e_j0f.c revision 50476 
1/* e_j0f.c -- float version of e_j0.c.
2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
3 */
4
5/*
6 * ====================================================
7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
8 *
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
12 * is preserved.
13 * ====================================================
14 */
15
16#ifndef lint
17static char rcsid[] = "$FreeBSD: head/lib/msun/src/e_j0f.c 50476 1999-08-28 00:22:10Z peter $";
18#endif
19
20#include "math.h"
21#include "math_private.h"
22
23#ifdef __STDC__
24static float pzerof(float), qzerof(float);
25#else
26static float pzerof(), qzerof();
27#endif
28
29#ifdef __STDC__
30static const float
31#else
32static float
33#endif
34huge 	= 1e30,
35one	= 1.0,
36invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
37tpi      =  6.3661974669e-01, /* 0x3f22f983 */
38 		/* R0/S0 on [0, 2.00] */
39R02  =  1.5625000000e-02, /* 0x3c800000 */
40R03  = -1.8997929874e-04, /* 0xb947352e */
41R04  =  1.8295404516e-06, /* 0x35f58e88 */
42R05  = -4.6183270541e-09, /* 0xb19eaf3c */
43S01  =  1.5619102865e-02, /* 0x3c7fe744 */
44S02  =  1.1692678527e-04, /* 0x38f53697 */
45S03  =  5.1354652442e-07, /* 0x3509daa6 */
46S04  =  1.1661400734e-09; /* 0x30a045e8 */
47
48#ifdef __STDC__
49static const float zero = 0.0;
50#else
51static float zero = 0.0;
52#endif
53
54#ifdef __STDC__
55	float __ieee754_j0f(float x)
56#else
57	float __ieee754_j0f(x)
58	float x;
59#endif
60{
61	float z, s,c,ss,cc,r,u,v;
62	int32_t hx,ix;
63
64	GET_FLOAT_WORD(hx,x);
65	ix = hx&0x7fffffff;
66	if(ix>=0x7f800000) return one/(x*x);
67	x = fabsf(x);
68	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
69		s = sinf(x);
70		c = cosf(x);
71		ss = s-c;
72		cc = s+c;
73		if(ix<0x7f000000) {  /* make sure x+x not overflow */
74		    z = -cosf(x+x);
75		    if ((s*c)<zero) cc = z/ss;
76		    else 	    ss = z/cc;
77		}
78	/*
79	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
80	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
81	 */
82		if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x);
83		else {
84		    u = pzerof(x); v = qzerof(x);
85		    z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
86		}
87		return z;
88	}
89	if(ix<0x39000000) {	/* |x| < 2**-13 */
90	    if(huge+x>one) {	/* raise inexact if x != 0 */
91	        if(ix<0x32000000) return one;	/* |x|<2**-27 */
92	        else 	      return one - (float)0.25*x*x;
93	    }
94	}
95	z = x*x;
96	r =  z*(R02+z*(R03+z*(R04+z*R05)));
97	s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
98	if(ix < 0x3F800000) {	/* |x| < 1.00 */
99	    return one + z*((float)-0.25+(r/s));
100	} else {
101	    u = (float)0.5*x;
102	    return((one+u)*(one-u)+z*(r/s));
103	}
104}
105
106#ifdef __STDC__
107static const float
108#else
109static float
110#endif
111u00  = -7.3804296553e-02, /* 0xbd9726b5 */
112u01  =  1.7666645348e-01, /* 0x3e34e80d */
113u02  = -1.3818567619e-02, /* 0xbc626746 */
114u03  =  3.4745343146e-04, /* 0x39b62a69 */
115u04  = -3.8140706238e-06, /* 0xb67ff53c */
116u05  =  1.9559013964e-08, /* 0x32a802ba */
117u06  = -3.9820518410e-11, /* 0xae2f21eb */
118v01  =  1.2730483897e-02, /* 0x3c509385 */
119v02  =  7.6006865129e-05, /* 0x389f65e0 */
120v03  =  2.5915085189e-07, /* 0x348b216c */
121v04  =  4.4111031494e-10; /* 0x2ff280c2 */
122
123#ifdef __STDC__
124	float __ieee754_y0f(float x)
125#else
126	float __ieee754_y0f(x)
127	float x;
128#endif
129{
130	float z, s,c,ss,cc,u,v;
131	int32_t hx,ix;
132
133	GET_FLOAT_WORD(hx,x);
134        ix = 0x7fffffff&hx;
135    /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
136	if(ix>=0x7f800000) return  one/(x+x*x);
137        if(ix==0) return -one/zero;
138        if(hx<0) return zero/zero;
139        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
140        /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
141         * where x0 = x-pi/4
142         *      Better formula:
143         *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/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) * (sin(x) - cos(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                s = sinf(x);
152                c = cosf(x);
153                ss = s-c;
154                cc = s+c;
155	/*
156	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
157	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
158	 */
159                if(ix<0x7f000000) {  /* make sure x+x not overflow */
160                    z = -cosf(x+x);
161                    if ((s*c)<zero) cc = z/ss;
162                    else            ss = z/cc;
163                }
164                if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x);
165                else {
166                    u = pzerof(x); v = qzerof(x);
167                    z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
168                }
169                return z;
170	}
171	if(ix<=0x32000000) {	/* x < 2**-27 */
172	    return(u00 + tpi*__ieee754_logf(x));
173	}
174	z = x*x;
175	u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
176	v = one+z*(v01+z*(v02+z*(v03+z*v04)));
177	return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
178}
179
180/* The asymptotic expansions of pzero is
181 *	1 - 9/128 s^2 + 11025/98304 s^4 - ...,	where s = 1/x.
182 * For x >= 2, We approximate pzero by
183 * 	pzero(x) = 1 + (R/S)
184 * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
185 * 	  S = 1 + pS0*s^2 + ... + pS4*s^10
186 * and
187 *	| pzero(x)-1-R/S | <= 2  ** ( -60.26)
188 */
189#ifdef __STDC__
190static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
191#else
192static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
193#endif
194  0.0000000000e+00, /* 0x00000000 */
195 -7.0312500000e-02, /* 0xbd900000 */
196 -8.0816707611e+00, /* 0xc1014e86 */
197 -2.5706311035e+02, /* 0xc3808814 */
198 -2.4852163086e+03, /* 0xc51b5376 */
199 -5.2530439453e+03, /* 0xc5a4285a */
200};
201#ifdef __STDC__
202static const float pS8[5] = {
203#else
204static float pS8[5] = {
205#endif
206  1.1653436279e+02, /* 0x42e91198 */
207  3.8337448730e+03, /* 0x456f9beb */
208  4.0597855469e+04, /* 0x471e95db */
209  1.1675296875e+05, /* 0x47e4087c */
210  4.7627726562e+04, /* 0x473a0bba */
211};
212#ifdef __STDC__
213static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
214#else
215static float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
216#endif
217 -1.1412546255e-11, /* 0xad48c58a */
218 -7.0312492549e-02, /* 0xbd8fffff */
219 -4.1596107483e+00, /* 0xc0851b88 */
220 -6.7674766541e+01, /* 0xc287597b */
221 -3.3123129272e+02, /* 0xc3a59d9b */
222 -3.4643338013e+02, /* 0xc3ad3779 */
223};
224#ifdef __STDC__
225static const float pS5[5] = {
226#else
227static float pS5[5] = {
228#endif
229  6.0753936768e+01, /* 0x42730408 */
230  1.0512523193e+03, /* 0x44836813 */
231  5.9789707031e+03, /* 0x45bad7c4 */
232  9.6254453125e+03, /* 0x461665c8 */
233  2.4060581055e+03, /* 0x451660ee */
234};
235
236#ifdef __STDC__
237static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
238#else
239static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
240#endif
241 -2.5470459075e-09, /* 0xb12f081b */
242 -7.0311963558e-02, /* 0xbd8fffb8 */
243 -2.4090321064e+00, /* 0xc01a2d95 */
244 -2.1965976715e+01, /* 0xc1afba52 */
245 -5.8079170227e+01, /* 0xc2685112 */
246 -3.1447946548e+01, /* 0xc1fb9565 */
247};
248#ifdef __STDC__
249static const float pS3[5] = {
250#else
251static float pS3[5] = {
252#endif
253  3.5856033325e+01, /* 0x420f6c94 */
254  3.6151397705e+02, /* 0x43b4c1ca */
255  1.1936077881e+03, /* 0x44953373 */
256  1.1279968262e+03, /* 0x448cffe6 */
257  1.7358093262e+02, /* 0x432d94b8 */
258};
259
260#ifdef __STDC__
261static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
262#else
263static float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
264#endif
265 -8.8753431271e-08, /* 0xb3be98b7 */
266 -7.0303097367e-02, /* 0xbd8ffb12 */
267 -1.4507384300e+00, /* 0xbfb9b1cc */
268 -7.6356959343e+00, /* 0xc0f4579f */
269 -1.1193166733e+01, /* 0xc1331736 */
270 -3.2336456776e+00, /* 0xc04ef40d */
271};
272#ifdef __STDC__
273static const float pS2[5] = {
274#else
275static float pS2[5] = {
276#endif
277  2.2220300674e+01, /* 0x41b1c32d */
278  1.3620678711e+02, /* 0x430834f0 */
279  2.7047027588e+02, /* 0x43873c32 */
280  1.5387539673e+02, /* 0x4319e01a */
281  1.4657617569e+01, /* 0x416a859a */
282};
283
284#ifdef __STDC__
285	static float pzerof(float x)
286#else
287	static float pzerof(x)
288	float x;
289#endif
290{
291#ifdef __STDC__
292	const float *p,*q;
293#else
294	float *p,*q;
295#endif
296	float z,r,s;
297	int32_t ix;
298	GET_FLOAT_WORD(ix,x);
299	ix &= 0x7fffffff;
300	if(ix>=0x41000000)     {p = pR8; q= pS8;}
301	else if(ix>=0x40f71c58){p = pR5; q= pS5;}
302	else if(ix>=0x4036db68){p = pR3; q= pS3;}
303	else if(ix>=0x40000000){p = pR2; q= pS2;}
304	z = one/(x*x);
305	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
306	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
307	return one+ r/s;
308}
309
310
311/* For x >= 8, the asymptotic expansions of qzero is
312 *	-1/8 s + 75/1024 s^3 - ..., where s = 1/x.
313 * We approximate pzero by
314 * 	qzero(x) = s*(-1.25 + (R/S))
315 * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
316 * 	  S = 1 + qS0*s^2 + ... + qS5*s^12
317 * and
318 *	| qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
319 */
320#ifdef __STDC__
321static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
322#else
323static float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
324#endif
325  0.0000000000e+00, /* 0x00000000 */
326  7.3242187500e-02, /* 0x3d960000 */
327  1.1768206596e+01, /* 0x413c4a93 */
328  5.5767340088e+02, /* 0x440b6b19 */
329  8.8591972656e+03, /* 0x460a6cca */
330  3.7014625000e+04, /* 0x471096a0 */
331};
332#ifdef __STDC__
333static const float qS8[6] = {
334#else
335static float qS8[6] = {
336#endif
337  1.6377603149e+02, /* 0x4323c6aa */
338  8.0983447266e+03, /* 0x45fd12c2 */
339  1.4253829688e+05, /* 0x480b3293 */
340  8.0330925000e+05, /* 0x49441ed4 */
341  8.4050156250e+05, /* 0x494d3359 */
342 -3.4389928125e+05, /* 0xc8a7eb69 */
343};
344
345#ifdef __STDC__
346static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
347#else
348static float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
349#endif
350  1.8408595828e-11, /* 0x2da1ec79 */
351  7.3242180049e-02, /* 0x3d95ffff */
352  5.8356351852e+00, /* 0x40babd86 */
353  1.3511157227e+02, /* 0x43071c90 */
354  1.0272437744e+03, /* 0x448067cd */
355  1.9899779053e+03, /* 0x44f8bf4b */
356};
357#ifdef __STDC__
358static const float qS5[6] = {
359#else
360static float qS5[6] = {
361#endif
362  8.2776611328e+01, /* 0x42a58da0 */
363  2.0778142090e+03, /* 0x4501dd07 */
364  1.8847289062e+04, /* 0x46933e94 */
365  5.6751113281e+04, /* 0x475daf1d */
366  3.5976753906e+04, /* 0x470c88c1 */
367 -5.3543427734e+03, /* 0xc5a752be */
368};
369
370#ifdef __STDC__
371static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
372#else
373static float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
374#endif
375  4.3774099900e-09, /* 0x3196681b */
376  7.3241114616e-02, /* 0x3d95ff70 */
377  3.3442313671e+00, /* 0x405607e3 */
378  4.2621845245e+01, /* 0x422a7cc5 */
379  1.7080809021e+02, /* 0x432acedf */
380  1.6673394775e+02, /* 0x4326bbe4 */
381};
382#ifdef __STDC__
383static const float qS3[6] = {
384#else
385static float qS3[6] = {
386#endif
387  4.8758872986e+01, /* 0x42430916 */
388  7.0968920898e+02, /* 0x44316c1c */
389  3.7041481934e+03, /* 0x4567825f */
390  6.4604252930e+03, /* 0x45c9e367 */
391  2.5163337402e+03, /* 0x451d4557 */
392 -1.4924745178e+02, /* 0xc3153f59 */
393};
394
395#ifdef __STDC__
396static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
397#else
398static float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
399#endif
400  1.5044444979e-07, /* 0x342189db */
401  7.3223426938e-02, /* 0x3d95f62a */
402  1.9981917143e+00, /* 0x3fffc4bf */
403  1.4495602608e+01, /* 0x4167edfd */
404  3.1666231155e+01, /* 0x41fd5471 */
405  1.6252708435e+01, /* 0x4182058c */
406};
407#ifdef __STDC__
408static const float qS2[6] = {
409#else
410static float qS2[6] = {
411#endif
412  3.0365585327e+01, /* 0x41f2ecb8 */
413  2.6934811401e+02, /* 0x4386ac8f */
414  8.4478375244e+02, /* 0x44533229 */
415  8.8293585205e+02, /* 0x445cbbe5 */
416  2.1266638184e+02, /* 0x4354aa98 */
417 -5.3109550476e+00, /* 0xc0a9f358 */
418};
419
420#ifdef __STDC__
421	static float qzerof(float x)
422#else
423	static float qzerof(x)
424	float x;
425#endif
426{
427#ifdef __STDC__
428	const float *p,*q;
429#else
430	float *p,*q;
431#endif
432	float s,r,z;
433	int32_t ix;
434	GET_FLOAT_WORD(ix,x);
435	ix &= 0x7fffffff;
436	if(ix>=0x41000000)     {p = qR8; q= qS8;}
437	else if(ix>=0x40f71c58){p = qR5; q= qS5;}
438	else if(ix>=0x4036db68){p = qR3; q= qS3;}
439	else if(ix>=0x40000000){p = qR2; q= qS2;}
440	z = one/(x*x);
441	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
442	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
443	return (-(float).125 + r/s)/x;
444}
445