summaryrefslogtreecommitdiff
path: root/src/math/__rem_pio2.c
blob: dcf672fbd70344d40b308cf0c0a11b862bea6dca (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
/* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 *
 * Optimized by Bruce D. Evans.
 */
/* __rem_pio2(x,y)
 *
 * return the remainder of x rem pi/2 in y[0]+y[1]
 * use __rem_pio2_large() for large x
 */

#include "libm.h"

#if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
#define EPS DBL_EPSILON
#elif FLT_EVAL_METHOD==2
#define EPS LDBL_EPSILON
#endif

/*
 * invpio2:  53 bits of 2/pi
 * pio2_1:   first  33 bit of pi/2
 * pio2_1t:  pi/2 - pio2_1
 * pio2_2:   second 33 bit of pi/2
 * pio2_2t:  pi/2 - (pio2_1+pio2_2)
 * pio2_3:   third  33 bit of pi/2
 * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
 */
static const double
toint   = 1.5/EPS,
pio4    = 0x1.921fb54442d18p-1,
invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
pio2_1  = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
pio2_2  = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
pio2_3  = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */

/* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
int __rem_pio2(double x, double *y)
{
	union {double f; uint64_t i;} u = {x};
	double_t z,w,t,r,fn;
	double tx[3],ty[2];
	uint32_t ix;
	int sign, n, ex, ey, i;

	sign = u.i>>63;
	ix = u.i>>32 & 0x7fffffff;
	if (ix <= 0x400f6a7a) {  /* |x| ~<= 5pi/4 */
		if ((ix & 0xfffff) == 0x921fb)  /* |x| ~= pi/2 or 2pi/2 */
			goto medium;  /* cancellation -- use medium case */
		if (ix <= 0x4002d97c) {  /* |x| ~<= 3pi/4 */
			if (!sign) {
				z = x - pio2_1;  /* one round good to 85 bits */
				y[0] = z - pio2_1t;
				y[1] = (z-y[0]) - pio2_1t;
				return 1;
			} else {
				z = x + pio2_1;
				y[0] = z + pio2_1t;
				y[1] = (z-y[0]) + pio2_1t;
				return -1;
			}
		} else {
			if (!sign) {
				z = x - 2*pio2_1;
				y[0] = z - 2*pio2_1t;
				y[1] = (z-y[0]) - 2*pio2_1t;
				return 2;
			} else {
				z = x + 2*pio2_1;
				y[0] = z + 2*pio2_1t;
				y[1] = (z-y[0]) + 2*pio2_1t;
				return -2;
			}
		}
	}
	if (ix <= 0x401c463b) {  /* |x| ~<= 9pi/4 */
		if (ix <= 0x4015fdbc) {  /* |x| ~<= 7pi/4 */
			if (ix == 0x4012d97c)  /* |x| ~= 3pi/2 */
				goto medium;
			if (!sign) {
				z = x - 3*pio2_1;
				y[0] = z - 3*pio2_1t;
				y[1] = (z-y[0]) - 3*pio2_1t;
				return 3;
			} else {
				z = x + 3*pio2_1;
				y[0] = z + 3*pio2_1t;
				y[1] = (z-y[0]) + 3*pio2_1t;
				return -3;
			}
		} else {
			if (ix == 0x401921fb)  /* |x| ~= 4pi/2 */
				goto medium;
			if (!sign) {
				z = x - 4*pio2_1;
				y[0] = z - 4*pio2_1t;
				y[1] = (z-y[0]) - 4*pio2_1t;
				return 4;
			} else {
				z = x + 4*pio2_1;
				y[0] = z + 4*pio2_1t;
				y[1] = (z-y[0]) + 4*pio2_1t;
				return -4;
			}
		}
	}
	if (ix < 0x413921fb) {  /* |x| ~< 2^20*(pi/2), medium size */
medium:
		/* rint(x/(pi/2)) */
		fn = (double_t)x*invpio2 + toint - toint;
		n = (int32_t)fn;
		r = x - fn*pio2_1;
		w = fn*pio2_1t;  /* 1st round, good to 85 bits */
		/* Matters with directed rounding. */
		if (predict_false(r - w < -pio4)) {
			n--;
			fn--;
			r = x - fn*pio2_1;
			w = fn*pio2_1t;
		} else if (predict_false(r - w > pio4)) {
			n++;
			fn++;
			r = x - fn*pio2_1;
			w = fn*pio2_1t;
		}
		y[0] = r - w;
		u.f = y[0];
		ey = u.i>>52 & 0x7ff;
		ex = ix>>20;
		if (ex - ey > 16) { /* 2nd round, good to 118 bits */
			t = r;
			w = fn*pio2_2;
			r = t - w;
			w = fn*pio2_2t - ((t-r)-w);
			y[0] = r - w;
			u.f = y[0];
			ey = u.i>>52 & 0x7ff;
			if (ex - ey > 49) {  /* 3rd round, good to 151 bits, covers all cases */
				t = r;
				w = fn*pio2_3;
				r = t - w;
				w = fn*pio2_3t - ((t-r)-w);
				y[0] = r - w;
			}
		}
		y[1] = (r - y[0]) - w;
		return n;
	}
	/*
	 * all other (large) arguments
	 */
	if (ix >= 0x7ff00000) {  /* x is inf or NaN */
		y[0] = y[1] = x - x;
		return 0;
	}
	/* set z = scalbn(|x|,-ilogb(x)+23) */
	u.f = x;
	u.i &= (uint64_t)-1>>12;
	u.i |= (uint64_t)(0x3ff + 23)<<52;
	z = u.f;
	for (i=0; i < 2; i++) {
		tx[i] = (double)(int32_t)z;
		z     = (z-tx[i])*0x1p24;
	}
	tx[i] = z;
	/* skip zero terms, first term is non-zero */
	while (tx[i] == 0.0)
		i--;
	n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
	if (sign) {
		y[0] = -ty[0];
		y[1] = -ty[1];
		return -n;
	}
	y[0] = ty[0];
	y[1] = ty[1];
	return n;
}