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e_j0f.c (10517B)
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1 /* e_j0f.c -- float version of e_j0.c.
2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
3 */
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 #ifndef lint
17 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_j0f.c,v 1.7 2002/05/28 18:15:03 alfred Exp $";
18 #endif
19
20 #include "math.h"
21 #include "math_private.h"
22
23 static float pzerof(float), qzerof(float);
24
25 static const float
26 huge = 1e30,
27 one = 1.0,
28 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
29 tpi = 6.3661974669e-01, /* 0x3f22f983 */
30 /* R0/S0 on [0, 2.00] */
31 R02 = 1.5625000000e-02, /* 0x3c800000 */
32 R03 = -1.8997929874e-04, /* 0xb947352e */
33 R04 = 1.8295404516e-06, /* 0x35f58e88 */
34 R05 = -4.6183270541e-09, /* 0xb19eaf3c */
35 S01 = 1.5619102865e-02, /* 0x3c7fe744 */
36 S02 = 1.1692678527e-04, /* 0x38f53697 */
37 S03 = 5.1354652442e-07, /* 0x3509daa6 */
38 S04 = 1.1661400734e-09; /* 0x30a045e8 */
39
40 static const float zero = 0.0;
41
42 float
43 __ieee754_j0f(float x)
44 {
45 float z, s,c,ss,cc,r,u,v;
46 int32_t hx,ix;
47
48 GET_FLOAT_WORD(hx,x);
49 ix = hx&0x7fffffff;
50 if(ix>=0x7f800000) return one/(x*x);
51 x = fabsf(x);
52 if(ix >= 0x40000000) { /* |x| >= 2.0 */
53 s = sinf(x);
54 c = cosf(x);
55 ss = s-c;
56 cc = s+c;
57 if(ix<0x7f000000) { /* make sure x+x not overflow */
58 z = -cosf(x+x);
59 if ((s*c)<zero) cc = z/ss;
60 else ss = z/cc;
61 }
62 /*
63 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
64 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
65 */
66 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x);
67 else {
68 u = pzerof(x); v = qzerof(x);
69 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
70 }
71 return z;
72 }
73 if(ix<0x39000000) { /* |x| < 2**-13 */
74 if(huge+x>one) { /* raise inexact if x != 0 */
75 if(ix<0x32000000) return one; /* |x|<2**-27 */
76 else return one - (float)0.25*x*x;
77 }
78 }
79 z = x*x;
80 r = z*(R02+z*(R03+z*(R04+z*R05)));
81 s = one+z*(S01+z*(S02+z*(S03+z*S04)));
82 if(ix < 0x3F800000) { /* |x| < 1.00 */
83 return one + z*((float)-0.25+(r/s));
84 } else {
85 u = (float)0.5*x;
86 return((one+u)*(one-u)+z*(r/s));
87 }
88 }
89
90 static const float
91 u00 = -7.3804296553e-02, /* 0xbd9726b5 */
92 u01 = 1.7666645348e-01, /* 0x3e34e80d */
93 u02 = -1.3818567619e-02, /* 0xbc626746 */
94 u03 = 3.4745343146e-04, /* 0x39b62a69 */
95 u04 = -3.8140706238e-06, /* 0xb67ff53c */
96 u05 = 1.9559013964e-08, /* 0x32a802ba */
97 u06 = -3.9820518410e-11, /* 0xae2f21eb */
98 v01 = 1.2730483897e-02, /* 0x3c509385 */
99 v02 = 7.6006865129e-05, /* 0x389f65e0 */
100 v03 = 2.5915085189e-07, /* 0x348b216c */
101 v04 = 4.4111031494e-10; /* 0x2ff280c2 */
102
103 float
104 __ieee754_y0f(float x)
105 {
106 float z, s,c,ss,cc,u,v;
107 int32_t hx,ix;
108
109 GET_FLOAT_WORD(hx,x);
110 ix = 0x7fffffff&hx;
111 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
112 if(ix>=0x7f800000) return one/(x+x*x);
113 if(ix==0) return -one/zero;
114 if(hx<0) return zero/zero;
115 if(ix >= 0x40000000) { /* |x| >= 2.0 */
116 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
117 * where x0 = x-pi/4
118 * Better formula:
119 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
120 * = 1/sqrt(2) * (sin(x) + cos(x))
121 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
122 * = 1/sqrt(2) * (sin(x) - cos(x))
123 * To avoid cancellation, use
124 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
125 * to compute the worse one.
126 */
127 s = sinf(x);
128 c = cosf(x);
129 ss = s-c;
130 cc = s+c;
131 /*
132 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
133 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
134 */
135 if(ix<0x7f000000) { /* make sure x+x not overflow */
136 z = -cosf(x+x);
137 if ((s*c)<zero) cc = z/ss;
138 else ss = z/cc;
139 }
140 if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x);
141 else {
142 u = pzerof(x); v = qzerof(x);
143 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
144 }
145 return z;
146 }
147 if(ix<=0x32000000) { /* x < 2**-27 */
148 return(u00 + tpi*__ieee754_logf(x));
149 }
150 z = x*x;
151 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
152 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
153 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
154 }
155
156 /* The asymptotic expansions of pzero is
157 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
158 * For x >= 2, We approximate pzero by
159 * pzero(x) = 1 + (R/S)
160 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
161 * S = 1 + pS0*s^2 + ... + pS4*s^10
162 * and
163 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
164 */
165 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
166 0.0000000000e+00, /* 0x00000000 */
167 -7.0312500000e-02, /* 0xbd900000 */
168 -8.0816707611e+00, /* 0xc1014e86 */
169 -2.5706311035e+02, /* 0xc3808814 */
170 -2.4852163086e+03, /* 0xc51b5376 */
171 -5.2530439453e+03, /* 0xc5a4285a */
172 };
173 static const float pS8[5] = {
174 1.1653436279e+02, /* 0x42e91198 */
175 3.8337448730e+03, /* 0x456f9beb */
176 4.0597855469e+04, /* 0x471e95db */
177 1.1675296875e+05, /* 0x47e4087c */
178 4.7627726562e+04, /* 0x473a0bba */
179 };
180 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
181 -1.1412546255e-11, /* 0xad48c58a */
182 -7.0312492549e-02, /* 0xbd8fffff */
183 -4.1596107483e+00, /* 0xc0851b88 */
184 -6.7674766541e+01, /* 0xc287597b */
185 -3.3123129272e+02, /* 0xc3a59d9b */
186 -3.4643338013e+02, /* 0xc3ad3779 */
187 };
188 static const float pS5[5] = {
189 6.0753936768e+01, /* 0x42730408 */
190 1.0512523193e+03, /* 0x44836813 */
191 5.9789707031e+03, /* 0x45bad7c4 */
192 9.6254453125e+03, /* 0x461665c8 */
193 2.4060581055e+03, /* 0x451660ee */
194 };
195
196 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
197 -2.5470459075e-09, /* 0xb12f081b */
198 -7.0311963558e-02, /* 0xbd8fffb8 */
199 -2.4090321064e+00, /* 0xc01a2d95 */
200 -2.1965976715e+01, /* 0xc1afba52 */
201 -5.8079170227e+01, /* 0xc2685112 */
202 -3.1447946548e+01, /* 0xc1fb9565 */
203 };
204 static const float pS3[5] = {
205 3.5856033325e+01, /* 0x420f6c94 */
206 3.6151397705e+02, /* 0x43b4c1ca */
207 1.1936077881e+03, /* 0x44953373 */
208 1.1279968262e+03, /* 0x448cffe6 */
209 1.7358093262e+02, /* 0x432d94b8 */
210 };
211
212 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
213 -8.8753431271e-08, /* 0xb3be98b7 */
214 -7.0303097367e-02, /* 0xbd8ffb12 */
215 -1.4507384300e+00, /* 0xbfb9b1cc */
216 -7.6356959343e+00, /* 0xc0f4579f */
217 -1.1193166733e+01, /* 0xc1331736 */
218 -3.2336456776e+00, /* 0xc04ef40d */
219 };
220 static const float pS2[5] = {
221 2.2220300674e+01, /* 0x41b1c32d */
222 1.3620678711e+02, /* 0x430834f0 */
223 2.7047027588e+02, /* 0x43873c32 */
224 1.5387539673e+02, /* 0x4319e01a */
225 1.4657617569e+01, /* 0x416a859a */
226 };
227
228 static float pzerof(float x)
229 {
230 const float *p,*q;
231 float z,r,s;
232 int32_t ix;
233 GET_FLOAT_WORD(ix,x);
234 ix &= 0x7fffffff;
235 if(ix>=0x41000000) {p = pR8; q= pS8;}
236 else if(ix>=0x40f71c58){p = pR5; q= pS5;}
237 else if(ix>=0x4036db68){p = pR3; q= pS3;}
238 else if(ix>=0x40000000){p = pR2; q= pS2;}
239 z = one/(x*x);
240 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
241 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
242 return one+ r/s;
243 }
244
245
246 /* For x >= 8, the asymptotic expansions of qzero is
247 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
248 * We approximate pzero by
249 * qzero(x) = s*(-1.25 + (R/S))
250 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
251 * S = 1 + qS0*s^2 + ... + qS5*s^12
252 * and
253 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
254 */
255 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
256 0.0000000000e+00, /* 0x00000000 */
257 7.3242187500e-02, /* 0x3d960000 */
258 1.1768206596e+01, /* 0x413c4a93 */
259 5.5767340088e+02, /* 0x440b6b19 */
260 8.8591972656e+03, /* 0x460a6cca */
261 3.7014625000e+04, /* 0x471096a0 */
262 };
263 static const float qS8[6] = {
264 1.6377603149e+02, /* 0x4323c6aa */
265 8.0983447266e+03, /* 0x45fd12c2 */
266 1.4253829688e+05, /* 0x480b3293 */
267 8.0330925000e+05, /* 0x49441ed4 */
268 8.4050156250e+05, /* 0x494d3359 */
269 -3.4389928125e+05, /* 0xc8a7eb69 */
270 };
271
272 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
273 1.8408595828e-11, /* 0x2da1ec79 */
274 7.3242180049e-02, /* 0x3d95ffff */
275 5.8356351852e+00, /* 0x40babd86 */
276 1.3511157227e+02, /* 0x43071c90 */
277 1.0272437744e+03, /* 0x448067cd */
278 1.9899779053e+03, /* 0x44f8bf4b */
279 };
280 static const float qS5[6] = {
281 8.2776611328e+01, /* 0x42a58da0 */
282 2.0778142090e+03, /* 0x4501dd07 */
283 1.8847289062e+04, /* 0x46933e94 */
284 5.6751113281e+04, /* 0x475daf1d */
285 3.5976753906e+04, /* 0x470c88c1 */
286 -5.3543427734e+03, /* 0xc5a752be */
287 };
288
289 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
290 4.3774099900e-09, /* 0x3196681b */
291 7.3241114616e-02, /* 0x3d95ff70 */
292 3.3442313671e+00, /* 0x405607e3 */
293 4.2621845245e+01, /* 0x422a7cc5 */
294 1.7080809021e+02, /* 0x432acedf */
295 1.6673394775e+02, /* 0x4326bbe4 */
296 };
297 static const float qS3[6] = {
298 4.8758872986e+01, /* 0x42430916 */
299 7.0968920898e+02, /* 0x44316c1c */
300 3.7041481934e+03, /* 0x4567825f */
301 6.4604252930e+03, /* 0x45c9e367 */
302 2.5163337402e+03, /* 0x451d4557 */
303 -1.4924745178e+02, /* 0xc3153f59 */
304 };
305
306 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
307 1.5044444979e-07, /* 0x342189db */
308 7.3223426938e-02, /* 0x3d95f62a */
309 1.9981917143e+00, /* 0x3fffc4bf */
310 1.4495602608e+01, /* 0x4167edfd */
311 3.1666231155e+01, /* 0x41fd5471 */
312 1.6252708435e+01, /* 0x4182058c */
313 };
314 static const float qS2[6] = {
315 3.0365585327e+01, /* 0x41f2ecb8 */
316 2.6934811401e+02, /* 0x4386ac8f */
317 8.4478375244e+02, /* 0x44533229 */
318 8.8293585205e+02, /* 0x445cbbe5 */
319 2.1266638184e+02, /* 0x4354aa98 */
320 -5.3109550476e+00, /* 0xc0a9f358 */
321 };
322
323 static float qzerof(float x)
324 {
325 const float *p,*q;
326 float s,r,z;
327 int32_t ix;
328 GET_FLOAT_WORD(ix,x);
329 ix &= 0x7fffffff;
330 if(ix>=0x41000000) {p = qR8; q= qS8;}
331 else if(ix>=0x40f71c58){p = qR5; q= qS5;}
332 else if(ix>=0x4036db68){p = qR3; q= qS3;}
333 else if(ix>=0x40000000){p = qR2; q= qS2;}
334 z = one/(x*x);
335 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
336 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
337 return (-(float).125 + r/s)/x;
338 }