Lines Matching +full:- +full:y
27 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
28 * Let z = x-ymin;
29 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
34 * s = x-2.0;
37 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
40 * zeta(2)-1 2 zeta(3)-1 3
41 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
48 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
50 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
52 * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
57 * |w - f(z)| < 2**-58.74
60 * -x*G(-x)*G(x) = pi/sin(pi*x),
62 * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
63 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
66 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
67 * Note: one should avoid compute pi*(-x) directly in the
68 * computation of sin(pi*(-x)).
71 * lgamma(2+s) ~ s*(1-Euler) for tiny s
73 * lgamma(x) ~ -log(|x|) for tiny x
74 * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
76 * lgamma(-inf) = inf (bug for bug compatible with C99!?)
88 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
91 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
92 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
93 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
94 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
95 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
96 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
97 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
98 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
99 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
100 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
101 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
102 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
104 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
105 /* tt = -(tail of tf) */
106 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
107 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
108 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
109 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
110 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
111 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
112 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
113 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
114 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
115 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
116 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
117 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
118 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
119 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
120 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
121 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
122 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
123 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
125 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
126 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
127 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
130 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
131 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
132 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
133 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
134 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
135 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
136 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
137 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
138 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
139 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
141 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
142 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
143 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
144 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
145 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
146 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
147 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
148 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
149 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
150 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
151 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
152 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
156 * sin_pi(x) is only called for x < 0 and |x| < 2**(p-1) where p is
163 double y,z; in sin_pi() local
166 y = -x; in sin_pi()
168 vz = y+0x1p52; /* depend on 0 <= y < 0x1p52 */ in sin_pi()
169 z = vz-0x1p52; /* rint(y) for the above range */ in sin_pi()
170 if (z == y) in sin_pi()
173 vz = y+0x1p50; in sin_pi()
174 GET_LOW_WORD(n,vz); /* bits for rounded y (units 0.25) */ in sin_pi()
175 z = vz-0x1p50; /* y rounded to a multiple of 0.25 */ in sin_pi()
176 if (z > y) { in sin_pi()
177 z -= 0.25; /* adjust to round down */ in sin_pi()
178 n--; in sin_pi()
180 n &= 7; /* octant of y mod 2 */ in sin_pi()
181 y = y - z + n * 0.25; /* y mod 2 */ in sin_pi()
184 case 0: y = __kernel_sin(pi*y,zero,0); break; in sin_pi()
186 case 2: y = __kernel_cos(pi*(0.5-y),zero); break; in sin_pi()
188 case 4: y = __kernel_sin(pi*(one-y),zero,0); break; in sin_pi()
190 case 6: y = -__kernel_cos(pi*(y-1.5),zero); break; in sin_pi()
191 default: y = __kernel_sin(pi*(y-2.0),zero,0); break; in sin_pi()
193 return -y; in sin_pi()
200 double nadj,p,p1,p2,p3,q,r,t,w,y,z; in lgamma_r() local
206 /* purge +-Inf and NaNs */ in lgamma_r()
211 /* purge +-0 and tiny arguments */ in lgamma_r()
212 *signgamp = 1-2*((uint32_t)hx>>31); in lgamma_r()
213 if(ix<0x3c700000) { /* |x|<2**-56, return -log(|x|) */ in lgamma_r()
216 return -log(fabs(x)); in lgamma_r()
222 if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ in lgamma_r()
225 if(t==zero) return one/vzero; /* -integer */ in lgamma_r()
227 if(t<zero) *signgamp = -1; in lgamma_r()
228 x = -x; in lgamma_r()
232 if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0; in lgamma_r()
235 if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ in lgamma_r()
236 r = -log(x); in lgamma_r()
237 if(ix>=0x3FE76944) {y = one-x; i= 0;} in lgamma_r()
238 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} in lgamma_r()
239 else {y = x; i=2;} in lgamma_r()
242 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ in lgamma_r()
243 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ in lgamma_r()
244 else {y=x-one;i=2;} in lgamma_r()
248 z = y*y; in lgamma_r()
251 p = y*p1+p2; in lgamma_r()
252 r += p-y/2; break; in lgamma_r()
254 z = y*y; in lgamma_r()
255 w = z*y; in lgamma_r()
259 p = z*p1-(tt-w*(p2+y*p3)); in lgamma_r()
262 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); in lgamma_r()
263 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); in lgamma_r()
264 r += p1/p2-y/2; in lgamma_r()
270 y = x-i; in lgamma_r()
271 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); in lgamma_r()
272 q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); in lgamma_r()
273 r = y/2+p/q; in lgamma_r()
276 case 7: z *= (y+6); /* FALLTHRU */ in lgamma_r()
277 case 6: z *= (y+5); /* FALLTHRU */ in lgamma_r()
278 case 5: z *= (y+4); /* FALLTHRU */ in lgamma_r()
279 case 4: z *= (y+3); /* FALLTHRU */ in lgamma_r()
280 case 3: z *= (y+2); /* FALLTHRU */ in lgamma_r()
287 y = z*z; in lgamma_r()
288 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); in lgamma_r()
289 r = (x-half)*(t-one)+w; in lgamma_r()
292 r = x*(log(x)-one); in lgamma_r()
293 if(hx<0) r = nadj - r; in lgamma_r()