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 * ====================================================
17 static char rcsid
[] = "$FreeBSD: src/lib/msun/src/e_j1f.c,v 1.7 2002/05/28 18:15:04 alfred Exp $";
21 * See e_j1.c for complete comments.
25 #include "math_private.h"
27 static __inline
float ponef(float), qonef(float);
29 static const volatile float vone
__attribute__ ((__section__(".rodata"))) = 1, vzero
__attribute__ ((__section__(".rodata"))) = 0;
34 invsqrtpi
= 5.6418961287e-01, /* 0x3f106ebb */
35 tpi
= 6.3661974669e-01, /* 0x3f22f983 */
37 r00
= -6.2500000000e-02, /* 0xbd800000 */
38 r01
= 1.4070566976e-03, /* 0x3ab86cfd */
39 r02
= -1.5995563444e-05, /* 0xb7862e36 */
40 r03
= 4.9672799207e-08, /* 0x335557d2 */
41 s01
= 1.9153760746e-02, /* 0x3c9ce859 */
42 s02
= 1.8594678841e-04, /* 0x3942fab6 */
43 s03
= 1.1771846857e-06, /* 0x359dffc2 */
44 s04
= 5.0463624390e-09, /* 0x31ad6446 */
45 s05
= 1.2354227016e-11; /* 0x2d59567e */
47 static const float zero
= 0.0;
50 __ieee754_j1f(float x
)
52 float z
, s
,c
,ss
,cc
,r
,u
,v
,y
;
57 if(ix
>=0x7f800000) return one
/x
;
59 if(ix
>= 0x40000000) { /* |x| >= 2.0 */
64 if(ix
<0x7f000000) { /* make sure y+y not overflow */
66 if ((s
*c
)>zero
) cc
= z
/ss
;
70 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
71 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
73 if(ix
>0x58000000) z
= (invsqrtpi
*cc
)/sqrtf(y
); /* |x|>2**49 */
75 u
= ponef(y
); v
= qonef(y
);
76 z
= invsqrtpi
*(u
*cc
-v
*ss
)/sqrtf(y
);
81 if(ix
<0x39000000) { /* |x|<2**-13 */
82 if(huge
+x
>one
) return (float)0.5*x
;/* inexact if x!=0 necessary */
85 r
= z
*(r00
+z
*(r01
+z
*(r02
+z
*r03
)));
86 s
= one
+z
*(s01
+z
*(s02
+z
*(s03
+z
*(s04
+z
*s05
))));
88 return(x
*(float)0.5+r
/s
);
91 static const float U0
[5] = {
92 -1.9605709612e-01, /* 0xbe48c331 */
93 5.0443872809e-02, /* 0x3d4e9e3c */
94 -1.9125689287e-03, /* 0xbafaaf2a */
95 2.3525259166e-05, /* 0x37c5581c */
96 -9.1909917899e-08, /* 0xb3c56003 */
98 static const float V0
[5] = {
99 1.9916731864e-02, /* 0x3ca3286a */
100 2.0255257550e-04, /* 0x3954644b */
101 1.3560879779e-06, /* 0x35b602d4 */
102 6.2274145840e-09, /* 0x31d5f8eb */
103 1.6655924903e-11, /* 0x2d9281cf */
107 __ieee754_y1f(float x
)
109 float z
, s
,c
,ss
,cc
,u
,v
;
112 GET_FLOAT_WORD(hx
,x
);
114 if(ix
>=0x7f800000) return vone
/(x
+x
*x
);
115 if(ix
==0) return -one
/vzero
;
116 if(hx
<0) return vzero
/vzero
;
117 if(ix
>= 0x40000000) { /* |x| >= 2.0 */
122 if(ix
<0x7f000000) { /* make sure x+x not overflow */
124 if ((s
*c
)>zero
) cc
= z
/ss
;
127 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
130 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
131 * = 1/sqrt(2) * (sin(x) - cos(x))
132 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
133 * = -1/sqrt(2) * (cos(x) + sin(x))
134 * To avoid cancellation, use
135 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
136 * to compute the worse one.
138 if(ix
>0x58000000) z
= (invsqrtpi
*ss
)/sqrtf(x
); /* |x|>2**49 */
140 u
= ponef(x
); v
= qonef(x
);
141 z
= invsqrtpi
*(u
*ss
+v
*cc
)/sqrtf(x
);
145 if(ix
<=0x33000000) { /* x < 2**-25 */
149 u
= U0
[0]+z
*(U0
[1]+z
*(U0
[2]+z
*(U0
[3]+z
*U0
[4])));
150 v
= one
+z
*(V0
[0]+z
*(V0
[1]+z
*(V0
[2]+z
*(V0
[3]+z
*V0
[4]))));
151 return(x
*(u
/v
) + tpi
*(__ieee754_j1f(x
)*__ieee754_logf(x
)-one
/x
));
154 /* For x >= 8, the asymptotic expansions of pone is
155 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
156 * We approximate pone by
157 * pone(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 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
164 static const float pr8
[6] = { /* for x in [inf, 8]=1/[0,0.125] */
165 0.0000000000e+00, /* 0x00000000 */
166 1.1718750000e-01, /* 0x3df00000 */
167 1.3239480972e+01, /* 0x4153d4ea */
168 4.1205184937e+02, /* 0x43ce06a3 */
169 3.8747453613e+03, /* 0x45722bed */
170 7.9144794922e+03, /* 0x45f753d6 */
172 static const float ps8
[5] = {
173 1.1420736694e+02, /* 0x42e46a2c */
174 3.6509309082e+03, /* 0x45642ee5 */
175 3.6956207031e+04, /* 0x47105c35 */
176 9.7602796875e+04, /* 0x47bea166 */
177 3.0804271484e+04, /* 0x46f0a88b */
180 static const float pr5
[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
181 1.3199052094e-11, /* 0x2d68333f */
182 1.1718749255e-01, /* 0x3defffff */
183 6.8027510643e+00, /* 0x40d9b023 */
184 1.0830818176e+02, /* 0x42d89dca */
185 5.1763616943e+02, /* 0x440168b7 */
186 5.2871520996e+02, /* 0x44042dc6 */
188 static const float ps5
[5] = {
189 5.9280597687e+01, /* 0x426d1f55 */
190 9.9140142822e+02, /* 0x4477d9b1 */
191 5.3532670898e+03, /* 0x45a74a23 */
192 7.8446904297e+03, /* 0x45f52586 */
193 1.5040468750e+03, /* 0x44bc0180 */
196 static const float pr3
[6] = {
197 3.0250391081e-09, /* 0x314fe10d */
198 1.1718686670e-01, /* 0x3defffab */
199 3.9329774380e+00, /* 0x407bb5e7 */
200 3.5119403839e+01, /* 0x420c7a45 */
201 9.1055007935e+01, /* 0x42b61c2a */
202 4.8559066772e+01, /* 0x42423c7c */
204 static const float ps3
[5] = {
205 3.4791309357e+01, /* 0x420b2a4d */
206 3.3676245117e+02, /* 0x43a86198 */
207 1.0468714600e+03, /* 0x4482dbe3 */
208 8.9081134033e+02, /* 0x445eb3ed */
209 1.0378793335e+02, /* 0x42cf936c */
212 static const float pr2
[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
213 1.0771083225e-07, /* 0x33e74ea8 */
214 1.1717621982e-01, /* 0x3deffa16 */
215 2.3685150146e+00, /* 0x401795c0 */
216 1.2242610931e+01, /* 0x4143e1bc */
217 1.7693971634e+01, /* 0x418d8d41 */
218 5.0735230446e+00, /* 0x40a25a4d */
220 static const float ps2
[5] = {
221 2.1436485291e+01, /* 0x41ab7dec */
222 1.2529022980e+02, /* 0x42fa9499 */
223 2.3227647400e+02, /* 0x436846c7 */
224 1.1767937469e+02, /* 0x42eb5bd7 */
225 8.3646392822e+00, /* 0x4105d590 */
228 static float ponef(float x
)
233 GET_FLOAT_WORD(ix
,x
);
235 if(ix
>=0x41000000) {p
= pr8
; q
= ps8
;}
236 else if(ix
>=0x409173eb){p
= pr5
; q
= ps5
;}
237 else if(ix
>=0x4036d917){p
= pr3
; q
= ps3
;}
238 else {p
= pr2
; q
= ps2
;} /* ix>=0x40000000 */
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]))));
246 /* For x >= 8, the asymptotic expansions of qone is
247 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
248 * We approximate pone by
249 * qone(x) = s*(0.375 + (R/S))
250 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
251 * S = 1 + qs1*s^2 + ... + qs6*s^12
253 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
256 static const float qr8
[6] = { /* for x in [inf, 8]=1/[0,0.125] */
257 0.0000000000e+00, /* 0x00000000 */
258 -1.0253906250e-01, /* 0xbdd20000 */
259 -1.6271753311e+01, /* 0xc1822c8d */
260 -7.5960174561e+02, /* 0xc43de683 */
261 -1.1849806641e+04, /* 0xc639273a */
262 -4.8438511719e+04, /* 0xc73d3683 */
264 static const float qs8
[6] = {
265 1.6139537048e+02, /* 0x43216537 */
266 7.8253862305e+03, /* 0x45f48b17 */
267 1.3387534375e+05, /* 0x4802bcd6 */
268 7.1965775000e+05, /* 0x492fb29c */
269 6.6660125000e+05, /* 0x4922be94 */
270 -2.9449025000e+05, /* 0xc88fcb48 */
273 static const float qr5
[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
274 -2.0897993405e-11, /* 0xadb7d219 */
275 -1.0253904760e-01, /* 0xbdd1fffe */
276 -8.0564479828e+00, /* 0xc100e736 */
277 -1.8366960144e+02, /* 0xc337ab6b */
278 -1.3731937256e+03, /* 0xc4aba633 */
279 -2.6124443359e+03, /* 0xc523471c */
281 static const float qs5
[6] = {
282 8.1276550293e+01, /* 0x42a28d98 */
283 1.9917987061e+03, /* 0x44f8f98f */
284 1.7468484375e+04, /* 0x468878f8 */
285 4.9851425781e+04, /* 0x4742bb6d */
286 2.7948074219e+04, /* 0x46da5826 */
287 -4.7191835938e+03, /* 0xc5937978 */
290 static const float qr3
[6] = {
291 -5.0783124372e-09, /* 0xb1ae7d4f */
292 -1.0253783315e-01, /* 0xbdd1ff5b */
293 -4.6101160049e+00, /* 0xc0938612 */
294 -5.7847221375e+01, /* 0xc267638e */
295 -2.2824453735e+02, /* 0xc3643e9a */
296 -2.1921012878e+02, /* 0xc35b35cb */
298 static const float qs3
[6] = {
299 4.7665153503e+01, /* 0x423ea91e */
300 6.7386511230e+02, /* 0x4428775e */
301 3.3801528320e+03, /* 0x45534272 */
302 5.5477290039e+03, /* 0x45ad5dd5 */
303 1.9031191406e+03, /* 0x44ede3d0 */
304 -1.3520118713e+02, /* 0xc3073381 */
307 static const float qr2
[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
308 -1.7838172539e-07, /* 0xb43f8932 */
309 -1.0251704603e-01, /* 0xbdd1f475 */
310 -2.7522056103e+00, /* 0xc0302423 */
311 -1.9663616180e+01, /* 0xc19d4f16 */
312 -4.2325313568e+01, /* 0xc2294d1f */
313 -2.1371921539e+01, /* 0xc1aaf9b2 */
315 static const float qs2
[6] = {
316 2.9533363342e+01, /* 0x41ec4454 */
317 2.5298155212e+02, /* 0x437cfb47 */
318 7.5750280762e+02, /* 0x443d602e */
319 7.3939318848e+02, /* 0x4438d92a */
320 1.5594900513e+02, /* 0x431bf2f2 */
321 -4.9594988823e+00, /* 0xc09eb437 */
324 static float qonef(float x
)
329 GET_FLOAT_WORD(ix
,x
);
331 if(ix
>=0x41000000) {p
= qr8
; q
= qs8
;}
332 else if(ix
>=0x409173eb){p
= qr5
; q
= qs5
;}
333 else if(ix
>=0x4036d917){p
= qr3
; q
= qs3
;}
334 else {p
= qr2
; q
= qs2
;} /* ix>=0x40000000 */
336 r
= p
[0]+z
*(p
[1]+z
*(p
[2]+z
*(p
[3]+z
*(p
[4]+z
*p
[5]))));
337 s
= one
+z
*(q
[0]+z
*(q
[1]+z
*(q
[2]+z
*(q
[3]+z
*(q
[4]+z
*q
[5])))));
338 return ((float).375 + r
/s
)/x
;