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1 /*
2  * Header for sinf, cosf and sincosf.
3  *
4  * Copyright (c) 2018, Arm Limited.
5  * SPDX-License-Identifier: MIT
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 double pio4 = 0x1.921FB54442D18p-1;
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