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/ld128/e_rem_pio2l.h 221234 2011-04-29 23:13:43Z kargl $");
|
18
| 18
|
19/* __ieee754_rem_pio2(x,y)
| 19/* ld128 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 * XXX need to verify that nonzero integer multiples of pi/2 within the
35 * range get no closer to a long double than 2**-140, or that
36 * ilogb(x) + ilogb(min_delta) < 45 - -140.
37 */
38/*
39 * invpio2: 113 bits of 2/pi
40 * pio2_1: first 68 bits of pi/2
|
33 * pio2_1t: pi/2 - pio2_1
| 41 * pio2_1t: pi/2 - pio2_1
|
34 * pio2_2: second 33 bit of pi/2
| 42 * pio2_2: second 68 bits of pi/2
|
35 * pio2_2t: pi/2 - (pio2_1+pio2_2)
| 43 * pio2_2t: pi/2 - (pio2_1+pio2_2)
|
36 * pio2_3: third 33 bit of pi/2
| 44 * pio2_3: third 68 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 */
| 45 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
46 */
47
48static const double
49zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
|
42two24 = 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 */
| 50two24 = 1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */
|
50
| 51
|
51#ifndef INLINE_REM_PIO2
52extern
| 52static const long double
53invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
54pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */
55pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */
56pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */
57pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */
58pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */
59pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
60
61#ifdef INLINE_REM_PIO2L
62static inline __always_inline
|
53#endif
| 63#endif
|
54__inline int
55__ieee754_rem_pio2(double x, double *y)
| 64int
65__ieee754_rem_pio2l(long double x, long double *y)
|
56{
| 66{
|
57 double z,w,t,r,fn;
58 double tx[3],ty[2];
59 int32_t e0,i,j,nx,n,ix,hx;
60 u_int32_t low;
| 67 union IEEEl2bits u,u1;
68 long double z,w,t,r,fn;
69 double tx[5],ty[3];
70 int64_t n;
71 int e0,ex,i,j,nx;
72 int16_t expsign;
|
61
| 73
|
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:
| 74 u.e = x;
75 expsign = u.xbits.expsign;
76 ex = expsign & 0x7fff;
77 if (ex < BIAS + 45 || ex == BIAS + 45 &&
78 u.bits.manh < 0x921fb54442d1LL) {
79 /* |x| ~< 2^45*(pi/2), medium size */
|
130 /* Use a specialized rint() to get fn. Assume round-to-nearest. */
| 80 /* 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;
133#ifdef HAVE_EFFICIENT_IRINT
134 n = irint(fn);
| 81 fn = x*invpio2+0x1.8p112;
82 fn = fn-0x1.8p112;
83#ifdef HAVE_EFFICIENT_I64RINT
84 n = i64rint(fn);
|
135#else
| 85#else
|
136 n = (int32_t)fn;
| 86 n = fn;
|
137#endif
138 r = x-fn*pio2_1;
| 87#endif
88 r = x-fn*pio2_1;
|
139 w = fn*pio2_1t; /* 1st round good to 85 bit */
| 89 w = fn*pio2_1t; /* 1st round good to 180 bit */
|
140 {
| 90 {
|
141 u_int32_t high;
142 j = ix>>20;
| 91 union IEEEl2bits u2;
92 int ex1;
93 j = ex;
|
143 y[0] = r-w;
| 94 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 */
| 95 u2.e = y[0];
96 ex1 = u2.xbits.expsign & 0x7fff;
97 i = j-ex1;
98 if(i>51) { /* 2nd iteration needed, good to 248 */
|
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;
| 99 t = r;
100 w = fn*pio2_2;
101 r = t-w;
102 w = fn*pio2_2t-((t-r)-w);
103 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 */
| 104 u2.e = y[0];
105 ex1 = u2.xbits.expsign & 0x7fff;
106 i = j-ex1;
107 if(i>119) { /* 3rd iteration need, 316 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 */
| 108 t = r; /* will cover all possible cases */
109 w = fn*pio2_3;
110 r = t-w;
111 w = fn*pio2_3t-((t-r)-w);
112 y[0] = r-w;
113 }
114 }
115 }
116 y[1] = (r-y[0])-w;
117 return n;
118 }
119 /*
120 * all other (large) arguments
121 */
|
169 if(ix>=0x7ff00000) { /* x is inf or NaN */
| 122 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) */
| 123 y[0]=y[1]=x-x; return 0;
124 }
125 /* 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)));
177 for(i=0;i<2;i++) {
| 126 u1.e = x;
127 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */
128 u1.xbits.expsign = ex - e0;
129 z = u1.e;
130 for(i=0;i<4;i++) {
|
178 tx[i] = (double)((int32_t)(z));
179 z = (z-tx[i])*two24;
180 }
| 131 tx[i] = (double)((int32_t)(z));
132 z = (z-tx[i])*two24;
133 }
|
181 tx[2] = z;
182 nx = 3;
| 134 tx[4] = z;
135 nx = 5;
|
183 while(tx[nx-1]==zero) nx--; /* skip zero term */
| 136 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;
| 137 n = __kernel_rem_pio2(tx,ty,e0,nx,3);
138 t = (long double)ty[2] + ty[1];
139 r = t + ty[0];
140 w = ty[0] - (r - t);
141 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
142 y[0] = r; y[1] = w; return n;
|
187}
| 143}
|