1 /* @(#)e_rem_pio2.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, 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
13 #include <sys/cdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: e_rem_pio2.c,v 1.12 2022/08/24 13:51:19 christos Exp $");
16 #endif
17
18 /* __ieee754_rem_pio2(x,y)
19 *
20 * return the remainder of x rem pi/2 in y[0]+y[1]
21 * use __kernel_rem_pio2()
22 */
23
24 #include "math.h"
25 #include "math_private.h"
26
27 static const int32_t npio2_hw[] = {
28 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
29 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
30 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
31 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
32 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
33 0x404858EB, 0x404921FB,
34 };
35
36 /*
37 * invpio2: 53 bits of 2/pi
38 * pio2_1: first 33 bit of pi/2
39 * pio2_1t: pi/2 - pio2_1
40 * pio2_2: second 33 bit of pi/2
41 * pio2_2t: pi/2 - (pio2_1+pio2_2)
42 * pio2_3: third 33 bit of pi/2
43 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
44 */
45
46 static const double
47 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
48 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
49 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
50 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
51 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
52 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
53 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
54 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
55 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
56 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
57
58 int32_t
__ieee754_rem_pio2(double x,double * y)59 __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 z = 0;
67 GET_HIGH_WORD(hx,x); /* high word of x */
68 ix = hx&0x7fffffff;
69 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
70 {y[0] = x; y[1] = 0; return 0;}
71 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
72 if(hx>0) {
73 z = x - pio2_1;
74 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
75 y[0] = z - pio2_1t;
76 y[1] = (z-y[0])-pio2_1t;
77 } else { /* near pi/2, use 33+33+53 bit pi */
78 z -= pio2_2;
79 y[0] = z - pio2_2t;
80 y[1] = (z-y[0])-pio2_2t;
81 }
82 return 1;
83 } else { /* negative x */
84 z = x + pio2_1;
85 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
86 y[0] = z + pio2_1t;
87 y[1] = (z-y[0])+pio2_1t;
88 } else { /* near pi/2, use 33+33+53 bit pi */
89 z += pio2_2;
90 y[0] = z + pio2_2t;
91 y[1] = (z-y[0])+pio2_2t;
92 }
93 return -1;
94 }
95 }
96 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
97 t = fabs(x);
98 n = (int32_t) (t*invpio2+half);
99 fn = (double)n;
100 r = t-fn*pio2_1;
101 w = fn*pio2_1t; /* 1st round good to 85 bit */
102 if(n<32&&ix!=npio2_hw[n-1]) {
103 y[0] = r-w; /* quick check no cancellation */
104 } else {
105 u_int32_t high;
106 j = ix>>20;
107 y[0] = r-w;
108 GET_HIGH_WORD(high,y[0]);
109 i = j-((high>>20)&0x7ff);
110 if(i>16) { /* 2nd iteration needed, good to 118 */
111 t = r;
112 w = fn*pio2_2;
113 r = t-w;
114 w = fn*pio2_2t-((t-r)-w);
115 y[0] = r-w;
116 GET_HIGH_WORD(high,y[0]);
117 i = j-((high>>20)&0x7ff);
118 if(i>49) { /* 3rd iteration need, 151 bits acc */
119 t = r; /* will cover all possible cases */
120 w = fn*pio2_3;
121 r = t-w;
122 w = fn*pio2_3t-((t-r)-w);
123 y[0] = r-w;
124 }
125 }
126 }
127 y[1] = (r-y[0])-w;
128 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
129 else return n;
130 }
131 /*
132 * all other (large) arguments
133 */
134 if(ix>=0x7ff00000) { /* x is inf or NaN */
135 y[0]=y[1]=x-x; return 0;
136 }
137 /* set z = scalbn(|x|,ilogb(x)-23) */
138 GET_LOW_WORD(low,x);
139 SET_LOW_WORD(z,low);
140 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
141 SET_HIGH_WORD(z, ix - ((int32_t)(e0<<20)));
142 for(i=0;i<2;i++) {
143 tx[i] = (double)((int32_t)(z));
144 z = (z-tx[i])*two24;
145 }
146 tx[2] = z;
147 nx = 3;
148 while(tx[nx-1]==zero) nx--; /* skip zero term */
149 n = __kernel_rem_pio2(tx,y,e0,nx,2);
150 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
151 return n;
152 }
153