1 /* e_j0f.c -- float version of e_j0.c.
2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
6 * ====================================================
7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
13 * ====================================================
15 * $FreeBSD: head/lib/msun/src/e_j0f.c 176451 2008-02-22 02:30:36Z das $
19 #include "math_private.h"
21 static float pzerof(float), qzerof(float);
26 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
27 tpi = 6.3661974669e-01, /* 0x3f22f983 */
28 /* R0/S0 on [0, 2.00] */
29 R02 = 1.5625000000e-02, /* 0x3c800000 */
30 R03 = -1.8997929874e-04, /* 0xb947352e */
31 R04 = 1.8295404516e-06, /* 0x35f58e88 */
32 R05 = -4.6183270541e-09, /* 0xb19eaf3c */
33 S01 = 1.5619102865e-02, /* 0x3c7fe744 */
34 S02 = 1.1692678527e-04, /* 0x38f53697 */
35 S03 = 5.1354652442e-07, /* 0x3509daa6 */
36 S04 = 1.1661400734e-09; /* 0x30a045e8 */
38 static const float zero = 0.0;
41 __ieee754_j0f(float x)
43 float z, s,c,ss,cc,r,u,v;
48 if(ix>=0x7f800000) return one/(x*x);
50 if(ix >= 0x40000000) { /* |x| >= 2.0 */
55 if(ix<0x7f000000) { /* make sure x+x not overflow */
57 if ((s*c)<zero) cc = z/ss;
61 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
62 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
64 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x);
66 u = pzerof(x); v = qzerof(x);
67 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
71 if(ix<0x39000000) { /* |x| < 2**-13 */
72 if(huge+x>one) { /* raise inexact if x != 0 */
73 if(ix<0x32000000) return one; /* |x|<2**-27 */
74 else return one - (float)0.25*x*x;
78 r = z*(R02+z*(R03+z*(R04+z*R05)));
79 s = one+z*(S01+z*(S02+z*(S03+z*S04)));
80 if(ix < 0x3F800000) { /* |x| < 1.00 */
81 return one + z*((float)-0.25+(r/s));
84 return((one+u)*(one-u)+z*(r/s));
89 u00 = -7.3804296553e-02, /* 0xbd9726b5 */
90 u01 = 1.7666645348e-01, /* 0x3e34e80d */
91 u02 = -1.3818567619e-02, /* 0xbc626746 */
92 u03 = 3.4745343146e-04, /* 0x39b62a69 */
93 u04 = -3.8140706238e-06, /* 0xb67ff53c */
94 u05 = 1.9559013964e-08, /* 0x32a802ba */
95 u06 = -3.9820518410e-11, /* 0xae2f21eb */
96 v01 = 1.2730483897e-02, /* 0x3c509385 */
97 v02 = 7.6006865129e-05, /* 0x389f65e0 */
98 v03 = 2.5915085189e-07, /* 0x348b216c */
99 v04 = 4.4111031494e-10; /* 0x2ff280c2 */
102 __ieee754_y0f(float x)
104 float z, s,c,ss,cc,u,v;
107 GET_FLOAT_WORD(hx,x);
109 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
110 if(ix>=0x7f800000) return one/(x+x*x);
111 if(ix==0) return -one/zero;
112 if(hx<0) return zero/zero;
113 if(ix >= 0x40000000) { /* |x| >= 2.0 */
114 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
117 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
118 * = 1/sqrt(2) * (sin(x) + cos(x))
119 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
120 * = 1/sqrt(2) * (sin(x) - cos(x))
121 * To avoid cancellation, use
122 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
123 * to compute the worse one.
130 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
131 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
133 if(ix<0x7f000000) { /* make sure x+x not overflow */
135 if ((s*c)<zero) cc = z/ss;
138 if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x);
140 u = pzerof(x); v = qzerof(x);
141 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
145 if(ix<=0x32000000) { /* x < 2**-27 */
146 return(u00 + tpi*__ieee754_logf(x));
149 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
150 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
151 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
154 /* The asymptotic expansions of pzero is
155 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
156 * For x >= 2, We approximate pzero by
157 * pzero(x) = 1 + (R/S)
158 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
159 * S = 1 + pS0*s^2 + ... + pS4*s^10
161 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
163 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
164 0.0000000000e+00, /* 0x00000000 */
165 -7.0312500000e-02, /* 0xbd900000 */
166 -8.0816707611e+00, /* 0xc1014e86 */
167 -2.5706311035e+02, /* 0xc3808814 */
168 -2.4852163086e+03, /* 0xc51b5376 */
169 -5.2530439453e+03, /* 0xc5a4285a */
171 static const float pS8[5] = {
172 1.1653436279e+02, /* 0x42e91198 */
173 3.8337448730e+03, /* 0x456f9beb */
174 4.0597855469e+04, /* 0x471e95db */
175 1.1675296875e+05, /* 0x47e4087c */
176 4.7627726562e+04, /* 0x473a0bba */
178 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
179 -1.1412546255e-11, /* 0xad48c58a */
180 -7.0312492549e-02, /* 0xbd8fffff */
181 -4.1596107483e+00, /* 0xc0851b88 */
182 -6.7674766541e+01, /* 0xc287597b */
183 -3.3123129272e+02, /* 0xc3a59d9b */
184 -3.4643338013e+02, /* 0xc3ad3779 */
186 static const float pS5[5] = {
187 6.0753936768e+01, /* 0x42730408 */
188 1.0512523193e+03, /* 0x44836813 */
189 5.9789707031e+03, /* 0x45bad7c4 */
190 9.6254453125e+03, /* 0x461665c8 */
191 2.4060581055e+03, /* 0x451660ee */
194 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
195 -2.5470459075e-09, /* 0xb12f081b */
196 -7.0311963558e-02, /* 0xbd8fffb8 */
197 -2.4090321064e+00, /* 0xc01a2d95 */
198 -2.1965976715e+01, /* 0xc1afba52 */
199 -5.8079170227e+01, /* 0xc2685112 */
200 -3.1447946548e+01, /* 0xc1fb9565 */
202 static const float pS3[5] = {
203 3.5856033325e+01, /* 0x420f6c94 */
204 3.6151397705e+02, /* 0x43b4c1ca */
205 1.1936077881e+03, /* 0x44953373 */
206 1.1279968262e+03, /* 0x448cffe6 */
207 1.7358093262e+02, /* 0x432d94b8 */
210 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
211 -8.8753431271e-08, /* 0xb3be98b7 */
212 -7.0303097367e-02, /* 0xbd8ffb12 */
213 -1.4507384300e+00, /* 0xbfb9b1cc */
214 -7.6356959343e+00, /* 0xc0f4579f */
215 -1.1193166733e+01, /* 0xc1331736 */
216 -3.2336456776e+00, /* 0xc04ef40d */
218 static const float pS2[5] = {
219 2.2220300674e+01, /* 0x41b1c32d */
220 1.3620678711e+02, /* 0x430834f0 */
221 2.7047027588e+02, /* 0x43873c32 */
222 1.5387539673e+02, /* 0x4319e01a */
223 1.4657617569e+01, /* 0x416a859a */
226 static float pzerof(float x)
231 GET_FLOAT_WORD(ix,x);
233 if(ix>=0x41000000) {p = pR8; q= pS8;}
234 else if(ix>=0x40f71c58){p = pR5; q= pS5;}
235 else if(ix>=0x4036db68){p = pR3; q= pS3;}
236 else if(ix>=0x40000000){p = pR2; q= pS2;}
238 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
239 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
244 /* For x >= 8, the asymptotic expansions of qzero is
245 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
246 * We approximate pzero by
247 * qzero(x) = s*(-1.25 + (R/S))
248 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
249 * S = 1 + qS0*s^2 + ... + qS5*s^12
251 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
253 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
254 0.0000000000e+00, /* 0x00000000 */
255 7.3242187500e-02, /* 0x3d960000 */
256 1.1768206596e+01, /* 0x413c4a93 */
257 5.5767340088e+02, /* 0x440b6b19 */
258 8.8591972656e+03, /* 0x460a6cca */
259 3.7014625000e+04, /* 0x471096a0 */
261 static const float qS8[6] = {
262 1.6377603149e+02, /* 0x4323c6aa */
263 8.0983447266e+03, /* 0x45fd12c2 */
264 1.4253829688e+05, /* 0x480b3293 */
265 8.0330925000e+05, /* 0x49441ed4 */
266 8.4050156250e+05, /* 0x494d3359 */
267 -3.4389928125e+05, /* 0xc8a7eb69 */
270 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
271 1.8408595828e-11, /* 0x2da1ec79 */
272 7.3242180049e-02, /* 0x3d95ffff */
273 5.8356351852e+00, /* 0x40babd86 */
274 1.3511157227e+02, /* 0x43071c90 */
275 1.0272437744e+03, /* 0x448067cd */
276 1.9899779053e+03, /* 0x44f8bf4b */
278 static const float qS5[6] = {
279 8.2776611328e+01, /* 0x42a58da0 */
280 2.0778142090e+03, /* 0x4501dd07 */
281 1.8847289062e+04, /* 0x46933e94 */
282 5.6751113281e+04, /* 0x475daf1d */
283 3.5976753906e+04, /* 0x470c88c1 */
284 -5.3543427734e+03, /* 0xc5a752be */
287 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
288 4.3774099900e-09, /* 0x3196681b */
289 7.3241114616e-02, /* 0x3d95ff70 */
290 3.3442313671e+00, /* 0x405607e3 */
291 4.2621845245e+01, /* 0x422a7cc5 */
292 1.7080809021e+02, /* 0x432acedf */
293 1.6673394775e+02, /* 0x4326bbe4 */
295 static const float qS3[6] = {
296 4.8758872986e+01, /* 0x42430916 */
297 7.0968920898e+02, /* 0x44316c1c */
298 3.7041481934e+03, /* 0x4567825f */
299 6.4604252930e+03, /* 0x45c9e367 */
300 2.5163337402e+03, /* 0x451d4557 */
301 -1.4924745178e+02, /* 0xc3153f59 */
304 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
305 1.5044444979e-07, /* 0x342189db */
306 7.3223426938e-02, /* 0x3d95f62a */
307 1.9981917143e+00, /* 0x3fffc4bf */
308 1.4495602608e+01, /* 0x4167edfd */
309 3.1666231155e+01, /* 0x41fd5471 */
310 1.6252708435e+01, /* 0x4182058c */
312 static const float qS2[6] = {
313 3.0365585327e+01, /* 0x41f2ecb8 */
314 2.6934811401e+02, /* 0x4386ac8f */
315 8.4478375244e+02, /* 0x44533229 */
316 8.8293585205e+02, /* 0x445cbbe5 */
317 2.1266638184e+02, /* 0x4354aa98 */
318 -5.3109550476e+00, /* 0xc0a9f358 */
321 static float qzerof(float x)
326 GET_FLOAT_WORD(ix,x);
328 if(ix>=0x41000000) {p = qR8; q= qS8;}
329 else if(ix>=0x40f71c58){p = qR5; q= qS5;}
330 else if(ix>=0x4036db68){p = qR3; q= qS3;}
331 else if(ix>=0x40000000){p = qR2; q= qS2;}
333 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
334 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
335 return (-(float).125 + r/s)/x;