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: head/lib/msun/src/e_j0f.c 283032 2015-05-17 16:27:06Z kargl $");
18
19 /*
20 * See e_j0.c for complete comments.
21 */
22
23 #include "math.h"
24 #include "math_private.h"
25
26 static __inline float pzerof(float), qzerof(float);
27
28 static const volatile float vone = 1, vzero = 0;
29
30 static const float
31 huge = 1e30,
32 one = 1.0,
33 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
34 tpi = 6.3661974669e-01, /* 0x3f22f983 */
35 /* R0/S0 on [0, 2.00] */
36 R02 = 1.5625000000e-02, /* 0x3c800000 */
37 R03 = -1.8997929874e-04, /* 0xb947352e */
38 R04 = 1.8295404516e-06, /* 0x35f58e88 */
39 R05 = -4.6183270541e-09, /* 0xb19eaf3c */
40 S01 = 1.5619102865e-02, /* 0x3c7fe744 */
41 S02 = 1.1692678527e-04, /* 0x38f53697 */
42 S03 = 5.1354652442e-07, /* 0x3509daa6 */
43 S04 = 1.1661400734e-09; /* 0x30a045e8 */
44
45 static const float zero = 0.0;
46
47 float
__ieee754_j0f(float x)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 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*((float)-0.25+(r/s));
89 } else {
90 u = (float)0.5*x;
91 return((one+u)*(one-u)+z*(r/s));
92 }
93 }
94
95 static const float
96 u00 = -7.3804296553e-02, /* 0xbd9726b5 */
97 u01 = 1.7666645348e-01, /* 0x3e34e80d */
98 u02 = -1.3818567619e-02, /* 0xbc626746 */
99 u03 = 3.4745343146e-04, /* 0x39b62a69 */
100 u04 = -3.8140706238e-06, /* 0xb67ff53c */
101 u05 = 1.9559013964e-08, /* 0x32a802ba */
102 u06 = -3.9820518410e-11, /* 0xae2f21eb */
103 v01 = 1.2730483897e-02, /* 0x3c509385 */
104 v02 = 7.6006865129e-05, /* 0x389f65e0 */
105 v03 = 2.5915085189e-07, /* 0x348b216c */
106 v04 = 4.4111031494e-10; /* 0x2ff280c2 */
107
108 float
__ieee754_y0f(float x)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 */
169 static 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 };
177 static 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 };
184 static 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 };
192 static 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
200 static 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 };
208 static 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
216 static 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 };
224 static 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
232 static __inline float
pzerof(float x)233 pzerof(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 */
260 static 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 };
268 static 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
277 static 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 };
285 static 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
294 static 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 };
302 static 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
311 static 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 };
319 static 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
328 static __inline float
qzerof(float x)329 qzerof(float x)
330 {
331 const float *p,*q;
332 float s,r,z;
333 int32_t ix;
334 GET_FLOAT_WORD(ix,x);
335 ix &= 0x7fffffff;
336 if(ix>=0x41000000) {p = qR8; q= qS8;}
337 else if(ix>=0x409173eb){p = qR5; q= qS5;}
338 else if(ix>=0x4036d917){p = qR3; q= qS3;}
339 else {p = qR2; q= qS2;} /* ix>=0x40000000 */
340 z = one/(x*x);
341 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
342 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
343 return (-(float).125 + r/s)/x;
344 }
345