1 /*
2 * Header for sinf, cosf and sincosf.
3 *
4 * Copyright (c) 2018-2021, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include <stdint.h>
9 #include <math.h>
10 #include "math_config.h"
11
12 /* 2PI * 2^-64. */
13 static const double pi63 = 0x1.921FB54442D18p-62;
14 /* PI / 4. */
15 static const float pio4f = 0x1.921FB6p-1f;
16
17 /* The constants and polynomials for sine and cosine. */
18 typedef struct
19 {
20 double sign[4]; /* Sign of sine in quadrants 0..3. */
21 double hpi_inv; /* 2 / PI ( * 2^24 if !TOINT_INTRINSICS). */
22 double hpi; /* PI / 2. */
23 double c0, c1, c2, c3, c4; /* Cosine polynomial. */
24 double s1, s2, s3; /* Sine polynomial. */
25 } sincos_t;
26
27 /* Polynomial data (the cosine polynomial is negated in the 2nd entry). */
28 extern const sincos_t __sincosf_table[2] HIDDEN;
29
30 /* Table with 4/PI to 192 bit precision. */
31 extern const uint32_t __inv_pio4[] HIDDEN;
32
33 /* Top 12 bits of the float representation with the sign bit cleared. */
34 static inline uint32_t
abstop12(float x)35 abstop12 (float x)
36 {
37 return (asuint (x) >> 20) & 0x7ff;
38 }
39
40 /* Compute the sine and cosine of inputs X and X2 (X squared), using the
41 polynomial P and store the results in SINP and COSP. N is the quadrant,
42 if odd the cosine and sine polynomials are swapped. */
43 static inline void
sincosf_poly(double x,double x2,const sincos_t * p,int n,float * sinp,float * cosp)44 sincosf_poly (double x, double x2, const sincos_t *p, int n, float *sinp,
45 float *cosp)
46 {
47 double x3, x4, x5, x6, s, c, c1, c2, s1;
48
49 x4 = x2 * x2;
50 x3 = x2 * x;
51 c2 = p->c3 + x2 * p->c4;
52 s1 = p->s2 + x2 * p->s3;
53
54 /* Swap sin/cos result based on quadrant. */
55 float *tmp = (n & 1 ? cosp : sinp);
56 cosp = (n & 1 ? sinp : cosp);
57 sinp = tmp;
58
59 c1 = p->c0 + x2 * p->c1;
60 x5 = x3 * x2;
61 x6 = x4 * x2;
62
63 s = x + x3 * p->s1;
64 c = c1 + x4 * p->c2;
65
66 *sinp = s + x5 * s1;
67 *cosp = c + x6 * c2;
68 }
69
70 /* Return the sine of inputs X and X2 (X squared) using the polynomial P.
71 N is the quadrant, and if odd the cosine polynomial is used. */
72 static inline float
sinf_poly(double x,double x2,const sincos_t * p,int n)73 sinf_poly (double x, double x2, const sincos_t *p, int n)
74 {
75 double x3, x4, x6, x7, s, c, c1, c2, s1;
76
77 if ((n & 1) == 0)
78 {
79 x3 = x * x2;
80 s1 = p->s2 + x2 * p->s3;
81
82 x7 = x3 * x2;
83 s = x + x3 * p->s1;
84
85 return s + x7 * s1;
86 }
87 else
88 {
89 x4 = x2 * x2;
90 c2 = p->c3 + x2 * p->c4;
91 c1 = p->c0 + x2 * p->c1;
92
93 x6 = x4 * x2;
94 c = c1 + x4 * p->c2;
95
96 return c + x6 * c2;
97 }
98 }
99
100 /* Fast range reduction using single multiply-subtract. Return the modulo of
101 X as a value between -PI/4 and PI/4 and store the quadrant in NP.
102 The values for PI/2 and 2/PI are accessed via P. Since PI/2 as a double
103 is accurate to 55 bits and the worst-case cancellation happens at 6 * PI/4,
104 the result is accurate for |X| <= 120.0. */
105 static inline double
reduce_fast(double x,const sincos_t * p,int * np)106 reduce_fast (double x, const sincos_t *p, int *np)
107 {
108 double r;
109 #if TOINT_INTRINSICS
110 /* Use fast round and lround instructions when available. */
111 r = x * p->hpi_inv;
112 *np = converttoint (r);
113 return x - roundtoint (r) * p->hpi;
114 #else
115 /* Use scaled float to int conversion with explicit rounding.
116 hpi_inv is prescaled by 2^24 so the quadrant ends up in bits 24..31.
117 This avoids inaccuracies introduced by truncating negative values. */
118 r = x * p->hpi_inv;
119 int n = ((int32_t)r + 0x800000) >> 24;
120 *np = n;
121 return x - n * p->hpi;
122 #endif
123 }
124
125 /* Reduce the range of XI to a multiple of PI/2 using fast integer arithmetic.
126 XI is a reinterpreted float and must be >= 2.0f (the sign bit is ignored).
127 Return the modulo between -PI/4 and PI/4 and store the quadrant in NP.
128 Reduction uses a table of 4/PI with 192 bits of precision. A 32x96->128 bit
129 multiply computes the exact 2.62-bit fixed-point modulo. Since the result
130 can have at most 29 leading zeros after the binary point, the double
131 precision result is accurate to 33 bits. */
132 static inline double
reduce_large(uint32_t xi,int * np)133 reduce_large (uint32_t xi, int *np)
134 {
135 const uint32_t *arr = &__inv_pio4[(xi >> 26) & 15];
136 int shift = (xi >> 23) & 7;
137 uint64_t n, res0, res1, res2;
138
139 xi = (xi & 0xffffff) | 0x800000;
140 xi <<= shift;
141
142 res0 = xi * arr[0];
143 res1 = (uint64_t)xi * arr[4];
144 res2 = (uint64_t)xi * arr[8];
145 res0 = (res2 >> 32) | (res0 << 32);
146 res0 += res1;
147
148 n = (res0 + (1ULL << 61)) >> 62;
149 res0 -= n << 62;
150 double x = (int64_t)res0;
151 *np = n;
152 return x * pi63;
153 }
154