1 /* e_j0f.c -- float version of e_j0.c.
2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
6 * ====================================================
7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
13 * ====================================================
17 #include <math_private.h>
19 static float pzerof(float), qzerof(float);
24 invsqrtpi
= 5.6418961287e-01, /* 0x3f106ebb */
25 tpi
= 6.3661974669e-01, /* 0x3f22f983 */
26 /* R0/S0 on [0, 2.00] */
27 R02
= 1.5625000000e-02, /* 0x3c800000 */
28 R03
= -1.8997929874e-04, /* 0xb947352e */
29 R04
= 1.8295404516e-06, /* 0x35f58e88 */
30 R05
= -4.6183270541e-09, /* 0xb19eaf3c */
31 S01
= 1.5619102865e-02, /* 0x3c7fe744 */
32 S02
= 1.1692678527e-04, /* 0x38f53697 */
33 S03
= 5.1354652442e-07, /* 0x3509daa6 */
34 S04
= 1.1661400734e-09; /* 0x30a045e8 */
36 static const float zero
= 0.0;
39 __ieee754_j0f(float x
)
41 float z
, s
,c
,ss
,cc
,r
,u
,v
;
46 if(ix
>=0x7f800000) return one
/(x
*x
);
48 if(ix
>= 0x40000000) { /* |x| >= 2.0 */
49 __sincosf (x
, &s
, &c
);
52 if(ix
<0x7f000000) { /* make sure x+x not overflow */
54 if ((s
*c
)<zero
) cc
= z
/ss
;
58 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
59 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
61 if(ix
>0x48000000) z
= (invsqrtpi
*cc
)/__ieee754_sqrtf(x
);
63 u
= pzerof(x
); v
= qzerof(x
);
64 z
= invsqrtpi
*(u
*cc
-v
*ss
)/__ieee754_sqrtf(x
);
68 if(ix
<0x39000000) { /* |x| < 2**-13 */
69 math_force_eval(huge
+x
); /* raise inexact if x != 0 */
70 if(ix
<0x32000000) return one
; /* |x|<2**-27 */
71 else return one
- (float)0.25*x
*x
;
74 r
= z
*(R02
+z
*(R03
+z
*(R04
+z
*R05
)));
75 s
= one
+z
*(S01
+z
*(S02
+z
*(S03
+z
*S04
)));
76 if(ix
< 0x3F800000) { /* |x| < 1.00 */
77 return one
+ z
*((float)-0.25+(r
/s
));
80 return((one
+u
)*(one
-u
)+z
*(r
/s
));
83 strong_alias (__ieee754_j0f
, __j0f_finite
)
86 u00
= -7.3804296553e-02, /* 0xbd9726b5 */
87 u01
= 1.7666645348e-01, /* 0x3e34e80d */
88 u02
= -1.3818567619e-02, /* 0xbc626746 */
89 u03
= 3.4745343146e-04, /* 0x39b62a69 */
90 u04
= -3.8140706238e-06, /* 0xb67ff53c */
91 u05
= 1.9559013964e-08, /* 0x32a802ba */
92 u06
= -3.9820518410e-11, /* 0xae2f21eb */
93 v01
= 1.2730483897e-02, /* 0x3c509385 */
94 v02
= 7.6006865129e-05, /* 0x389f65e0 */
95 v03
= 2.5915085189e-07, /* 0x348b216c */
96 v04
= 4.4111031494e-10; /* 0x2ff280c2 */
99 __ieee754_y0f(float x
)
101 float z
, s
,c
,ss
,cc
,u
,v
;
104 GET_FLOAT_WORD(hx
,x
);
106 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */
107 if(ix
>=0x7f800000) return one
/(x
+x
*x
);
108 if(ix
==0) return -1/zero
; /* -inf and divide by zero exception. */
109 if(hx
<0) return zero
/(zero
*x
);
110 if(ix
>= 0x40000000) { /* |x| >= 2.0 */
111 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
114 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
115 * = 1/sqrt(2) * (sin(x) + cos(x))
116 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
117 * = 1/sqrt(2) * (sin(x) - cos(x))
118 * To avoid cancellation, use
119 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
120 * to compute the worse one.
122 __sincosf (x
, &s
, &c
);
126 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
127 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
129 if(ix
<0x7f000000) { /* make sure x+x not overflow */
131 if ((s
*c
)<zero
) cc
= z
/ss
;
134 if(ix
>0x48000000) z
= (invsqrtpi
*ss
)/__ieee754_sqrtf(x
);
136 u
= pzerof(x
); v
= qzerof(x
);
137 z
= invsqrtpi
*(u
*ss
+v
*cc
)/__ieee754_sqrtf(x
);
141 if(ix
<=0x39800000) { /* x < 2**-13 */
142 return(u00
+ tpi
*__ieee754_logf(x
));
145 u
= u00
+z
*(u01
+z
*(u02
+z
*(u03
+z
*(u04
+z
*(u05
+z
*u06
)))));
146 v
= one
+z
*(v01
+z
*(v02
+z
*(v03
+z
*v04
)));
147 return(u
/v
+ tpi
*(__ieee754_j0f(x
)*__ieee754_logf(x
)));
149 strong_alias (__ieee754_y0f
, __y0f_finite
)
151 /* The asymptotic expansions of pzero is
152 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
153 * For x >= 2, We approximate pzero by
154 * pzero(x) = 1 + (R/S)
155 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
156 * S = 1 + pS0*s^2 + ... + pS4*s^10
158 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
160 static const float pR8
[6] = { /* for x in [inf, 8]=1/[0,0.125] */
161 0.0000000000e+00, /* 0x00000000 */
162 -7.0312500000e-02, /* 0xbd900000 */
163 -8.0816707611e+00, /* 0xc1014e86 */
164 -2.5706311035e+02, /* 0xc3808814 */
165 -2.4852163086e+03, /* 0xc51b5376 */
166 -5.2530439453e+03, /* 0xc5a4285a */
168 static const float pS8
[5] = {
169 1.1653436279e+02, /* 0x42e91198 */
170 3.8337448730e+03, /* 0x456f9beb */
171 4.0597855469e+04, /* 0x471e95db */
172 1.1675296875e+05, /* 0x47e4087c */
173 4.7627726562e+04, /* 0x473a0bba */
175 static const float pR5
[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
176 -1.1412546255e-11, /* 0xad48c58a */
177 -7.0312492549e-02, /* 0xbd8fffff */
178 -4.1596107483e+00, /* 0xc0851b88 */
179 -6.7674766541e+01, /* 0xc287597b */
180 -3.3123129272e+02, /* 0xc3a59d9b */
181 -3.4643338013e+02, /* 0xc3ad3779 */
183 static const float pS5
[5] = {
184 6.0753936768e+01, /* 0x42730408 */
185 1.0512523193e+03, /* 0x44836813 */
186 5.9789707031e+03, /* 0x45bad7c4 */
187 9.6254453125e+03, /* 0x461665c8 */
188 2.4060581055e+03, /* 0x451660ee */
191 static const float pR3
[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
192 -2.5470459075e-09, /* 0xb12f081b */
193 -7.0311963558e-02, /* 0xbd8fffb8 */
194 -2.4090321064e+00, /* 0xc01a2d95 */
195 -2.1965976715e+01, /* 0xc1afba52 */
196 -5.8079170227e+01, /* 0xc2685112 */
197 -3.1447946548e+01, /* 0xc1fb9565 */
199 static const float pS3
[5] = {
200 3.5856033325e+01, /* 0x420f6c94 */
201 3.6151397705e+02, /* 0x43b4c1ca */
202 1.1936077881e+03, /* 0x44953373 */
203 1.1279968262e+03, /* 0x448cffe6 */
204 1.7358093262e+02, /* 0x432d94b8 */
207 static const float pR2
[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
208 -8.8753431271e-08, /* 0xb3be98b7 */
209 -7.0303097367e-02, /* 0xbd8ffb12 */
210 -1.4507384300e+00, /* 0xbfb9b1cc */
211 -7.6356959343e+00, /* 0xc0f4579f */
212 -1.1193166733e+01, /* 0xc1331736 */
213 -3.2336456776e+00, /* 0xc04ef40d */
215 static const float pS2
[5] = {
216 2.2220300674e+01, /* 0x41b1c32d */
217 1.3620678711e+02, /* 0x430834f0 */
218 2.7047027588e+02, /* 0x43873c32 */
219 1.5387539673e+02, /* 0x4319e01a */
220 1.4657617569e+01, /* 0x416a859a */
229 GET_FLOAT_WORD(ix
,x
);
231 /* ix >= 0x40000000 for all calls to this function. */
232 if(ix
>=0x41000000) {p
= pR8
; q
= pS8
;}
233 else if(ix
>=0x40f71c58){p
= pR5
; q
= pS5
;}
234 else if(ix
>=0x4036db68){p
= pR3
; q
= pS3
;}
235 else {p
= pR2
; q
= pS2
;}
237 r
= p
[0]+z
*(p
[1]+z
*(p
[2]+z
*(p
[3]+z
*(p
[4]+z
*p
[5]))));
238 s
= one
+z
*(q
[0]+z
*(q
[1]+z
*(q
[2]+z
*(q
[3]+z
*q
[4]))));
243 /* For x >= 8, the asymptotic expansions of qzero is
244 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
245 * We approximate pzero by
246 * qzero(x) = s*(-1.25 + (R/S))
247 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
248 * S = 1 + qS0*s^2 + ... + qS5*s^12
250 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
252 static const float qR8
[6] = { /* for x in [inf, 8]=1/[0,0.125] */
253 0.0000000000e+00, /* 0x00000000 */
254 7.3242187500e-02, /* 0x3d960000 */
255 1.1768206596e+01, /* 0x413c4a93 */
256 5.5767340088e+02, /* 0x440b6b19 */
257 8.8591972656e+03, /* 0x460a6cca */
258 3.7014625000e+04, /* 0x471096a0 */
260 static const float qS8
[6] = {
261 1.6377603149e+02, /* 0x4323c6aa */
262 8.0983447266e+03, /* 0x45fd12c2 */
263 1.4253829688e+05, /* 0x480b3293 */
264 8.0330925000e+05, /* 0x49441ed4 */
265 8.4050156250e+05, /* 0x494d3359 */
266 -3.4389928125e+05, /* 0xc8a7eb69 */
269 static const float qR5
[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
270 1.8408595828e-11, /* 0x2da1ec79 */
271 7.3242180049e-02, /* 0x3d95ffff */
272 5.8356351852e+00, /* 0x40babd86 */
273 1.3511157227e+02, /* 0x43071c90 */
274 1.0272437744e+03, /* 0x448067cd */
275 1.9899779053e+03, /* 0x44f8bf4b */
277 static const float qS5
[6] = {
278 8.2776611328e+01, /* 0x42a58da0 */
279 2.0778142090e+03, /* 0x4501dd07 */
280 1.8847289062e+04, /* 0x46933e94 */
281 5.6751113281e+04, /* 0x475daf1d */
282 3.5976753906e+04, /* 0x470c88c1 */
283 -5.3543427734e+03, /* 0xc5a752be */
286 static const float qR3
[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
287 4.3774099900e-09, /* 0x3196681b */
288 7.3241114616e-02, /* 0x3d95ff70 */
289 3.3442313671e+00, /* 0x405607e3 */
290 4.2621845245e+01, /* 0x422a7cc5 */
291 1.7080809021e+02, /* 0x432acedf */
292 1.6673394775e+02, /* 0x4326bbe4 */
294 static const float qS3
[6] = {
295 4.8758872986e+01, /* 0x42430916 */
296 7.0968920898e+02, /* 0x44316c1c */
297 3.7041481934e+03, /* 0x4567825f */
298 6.4604252930e+03, /* 0x45c9e367 */
299 2.5163337402e+03, /* 0x451d4557 */
300 -1.4924745178e+02, /* 0xc3153f59 */
303 static const float qR2
[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
304 1.5044444979e-07, /* 0x342189db */
305 7.3223426938e-02, /* 0x3d95f62a */
306 1.9981917143e+00, /* 0x3fffc4bf */
307 1.4495602608e+01, /* 0x4167edfd */
308 3.1666231155e+01, /* 0x41fd5471 */
309 1.6252708435e+01, /* 0x4182058c */
311 static const float qS2
[6] = {
312 3.0365585327e+01, /* 0x41f2ecb8 */
313 2.6934811401e+02, /* 0x4386ac8f */
314 8.4478375244e+02, /* 0x44533229 */
315 8.8293585205e+02, /* 0x445cbbe5 */
316 2.1266638184e+02, /* 0x4354aa98 */
317 -5.3109550476e+00, /* 0xc0a9f358 */
326 GET_FLOAT_WORD(ix
,x
);
328 /* ix >= 0x40000000 for all calls to this function. */
329 if(ix
>=0x41000000) {p
= qR8
; q
= qS8
;}
330 else if(ix
>=0x40f71c58){p
= qR5
; q
= qS5
;}
331 else if(ix
>=0x4036db68){p
= qR3
; q
= qS3
;}
332 else {p
= qR2
; q
= qS2
;}
334 r
= p
[0]+z
*(p
[1]+z
*(p
[2]+z
*(p
[3]+z
*(p
[4]+z
*p
[5]))));
335 s
= one
+z
*(q
[0]+z
*(q
[1]+z
*(q
[2]+z
*(q
[3]+z
*(q
[4]+z
*q
[5])))));
336 return (-(float).125 + r
/s
)/x
;