)
97 PL_TEST_INTERVAL (acos, 0.5, 1.0, 50000)
98 PL_TEST_INTERVAL (acos, 1.0, 0x1p11, 50000)
99 PL_TEST_INTERVAL (acos, 0x1p11, inf, 20000)
100 PL_TEST_INTERVAL (acos, -0, -inf, 20000)
101