1 /* e_j1f.c -- float version of e_j1.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 * ====================================================
16 #include <sys/cdefs.h>
18 * See e_j1.c for complete comments.
22 #include "math_private.h"
24 static __inline float ponef(float), qonef(float);
26 static const volatile float vone = 1, vzero = 0;
31 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
32 tpi = 6.3661974669e-01, /* 0x3f22f983 */
34 r00 = -6.2500000000e-02, /* 0xbd800000 */
35 r01 = 1.4070566976e-03, /* 0x3ab86cfd */
36 r02 = -1.5995563444e-05, /* 0xb7862e36 */
37 r03 = 4.9672799207e-08, /* 0x335557d2 */
38 s01 = 1.9153760746e-02, /* 0x3c9ce859 */
39 s02 = 1.8594678841e-04, /* 0x3942fab6 */
40 s03 = 1.1771846857e-06, /* 0x359dffc2 */
41 s04 = 5.0463624390e-09, /* 0x31ad6446 */
42 s05 = 1.2354227016e-11; /* 0x2d59567e */
44 static const float zero = 0.0;
49 float z, s,c,ss,cc,r,u,v,y;
54 if(ix>=0x7f800000) return one/x;
56 if(ix >= 0x40000000) { /* |x| >= 2.0 */
60 if(ix<0x7f000000) { /* make sure y+y not overflow */
62 if ((s*c)>zero) cc = z/ss;
66 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
67 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
69 if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(y); /* |x|>2**49 */
71 u = ponef(y); v = qonef(y);
72 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
77 if(ix<0x39000000) { /* |x|<2**-13 */
78 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
81 r = z*(r00+z*(r01+z*(r02+z*r03)));
82 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
84 return(x*(float)0.5+r/s);
87 static const float U0[5] = {
88 -1.9605709612e-01, /* 0xbe48c331 */
89 5.0443872809e-02, /* 0x3d4e9e3c */
90 -1.9125689287e-03, /* 0xbafaaf2a */
91 2.3525259166e-05, /* 0x37c5581c */
92 -9.1909917899e-08, /* 0xb3c56003 */
94 static const float V0[5] = {
95 1.9916731864e-02, /* 0x3ca3286a */
96 2.0255257550e-04, /* 0x3954644b */
97 1.3560879779e-06, /* 0x35b602d4 */
98 6.2274145840e-09, /* 0x31d5f8eb */
99 1.6655924903e-11, /* 0x2d9281cf */
105 float z, s,c,ss,cc,u,v;
108 GET_FLOAT_WORD(hx,x);
110 if(ix>=0x7f800000) return vone/(x+x*x);
111 if(ix==0) return -one/vzero;
112 if(hx<0) return vzero/vzero;
113 if(ix >= 0x40000000) { /* |x| >= 2.0 */
117 if(ix<0x7f000000) { /* make sure x+x not overflow */
119 if ((s*c)>zero) cc = z/ss;
122 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
125 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
126 * = 1/sqrt(2) * (sin(x) - cos(x))
127 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
128 * = -1/sqrt(2) * (cos(x) + sin(x))
129 * To avoid cancellation, use
130 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
131 * to compute the worse one.
133 if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */
135 u = ponef(x); v = qonef(x);
136 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
140 if(ix<=0x33000000) { /* x < 2**-25 */
144 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
145 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
146 return(x*(u/v) + tpi*(j1f(x)*logf(x)-one/x));
149 /* For x >= 8, the asymptotic expansions of pone is
150 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
151 * We approximate pone by
152 * pone(x) = 1 + (R/S)
153 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
154 * S = 1 + ps0*s^2 + ... + ps4*s^10
156 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
159 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
160 0.0000000000e+00, /* 0x00000000 */
161 1.1718750000e-01, /* 0x3df00000 */
162 1.3239480972e+01, /* 0x4153d4ea */
163 4.1205184937e+02, /* 0x43ce06a3 */
164 3.8747453613e+03, /* 0x45722bed */
165 7.9144794922e+03, /* 0x45f753d6 */
167 static const float ps8[5] = {
168 1.1420736694e+02, /* 0x42e46a2c */
169 3.6509309082e+03, /* 0x45642ee5 */
170 3.6956207031e+04, /* 0x47105c35 */
171 9.7602796875e+04, /* 0x47bea166 */
172 3.0804271484e+04, /* 0x46f0a88b */
175 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
176 1.3199052094e-11, /* 0x2d68333f */
177 1.1718749255e-01, /* 0x3defffff */
178 6.8027510643e+00, /* 0x40d9b023 */
179 1.0830818176e+02, /* 0x42d89dca */
180 5.1763616943e+02, /* 0x440168b7 */
181 5.2871520996e+02, /* 0x44042dc6 */
183 static const float ps5[5] = {
184 5.9280597687e+01, /* 0x426d1f55 */
185 9.9140142822e+02, /* 0x4477d9b1 */
186 5.3532670898e+03, /* 0x45a74a23 */
187 7.8446904297e+03, /* 0x45f52586 */
188 1.5040468750e+03, /* 0x44bc0180 */
191 static const float pr3[6] = {
192 3.0250391081e-09, /* 0x314fe10d */
193 1.1718686670e-01, /* 0x3defffab */
194 3.9329774380e+00, /* 0x407bb5e7 */
195 3.5119403839e+01, /* 0x420c7a45 */
196 9.1055007935e+01, /* 0x42b61c2a */
197 4.8559066772e+01, /* 0x42423c7c */
199 static const float ps3[5] = {
200 3.4791309357e+01, /* 0x420b2a4d */
201 3.3676245117e+02, /* 0x43a86198 */
202 1.0468714600e+03, /* 0x4482dbe3 */
203 8.9081134033e+02, /* 0x445eb3ed */
204 1.0378793335e+02, /* 0x42cf936c */
207 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
208 1.0771083225e-07, /* 0x33e74ea8 */
209 1.1717621982e-01, /* 0x3deffa16 */
210 2.3685150146e+00, /* 0x401795c0 */
211 1.2242610931e+01, /* 0x4143e1bc */
212 1.7693971634e+01, /* 0x418d8d41 */
213 5.0735230446e+00, /* 0x40a25a4d */
215 static const float ps2[5] = {
216 2.1436485291e+01, /* 0x41ab7dec */
217 1.2529022980e+02, /* 0x42fa9499 */
218 2.3227647400e+02, /* 0x436846c7 */
219 1.1767937469e+02, /* 0x42eb5bd7 */
220 8.3646392822e+00, /* 0x4105d590 */
223 static __inline float
229 GET_FLOAT_WORD(ix,x);
231 if(ix>=0x41000000) {p = pr8; q= ps8;}
232 else if(ix>=0x409173eb){p = pr5; q= ps5;}
233 else if(ix>=0x4036d917){p = pr3; q= ps3;}
234 else {p = pr2; q= ps2;} /* ix>=0x40000000 */
236 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
237 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
242 /* For x >= 8, the asymptotic expansions of qone is
243 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
244 * We approximate pone by
245 * qone(x) = s*(0.375 + (R/S))
246 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
247 * S = 1 + qs1*s^2 + ... + qs6*s^12
249 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
252 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
253 0.0000000000e+00, /* 0x00000000 */
254 -1.0253906250e-01, /* 0xbdd20000 */
255 -1.6271753311e+01, /* 0xc1822c8d */
256 -7.5960174561e+02, /* 0xc43de683 */
257 -1.1849806641e+04, /* 0xc639273a */
258 -4.8438511719e+04, /* 0xc73d3683 */
260 static const float qs8[6] = {
261 1.6139537048e+02, /* 0x43216537 */
262 7.8253862305e+03, /* 0x45f48b17 */
263 1.3387534375e+05, /* 0x4802bcd6 */
264 7.1965775000e+05, /* 0x492fb29c */
265 6.6660125000e+05, /* 0x4922be94 */
266 -2.9449025000e+05, /* 0xc88fcb48 */
269 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
270 -2.0897993405e-11, /* 0xadb7d219 */
271 -1.0253904760e-01, /* 0xbdd1fffe */
272 -8.0564479828e+00, /* 0xc100e736 */
273 -1.8366960144e+02, /* 0xc337ab6b */
274 -1.3731937256e+03, /* 0xc4aba633 */
275 -2.6124443359e+03, /* 0xc523471c */
277 static const float qs5[6] = {
278 8.1276550293e+01, /* 0x42a28d98 */
279 1.9917987061e+03, /* 0x44f8f98f */
280 1.7468484375e+04, /* 0x468878f8 */
281 4.9851425781e+04, /* 0x4742bb6d */
282 2.7948074219e+04, /* 0x46da5826 */
283 -4.7191835938e+03, /* 0xc5937978 */
286 static const float qr3[6] = {
287 -5.0783124372e-09, /* 0xb1ae7d4f */
288 -1.0253783315e-01, /* 0xbdd1ff5b */
289 -4.6101160049e+00, /* 0xc0938612 */
290 -5.7847221375e+01, /* 0xc267638e */
291 -2.2824453735e+02, /* 0xc3643e9a */
292 -2.1921012878e+02, /* 0xc35b35cb */
294 static const float qs3[6] = {
295 4.7665153503e+01, /* 0x423ea91e */
296 6.7386511230e+02, /* 0x4428775e */
297 3.3801528320e+03, /* 0x45534272 */
298 5.5477290039e+03, /* 0x45ad5dd5 */
299 1.9031191406e+03, /* 0x44ede3d0 */
300 -1.3520118713e+02, /* 0xc3073381 */
303 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
304 -1.7838172539e-07, /* 0xb43f8932 */
305 -1.0251704603e-01, /* 0xbdd1f475 */
306 -2.7522056103e+00, /* 0xc0302423 */
307 -1.9663616180e+01, /* 0xc19d4f16 */
308 -4.2325313568e+01, /* 0xc2294d1f */
309 -2.1371921539e+01, /* 0xc1aaf9b2 */
311 static const float qs2[6] = {
312 2.9533363342e+01, /* 0x41ec4454 */
313 2.5298155212e+02, /* 0x437cfb47 */
314 7.5750280762e+02, /* 0x443d602e */
315 7.3939318848e+02, /* 0x4438d92a */
316 1.5594900513e+02, /* 0x431bf2f2 */
317 -4.9594988823e+00, /* 0xc09eb437 */
320 static __inline float
326 GET_FLOAT_WORD(ix,x);
328 if(ix>=0x41000000) {p = qr8; q= qs8;}
329 else if(ix>=0x409173eb){p = qr5; q= qs5;}
330 else if(ix>=0x4036d917){p = qr3; q= qs3;}
331 else {p = qr2; q= qs2;} /* ix>=0x40000000 */
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).375 + r/s)/x;