e_j0f.c revision 343258 
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#include <sys/cdefs.h>
17__FBSDID("$FreeBSD: stable/10/lib/msun/src/e_j0f.c 343258 2019-01-21 14:55:10Z pfg $");
18
19/*
20 * See e_j0.c for complete comments.
21 */
22
23#include "math.h"
24#include "math_private.h"
25
26static __inline float pzerof(float), qzerof(float);
27
28static const volatile float vone = 1,  vzero = 0;
29
30static const float
31huge 	= 1e30,
32one	= 1.0,
33invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
34tpi      =  6.3661974669e-01, /* 0x3f22f983 */
35 		/* R0/S0 on [0, 2.00] */
36R02  =  1.5625000000e-02, /* 0x3c800000 */
37R03  = -1.8997929874e-04, /* 0xb947352e */
38R04  =  1.8295404516e-06, /* 0x35f58e88 */
39R05  = -4.6183270541e-09, /* 0xb19eaf3c */
40S01  =  1.5619102865e-02, /* 0x3c7fe744 */
41S02  =  1.1692678527e-04, /* 0x38f53697 */
42S03  =  5.1354652442e-07, /* 0x3509daa6 */
43S04  =  1.1661400734e-09; /* 0x30a045e8 */
44
45static const float zero = 0, qrtr = 0.25;
46
47float
48__ieee754_j0f(float x)
49{
50	float z, s,c,ss,cc,r,u,v;
51	int32_t hx,ix;
52
53	GET_FLOAT_WORD(hx,x);
54	ix = hx&0x7fffffff;
55	if(ix>=0x7f800000) return one/(x*x);
56	x = fabsf(x);
57	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
58		s = sinf(x);
59		c = cosf(x);
60		ss = s-c;
61		cc = s+c;
62		if(ix<0x7f000000) {  /* Make sure x+x does not overflow. */
63		    z = -cosf(x+x);
64		    if ((s*c)<zero) cc = z/ss;
65		    else 	    ss = z/cc;
66		}
67	/*
68	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
69	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
70	 */
71		if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(x); /* |x|>2**49 */
72		else {
73		    u = pzerof(x); v = qzerof(x);
74		    z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
75		}
76		return z;
77	}
78	if(ix<0x3b000000) {	/* |x| < 2**-9 */
79	    if(huge+x>one) {	/* raise inexact if x != 0 */
80	        if(ix<0x39800000) return one;	/* |x|<2**-12 */
81	        else 	      return one - x*x/4;
82	    }
83	}
84	z = x*x;
85	r =  z*(R02+z*(R03+z*(R04+z*R05)));
86	s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
87	if(ix < 0x3F800000) {	/* |x| < 1.00 */
88	    return one + z*((r/s)-qrtr);
89	} else {
90	    u = x/2;
91	    return((one+u)*(one-u)+z*(r/s));
92	}
93}
94
95static const float
96u00  = -7.3804296553e-02, /* 0xbd9726b5 */
97u01  =  1.7666645348e-01, /* 0x3e34e80d */
98u02  = -1.3818567619e-02, /* 0xbc626746 */
99u03  =  3.4745343146e-04, /* 0x39b62a69 */
100u04  = -3.8140706238e-06, /* 0xb67ff53c */
101u05  =  1.9559013964e-08, /* 0x32a802ba */
102u06  = -3.9820518410e-11, /* 0xae2f21eb */
103v01  =  1.2730483897e-02, /* 0x3c509385 */
104v02  =  7.6006865129e-05, /* 0x389f65e0 */
105v03  =  2.5915085189e-07, /* 0x348b216c */
106v04  =  4.4111031494e-10; /* 0x2ff280c2 */
107
108float
109__ieee754_y0f(float x)
110{
111	float z, s,c,ss,cc,u,v;
112	int32_t hx,ix;
113
114	GET_FLOAT_WORD(hx,x);
115        ix = 0x7fffffff&hx;
116	if(ix>=0x7f800000) return  vone/(x+x*x);
117	if(ix==0) return -one/vzero;
118	if(hx<0) return vzero/vzero;
119        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
120        /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
121         * where x0 = x-pi/4
122         *      Better formula:
123         *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
124         *                      =  1/sqrt(2) * (sin(x) + cos(x))
125         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
126         *                      =  1/sqrt(2) * (sin(x) - cos(x))
127         * To avoid cancellation, use
128         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
129         * to compute the worse one.
130         */
131                s = sinf(x);
132                c = cosf(x);
133                ss = s-c;
134                cc = s+c;
135	/*
136	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
137	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
138	 */
139                if(ix<0x7f000000) {  /* make sure x+x not overflow */
140                    z = -cosf(x+x);
141                    if ((s*c)<zero) cc = z/ss;
142                    else            ss = z/cc;
143                }
144                if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */
145                else {
146                    u = pzerof(x); v = qzerof(x);
147                    z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
148                }
149                return z;
150	}
151	if(ix<=0x39000000) {	/* x < 2**-13 */
152	    return(u00 + tpi*__ieee754_logf(x));
153	}
154	z = x*x;
155	u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
156	v = one+z*(v01+z*(v02+z*(v03+z*v04)));
157	return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
158}
159
160/* The asymptotic expansions of pzero is
161 *	1 - 9/128 s^2 + 11025/98304 s^4 - ...,	where s = 1/x.
162 * For x >= 2, We approximate pzero by
163 * 	pzero(x) = 1 + (R/S)
164 * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
165 * 	  S = 1 + pS0*s^2 + ... + pS4*s^10
166 * and
167 *	| pzero(x)-1-R/S | <= 2  ** ( -60.26)
168 */
169static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
170  0.0000000000e+00, /* 0x00000000 */
171 -7.0312500000e-02, /* 0xbd900000 */
172 -8.0816707611e+00, /* 0xc1014e86 */
173 -2.5706311035e+02, /* 0xc3808814 */
174 -2.4852163086e+03, /* 0xc51b5376 */
175 -5.2530439453e+03, /* 0xc5a4285a */
176};
177static const float pS8[5] = {
178  1.1653436279e+02, /* 0x42e91198 */
179  3.8337448730e+03, /* 0x456f9beb */
180  4.0597855469e+04, /* 0x471e95db */
181  1.1675296875e+05, /* 0x47e4087c */
182  4.7627726562e+04, /* 0x473a0bba */
183};
184static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
185 -1.1412546255e-11, /* 0xad48c58a */
186 -7.0312492549e-02, /* 0xbd8fffff */
187 -4.1596107483e+00, /* 0xc0851b88 */
188 -6.7674766541e+01, /* 0xc287597b */
189 -3.3123129272e+02, /* 0xc3a59d9b */
190 -3.4643338013e+02, /* 0xc3ad3779 */
191};
192static const float pS5[5] = {
193  6.0753936768e+01, /* 0x42730408 */
194  1.0512523193e+03, /* 0x44836813 */
195  5.9789707031e+03, /* 0x45bad7c4 */
196  9.6254453125e+03, /* 0x461665c8 */
197  2.4060581055e+03, /* 0x451660ee */
198};
199
200static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
201 -2.5470459075e-09, /* 0xb12f081b */
202 -7.0311963558e-02, /* 0xbd8fffb8 */
203 -2.4090321064e+00, /* 0xc01a2d95 */
204 -2.1965976715e+01, /* 0xc1afba52 */
205 -5.8079170227e+01, /* 0xc2685112 */
206 -3.1447946548e+01, /* 0xc1fb9565 */
207};
208static const float pS3[5] = {
209  3.5856033325e+01, /* 0x420f6c94 */
210  3.6151397705e+02, /* 0x43b4c1ca */
211  1.1936077881e+03, /* 0x44953373 */
212  1.1279968262e+03, /* 0x448cffe6 */
213  1.7358093262e+02, /* 0x432d94b8 */
214};
215
216static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
217 -8.8753431271e-08, /* 0xb3be98b7 */
218 -7.0303097367e-02, /* 0xbd8ffb12 */
219 -1.4507384300e+00, /* 0xbfb9b1cc */
220 -7.6356959343e+00, /* 0xc0f4579f */
221 -1.1193166733e+01, /* 0xc1331736 */
222 -3.2336456776e+00, /* 0xc04ef40d */
223};
224static const float pS2[5] = {
225  2.2220300674e+01, /* 0x41b1c32d */
226  1.3620678711e+02, /* 0x430834f0 */
227  2.7047027588e+02, /* 0x43873c32 */
228  1.5387539673e+02, /* 0x4319e01a */
229  1.4657617569e+01, /* 0x416a859a */
230};
231
232static __inline float
233pzerof(float x)
234{
235	const float *p,*q;
236	float z,r,s;
237	int32_t ix;
238	GET_FLOAT_WORD(ix,x);
239	ix &= 0x7fffffff;
240	if(ix>=0x41000000)     {p = pR8; q= pS8;}
241	else if(ix>=0x409173eb){p = pR5; q= pS5;}
242	else if(ix>=0x4036d917){p = pR3; q= pS3;}
243	else                   {p = pR2; q= pS2;}	/* ix>=0x40000000 */
244	z = one/(x*x);
245	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
246	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
247	return one+ r/s;
248}
249
250
251/* For x >= 8, the asymptotic expansions of qzero is
252 *	-1/8 s + 75/1024 s^3 - ..., where s = 1/x.
253 * We approximate pzero by
254 * 	qzero(x) = s*(-1.25 + (R/S))
255 * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
256 * 	  S = 1 + qS0*s^2 + ... + qS5*s^12
257 * and
258 *	| qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
259 */
260static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
261  0.0000000000e+00, /* 0x00000000 */
262  7.3242187500e-02, /* 0x3d960000 */
263  1.1768206596e+01, /* 0x413c4a93 */
264  5.5767340088e+02, /* 0x440b6b19 */
265  8.8591972656e+03, /* 0x460a6cca */
266  3.7014625000e+04, /* 0x471096a0 */
267};
268static const float qS8[6] = {
269  1.6377603149e+02, /* 0x4323c6aa */
270  8.0983447266e+03, /* 0x45fd12c2 */
271  1.4253829688e+05, /* 0x480b3293 */
272  8.0330925000e+05, /* 0x49441ed4 */
273  8.4050156250e+05, /* 0x494d3359 */
274 -3.4389928125e+05, /* 0xc8a7eb69 */
275};
276
277static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
278  1.8408595828e-11, /* 0x2da1ec79 */
279  7.3242180049e-02, /* 0x3d95ffff */
280  5.8356351852e+00, /* 0x40babd86 */
281  1.3511157227e+02, /* 0x43071c90 */
282  1.0272437744e+03, /* 0x448067cd */
283  1.9899779053e+03, /* 0x44f8bf4b */
284};
285static const float qS5[6] = {
286  8.2776611328e+01, /* 0x42a58da0 */
287  2.0778142090e+03, /* 0x4501dd07 */
288  1.8847289062e+04, /* 0x46933e94 */
289  5.6751113281e+04, /* 0x475daf1d */
290  3.5976753906e+04, /* 0x470c88c1 */
291 -5.3543427734e+03, /* 0xc5a752be */
292};
293
294static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
295  4.3774099900e-09, /* 0x3196681b */
296  7.3241114616e-02, /* 0x3d95ff70 */
297  3.3442313671e+00, /* 0x405607e3 */
298  4.2621845245e+01, /* 0x422a7cc5 */
299  1.7080809021e+02, /* 0x432acedf */
300  1.6673394775e+02, /* 0x4326bbe4 */
301};
302static const float qS3[6] = {
303  4.8758872986e+01, /* 0x42430916 */
304  7.0968920898e+02, /* 0x44316c1c */
305  3.7041481934e+03, /* 0x4567825f */
306  6.4604252930e+03, /* 0x45c9e367 */
307  2.5163337402e+03, /* 0x451d4557 */
308 -1.4924745178e+02, /* 0xc3153f59 */
309};
310
311static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
312  1.5044444979e-07, /* 0x342189db */
313  7.3223426938e-02, /* 0x3d95f62a */
314  1.9981917143e+00, /* 0x3fffc4bf */
315  1.4495602608e+01, /* 0x4167edfd */
316  3.1666231155e+01, /* 0x41fd5471 */
317  1.6252708435e+01, /* 0x4182058c */
318};
319static const float qS2[6] = {
320  3.0365585327e+01, /* 0x41f2ecb8 */
321  2.6934811401e+02, /* 0x4386ac8f */
322  8.4478375244e+02, /* 0x44533229 */
323  8.8293585205e+02, /* 0x445cbbe5 */
324  2.1266638184e+02, /* 0x4354aa98 */
325 -5.3109550476e+00, /* 0xc0a9f358 */
326};
327
328static __inline float
329qzerof(float x)
330{
331	static const float eighth = 0.125;
332	const float *p,*q;
333	float s,r,z;
334	int32_t ix;
335	GET_FLOAT_WORD(ix,x);
336	ix &= 0x7fffffff;
337	if(ix>=0x41000000)     {p = qR8; q= qS8;}
338	else if(ix>=0x409173eb){p = qR5; q= qS5;}
339	else if(ix>=0x4036d917){p = qR3; q= qS3;}
340	else                   {p = qR2; q= qS2;}	/* ix>=0x40000000 */
341	z = one/(x*x);
342	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
343	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
344	return (r/s-eighth)/x;
345}
346