POK
e_j0f.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_j0f.c -- float version of e_j0.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 "math_private.h"
36 
37 static float pzerof(float), qzerof(float);
38 
39 static const float
40 huge = 1e30,
41 one = 1.0,
42 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
43 tpi = 6.3661974669e-01, /* 0x3f22f983 */
44  /* R0/S0 on [0, 2.00] */
45 R02 = 1.5625000000e-02, /* 0x3c800000 */
46 R03 = -1.8997929874e-04, /* 0xb947352e */
47 R04 = 1.8295404516e-06, /* 0x35f58e88 */
48 R05 = -4.6183270541e-09, /* 0xb19eaf3c */
49 S01 = 1.5619102865e-02, /* 0x3c7fe744 */
50 S02 = 1.1692678527e-04, /* 0x38f53697 */
51 S03 = 5.1354652442e-07, /* 0x3509daa6 */
52 S04 = 1.1661400734e-09; /* 0x30a045e8 */
53 
54 static const float zero = 0.0;
55 
56 float
57 __ieee754_j0f(float x)
58 {
59  float z, s,c,ss,cc,r,u,v;
60  int32_t hx,ix;
61 
62  GET_FLOAT_WORD(hx,x);
63  ix = hx&0x7fffffff;
64  if(ix>=0x7f800000) return one/(x*x);
65  x = fabsf(x);
66  if(ix >= 0x40000000) { /* |x| >= 2.0 */
67  s = sinf(x);
68  c = cosf(x);
69  ss = s-c;
70  cc = s+c;
71  if(ix<0x7f000000) { /* make sure x+x not overflow */
72  z = -cosf(x+x);
73  if ((s*c)<zero) cc = z/ss;
74  else ss = z/cc;
75  }
76  /*
77  * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
78  * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
79  */
80 #ifdef DEAD_CODE
81  if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x);
82  else
83 #endif
84  {
85  u = pzerof(x); v = qzerof(x);
86  z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
87  }
88  return z;
89  }
90  if(ix<0x39000000) { /* |x| < 2**-13 */
91  if(huge+x>one) { /* raise inexact if x != 0 */
92  if(ix<0x32000000) return one; /* |x|<2**-27 */
93  else return one - (float)0.25*x*x;
94  }
95  }
96  z = x*x;
97  r = z*(R02+z*(R03+z*(R04+z*R05)));
98  s = one+z*(S01+z*(S02+z*(S03+z*S04)));
99  if(ix < 0x3F800000) { /* |x| < 1.00 */
100  return one + z*((float)-0.25+(r/s));
101  } else {
102  u = (float)0.5*x;
103  return((one+u)*(one-u)+z*(r/s));
104  }
105 }
106 
107 static const float
108 u00 = -7.3804296553e-02, /* 0xbd9726b5 */
109 u01 = 1.7666645348e-01, /* 0x3e34e80d */
110 u02 = -1.3818567619e-02, /* 0xbc626746 */
111 u03 = 3.4745343146e-04, /* 0x39b62a69 */
112 u04 = -3.8140706238e-06, /* 0xb67ff53c */
113 u05 = 1.9559013964e-08, /* 0x32a802ba */
114 u06 = -3.9820518410e-11, /* 0xae2f21eb */
115 v01 = 1.2730483897e-02, /* 0x3c509385 */
116 v02 = 7.6006865129e-05, /* 0x389f65e0 */
117 v03 = 2.5915085189e-07, /* 0x348b216c */
118 v04 = 4.4111031494e-10; /* 0x2ff280c2 */
119 
120 float
121 __ieee754_y0f(float x)
122 {
123  float z, s,c,ss,cc,u,v;
124  int32_t hx,ix;
125 
126  GET_FLOAT_WORD(hx,x);
127  ix = 0x7fffffff&hx;
128  /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
129  if(ix>=0x7f800000) return one/(x+x*x);
130  if(ix==0) return -one/zero;
131  if(hx<0) return zero/zero;
132  if(ix >= 0x40000000) { /* |x| >= 2.0 */
133  /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
134  * where x0 = x-pi/4
135  * Better formula:
136  * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
137  * = 1/sqrt(2) * (sin(x) + cos(x))
138  * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
139  * = 1/sqrt(2) * (sin(x) - cos(x))
140  * To avoid cancellation, use
141  * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
142  * to compute the worse one.
143  */
144  s = sinf(x);
145  c = cosf(x);
146  ss = s-c;
147  cc = s+c;
148  /*
149  * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
150  * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
151  */
152  if(ix<0x7f000000) { /* make sure x+x not overflow */
153  z = -cosf(x+x);
154  if ((s*c)<zero) cc = z/ss;
155  else ss = z/cc;
156  }
157 #ifdef DEAD_CODE
158  if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x);
159  else
160 #endif
161  {
162  u = pzerof(x); v = qzerof(x);
163  z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
164  }
165  return z;
166  }
167  if(ix<=0x32000000) { /* x < 2**-27 */
168  return(u00 + tpi*__ieee754_logf(x));
169  }
170  z = x*x;
171  u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
172  v = one+z*(v01+z*(v02+z*(v03+z*v04)));
173  return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
174 }
175 
176 /* The asymptotic expansions of pzero is
177  * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
178  * For x >= 2, We approximate pzero by
179  * pzero(x) = 1 + (R/S)
180  * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
181  * S = 1 + pS0*s^2 + ... + pS4*s^10
182  * and
183  * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
184  */
185 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
186  0.0000000000e+00, /* 0x00000000 */
187  -7.0312500000e-02, /* 0xbd900000 */
188  -8.0816707611e+00, /* 0xc1014e86 */
189  -2.5706311035e+02, /* 0xc3808814 */
190  -2.4852163086e+03, /* 0xc51b5376 */
191  -5.2530439453e+03, /* 0xc5a4285a */
192 };
193 static const float pS8[5] = {
194  1.1653436279e+02, /* 0x42e91198 */
195  3.8337448730e+03, /* 0x456f9beb */
196  4.0597855469e+04, /* 0x471e95db */
197  1.1675296875e+05, /* 0x47e4087c */
198  4.7627726562e+04, /* 0x473a0bba */
199 };
200 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
201  -1.1412546255e-11, /* 0xad48c58a */
202  -7.0312492549e-02, /* 0xbd8fffff */
203  -4.1596107483e+00, /* 0xc0851b88 */
204  -6.7674766541e+01, /* 0xc287597b */
205  -3.3123129272e+02, /* 0xc3a59d9b */
206  -3.4643338013e+02, /* 0xc3ad3779 */
207 };
208 static const float pS5[5] = {
209  6.0753936768e+01, /* 0x42730408 */
210  1.0512523193e+03, /* 0x44836813 */
211  5.9789707031e+03, /* 0x45bad7c4 */
212  9.6254453125e+03, /* 0x461665c8 */
213  2.4060581055e+03, /* 0x451660ee */
214 };
215 
216 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
217  -2.5470459075e-09, /* 0xb12f081b */
218  -7.0311963558e-02, /* 0xbd8fffb8 */
219  -2.4090321064e+00, /* 0xc01a2d95 */
220  -2.1965976715e+01, /* 0xc1afba52 */
221  -5.8079170227e+01, /* 0xc2685112 */
222  -3.1447946548e+01, /* 0xc1fb9565 */
223 };
224 static const float pS3[5] = {
225  3.5856033325e+01, /* 0x420f6c94 */
226  3.6151397705e+02, /* 0x43b4c1ca */
227  1.1936077881e+03, /* 0x44953373 */
228  1.1279968262e+03, /* 0x448cffe6 */
229  1.7358093262e+02, /* 0x432d94b8 */
230 };
231 
232 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
233  -8.8753431271e-08, /* 0xb3be98b7 */
234  -7.0303097367e-02, /* 0xbd8ffb12 */
235  -1.4507384300e+00, /* 0xbfb9b1cc */
236  -7.6356959343e+00, /* 0xc0f4579f */
237  -1.1193166733e+01, /* 0xc1331736 */
238  -3.2336456776e+00, /* 0xc04ef40d */
239 };
240 static const float pS2[5] = {
241  2.2220300674e+01, /* 0x41b1c32d */
242  1.3620678711e+02, /* 0x430834f0 */
243  2.7047027588e+02, /* 0x43873c32 */
244  1.5387539673e+02, /* 0x4319e01a */
245  1.4657617569e+01, /* 0x416a859a */
246 };
247 
248 static float
249 pzerof(float x)
250 {
251  const float *p,*q;
252  float z,r,s;
253  int32_t ix;
254 
255  p = q = 0;
256  GET_FLOAT_WORD(ix,x);
257  ix &= 0x7fffffff;
258  if(ix>=0x41000000) {p = pR8; q= pS8;}
259  else if(ix>=0x40f71c58){p = pR5; q= pS5;}
260  else if(ix>=0x4036db68){p = pR3; q= pS3;}
261  else if(ix>=0x40000000){p = pR2; q= pS2;}
262  z = one/(x*x);
263  r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
264  s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
265  return one+ r/s;
266 }
267 
268 
269 /* For x >= 8, the asymptotic expansions of qzero is
270  * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
271  * We approximate pzero by
272  * qzero(x) = s*(-1.25 + (R/S))
273  * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
274  * S = 1 + qS0*s^2 + ... + qS5*s^12
275  * and
276  * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
277  */
278 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
279  0.0000000000e+00, /* 0x00000000 */
280  7.3242187500e-02, /* 0x3d960000 */
281  1.1768206596e+01, /* 0x413c4a93 */
282  5.5767340088e+02, /* 0x440b6b19 */
283  8.8591972656e+03, /* 0x460a6cca */
284  3.7014625000e+04, /* 0x471096a0 */
285 };
286 static const float qS8[6] = {
287  1.6377603149e+02, /* 0x4323c6aa */
288  8.0983447266e+03, /* 0x45fd12c2 */
289  1.4253829688e+05, /* 0x480b3293 */
290  8.0330925000e+05, /* 0x49441ed4 */
291  8.4050156250e+05, /* 0x494d3359 */
292  -3.4389928125e+05, /* 0xc8a7eb69 */
293 };
294 
295 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
296  1.8408595828e-11, /* 0x2da1ec79 */
297  7.3242180049e-02, /* 0x3d95ffff */
298  5.8356351852e+00, /* 0x40babd86 */
299  1.3511157227e+02, /* 0x43071c90 */
300  1.0272437744e+03, /* 0x448067cd */
301  1.9899779053e+03, /* 0x44f8bf4b */
302 };
303 static const float qS5[6] = {
304  8.2776611328e+01, /* 0x42a58da0 */
305  2.0778142090e+03, /* 0x4501dd07 */
306  1.8847289062e+04, /* 0x46933e94 */
307  5.6751113281e+04, /* 0x475daf1d */
308  3.5976753906e+04, /* 0x470c88c1 */
309  -5.3543427734e+03, /* 0xc5a752be */
310 };
311 
312 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
313  4.3774099900e-09, /* 0x3196681b */
314  7.3241114616e-02, /* 0x3d95ff70 */
315  3.3442313671e+00, /* 0x405607e3 */
316  4.2621845245e+01, /* 0x422a7cc5 */
317  1.7080809021e+02, /* 0x432acedf */
318  1.6673394775e+02, /* 0x4326bbe4 */
319 };
320 static const float qS3[6] = {
321  4.8758872986e+01, /* 0x42430916 */
322  7.0968920898e+02, /* 0x44316c1c */
323  3.7041481934e+03, /* 0x4567825f */
324  6.4604252930e+03, /* 0x45c9e367 */
325  2.5163337402e+03, /* 0x451d4557 */
326  -1.4924745178e+02, /* 0xc3153f59 */
327 };
328 
329 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
330  1.5044444979e-07, /* 0x342189db */
331  7.3223426938e-02, /* 0x3d95f62a */
332  1.9981917143e+00, /* 0x3fffc4bf */
333  1.4495602608e+01, /* 0x4167edfd */
334  3.1666231155e+01, /* 0x41fd5471 */
335  1.6252708435e+01, /* 0x4182058c */
336 };
337 static const float qS2[6] = {
338  3.0365585327e+01, /* 0x41f2ecb8 */
339  2.6934811401e+02, /* 0x4386ac8f */
340  8.4478375244e+02, /* 0x44533229 */
341  8.8293585205e+02, /* 0x445cbbe5 */
342  2.1266638184e+02, /* 0x4354aa98 */
343  -5.3109550476e+00, /* 0xc0a9f358 */
344 };
345 
346 static float
347 qzerof(float x)
348 {
349  const float *p,*q;
350  float s,r,z;
351  int32_t ix;
352 
353  p = q = 0;
354  GET_FLOAT_WORD(ix,x);
355  ix &= 0x7fffffff;
356  if(ix>=0x41000000) {p = qR8; q= qS8;}
357  else if(ix>=0x40f71c58){p = qR5; q= qS5;}
358  else if(ix>=0x4036db68){p = qR3; q= qS3;}
359  else if(ix>=0x40000000){p = qR2; q= qS2;}
360  z = one/(x*x);
361  r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
362  s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
363  return (-(float).125 + r/s)/x;
364 }
365 #endif
366