1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
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20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25 /*
26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 * Use is subject to license terms.
28 */
29
30 #if defined(ELFOBJ)
31 #pragma weak tgamma = __tgamma
32 #endif
33
34 /* INDENT OFF */
35 /*
36 * True gamma function
37 * double tgamma(double x)
38 *
39 * Error:
40 * ------
41 * Less that one ulp for both positive and negative arguments.
42 *
43 * Algorithm:
44 * ---------
45 * A: For negative argument
46 * (1) gamma(-n or -inf) is NaN
47 * (2) Underflow Threshold
48 * (3) Reduction to gamma(1+x)
49 * B: For x between 1 and 2
50 * C: For x between 0 and 1
51 * D: For x between 2 and 8
52 * E: Overflow thresold {see over.c}
53 * F: For overflow_threshold >= x >= 8
54 *
55 * Implementation details
56 * -----------------------
57 * -pi
58 * (A) For negative argument, use gamma(-x) = ------------------------.
59 * (sin(pi*x)*gamma(1+x))
60 *
61 * (1) gamma(-n or -inf) is NaN with invalid signal by SUSv3 spec.
62 * (Ideally, gamma(-n) = 1/sinpi(n) = (-1)**(n+1) * inf.)
63 *
64 * (2) Underflow Threshold. For each precision, there is a value T
65 * such that when x>T and when x is not an integer, gamma(-x) will
66 * always underflow. A table of the underflow threshold value is given
67 * below. For proof, see file "under.c".
68 *
69 * Precision underflow threshold T =
70 * ----------------------------------------------------------------------
71 * single 41.000041962 = 41 + 11 ULP
72 * (machine format) 4224000B
73 * double 183.000000000000312639 = 183 + 11 ULP
74 * (machine format) 4066E000 0000000B
75 * quad 1774.0000000000000000000000000000017749370 = 1774 + 9 ULP
76 * (machine format) 4009BB80000000000000000000000009
77 * ----------------------------------------------------------------------
78 *
79 * (3) Reduction to gamma(1+x).
80 * Because of (1) and (2), we need only consider non-integral x
81 * such that 0<x<T. Let k = [x] and z = x-[x]. Define
82 * sin(x*pi) cos(x*pi)
83 * kpsin(x) = --------- and kpcos(x) = --------- . Then
84 * pi pi
85 * 1
86 * gamma(-x) = --------------------.
87 * -kpsin(x)*gamma(1+x)
88 * Since x = k+z,
89 * k+1
90 * -sin(x*pi) = -sin(k*pi+z*pi) = (-1) *sin(z*pi),
91 * k+1
92 * we have -kpsin(x) = (-1) * kpsin(z). We can further
93 * reduce z to t by
94 * (I) t = z when 0.00000 <= z < 0.31830...
95 * (II) t = 0.5-z when 0.31830... <= z < 0.681690...
96 * (III) t = 1-z when 0.681690... <= z < 1.00000
97 * and correspondingly
98 * (I) kpsin(z) = kpsin(t) ... 0<= z < 0.3184
99 * (II) kpsin(z) = kpcos(t) ... |t| < 0.182
100 * (III) kpsin(z) = kpsin(t) ... 0<= t < 0.3184
101 *
102 * Using a special Remez algorithm, we obtain the following polynomial
103 * approximation for kpsin(t) for 0<=t<0.3184:
104 *
105 * Computation note: in simulating higher precision arithmetic, kcpsin
106 * return head = t and tail = ks[0]*t^3 + (...) to maintain extra bits.
107 *
108 * Quad precision, remez error <= 2**(-129.74)
109 * 3 5 27
110 * kpsin(t) = t + ks[0] * t + ks[1] * t + ... + ks[12] * t
111 *
112 * ks[ 0] = -1.64493406684822643647241516664602518705158902870e+0000
113 * ks[ 1] = 8.11742425283353643637002772405874238094995726160e-0001
114 * ks[ 2] = -1.90751824122084213696472111835337366232282723933e-0001
115 * ks[ 3] = 2.61478478176548005046532613563241288115395517084e-0002
116 * ks[ 4] = -2.34608103545582363750893072647117829448016479971e-0003
117 * ks[ 5] = 1.48428793031071003684606647212534027556262040158e-0004
118 * ks[ 6] = -6.97587366165638046518462722252768122615952898698e-0006
119 * ks[ 7] = 2.53121740413702536928659271747187500934840057929e-0007
120 * ks[ 8] = -7.30471182221385990397683641695766121301933621956e-0009
121 * ks[ 9] = 1.71653847451163495739958249695549313987973589884e-0010
122 * ks[10] = -3.34813314714560776122245796929054813458341420565e-0012
123 * ks[11] = 5.50724992262622033449487808306969135431411753047e-0014
124 * ks[12] = -7.67678132753577998601234393215802221104236979928e-0016
125 *
126 * Double precision, Remez error <= 2**(-62.9)
127 * 3 5 15
128 * kpsin(t) = t + ks[0] * t + ks[1] * t + ... + ks[6] * t
129 *
130 * ks[0] = -1.644934066848226406065691 (0x3ffa51a6 625307d3)
131 * ks[1] = 8.11742425283341655883668741874008920850698590621e-0001
132 * ks[2] = -1.90751824120862873825597279118304943994042258291e-0001
133 * ks[3] = 2.61478477632554278317289628332654539353521911570e-0002
134 * ks[4] = -2.34607978510202710377617190278735525354347705866e-0003
135 * ks[5] = 1.48413292290051695897242899977121846763824221705e-0004
136 * ks[6] = -6.87730769637543488108688726777687262485357072242e-0006
137 *
138 * Single precision, Remez error <= 2**(-34.09)
139 * 3 5 9
140 * kpsin(t) = t + ks[0] * t + ks[1] * t + ... + ks[3] * t
141 *
142 * ks[0] = -1.64493404985645811354476665052005342839447790544e+0000
143 * ks[1] = 8.11740794458351064092797249069438269367389272270e-0001
144 * ks[2] = -1.90703144603551216933075809162889536878854055202e-0001
145 * ks[3] = 2.55742333994264563281155312271481108635575331201e-0002
146 *
147 * Computation note: in simulating higher precision arithmetic, kcpsin
148 * return head = t and tail = kc[0]*t^3 + (...) to maintain extra bits
149 * precision.
150 *
151 * And for kpcos(t) for |t|< 0.183:
152 *
153 * Quad precision, remez <= 2**(-122.48)
154 * 2 4 22
155 * kpcos(t) = 1/pi + pi/2 * t + kc[2] * t + ... + kc[11] * t
156 *
157 * kc[2] = 1.29192819501249250731151312779548918765320728489e+0000
158 * kc[3] = -4.25027339979557573976029596929319207009444090366e-0001
159 * kc[4] = 7.49080661650990096109672954618317623888421628613e-0002
160 * kc[5] = -8.21458866111282287985539464173976555436050215120e-0003
161 * kc[6] = 6.14202578809529228503205255165761204750211603402e-0004
162 * kc[7] = -3.33073432691149607007217330302595267179545908740e-0005
163 * kc[8] = 1.36970959047832085796809745461530865597993680204e-0006
164 * kc[9] = -4.41780774262583514450246512727201806217271097336e-0008
165 * kc[10]= 1.14741409212381858820016567664488123478660705759e-0009
166 * kc[11]= -2.44261236114707374558437500654381006300502749632e-0011
167 *
168 * Double precision, remez < 2**(61.91)
169 * 2 4 12
170 * kpcos(t) = 1/pi + pi/2 *t + kc[2] * t + ... + kc[6] * t
171 *
172 * kc[2] = 1.29192819501230224953283586722575766189551966008e+0000
173 * kc[3] = -4.25027339940149518500158850753393173519732149213e-0001
174 * kc[4] = 7.49080625187015312373925142219429422375556727752e-0002
175 * kc[5] = -8.21442040906099210866977352284054849051348692715e-0003
176 * kc[6] = 6.10411356829515414575566564733632532333904115968e-0004
177 *
178 * Single precision, remez < 2**(-30.13)
179 * 2 6
180 * kpcos(t) = kc[0] + kc[1] * t + ... + kc[3] * t
181 *
182 * kc[0] = 3.18309886183790671537767526745028724068919291480e-0001
183 * kc[1] = -1.57079581447762568199467875065854538626594937791e+0000
184 * kc[2] = 1.29183528092558692844073004029568674027807393862e+0000
185 * kc[3] = -4.20232949771307685981015914425195471602739075537e-0001
186 *
187 * Computation note: in simulating higher precision arithmetic, kcpcos
188 * return head = 1/pi chopped, and tail = pi/2 *t^2 + (tail part of 1/pi
189 * + ...) to maintain extra bits precision. In particular, pi/2 * t^2
190 * is calculated with great care.
191 *
192 * Thus, the computation of gamma(-x), x>0, is:
193 * Let k = int(x), z = x-k.
194 * For z in (I)
195 * k+1
196 * (-1)
197 * gamma(-x) = ------------------- ;
198 * kpsin(z)*gamma(1+x)
199 *
200 * otherwise, for z in (II),
201 * k+1
202 * (-1)
203 * gamma(-x) = ----------------------- ;
204 * kpcos(0.5-z)*gamma(1+x)
205 *
206 * otherwise, for z in (III),
207 * k+1
208 * (-1)
209 * gamma(-x) = --------------------- .
210 * kpsin(1-z)*gamma(1+x)
211 *
212 * Thus, the computation of gamma(-x) reduced to the computation of
213 * gamma(1+x) and kpsin(), kpcos().
214 *
215 * (B) For x between 1 and 2. We break [1,2] into three parts:
216 * GT1 = [1.0000, 1.2845]
217 * GT2 = [1.2844, 1.6374]
218 * GT3 = [1.6373, 2.0000]
219 *
220 * For x in GTi, i=1,2,3, let
221 * z1 = 1.134861805732790769689793935774652917006
222 * gz1 = gamma(z1) = 0.9382046279096824494097535615803269576988
223 * tz1 = gamma'(z1) = -0.3517214357852935791015625000000000000000
224 *
225 * z2 = 1.461632144968362341262659542325721328468e+0000
226 * gz2 = gamma(z2) = 0.8856031944108887002788159005825887332080
227 * tz2 = gamma'(z2) = 0.00
228 *
229 * z3 = 1.819773101100500601787868704921606996312e+0000
230 * gz3 = gamma(z3) = 0.9367814114636523216188468970808378497426
231 * tz3 = gamma'(z3) = 0.2805306315422058105468750000000000000000
232 *
233 * and
234 * y = x-zi ... for extra precision, write y = y.h + y.l
235 * Then
236 * gamma(x) = gzi + tzi*(y.h+y.l) + y*y*Ri(y),
237 * = gzi.h + (tzi*y.h + ((tzi*y.l+gzi.l) + y*y*Ri(y)))
238 * = gy.h + gy.l
239 * where
240 * (I) For double precision
241 *
242 * Ri(y) = Pi(y)/Qi(y), i=1,2,3;
243 *
244 * P1(y) = p1[0] + p1[1]*y + ... + p1[4]*y^4
245 * Q1(y) = q1[0] + q1[1]*y + ... + q1[5]*y^5
246 *
247 * P2(y) = p2[0] + p2[1]*y + ... + p2[3]*y^3
248 * Q2(y) = q2[0] + q2[1]*y + ... + q2[6]*y^6
249 *
250 * P3(y) = p3[0] + p3[1]*y + ... + p3[4]*y^4
251 * Q3(y) = q3[0] + q3[1]*y + ... + q3[5]*y^5
252 *
253 * Remez precision of Ri(y):
254 * |gamma(x)-(gzi+tzi*y) - y*y*Ri(y)| <= 2**-62.3 ... for i = 1
255 * <= 2**-59.4 ... for i = 2
256 * <= 2**-62.1 ... for i = 3
257 *
258 * (II) For quad precision
259 *
260 * Ri(y) = Pi(y)/Qi(y), i=1,2,3;
261 *
262 * P1(y) = p1[0] + p1[1]*y + ... + p1[9]*y^9
263 * Q1(y) = q1[0] + q1[1]*y + ... + q1[8]*y^8
264 *
265 * P2(y) = p2[0] + p2[1]*y + ... + p2[9]*y^9
266 * Q2(y) = q2[0] + q2[1]*y + ... + q2[9]*y^9
267 *
268 * P3(y) = p3[0] + p3[1]*y + ... + p3[9]*y^9
269 * Q3(y) = q3[0] + q3[1]*y + ... + q3[9]*y^9
270 *
271 * Remez precision of Ri(y):
272 * |gamma(x)-(gzi+tzi*y) - y*y*Ri(y)| <= 2**-118.2 ... for i = 1
273 * <= 2**-126.8 ... for i = 2
274 * <= 2**-119.5 ... for i = 3
275 *
276 * (III) For single precision
277 *
278 * Ri(y) = Pi(y), i=1,2,3;
279 *
280 * P1(y) = p1[0] + p1[1]*y + ... + p1[5]*y^5
281 *
282 * P2(y) = p2[0] + p2[1]*y + ... + p2[5]*y^5
283 *
284 * P3(y) = p3[0] + p3[1]*y + ... + p3[4]*y^4
285 *
286 * Remez precision of Ri(y):
287 * |gamma(x)-(gzi+tzi*y) - y*y*Ri(y)| <= 2**-30.8 ... for i = 1
288 * <= 2**-31.6 ... for i = 2
289 * <= 2**-29.5 ... for i = 3
290 *
291 * Notes. (1) GTi and zi are choosen to balance the interval width and
292 * minimize the distant between gamma(x) and the tangent line at
293 * zi. In particular, we have
294 * |gamma(x)-(gzi+tzi*(x-zi))| <= 0.01436... for x in [1,z2]
295 * <= 0.01265... for x in [z2,2]
296 *
297 * (2) zi are slightly adjusted so that tzi=gamma'(zi) is very
298 * close to a single precision value.
299 *
300 * Coefficents: Single precision
301 * i= 1:
302 * P1[0] = 7.09087253435088360271451613398019280077561279443e-0001
303 * P1[1] = -5.17229560788652108545141978238701790105241761089e-0001
304 * P1[2] = 5.23403394528150789405825222323770647162337764327e-0001
305 * P1[3] = -4.54586308717075010784041566069480411732634814899e-0001
306 * P1[4] = 4.20596490915239085459964590559256913498190955233e-0001
307 * P1[5] = -3.57307589712377520978332185838241458642142185789e-0001
308 *
309 * i = 2:
310 * p2[0] = 4.28486983980295198166056119223984284434264344578e-0001
311 * p2[1] = -1.30704539487709138528680121627899735386650103914e-0001
312 * p2[2] = 1.60856285038051955072861219352655851542955430871e-0001
313 * p2[3] = -9.22285161346010583774458802067371182158937943507e-0002
314 * p2[4] = 7.19240511767225260740890292605070595560626179357e-0002
315 * p2[5] = -4.88158265593355093703112238534484636193260459574e-0002
316 *
317 * i = 3
318 * p3[0] = 3.82409531118807759081121479786092134814808872880e-0001
319 * p3[1] = 2.65309888180188647956400403013495759365167853426e-0002
320 * p3[2] = 8.06815109775079171923561169415370309376296739835e-0002
321 * p3[3] = -1.54821591666137613928840890835174351674007764799e-0002
322 * p3[4] = 1.76308239242717268530498313416899188157165183405e-0002
323 *
324 * Coefficents: Double precision
325 * i = 1:
326 * p1[0] = 0.70908683619977797008004927192814648151397705078125000
327 * p1[1] = 1.71987061393048558089579513384356441668351720061e-0001
328 * p1[2] = -3.19273345791990970293320316122813960527705450671e-0002
329 * p1[3] = 8.36172645419110036267169600390549973563534476989e-0003
330 * p1[4] = 1.13745336648572838333152213474277971244629758101e-0003
331 * q1[0] = 1.0
332 * q1[1] = 9.71980217826032937526460731778472389791321968082e-0001
333 * q1[2] = -7.43576743326756176594084137256042653497087666030e-0002
334 * q1[3] = -1.19345944932265559769719470515102012246995255372e-0001
335 * q1[4] = 1.59913445751425002620935120470781382215050284762e-0002
336 * q1[5] = 1.12601136853374984566572691306402321911547550783e-0003
337 * i = 2:
338 * p2[0] = 0.42848681585558601181418225678498856723308563232421875
339 * p2[1] = 6.53596762668970816023718845105667418483122103629e-0002
340 * p2[2] = -6.97280829631212931321050770925128264272768936731e-0003
341 * p2[3] = 6.46342359021981718947208605674813260166116632899e-0003
342 * q2[0] = 1.0
343 * q2[1] = 4.57572620560506047062553957454062012327519313936e-0001
344 * q2[2] = -2.52182594886075452859655003407796103083422572036e-0001
345 * q2[3] = -1.82970945407778594681348166040103197178711552827e-0002
346 * q2[4] = 2.43574726993169566475227642128830141304953840502e-0002
347 * q2[5] = -5.20390406466942525358645957564897411258667085501e-0003
348 * q2[6] = 4.79520251383279837635552431988023256031951133885e-0004
349 * i = 3:
350 * p3[0] = 0.382409479734567459008331979930517263710498809814453125
351 * p3[1] = 1.42876048697668161599069814043449301572928034140e-0001
352 * p3[2] = 3.42157571052250536817923866013561760785748899071e-0003
353 * p3[3] = -5.01542621710067521405087887856991700987709272937e-0004
354 * p3[4] = 8.89285814866740910123834688163838287618332122670e-0004
355 * q3[0] = 1.0
356 * q3[1] = 3.04253086629444201002215640948957897906299633168e-0001
357 * q3[2] = -2.23162407379999477282555672834881213873185520006e-0001
358 * q3[3] = -1.05060867741952065921809811933670131427552903636e-0002
359 * q3[4] = 1.70511763916186982473301861980856352005926669320e-0002
360 * q3[5] = -2.12950201683609187927899416700094630764182477464e-0003
361 *
362 * Note that all pi0 are exact in double, which is obtained by a
363 * special Remez Algorithm.
364 *
365 * Coefficents: Quad precision
366 * i = 1:
367 * p1[0] = 0.709086836199777919037185741507610124611513720557
368 * p1[1] = 4.45754781206489035827915969367354835667391606951e-0001
369 * p1[2] = 3.21049298735832382311662273882632210062918153852e-0002
370 * p1[3] = -5.71296796342106617651765245858289197369688864350e-0003
371 * p1[4] = 6.04666892891998977081619174969855831606965352773e-0003
372 * p1[5] = 8.99106186996888711939627812174765258822658645168e-0004
373 * p1[6] = -6.96496846144407741431207008527018441810175568949e-0005
374 * p1[7] = 1.52597046118984020814225409300131445070213882429e-0005
375 * p1[8] = 5.68521076168495673844711465407432189190681541547e-0007
376 * p1[9] = 3.30749673519634895220582062520286565610418952979e-0008
377 * q1[0] = 1.0+0000
378 * q1[1] = 1.35806511721671070408570853537257079579490650668e+0000
379 * q1[2] = 2.97567810153429553405327140096063086994072952961e-0001
380 * q1[3] = -1.52956835982588571502954372821681851681118097870e-0001
381 * q1[4] = -2.88248519561420109768781615289082053597954521218e-0002
382 * q1[5] = 1.03475311719937405219789948456313936302378395955e-0002
383 * q1[6] = 4.12310203243891222368965360124391297374822742313e-0004
384 * q1[7] = -3.12653708152290867248931925120380729518332507388e-0004
385 * q1[8] = 2.36672170850409745237358105667757760527014332458e-0005
386 *
387 * i = 2:
388 * p2[0] = 0.428486815855585429730209907810650616737756697477
389 * p2[1] = 2.63622124067885222919192651151581541943362617352e-0001
390 * p2[2] = 3.85520683670028865731877276741390421744971446855e-0002
391 * p2[3] = 3.05065978278128549958897133190295325258023525862e-0003
392 * p2[4] = 2.48232934951723128892080415054084339152450445081e-0003
393 * p2[5] = 3.67092777065632360693313762221411547741550105407e-0004
394 * p2[6] = 3.81228045616085789674530902563145250532194518946e-0006
395 * p2[7] = 4.61677225867087554059531455133839175822537617677e-0006
396 * p2[8] = 2.18209052385703200438239200991201916609364872993e-0007
397 * p2[9] = 1.00490538985245846460006244065624754421022542454e-0008
398 * q2[0] = 1.0
399 * q2[1] = 9.20276350207639290567783725273128544224570775056e-0001
400 * q2[2] = -4.79533683654165107448020515733883781138947771495e-0003
401 * q2[3] = -1.24538337585899300494444600248687901947684291683e-0001
402 * q2[4] = 4.49866050763472358547524708431719114204535491412e-0003
403 * q2[5] = 7.20715455697920560621638325356292640604078591907e-0003
404 * q2[6] = -8.68513169029126780280798337091982780598228096116e-0004
405 * q2[7] = -1.25104431629401181525027098222745544809974229874e-0004
406 * q2[8] = 3.10558344839000038489191304550998047521253437464e-0005
407 * q2[9] = -1.76829227852852176018537139573609433652506765712e-0006
408 *
409 * i = 3
410 * p3[0] = 0.3824094797345675048502747661075355640070439388902
411 * p3[1] = 3.42198093076618495415854906335908427159833377774e-0001
412 * p3[2] = 9.63828189500585568303961406863153237440702754858e-0002
413 * p3[3] = 8.76069421042696384852462044188520252156846768667e-0003
414 * p3[4] = 1.86477890389161491224872014149309015261897537488e-0003
415 * p3[5] = 8.16871354540309895879974742853701311541286944191e-0004
416 * p3[6] = 6.83783483674600322518695090864659381650125625216e-0005
417 * p3[7] = -1.10168269719261574708565935172719209272190828456e-0006
418 * p3[8] = 9.66243228508380420159234853278906717065629721016e-0007
419 * p3[9] = 2.31858885579177250541163820671121664974334728142e-0008
420 * q3[0] = 1.0
421 * q3[1] = 8.25479821168813634632437430090376252512793067339e-0001
422 * q3[2] = -1.62251363073937769739639623669295110346015576320e-0002
423 * q3[3] = -1.10621286905916732758745130629426559691187579852e-0001
424 * q3[4] = 3.48309693970985612644446415789230015515365291459e-0003
425 * q3[5] = 6.73553737487488333032431261131289672347043401328e-0003
426 * q3[6] = -7.63222008393372630162743587811004613050245128051e-0004
427 * q3[7] = -1.35792670669190631476784768961953711773073251336e-0004
428 * q3[8] = 3.19610150954223587006220730065608156460205690618e-0005
429 * q3[9] = -1.82096553862822346610109522015129585693354348322e-0006
430 *
431 * (C) For x between 0 and 1.
432 * Let P stand for the number of significant bits in the working precision.
433 * -P 1
434 * (1)For 0 <= x <= 2 , gamma(x) is computed by --- rounded to nearest.
435 * x
436 * The error is bound by 0.739 ulp(gamma(x)) in IEEE double precision.
437 * Proof.
438 * 1 2
439 * Since -------- ~ x + 0.577...*x - ..., we have, for small x,
440 * gamma(x)
441 * 1 1
442 * ----------- < gamma(x) < --- and
443 * x(1+0.578x) x
444 * 1 1 1
445 * 0 < --- - gamma(x) <= --- - ----------- < 0.578
446 * x x x(1+0.578x)
447 * 1 1 -P
448 * The error is thus bounded by --- ulp(---) + 0.578. Since x <= 2 ,
449 * 2 x
450 * 1 P 1 P 1
451 * --- >= 2 , ulp(---) >= ulp(2 ) >= 2. Thus 0.578=0.289*2<=0.289ulp(-)
452 * x x x
453 * Thus
454 * 1 1
455 * | gamma(x) - [---] rounded | <= (0.5+0.289)*ulp(---).
456 * x x
457 * -P 1
458 * Note that for x<= 2 , it is easy to see that ulp(---)=ulp(gamma(x))
459 * x
460 * n 1
461 * except only when x = 2 , (n<= -53). In such cases, --- is exact
462 * x
463 * and therefore the error is bounded by
464 * 1
465 * 0.298*ulp(---) = 0.298*2*ulp(gamma(x)) = 0.578ulp(gamma(x)).
466 * x
467 * Thus we conclude that the error in gamma is less than 0.739 ulp.
468 *
469 * (2)Otherwise, for x in GTi-1 (see B), let y = x-(zi-1). From (B) we obtain
470 * gamma(1+x)
471 * gamma(1+x) = gy.h + gy.l, then compute gamma(x) by -----------.
472 * x
473 * gy.h
474 * Implementaion note. Write x = x.h+x.l, and Let th = ----- chopped to
475 * x
476 * 20 bits, then
477 * gy.h+gy.l
478 * gamma(x) = th + (---------- - th )
479 * x
480 * 1
481 * = th + ---*(gy.h-th*x.h+gy.l-th*x.l)
482 * x
483 *
484 * (D) For x between 2 and 8. Let n = 1+x chopped to an integer. Then
485 *
486 * gamma(x)=(x-1)*(x-2)*...*(x-n)*gamma(x-n)
487 *
488 * Since x-n is between 1 and 2, we can apply (B) to compute gamma(x).
489 *
490 * Implementation detail. The computation of (x-1)(x-2)...(x-n) in simulated
491 * higher precision arithmetic can be somewhat optimized. For example, in
492 * computing (x-1)*(x-2)*(x-3)*(x-4), if we compute (x-1)*(x-4) = z.h+z.l,
493 * then (x-2)(x-3) = z.h+2+z.l readily. In below, we list the expression
494 * of the formula to compute gamma(x).
495 *
496 * Assume x-n is in GTi (i=1,2, or 3, see B for detail). Let y = x - n - zi.
497 * By (B) we have gamma(x-n) = gy.h+gy.l. If x = x.h+x.l, then we have
498 * n=1 (x in [2,3]):
499 * gamma(x) = (x-1)*gamma(x-1) = (x-1)*(gy.h+gy.l)
500 * = [(x.h-1)+x.l]*(gy.h+gy.l)
501 * n=2 (x in [3,4]):
502 * gamma(x) = (x-1)(x-2)*gamma(x-2) = (x-1)*(x-2)*(gy.h+gy.l)
503 * = ((x.h-2)+x.l)*((x.h-1)+x.l)*(gy.h+gy.l)
504 * = [x.h*(x.h-3)+2+x.l*(x+(x.h-3))]*(gy.h+gy.l)
505 * n=3 (x in [4,5])
506 * gamma(x) = (x-1)(x-2)(x-3)*(gy.h+gy.l)
507 * = (x.h*(x.h-3)+2+x.l*(x+(x.h-3)))*[((x.h-3)+x.l)(gy.h+gy.l)]
508 * n=4 (x in [5,6])
509 * gamma(x) = [(x-1)(x-4)]*[(x-2)(x-3)]*(gy.h+gy.l)
510 * = [(x.h*(x.h-5)+4+x.l(x+(x.h-5)))]*[(x-2)*(x-3)]*(gy.h+gy.l)
511 * = (y.h+y.l)*(y.h+1+y.l)*(gy.h+gy.l)
512 * n=5 (x in [6,7])
513 * gamma(x) = [(x-1)(x-4)]*[(x-2)(x-3)]*[(x-5)*(gy.h+gy.l)]
514 * n=6 (x in [7,8])
515 * gamma(x) = [(x-1)(x-6)]*[(x-2)(x-5)]*[(x-3)(x-4)]*(gy.h+gy.l)]
516 * = [(y.h+y.l)(y.h+4+y.l)][(y.h+6+y.l)(gy.h+gy.l)]
517 *
518 * (E)Overflow Thresold. For x > Overflow thresold of gamma,
519 * return huge*huge (overflow).
520 *
521 * By checking whether lgamma(x) >= 2**{128,1024,16384}, one can
522 * determine the overflow threshold for x in single, double, and
523 * quad precision. See over.c for details.
524 *
525 * The overflow threshold of gamma(x) are
526 *
527 * single: x = 3.5040096283e+01
528 * = 0x420C290F (IEEE single)
529 * double: x = 1.71624376956302711505e+02
530 * = 0x406573FAE561F647 (IEEE double)
531 * quad: x = 1.7555483429044629170038892160702032034177e+03
532 * = 0x4009B6E3180CD66A5C4206F128BA77F4 (quad)
533 *
534 * (F)For overflow_threshold >= x >= 8, we use asymptotic approximation.
535 * (1) Stirling's formula
536 *
537 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
538 * = L1 + L2 + L3,
539 * where
540 * L1(x) = (x-.5)*(log(x)-1),
541 * L2 = .5(log(2pi)-1) = 0.41893853....,
542 * L3(x) = (1/x)P(1/(x*x)),
543 *
544 * The range of L1,L2, and L3 are as follows:
545 *
546 * ------------------------------------------------------------------
547 * Range(L1) = (single) [8.09..,88.30..] =[2** 3.01..,2** 6.46..]
548 * (double) [8.09..,709.3..] =[2** 3.01..,2** 9.47..]
549 * (quad) [8.09..,11356.10..]=[2** 3.01..,2** 13.47..]
550 * Range(L2) = 0.41893853.....
551 * Range(L3) = [0.0104...., 0.00048....] =[2**-6.58..,2**-11.02..]
552 * ------------------------------------------------------------------
553 *
554 * Gamma(x) is then computed by exp(L1+L2+L3).
555 *
556 * (2) Error analysis of (F):
557 * --------------------------
558 * The error in Gamma(x) depends on the error inherited in the computation
559 * of L= L1+L2+L3. Let L' be the computed value of L. The absolute error
560 * in L' is t = L-L'. Since exp(L') = exp(L-t) = exp(L)*exp(t) ~
561 * (1+t)*exp(L), the relative error in exp(L') is approximately t.
562 *
563 * To guarantee the relatively accuracy in exp(L'), we would like
564 * |t| < 2**(-P-5) where P denotes for the number of significant bits
565 * of the working precision. Consequently, each of the L1,L2, and L3
566 * must be computed with absolute error bounded by 2**(-P-5) in absolute
567 * value.
568 *
569 * Since L2 is a constant, it can be pre-computed to the desired accuracy.
570 * Also |L3| < 2**-6; therefore, it suffices to compute L3 with the
571 * working precision. That is,
572 * L3(x) approxmiate log(G(x))-(x-.5)(log(x)-1)-.5(log(2pi)-1)
573 * to a precision bounded by 2**(-P-5).
574 *
575 * 2**(-6)
576 * _________V___________________
577 * L1(x): |_________|___________________|
578 * __ ________________________
579 * L2: |__|________________________|
580 * __________________________
581 * + L3(x): |__________________________|
582 * -------------------------------------------
583 * [leading] + [Trailing]
584 *
585 * For L1(x)=(x-0.5)*(log(x)-1), we need ilogb(L1(x))+5 extra bits for
586 * both multiplicants to guarantee L1(x)'s absolute error is bounded by
587 * 2**(-P-5) in absolute value. Here ilogb(y) is defined to be the unbias
588 * binary exponent of y in IEEE format. We can get x-0.5 to the desire
589 * accuracy easily. It remains to compute log(x)-1 with ilogb(L1(x))+5
590 * extra bits accracy. Note that the range of L1 is 88.30.., 709.3.., and
591 * 11356.10... for single, double, and quadruple precision, we have
592 *
593 * single double quadruple
594 * ------------------------------------
595 * ilogb(L1(x))+5 <= 11 14 18
596 * ------------------------------------
597 *
598 * (3) Table Driven Method for log(x)-1:
599 * --------------------------------------
600 * Let x = 2**n * y, where 1 <= y < 2. Let Z={z(i),i=1,...,m}
601 * be a set of predetermined evenly distributed floating point numbers
602 * in [1, 2]. Let z(j) be the closest one to y, then
603 * log(x)-1 = n*log(2)-1 + log(y)
604 * = n*log(2)-1 + log(z(j)*y/z(j))
605 * = n*log(2)-1 + log(z(j)) + log(y/z(j))
606 * = T1(n) + T2(j) + T3,
607 *
608 * where T1(n) = n*log(2)-1 and T2(j) = log(z(j)). Both T1 and T2 can be
609 * pre-calculated and be looked-up in a table. Note that 8 <= x < 1756
610 * implies 3<=n<=10 implies 1.079.. < T1(n) < 6.931.
611 *
612 *
613 * y-z(i) y 1+s
614 * For T3, let s = --------; then ----- = ----- and
615 * y+z(i) z(i) 1-s
616 * 1+s 2 3 2 5
617 * T3 = log(-----) = 2s + --- s + --- s + ....
618 * 1-s 3 5
619 *
620 * Suppose the first term 2s is compute in extra precision. The
621 * dominating error in T3 would then be the rounding error of the
622 * second term 2/3*s**3. To force the rounding bounded by
623 * the required accuracy, we have
624 * single: |2/3*s**3| < 2**-11 ==> |s|<0.09014...
625 * double: |2/3*s**3| < 2**-14 ==> |s|<0.04507...
626 * quad : |2/3*s**3| < 2**-18 ==> |s|<0.01788... = 2**(-5.80..)
627 *
628 * Base on this analysis, we choose Z = {z(i)|z(i)=1+i/64+1/128, 0<=i<=63}.
629 * For any y in [1,2), let j = [64*y] chopped to integer, then z(j) is
630 * the closest to y, and it is not difficult to see that |s| < 2**(-8).
631 * Please note that the polynomial approximation of T3 must be accurate
632 * -24-11 -35 -53-14 -67 -113-18 -131
633 * to 2 =2 , 2 = 2 , and 2 =2
634 * for single, double, and quadruple precision respectively.
635 *
636 * Inplementation notes.
637 * (1) Table look-up entries for T1(n) and T2(j), as well as the calculation
638 * of the leading term 2s in T3, are broken up into leading and trailing
639 * part such that (leading part)* 2**24 will always be an integer. That
640 * will guarantee the addition of the leading parts will be exact.
641 *
642 * 2**(-24)
643 * _________V___________________
644 * T1(n): |_________|___________________|
645 * _______ ______________________
646 * T2(j): |_______|______________________|
647 * ____ _______________________
648 * 2s: |____|_______________________|
649 * __________________________
650 * + T3(s)-2s: |__________________________|
651 * -------------------------------------------
652 * [leading] + [Trailing]
653 *
654 * (2) How to compute 2s accurately.
655 * (A) Compute v = 2s to the working precision. If |v| < 2**(-18),
656 * stop.
657 * (B) chopped v to 2**(-24): v = ((int)(v*2**24))/2**24
658 * (C) 2s = v + (2s - v), where
659 * 1
660 * 2s - v = --- * (2(y-z) - v*(y+z) )
661 * y+z
662 * 1
663 * = --- * ( [2(y-z) - v*(y+z)_h ] - v*(y+z)_l )
664 * y+z
665 * where (y+z)_h = (y+z) rounded to 24 bits by (double)(float),
666 * and (y+z)_l = ((z+z)-(y+z)_h)+(y-z). Note the the quantity
667 * in [] is exact.
668 * 2 4
669 * (3) Remez approximation for (T3(s)-2s)/s = T3[0]*s + T3[1]*s + ...:
670 * Single precision: 1 term (compute in double precision arithmetic)
671 * T3(s) = 2s + S1*s^3, S1 = 0.6666717231848518054693623697539230
672 * Remez error: |T3(s)/s - (2s+S1*s^3)| < 2**(-35.87)
673 * Double precision: 3 terms, Remez error is bounded by 2**(-72.40),
674 * see "tgamma_log"
675 * Quad precision: 7 terms, Remez error is bounded by 2**(-136.54),
676 * see "tgammal_log"
677 *
678 * The computation of 0.5*(ln(2pi)-1):
679 * 0.5*(ln(2pi)-1) = 0.4189385332046727417803297364056176398614...
680 * split 0.5*(ln(2pi)-1) to hln2pi_h + hln2pi_l, where hln2pi_h is the
681 * leading 21 bits of the constant.
682 * hln2pi_h= 0.4189383983612060546875
683 * hln2pi_l= 1.348434666870928297364056176398612173648e-07
684 *
685 * The computation of 1/x*P(1/x^2) = log(G(x))-(x-.5)(ln(x)-1)-(.5ln(2pi)-1):
686 * Let s = 1/x <= 1/8 < 0.125. We have
687 * quad precision
688 * |GP(s) - s*P(s^2)| <= 2**(-120.6), where
689 * 3 5 39
690 * GP(s) = GP0*s+GP1*s +GP2*s +... +GP19*s ,
691 * GP0 = 0.083333333333333333333333333333333172839171301
692 * hex 0x3ffe5555 55555555 55555555 55555548
693 * GP1 = -2.77777777777777777777777777492501211999399424104e-0003
694 * GP2 = 7.93650793650793650793635650541638236350020883243e-0004
695 * GP3 = -5.95238095238095238057299772679324503339241961704e-0004
696 * GP4 = 8.41750841750841696138422987977683524926142600321e-0004
697 * GP5 = -1.91752691752686682825032547823699662178842123308e-0003
698 * GP6 = 6.41025641022403480921891559356473451161279359322e-0003
699 * GP7 = -2.95506535798414019189819587455577003732808185071e-0002
700 * GP8 = 1.79644367229970031486079180060923073476568732136e-0001
701 * GP9 = -1.39243086487274662174562872567057200255649290646e+0000
702 * GP10 = 1.34025874044417962188677816477842265259608269775e+0001
703 * GP11 = -1.56803713480127469414495545399982508700748274318e+0002
704 * GP12 = 2.18739841656201561694927630335099313968924493891e+0003
705 * GP13 = -3.55249848644100338419187038090925410976237921269e+0004
706 * GP14 = 6.43464880437835286216768959439484376449179576452e+0005
707 * GP15 = -1.20459154385577014992600342782821389605893904624e+0007
708 * GP16 = 2.09263249637351298563934942349749718491071093210e+0008
709 * GP17 = -2.96247483183169219343745316433899599834685703457e+0009
710 * GP18 = 2.88984933605896033154727626086506756972327292981e+0010
711 * GP19 = -1.40960434146030007732838382416230610302678063984e+0011
712 *
713 * double precision
714 * |GP(s) - s*P(s^2)| <= 2**(-63.5), where
715 * 3 5 7 9 11 13 15
716 * GP(s) = GP0*s+GP1*s +GP2*s +GP3*s +GP4*s +GP5*s +GP6*s +GP7*s ,
717 *
718 * GP0= 0.0833333333333333287074040640618477 (3FB55555 55555555)
719 * GP1= -2.77777777776649355200565611114627670089130772843e-0003
720 * GP2= 7.93650787486083724805476194170211775784158551509e-0004
721 * GP3= -5.95236628558314928757811419580281294593903582971e-0004
722 * GP4= 8.41566473999853451983137162780427812781178932540e-0004
723 * GP5= -1.90424776670441373564512942038926168175921303212e-0003
724 * GP6= 5.84933161530949666312333949534482303007354299178e-0003
725 * GP7= -1.59453228931082030262124832506144392496561694550e-0002
726 * single precision
727 * |GP(s) - s*P(s^2)| <= 2**(-37.78), where
728 * 3 5
729 * GP(s) = GP0*s+GP1*s +GP2*s
730 * GP0 = 8.33333330959694065245736888749042811909994573178e-0002
731 * GP1 = -2.77765545601667179767706600890361535225507762168e-0003
732 * GP2 = 7.77830853479775281781085278324621033523037489883e-0004
733 *
734 *
735 * Implementation note:
736 * z = (1/x), z2 = z*z, z4 = z2*z2;
737 * p = z*(GP0+z2*(GP1+....+z2*GP7))
738 * = z*(GP0+(z4*(GP2+z4*(GP4+z4*GP6))+z2*(GP1+z4*(GP3+z4*(GP5+z4*GP7)))))
739 *
740 * Adding everything up:
741 * t = rr.h*ww.h+hln2pi_h ... exact
742 * w = (hln2pi_l + ((x-0.5)*ww.l+rr.l*ww.h)) + p
743 *
744 * Computing exp(t+w):
745 * s = t+w; write s = (n+j/32)*ln2+r, |r|<=(1/64)*ln2, then
746 * exp(s) = 2**n * (2**(j/32) + 2**(j/32)*expm1(r)), where
747 * expm1(r) = r + Et1*r^2 + Et2*r^3 + ... + Et5*r^6, and
748 * 2**(j/32) is obtained by table look-up S[j]+S_trail[j].
749 * Remez error bound:
750 * |exp(r) - (1+r+Et1*r^2+...+Et5*r^6)| <= 2^(-63).
751 */
752
753 #include "libm.h"
754
755 #define __HI(x) ((int *) &x)[HIWORD]
756 #define __LO(x) ((unsigned *) &x)[LOWORD]
757
758 struct Double {
759 double h;
760 double l;
761 };
762
763 /* Hex value of GP0 shoule be 3FB55555 55555555 */
764 static const double c[] = {
765 +1.0,
766 +2.0,
767 +0.5,
768 +1.0e-300,
769 +6.66666666666666740682e-01, /* A1=T3[0] */
770 +3.99999999955626478023093908674902212920e-01, /* A2=T3[1] */
771 +2.85720221533145659809237398709372330980e-01, /* A3=T3[2] */
772 +0.0833333333333333287074040640618477, /* GP[0] */
773 -2.77777777776649355200565611114627670089130772843e-03,
774 +7.93650787486083724805476194170211775784158551509e-04,
775 -5.95236628558314928757811419580281294593903582971e-04,
776 +8.41566473999853451983137162780427812781178932540e-04,
777 -1.90424776670441373564512942038926168175921303212e-03,
778 +5.84933161530949666312333949534482303007354299178e-03,
779 -1.59453228931082030262124832506144392496561694550e-02,
780 +4.18937683105468750000e-01, /* hln2pi_h */
781 +8.50099203991780279640e-07, /* hln2pi_l */
782 +4.18938533204672741744150788368695779923320328369e-01, /* hln2pi */
783 +2.16608493865351192653e-02, /* ln2_32hi */
784 +5.96317165397058656257e-12, /* ln2_32lo */
785 +4.61662413084468283841e+01, /* invln2_32 */
786 +5.0000000000000000000e-1, /* Et1 */
787 +1.66666666665223585560605991943703896196054020060e-01, /* Et2 */
788 +4.16666666665895103520154073534275286743788421687e-02, /* Et3 */
789 +8.33336844093536520775865096538773197505523826029e-03, /* Et4 */
790 +1.38889201930843436040204096950052984793587640227e-03, /* Et5 */
791 };
792
793 #define one c[0]
794 #define two c[1]
795 #define half c[2]
796 #define tiny c[3]
797 #define A1 c[4]
798 #define A2 c[5]
799 #define A3 c[6]
800 #define GP0 c[7]
801 #define GP1 c[8]
802 #define GP2 c[9]
803 #define GP3 c[10]
804 #define GP4 c[11]
805 #define GP5 c[12]
806 #define GP6 c[13]
807 #define GP7 c[14]
808 #define hln2pi_h c[15]
809 #define hln2pi_l c[16]
810 #define hln2pi c[17]
811 #define ln2_32hi c[18]
812 #define ln2_32lo c[19]
813 #define invln2_32 c[20]
814 #define Et1 c[21]
815 #define Et2 c[22]
816 #define Et3 c[23]
817 #define Et4 c[24]
818 #define Et5 c[25]
819
820 /*
821 * double precision coefficients for computing log(x)-1 in tgamma.
822 * See "algorithm" for details
823 *
824 * log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
825 * j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
826 * T1(n) = T1[2n,2n+1] = n*log(2)-1,
827 * T2(j) = T2[2j,2j+1] = log(z[j]),
828 * T3(s) = 2s + T3[0]s^3 + T3[1]s^5 + T3[2]s^7
829 * = 2s + A1*s^3 + A2*s^5 + A3*s^7 (see const A1,A2,A3)
830 * Note
831 * (1) the leading entries are truncated to 24 binary point.
832 * See Remezpak/sun/tgamma_log_64.c
833 * (2) Remez error for T3(s) is bounded by 2**(-72.4)
834 * See mpremez/work/Log/tgamma_log_4_outr2
835 */
836
837 static const double T1[] = {
838 -1.00000000000000000000e+00, /* 0xBFF00000 0x00000000 */
839 +0.00000000000000000000e+00, /* 0x00000000 0x00000000 */
840 -3.06852817535400390625e-01, /* 0xBFD3A37A 0x00000000 */
841 -1.90465429995776763166e-09, /* 0xBE205C61 0x0CA86C38 */
842 +3.86294305324554443359e-01, /* 0x3FD8B90B 0xC0000000 */
843 +5.57953361754750897367e-08, /* 0x3E6DF473 0xDE6AF279 */
844 +1.07944148778915405273e+00, /* 0x3FF14564 0x70000000 */
845 +5.38906818755173187963e-08, /* 0x3E6CEEAD 0xCDA06BB5 */
846 +1.77258867025375366211e+00, /* 0x3FFC5C85 0xF0000000 */
847 +5.19860275755595544734e-08, /* 0x3E6BE8E7 0xBCD5E4F2 */
848 +2.46573585271835327148e+00, /* 0x4003B9D3 0xB8000000 */
849 +5.00813732756017835330e-08, /* 0x3E6AE321 0xAC0B5E2E */
850 +3.15888303518295288086e+00, /* 0x40094564 0x78000000 */
851 +4.81767189756440192100e-08, /* 0x3E69DD5B 0x9B40D76B */
852 +3.85203021764755249023e+00, /* 0x400ED0F5 0x38000000 */
853 +4.62720646756862482697e-08, /* 0x3E68D795 0x8A7650A7 */
854 +4.54517740011215209961e+00, /* 0x40122E42 0xFC000000 */
855 +4.43674103757284839467e-08, /* 0x3E67D1CF 0x79ABC9E4 */
856 +5.23832458257675170898e+00, /* 0x4014F40B 0x5C000000 */
857 +4.24627560757707130063e-08, /* 0x3E66CC09 0x68E14320 */
858 +5.93147176504135131836e+00, /* 0x4017B9D3 0xBC000000 */
859 +4.05581017758129486834e-08, /* 0x3E65C643 0x5816BC5D */
860 };
861
862 static const double T2[] = {
863 +7.78210163116455078125e-03, /* 0x3F7FE020 0x00000000 */
864 +3.88108903981662140884e-08, /* 0x3E64D620 0xCF11F86F */
865 +2.31670141220092773438e-02, /* 0x3F97B918 0x00000000 */
866 +4.51595251008850513740e-08, /* 0x3E683EAD 0x88D54940 */
867 +3.83188128471374511719e-02, /* 0x3FA39E86 0x00000000 */
868 +5.14549991480218823411e-08, /* 0x3E6B9FEB 0xD5FA9016 */
869 +5.32444715499877929688e-02, /* 0x3FAB42DC 0x00000000 */
870 +4.29688244898971182165e-08, /* 0x3E671197 0x1BEC28D1 */
871 +6.79506063461303710938e-02, /* 0x3FB16536 0x00000000 */
872 +5.55623773783008185114e-08, /* 0x3E6DD46F 0x5C1D0C4C */
873 +8.24436545372009277344e-02, /* 0x3FB51B07 0x00000000 */
874 +1.46738736635337847313e-08, /* 0x3E4F830C 0x1FB493C7 */
875 +9.67295765876770019531e-02, /* 0x3FB8C345 0x00000000 */
876 +4.98708741103424492282e-08, /* 0x3E6AC633 0x641EB597 */
877 +1.10814332962036132812e-01, /* 0x3FBC5E54 0x00000000 */
878 +3.33782539813823062226e-08, /* 0x3E61EB78 0xE862BAC3 */
879 +1.24703466892242431641e-01, /* 0x3FBFEC91 0x00000000 */
880 +1.16087148042227818450e-08, /* 0x3E48EDF5 0x5D551729 */
881 +1.38402283191680908203e-01, /* 0x3FC1B72A 0x80000000 */
882 +3.96674382274822001957e-08, /* 0x3E654BD9 0xE80A4181 */
883 +1.51916027069091796875e-01, /* 0x3FC371FC 0x00000000 */
884 +1.49567501781968021494e-08, /* 0x3E500F47 0xBA1DE6CB */
885 +1.65249526500701904297e-01, /* 0x3FC526E5 0x80000000 */
886 +4.63946052585787334062e-08, /* 0x3E68E86D 0x0DE8B900 */
887 +1.78407609462738037109e-01, /* 0x3FC6D60F 0x80000000 */
888 +4.80100802600100279538e-08, /* 0x3E69C674 0x8723551E */
889 +1.91394805908203125000e-01, /* 0x3FC87FA0 0x00000000 */
890 +4.70914263296092971436e-08, /* 0x3E694832 0x44240802 */
891 +2.04215526580810546875e-01, /* 0x3FCA23BC 0x00000000 */
892 +1.48478803446288209001e-08, /* 0x3E4FE2B5 0x63193712 */
893 +2.16873884201049804688e-01, /* 0x3FCBC286 0x00000000 */
894 +5.40995645549315919488e-08, /* 0x3E6D0B63 0x358A7E74 */
895 +2.29374051094055175781e-01, /* 0x3FCD5C21 0x00000000 */
896 +4.99707906542102284117e-08, /* 0x3E6AD3EE 0xE456E443 */
897 +2.41719901561737060547e-01, /* 0x3FCEF0AD 0x80000000 */
898 +3.53254081075974352804e-08, /* 0x3E62F716 0x4D948638 */
899 +2.53915190696716308594e-01, /* 0x3FD04025 0x80000000 */
900 +1.92842471355435739091e-08, /* 0x3E54B4D0 0x40DAE27C */
901 +2.65963494777679443359e-01, /* 0x3FD1058B 0xC0000000 */
902 +5.37194584979797487125e-08, /* 0x3E6CD725 0x6A8C4FD0 */
903 +2.77868449687957763672e-01, /* 0x3FD1C898 0xC0000000 */
904 +1.31549854251447496506e-09, /* 0x3E16999F 0xAFBC68E7 */
905 +2.89633274078369140625e-01, /* 0x3FD2895A 0x00000000 */
906 +1.85046735362538929911e-08, /* 0x3E53DE86 0xA35EB493 */
907 +3.01261305809020996094e-01, /* 0x3FD347DD 0x80000000 */
908 +2.47691407849191245052e-08, /* 0x3E5A987D 0x54D64567 */
909 +3.12755703926086425781e-01, /* 0x3FD40430 0x80000000 */
910 +6.07781046260499658610e-09, /* 0x3E3A1A9F 0x8EF4304A */
911 +3.24119448661804199219e-01, /* 0x3FD4BE5F 0x80000000 */
912 +1.99924077768719198045e-08, /* 0x3E557778 0xA0DB4C99 */
913 +3.35355520248413085938e-01, /* 0x3FD57677 0x00000000 */
914 +2.16727247443196802771e-08, /* 0x3E57455A 0x6C549AB7 */
915 +3.46466720104217529297e-01, /* 0x3FD62C82 0xC0000000 */
916 +4.72419910516215900493e-08, /* 0x3E695CE3 0xCA97B7B0 */
917 +3.57455849647521972656e-01, /* 0x3FD6E08E 0x80000000 */
918 +3.92742818015697624778e-08, /* 0x3E6515D0 0xF1C609CA */
919 +3.68325531482696533203e-01, /* 0x3FD792A5 0x40000000 */
920 +2.96760111198451042238e-08, /* 0x3E5FDD47 0xA27C15DA */
921 +3.79078328609466552734e-01, /* 0x3FD842D1 0xC0000000 */
922 +2.43255029056564770289e-08, /* 0x3E5A1E8B 0x17493B14 */
923 +3.89716744422912597656e-01, /* 0x3FD8F11E 0x80000000 */
924 +6.71711261571421332726e-09, /* 0x3E3CD98B 0x1DF85DA7 */
925 +4.00243163108825683594e-01, /* 0x3FD99D95 0x80000000 */
926 +1.01818702333557515008e-09, /* 0x3E117E08 0xACBA92EF */
927 +4.10659909248352050781e-01, /* 0x3FDA4840 0x80000000 */
928 +1.57369163351530571459e-08, /* 0x3E50E5BB 0x0A2BFCA7 */
929 +4.20969247817993164062e-01, /* 0x3FDAF129 0x00000000 */
930 +4.68261364720663662040e-08, /* 0x3E6923BC 0x358899C2 */
931 +4.31173443794250488281e-01, /* 0x3FDB9858 0x80000000 */
932 +2.10241208525779214510e-08, /* 0x3E569310 0xFB598FB1 */
933 +4.41274523735046386719e-01, /* 0x3FDC3DD7 0x80000000 */
934 +3.70698288427707487748e-08, /* 0x3E63E6D6 0xA6B9D9E1 */
935 +4.51274633407592773438e-01, /* 0x3FDCE1AF 0x00000000 */
936 +1.07318658117071930723e-08, /* 0x3E470BE7 0xD6F6FA58 */
937 +4.61175680160522460938e-01, /* 0x3FDD83E7 0x00000000 */
938 +3.49616477054305011286e-08, /* 0x3E62C517 0x9F2828AE */
939 +4.70979690551757812500e-01, /* 0x3FDE2488 0x00000000 */
940 +2.46670332000468969567e-08, /* 0x3E5A7C6C 0x261CBD8F */
941 +4.80688512325286865234e-01, /* 0x3FDEC399 0xC0000000 */
942 +1.70204650424422423704e-08, /* 0x3E52468C 0xC0175CEE */
943 +4.90303933620452880859e-01, /* 0x3FDF6123 0xC0000000 */
944 +5.44247409572909703749e-08, /* 0x3E6D3814 0x5630A2B6 */
945 +4.99827861785888671875e-01, /* 0x3FDFFD2E 0x00000000 */
946 +7.77056065794633071345e-09, /* 0x3E40AFE9 0x30AB2FA0 */
947 +5.09261846542358398438e-01, /* 0x3FE04BDF 0x80000000 */
948 +5.52474495483665749052e-08, /* 0x3E6DA926 0xD265FCC1 */
949 +5.18607735633850097656e-01, /* 0x3FE0986F 0x40000000 */
950 +2.85741955344967264536e-08, /* 0x3E5EAE6A 0x41723FB5 */
951 +5.27867078781127929688e-01, /* 0x3FE0E449 0x80000000 */
952 +1.08397144554263914271e-08, /* 0x3E474732 0x2FDBAB97 */
953 +5.37041425704956054688e-01, /* 0x3FE12F71 0x80000000 */
954 +4.01919275998792285777e-08, /* 0x3E6593EF 0xBC530123 */
955 +5.46132385730743408203e-01, /* 0x3FE179EA 0xA0000000 */
956 +5.18673922421792693237e-08, /* 0x3E6BD899 0xA0BFC60E */
957 +5.55141448974609375000e-01, /* 0x3FE1C3B8 0x00000000 */
958 +5.85658922177154808539e-08, /* 0x3E6F713C 0x24BC94F9 */
959 +5.64070105552673339844e-01, /* 0x3FE20CDC 0xC0000000 */
960 +3.27321296262276338905e-08, /* 0x3E6192AB 0x6D93503D */
961 +5.72919726371765136719e-01, /* 0x3FE2555B 0xC0000000 */
962 +2.71900203723740076878e-08, /* 0x3E5D31EF 0x96780876 */
963 +5.81691682338714599609e-01, /* 0x3FE29D37 0xE0000000 */
964 +5.72959078829112371070e-08, /* 0x3E6EC2B0 0x8AC85CD7 */
965 +5.90387403964996337891e-01, /* 0x3FE2E474 0x20000000 */
966 +4.26371800367512948470e-08, /* 0x3E66E402 0x68405422 */
967 +5.99008142948150634766e-01, /* 0x3FE32B13 0x20000000 */
968 +4.66979327646159769249e-08, /* 0x3E69121D 0x71320557 */
969 +6.07555210590362548828e-01, /* 0x3FE37117 0xA0000000 */
970 +3.96341792466729582847e-08, /* 0x3E654747 0xB5C5DD02 */
971 +6.16029858589172363281e-01, /* 0x3FE3B684 0x40000000 */
972 +1.86263416563663175432e-08, /* 0x3E53FFF8 0x455F1DBE */
973 +6.24433279037475585938e-01, /* 0x3FE3FB5B 0x80000000 */
974 +8.97441791510503832111e-09, /* 0x3E4345BD 0x096D3A75 */
975 +6.32766664028167724609e-01, /* 0x3FE43F9F 0xE0000000 */
976 +5.54287010493641158796e-09, /* 0x3E37CE73 0x3BD393DD */
977 +6.41031146049499511719e-01, /* 0x3FE48353 0xC0000000 */
978 +3.33714317793368531132e-08, /* 0x3E61EA88 0xDF73D5E9 */
979 +6.49227917194366455078e-01, /* 0x3FE4C679 0xA0000000 */
980 +2.94307433638127158696e-08, /* 0x3E5F99DC 0x7362D1DA */
981 +6.57358050346374511719e-01, /* 0x3FE50913 0xC0000000 */
982 +2.23619855184231409785e-08, /* 0x3E5802D0 0xD6979675 */
983 +6.65422618389129638672e-01, /* 0x3FE54B24 0x60000000 */
984 +1.41559608102782173188e-08, /* 0x3E4E6652 0x5EA4550A */
985 +6.73422634601593017578e-01, /* 0x3FE58CAD 0xA0000000 */
986 +4.06105737027198329700e-08, /* 0x3E65CD79 0x893092F2 */
987 +6.81359171867370605469e-01, /* 0x3FE5CDB1 0xC0000000 */
988 +5.29405324634793230630e-08, /* 0x3E6C6C17 0x648CF6E4 */
989 +6.89233243465423583984e-01, /* 0x3FE60E32 0xE0000000 */
990 +3.77733853963405370102e-08, /* 0x3E644788 0xD8CA7C89 */
991 };
992
993 /* S[j],S_trail[j] = 2**(j/32.) for the final computation of exp(t+w) */
994 static const double S[] = {
995 +1.00000000000000000000e+00, /* 3FF0000000000000 */
996 +1.02189714865411662714e+00, /* 3FF059B0D3158574 */
997 +1.04427378242741375480e+00, /* 3FF0B5586CF9890F */
998 +1.06714040067682369717e+00, /* 3FF11301D0125B51 */
999 +1.09050773266525768967e+00, /* 3FF172B83C7D517B */
1000 +1.11438674259589243221e+00, /* 3FF1D4873168B9AA */
1001 +1.13878863475669156458e+00, /* 3FF2387A6E756238 */
1002 +1.16372485877757747552e+00, /* 3FF29E9DF51FDEE1 */
1003 +1.18920711500272102690e+00, /* 3FF306FE0A31B715 */
1004 +1.21524735998046895524e+00, /* 3FF371A7373AA9CB */
1005 +1.24185781207348400201e+00, /* 3FF3DEA64C123422 */
1006 +1.26905095719173321989e+00, /* 3FF44E086061892D */
1007 +1.29683955465100964055e+00, /* 3FF4BFDAD5362A27 */
1008 +1.32523664315974132322e+00, /* 3FF5342B569D4F82 */
1009 +1.35425554693689265129e+00, /* 3FF5AB07DD485429 */
1010 +1.38390988196383202258e+00, /* 3FF6247EB03A5585 */
1011 +1.41421356237309514547e+00, /* 3FF6A09E667F3BCD */
1012 +1.44518080697704665027e+00, /* 3FF71F75E8EC5F74 */
1013 +1.47682614593949934623e+00, /* 3FF7A11473EB0187 */
1014 +1.50916442759342284141e+00, /* 3FF82589994CCE13 */
1015 +1.54221082540794074411e+00, /* 3FF8ACE5422AA0DB */
1016 +1.57598084510788649659e+00, /* 3FF93737B0CDC5E5 */
1017 +1.61049033194925428347e+00, /* 3FF9C49182A3F090 */
1018 +1.64575547815396494578e+00, /* 3FFA5503B23E255D */
1019 +1.68179283050742900407e+00, /* 3FFAE89F995AD3AD */
1020 +1.71861929812247793414e+00, /* 3FFB7F76F2FB5E47 */
1021 +1.75625216037329945351e+00, /* 3FFC199BDD85529C */
1022 +1.79470907500310716820e+00, /* 3FFCB720DCEF9069 */
1023 +1.83400808640934243066e+00, /* 3FFD5818DCFBA487 */
1024 +1.87416763411029996256e+00, /* 3FFDFC97337B9B5F */
1025 +1.91520656139714740007e+00, /* 3FFEA4AFA2A490DA */
1026 +1.95714412417540017941e+00, /* 3FFF50765B6E4540 */
1027 };
1028
1029 static const double S_trail[] = {
1030 +0.00000000000000000000e+00,
1031 +5.10922502897344389359e-17, /* 3C8D73E2A475B465 */
1032 +8.55188970553796365958e-17, /* 3C98A62E4ADC610A */
1033 -7.89985396684158212226e-17, /* BC96C51039449B3A */
1034 -3.04678207981247114697e-17, /* BC819041B9D78A76 */
1035 +1.04102784568455709549e-16, /* 3C9E016E00A2643C */
1036 +8.91281267602540777782e-17, /* 3C99B07EB6C70573 */
1037 +3.82920483692409349872e-17, /* 3C8612E8AFAD1255 */
1038 +3.98201523146564611098e-17, /* 3C86F46AD23182E4 */
1039 -7.71263069268148813091e-17, /* BC963AEABF42EAE2 */
1040 +4.65802759183693679123e-17, /* 3C8ADA0911F09EBC */
1041 +2.66793213134218609523e-18, /* 3C489B7A04EF80D0 */
1042 +2.53825027948883149593e-17, /* 3C7D4397AFEC42E2 */
1043 -2.85873121003886075697e-17, /* BC807ABE1DB13CAC */
1044 +7.70094837980298946162e-17, /* 3C96324C054647AD */
1045 -6.77051165879478628716e-17, /* BC9383C17E40B497 */
1046 -9.66729331345291345105e-17, /* BC9BDD3413B26456 */
1047 -3.02375813499398731940e-17, /* BC816E4786887A99 */
1048 -3.48399455689279579579e-17, /* BC841577EE04992F */
1049 -1.01645532775429503911e-16, /* BC9D4C1DD41532D8 */
1050 +7.94983480969762085616e-17, /* 3C96E9F156864B27 */
1051 -1.01369164712783039808e-17, /* BC675FC781B57EBC */
1052 +2.47071925697978878522e-17, /* 3C7C7C46B071F2BE */
1053 -1.01256799136747726038e-16, /* BC9D2F6EDB8D41E1 */
1054 +8.19901002058149652013e-17, /* 3C97A1CD345DCC81 */
1055 -1.85138041826311098821e-17, /* BC75584F7E54AC3B */
1056 +2.96014069544887330703e-17, /* 3C811065895048DD */
1057 +1.82274584279120867698e-17, /* 3C7503CBD1E949DB */
1058 +3.28310722424562658722e-17, /* 3C82ED02D75B3706 */
1059 -6.12276341300414256164e-17, /* BC91A5CD4F184B5C */
1060 -1.06199460561959626376e-16, /* BC9E9C23179C2893 */
1061 +8.96076779103666776760e-17, /* 3C99D3E12DD8A18B */
1062 };
1063
1064 /* Primary interval GTi() */
1065 static const double cr[] = {
1066 /* p1, q1 */
1067 +0.70908683619977797008004927192814648151397705078125000,
1068 +1.71987061393048558089579513384356441668351720061e-0001,
1069 -3.19273345791990970293320316122813960527705450671e-0002,
1070 +8.36172645419110036267169600390549973563534476989e-0003,
1071 +1.13745336648572838333152213474277971244629758101e-0003,
1072 +1.0,
1073 +9.71980217826032937526460731778472389791321968082e-0001,
1074 -7.43576743326756176594084137256042653497087666030e-0002,
1075 -1.19345944932265559769719470515102012246995255372e-0001,
1076 +1.59913445751425002620935120470781382215050284762e-0002,
1077 +1.12601136853374984566572691306402321911547550783e-0003,
1078 /* p2, q2 */
1079 +0.42848681585558601181418225678498856723308563232421875,
1080 +6.53596762668970816023718845105667418483122103629e-0002,
1081 -6.97280829631212931321050770925128264272768936731e-0003,
1082 +6.46342359021981718947208605674813260166116632899e-0003,
1083 +1.0,
1084 +4.57572620560506047062553957454062012327519313936e-0001,
1085 -2.52182594886075452859655003407796103083422572036e-0001,
1086 -1.82970945407778594681348166040103197178711552827e-0002,
1087 +2.43574726993169566475227642128830141304953840502e-0002,
1088 -5.20390406466942525358645957564897411258667085501e-0003,
1089 +4.79520251383279837635552431988023256031951133885e-0004,
1090 /* p3, q3 */
1091 +0.382409479734567459008331979930517263710498809814453125,
1092 +1.42876048697668161599069814043449301572928034140e-0001,
1093 +3.42157571052250536817923866013561760785748899071e-0003,
1094 -5.01542621710067521405087887856991700987709272937e-0004,
1095 +8.89285814866740910123834688163838287618332122670e-0004,
1096 +1.0,
1097 +3.04253086629444201002215640948957897906299633168e-0001,
1098 -2.23162407379999477282555672834881213873185520006e-0001,
1099 -1.05060867741952065921809811933670131427552903636e-0002,
1100 +1.70511763916186982473301861980856352005926669320e-0002,
1101 -2.12950201683609187927899416700094630764182477464e-0003,
1102 };
1103
1104 #define P10 cr[0]
1105 #define P11 cr[1]
1106 #define P12 cr[2]
1107 #define P13 cr[3]
1108 #define P14 cr[4]
1109 #define Q10 cr[5]
1110 #define Q11 cr[6]
1111 #define Q12 cr[7]
1112 #define Q13 cr[8]
1113 #define Q14 cr[9]
1114 #define Q15 cr[10]
1115 #define P20 cr[11]
1116 #define P21 cr[12]
1117 #define P22 cr[13]
1118 #define P23 cr[14]
1119 #define Q20 cr[15]
1120 #define Q21 cr[16]
1121 #define Q22 cr[17]
1122 #define Q23 cr[18]
1123 #define Q24 cr[19]
1124 #define Q25 cr[20]
1125 #define Q26 cr[21]
1126 #define P30 cr[22]
1127 #define P31 cr[23]
1128 #define P32 cr[24]
1129 #define P33 cr[25]
1130 #define P34 cr[26]
1131 #define Q30 cr[27]
1132 #define Q31 cr[28]
1133 #define Q32 cr[29]
1134 #define Q33 cr[30]
1135 #define Q34 cr[31]
1136 #define Q35 cr[32]
1137
1138 static const double
1139 GZ1_h = +0.938204627909682398190,
1140 GZ1_l = +5.121952600248205157935e-17,
1141 GZ2_h = +0.885603194410888749921,
1142 GZ2_l = -4.964236872556339810692e-17,
1143 GZ3_h = +0.936781411463652347038,
1144 GZ3_l = -2.541923110834479415023e-17,
1145 TZ1 = -0.3517214357852935791015625,
1146 TZ3 = +0.280530631542205810546875;
1147 /* INDENT ON */
1148
1149 /* compute gamma(y=yh+yl) for y in GT1 = [1.0000, 1.2845] */
1150 /* assume yh got 20 significant bits */
1151 static struct Double
1152 GT1(double yh, double yl) {
1153 double t3, t4, y, z;
1154 struct Double r;
1155
1156 y = yh + yl;
1157 z = y * y;
1158 t3 = (z * (P10 + y * ((P11 + y * P12) + z * (P13 + y * P14)))) /
1159 (Q10 + y * ((Q11 + y * Q12) + z * ((Q13 + Q14 * y) + z * Q15)));
1160 t3 += (TZ1 * yl + GZ1_l);
1161 t4 = TZ1 * yh;
1162 r.h = (double) ((float) (t4 + GZ1_h + t3));
1163 t3 += (t4 - (r.h - GZ1_h));
1164 r.l = t3;
1165 return (r);
1166 }
1167
1168 /* compute gamma(y=yh+yl) for y in GT2 = [1.2844, 1.6374] */
1169 /* assume yh got 20 significant bits */
1170 static struct Double
1171 GT2(double yh, double yl) {
1172 double t3, y, z;
1173 struct Double r;
1174
1175 y = yh + yl;
1176 z = y * y;
1177 t3 = (z * (P20 + y * P21 + z * (P22 + y * P23))) /
1178 (Q20 + (y * ((Q21 + Q22 * y) + z * Q23) +
1179 (z * z) * ((Q24 + Q25 * y) + z * Q26))) + GZ2_l;
1180 r.h = (double) ((float) (GZ2_h + t3));
1181 r.l = t3 - (r.h - GZ2_h);
1182 return (r);
1183 }
1184
1185 /* compute gamma(y=yh+yl) for y in GT3 = [1.6373, 2.0000] */
1186 /* assume yh got 20 significant bits */
1187 static struct Double
1188 GT3(double yh, double yl) {
1189 double t3, t4, y, z;
1190 struct Double r;
1191
1192 y = yh + yl;
1193 z = y * y;
1194 t3 = (z * (P30 + y * ((P31 + y * P32) + z * (P33 + y * P34)))) /
1195 (Q30 + y * ((Q31 + y * Q32) + z * ((Q33 + Q34 * y) + z * Q35)));
1196 t3 += (TZ3 * yl + GZ3_l);
1197 t4 = TZ3 * yh;
1198 r.h = (double) ((float) (t4 + GZ3_h + t3));
1199 t3 += (t4 - (r.h - GZ3_h));
1200 r.l = t3;
1201 return (r);
1202 }
1203
1204 /* INDENT OFF */
1205 /*
1206 * return tgamma(x) scaled by 2**-m for 8<x<=171.62... using Stirling's formula
1207 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
1208 * = L1 + L2 + L3,
1209 */
1210 /* INDENT ON */
1211 static struct Double
1212 large_gam(double x, int *m) {
1213 double z, t1, t2, t3, z2, t5, w, y, u, r, z4, v, t24 = 16777216.0,
1214 p24 = 1.0 / 16777216.0;
1215 int n2, j2, k, ix, j;
1216 unsigned lx;
1217 struct Double zz;
1218 double u2, ss_h, ss_l, r_h, w_h, w_l, t4;
1219
1220 /* INDENT OFF */
1221 /*
1222 * compute ss = ss.h+ss.l = log(x)-1 (see tgamma_log.h for details)
1223 *
1224 * log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
1225 * j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
1226 * T1(n) = T1[2n,2n+1] = n*log(2)-1,
1227 * T2(j) = T2[2j,2j+1] = log(z[j]),
1228 * T3(s) = 2s + A1[0]s^3 + A2[1]s^5 + A3[2]s^7
1229 * Note
1230 * (1) the leading entries are truncated to 24 binary point.
1231 * (2) Remez error for T3(s) is bounded by 2**(-72.4)
1232 * 2**(-24)
1233 * _________V___________________
1234 * T1(n): |_________|___________________|
1235 * _______ ______________________
1236 * T2(j): |_______|______________________|
1237 * ____ _______________________
1238 * 2s: |____|_______________________|
1239 * __________________________
1240 * + T3(s)-2s: |__________________________|
1241 * -------------------------------------------
1242 * [leading] + [Trailing]
1243 */
1244 /* INDENT ON */
1245 ix = __HI(x);
1246 lx = __LO(x);
1247 n2 = (ix >> 20) - 0x3ff; /* exponent of x, range:3-7 */
1248 n2 += n2; /* 2n */
1249 ix = (ix & 0x000fffff) | 0x3ff00000; /* y = scale x to [1,2] */
1250 __HI(y) = ix;
1251 __LO(y) = lx;
1252 __HI(z) = (ix & 0xffffc000) | 0x2000; /* z[j]=1+j/64+1/128 */
1253 __LO(z) = 0;
1254 j2 = (ix >> 13) & 0x7e; /* 2j */
1255 t1 = y + z;
1256 t2 = y - z;
1257 r = one / t1;
1258 t1 = (double) ((float) t1);
1259 u = r * t2; /* u = (y-z)/(y+z) */
1260 t4 = T2[j2 + 1] + T1[n2 + 1];
1261 z2 = u * u;
1262 k = __HI(u) & 0x7fffffff;
1263 t3 = T2[j2] + T1[n2];
1264 if ((k >> 20) < 0x3ec) { /* |u|<2**-19 */
1265 t2 = t4 + u * ((two + z2 * A1) + (z2 * z2) * (A2 + z2 * A3));
1266 } else {
1267 t5 = t4 + u * (z2 * A1 + (z2 * z2) * (A2 + z2 * A3));
1268 u2 = u + u;
1269 v = (double) ((int) (u2 * t24)) * p24;
1270 t2 = t5 + r * ((two * t2 - v * t1) - v * (y - (t1 - z)));
1271 t3 += v;
1272 }
1273 ss_h = (double) ((float) (t2 + t3));
1274 ss_l = t2 - (ss_h - t3);
1275
1276 /*
1277 * compute ww = (x-.5)*(log(x)-1) + .5*(log(2pi)-1) + 1/x*(P(1/x^2)))
1278 * where ss = log(x) - 1 in already in extra precision
1279 */
1280 z = one / x;
1281 r = x - half;
1282 r_h = (double) ((float) r);
1283 w_h = r_h * ss_h + hln2pi_h;
1284 z2 = z * z;
1285 w = (r - r_h) * ss_h + r * ss_l;
1286 z4 = z2 * z2;
1287 t1 = z2 * (GP1 + z4 * (GP3 + z4 * (GP5 + z4 * GP7)));
1288 t2 = z4 * (GP2 + z4 * (GP4 + z4 * GP6));
1289 t1 += t2;
1290 w += hln2pi_l;
1291 w_l = z * (GP0 + t1) + w;
1292 k = (int) ((w_h + w_l) * invln2_32 + half);
1293
1294 /* compute the exponential of w_h+w_l */
1295 j = k & 0x1f;
1296 *m = (k >> 5);
1297 t3 = (double) k;
1298
1299 /* perform w - k*ln2_32 (represent as w_h - w_l) */
1300 t1 = w_h - t3 * ln2_32hi;
1301 t2 = t3 * ln2_32lo;
1302 w = w_l - t2;
1303 w_h = t1 + w_l;
1304 w_l = t2 - (w_l - (w_h - t1));
1305
1306 /* compute exp(w_h+w_l) */
1307 z = w_h - w_l;
1308 z2 = z * z;
1309 t1 = z2 * (Et1 + z2 * (Et3 + z2 * Et5));
1310 t2 = z2 * (Et2 + z2 * Et4);
1311 t3 = w_h - (w_l - (t1 + z * t2));
1312 zz.l = S_trail[j] * (one + t3) + S[j] * t3;
1313 zz.h = S[j];
1314 return (zz);
1315 }
1316
1317 /* INDENT OFF */
1318 /*
1319 * kpsin(x)= sin(pi*x)/pi
1320 * 3 5 7 9 11 13 15
1321 * = x+ks[0]*x +ks[1]*x +ks[2]*x +ks[3]*x +ks[4]*x +ks[5]*x +ks[6]*x
1322 */
1323 static const double ks[] = {
1324 -1.64493406684822640606569,
1325 +8.11742425283341655883668741874008920850698590621e-0001,
1326 -1.90751824120862873825597279118304943994042258291e-0001,
1327 +2.61478477632554278317289628332654539353521911570e-0002,
1328 -2.34607978510202710377617190278735525354347705866e-0003,
1329 +1.48413292290051695897242899977121846763824221705e-0004,
1330 -6.87730769637543488108688726777687262485357072242e-0006,
1331 };
1332 /* INDENT ON */
1333
1334 /* assume x is not tiny and positive */
1335 static struct Double
1336 kpsin(double x) {
1337 double z, t1, t2, t3, t4;
1338 struct Double xx;
1339
1340 z = x * x;
1341 xx.h = x;
1342 t1 = z * x;
1343 t2 = z * z;
1344 t4 = t1 * ks[0];
1345 t3 = (t1 * z) * ((ks[1] + z * ks[2] + t2 * ks[3]) + (z * t2) *
1346 (ks[4] + z * ks[5] + t2 * ks[6]));
1347 xx.l = t4 + t3;
1348 return (xx);
1349 }
1350
1351 /* INDENT OFF */
1352 /*
1353 * kpcos(x)= cos(pi*x)/pi
1354 * 2 4 6 8 10 12
1355 * = 1/pi +kc[0]*x +kc[1]*x +kc[2]*x +kc[3]*x +kc[4]*x +kc[5]*x
1356 */
1357
1358 static const double one_pi_h = 0.318309886183790635705292970,
1359 one_pi_l = 3.583247455607534006714276420e-17;
1360 static const double npi_2_h = -1.5625,
1361 npi_2_l = -0.00829632679489661923132169163975055099555883223;
1362 static const double kc[] = {
1363 -1.57079632679489661923132169163975055099555883223e+0000,
1364 +1.29192819501230224953283586722575766189551966008e+0000,
1365 -4.25027339940149518500158850753393173519732149213e-0001,
1366 +7.49080625187015312373925142219429422375556727752e-0002,
1367 -8.21442040906099210866977352284054849051348692715e-0003,
1368 +6.10411356829515414575566564733632532333904115968e-0004,
1369 };
1370 /* INDENT ON */
1371
1372 /* assume x is not tiny and positive */
1373 static struct Double
1374 kpcos(double x) {
1375 double z, t1, t2, t3, t4, x4, x8;
1376 struct Double xx;
1377
1378 z = x * x;
1379 xx.h = one_pi_h;
1380 t1 = (double) ((float) x);
1381 x4 = z * z;
1382 t2 = npi_2_l * z + npi_2_h * (x + t1) * (x - t1);
1383 t3 = one_pi_l + x4 * ((kc[1] + z * kc[2]) + x4 * (kc[3] + z *
1384 kc[4] + x4 * kc[5]));
1385 t4 = t1 * t1; /* 48 bits mantissa */
1386 x8 = t2 + t3;
1387 t4 *= npi_2_h; /* npi_2_h is 5 bits const. The product is exact */
1388 xx.l = x8 + t4; /* that will minimized the rounding error in xx.l */
1389 return (xx);
1390 }
1391
1392 /* INDENT OFF */
1393 static const double
1394 /* 0.134861805732790769689793935774652917006 */
1395 t0z1 = 0.1348618057327907737708,
1396 t0z1_l = -4.0810077708578299022531e-18,
1397 /* 0.461632144968362341262659542325721328468 */
1398 t0z2 = 0.4616321449683623567850,
1399 t0z2_l = -1.5522348162858676890521e-17,
1400 /* 0.819773101100500601787868704921606996312 */
1401 t0z3 = 0.8197731011005006118708,
1402 t0z3_l = -1.0082945122487103498325e-17;
1403 /* 1.134861805732790769689793935774652917006 */
1404 /* INDENT ON */
1405
1406 /* gamma(x+i) for 0 <= x < 1 */
1407 static struct Double
1408 gam_n(int i, double x) {
1409 struct Double rr = {0.0L, 0.0L}, yy;
1410 double r1, r2, t2, z, xh, xl, yh, yl, zh, z1, z2, zl, x5, wh, wl;
1411
1412 /* compute yy = gamma(x+1) */
1413 if (x > 0.2845) {
1414 if (x > 0.6374) {
1415 r1 = x - t0z3;
1416 r2 = (double) ((float) (r1 - t0z3_l));
1417 t2 = r1 - r2;
1418 yy = GT3(r2, t2 - t0z3_l);
1419 } else {
1420 r1 = x - t0z2;
1421 r2 = (double) ((float) (r1 - t0z2_l));
1422 t2 = r1 - r2;
1423 yy = GT2(r2, t2 - t0z2_l);
1424 }
1425 } else {
1426 r1 = x - t0z1;
1427 r2 = (double) ((float) (r1 - t0z1_l));
1428 t2 = r1 - r2;
1429 yy = GT1(r2, t2 - t0z1_l);
1430 }
1431
1432 /* compute gamma(x+i) = (x+i-1)*...*(x+1)*yy, 0<i<8 */
1433 switch (i) {
1434 case 0: /* yy/x */
1435 r1 = one / x;
1436 xh = (double) ((float) x); /* x is not tiny */
1437 rr.h = (double) ((float) ((yy.h + yy.l) * r1));
1438 rr.l = r1 * (yy.h - rr.h * xh) -
1439 ((r1 * rr.h) * (x - xh) - r1 * yy.l);
1440 break;
1441 case 1: /* yy */
1442 rr.h = yy.h;
1443 rr.l = yy.l;
1444 break;
1445 case 2: /* (x+1)*yy */
1446 z = x + one; /* may not be exact */
1447 zh = (double) ((float) z);
1448 rr.h = zh * yy.h;
1449 rr.l = z * yy.l + (x - (zh - one)) * yy.h;
1450 break;
1451 case 3: /* (x+2)*(x+1)*yy */
1452 z1 = x + one;
1453 z2 = x + 2.0;
1454 z = z1 * z2;
1455 xh = (double) ((float) z);
1456 zh = (double) ((float) z1);
1457 xl = (x - (zh - one)) * (z2 + zh) - (xh - zh * (zh + one));
1458 rr.h = xh * yy.h;
1459 rr.l = z * yy.l + xl * yy.h;
1460 break;
1461
1462 case 4: /* (x+1)*(x+3)*(x+2)*yy */
1463 z1 = x + 2.0;
1464 z2 = (x + one) * (x + 3.0);
1465 zh = z1;
1466 __LO(zh) = 0;
1467 __HI(zh) &= 0xfffffff8; /* zh 18 bits mantissa */
1468 zl = x - (zh - 2.0);
1469 z = z1 * z2;
1470 xh = (double) ((float) z);
1471 xl = zl * (z2 + zh * (z1 + zh)) - (xh - zh * (zh * zh - one));
1472 rr.h = xh * yy.h;
1473 rr.l = z * yy.l + xl * yy.h;
1474 break;
1475 case 5: /* ((x+1)*(x+4)*(x+2)*(x+3))*yy */
1476 z1 = x + 2.0;
1477 z2 = x + 3.0;
1478 z = z1 * z2;
1479 zh = (double) ((float) z1);
1480 yh = (double) ((float) z);
1481 yl = (x - (zh - 2.0)) * (z2 + zh) - (yh - zh * (zh + one));
1482 z2 = z - 2.0;
1483 z *= z2;
1484 xh = (double) ((float) z);
1485 xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0));
1486 rr.h = xh * yy.h;
1487 rr.l = z * yy.l + xl * yy.h;
1488 break;
1489 case 6: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5))*yy */
1490 z1 = x + 2.0;
1491 z2 = x + 3.0;
1492 z = z1 * z2;
1493 zh = (double) ((float) z1);
1494 yh = (double) ((float) z);
1495 z1 = x - (zh - 2.0);
1496 yl = z1 * (z2 + zh) - (yh - zh * (zh + one));
1497 z2 = z - 2.0;
1498 x5 = x + 5.0;
1499 z *= z2;
1500 xh = (double) ((float) z);
1501 zh += 3.0;
1502 xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0));
1503 /* xh+xl=(x+1)*...*(x+4) */
1504 /* wh+wl=(x+5)*yy */
1505 wh = (double) ((float) (x5 * (yy.h + yy.l)));
1506 wl = (z1 * yy.h + x5 * yy.l) - (wh - zh * yy.h);
1507 rr.h = wh * xh;
1508 rr.l = z * wl + xl * wh;
1509 break;
1510 case 7: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6))*yy */
1511 z1 = x + 3.0;
1512 z2 = x + 4.0;
1513 z = z2 * z1;
1514 zh = (double) ((float) z1);
1515 yh = (double) ((float) z); /* yh+yl = (x+3)(x+4) */
1516 yl = (x - (zh - 3.0)) * (z2 + zh) - (yh - (zh * (zh + one)));
1517 z1 = x + 6.0;
1518 z2 = z - 2.0; /* z2 = (x+2)*(x+5) */
1519 z *= z2;
1520 xh = (double) ((float) z);
1521 xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0));
1522 /* xh+xl=(x+2)*...*(x+5) */
1523 /* wh+wl=(x+1)(x+6)*yy */
1524 z2 -= 4.0; /* z2 = (x+1)(x+6) */
1525 wh = (double) ((float) (z2 * (yy.h + yy.l)));
1526 wl = (z2 * yy.l + yl * yy.h) - (wh - (yh - 6.0) * yy.h);
1527 rr.h = wh * xh;
1528 rr.l = z * wl + xl * wh;
1529 }
1530 return (rr);
1531 }
1532
1533 double
1534 tgamma(double x) {
1535 struct Double ss, ww;
1536 double t, t1, t2, t3, t4, t5, w, y, z, z1, z2, z3, z5;
1537 int i, j, k, m, ix, hx, xk;
1538 unsigned lx;
1539
1540 hx = __HI(x);
1541 lx = __LO(x);
1542 ix = hx & 0x7fffffff;
1543 y = x;
1544
1545 if (ix < 0x3ca00000)
1546 return (one / x); /* |x| < 2**-53 */
1547 if (ix >= 0x7ff00000)
1548 /* +Inf -> +Inf, -Inf or NaN -> NaN */
1549 return (x * ((hx < 0)? 0.0 : x));
1550 if (hx > 0x406573fa || /* x > 171.62... overflow to +inf */
1551 (hx == 0x406573fa && lx > 0xE561F647)) {
1552 z = x / tiny;
1553 return (z * z);
1554 }
1555 if (hx >= 0x40200000) { /* x >= 8 */
1556 ww = large_gam(x, &m);
1557 w = ww.h + ww.l;
1558 __HI(w) += m << 20;
1559 return (w);
1560 }
1561 if (hx > 0) { /* 0 < x < 8 */
1562 i = (int) x;
1563 ww = gam_n(i, x - (double) i);
1564 return (ww.h + ww.l);
1565 }
1566
1567 /* negative x */
1568 /* INDENT OFF */
1569 /*
1570 * compute: xk =
1571 * -2 ... x is an even int (-inf is even)
1572 * -1 ... x is an odd int
1573 * +0 ... x is not an int but chopped to an even int
1574 * +1 ... x is not an int but chopped to an odd int
1575 */
1576 /* INDENT ON */
1577 xk = 0;
1578 if (ix >= 0x43300000) {
1579 if (ix >= 0x43400000)
1580 xk = -2;
1581 else
1582 xk = -2 + (lx & 1);
1583 } else if (ix >= 0x3ff00000) {
1584 k = (ix >> 20) - 0x3ff;
1585 if (k > 20) {
1586 j = lx >> (52 - k);
1587 if ((j << (52 - k)) == lx)
1588 xk = -2 + (j & 1);
1589 else
1590 xk = j & 1;
1591 } else {
1592 j = ix >> (20 - k);
1593 if ((j << (20 - k)) == ix && lx == 0)
1594 xk = -2 + (j & 1);
1595 else
1596 xk = j & 1;
1597 }
1598 }
1599 if (xk < 0)
1600 /* ideally gamma(-n)= (-1)**(n+1) * inf, but c99 expect NaN */
1601 return ((x - x) / (x - x)); /* 0/0 = NaN */
1602
1603
1604 /* negative underflow thresold */
1605 if (ix > 0x4066e000 || (ix == 0x4066e000 && lx > 11)) {
1606 /* x < -183.0 - 11ulp */
1607 z = tiny / x;
1608 if (xk == 1)
1609 z = -z;
1610 return (z * tiny);
1611 }
1612
1613 /* now compute gamma(x) by -1/((sin(pi*y)/pi)*gamma(1+y)), y = -x */
1614
1615 /*
1616 * First compute ss = -sin(pi*y)/pi , so that
1617 * gamma(x) = 1/(ss*gamma(1+y))
1618 */
1619 y = -x;
1620 j = (int) y;
1621 z = y - (double) j;
1622 if (z > 0.3183098861837906715377675)
1623 if (z > 0.6816901138162093284622325)
1624 ss = kpsin(one - z);
1625 else
1626 ss = kpcos(0.5 - z);
1627 else
1628 ss = kpsin(z);
1629 if (xk == 0) {
1630 ss.h = -ss.h;
1631 ss.l = -ss.l;
1632 }
1633
1634 /* Then compute ww = gamma(1+y), note that result scale to 2**m */
1635 m = 0;
1636 if (j < 7) {
1637 ww = gam_n(j + 1, z);
1638 } else {
1639 w = y + one;
1640 if ((lx & 1) == 0) { /* y+1 exact (note that y<184) */
1641 ww = large_gam(w, &m);
1642 } else {
1643 t = w - one;
1644 if (t == y) { /* y+one exact */
1645 ww = large_gam(w, &m);
1646 } else { /* use y*gamma(y) */
1647 if (j == 7)
1648 ww = gam_n(j, z);
1649 else
1650 ww = large_gam(y, &m);
1651 t4 = ww.h + ww.l;
1652 t1 = (double) ((float) y);
1653 t2 = (double) ((float) t4);
1654 /* t4 will not be too large */
1655 ww.l = y * (ww.l - (t2 - ww.h)) + (y - t1) * t2;
1656 ww.h = t1 * t2;
1657 }
1658 }
1659 }
1660
1661 /* compute 1/(ss*ww) */
1662 t3 = ss.h + ss.l;
1663 t4 = ww.h + ww.l;
1664 t1 = (double) ((float) t3);
1665 t2 = (double) ((float) t4);
1666 z1 = ss.l - (t1 - ss.h); /* (t1,z1) = ss */
1667 z2 = ww.l - (t2 - ww.h); /* (t2,z2) = ww */
1668 t3 = t3 * t4; /* t3 = ss*ww */
1669 z3 = one / t3; /* z3 = 1/(ss*ww) */
1670 t5 = t1 * t2;
1671 z5 = z1 * t4 + t1 * z2; /* (t5,z5) = ss*ww */
1672 t1 = (double) ((float) t3); /* (t1,z1) = ss*ww */
1673 z1 = z5 - (t1 - t5);
1674 t2 = (double) ((float) z3); /* leading 1/(ss*ww) */
1675 z2 = z3 * (t2 * z1 - (one - t2 * t1));
1676 z = t2 - z2;
1677
1678 /* check whether z*2**-m underflow */
1679 if (m != 0) {
1680 hx = __HI(z);
1681 i = hx & 0x80000000;
1682 ix = hx ^ i;
1683 j = ix >> 20;
1684 if (j > m) {
1685 ix -= m << 20;
1686 __HI(z) = ix ^ i;
1687 } else if ((m - j) > 52) {
1688 /* underflow */
1689 if (xk == 0)
1690 z = -tiny * tiny;
1691 else
1692 z = tiny * tiny;
1693 } else {
1694 /* subnormal */
1695 m -= 60;
1696 t = one;
1697 __HI(t) -= 60 << 20;
1698 ix -= m << 20;
1699 __HI(z) = ix ^ i;
1700 z *= t;
1701 }
1702 }
1703 return (z);
1704 }