1
2/* @(#)e_rem_pio2.c 1.4 95/01/18 */
| 1/* From: @(#)e_rem_pio2.c 1.4 95/01/18 */
|
3/*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
| 2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
| 5 * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
|
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 * Optimized by Bruce D. Evans.
14 */
15
16#include <sys/cdefs.h>
| 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 * Optimized by Bruce D. Evans.
14 */
15
16#include <sys/cdefs.h>
|
17__FBSDID("$FreeBSD: head/lib/msun/src/e_rem_pio2.c 193368 2009-06-03 08:16:34Z ed $");
| 17__FBSDID("$FreeBSD: head/lib/msun/ld80/e_rem_pio2l.h 221234 2011-04-29 23:13:43Z kargl $");
|
18
| 18
|
19/* __ieee754_rem_pio2(x,y)
| 19/* ld80 version of __ieee754_rem_pio2l(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 <float.h>
26
27#include "math.h"
28#include "math_private.h"
| 20 *
21 * return the remainder of x rem pi/2 in y[0]+y[1]
22 * use __kernel_rem_pio2()
23 */
24
25#include <float.h>
26
27#include "math.h"
28#include "math_private.h"
|
| 29#include "fpmath.h"
|
29
| 30
|
| 31#define BIAS (LDBL_MAX_EXP - 1)
32
|
30/*
| 33/*
|
31 * invpio2: 53 bits of 2/pi
32 * pio2_1: first 33 bit of pi/2
| 34 * invpio2: 64 bits of 2/pi
35 * pio2_1: first 39 bits of pi/2
|
33 * pio2_1t: pi/2 - pio2_1
| 36 * pio2_1t: pi/2 - pio2_1
|
34 * pio2_2: second 33 bit of pi/2
| 37 * pio2_2: second 39 bits of pi/2
|
35 * pio2_2t: pi/2 - (pio2_1+pio2_2)
| 38 * pio2_2t: pi/2 - (pio2_1+pio2_2)
|
36 * pio2_3: third 33 bit of pi/2
| 39 * pio2_3: third 39 bits of pi/2
|
37 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
38 */
39
40static const double
41zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
42two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
| 40 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
41 */
42
43static const double
44zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
45two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
|
43invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
44pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
45pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
46pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
47pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
48pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
49pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
| 46pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */
47pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */
48pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */
|
50
| 49
|
51#ifndef INLINE_REM_PIO2
52extern
| 50#if defined(__amd64__) || defined(__i386__)
51/* Long double constants are slow on these arches, and broken on i386. */
52static const volatile double
53invpio2hi = 6.3661977236758138e-01, /* 0x145f306dc9c883.0p-53 */
54invpio2lo = -3.9356538861223811e-17, /* -0x16b00000000000.0p-107 */
55pio2_1thi = -1.0746346554971943e-12, /* -0x12e7b9676733af.0p-92 */
56pio2_1tlo = 8.8451028997905949e-29, /* 0x1c080000000000.0p-146 */
57pio2_2thi = 6.3683171635109499e-25, /* 0x18a2e03707344a.0p-133 */
58pio2_2tlo = 2.3183081793789774e-41, /* 0x10280000000000.0p-187 */
59pio2_3thi = -2.7529965190440717e-37, /* -0x176b7ed8fbbacc.0p-174 */
60pio2_3tlo = -4.2006647512740502e-54; /* -0x19c00000000000.0p-230 */
61#define invpio2 ((long double)invpio2hi + invpio2lo)
62#define pio2_1t ((long double)pio2_1thi + pio2_1tlo)
63#define pio2_2t ((long double)pio2_2thi + pio2_2tlo)
64#define pio2_3t ((long double)pio2_3thi + pio2_3tlo)
65#else
66static const long double
67invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */
68pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */
69pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */
70pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */
|
53#endif
| 71#endif
|
54__inline int
55__ieee754_rem_pio2(double x, double *y)
| 72
73#ifdef INLINE_REM_PIO2L
74static inline __always_inline
75#endif
76int
77__ieee754_rem_pio2l(long double x, long double *y)
|
56{
| 78{
|
57 double z,w,t,r,fn;
| 79 union IEEEl2bits u,u1;
80 long double z,w,t,r,fn;
|
58 double tx[3],ty[2];
| 81 double tx[3],ty[2];
|
59 int32_t e0,i,j,nx,n,ix,hx;
60 u_int32_t low;
| 82 int e0,ex,i,j,nx,n;
83 int16_t expsign;
|
61
| 84
|
62 GET_HIGH_WORD(hx,x); /* high word of x */
63 ix = hx&0x7fffffff;
64#if 0 /* Must be handled in caller. */
65 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
66 {y[0] = x; y[1] = 0; return 0;}
67#endif
68 if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
69 if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
70 goto medium; /* cancellation -- use medium case */
71 if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */
72 if (hx > 0) {
73 z = x - pio2_1; /* one round good to 85 bits */
74 y[0] = z - pio2_1t;
75 y[1] = (z-y[0])-pio2_1t;
76 return 1;
77 } else {
78 z = x + pio2_1;
79 y[0] = z + pio2_1t;
80 y[1] = (z-y[0])+pio2_1t;
81 return -1;
82 }
83 } else {
84 if (hx > 0) {
85 z = x - 2*pio2_1;
86 y[0] = z - 2*pio2_1t;
87 y[1] = (z-y[0])-2*pio2_1t;
88 return 2;
89 } else {
90 z = x + 2*pio2_1;
91 y[0] = z + 2*pio2_1t;
92 y[1] = (z-y[0])+2*pio2_1t;
93 return -2;
94 }
95 }
96 }
97 if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */
98 if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
99 if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
100 goto medium;
101 if (hx > 0) {
102 z = x - 3*pio2_1;
103 y[0] = z - 3*pio2_1t;
104 y[1] = (z-y[0])-3*pio2_1t;
105 return 3;
106 } else {
107 z = x + 3*pio2_1;
108 y[0] = z + 3*pio2_1t;
109 y[1] = (z-y[0])+3*pio2_1t;
110 return -3;
111 }
112 } else {
113 if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
114 goto medium;
115 if (hx > 0) {
116 z = x - 4*pio2_1;
117 y[0] = z - 4*pio2_1t;
118 y[1] = (z-y[0])-4*pio2_1t;
119 return 4;
120 } else {
121 z = x + 4*pio2_1;
122 y[0] = z + 4*pio2_1t;
123 y[1] = (z-y[0])+4*pio2_1t;
124 return -4;
125 }
126 }
127 }
128 if(ix<0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
129medium:
| 85 u.e = x;
86 expsign = u.xbits.expsign;
87 ex = expsign & 0x7fff;
88 if (ex < BIAS + 25 || ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2) {
89 /* |x| ~< 2^25*(pi/2), medium size */
|
130 /* Use a specialized rint() to get fn. Assume round-to-nearest. */
| 90 /* Use a specialized rint() to get fn. Assume round-to-nearest. */
|
131 STRICT_ASSIGN(double,fn,x*invpio2+0x1.8p52);
132 fn = fn-0x1.8p52;
| 91 fn = x*invpio2+0x1.8p63;
92 fn = fn-0x1.8p63;
|
133#ifdef HAVE_EFFICIENT_IRINT
134 n = irint(fn);
135#else
| 93#ifdef HAVE_EFFICIENT_IRINT
94 n = irint(fn);
95#else
|
136 n = (int32_t)fn;
| 96 n = fn;
|
137#endif
138 r = x-fn*pio2_1;
| 97#endif
98 r = x-fn*pio2_1;
|
139 w = fn*pio2_1t; /* 1st round good to 85 bit */
| 99 w = fn*pio2_1t; /* 1st round good to 102 bit */
|
140 {
| 100 {
|
141 u_int32_t high;
142 j = ix>>20;
| 101 union IEEEl2bits u2;
102 int ex1;
103 j = ex;
|
143 y[0] = r-w;
| 104 y[0] = r-w;
|
144 GET_HIGH_WORD(high,y[0]);
145 i = j-((high>>20)&0x7ff);
146 if(i>16) { /* 2nd iteration needed, good to 118 */
| 105 u2.e = y[0];
106 ex1 = u2.xbits.expsign & 0x7fff;
107 i = j-ex1;
108 if(i>22) { /* 2nd iteration needed, good to 141 */
|
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;
| 109 t = r;
110 w = fn*pio2_2;
111 r = t-w;
112 w = fn*pio2_2t-((t-r)-w);
113 y[0] = r-w;
|
152 GET_HIGH_WORD(high,y[0]);
153 i = j-((high>>20)&0x7ff);
154 if(i>49) { /* 3rd iteration need, 151 bits acc */
| 114 u2.e = y[0];
115 ex1 = u2.xbits.expsign & 0x7fff;
116 i = j-ex1;
117 if(i>61) { /* 3rd iteration need, 180 bits acc */
|
155 t = r; /* will cover all possible cases */
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 }
163 y[1] = (r-y[0])-w;
164 return n;
165 }
166 /*
167 * all other (large) arguments
168 */
| 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 return n;
128 }
129 /*
130 * all other (large) arguments
131 */
|
169 if(ix>=0x7ff00000) { /* x is inf or NaN */
| 132 if(ex==0x7fff) { /* x is inf or NaN */
|
170 y[0]=y[1]=x-x; return 0;
171 }
172 /* set z = scalbn(|x|,ilogb(x)-23) */
| 133 y[0]=y[1]=x-x; return 0;
134 }
135 /* set z = scalbn(|x|,ilogb(x)-23) */
|
173 GET_LOW_WORD(low,x);
174 SET_LOW_WORD(z,low);
175 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
176 SET_HIGH_WORD(z, ix - ((int32_t)(e0<<20)));
| 136 u1.e = x;
137 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */
138 u1.xbits.expsign = ex - e0;
139 z = u1.e;
|
177 for(i=0;i<2;i++) {
178 tx[i] = (double)((int32_t)(z));
179 z = (z-tx[i])*two24;
180 }
181 tx[2] = z;
182 nx = 3;
183 while(tx[nx-1]==zero) nx--; /* skip zero term */
| 140 for(i=0;i<2;i++) {
141 tx[i] = (double)((int32_t)(z));
142 z = (z-tx[i])*two24;
143 }
144 tx[2] = z;
145 nx = 3;
146 while(tx[nx-1]==zero) nx--; /* skip zero term */
|
184 n = __kernel_rem_pio2(tx,ty,e0,nx,1);
185 if(hx<0) {y[0] = -ty[0]; y[1] = -ty[1]; return -n;}
186 y[0] = ty[0]; y[1] = ty[1]; return n;
| 147 n = __kernel_rem_pio2(tx,ty,e0,nx,2);
148 r = (long double)ty[0] + ty[1];
149 w = ty[1] - (r - ty[0]);
150 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
151 y[0] = r; y[1] = w; return n;
|
187}
| 152}
|