1 /* From: @(#)e_rem_pio2.c 1.4 95/01/18 */
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>
17 #if 0
18 __FBSDID("$FreeBSD: head/lib/msun/ld80/e_rem_pio2l.h 336545 2018-07-20 12:42:24Z bde $");
19 #endif
20
21 /* ld80 version of __ieee754_rem_pio2l(x,y)
22 *
23 * return the remainder of x rem pi/2 in y[0]+y[1]
24 * use __kernel_rem_pio2()
25 */
26
27 #include <float.h>
28 #include <machine/ieee.h>
29
30 #include "math.h"
31 #include "math_private.h"
32
33 #define BIAS (LDBL_MAX_EXP - 1)
34
35 /*
36 * invpio2: 64 bits of 2/pi
37 * pio2_1: first 39 bits of pi/2
38 * pio2_1t: pi/2 - pio2_1
39 * pio2_2: second 39 bits of pi/2
40 * pio2_2t: pi/2 - (pio2_1+pio2_2)
41 * pio2_3: third 39 bits of pi/2
42 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
43 */
44
45 static const double
46 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
47 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
48 pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */
49 pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */
50 pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */
51
52 #if defined(__amd64__) || defined(__i386__)
53 /* Long double constants are slow on these arches, and broken on i386. */
54 static const volatile double
55 invpio2hi = 6.3661977236758138e-01, /* 0x145f306dc9c883.0p-53 */
56 invpio2lo = -3.9356538861223811e-17, /* -0x16b00000000000.0p-107 */
57 pio2_1thi = -1.0746346554971943e-12, /* -0x12e7b9676733af.0p-92 */
58 pio2_1tlo = 8.8451028997905949e-29, /* 0x1c080000000000.0p-146 */
59 pio2_2thi = 6.3683171635109499e-25, /* 0x18a2e03707344a.0p-133 */
60 pio2_2tlo = 2.3183081793789774e-41, /* 0x10280000000000.0p-187 */
61 pio2_3thi = -2.7529965190440717e-37, /* -0x176b7ed8fbbacc.0p-174 */
62 pio2_3tlo = -4.2006647512740502e-54; /* -0x19c00000000000.0p-230 */
63 #define invpio2 ((long double)invpio2hi + invpio2lo)
64 #define pio2_1t ((long double)pio2_1thi + pio2_1tlo)
65 #define pio2_2t ((long double)pio2_2thi + pio2_2tlo)
66 #define pio2_3t ((long double)pio2_3thi + pio2_3tlo)
67 #else
68 static const long double
69 invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */
70 pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */
71 pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */
72 pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */
73 #endif
74
75 static inline __always_inline int
__ieee754_rem_pio2l(long double x,long double * y)76 __ieee754_rem_pio2l(long double x, long double *y)
77 {
78 union ieee_ext_u u, u1;
79 long double z,w,t,r,fn;
80 double tx[3],ty[2];
81 int e0,ex,i,j,nx,n;
82 int16_t expsign, expsign1;
83
84 u.extu_ld = x;
85 ex = u.extu_exp;
86 expsign = GET_EXPSIGN(&u);
87 if (ex < BIAS + 25 || (ex == BIAS + 25 && u.extu_frach < 0xc90fdaa2U)) {
88 /* |x| ~< 2^25*(pi/2), medium size */
89 fn = rnintl(x*invpio2);
90 n = irint(fn);
91 r = x-fn*pio2_1;
92 w = fn*pio2_1t; /* 1st round good to 102 bit */
93 {
94 union ieee_ext_u u2;
95 int ex1;
96 j = ex;
97 y[0] = r-w;
98 u2.extu_ld = y[0];
99 ex1 = u2.extu_exp;
100 i = j-ex1;
101 if(i>22) { /* 2nd iteration needed, good to 141 */
102 t = r;
103 w = fn*pio2_2;
104 r = t-w;
105 w = fn*pio2_2t-((t-r)-w);
106 y[0] = r-w;
107 u2.extu_ld = y[0];
108 ex1 = u2.extu_exp;
109 i = j-ex1;
110 if(i>61) { /* 3rd iteration need, 180 bits acc */
111 t = r; /* will cover all possible cases */
112 w = fn*pio2_3;
113 r = t-w;
114 w = fn*pio2_3t-((t-r)-w);
115 y[0] = r-w;
116 }
117 }
118 }
119 y[1] = (r-y[0])-w;
120 return n;
121 }
122 /*
123 * all other (large) arguments
124 */
125 if(ex==0x7fff) { /* x is inf or NaN */
126 y[0]=y[1]=x-x; return 0;
127 }
128 /* set z = scalbn(|x|,ilogb(x)-23) */
129 u1.extu_ld = x;
130 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */
131 expsign1 = ex - e0;
132 SET_EXPSIGN(&u1, expsign1);
133 z = u1.extu_ld;
134 for(i=0;i<2;i++) {
135 tx[i] = (double)((int32_t)(z));
136 z = (z-tx[i])*two24;
137 }
138 tx[2] = z;
139 nx = 3;
140 while(tx[nx-1]==zero) nx--; /* skip zero term */
141 n = __kernel_rem_pio2(tx,ty,e0,nx,2);
142 r = (long double)ty[0] + ty[1];
143 w = ty[1] - (r - ty[0]);
144 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
145 y[0] = r; y[1] = w; return n;
146 }
147