1 /* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
10 * ====================================================
12 * Optimized by Bruce D. Evans.
16 * return the remainder of x rem pi/2 in y[0]+y[1]
17 * use __rem_pio2_large() for large x
23 * invpio2: 53 bits of 2/pi
24 * pio2_1: first 33 bit of pi/2
25 * pio2_1t: pi/2 - pio2_1
26 * pio2_2: second 33 bit of pi/2
27 * pio2_2t: pi/2 - (pio2_1+pio2_2)
28 * pio2_3: third 33 bit of pi/2
29 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
32 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
33 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
34 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
35 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
36 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
37 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
38 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
40 /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
41 int __rem_pio2(double x, double *y)
43 union {double f; uint64_t i;} u = {x};
45 double tx[3],ty[2],fn;
47 int sign, n, ex, ey, i;
50 ix = u.i>>32 & 0x7fffffff;
51 if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
52 if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
53 goto medium; /* cancellation -- use medium case */
54 if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */
56 z = x - pio2_1; /* one round good to 85 bits */
58 y[1] = (z-y[0]) - pio2_1t;
63 y[1] = (z-y[0]) + pio2_1t;
70 y[1] = (z-y[0]) - 2*pio2_1t;
75 y[1] = (z-y[0]) + 2*pio2_1t;
80 if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */
81 if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
82 if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
87 y[1] = (z-y[0]) - 3*pio2_1t;
92 y[1] = (z-y[0]) + 3*pio2_1t;
96 if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
100 y[0] = z - 4*pio2_1t;
101 y[1] = (z-y[0]) - 4*pio2_1t;
105 y[0] = z + 4*pio2_1t;
106 y[1] = (z-y[0]) + 4*pio2_1t;
111 if (ix < 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
113 /* rint(x/(pi/2)), Assume round-to-nearest. */
114 fn = x*invpio2 + 0x1.8p52;
118 w = fn*pio2_1t; /* 1st round, good to 85 bits */
121 ey = u.i>>52 & 0x7ff;
123 if (ex - ey > 16) { /* 2nd round, good to 118 bits */
127 w = fn*pio2_2t - ((t-r)-w);
130 ey = u.i>>52 & 0x7ff;
131 if (ex - ey > 49) { /* 3rd round, good to 151 bits, covers all cases */
135 w = fn*pio2_3t - ((t-r)-w);
139 y[1] = (r - y[0]) - w;
143 * all other (large) arguments
145 if (ix >= 0x7ff00000) { /* x is inf or NaN */
149 /* set z = scalbn(|x|,-ilogb(x)+23) */
151 u.i &= (uint64_t)-1>>12;
152 u.i |= (uint64_t)(0x3ff + 23)<<52;
154 for (i=0; i < 2; i++) {
155 tx[i] = (double)(int32_t)z;
156 z = (z-tx[i])*0x1p24;
159 /* skip zero terms, first term is non-zero */
162 n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);