18
19#include "math.h"
20#include "math_private.h"
21
22static float pzerof(float), qzerof(float);
23
24static const float
25huge = 1e30,
26one = 1.0,
27invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
28tpi = 6.3661974669e-01, /* 0x3f22f983 */
29 /* R0/S0 on [0, 2.00] */
30R02 = 1.5625000000e-02, /* 0x3c800000 */
31R03 = -1.8997929874e-04, /* 0xb947352e */
32R04 = 1.8295404516e-06, /* 0x35f58e88 */
33R05 = -4.6183270541e-09, /* 0xb19eaf3c */
34S01 = 1.5619102865e-02, /* 0x3c7fe744 */
35S02 = 1.1692678527e-04, /* 0x38f53697 */
36S03 = 5.1354652442e-07, /* 0x3509daa6 */
37S04 = 1.1661400734e-09; /* 0x30a045e8 */
38
39static const float zero = 0.0;
40
41float
42__ieee754_j0f(float x)
43{
44 float z, s,c,ss,cc,r,u,v;
45 int32_t hx,ix;
46
47 GET_FLOAT_WORD(hx,x);
48 ix = hx&0x7fffffff;
49 if(ix>=0x7f800000) return one/(x*x);
50 x = fabsf(x);
51 if(ix >= 0x40000000) { /* |x| >= 2.0 */
52 s = sinf(x);
53 c = cosf(x);
54 ss = s-c;
55 cc = s+c;
56 if(ix<0x7f000000) { /* make sure x+x not overflow */
57 z = -cosf(x+x);
58 if ((s*c)<zero) cc = z/ss;
59 else ss = z/cc;
60 }
61 /*
62 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
63 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
64 */
65 if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(x); /* |x|>2**49 */
66 else {
67 u = pzerof(x); v = qzerof(x);
68 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
69 }
70 return z;
71 }
72 if(ix<0x3b000000) { /* |x| < 2**-9 */
73 if(huge+x>one) { /* raise inexact if x != 0 */
74 if(ix<0x39800000) return one; /* |x|<2**-12 */
75 else return one - x*x/4;
76 }
77 }
78 z = x*x;
79 r = z*(R02+z*(R03+z*(R04+z*R05)));
80 s = one+z*(S01+z*(S02+z*(S03+z*S04)));
81 if(ix < 0x3F800000) { /* |x| < 1.00 */
82 return one + z*((float)-0.25+(r/s));
83 } else {
84 u = (float)0.5*x;
85 return((one+u)*(one-u)+z*(r/s));
86 }
87}
88
89static const float
90u00 = -7.3804296553e-02, /* 0xbd9726b5 */
91u01 = 1.7666645348e-01, /* 0x3e34e80d */
92u02 = -1.3818567619e-02, /* 0xbc626746 */
93u03 = 3.4745343146e-04, /* 0x39b62a69 */
94u04 = -3.8140706238e-06, /* 0xb67ff53c */
95u05 = 1.9559013964e-08, /* 0x32a802ba */
96u06 = -3.9820518410e-11, /* 0xae2f21eb */
97v01 = 1.2730483897e-02, /* 0x3c509385 */
98v02 = 7.6006865129e-05, /* 0x389f65e0 */
99v03 = 2.5915085189e-07, /* 0x348b216c */
100v04 = 4.4111031494e-10; /* 0x2ff280c2 */
101
102float
103__ieee754_y0f(float x)
104{
105 float z, s,c,ss,cc,u,v;
106 int32_t hx,ix;
107
108 GET_FLOAT_WORD(hx,x);
109 ix = 0x7fffffff&hx;
110 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
111 if(ix>=0x7f800000) return one/(x+x*x);
112 if(ix==0) return -one/zero;
113 if(hx<0) return zero/zero;
114 if(ix >= 0x40000000) { /* |x| >= 2.0 */
115 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
116 * where x0 = x-pi/4
117 * Better formula:
118 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
119 * = 1/sqrt(2) * (sin(x) + cos(x))
120 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
121 * = 1/sqrt(2) * (sin(x) - cos(x))
122 * To avoid cancellation, use
123 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
124 * to compute the worse one.
125 */
126 s = sinf(x);
127 c = cosf(x);
128 ss = s-c;
129 cc = s+c;
130 /*
131 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
132 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
133 */
134 if(ix<0x7f000000) { /* make sure x+x not overflow */
135 z = -cosf(x+x);
136 if ((s*c)<zero) cc = z/ss;
137 else ss = z/cc;
138 }
139 if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */
140 else {
141 u = pzerof(x); v = qzerof(x);
142 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
143 }
144 return z;
145 }
146 if(ix<=0x39000000) { /* x < 2**-13 */
147 return(u00 + tpi*__ieee754_logf(x));
148 }
149 z = x*x;
150 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
151 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
152 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
153}
154
155/* The asymptotic expansions of pzero is
156 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
157 * For x >= 2, We approximate pzero by
158 * pzero(x) = 1 + (R/S)
159 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
160 * S = 1 + pS0*s^2 + ... + pS4*s^10
161 * and
162 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
163 */
164static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
165 0.0000000000e+00, /* 0x00000000 */
166 -7.0312500000e-02, /* 0xbd900000 */
167 -8.0816707611e+00, /* 0xc1014e86 */
168 -2.5706311035e+02, /* 0xc3808814 */
169 -2.4852163086e+03, /* 0xc51b5376 */
170 -5.2530439453e+03, /* 0xc5a4285a */
171};
172static const float pS8[5] = {
173 1.1653436279e+02, /* 0x42e91198 */
174 3.8337448730e+03, /* 0x456f9beb */
175 4.0597855469e+04, /* 0x471e95db */
176 1.1675296875e+05, /* 0x47e4087c */
177 4.7627726562e+04, /* 0x473a0bba */
178};
179static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
180 -1.1412546255e-11, /* 0xad48c58a */
181 -7.0312492549e-02, /* 0xbd8fffff */
182 -4.1596107483e+00, /* 0xc0851b88 */
183 -6.7674766541e+01, /* 0xc287597b */
184 -3.3123129272e+02, /* 0xc3a59d9b */
185 -3.4643338013e+02, /* 0xc3ad3779 */
186};
187static const float pS5[5] = {
188 6.0753936768e+01, /* 0x42730408 */
189 1.0512523193e+03, /* 0x44836813 */
190 5.9789707031e+03, /* 0x45bad7c4 */
191 9.6254453125e+03, /* 0x461665c8 */
192 2.4060581055e+03, /* 0x451660ee */
193};
194
195static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
196 -2.5470459075e-09, /* 0xb12f081b */
197 -7.0311963558e-02, /* 0xbd8fffb8 */
198 -2.4090321064e+00, /* 0xc01a2d95 */
199 -2.1965976715e+01, /* 0xc1afba52 */
200 -5.8079170227e+01, /* 0xc2685112 */
201 -3.1447946548e+01, /* 0xc1fb9565 */
202};
203static const float pS3[5] = {
204 3.5856033325e+01, /* 0x420f6c94 */
205 3.6151397705e+02, /* 0x43b4c1ca */
206 1.1936077881e+03, /* 0x44953373 */
207 1.1279968262e+03, /* 0x448cffe6 */
208 1.7358093262e+02, /* 0x432d94b8 */
209};
210
211static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
212 -8.8753431271e-08, /* 0xb3be98b7 */
213 -7.0303097367e-02, /* 0xbd8ffb12 */
214 -1.4507384300e+00, /* 0xbfb9b1cc */
215 -7.6356959343e+00, /* 0xc0f4579f */
216 -1.1193166733e+01, /* 0xc1331736 */
217 -3.2336456776e+00, /* 0xc04ef40d */
218};
219static const float pS2[5] = {
220 2.2220300674e+01, /* 0x41b1c32d */
221 1.3620678711e+02, /* 0x430834f0 */
222 2.7047027588e+02, /* 0x43873c32 */
223 1.5387539673e+02, /* 0x4319e01a */
224 1.4657617569e+01, /* 0x416a859a */
225};
226
| 18
19#include "math.h"
20#include "math_private.h"
21
22static float pzerof(float), qzerof(float);
23
24static const float
25huge = 1e30,
26one = 1.0,
27invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
28tpi = 6.3661974669e-01, /* 0x3f22f983 */
29 /* R0/S0 on [0, 2.00] */
30R02 = 1.5625000000e-02, /* 0x3c800000 */
31R03 = -1.8997929874e-04, /* 0xb947352e */
32R04 = 1.8295404516e-06, /* 0x35f58e88 */
33R05 = -4.6183270541e-09, /* 0xb19eaf3c */
34S01 = 1.5619102865e-02, /* 0x3c7fe744 */
35S02 = 1.1692678527e-04, /* 0x38f53697 */
36S03 = 5.1354652442e-07, /* 0x3509daa6 */
37S04 = 1.1661400734e-09; /* 0x30a045e8 */
38
39static const float zero = 0.0;
40
41float
42__ieee754_j0f(float x)
43{
44 float z, s,c,ss,cc,r,u,v;
45 int32_t hx,ix;
46
47 GET_FLOAT_WORD(hx,x);
48 ix = hx&0x7fffffff;
49 if(ix>=0x7f800000) return one/(x*x);
50 x = fabsf(x);
51 if(ix >= 0x40000000) { /* |x| >= 2.0 */
52 s = sinf(x);
53 c = cosf(x);
54 ss = s-c;
55 cc = s+c;
56 if(ix<0x7f000000) { /* make sure x+x not overflow */
57 z = -cosf(x+x);
58 if ((s*c)<zero) cc = z/ss;
59 else ss = z/cc;
60 }
61 /*
62 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
63 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
64 */
65 if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(x); /* |x|>2**49 */
66 else {
67 u = pzerof(x); v = qzerof(x);
68 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
69 }
70 return z;
71 }
72 if(ix<0x3b000000) { /* |x| < 2**-9 */
73 if(huge+x>one) { /* raise inexact if x != 0 */
74 if(ix<0x39800000) return one; /* |x|<2**-12 */
75 else return one - x*x/4;
76 }
77 }
78 z = x*x;
79 r = z*(R02+z*(R03+z*(R04+z*R05)));
80 s = one+z*(S01+z*(S02+z*(S03+z*S04)));
81 if(ix < 0x3F800000) { /* |x| < 1.00 */
82 return one + z*((float)-0.25+(r/s));
83 } else {
84 u = (float)0.5*x;
85 return((one+u)*(one-u)+z*(r/s));
86 }
87}
88
89static const float
90u00 = -7.3804296553e-02, /* 0xbd9726b5 */
91u01 = 1.7666645348e-01, /* 0x3e34e80d */
92u02 = -1.3818567619e-02, /* 0xbc626746 */
93u03 = 3.4745343146e-04, /* 0x39b62a69 */
94u04 = -3.8140706238e-06, /* 0xb67ff53c */
95u05 = 1.9559013964e-08, /* 0x32a802ba */
96u06 = -3.9820518410e-11, /* 0xae2f21eb */
97v01 = 1.2730483897e-02, /* 0x3c509385 */
98v02 = 7.6006865129e-05, /* 0x389f65e0 */
99v03 = 2.5915085189e-07, /* 0x348b216c */
100v04 = 4.4111031494e-10; /* 0x2ff280c2 */
101
102float
103__ieee754_y0f(float x)
104{
105 float z, s,c,ss,cc,u,v;
106 int32_t hx,ix;
107
108 GET_FLOAT_WORD(hx,x);
109 ix = 0x7fffffff&hx;
110 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
111 if(ix>=0x7f800000) return one/(x+x*x);
112 if(ix==0) return -one/zero;
113 if(hx<0) return zero/zero;
114 if(ix >= 0x40000000) { /* |x| >= 2.0 */
115 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
116 * where x0 = x-pi/4
117 * Better formula:
118 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
119 * = 1/sqrt(2) * (sin(x) + cos(x))
120 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
121 * = 1/sqrt(2) * (sin(x) - cos(x))
122 * To avoid cancellation, use
123 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
124 * to compute the worse one.
125 */
126 s = sinf(x);
127 c = cosf(x);
128 ss = s-c;
129 cc = s+c;
130 /*
131 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
132 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
133 */
134 if(ix<0x7f000000) { /* make sure x+x not overflow */
135 z = -cosf(x+x);
136 if ((s*c)<zero) cc = z/ss;
137 else ss = z/cc;
138 }
139 if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */
140 else {
141 u = pzerof(x); v = qzerof(x);
142 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
143 }
144 return z;
145 }
146 if(ix<=0x39000000) { /* x < 2**-13 */
147 return(u00 + tpi*__ieee754_logf(x));
148 }
149 z = x*x;
150 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
151 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
152 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
153}
154
155/* The asymptotic expansions of pzero is
156 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
157 * For x >= 2, We approximate pzero by
158 * pzero(x) = 1 + (R/S)
159 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
160 * S = 1 + pS0*s^2 + ... + pS4*s^10
161 * and
162 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
163 */
164static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
165 0.0000000000e+00, /* 0x00000000 */
166 -7.0312500000e-02, /* 0xbd900000 */
167 -8.0816707611e+00, /* 0xc1014e86 */
168 -2.5706311035e+02, /* 0xc3808814 */
169 -2.4852163086e+03, /* 0xc51b5376 */
170 -5.2530439453e+03, /* 0xc5a4285a */
171};
172static const float pS8[5] = {
173 1.1653436279e+02, /* 0x42e91198 */
174 3.8337448730e+03, /* 0x456f9beb */
175 4.0597855469e+04, /* 0x471e95db */
176 1.1675296875e+05, /* 0x47e4087c */
177 4.7627726562e+04, /* 0x473a0bba */
178};
179static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
180 -1.1412546255e-11, /* 0xad48c58a */
181 -7.0312492549e-02, /* 0xbd8fffff */
182 -4.1596107483e+00, /* 0xc0851b88 */
183 -6.7674766541e+01, /* 0xc287597b */
184 -3.3123129272e+02, /* 0xc3a59d9b */
185 -3.4643338013e+02, /* 0xc3ad3779 */
186};
187static const float pS5[5] = {
188 6.0753936768e+01, /* 0x42730408 */
189 1.0512523193e+03, /* 0x44836813 */
190 5.9789707031e+03, /* 0x45bad7c4 */
191 9.6254453125e+03, /* 0x461665c8 */
192 2.4060581055e+03, /* 0x451660ee */
193};
194
195static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
196 -2.5470459075e-09, /* 0xb12f081b */
197 -7.0311963558e-02, /* 0xbd8fffb8 */
198 -2.4090321064e+00, /* 0xc01a2d95 */
199 -2.1965976715e+01, /* 0xc1afba52 */
200 -5.8079170227e+01, /* 0xc2685112 */
201 -3.1447946548e+01, /* 0xc1fb9565 */
202};
203static const float pS3[5] = {
204 3.5856033325e+01, /* 0x420f6c94 */
205 3.6151397705e+02, /* 0x43b4c1ca */
206 1.1936077881e+03, /* 0x44953373 */
207 1.1279968262e+03, /* 0x448cffe6 */
208 1.7358093262e+02, /* 0x432d94b8 */
209};
210
211static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
212 -8.8753431271e-08, /* 0xb3be98b7 */
213 -7.0303097367e-02, /* 0xbd8ffb12 */
214 -1.4507384300e+00, /* 0xbfb9b1cc */
215 -7.6356959343e+00, /* 0xc0f4579f */
216 -1.1193166733e+01, /* 0xc1331736 */
217 -3.2336456776e+00, /* 0xc04ef40d */
218};
219static const float pS2[5] = {
220 2.2220300674e+01, /* 0x41b1c32d */
221 1.3620678711e+02, /* 0x430834f0 */
222 2.7047027588e+02, /* 0x43873c32 */
223 1.5387539673e+02, /* 0x4319e01a */
224 1.4657617569e+01, /* 0x416a859a */
225};
226
|
228{
229 const float *p,*q;
230 float z,r,s;
231 int32_t ix;
232 GET_FLOAT_WORD(ix,x);
233 ix &= 0x7fffffff;
234 if(ix>=0x41000000) {p = pR8; q= pS8;}
235 else if(ix>=0x409173eb){p = pR5; q= pS5;}
236 else if(ix>=0x4036d917){p = pR3; q= pS3;}
237 else {p = pR2; q= pS2;} /* ix>=0x40000000 */
238 z = one/(x*x);
239 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
240 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
241 return one+ r/s;
242}
243
244
245/* For x >= 8, the asymptotic expansions of qzero is
246 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
247 * We approximate pzero by
248 * qzero(x) = s*(-1.25 + (R/S))
249 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
250 * S = 1 + qS0*s^2 + ... + qS5*s^12
251 * and
252 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
253 */
254static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
255 0.0000000000e+00, /* 0x00000000 */
256 7.3242187500e-02, /* 0x3d960000 */
257 1.1768206596e+01, /* 0x413c4a93 */
258 5.5767340088e+02, /* 0x440b6b19 */
259 8.8591972656e+03, /* 0x460a6cca */
260 3.7014625000e+04, /* 0x471096a0 */
261};
262static const float qS8[6] = {
263 1.6377603149e+02, /* 0x4323c6aa */
264 8.0983447266e+03, /* 0x45fd12c2 */
265 1.4253829688e+05, /* 0x480b3293 */
266 8.0330925000e+05, /* 0x49441ed4 */
267 8.4050156250e+05, /* 0x494d3359 */
268 -3.4389928125e+05, /* 0xc8a7eb69 */
269};
270
271static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
272 1.8408595828e-11, /* 0x2da1ec79 */
273 7.3242180049e-02, /* 0x3d95ffff */
274 5.8356351852e+00, /* 0x40babd86 */
275 1.3511157227e+02, /* 0x43071c90 */
276 1.0272437744e+03, /* 0x448067cd */
277 1.9899779053e+03, /* 0x44f8bf4b */
278};
279static const float qS5[6] = {
280 8.2776611328e+01, /* 0x42a58da0 */
281 2.0778142090e+03, /* 0x4501dd07 */
282 1.8847289062e+04, /* 0x46933e94 */
283 5.6751113281e+04, /* 0x475daf1d */
284 3.5976753906e+04, /* 0x470c88c1 */
285 -5.3543427734e+03, /* 0xc5a752be */
286};
287
288static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
289 4.3774099900e-09, /* 0x3196681b */
290 7.3241114616e-02, /* 0x3d95ff70 */
291 3.3442313671e+00, /* 0x405607e3 */
292 4.2621845245e+01, /* 0x422a7cc5 */
293 1.7080809021e+02, /* 0x432acedf */
294 1.6673394775e+02, /* 0x4326bbe4 */
295};
296static const float qS3[6] = {
297 4.8758872986e+01, /* 0x42430916 */
298 7.0968920898e+02, /* 0x44316c1c */
299 3.7041481934e+03, /* 0x4567825f */
300 6.4604252930e+03, /* 0x45c9e367 */
301 2.5163337402e+03, /* 0x451d4557 */
302 -1.4924745178e+02, /* 0xc3153f59 */
303};
304
305static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
306 1.5044444979e-07, /* 0x342189db */
307 7.3223426938e-02, /* 0x3d95f62a */
308 1.9981917143e+00, /* 0x3fffc4bf */
309 1.4495602608e+01, /* 0x4167edfd */
310 3.1666231155e+01, /* 0x41fd5471 */
311 1.6252708435e+01, /* 0x4182058c */
312};
313static const float qS2[6] = {
314 3.0365585327e+01, /* 0x41f2ecb8 */
315 2.6934811401e+02, /* 0x4386ac8f */
316 8.4478375244e+02, /* 0x44533229 */
317 8.8293585205e+02, /* 0x445cbbe5 */
318 2.1266638184e+02, /* 0x4354aa98 */
319 -5.3109550476e+00, /* 0xc0a9f358 */
320};
321
| 229{
230 const float *p,*q;
231 float z,r,s;
232 int32_t ix;
233 GET_FLOAT_WORD(ix,x);
234 ix &= 0x7fffffff;
235 if(ix>=0x41000000) {p = pR8; q= pS8;}
236 else if(ix>=0x409173eb){p = pR5; q= pS5;}
237 else if(ix>=0x4036d917){p = pR3; q= pS3;}
238 else {p = pR2; q= pS2;} /* ix>=0x40000000 */
239 z = one/(x*x);
240 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
241 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
242 return one+ r/s;
243}
244
245
246/* For x >= 8, the asymptotic expansions of qzero is
247 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
248 * We approximate pzero by
249 * qzero(x) = s*(-1.25 + (R/S))
250 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
251 * S = 1 + qS0*s^2 + ... + qS5*s^12
252 * and
253 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
254 */
255static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
256 0.0000000000e+00, /* 0x00000000 */
257 7.3242187500e-02, /* 0x3d960000 */
258 1.1768206596e+01, /* 0x413c4a93 */
259 5.5767340088e+02, /* 0x440b6b19 */
260 8.8591972656e+03, /* 0x460a6cca */
261 3.7014625000e+04, /* 0x471096a0 */
262};
263static const float qS8[6] = {
264 1.6377603149e+02, /* 0x4323c6aa */
265 8.0983447266e+03, /* 0x45fd12c2 */
266 1.4253829688e+05, /* 0x480b3293 */
267 8.0330925000e+05, /* 0x49441ed4 */
268 8.4050156250e+05, /* 0x494d3359 */
269 -3.4389928125e+05, /* 0xc8a7eb69 */
270};
271
272static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
273 1.8408595828e-11, /* 0x2da1ec79 */
274 7.3242180049e-02, /* 0x3d95ffff */
275 5.8356351852e+00, /* 0x40babd86 */
276 1.3511157227e+02, /* 0x43071c90 */
277 1.0272437744e+03, /* 0x448067cd */
278 1.9899779053e+03, /* 0x44f8bf4b */
279};
280static const float qS5[6] = {
281 8.2776611328e+01, /* 0x42a58da0 */
282 2.0778142090e+03, /* 0x4501dd07 */
283 1.8847289062e+04, /* 0x46933e94 */
284 5.6751113281e+04, /* 0x475daf1d */
285 3.5976753906e+04, /* 0x470c88c1 */
286 -5.3543427734e+03, /* 0xc5a752be */
287};
288
289static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
290 4.3774099900e-09, /* 0x3196681b */
291 7.3241114616e-02, /* 0x3d95ff70 */
292 3.3442313671e+00, /* 0x405607e3 */
293 4.2621845245e+01, /* 0x422a7cc5 */
294 1.7080809021e+02, /* 0x432acedf */
295 1.6673394775e+02, /* 0x4326bbe4 */
296};
297static const float qS3[6] = {
298 4.8758872986e+01, /* 0x42430916 */
299 7.0968920898e+02, /* 0x44316c1c */
300 3.7041481934e+03, /* 0x4567825f */
301 6.4604252930e+03, /* 0x45c9e367 */
302 2.5163337402e+03, /* 0x451d4557 */
303 -1.4924745178e+02, /* 0xc3153f59 */
304};
305
306static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
307 1.5044444979e-07, /* 0x342189db */
308 7.3223426938e-02, /* 0x3d95f62a */
309 1.9981917143e+00, /* 0x3fffc4bf */
310 1.4495602608e+01, /* 0x4167edfd */
311 3.1666231155e+01, /* 0x41fd5471 */
312 1.6252708435e+01, /* 0x4182058c */
313};
314static const float qS2[6] = {
315 3.0365585327e+01, /* 0x41f2ecb8 */
316 2.6934811401e+02, /* 0x4386ac8f */
317 8.4478375244e+02, /* 0x44533229 */
318 8.8293585205e+02, /* 0x445cbbe5 */
319 2.1266638184e+02, /* 0x4354aa98 */
320 -5.3109550476e+00, /* 0xc0a9f358 */
321};
322
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