1 /* Header for single-precision sin/cos/sincos functions.
2 Copyright (c) 2018 Arm Ltd. All rights reserved.
3
4 SPDX-License-Identifier: BSD-3-Clause
5
6 Redistribution and use in source and binary forms, with or without
7 modification, are permitted provided that the following conditions
8 are met:
9 1. Redistributions of source code must retain the above copyright
10 notice, this list of conditions and the following disclaimer.
11 2. Redistributions in binary form must reproduce the above copyright
12 notice, this list of conditions and the following disclaimer in the
13 documentation and/or other materials provided with the distribution.
14 3. The name of the company may not be used to endorse or promote
15 products derived from this software without specific prior written
16 permission.
17
18 THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS IS'' AND ANY EXPRESS OR IMPLIED
19 WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
20 MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
21 IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
22 SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
23 TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
24 PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
25 LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
26 NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
27 SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
28
29 #include <stdint.h>
30 #include <math.h>
31 #include "math_config.h"
32
33 /* 2PI * 2^-64. */
34 static const double pi63 = 0x1.921FB54442D18p-62;
35 /* PI / 4. */
36 static const double pio4 = 0x1.921FB54442D18p-1;
37
38 /* The constants and polynomials for sine and cosine. */
39 typedef struct
40 {
41 double sign[4]; /* Sign of sine in quadrants 0..3. */
42 double hpi_inv; /* 2 / PI ( * 2^24 if !TOINT_INTRINSICS). */
43 double hpi; /* PI / 2. */
44 double c0, c1, c2, c3, c4; /* Cosine polynomial. */
45 double s1, s2, s3; /* Sine polynomial. */
46 } sincos_t;
47
48 /* Polynomial data (the cosine polynomial is negated in the 2nd entry). */
49 extern const sincos_t __sincosf_table[2] HIDDEN;
50
51 /* Table with 4/PI to 192 bit precision. */
52 extern const uint32_t __inv_pio4[] HIDDEN;
53
54 /* Top 12 bits of the float representation with the sign bit cleared. */
55 static inline uint32_t
abstop12(float x)56 abstop12 (float x)
57 {
58 return (asuint (x) >> 20) & 0x7ff;
59 }
60
61 /* Compute the sine and cosine of inputs X and X2 (X squared), using the
62 polynomial P and store the results in SINP and COSP. N is the quadrant,
63 if odd the cosine and sine polynomials are swapped. */
64 static inline void
sincosf_poly(double x,double x2,const sincos_t * p,int n,float * sinp,float * cosp)65 sincosf_poly (double x, double x2, const sincos_t *p, int n, float *sinp,
66 float *cosp)
67 {
68 double x3, x4, x5, x6, s, c, c1, c2, s1;
69
70 x4 = x2 * x2;
71 x3 = x2 * x;
72 c2 = p->c3 + x2 * p->c4;
73 s1 = p->s2 + x2 * p->s3;
74
75 /* Swap sin/cos result based on quadrant. */
76 float *tmp = (n & 1 ? cosp : sinp);
77 cosp = (n & 1 ? sinp : cosp);
78 sinp = tmp;
79
80 c1 = p->c0 + x2 * p->c1;
81 x5 = x3 * x2;
82 x6 = x4 * x2;
83
84 s = x + x3 * p->s1;
85 c = c1 + x4 * p->c2;
86
87 *sinp = s + x5 * s1;
88 *cosp = c + x6 * c2;
89 }
90
91 /* Return the sine of inputs X and X2 (X squared) using the polynomial P.
92 N is the quadrant, and if odd the cosine polynomial is used. */
93 static inline float
sinf_poly(double x,double x2,const sincos_t * p,int n)94 sinf_poly (double x, double x2, const sincos_t *p, int n)
95 {
96 double x3, x4, x6, x7, s, c, c1, c2, s1;
97
98 if ((n & 1) == 0)
99 {
100 x3 = x * x2;
101 s1 = p->s2 + x2 * p->s3;
102
103 x7 = x3 * x2;
104 s = x + x3 * p->s1;
105
106 return s + x7 * s1;
107 }
108 else
109 {
110 x4 = x2 * x2;
111 c2 = p->c3 + x2 * p->c4;
112 c1 = p->c0 + x2 * p->c1;
113
114 x6 = x4 * x2;
115 c = c1 + x4 * p->c2;
116
117 return c + x6 * c2;
118 }
119 }
120
121 /* Fast range reduction using single multiply-subtract. Return the modulo of
122 X as a value between -PI/4 and PI/4 and store the quadrant in NP.
123 The values for PI/2 and 2/PI are accessed via P. Since PI/2 as a double
124 is accurate to 55 bits and the worst-case cancellation happens at 6 * PI/4,
125 the result is accurate for |X| <= 120.0. */
126 static inline double
reduce_fast(double x,const sincos_t * p,int * np)127 reduce_fast (double x, const sincos_t *p, int *np)
128 {
129 double r;
130 #if TOINT_INTRINSICS
131 /* Use fast round and lround instructions when available. */
132 r = x * p->hpi_inv;
133 *np = converttoint (r);
134 return x - roundtoint (r) * p->hpi;
135 #else
136 /* Use scaled float to int conversion with explicit rounding.
137 hpi_inv is prescaled by 2^24 so the quadrant ends up in bits 24..31.
138 This avoids inaccuracies introduced by truncating negative values. */
139 r = x * p->hpi_inv;
140 int n = ((int32_t)r + 0x800000) >> 24;
141 *np = n;
142 return x - n * p->hpi;
143 #endif
144 }
145
146 /* Reduce the range of XI to a multiple of PI/2 using fast integer arithmetic.
147 XI is a reinterpreted float and must be >= 2.0f (the sign bit is ignored).
148 Return the modulo between -PI/4 and PI/4 and store the quadrant in NP.
149 Reduction uses a table of 4/PI with 192 bits of precision. A 32x96->128 bit
150 multiply computes the exact 2.62-bit fixed-point modulo. Since the result
151 can have at most 29 leading zeros after the binary point, the double
152 precision result is accurate to 33 bits. */
153 static inline double
reduce_large(uint32_t xi,int * np)154 reduce_large (uint32_t xi, int *np)
155 {
156 const uint32_t *arr = &__inv_pio4[(xi >> 26) & 15];
157 int shift = (xi >> 23) & 7;
158 uint64_t n, res0, res1, res2;
159
160 xi = (xi & 0xffffff) | 0x800000;
161 xi <<= shift;
162
163 res0 = xi * arr[0];
164 res1 = (uint64_t)xi * arr[4];
165 res2 = (uint64_t)xi * arr[8];
166 res0 = (res2 >> 32) | (res0 << 32);
167 res0 += res1;
168
169 n = (res0 + (1ULL << 61)) >> 62;
170 res0 -= n << 62;
171 double x = (int64_t)res0;
172 *np = n;
173 return x * pi63;
174 }
175
