1 /* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 *
12 * Optimized by Bruce D. Evans.
13 */
14 /* __rem_pio2(x,y)
15 *
16 * return the remainder of x rem pi/2 in y[0]+y[1]
17 * use __rem_pio2_large() for large x
18 */
19
20 #include "libm.h"
21
22 #if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
23 #define EPS DBL_EPSILON
24 #elif FLT_EVAL_METHOD==2
25 #define EPS LDBL_EPSILON
26 #endif
27
28 /*
29 * invpio2: 53 bits of 2/pi
30 * pio2_1: first 33 bit of pi/2
31 * pio2_1t: pi/2 - pio2_1
32 * pio2_2: second 33 bit of pi/2
33 * pio2_2t: pi/2 - (pio2_1+pio2_2)
34 * pio2_3: third 33 bit of pi/2
35 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
36 */
37 static const double
38 toint = 1.5/EPS,
39 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
40 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
41 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
42 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
43 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
44 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
45 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
46
47 /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
__rem_pio2(double x,double * y)48 int __rem_pio2(double x, double *y)
49 {
50 union {double f; uint64_t i;} u = {x};
51 double_t z,w,t,r,fn;
52 double tx[3],ty[2];
53 uint32_t ix;
54 int sign, n, ex, ey, i;
55
56 sign = u.i>>63;
57 ix = u.i>>32 & 0x7fffffff;
58 if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
59 if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
60 goto medium; /* cancellation -- use medium case */
61 if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */
62 if (!sign) {
63 z = x - pio2_1; /* one round good to 85 bits */
64 y[0] = z - pio2_1t;
65 y[1] = (z-y[0]) - pio2_1t;
66 return 1;
67 } else {
68 z = x + pio2_1;
69 y[0] = z + pio2_1t;
70 y[1] = (z-y[0]) + pio2_1t;
71 return -1;
72 }
73 } else {
74 if (!sign) {
75 z = x - 2*pio2_1;
76 y[0] = z - 2*pio2_1t;
77 y[1] = (z-y[0]) - 2*pio2_1t;
78 return 2;
79 } else {
80 z = x + 2*pio2_1;
81 y[0] = z + 2*pio2_1t;
82 y[1] = (z-y[0]) + 2*pio2_1t;
83 return -2;
84 }
85 }
86 }
87 if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */
88 if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
89 if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
90 goto medium;
91 if (!sign) {
92 z = x - 3*pio2_1;
93 y[0] = z - 3*pio2_1t;
94 y[1] = (z-y[0]) - 3*pio2_1t;
95 return 3;
96 } else {
97 z = x + 3*pio2_1;
98 y[0] = z + 3*pio2_1t;
99 y[1] = (z-y[0]) + 3*pio2_1t;
100 return -3;
101 }
102 } else {
103 if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
104 goto medium;
105 if (!sign) {
106 z = x - 4*pio2_1;
107 y[0] = z - 4*pio2_1t;
108 y[1] = (z-y[0]) - 4*pio2_1t;
109 return 4;
110 } else {
111 z = x + 4*pio2_1;
112 y[0] = z + 4*pio2_1t;
113 y[1] = (z-y[0]) + 4*pio2_1t;
114 return -4;
115 }
116 }
117 }
118 if (ix < 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
119 medium:
120 /* rint(x/(pi/2)), Assume round-to-nearest. */
121 fn = (double_t)x*invpio2 + toint - toint;
122 n = (int32_t)fn;
123 r = x - fn*pio2_1;
124 w = fn*pio2_1t; /* 1st round, good to 85 bits */
125 y[0] = r - w;
126 u.f = y[0];
127 ey = u.i>>52 & 0x7ff;
128 ex = ix>>20;
129 if (ex - ey > 16) { /* 2nd round, good to 118 bits */
130 t = r;
131 w = fn*pio2_2;
132 r = t - w;
133 w = fn*pio2_2t - ((t-r)-w);
134 y[0] = r - w;
135 u.f = y[0];
136 ey = u.i>>52 & 0x7ff;
137 if (ex - ey > 49) { /* 3rd round, good to 151 bits, covers all cases */
138 t = r;
139 w = fn*pio2_3;
140 r = t - w;
141 w = fn*pio2_3t - ((t-r)-w);
142 y[0] = r - w;
143 }
144 }
145 y[1] = (r - y[0]) - w;
146 return n;
147 }
148 /*
149 * all other (large) arguments
150 */
151 if (ix >= 0x7ff00000) { /* x is inf or NaN */
152 y[0] = y[1] = x - x;
153 return 0;
154 }
155 /* set z = scalbn(|x|,-ilogb(x)+23) */
156 u.f = x;
157 u.i &= (uint64_t)-1>>12;
158 u.i |= (uint64_t)(0x3ff + 23)<<52;
159 z = u.f;
160 for (i=0; i < 2; i++) {
161 tx[i] = (double)(int32_t)z;
162 z = (z-tx[i])*0x1p24;
163 }
164 tx[i] = z;
165 /* skip zero terms, first term is non-zero */
166 while (tx[i] == 0.0)
167 i--;
168 n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
169 if (sign) {
170 y[0] = -ty[0];
171 y[1] = -ty[1];
172 return -n;
173 }
174 y[0] = ty[0];
175 y[1] = ty[1];
176 return n;
177 }
178