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/ld128/e_rem_pio2l.h 336545 2018-07-20 12:42:24Z bde $");
19 #endif
20
21 /* ld128 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 * XXX need to verify that nonzero integer multiples of pi/2 within the
37 * range get no closer to a long double than 2**-140, or that
38 * ilogb(x) + ilogb(min_delta) < 45 - -140.
39 */
40 /*
41 * invpio2: 113 bits of 2/pi
42 * pio2_1: first 68 bits of pi/2
43 * pio2_1t: pi/2 - pio2_1
44 * pio2_2: second 68 bits of pi/2
45 * pio2_2t: pi/2 - (pio2_1+pio2_2)
46 * pio2_3: third 68 bits of pi/2
47 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
48 */
49
50 static const double
51 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
52 two24 = 1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */
53
54 static const long double
55 invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
56 pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */
57 pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */
58 pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */
59 pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */
60 pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */
61 pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
62
63 static inline __always_inline int
__ieee754_rem_pio2l(long double x,long double * y)64 __ieee754_rem_pio2l(long double x, long double *y)
65 {
66 union ieee_ext_u u,u1;
67 long double z,w,t,r,fn;
68 double tx[5],ty[3];
69 int64_t n;
70 int e0,ex,i,j,nx;
71 int16_t expsign, expsign1;
72
73 u.extu_ld = x;
74 ex = u.extu_exp;
75 expsign = u.extu_exp | (u.extu_sign << 15);
76 if (ex < BIAS + 45 || (ex == BIAS + 45 &&
77 u.extu_frach < 0x921fb54442d1LL)) {
78 /* |x| ~< 2^45*(pi/2), medium size */
79 /* TODO: use only double precision for fn, as in expl(). */
80 fn = rnintl(x * invpio2);
81 n = i64rint(fn);
82 r = x-fn*pio2_1;
83 w = fn*pio2_1t; /* 1st round good to 180 bit */
84 {
85 union ieee_ext_u u2;
86 int ex1;
87 j = ex;
88 y[0] = r-w;
89 u2.extu_ld = y[0];
90 ex1 = u2.extu_exp;
91 i = j-ex1;
92 if(i>51) { /* 2nd iteration needed, good to 248 */
93 t = r;
94 w = fn*pio2_2;
95 r = t-w;
96 w = fn*pio2_2t-((t-r)-w);
97 y[0] = r-w;
98 u2.extu_ld = y[0];
99 ex1 = u2.extu_exp;
100 i = j-ex1;
101 if(i>119) { /* 3rd iteration need, 316 bits acc */
102 t = r; /* will cover all possible cases */
103 w = fn*pio2_3;
104 r = t-w;
105 w = fn*pio2_3t-((t-r)-w);
106 y[0] = r-w;
107 }
108 }
109 }
110 y[1] = (r-y[0])-w;
111 return n;
112 }
113 /*
114 * all other (large) arguments
115 */
116 if(ex==0x7fff) { /* x is inf or NaN */
117 y[0]=y[1]=x-x; return 0;
118 }
119 /* set z = scalbn(|x|,ilogb(x)-23) */
120 u1.extu_ld = x;
121 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */
122 expsign1 = ex - e0;
123 u1.extu_exp = expsign1;
124 u1.extu_sign = expsign1 >> 15;
125 z = u1.extu_ld;
126 for(i=0;i<4;i++) {
127 tx[i] = (double)((int32_t)(z));
128 z = (z-tx[i])*two24;
129 }
130 tx[4] = z;
131 nx = 5;
132 while(tx[nx-1]==zero) nx--; /* skip zero term */
133 n = __kernel_rem_pio2(tx,ty,e0,nx,3);
134 t = (long double)ty[2] + ty[1];
135 r = t + ty[0];
136 w = ty[0] - (r - t);
137 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
138 y[0] = r; y[1] = w; return n;
139 }
140