1 |
|
|
/* @(#)k_rem_pio2.c 5.1 93/09/24 */ |
2 |
|
|
/* |
3 |
|
|
* ==================================================== |
4 |
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
5 |
|
|
* |
6 |
|
|
* Developed at SunPro, a Sun Microsystems, Inc. business. |
7 |
|
|
* Permission to use, copy, modify, and distribute this |
8 |
|
|
* software is freely granted, provided that this notice |
9 |
|
|
* is preserved. |
10 |
|
|
* ==================================================== |
11 |
|
|
*/ |
12 |
|
|
|
13 |
|
|
/* |
14 |
|
|
* __kernel_rem_pio2(x,y,e0,nx,prec) |
15 |
|
|
* double x[],y[]; int e0,nx,prec; |
16 |
|
|
* |
17 |
|
|
* __kernel_rem_pio2 return the last three digits of N with |
18 |
|
|
* y = x - N*pi/2 |
19 |
|
|
* so that |y| < pi/2. |
20 |
|
|
* |
21 |
|
|
* The method is to compute the integer (mod 8) and fraction parts of |
22 |
|
|
* (2/pi)*x without doing the full multiplication. In general we |
23 |
|
|
* skip the part of the product that are known to be a huge integer ( |
24 |
|
|
* more accurately, = 0 mod 8 ). Thus the number of operations are |
25 |
|
|
* independent of the exponent of the input. |
26 |
|
|
* |
27 |
|
|
* (2/pi) is represented by an array of 24-bit integers in ipio2[]. |
28 |
|
|
* |
29 |
|
|
* Input parameters: |
30 |
|
|
* x[] The input value (must be positive) is broken into nx |
31 |
|
|
* pieces of 24-bit integers in double precision format. |
32 |
|
|
* x[i] will be the i-th 24 bit of x. The scaled exponent |
33 |
|
|
* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 |
34 |
|
|
* match x's up to 24 bits. |
35 |
|
|
* |
36 |
|
|
* Example of breaking a double positive z into x[0]+x[1]+x[2]: |
37 |
|
|
* e0 = ilogb(z)-23 |
38 |
|
|
* z = scalbn(z,-e0) |
39 |
|
|
* for i = 0,1,2 |
40 |
|
|
* x[i] = floor(z) |
41 |
|
|
* z = (z-x[i])*2**24 |
42 |
|
|
* |
43 |
|
|
* |
44 |
|
|
* y[] output result in an array of double precision numbers. |
45 |
|
|
* The dimension of y[] is: |
46 |
|
|
* 24-bit precision 1 |
47 |
|
|
* 53-bit precision 2 |
48 |
|
|
* 64-bit precision 2 |
49 |
|
|
* 113-bit precision 3 |
50 |
|
|
* The actual value is the sum of them. Thus for 113-bit |
51 |
|
|
* precison, one may have to do something like: |
52 |
|
|
* |
53 |
|
|
* long double t,w,r_head, r_tail; |
54 |
|
|
* t = (long double)y[2] + (long double)y[1]; |
55 |
|
|
* w = (long double)y[0]; |
56 |
|
|
* r_head = t+w; |
57 |
|
|
* r_tail = w - (r_head - t); |
58 |
|
|
* |
59 |
|
|
* e0 The exponent of x[0]. Must be <= 16360 or you need to |
60 |
|
|
* expand the ipio2 table. |
61 |
|
|
* |
62 |
|
|
* nx dimension of x[] |
63 |
|
|
* |
64 |
|
|
* prec an integer indicating the precision: |
65 |
|
|
* 0 24 bits (single) |
66 |
|
|
* 1 53 bits (double) |
67 |
|
|
* 2 64 bits (extended) |
68 |
|
|
* 3 113 bits (quad) |
69 |
|
|
* |
70 |
|
|
* External function: |
71 |
|
|
* double scalbn(), floor(); |
72 |
|
|
* |
73 |
|
|
* |
74 |
|
|
* Here is the description of some local variables: |
75 |
|
|
* |
76 |
|
|
* jk jk+1 is the initial number of terms of ipio2[] needed |
77 |
|
|
* in the computation. The recommended value is 2,3,4, |
78 |
|
|
* 6 for single, double, extended,and quad. |
79 |
|
|
* |
80 |
|
|
* jz local integer variable indicating the number of |
81 |
|
|
* terms of ipio2[] used. |
82 |
|
|
* |
83 |
|
|
* jx nx - 1 |
84 |
|
|
* |
85 |
|
|
* jv index for pointing to the suitable ipio2[] for the |
86 |
|
|
* computation. In general, we want |
87 |
|
|
* ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 |
88 |
|
|
* is an integer. Thus |
89 |
|
|
* e0-3-24*jv >= 0 or (e0-3)/24 >= jv |
90 |
|
|
* Hence jv = max(0,(e0-3)/24). |
91 |
|
|
* |
92 |
|
|
* jp jp+1 is the number of terms in PIo2[] needed, jp = jk. |
93 |
|
|
* |
94 |
|
|
* q[] double array with integral value, representing the |
95 |
|
|
* 24-bits chunk of the product of x and 2/pi. |
96 |
|
|
* |
97 |
|
|
* q0 the corresponding exponent of q[0]. Note that the |
98 |
|
|
* exponent for q[i] would be q0-24*i. |
99 |
|
|
* |
100 |
|
|
* PIo2[] double precision array, obtained by cutting pi/2 |
101 |
|
|
* into 24 bits chunks. |
102 |
|
|
* |
103 |
|
|
* f[] ipio2[] in floating point |
104 |
|
|
* |
105 |
|
|
* iq[] integer array by breaking up q[] in 24-bits chunk. |
106 |
|
|
* |
107 |
|
|
* fq[] final product of x*(2/pi) in fq[0],..,fq[jk] |
108 |
|
|
* |
109 |
|
|
* ih integer. If >0 it indicates q[] is >= 0.5, hence |
110 |
|
|
* it also indicates the *sign* of the result. |
111 |
|
|
* |
112 |
|
|
*/ |
113 |
|
|
|
114 |
|
|
|
115 |
|
|
/* |
116 |
|
|
* Constants: |
117 |
|
|
* The hexadecimal values are the intended ones for the following |
118 |
|
|
* constants. The decimal values may be used, provided that the |
119 |
|
|
* compiler will convert from decimal to binary accurately enough |
120 |
|
|
* to produce the hexadecimal values shown. |
121 |
|
|
*/ |
122 |
|
|
|
123 |
|
|
#include <float.h> |
124 |
|
|
#include <math.h> |
125 |
|
|
|
126 |
|
|
#include "math_private.h" |
127 |
|
|
|
128 |
|
|
static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ |
129 |
|
|
|
130 |
|
|
/* |
131 |
|
|
* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi |
132 |
|
|
* |
133 |
|
|
* integer array, contains the (24*i)-th to (24*i+23)-th |
134 |
|
|
* bit of 2/pi after binary point. The corresponding |
135 |
|
|
* floating value is |
136 |
|
|
* |
137 |
|
|
* ipio2[i] * 2^(-24(i+1)). |
138 |
|
|
* |
139 |
|
|
* NB: This table must have at least (e0-3)/24 + jk terms. |
140 |
|
|
* For quad precision (e0 <= 16360, jk = 6), this is 686. |
141 |
|
|
*/ |
142 |
|
|
static const int32_t ipio2[] = { |
143 |
|
|
0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, |
144 |
|
|
0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, |
145 |
|
|
0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, |
146 |
|
|
0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, |
147 |
|
|
0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, |
148 |
|
|
0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, |
149 |
|
|
0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, |
150 |
|
|
0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, |
151 |
|
|
0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, |
152 |
|
|
0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, |
153 |
|
|
0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, |
154 |
|
|
|
155 |
|
|
#if LDBL_MAX_EXP > 1024 |
156 |
|
|
#if LDBL_MAX_EXP > 16384 |
157 |
|
|
#error "ipio2 table needs to be expanded" |
158 |
|
|
#endif |
159 |
|
|
0x47C419, 0xC367CD, 0xDCE809, 0x2A8359, 0xC4768B, 0x961CA6, |
160 |
|
|
0xDDAF44, 0xD15719, 0x053EA5, 0xFF0705, 0x3F7E33, 0xE832C2, |
161 |
|
|
0xDE4F98, 0x327DBB, 0xC33D26, 0xEF6B1E, 0x5EF89F, 0x3A1F35, |
162 |
|
|
0xCAF27F, 0x1D87F1, 0x21907C, 0x7C246A, 0xFA6ED5, 0x772D30, |
163 |
|
|
0x433B15, 0xC614B5, 0x9D19C3, 0xC2C4AD, 0x414D2C, 0x5D000C, |
164 |
|
|
0x467D86, 0x2D71E3, 0x9AC69B, 0x006233, 0x7CD2B4, 0x97A7B4, |
165 |
|
|
0xD55537, 0xF63ED7, 0x1810A3, 0xFC764D, 0x2A9D64, 0xABD770, |
166 |
|
|
0xF87C63, 0x57B07A, 0xE71517, 0x5649C0, 0xD9D63B, 0x3884A7, |
167 |
|
|
0xCB2324, 0x778AD6, 0x23545A, 0xB91F00, 0x1B0AF1, 0xDFCE19, |
168 |
|
|
0xFF319F, 0x6A1E66, 0x615799, 0x47FBAC, 0xD87F7E, 0xB76522, |
169 |
|
|
0x89E832, 0x60BFE6, 0xCDC4EF, 0x09366C, 0xD43F5D, 0xD7DE16, |
170 |
|
|
0xDE3B58, 0x929BDE, 0x2822D2, 0xE88628, 0x4D58E2, 0x32CAC6, |
171 |
|
|
0x16E308, 0xCB7DE0, 0x50C017, 0xA71DF3, 0x5BE018, 0x34132E, |
172 |
|
|
0x621283, 0x014883, 0x5B8EF5, 0x7FB0AD, 0xF2E91E, 0x434A48, |
173 |
|
|
0xD36710, 0xD8DDAA, 0x425FAE, 0xCE616A, 0xA4280A, 0xB499D3, |
174 |
|
|
0xF2A606, 0x7F775C, 0x83C2A3, 0x883C61, 0x78738A, 0x5A8CAF, |
175 |
|
|
0xBDD76F, 0x63A62D, 0xCBBFF4, 0xEF818D, 0x67C126, 0x45CA55, |
176 |
|
|
0x36D9CA, 0xD2A828, 0x8D61C2, 0x77C912, 0x142604, 0x9B4612, |
177 |
|
|
0xC459C4, 0x44C5C8, 0x91B24D, 0xF31700, 0xAD43D4, 0xE54929, |
178 |
|
|
0x10D5FD, 0xFCBE00, 0xCC941E, 0xEECE70, 0xF53E13, 0x80F1EC, |
179 |
|
|
0xC3E7B3, 0x28F8C7, 0x940593, 0x3E71C1, 0xB3092E, 0xF3450B, |
180 |
|
|
0x9C1288, 0x7B20AB, 0x9FB52E, 0xC29247, 0x2F327B, 0x6D550C, |
181 |
|
|
0x90A772, 0x1FE76B, 0x96CB31, 0x4A1679, 0xE27941, 0x89DFF4, |
182 |
|
|
0x9794E8, 0x84E6E2, 0x973199, 0x6BED88, 0x365F5F, 0x0EFDBB, |
183 |
|
|
0xB49A48, 0x6CA467, 0x427271, 0x325D8D, 0xB8159F, 0x09E5BC, |
184 |
|
|
0x25318D, 0x3974F7, 0x1C0530, 0x010C0D, 0x68084B, 0x58EE2C, |
185 |
|
|
0x90AA47, 0x02E774, 0x24D6BD, 0xA67DF7, 0x72486E, 0xEF169F, |
186 |
|
|
0xA6948E, 0xF691B4, 0x5153D1, 0xF20ACF, 0x339820, 0x7E4BF5, |
187 |
|
|
0x6863B2, 0x5F3EDD, 0x035D40, 0x7F8985, 0x295255, 0xC06437, |
188 |
|
|
0x10D86D, 0x324832, 0x754C5B, 0xD4714E, 0x6E5445, 0xC1090B, |
189 |
|
|
0x69F52A, 0xD56614, 0x9D0727, 0x50045D, 0xDB3BB4, 0xC576EA, |
190 |
|
|
0x17F987, 0x7D6B49, 0xBA271D, 0x296996, 0xACCCC6, 0x5414AD, |
191 |
|
|
0x6AE290, 0x89D988, 0x50722C, 0xBEA404, 0x940777, 0x7030F3, |
192 |
|
|
0x27FC00, 0xA871EA, 0x49C266, 0x3DE064, 0x83DD97, 0x973FA3, |
193 |
|
|
0xFD9443, 0x8C860D, 0xDE4131, 0x9D3992, 0x8C70DD, 0xE7B717, |
194 |
|
|
0x3BDF08, 0x2B3715, 0xA0805C, 0x93805A, 0x921110, 0xD8E80F, |
195 |
|
|
0xAF806C, 0x4BFFDB, 0x0F9038, 0x761859, 0x15A562, 0xBBCB61, |
196 |
|
|
0xB989C7, 0xBD4010, 0x04F2D2, 0x277549, 0xF6B6EB, 0xBB22DB, |
197 |
|
|
0xAA140A, 0x2F2689, 0x768364, 0x333B09, 0x1A940E, 0xAA3A51, |
198 |
|
|
0xC2A31D, 0xAEEDAF, 0x12265C, 0x4DC26D, 0x9C7A2D, 0x9756C0, |
199 |
|
|
0x833F03, 0xF6F009, 0x8C402B, 0x99316D, 0x07B439, 0x15200C, |
200 |
|
|
0x5BC3D8, 0xC492F5, 0x4BADC6, 0xA5CA4E, 0xCD37A7, 0x36A9E6, |
201 |
|
|
0x9492AB, 0x6842DD, 0xDE6319, 0xEF8C76, 0x528B68, 0x37DBFC, |
202 |
|
|
0xABA1AE, 0x3115DF, 0xA1AE00, 0xDAFB0C, 0x664D64, 0xB705ED, |
203 |
|
|
0x306529, 0xBF5657, 0x3AFF47, 0xB9F96A, 0xF3BE75, 0xDF9328, |
204 |
|
|
0x3080AB, 0xF68C66, 0x15CB04, 0x0622FA, 0x1DE4D9, 0xA4B33D, |
205 |
|
|
0x8F1B57, 0x09CD36, 0xE9424E, 0xA4BE13, 0xB52333, 0x1AAAF0, |
206 |
|
|
0xA8654F, 0xA5C1D2, 0x0F3F0B, 0xCD785B, 0x76F923, 0x048B7B, |
207 |
|
|
0x721789, 0x53A6C6, 0xE26E6F, 0x00EBEF, 0x584A9B, 0xB7DAC4, |
208 |
|
|
0xBA66AA, 0xCFCF76, 0x1D02D1, 0x2DF1B1, 0xC1998C, 0x77ADC3, |
209 |
|
|
0xDA4886, 0xA05DF7, 0xF480C6, 0x2FF0AC, 0x9AECDD, 0xBC5C3F, |
210 |
|
|
0x6DDED0, 0x1FC790, 0xB6DB2A, 0x3A25A3, 0x9AAF00, 0x9353AD, |
211 |
|
|
0x0457B6, 0xB42D29, 0x7E804B, 0xA707DA, 0x0EAA76, 0xA1597B, |
212 |
|
|
0x2A1216, 0x2DB7DC, 0xFDE5FA, 0xFEDB89, 0xFDBE89, 0x6C76E4, |
213 |
|
|
0xFCA906, 0x70803E, 0x156E85, 0xFF87FD, 0x073E28, 0x336761, |
214 |
|
|
0x86182A, 0xEABD4D, 0xAFE7B3, 0x6E6D8F, 0x396795, 0x5BBF31, |
215 |
|
|
0x48D784, 0x16DF30, 0x432DC7, 0x356125, 0xCE70C9, 0xB8CB30, |
216 |
|
|
0xFD6CBF, 0xA200A4, 0xE46C05, 0xA0DD5A, 0x476F21, 0xD21262, |
217 |
|
|
0x845CB9, 0x496170, 0xE0566B, 0x015299, 0x375550, 0xB7D51E, |
218 |
|
|
0xC4F133, 0x5F6E13, 0xE4305D, 0xA92E85, 0xC3B21D, 0x3632A1, |
219 |
|
|
0xA4B708, 0xD4B1EA, 0x21F716, 0xE4698F, 0x77FF27, 0x80030C, |
220 |
|
|
0x2D408D, 0xA0CD4F, 0x99A520, 0xD3A2B3, 0x0A5D2F, 0x42F9B4, |
221 |
|
|
0xCBDA11, 0xD0BE7D, 0xC1DB9B, 0xBD17AB, 0x81A2CA, 0x5C6A08, |
222 |
|
|
0x17552E, 0x550027, 0xF0147F, 0x8607E1, 0x640B14, 0x8D4196, |
223 |
|
|
0xDEBE87, 0x2AFDDA, 0xB6256B, 0x34897B, 0xFEF305, 0x9EBFB9, |
224 |
|
|
0x4F6A68, 0xA82A4A, 0x5AC44F, 0xBCF82D, 0x985AD7, 0x95C7F4, |
225 |
|
|
0x8D4D0D, 0xA63A20, 0x5F57A4, 0xB13F14, 0x953880, 0x0120CC, |
226 |
|
|
0x86DD71, 0xB6DEC9, 0xF560BF, 0x11654D, 0x6B0701, 0xACB08C, |
227 |
|
|
0xD0C0B2, 0x485551, 0x0EFB1E, 0xC37295, 0x3B06A3, 0x3540C0, |
228 |
|
|
0x7BDC06, 0xCC45E0, 0xFA294E, 0xC8CAD6, 0x41F3E8, 0xDE647C, |
229 |
|
|
0xD8649B, 0x31BED9, 0xC397A4, 0xD45877, 0xC5E369, 0x13DAF0, |
230 |
|
|
0x3C3ABA, 0x461846, 0x5F7555, 0xF5BDD2, 0xC6926E, 0x5D2EAC, |
231 |
|
|
0xED440E, 0x423E1C, 0x87C461, 0xE9FD29, 0xF3D6E7, 0xCA7C22, |
232 |
|
|
0x35916F, 0xC5E008, 0x8DD7FF, 0xE26A6E, 0xC6FDB0, 0xC10893, |
233 |
|
|
0x745D7C, 0xB2AD6B, 0x9D6ECD, 0x7B723E, 0x6A11C6, 0xA9CFF7, |
234 |
|
|
0xDF7329, 0xBAC9B5, 0x5100B7, 0x0DB2E2, 0x24BA74, 0x607DE5, |
235 |
|
|
0x8AD874, 0x2C150D, 0x0C1881, 0x94667E, 0x162901, 0x767A9F, |
236 |
|
|
0xBEFDFD, 0xEF4556, 0x367ED9, 0x13D9EC, 0xB9BA8B, 0xFC97C4, |
237 |
|
|
0x27A831, 0xC36EF1, 0x36C594, 0x56A8D8, 0xB5A8B4, 0x0ECCCF, |
238 |
|
|
0x2D8912, 0x34576F, 0x89562C, 0xE3CE99, 0xB920D6, 0xAA5E6B, |
239 |
|
|
0x9C2A3E, 0xCC5F11, 0x4A0BFD, 0xFBF4E1, 0x6D3B8E, 0x2C86E2, |
240 |
|
|
0x84D4E9, 0xA9B4FC, 0xD1EEEF, 0xC9352E, 0x61392F, 0x442138, |
241 |
|
|
0xC8D91B, 0x0AFC81, 0x6A4AFB, 0xD81C2F, 0x84B453, 0x8C994E, |
242 |
|
|
0xCC2254, 0xDC552A, 0xD6C6C0, 0x96190B, 0xB8701A, 0x649569, |
243 |
|
|
0x605A26, 0xEE523F, 0x0F117F, 0x11B5F4, 0xF5CBFC, 0x2DBC34, |
244 |
|
|
0xEEBC34, 0xCC5DE8, 0x605EDD, 0x9B8E67, 0xEF3392, 0xB817C9, |
245 |
|
|
0x9B5861, 0xBC57E1, 0xC68351, 0x103ED8, 0x4871DD, 0xDD1C2D, |
246 |
|
|
0xA118AF, 0x462C21, 0xD7F359, 0x987AD9, 0xC0549E, 0xFA864F, |
247 |
|
|
0xFC0656, 0xAE79E5, 0x362289, 0x22AD38, 0xDC9367, 0xAAE855, |
248 |
|
|
0x382682, 0x9BE7CA, 0xA40D51, 0xB13399, 0x0ED7A9, 0x480569, |
249 |
|
|
0xF0B265, 0xA7887F, 0x974C88, 0x36D1F9, 0xB39221, 0x4A827B, |
250 |
|
|
0x21CF98, 0xDC9F40, 0x5547DC, 0x3A74E1, 0x42EB67, 0xDF9DFE, |
251 |
|
|
0x5FD45E, 0xA4677B, 0x7AACBA, 0xA2F655, 0x23882B, 0x55BA41, |
252 |
|
|
0x086E59, 0x862A21, 0x834739, 0xE6E389, 0xD49EE5, 0x40FB49, |
253 |
|
|
0xE956FF, 0xCA0F1C, 0x8A59C5, 0x2BFA94, 0xC5C1D3, 0xCFC50F, |
254 |
|
|
0xAE5ADB, 0x86C547, 0x624385, 0x3B8621, 0x94792C, 0x876110, |
255 |
|
|
0x7B4C2A, 0x1A2C80, 0x12BF43, 0x902688, 0x893C78, 0xE4C4A8, |
256 |
|
|
0x7BDBE5, 0xC23AC4, 0xEAF426, 0x8A67F7, 0xBF920D, 0x2BA365, |
257 |
|
|
0xB1933D, 0x0B7CBD, 0xDC51A4, 0x63DD27, 0xDDE169, 0x19949A, |
258 |
|
|
0x9529A8, 0x28CE68, 0xB4ED09, 0x209F44, 0xCA984E, 0x638270, |
259 |
|
|
0x237C7E, 0x32B90F, 0x8EF5A7, 0xE75614, 0x08F121, 0x2A9DB5, |
260 |
|
|
0x4D7E6F, 0x5119A5, 0xABF9B5, 0xD6DF82, 0x61DD96, 0x023616, |
261 |
|
|
0x9F3AC4, 0xA1A283, 0x6DED72, 0x7A8D39, 0xA9B882, 0x5C326B, |
262 |
|
|
0x5B2746, 0xED3400, 0x7700D2, 0x55F4FC, 0x4D5901, 0x8071E0, |
263 |
|
|
#endif |
264 |
|
|
|
265 |
|
|
}; |
266 |
|
|
|
267 |
|
|
static const double PIo2[] = { |
268 |
|
|
1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ |
269 |
|
|
7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ |
270 |
|
|
5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ |
271 |
|
|
3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ |
272 |
|
|
1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ |
273 |
|
|
1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ |
274 |
|
|
2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ |
275 |
|
|
2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ |
276 |
|
|
}; |
277 |
|
|
|
278 |
|
|
static const double |
279 |
|
|
zero = 0.0, |
280 |
|
|
one = 1.0, |
281 |
|
|
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
282 |
|
|
twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ |
283 |
|
|
|
284 |
|
|
int |
285 |
|
|
__kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec) |
286 |
|
|
{ |
287 |
|
560 |
int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; |
288 |
|
280 |
double z,fw,f[20],fq[20],q[20]; |
289 |
|
|
|
290 |
|
|
/* initialize jk*/ |
291 |
|
280 |
jk = init_jk[prec]; |
292 |
|
|
jp = jk; |
293 |
|
|
|
294 |
|
|
/* determine jx,jv,q0, note that 3>q0 */ |
295 |
|
280 |
jx = nx-1; |
296 |
|
280 |
jv = (e0-3)/24; if(jv<0) jv=0; |
297 |
|
280 |
q0 = e0-24*(jv+1); |
298 |
|
|
|
299 |
|
|
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ |
300 |
|
280 |
j = jv-jx; m = jx+jk; |
301 |
✓✓✓✓
|
7840 |
for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; |
302 |
|
|
|
303 |
|
|
/* compute q[0],q[1],...q[jk] */ |
304 |
✓✓ |
3360 |
for (i=0;i<=jk;i++) { |
305 |
✓✓ |
12600 |
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; |
306 |
|
|
} |
307 |
|
|
|
308 |
|
280 |
jz = jk; |
309 |
|
|
recompute: |
310 |
|
|
/* distill q[] into iq[] reversingly */ |
311 |
✓✓ |
3472 |
for(i=0,j=jz,z=q[jz];j>0;i++,j--) { |
312 |
|
1400 |
fw = (double)((int32_t)(twon24* z)); |
313 |
|
1400 |
iq[i] = (int32_t)(z-two24*fw); |
314 |
|
1400 |
z = q[j-1]+fw; |
315 |
|
|
} |
316 |
|
|
|
317 |
|
|
/* compute n */ |
318 |
|
336 |
z = scalbn(z,q0); /* actual value of z */ |
319 |
|
336 |
z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ |
320 |
|
336 |
n = (int32_t) z; |
321 |
|
336 |
z -= (double)n; |
322 |
|
|
ih = 0; |
323 |
✗✓ |
336 |
if(q0>0) { /* need iq[jz-1] to determine n */ |
324 |
|
|
i = (iq[jz-1]>>(24-q0)); n += i; |
325 |
|
|
iq[jz-1] -= i<<(24-q0); |
326 |
|
|
ih = iq[jz-1]>>(23-q0); |
327 |
|
|
} |
328 |
✗✓ |
336 |
else if(q0==0) ih = iq[jz-1]>>23; |
329 |
✓✓ |
560 |
else if(z>=0.5) ih=2; |
330 |
|
|
|
331 |
✓✓ |
336 |
if(ih>0) { /* q > 0.5 */ |
332 |
|
224 |
n += 1; carry = 0; |
333 |
✓✓ |
2352 |
for(i=0;i<jz ;i++) { /* compute 1-q */ |
334 |
|
952 |
j = iq[i]; |
335 |
✓✓ |
952 |
if(carry==0) { |
336 |
✓✗ |
224 |
if(j!=0) { |
337 |
|
224 |
carry = 1; iq[i] = 0x1000000- j; |
338 |
|
224 |
} |
339 |
|
728 |
} else iq[i] = 0xffffff - j; |
340 |
|
952 |
} |
341 |
✗✓ |
224 |
if(q0>0) { /* rare case: chance is 1 in 12 */ |
342 |
|
|
switch(q0) { |
343 |
|
|
case 1: |
344 |
|
|
iq[jz-1] &= 0x7fffff; break; |
345 |
|
|
case 2: |
346 |
|
|
iq[jz-1] &= 0x3fffff; break; |
347 |
|
|
} |
348 |
|
|
} |
349 |
✓✗ |
224 |
if(ih==2) { |
350 |
|
224 |
z = one - z; |
351 |
✓✗ |
448 |
if(carry!=0) z -= scalbn(one,q0); |
352 |
|
|
} |
353 |
|
|
} |
354 |
|
|
|
355 |
|
|
/* check if recomputation is needed */ |
356 |
✓✓ |
336 |
if(z==zero) { |
357 |
|
|
j = 0; |
358 |
✓✓ |
336 |
for (i=jz-1;i>=jk;i--) j |= iq[i]; |
359 |
✓✓ |
112 |
if(j==0) { /* need recomputation */ |
360 |
✗✓ |
112 |
for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ |
361 |
|
|
|
362 |
✓✓ |
224 |
for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ |
363 |
|
56 |
f[jx+i] = (double) ipio2[jv+i]; |
364 |
✓✓ |
448 |
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; |
365 |
|
56 |
q[i] = fw; |
366 |
|
|
} |
367 |
|
|
jz += k; |
368 |
|
56 |
goto recompute; |
369 |
|
|
} |
370 |
|
|
} |
371 |
|
|
|
372 |
|
|
/* chop off zero terms */ |
373 |
✓✓ |
280 |
if(z==0.0) { |
374 |
|
56 |
jz -= 1; q0 -= 24; |
375 |
✗✓ |
112 |
while(iq[jz]==0) { jz--; q0-=24;} |
376 |
|
|
} else { /* break z into 24-bit if necessary */ |
377 |
|
224 |
z = scalbn(z,-q0); |
378 |
✓✗ |
224 |
if(z>=two24) { |
379 |
|
224 |
fw = (double)((int32_t)(twon24*z)); |
380 |
|
224 |
iq[jz] = (int32_t)(z-two24*fw); |
381 |
|
224 |
jz += 1; q0 += 24; |
382 |
|
224 |
iq[jz] = (int32_t) fw; |
383 |
|
224 |
} else iq[jz] = (int32_t) z ; |
384 |
|
|
} |
385 |
|
|
|
386 |
|
|
/* convert integer "bit" chunk to floating-point value */ |
387 |
|
280 |
fw = scalbn(one,q0); |
388 |
✓✓ |
3808 |
for(i=jz;i>=0;i--) { |
389 |
|
1624 |
q[i] = fw*(double)iq[i]; fw*=twon24; |
390 |
|
|
} |
391 |
|
|
|
392 |
|
|
/* compute PIo2[0,...,jp]*q[jz,...,0] */ |
393 |
✓✓ |
3808 |
for(i=jz;i>=0;i--) { |
394 |
✓✓✓✓
|
20328 |
for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; |
395 |
|
1624 |
fq[jz-i] = fw; |
396 |
|
|
} |
397 |
|
|
|
398 |
|
|
/* compress fq[] into y[] */ |
399 |
✗✗✓✗
✓ |
560 |
switch(prec) { |
400 |
|
|
case 0: |
401 |
|
|
fw = 0.0; |
402 |
|
|
for (i=jz;i>=0;i--) fw += fq[i]; |
403 |
|
|
y[0] = (ih==0)? fw: -fw; |
404 |
|
|
break; |
405 |
|
|
case 1: |
406 |
|
|
case 2: |
407 |
|
|
fw = 0.0; |
408 |
✓✓ |
3808 |
for (i=jz;i>=0;i--) fw += fq[i]; |
409 |
|
|
STRICT_ASSIGN(double,fw,fw); |
410 |
|
280 |
y[0] = (ih==0)? fw: -fw; |
411 |
|
280 |
fw = fq[0]-fw; |
412 |
✓✓ |
3248 |
for (i=1;i<=jz;i++) fw += fq[i]; |
413 |
|
280 |
y[1] = (ih==0)? fw: -fw; |
414 |
|
280 |
break; |
415 |
|
|
case 3: /* painful */ |
416 |
|
|
for (i=jz;i>0;i--) { |
417 |
|
|
fw = fq[i-1]+fq[i]; |
418 |
|
|
fq[i] += fq[i-1]-fw; |
419 |
|
|
fq[i-1] = fw; |
420 |
|
|
} |
421 |
|
|
for (i=jz;i>1;i--) { |
422 |
|
|
fw = fq[i-1]+fq[i]; |
423 |
|
|
fq[i] += fq[i-1]-fw; |
424 |
|
|
fq[i-1] = fw; |
425 |
|
|
} |
426 |
|
|
for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; |
427 |
|
|
if(ih==0) { |
428 |
|
|
y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; |
429 |
|
|
} else { |
430 |
|
|
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; |
431 |
|
|
} |
432 |
|
|
} |
433 |
|
840 |
return n&7; |
434 |
|
280 |
} |