1
2/* @(#)e_rem_pio2.c 1.4 95/01/18 */
3/*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunSoft, 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#ifndef lint
16static char rcsid[] = "$FreeBSD: head/lib/msun/src/e_rem_pio2.c 176356 2008-02-17 07:31:59Z das $";
17#endif
18
19/* __ieee754_rem_pio2(x,y)
20 *
21 * return the remainder of x rem pi/2 in y[0]+y[1]
22 * use __kernel_rem_pio2()
23 */
24
25#include "math.h"
26#include "math_private.h"
27
28static const int32_t npio2_hw[] = {
290x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
300x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
310x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
320x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
330x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
340x404858EB, 0x404921FB,
35};
36
37/*
38 * invpio2: 53 bits of 2/pi
39 * pio2_1: first 33 bit of pi/2
40 * pio2_1t: pi/2 - pio2_1
41 * pio2_2: second 33 bit of pi/2
42 * pio2_2t: pi/2 - (pio2_1+pio2_2)
43 * pio2_3: third 33 bit of pi/2
44 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
45 */
46
47static const double
48zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
49half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
50two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
51invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
52pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
53pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
54pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
55pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
56pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
57pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
58
59 int32_t __ieee754_rem_pio2(double x, double *y)
60{
61 double z,w,t,r,fn;
62 double tx[3];
63 int32_t e0,i,j,nx,n,ix,hx;
64 u_int32_t low;
65
66 GET_HIGH_WORD(hx,x); /* high word of x */
67 ix = hx&0x7fffffff;
68 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
69 {y[0] = x; y[1] = 0; return 0;}
70 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
71 if(hx>0) {
72 z = x - pio2_1;
73 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
74 y[0] = z - pio2_1t;
75 y[1] = (z-y[0])-pio2_1t;
76 } else { /* near pi/2, use 33+33+53 bit pi */
77 z -= pio2_2;
78 y[0] = z - pio2_2t;
79 y[1] = (z-y[0])-pio2_2t;
80 }
81 return 1;
82 } else { /* negative x */
83 z = x + pio2_1;
84 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
85 y[0] = z + pio2_1t;
86 y[1] = (z-y[0])+pio2_1t;
87 } else { /* near pi/2, use 33+33+53 bit pi */
88 z += pio2_2;
89 y[0] = z + pio2_2t;
90 y[1] = (z-y[0])+pio2_2t;
91 }
92 return -1;
93 }
94 }
95 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
96 t = fabs(x);
97 n = (int32_t) (t*invpio2+half);
98 fn = (double)n;
99 r = t-fn*pio2_1;
100 w = fn*pio2_1t; /* 1st round good to 85 bit */
101 if(n<32&&ix!=npio2_hw[n-1]) {
102 y[0] = r-w; /* quick check no cancellation */
103 } else {
104 u_int32_t high;
105 j = ix>>20;
106 y[0] = r-w;
107 GET_HIGH_WORD(high,y[0]);
108 i = j-((high>>20)&0x7ff);
109 if(i>16) { /* 2nd iteration needed, good to 118 */
110 t = r;
111 w = fn*pio2_2;
112 r = t-w;
113 w = fn*pio2_2t-((t-r)-w);
114 y[0] = r-w;
115 GET_HIGH_WORD(high,y[0]);
116 i = j-((high>>20)&0x7ff);
117 if(i>49) { /* 3rd iteration need, 151 bits acc */
118 t = r; /* will cover all possible cases */
119 w = fn*pio2_3;
120 r = t-w;
121 w = fn*pio2_3t-((t-r)-w);
122 y[0] = r-w;
123 }
124 }
125 }
126 y[1] = (r-y[0])-w;
127 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
128 else return n;
129 }
130 /*
131 * all other (large) arguments
132 */
133 if(ix>=0x7ff00000) { /* x is inf or NaN */
134 y[0]=y[1]=x-x; return 0;
135 }
136 /* set z = scalbn(|x|,ilogb(x)-23) */
137 GET_LOW_WORD(low,x);
138 SET_LOW_WORD(z,low);
139 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
140 SET_HIGH_WORD(z, ix - ((int32_t)(e0<<20)));
141 for(i=0;i<2;i++) {
142 tx[i] = (double)((int32_t)(z));
143 z = (z-tx[i])*two24;
144 }
145 tx[2] = z;
146 nx = 3;
147 while(tx[nx-1]==zero) nx--; /* skip zero term */
148 n = __kernel_rem_pio2(tx,y,e0,nx,2);
149 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
150 return n;
151}