1
2 /* @(#)e_rem_pio2.c 5.1 93/09/24 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 *
13 */
14
15 /* __rem_pio2(x,y)
16 *
17 * return the remainder of x rem pi/2 in y[0]+y[1]
18 * use __kernel_rem_pio2()
19 */
20
21 #include "fdlibm.h"
22
23 #ifdef _NEED_FLOAT64
24
25 /*
26 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
27 */
28 static const __int32_t two_over_pi[] = {
29 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C,
30 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 0x424DD2, 0xE00649,
31 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 0xA73EE8, 0x8235F5, 0x2EBB44,
32 0x84E99C, 0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C, 0x845F8B,
33 0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D,
34 0x367ECF, 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
35 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330,
36 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 0x91615E, 0xE61B08,
37 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA,
38 0x73A8C9, 0x60E27B, 0xC08C6B,
39 };
40
41 static const __int32_t npio2_hw[] = {
42 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
43 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
44 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
45 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
46 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
47 0x404858EB, 0x404921FB,
48 };
49
50 /*
51 * invpio2: 53 bits of 2/pi
52 * pio2_1: first 33 bit of pi/2
53 * pio2_1t: pi/2 - pio2_1
54 * pio2_2: second 33 bit of pi/2
55 * pio2_2t: pi/2 - (pio2_1+pio2_2)
56 * pio2_3: third 33 bit of pi/2
57 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
58 */
59
60 static const __float64
61 zero = _F_64(0.00000000000000000000e+00), /* 0x00000000, 0x00000000 */
62 half = _F_64(5.00000000000000000000e-01), /* 0x3FE00000, 0x00000000 */
63 two24 = _F_64(1.67772160000000000000e+07), /* 0x41700000, 0x00000000 */
64 invpio2 = _F_64(6.36619772367581382433e-01), /* 0x3FE45F30, 0x6DC9C883 */
65 pio2_1 = _F_64(1.57079632673412561417e+00), /* 0x3FF921FB, 0x54400000 */
66 pio2_1t = _F_64(6.07710050650619224932e-11), /* 0x3DD0B461, 0x1A626331 */
67 pio2_2 = _F_64(6.07710050630396597660e-11), /* 0x3DD0B461, 0x1A600000 */
68 pio2_2t = _F_64(2.02226624879595063154e-21), /* 0x3BA3198A, 0x2E037073 */
69 pio2_3 = _F_64(2.02226624871116645580e-21), /* 0x3BA3198A, 0x2E000000 */
70 pio2_3t = _F_64(8.47842766036889956997e-32); /* 0x397B839A, 0x252049C1 */
71
72 __int32_t
__rem_pio2(__float64 x,__float64 * y)73 __rem_pio2(__float64 x, __float64 *y)
74 {
75 __float64 z = _F_64(0.0), w, t, r, fn;
76 __float64 tx[3];
77 __int32_t i, j, n, ix, hx;
78 int e0, nx;
79 __uint32_t low;
80
81 GET_HIGH_WORD(hx, x); /* high word of x */
82 ix = hx & 0x7fffffff;
83 if (ix <= 0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
84 {
85 y[0] = x;
86 y[1] = 0;
87 return 0;
88 }
89 if (ix < 0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
90 if (hx > 0) {
91 z = x - pio2_1;
92 if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */
93 y[0] = z - pio2_1t;
94 y[1] = (z - y[0]) - pio2_1t;
95 } else { /* near pi/2, use 33+33+53 bit pi */
96 z -= pio2_2;
97 y[0] = z - pio2_2t;
98 y[1] = (z - y[0]) - pio2_2t;
99 }
100 return 1;
101 } else { /* negative x */
102 z = x + pio2_1;
103 if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */
104 y[0] = z + pio2_1t;
105 y[1] = (z - y[0]) + pio2_1t;
106 } else { /* near pi/2, use 33+33+53 bit pi */
107 z += pio2_2;
108 y[0] = z + pio2_2t;
109 y[1] = (z - y[0]) + pio2_2t;
110 }
111 return -1;
112 }
113 }
114 if (ix <= 0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
115 t = _NAME_64(fabs)(x);
116 n = (__int32_t)(t * invpio2 + half);
117 fn = (__float64)n;
118 r = t - fn * pio2_1;
119 w = fn * pio2_1t; /* 1st round good to 85 bit */
120 if (n < 32 && ix != npio2_hw[n - 1]) {
121 y[0] = r - w; /* quick check no cancellation */
122 } else {
123 __uint32_t high;
124 j = ix >> 20;
125 y[0] = r - w;
126 GET_HIGH_WORD(high, y[0]);
127 i = j - ((high >> 20) & 0x7ff);
128 if (i > 16) { /* 2nd iteration needed, good to 118 */
129 t = r;
130 w = fn * pio2_2;
131 r = t - w;
132 w = fn * pio2_2t - ((t - r) - w);
133 y[0] = r - w;
134 GET_HIGH_WORD(high, y[0]);
135 i = j - ((high >> 20) & 0x7ff);
136 if (i > 49) { /* 3rd iteration need, 151 bits acc */
137 t = r; /* will cover all possible cases */
138 w = fn * pio2_3;
139 r = t - w;
140 w = fn * pio2_3t - ((t - r) - w);
141 y[0] = r - w;
142 }
143 }
144 }
145 y[1] = (r - y[0]) - w;
146 if (hx < 0) {
147 y[0] = -y[0];
148 y[1] = -y[1];
149 return -n;
150 } else
151 return n;
152 }
153 /*
154 * all other (large) arguments
155 */
156 if (ix >= 0x7ff00000) { /* x is inf or NaN */
157 y[0] = y[1] = x - x;
158 return 0;
159 }
160 /* set z = scalbn(|x|,ilogb(x)-23) */
161 GET_LOW_WORD(low, x);
162 SET_LOW_WORD(z, low);
163 e0 = (int)((ix >> 20) - 1046); /* e0 = ilogb(z)-23; */
164 SET_HIGH_WORD(z, ix - ((__int32_t)e0 << 20));
165 for (i = 0; i < 2; i++) {
166 tx[i] = (__float64)((__int32_t)(z));
167 z = (z - tx[i]) * two24;
168 }
169 tx[2] = z;
170 nx = 3;
171 while (tx[nx - 1] == zero)
172 nx--; /* skip zero term */
173 n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
174 if (hx < 0) {
175 y[0] = -y[0];
176 y[1] = -y[1];
177 return -n;
178 }
179 return n;
180 }
181
182 #endif /* _NEED_FLOAT64 */
183
