1/*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12/* __ieee754_rem_pio2(x,y)
13 *
14 * return the remainder of x rem pi/2 in y[0]+y[1]
15 * use __kernel_rem_pio2()
16 */
17
18#include "math_libm.h"
19#include "math_private.h"
20
21/*
22 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
23 */
24static const int32_t two_over_pi[] = {
250xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
260x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
270x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
280xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
290x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
300x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
310x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
320xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
330x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
340x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
350x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
36};
37
38static const int32_t npio2_hw[] = {
390x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
400x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
410x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
420x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
430x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
440x404858EB, 0x404921FB,
45};
46
47/*
48 * invpio2: 53 bits of 2/pi
49 * pio2_1: first 33 bit of pi/2
50 * pio2_1t: pi/2 - pio2_1
51 * pio2_2: second 33 bit of pi/2
52 * pio2_2t: pi/2 - (pio2_1+pio2_2)
53 * pio2_3: third 33 bit of pi/2
54 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
55 */
56
57static const double
58zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
59half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
60two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
61invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
62pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
63pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
64pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
65pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
66pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
67pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
68
69int32_t attribute_hidden __ieee754_rem_pio2(double x, double *y)
70{
71 double z=0.0,w,t,r,fn;
72 double tx[3];
73 int32_t e0,i,j,nx,n,ix,hx;
74 u_int32_t low;
75
76 GET_HIGH_WORD(hx,x); /* high word of x */
77 ix = hx&0x7fffffff;
78 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
79 {y[0] = x; y[1] = 0; return 0;}
80 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
81 if(hx>0) {
82 z = x - pio2_1;
83 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
84 y[0] = z - pio2_1t;
85 y[1] = (z-y[0])-pio2_1t;
86 } else { /* near pi/2, use 33+33+53 bit pi */
87 z -= pio2_2;
88 y[0] = z - pio2_2t;
89 y[1] = (z-y[0])-pio2_2t;
90 }
91 return 1;
92 } else { /* negative x */
93 z = x + pio2_1;
94 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
95 y[0] = z + pio2_1t;
96 y[1] = (z-y[0])+pio2_1t;
97 } else { /* near pi/2, use 33+33+53 bit pi */
98 z += pio2_2;
99 y[0] = z + pio2_2t;
100 y[1] = (z-y[0])+pio2_2t;
101 }
102 return -1;
103 }
104 }
105 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
106 t = fabs(x);
107 n = (int32_t) (t*invpio2+half);
108 fn = (double)n;
109 r = t-fn*pio2_1;
110 w = fn*pio2_1t; /* 1st round good to 85 bit */
111 if(n<32&&ix!=npio2_hw[n-1]) {
112 y[0] = r-w; /* quick check no cancellation */
113 } else {
114 u_int32_t high;
115 j = ix>>20;
116 y[0] = r-w;
117 GET_HIGH_WORD(high,y[0]);
118 i = j-((high>>20)&0x7ff);
119 if(i>16) { /* 2nd iteration needed, good to 118 */
120 t = r;
121 w = fn*pio2_2;
122 r = t-w;
123 w = fn*pio2_2t-((t-r)-w);
124 y[0] = r-w;
125 GET_HIGH_WORD(high,y[0]);
126 i = j-((high>>20)&0x7ff);
127 if(i>49) { /* 3rd iteration need, 151 bits acc */
128 t = r; /* will cover all possible cases */
129 w = fn*pio2_3;
130 r = t-w;
131 w = fn*pio2_3t-((t-r)-w);
132 y[0] = r-w;
133 }
134 }
135 }
136 y[1] = (r-y[0])-w;
137 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
138 else return n;
139 }
140 /*
141 * all other (large) arguments
142 */
143 if(ix>=0x7ff00000) { /* x is inf or NaN */
144 y[0]=y[1]=x-x; return 0;
145 }
146 /* set z = scalbn(|x|,ilogb(x)-23) */
147 GET_LOW_WORD(low,x);
148 SET_LOW_WORD(z,low);
149 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
150 SET_HIGH_WORD(z, ix - ((int32_t)(e0<<20)));
151 for(i=0;i<2;i++) {
152 tx[i] = (double)((int32_t)(z));
153 z = (z-tx[i])*two24;
154 }
155 tx[2] = z;
156 nx = 3;
157 while((nx > 0) && tx[nx-1]==zero) nx--; /* skip zero term */
158 n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
159 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
160 return n;
161}
162