master
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 */
37static const double
38toint = 1.5/EPS,
39#ifdef __wasilibc_unmodified_upstream // Wasm doesn't have alternate rounding modes
40pio4 = 0x1.921fb54442d18p-1,
41#endif
42invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
43pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
44pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
45pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
46pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
47pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
48pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
49
50/* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
51int __rem_pio2(double x, double *y)
52{
53 union {double f; uint64_t i;} u = {x};
54 double_t z,w,t,r,fn;
55 double tx[3],ty[2];
56 uint32_t ix;
57 int sign, n, ex, ey, i;
58
59 sign = u.i>>63;
60 ix = u.i>>32 & 0x7fffffff;
61 if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
62 if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
63 goto medium; /* cancellation -- use medium case */
64 if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */
65 if (!sign) {
66 z = x - pio2_1; /* one round good to 85 bits */
67 y[0] = z - pio2_1t;
68 y[1] = (z-y[0]) - pio2_1t;
69 return 1;
70 } else {
71 z = x + pio2_1;
72 y[0] = z + pio2_1t;
73 y[1] = (z-y[0]) + pio2_1t;
74 return -1;
75 }
76 } else {
77 if (!sign) {
78 z = x - 2*pio2_1;
79 y[0] = z - 2*pio2_1t;
80 y[1] = (z-y[0]) - 2*pio2_1t;
81 return 2;
82 } else {
83 z = x + 2*pio2_1;
84 y[0] = z + 2*pio2_1t;
85 y[1] = (z-y[0]) + 2*pio2_1t;
86 return -2;
87 }
88 }
89 }
90 if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */
91 if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
92 if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
93 goto medium;
94 if (!sign) {
95 z = x - 3*pio2_1;
96 y[0] = z - 3*pio2_1t;
97 y[1] = (z-y[0]) - 3*pio2_1t;
98 return 3;
99 } else {
100 z = x + 3*pio2_1;
101 y[0] = z + 3*pio2_1t;
102 y[1] = (z-y[0]) + 3*pio2_1t;
103 return -3;
104 }
105 } else {
106 if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
107 goto medium;
108 if (!sign) {
109 z = x - 4*pio2_1;
110 y[0] = z - 4*pio2_1t;
111 y[1] = (z-y[0]) - 4*pio2_1t;
112 return 4;
113 } else {
114 z = x + 4*pio2_1;
115 y[0] = z + 4*pio2_1t;
116 y[1] = (z-y[0]) + 4*pio2_1t;
117 return -4;
118 }
119 }
120 }
121 if (ix < 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
122medium:
123 /* rint(x/(pi/2)) */
124 fn = (double_t)x*invpio2 + toint - toint;
125 n = (int32_t)fn;
126 r = x - fn*pio2_1;
127 w = fn*pio2_1t; /* 1st round, good to 85 bits */
128#ifdef __wasilibc_unmodified_upstream // Wasm doesn't have alternate rounding modes
129 /* Matters with directed rounding. */
130 if (predict_false(r - w < -pio4)) {
131 n--;
132 fn--;
133 r = x - fn*pio2_1;
134 w = fn*pio2_1t;
135 } else if (predict_false(r - w > pio4)) {
136 n++;
137 fn++;
138 r = x - fn*pio2_1;
139 w = fn*pio2_1t;
140 }
141#endif
142 y[0] = r - w;
143 u.f = y[0];
144 ey = u.i>>52 & 0x7ff;
145 ex = ix>>20;
146 if (ex - ey > 16) { /* 2nd round, good to 118 bits */
147 t = r;
148 w = fn*pio2_2;
149 r = t - w;
150 w = fn*pio2_2t - ((t-r)-w);
151 y[0] = r - w;
152 u.f = y[0];
153 ey = u.i>>52 & 0x7ff;
154 if (ex - ey > 49) { /* 3rd round, good to 151 bits, covers all cases */
155 t = r;
156 w = fn*pio2_3;
157 r = t - w;
158 w = fn*pio2_3t - ((t-r)-w);
159 y[0] = r - w;
160 }
161 }
162 y[1] = (r - y[0]) - w;
163 return n;
164 }
165 /*
166 * all other (large) arguments
167 */
168 if (ix >= 0x7ff00000) { /* x is inf or NaN */
169 y[0] = y[1] = x - x;
170 return 0;
171 }
172 /* set z = scalbn(|x|,-ilogb(x)+23) */
173 u.f = x;
174 u.i &= (uint64_t)-1>>12;
175 u.i |= (uint64_t)(0x3ff + 23)<<52;
176 z = u.f;
177 for (i=0; i < 2; i++) {
178 tx[i] = (double)(int32_t)z;
179 z = (z-tx[i])*0x1p24;
180 }
181 tx[i] = z;
182 /* skip zero terms, first term is non-zero */
183 while (tx[i] == 0.0)
184 i--;
185 n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
186 if (sign) {
187 y[0] = -ty[0];
188 y[1] = -ty[1];
189 return -n;
190 }
191 y[0] = ty[0];
192 y[1] = ty[1];
193 return n;
194}