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 * Optimized by Bruce D. Evans.
14 */
15
16 #include <sys/cdefs.h>
17 __FBSDID("$FreeBSD$");
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 <float.h>
26
27 #include "math.h"
28 #include "math_private.h"
29
30 /*
31 * invpio2: 53 bits of 2/pi
32 * pio2_1: first 33 bit of pi/2
33 * pio2_1t: pi/2 - pio2_1
34 * pio2_2: second 33 bit of pi/2
35 * pio2_2t: pi/2 - (pio2_1+pio2_2)
36 * pio2_3: third 33 bit of pi/2
37 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
38 */
39
40 static const double
41 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
42 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
43 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
44 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
45 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
46 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
47 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
48 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
49 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
50
51 #ifdef INLINE_REM_PIO2
52 static __inline __always_inline
53 #endif
54 int
__ieee754_rem_pio2(double x,double * y)55 __ieee754_rem_pio2(double x, double *y)
56 {
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;
61
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 */
129 medium:
130 fn = rnint((double_t)x*invpio2);
131 n = irint(fn);
132 r = x-fn*pio2_1;
133 w = fn*pio2_1t; /* 1st round good to 85 bit */
134 {
135 u_int32_t high;
136 j = ix>>20;
137 y[0] = r-w;
138 GET_HIGH_WORD(high,y[0]);
139 i = j-((high>>20)&0x7ff);
140 if(i>16) { /* 2nd iteration needed, good to 118 */
141 t = r;
142 w = fn*pio2_2;
143 r = t-w;
144 w = fn*pio2_2t-((t-r)-w);
145 y[0] = r-w;
146 GET_HIGH_WORD(high,y[0]);
147 i = j-((high>>20)&0x7ff);
148 if(i>49) { /* 3rd iteration need, 151 bits acc */
149 t = r; /* will cover all possible cases */
150 w = fn*pio2_3;
151 r = t-w;
152 w = fn*pio2_3t-((t-r)-w);
153 y[0] = r-w;
154 }
155 }
156 }
157 y[1] = (r-y[0])-w;
158 return n;
159 }
160 /*
161 * all other (large) arguments
162 */
163 if(ix>=0x7ff00000) { /* x is inf or NaN */
164 y[0]=y[1]=x-x; return 0;
165 }
166 /* set z = scalbn(|x|,ilogb(x)-23) */
167 GET_LOW_WORD(low,x);
168 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
169 INSERT_WORDS(z, ix - ((int32_t)(e0<<20)), low);
170 for(i=0;i<2;i++) {
171 tx[i] = (double)((int32_t)(z));
172 z = (z-tx[i])*two24;
173 }
174 tx[2] = z;
175 nx = 3;
176 while(tx[nx-1]==zero) nx--; /* skip zero term */
177 n = __kernel_rem_pio2(tx,ty,e0,nx,1);
178 if(hx<0) {y[0] = -ty[0]; y[1] = -ty[1]; return -n;}
179 y[0] = ty[0]; y[1] = ty[1]; return n;
180 }
181