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