1 /*-
2 * SPDX-License-Identifier: BSD-2-Clause
3 *
4 * Copyright (c) 2026 Steven G. Kargl
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice unmodified, this list of conditions, and the following
12 * disclaimer.
13 * 2. Redistributions in binary form must reproduce the above copyright
14 * notice, this list of conditions and the following disclaimer in the
15 * documentation and/or other materials provided with the distribution.
16 *
17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
18 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
21 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
23 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
24 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
25 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
26 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 */
28
29 /**
30 * src/s_asinpi.c for implementation details.
31 */
32
33 #include "math.h"
34 #include "math_private.h"
35
36 #define _CC (0x1p12F + 1)
37 #define _ROOT sqrtf
38
39 volatile static const float tiny = 1.e-30;
40 static const float half = 0.5f, one = 1.f;
41
42 /* Full precision high and low parts of 1 / pi. */
43 static const float
44 invpihi = 3.18309873e-01f,
45 invpilo = 1.28412765e-08f;
46
47 /*
48 * Prior to the leading multiplication by x^2, the rational approximation
49 * has an absolute minimax error less than 2.8e-10 over the [0x1p-12,0.5]
50 * domain (or log2(error) = -31.7).
51 */
52 static inline float
__r(float xs)53 __r(float xs)
54 {
55 static const float
56 R0 = 5.30516468e-02f,
57 R1 = -3.80413085e-02f,
58 R2 = 1.74116367e-03f,
59 S1 = -1.16706085e+00f,
60 S2 = 2.90115148e-01f;
61 float r, s;
62 r = R0 + (R1 + R2 * xs) * xs;
63 s = 1 + (S1 + S2 * xs) * xs;
64 return (xs * (r / s));
65 }
66
67 float
asinpif(float x)68 asinpif(float x)
69 {
70 float ax, hi, lo, xh, xl, y, zh, zl;
71 uint32_t hx, ix;
72
73 GET_FLOAT_WORD(hx, x);
74 ix = hx & 0x7fffffff;
75
76 /* |x| > 1 */
77 if (ix > 0x3f800000)
78 return ((x - x) / (x - x));
79
80 SET_FLOAT_WORD(ax, ix);
81
82 if (ix <= 0x3f000000) { /* |x| <= 0.5 */
83 if (ix < 0x39800000) { /* |x| < 0x1p-12 */
84 if (ix < 0x03800000) { /* |x| < 0x1p-120 */
85 if (ix == 0)
86 return (x);
87 /* Scale for near subnormal. */
88 ax *= 0x1p25f;
89 _XMUL(ax, 0, invpihi, invpilo, hi, lo);
90 y = (hi + lo) * 0x1p-25f;
91 } else {
92 _XMUL(ax, 0, invpihi, invpilo, hi, lo);
93 y = hi + lo;
94 }
95 } else {
96 y = __r(ax * ax);
97 _XADD(invpihi, invpilo, y, 0, xh, xl);
98 _XMUL(ax, 0, xh, xl, hi, lo);
99 y = hi + lo;
100 }
101 } else if (ix < 0x3f800000) { /* |x| < 1 */
102 y = 1 - ax;
103 x = __r(y / 2);
104 _XADD(invpihi, invpilo, x, 0, xh, xl);
105 _SQRT(2 * y, zh, zl);
106 _XMUL(xh, xl, zh, zl, hi, lo);
107 _XADD(half, 0, -hi, -lo, y, x);
108 } else /* |x| == 1 */
109 y = half;
110
111 return ((hx & 0x80000000) ? -y : y);
112 }
113
114
115 /*
116 * See src/s_asinpi.c for implementation details.
117 */
118
119 float
acospif(float x)120 acospif(float x)
121 {
122 float ax, hi, lo, xh, xl, y, zh, zl;
123 uint32_t hx, ix;
124
125 GET_FLOAT_WORD(hx, x);
126 ix = hx & 0x7fffffff;
127
128 /* |x| > 1 */
129 if (ix > 0x3f800000)
130 return ((x - x) / (x - x));
131
132 if (ix <= 0x3f000000) { /* |x| <= 0.5 */
133 if (ix <= 0x3a800000) { /* |x| <= 0x1p-10 */
134 y = (ix == 0) ? half : ((ix < 0x33800000) ?
135 half - tiny : half - x * invpihi);
136 } else {
137 y = __r(x * x);
138 _XADD(invpihi, invpilo, y, 0, xh, xl);
139 _XMUL(x, 0, xh, xl, hi, lo);
140 _XADD(half, 0, -hi, -lo, y, ax);
141 }
142 } else if (ix < 0x3f800000) { /* |x| < 1 */
143 SET_FLOAT_WORD(ax, ix);
144 y = 1 - ax;
145 ax = __r(y / 2);
146 _XADD(invpihi, invpilo, ax, 0, xh, xl);
147 _SQRT(2 * y, zh, zl);
148 _XMUL(xh, xl, zh, zl, hi, lo);
149 if (hx & 0x80000000)
150 _XADD(one, 0, -hi, -lo, y, ax);
151 else
152 y = hi + lo;
153 } else /* |x| == 1 */
154 y = hx & 0x80000000 ? 1 : 0;
155
156 return (y);
157 }
158