POK
e_j1f.c
1 /*
2  * POK header
3  *
4  * The following file is a part of the POK project. Any modification should
5  * made according to the POK licence. You CANNOT use this file or a part of
6  * this file is this part of a file for your own project
7  *
8  * For more information on the POK licence, please see our LICENCE FILE
9  *
10  * Please follow the coding guidelines described in doc/CODING_GUIDELINES
11  *
12  * Copyright (c) 2007-2009 POK team
13  *
14  * Created by julien on Fri Jan 30 14:41:34 2009
15  */
16 
17 /* e_j1f.c -- float version of e_j1.c.
18  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
19  */
20 
21 /*
22  * ====================================================
23  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
24  *
25  * Developed at SunPro, a Sun Microsystems, Inc. business.
26  * Permission to use, copy, modify, and distribute this
27  * software is freely granted, provided that this notice
28  * is preserved.
29  * ====================================================
30  */
31 
32 #ifdef POK_NEEDS_LIBMATH
33 
34 #include <libm.h>
35 #include "namespace.h"
36 #include "math_private.h"
37 
38 static float ponef(float), qonef(float);
39 
40 static const float
41 huge = 1e30,
42 one = 1.0,
43 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
44 tpi = 6.3661974669e-01, /* 0x3f22f983 */
45  /* R0/S0 on [0,2] */
46 r00 = -6.2500000000e-02, /* 0xbd800000 */
47 r01 = 1.4070566976e-03, /* 0x3ab86cfd */
48 r02 = -1.5995563444e-05, /* 0xb7862e36 */
49 r03 = 4.9672799207e-08, /* 0x335557d2 */
50 s01 = 1.9153760746e-02, /* 0x3c9ce859 */
51 s02 = 1.8594678841e-04, /* 0x3942fab6 */
52 s03 = 1.1771846857e-06, /* 0x359dffc2 */
53 s04 = 5.0463624390e-09, /* 0x31ad6446 */
54 s05 = 1.2354227016e-11; /* 0x2d59567e */
55 
56 static const float zero = 0.0;
57 
58 float
59 __ieee754_j1f(float x)
60 {
61  float z, s,c,ss,cc,r,u,v,y;
62  int32_t hx,ix;
63 
64  GET_FLOAT_WORD(hx,x);
65  ix = hx&0x7fffffff;
66  if(ix>=0x7f800000) return one/x;
67  y = fabsf(x);
68  if(ix >= 0x40000000) { /* |x| >= 2.0 */
69  s = sinf(y);
70  c = cosf(y);
71  ss = -s-c;
72  cc = s-c;
73  if(ix<0x7f000000) { /* make sure y+y not overflow */
74  z = cosf(y+y);
75  if ((s*c)>zero) cc = z/ss;
76  else ss = z/cc;
77  }
78  /*
79  * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
80  * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
81  */
82 #ifdef DEAD_CODE
83  if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y);
84  else
85 #endif
86  {
87  u = ponef(y); v = qonef(y);
88  z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
89  }
90  if(hx<0) return -z;
91  else return z;
92  }
93  if(ix<0x32000000) { /* |x|<2**-27 */
94  if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
95  }
96  z = x*x;
97  r = z*(r00+z*(r01+z*(r02+z*r03)));
98  s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
99  r *= x;
100  return(x*(float)0.5+r/s);
101 }
102 
103 static const float U0[5] = {
104  -1.9605709612e-01, /* 0xbe48c331 */
105  5.0443872809e-02, /* 0x3d4e9e3c */
106  -1.9125689287e-03, /* 0xbafaaf2a */
107  2.3525259166e-05, /* 0x37c5581c */
108  -9.1909917899e-08, /* 0xb3c56003 */
109 };
110 static const float V0[5] = {
111  1.9916731864e-02, /* 0x3ca3286a */
112  2.0255257550e-04, /* 0x3954644b */
113  1.3560879779e-06, /* 0x35b602d4 */
114  6.2274145840e-09, /* 0x31d5f8eb */
115  1.6655924903e-11, /* 0x2d9281cf */
116 };
117 
118 float
119 __ieee754_y1f(float x)
120 {
121  float z, s,c,ss,cc,u,v;
122  int32_t hx,ix;
123 
124  GET_FLOAT_WORD(hx,x);
125  ix = 0x7fffffff&hx;
126  /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
127  if(ix>=0x7f800000) return one/(x+x*x);
128  if(ix==0) return -one/zero;
129  if(hx<0) return zero/zero;
130  if(ix >= 0x40000000) { /* |x| >= 2.0 */
131  s = sinf(x);
132  c = cosf(x);
133  ss = -s-c;
134  cc = s-c;
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  /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
141  * where x0 = x-3pi/4
142  * Better formula:
143  * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
144  * = 1/sqrt(2) * (sin(x) - cos(x))
145  * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
146  * = -1/sqrt(2) * (cos(x) + sin(x))
147  * To avoid cancellation, use
148  * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
149  * to compute the worse one.
150  */
151  if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
152  else {
153  u = ponef(x); v = qonef(x);
154  z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
155  }
156  return z;
157  }
158  if(ix<=0x24800000) { /* x < 2**-54 */
159  return(-tpi/x);
160  }
161  z = x*x;
162  u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
163  v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
164  return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
165 }
166 
167 /* For x >= 8, the asymptotic expansions of pone is
168  * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
169  * We approximate pone by
170  * pone(x) = 1 + (R/S)
171  * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
172  * S = 1 + ps0*s^2 + ... + ps4*s^10
173  * and
174  * | pone(x)-1-R/S | <= 2 ** ( -60.06)
175  */
176 
177 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
178  0.0000000000e+00, /* 0x00000000 */
179  1.1718750000e-01, /* 0x3df00000 */
180  1.3239480972e+01, /* 0x4153d4ea */
181  4.1205184937e+02, /* 0x43ce06a3 */
182  3.8747453613e+03, /* 0x45722bed */
183  7.9144794922e+03, /* 0x45f753d6 */
184 };
185 static const float ps8[5] = {
186  1.1420736694e+02, /* 0x42e46a2c */
187  3.6509309082e+03, /* 0x45642ee5 */
188  3.6956207031e+04, /* 0x47105c35 */
189  9.7602796875e+04, /* 0x47bea166 */
190  3.0804271484e+04, /* 0x46f0a88b */
191 };
192 
193 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
194  1.3199052094e-11, /* 0x2d68333f */
195  1.1718749255e-01, /* 0x3defffff */
196  6.8027510643e+00, /* 0x40d9b023 */
197  1.0830818176e+02, /* 0x42d89dca */
198  5.1763616943e+02, /* 0x440168b7 */
199  5.2871520996e+02, /* 0x44042dc6 */
200 };
201 static const float ps5[5] = {
202  5.9280597687e+01, /* 0x426d1f55 */
203  9.9140142822e+02, /* 0x4477d9b1 */
204  5.3532670898e+03, /* 0x45a74a23 */
205  7.8446904297e+03, /* 0x45f52586 */
206  1.5040468750e+03, /* 0x44bc0180 */
207 };
208 
209 static const float pr3[6] = {
210  3.0250391081e-09, /* 0x314fe10d */
211  1.1718686670e-01, /* 0x3defffab */
212  3.9329774380e+00, /* 0x407bb5e7 */
213  3.5119403839e+01, /* 0x420c7a45 */
214  9.1055007935e+01, /* 0x42b61c2a */
215  4.8559066772e+01, /* 0x42423c7c */
216 };
217 static const float ps3[5] = {
218  3.4791309357e+01, /* 0x420b2a4d */
219  3.3676245117e+02, /* 0x43a86198 */
220  1.0468714600e+03, /* 0x4482dbe3 */
221  8.9081134033e+02, /* 0x445eb3ed */
222  1.0378793335e+02, /* 0x42cf936c */
223 };
224 
225 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
226  1.0771083225e-07, /* 0x33e74ea8 */
227  1.1717621982e-01, /* 0x3deffa16 */
228  2.3685150146e+00, /* 0x401795c0 */
229  1.2242610931e+01, /* 0x4143e1bc */
230  1.7693971634e+01, /* 0x418d8d41 */
231  5.0735230446e+00, /* 0x40a25a4d */
232 };
233 static const float ps2[5] = {
234  2.1436485291e+01, /* 0x41ab7dec */
235  1.2529022980e+02, /* 0x42fa9499 */
236  2.3227647400e+02, /* 0x436846c7 */
237  1.1767937469e+02, /* 0x42eb5bd7 */
238  8.3646392822e+00, /* 0x4105d590 */
239 };
240 
241 static float
242 ponef(float x)
243 {
244  const float *p,*q;
245  float z,r,s;
246  int32_t ix;
247 
248  p = q = 0;
249  GET_FLOAT_WORD(ix,x);
250  ix &= 0x7fffffff;
251  if(ix>=0x41000000) {p = pr8; q= ps8;}
252  else if(ix>=0x40f71c58){p = pr5; q= ps5;}
253  else if(ix>=0x4036db68){p = pr3; q= ps3;}
254  else if(ix>=0x40000000){p = pr2; q= ps2;}
255  z = one/(x*x);
256  r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
257  s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
258  return one+ r/s;
259 }
260 
261 
262 /* For x >= 8, the asymptotic expansions of qone is
263  * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
264  * We approximate pone by
265  * qone(x) = s*(0.375 + (R/S))
266  * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
267  * S = 1 + qs1*s^2 + ... + qs6*s^12
268  * and
269  * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
270  */
271 
272 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
273  0.0000000000e+00, /* 0x00000000 */
274  -1.0253906250e-01, /* 0xbdd20000 */
275  -1.6271753311e+01, /* 0xc1822c8d */
276  -7.5960174561e+02, /* 0xc43de683 */
277  -1.1849806641e+04, /* 0xc639273a */
278  -4.8438511719e+04, /* 0xc73d3683 */
279 };
280 static const float qs8[6] = {
281  1.6139537048e+02, /* 0x43216537 */
282  7.8253862305e+03, /* 0x45f48b17 */
283  1.3387534375e+05, /* 0x4802bcd6 */
284  7.1965775000e+05, /* 0x492fb29c */
285  6.6660125000e+05, /* 0x4922be94 */
286  -2.9449025000e+05, /* 0xc88fcb48 */
287 };
288 
289 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
290  -2.0897993405e-11, /* 0xadb7d219 */
291  -1.0253904760e-01, /* 0xbdd1fffe */
292  -8.0564479828e+00, /* 0xc100e736 */
293  -1.8366960144e+02, /* 0xc337ab6b */
294  -1.3731937256e+03, /* 0xc4aba633 */
295  -2.6124443359e+03, /* 0xc523471c */
296 };
297 static const float qs5[6] = {
298  8.1276550293e+01, /* 0x42a28d98 */
299  1.9917987061e+03, /* 0x44f8f98f */
300  1.7468484375e+04, /* 0x468878f8 */
301  4.9851425781e+04, /* 0x4742bb6d */
302  2.7948074219e+04, /* 0x46da5826 */
303  -4.7191835938e+03, /* 0xc5937978 */
304 };
305 
306 static const float qr3[6] = { /* for x in [4.5454,2.8570]=1/[0.22001,0.3499] */
307  -5.0783124372e-09, /* 0xb1ae7d4f */
308  -1.0253783315e-01, /* 0xbdd1ff5b */
309  -4.6101160049e+00, /* 0xc0938612 */
310  -5.7847221375e+01, /* 0xc267638e */
311  -2.2824453735e+02, /* 0xc3643e9a */
312  -2.1921012878e+02, /* 0xc35b35cb */
313 };
314 static const float qs3[6] = {
315  4.7665153503e+01, /* 0x423ea91e */
316  6.7386511230e+02, /* 0x4428775e */
317  3.3801528320e+03, /* 0x45534272 */
318  5.5477290039e+03, /* 0x45ad5dd5 */
319  1.9031191406e+03, /* 0x44ede3d0 */
320  -1.3520118713e+02, /* 0xc3073381 */
321 };
322 
323 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
324  -1.7838172539e-07, /* 0xb43f8932 */
325  -1.0251704603e-01, /* 0xbdd1f475 */
326  -2.7522056103e+00, /* 0xc0302423 */
327  -1.9663616180e+01, /* 0xc19d4f16 */
328  -4.2325313568e+01, /* 0xc2294d1f */
329  -2.1371921539e+01, /* 0xc1aaf9b2 */
330 };
331 static const float qs2[6] = {
332  2.9533363342e+01, /* 0x41ec4454 */
333  2.5298155212e+02, /* 0x437cfb47 */
334  7.5750280762e+02, /* 0x443d602e */
335  7.3939318848e+02, /* 0x4438d92a */
336  1.5594900513e+02, /* 0x431bf2f2 */
337  -4.9594988823e+00, /* 0xc09eb437 */
338 };
339 
340 static float
341 qonef(float x)
342 {
343  const float *p,*q;
344  float s,r,z;
345  int32_t ix;
346 
347  p = q = 0;
348  GET_FLOAT_WORD(ix,x);
349  ix &= 0x7fffffff;
350  /* [inf, 8] (8 41000000) */
351  if(ix>=0x41000000) {p = qr8; q= qs8;}
352  /* [8, 4.5454] (4.5454 409173eb) */
353  else if(ix>=0x409173eb){p = qr5; q= qs5;}
354  /* [4.5454, 2.8570] (2.8570 4036d917) */
355  else if(ix>=0x4036d917){p = qr3; q= qs3;}
356  /* [2.8570, 2] (2 40000000) */
357  else if(ix>=0x40000000){p = qr2; q= qs2;}
358  z = one/(x*x);
359  r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
360  s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
361  return ((float).375 + r/s)/x;
362 }
363 #endif
364