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 __FBSDID("$FreeBSD$");
18
19 /* ld128 version of __ieee754_rem_pio2l(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 #include "fpmath.h"
30
31 #define BIAS (LDBL_MAX_EXP - 1)
32
33 /*
34 * XXX need to verify that nonzero integer multiples of pi/2 within the
35 * range get no closer to a long double than 2**-140, or that
36 * ilogb(x) + ilogb(min_delta) < 45 - -140.
37 */
38 /*
39 * invpio2: 113 bits of 2/pi
40 * pio2_1: first 68 bits of pi/2
41 * pio2_1t: pi/2 - pio2_1
42 * pio2_2: second 68 bits of pi/2
43 * pio2_2t: pi/2 - (pio2_1+pio2_2)
44 * pio2_3: third 68 bits of pi/2
45 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
46 */
47
48 static const double
49 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
50 two24 = 1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */
51
52 static const long double
53 invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
54 pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */
55 pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */
56 pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */
57 pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */
58 pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */
59 pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
60
61 static inline __always_inline int
__ieee754_rem_pio2l(long double x,long double * y)62 __ieee754_rem_pio2l(long double x, long double *y)
63 {
64 union IEEEl2bits u,u1;
65 long double z,w,t,r,fn;
66 double tx[5],ty[3];
67 int64_t n;
68 int e0,ex,i,j,nx;
69 int16_t expsign;
70
71 u.e = x;
72 expsign = u.xbits.expsign;
73 ex = expsign & 0x7fff;
74 if (ex < BIAS + 45 || ex == BIAS + 45 &&
75 u.bits.manh < 0x921fb54442d1LL) {
76 /* |x| ~< 2^45*(pi/2), medium size */
77 /* TODO: use only double precision for fn, as in expl(). */
78 fn = rnintl(x * invpio2);
79 n = i64rint(fn);
80 r = x-fn*pio2_1;
81 w = fn*pio2_1t; /* 1st round good to 180 bit */
82 {
83 union IEEEl2bits u2;
84 int ex1;
85 j = ex;
86 y[0] = r-w;
87 u2.e = y[0];
88 ex1 = u2.xbits.expsign & 0x7fff;
89 i = j-ex1;
90 if(i>51) { /* 2nd iteration needed, good to 248 */
91 t = r;
92 w = fn*pio2_2;
93 r = t-w;
94 w = fn*pio2_2t-((t-r)-w);
95 y[0] = r-w;
96 u2.e = y[0];
97 ex1 = u2.xbits.expsign & 0x7fff;
98 i = j-ex1;
99 if(i>119) { /* 3rd iteration need, 316 bits acc */
100 t = r; /* will cover all possible cases */
101 w = fn*pio2_3;
102 r = t-w;
103 w = fn*pio2_3t-((t-r)-w);
104 y[0] = r-w;
105 }
106 }
107 }
108 y[1] = (r-y[0])-w;
109 return n;
110 }
111 /*
112 * all other (large) arguments
113 */
114 if(ex==0x7fff) { /* x is inf or NaN */
115 y[0]=y[1]=x-x; return 0;
116 }
117 /* set z = scalbn(|x|,ilogb(x)-23) */
118 u1.e = x;
119 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */
120 u1.xbits.expsign = ex - e0;
121 z = u1.e;
122 for(i=0;i<4;i++) {
123 tx[i] = (double)((int32_t)(z));
124 z = (z-tx[i])*two24;
125 }
126 tx[4] = z;
127 nx = 5;
128 while(tx[nx-1]==zero) nx--; /* skip zero term */
129 n = __kernel_rem_pio2(tx,ty,e0,nx,3);
130 t = (long double)ty[2] + ty[1];
131 r = t + ty[0];
132 w = ty[0] - (r - t);
133 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
134 y[0] = r; y[1] = w; return n;
135 }
136