1 /* origin: FreeBSD /usr/src/lib/msun/ld80/e_rem_pio2.c */
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 #include "libm.h"
16 #if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
17 /* ld80 and ld128 version of __rem_pio2(x,y)
18 *
19 * return the remainder of x rem pi/2 in y[0]+y[1]
20 * use __rem_pio2_large() for large x
21 */
22
23 static const long double toint = 1.5/LDBL_EPSILON;
24
25 #if LDBL_MANT_DIG == 64
26 /* u ~< 0x1p25*pi/2 */
27 #define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.m>>48) < ((0x3fff + 25)<<16 | 0x921f>>1 | 0x8000))
28 #define QUOBITS(x) ((uint32_t)(int32_t)x & 0x7fffffff)
29 #define ROUND1 22
30 #define ROUND2 61
31 #define NX 3
32 #define NY 2
33 /*
34 * invpio2: 64 bits of 2/pi
35 * pio2_1: first 39 bits of pi/2
36 * pio2_1t: pi/2 - pio2_1
37 * pio2_2: second 39 bits of pi/2
38 * pio2_2t: pi/2 - (pio2_1+pio2_2)
39 * pio2_3: third 39 bits of pi/2
40 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
41 */
42 static const double
43 pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */
44 pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */
45 pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */
46 static const long double
47 invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */
48 pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */
49 pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */
50 pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */
51 #elif LDBL_MANT_DIG == 113
52 /* u ~< 0x1p45*pi/2 */
53 #define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.top) < ((0x3fff + 45)<<16 | 0x921f))
54 #define QUOBITS(x) ((uint32_t)(int64_t)x & 0x7fffffff)
55 #define ROUND1 51
56 #define ROUND2 119
57 #define NX 5
58 #define NY 3
59 static const long double
60 invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
61 pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */
62 pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */
63 pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */
64 pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */
65 pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */
66 pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
67 #endif
68
__rem_pio2l(long double x,long double * y)69 int __rem_pio2l(long double x, long double *y)
70 {
71 union ldshape u,uz;
72 long double z,w,t,r,fn;
73 double tx[NX],ty[NY];
74 int ex,ey,n,i;
75
76 u.f = x;
77 ex = u.i.se & 0x7fff;
78 if (SMALL(u)) {
79 /* rint(x/(pi/2)), Assume round-to-nearest. */
80 fn = x*invpio2 + toint - toint;
81 n = QUOBITS(fn);
82 r = x-fn*pio2_1;
83 w = fn*pio2_1t; /* 1st round good to 102/180 bits (ld80/ld128) */
84 y[0] = r-w;
85 u.f = y[0];
86 ey = u.i.se & 0x7fff;
87 if (ex - ey > ROUND1) { /* 2nd iteration needed, good to 141/248 (ld80/ld128) */
88 t = r;
89 w = fn*pio2_2;
90 r = t-w;
91 w = fn*pio2_2t-((t-r)-w);
92 y[0] = r-w;
93 u.f = y[0];
94 ey = u.i.se & 0x7fff;
95 if (ex - ey > ROUND2) { /* 3rd iteration, good to 180/316 bits */
96 t = r; /* will cover all possible cases (not verified for ld128) */
97 w = fn*pio2_3;
98 r = t-w;
99 w = fn*pio2_3t-((t-r)-w);
100 y[0] = r-w;
101 }
102 }
103 y[1] = (r - y[0]) - w;
104 return n;
105 }
106 /*
107 * all other (large) arguments
108 */
109 if (ex == 0x7fff) { /* x is inf or NaN */
110 y[0] = y[1] = x - x;
111 return 0;
112 }
113 /* set z = scalbn(|x|,-ilogb(x)+23) */
114 uz.f = x;
115 uz.i.se = 0x3fff + 23;
116 z = uz.f;
117 for (i=0; i < NX - 1; i++) {
118 tx[i] = (double)(int32_t)z;
119 z = (z-tx[i])*0x1p24;
120 }
121 tx[i] = z;
122 while (tx[i] == 0)
123 i--;
124 n = __rem_pio2_large(tx, ty, ex-0x3fff-23, i+1, NY);
125 w = ty[1];
126 if (NY == 3)
127 w += ty[2];
128 r = ty[0] + w;
129 /* TODO: for ld128 this does not follow the recommendation of the
130 comments of __rem_pio2_large which seem wrong if |ty[0]| > |ty[1]+ty[2]| */
131 w -= r - ty[0];
132 if (u.i.se >> 15) {
133 y[0] = -r;
134 y[1] = -w;
135 return -n;
136 }
137 y[0] = r;
138 y[1] = w;
139 return n;
140 }
141 #endif
142