1
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
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
9 * is preserved.
10 * ====================================================
11 *
12 * Optimized by Bruce D. Evans.
13 */
14
15 /* __ieee754_rem_pio2(x,y)
16 *
17 * return the remainder of x rem pi/2 in y[0]+y[1]
18 * use __kernel_rem_pio2()
19 */
20
21 #include <float.h>
22
23 #include "math.h"
24 #include "math_private.h"
25
26 /*
27 * invpio2: 53 bits of 2/pi
28 * pio2_1: first 33 bit of pi/2
29 * pio2_1t: pi/2 - pio2_1
30 * pio2_2: second 33 bit of pi/2
31 * pio2_2t: pi/2 - (pio2_1+pio2_2)
32 * pio2_3: third 33 bit of pi/2
33 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
34 */
35
36 static const double
37 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
38 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
39 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
40 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
41 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
42 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
43 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
44 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
45 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
46
47 #ifdef INLINE_REM_PIO2
48 static __always_inline
49 #endif
50 int
__ieee754_rem_pio2(double x,double * y)51 __ieee754_rem_pio2(double x, double *y)
52 {
53 double z,w,t,r,fn;
54 double tx[3],ty[2];
55 int32_t e0,i,j,nx,n,ix,hx;
56 u_int32_t low;
57
58 GET_HIGH_WORD(hx,x); /* high word of x */
59 ix = hx&0x7fffffff;
60 #if 0 /* Must be handled in caller. */
61 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
62 {y[0] = x; y[1] = 0; return 0;}
63 #endif
64 if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
65 if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
66 goto medium; /* cancellation -- use medium case */
67 if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */
68 if (hx > 0) {
69 z = x - pio2_1; /* one round good to 85 bits */
70 y[0] = z - pio2_1t;
71 y[1] = (z-y[0])-pio2_1t;
72 return 1;
73 } else {
74 z = x + pio2_1;
75 y[0] = z + pio2_1t;
76 y[1] = (z-y[0])+pio2_1t;
77 return -1;
78 }
79 } else {
80 if (hx > 0) {
81 z = x - 2*pio2_1;
82 y[0] = z - 2*pio2_1t;
83 y[1] = (z-y[0])-2*pio2_1t;
84 return 2;
85 } else {
86 z = x + 2*pio2_1;
87 y[0] = z + 2*pio2_1t;
88 y[1] = (z-y[0])+2*pio2_1t;
89 return -2;
90 }
91 }
92 }
93 if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */
94 if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
95 if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
96 goto medium;
97 if (hx > 0) {
98 z = x - 3*pio2_1;
99 y[0] = z - 3*pio2_1t;
100 y[1] = (z-y[0])-3*pio2_1t;
101 return 3;
102 } else {
103 z = x + 3*pio2_1;
104 y[0] = z + 3*pio2_1t;
105 y[1] = (z-y[0])+3*pio2_1t;
106 return -3;
107 }
108 } else {
109 if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
110 goto medium;
111 if (hx > 0) {
112 z = x - 4*pio2_1;
113 y[0] = z - 4*pio2_1t;
114 y[1] = (z-y[0])-4*pio2_1t;
115 return 4;
116 } else {
117 z = x + 4*pio2_1;
118 y[0] = z + 4*pio2_1t;
119 y[1] = (z-y[0])+4*pio2_1t;
120 return -4;
121 }
122 }
123 }
124 if(ix<0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
125 medium:
126 fn = rnint((double_t)x*invpio2);
127 n = irint(fn);
128 r = x-fn*pio2_1;
129 w = fn*pio2_1t; /* 1st round good to 85 bit */
130 {
131 u_int32_t high;
132 j = ix>>20;
133 y[0] = r-w;
134 GET_HIGH_WORD(high,y[0]);
135 i = j-((high>>20)&0x7ff);
136 if(i>16) { /* 2nd iteration needed, good to 118 */
137 t = r;
138 w = fn*pio2_2;
139 r = t-w;
140 w = fn*pio2_2t-((t-r)-w);
141 y[0] = r-w;
142 GET_HIGH_WORD(high,y[0]);
143 i = j-((high>>20)&0x7ff);
144 if(i>49) { /* 3rd iteration need, 151 bits acc */
145 t = r; /* will cover all possible cases */
146 w = fn*pio2_3;
147 r = t-w;
148 w = fn*pio2_3t-((t-r)-w);
149 y[0] = r-w;
150 }
151 }
152 }
153 y[1] = (r-y[0])-w;
154 return n;
155 }
156 /*
157 * all other (large) arguments
158 */
159 if(ix>=0x7ff00000) { /* x is inf or NaN */
160 y[0]=y[1]=x-x; return 0;
161 }
162 /* set z = scalbn(|x|,ilogb(x)-23) */
163 GET_LOW_WORD(low,x);
164 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
165 INSERT_WORDS(z, ix - ((int32_t)((u_int32_t)e0<<20)), low);
166 for(i=0;i<2;i++) {
167 tx[i] = (double)((int32_t)(z));
168 z = (z-tx[i])*two24;
169 }
170 tx[2] = z;
171 nx = 3;
172 while(tx[nx-1]==zero) nx--; /* skip zero term */
173 n = __kernel_rem_pio2(tx,ty,e0,nx,1);
174 if(hx<0) {y[0] = -ty[0]; y[1] = -ty[1]; return -n;}
175 y[0] = ty[0]; y[1] = ty[1]; return n;
176 }
177