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 |
|
---|
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
|
---|