1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
|
/* origin: FreeBSD /usr/src/lib/msun/src/e_j0f.c */
/*
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#define _GNU_SOURCE
#include "libm.h"
static float pzerof(float), qzerof(float);
static const float
invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
tpi = 6.3661974669e-01; /* 0x3f22f983 */
static float common(uint32_t ix, float x, int y0)
{
float z,s,c,ss,cc;
/*
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
*/
s = sinf(x);
c = cosf(x);
if (y0)
c = -c;
cc = s+c;
if (ix < 0x7f000000) {
ss = s-c;
z = -cosf(2*x);
if (s*c < 0)
cc = z/ss;
else
ss = z/cc;
if (ix < 0x58800000) {
if (y0)
ss = -ss;
cc = pzerof(x)*cc-qzerof(x)*ss;
}
}
return invsqrtpi*cc/sqrtf(x);
}
/* R0/S0 on [0, 2.00] */
static const float
R02 = 1.5625000000e-02, /* 0x3c800000 */
R03 = -1.8997929874e-04, /* 0xb947352e */
R04 = 1.8295404516e-06, /* 0x35f58e88 */
R05 = -4.6183270541e-09, /* 0xb19eaf3c */
S01 = 1.5619102865e-02, /* 0x3c7fe744 */
S02 = 1.1692678527e-04, /* 0x38f53697 */
S03 = 5.1354652442e-07, /* 0x3509daa6 */
S04 = 1.1661400734e-09; /* 0x30a045e8 */
float j0f(float x)
{
float z,r,s;
uint32_t ix;
GET_FLOAT_WORD(ix, x);
ix &= 0x7fffffff;
if (ix >= 0x7f800000)
return 1/(x*x);
x = fabsf(x);
if (ix >= 0x40000000) { /* |x| >= 2 */
/* large ulp error near zeros */
return common(ix, x, 0);
}
if (ix >= 0x3a000000) { /* |x| >= 2**-11 */
/* up to 4ulp error near 2 */
z = x*x;
r = z*(R02+z*(R03+z*(R04+z*R05)));
s = 1+z*(S01+z*(S02+z*(S03+z*S04)));
return (1+x/2)*(1-x/2) + z*(r/s);
}
if (ix >= 0x21800000) /* |x| >= 2**-60 */
x = 0.25f*x*x;
return 1 - x;
}
static const float
u00 = -7.3804296553e-02, /* 0xbd9726b5 */
u01 = 1.7666645348e-01, /* 0x3e34e80d */
u02 = -1.3818567619e-02, /* 0xbc626746 */
u03 = 3.4745343146e-04, /* 0x39b62a69 */
u04 = -3.8140706238e-06, /* 0xb67ff53c */
u05 = 1.9559013964e-08, /* 0x32a802ba */
u06 = -3.9820518410e-11, /* 0xae2f21eb */
v01 = 1.2730483897e-02, /* 0x3c509385 */
v02 = 7.6006865129e-05, /* 0x389f65e0 */
v03 = 2.5915085189e-07, /* 0x348b216c */
v04 = 4.4111031494e-10; /* 0x2ff280c2 */
float y0f(float x)
{
float z,u,v;
uint32_t ix;
GET_FLOAT_WORD(ix, x);
if ((ix & 0x7fffffff) == 0)
return -1/0.0f;
if (ix>>31)
return 0/0.0f;
if (ix >= 0x7f800000)
return 1/x;
if (ix >= 0x40000000) { /* |x| >= 2.0 */
/* large ulp error near zeros */
return common(ix,x,1);
}
if (ix >= 0x39000000) { /* x >= 2**-13 */
/* large ulp error at x ~= 0.89 */
z = x*x;
u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
v = 1+z*(v01+z*(v02+z*(v03+z*v04)));
return u/v + tpi*(j0f(x)*logf(x));
}
return u00 + tpi*logf(x);
}
/* The asymptotic expansions of pzero is
* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
* For x >= 2, We approximate pzero by
* pzero(x) = 1 + (R/S)
* where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
* S = 1 + pS0*s^2 + ... + pS4*s^10
* and
* | pzero(x)-1-R/S | <= 2 ** ( -60.26)
*/
static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
0.0000000000e+00, /* 0x00000000 */
-7.0312500000e-02, /* 0xbd900000 */
-8.0816707611e+00, /* 0xc1014e86 */
-2.5706311035e+02, /* 0xc3808814 */
-2.4852163086e+03, /* 0xc51b5376 */
-5.2530439453e+03, /* 0xc5a4285a */
};
static const float pS8[5] = {
1.1653436279e+02, /* 0x42e91198 */
3.8337448730e+03, /* 0x456f9beb */
4.0597855469e+04, /* 0x471e95db */
1.1675296875e+05, /* 0x47e4087c */
4.7627726562e+04, /* 0x473a0bba */
};
static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
-1.1412546255e-11, /* 0xad48c58a */
-7.0312492549e-02, /* 0xbd8fffff */
-4.1596107483e+00, /* 0xc0851b88 */
-6.7674766541e+01, /* 0xc287597b */
-3.3123129272e+02, /* 0xc3a59d9b */
-3.4643338013e+02, /* 0xc3ad3779 */
};
static const float pS5[5] = {
6.0753936768e+01, /* 0x42730408 */
1.0512523193e+03, /* 0x44836813 */
5.9789707031e+03, /* 0x45bad7c4 */
9.6254453125e+03, /* 0x461665c8 */
2.4060581055e+03, /* 0x451660ee */
};
static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
-2.5470459075e-09, /* 0xb12f081b */
-7.0311963558e-02, /* 0xbd8fffb8 */
-2.4090321064e+00, /* 0xc01a2d95 */
-2.1965976715e+01, /* 0xc1afba52 */
-5.8079170227e+01, /* 0xc2685112 */
-3.1447946548e+01, /* 0xc1fb9565 */
};
static const float pS3[5] = {
3.5856033325e+01, /* 0x420f6c94 */
3.6151397705e+02, /* 0x43b4c1ca */
1.1936077881e+03, /* 0x44953373 */
1.1279968262e+03, /* 0x448cffe6 */
1.7358093262e+02, /* 0x432d94b8 */
};
static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
-8.8753431271e-08, /* 0xb3be98b7 */
-7.0303097367e-02, /* 0xbd8ffb12 */
-1.4507384300e+00, /* 0xbfb9b1cc */
-7.6356959343e+00, /* 0xc0f4579f */
-1.1193166733e+01, /* 0xc1331736 */
-3.2336456776e+00, /* 0xc04ef40d */
};
static const float pS2[5] = {
2.2220300674e+01, /* 0x41b1c32d */
1.3620678711e+02, /* 0x430834f0 */
2.7047027588e+02, /* 0x43873c32 */
1.5387539673e+02, /* 0x4319e01a */
1.4657617569e+01, /* 0x416a859a */
};
static float pzerof(float x)
{
const float *p,*q;
float_t z,r,s;
uint32_t ix;
GET_FLOAT_WORD(ix, x);
ix &= 0x7fffffff;
if (ix >= 0x41000000){p = pR8; q = pS8;}
else if (ix >= 0x409173eb){p = pR5; q = pS5;}
else if (ix >= 0x4036d917){p = pR3; q = pS3;}
else /*ix >= 0x40000000*/ {p = pR2; q = pS2;}
z = 1.0f/(x*x);
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
return 1.0f + r/s;
}
/* For x >= 8, the asymptotic expansions of qzero is
* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
* We approximate pzero by
* qzero(x) = s*(-1.25 + (R/S))
* where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
* S = 1 + qS0*s^2 + ... + qS5*s^12
* and
* | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
*/
static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
0.0000000000e+00, /* 0x00000000 */
7.3242187500e-02, /* 0x3d960000 */
1.1768206596e+01, /* 0x413c4a93 */
5.5767340088e+02, /* 0x440b6b19 */
8.8591972656e+03, /* 0x460a6cca */
3.7014625000e+04, /* 0x471096a0 */
};
static const float qS8[6] = {
1.6377603149e+02, /* 0x4323c6aa */
8.0983447266e+03, /* 0x45fd12c2 */
1.4253829688e+05, /* 0x480b3293 */
8.0330925000e+05, /* 0x49441ed4 */
8.4050156250e+05, /* 0x494d3359 */
-3.4389928125e+05, /* 0xc8a7eb69 */
};
static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
1.8408595828e-11, /* 0x2da1ec79 */
7.3242180049e-02, /* 0x3d95ffff */
5.8356351852e+00, /* 0x40babd86 */
1.3511157227e+02, /* 0x43071c90 */
1.0272437744e+03, /* 0x448067cd */
1.9899779053e+03, /* 0x44f8bf4b */
};
static const float qS5[6] = {
8.2776611328e+01, /* 0x42a58da0 */
2.0778142090e+03, /* 0x4501dd07 */
1.8847289062e+04, /* 0x46933e94 */
5.6751113281e+04, /* 0x475daf1d */
3.5976753906e+04, /* 0x470c88c1 */
-5.3543427734e+03, /* 0xc5a752be */
};
static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
4.3774099900e-09, /* 0x3196681b */
7.3241114616e-02, /* 0x3d95ff70 */
3.3442313671e+00, /* 0x405607e3 */
4.2621845245e+01, /* 0x422a7cc5 */
1.7080809021e+02, /* 0x432acedf */
1.6673394775e+02, /* 0x4326bbe4 */
};
static const float qS3[6] = {
4.8758872986e+01, /* 0x42430916 */
7.0968920898e+02, /* 0x44316c1c */
3.7041481934e+03, /* 0x4567825f */
6.4604252930e+03, /* 0x45c9e367 */
2.5163337402e+03, /* 0x451d4557 */
-1.4924745178e+02, /* 0xc3153f59 */
};
static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
1.5044444979e-07, /* 0x342189db */
7.3223426938e-02, /* 0x3d95f62a */
1.9981917143e+00, /* 0x3fffc4bf */
1.4495602608e+01, /* 0x4167edfd */
3.1666231155e+01, /* 0x41fd5471 */
1.6252708435e+01, /* 0x4182058c */
};
static const float qS2[6] = {
3.0365585327e+01, /* 0x41f2ecb8 */
2.6934811401e+02, /* 0x4386ac8f */
8.4478375244e+02, /* 0x44533229 */
8.8293585205e+02, /* 0x445cbbe5 */
2.1266638184e+02, /* 0x4354aa98 */
-5.3109550476e+00, /* 0xc0a9f358 */
};
static float qzerof(float x)
{
const float *p,*q;
float_t s,r,z;
uint32_t ix;
GET_FLOAT_WORD(ix, x);
ix &= 0x7fffffff;
if (ix >= 0x41000000){p = qR8; q = qS8;}
else if (ix >= 0x409173eb){p = qR5; q = qS5;}
else if (ix >= 0x4036d917){p = qR3; q = qS3;}
else /*ix >= 0x40000000*/ {p = qR2; q = qS2;}
z = 1.0f/(x*x);
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
return (-.125f + r/s)/x;
}
|
