/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* or http://www.opensolaris.org/os/licensing.
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
#pragma weak __tgamma = tgamma
/* INDENT OFF */
/*
* True gamma function
* double tgamma(double x)
*
* Error:
* ------
* Less that one ulp for both positive and negative arguments.
*
* Algorithm:
* ---------
* A: For negative argument
* (1) gamma(-n or -inf) is NaN
* (2) Underflow Threshold
* (3) Reduction to gamma(1+x)
* B: For x between 1 and 2
* C: For x between 0 and 1
* D: For x between 2 and 8
* E: Overflow thresold {see over.c}
* F: For overflow_threshold >= x >= 8
*
* Implementation details
* -----------------------
* -pi
* (A) For negative argument, use gamma(-x) = ------------------------.
* (sin(pi*x)*gamma(1+x))
*
* (1) gamma(-n or -inf) is NaN with invalid signal by SUSv3 spec.
* (Ideally, gamma(-n) = 1/sinpi(n) = (-1)**(n+1) * inf.)
*
* (2) Underflow Threshold. For each precision, there is a value T
* such that when x>T and when x is not an integer, gamma(-x) will
* always underflow. A table of the underflow threshold value is given
* below. For proof, see file "under.c".
*
* Precision underflow threshold T =
* ----------------------------------------------------------------------
* single 41.000041962 = 41 + 11 ULP
* (machine format) 4224000B
* double 183.000000000000312639 = 183 + 11 ULP
* (machine format) 4066E000 0000000B
* quad 1774.0000000000000000000000000000017749370 = 1774 + 9 ULP
* (machine format) 4009BB80000000000000000000000009
* ----------------------------------------------------------------------
*
* (3) Reduction to gamma(1+x).
* Because of (1) and (2), we need only consider non-integral x
* such that 0<x<T. Let k = [x] and z = x-[x]. Define
* sin(x*pi) cos(x*pi)
* kpsin(x) = --------- and kpcos(x) = --------- . Then
* pi pi
* 1
* gamma(-x) = --------------------.
* -kpsin(x)*gamma(1+x)
* Since x = k+z,
* k+1
* -sin(x*pi) = -sin(k*pi+z*pi) = (-1) *sin(z*pi),
* k+1
* we have -kpsin(x) = (-1) * kpsin(z). We can further
* reduce z to t by
* (I) t = z when 0.00000 <= z < 0.31830...
* (II) t = 0.5-z when 0.31830... <= z < 0.681690...
* (III) t = 1-z when 0.681690... <= z < 1.00000
* and correspondingly
* (I) kpsin(z) = kpsin(t) ... 0<= z < 0.3184
* (II) kpsin(z) = kpcos(t) ... |t| < 0.182
* (III) kpsin(z) = kpsin(t) ... 0<= t < 0.3184
*
* Using a special Remez algorithm, we obtain the following polynomial
* approximation for kpsin(t) for 0<=t<0.3184:
*
* Computation note: in simulating higher precision arithmetic, kcpsin
* return head = t and tail = ks[0]*t^3 + (...) to maintain extra bits.
*
* Quad precision, remez error <= 2**(-129.74)
* 3 5 27
* kpsin(t) = t + ks[0] * t + ks[1] * t + ... + ks[12] * t
*
* ks[ 0] = -1.64493406684822643647241516664602518705158902870e+0000
* ks[ 1] = 8.11742425283353643637002772405874238094995726160e-0001
* ks[ 2] = -1.90751824122084213696472111835337366232282723933e-0001
* ks[ 3] = 2.61478478176548005046532613563241288115395517084e-0002
* ks[ 4] = -2.34608103545582363750893072647117829448016479971e-0003
* ks[ 5] = 1.48428793031071003684606647212534027556262040158e-0004
* ks[ 6] = -6.97587366165638046518462722252768122615952898698e-0006
* ks[ 7] = 2.53121740413702536928659271747187500934840057929e-0007
* ks[ 8] = -7.30471182221385990397683641695766121301933621956e-0009
* ks[ 9] = 1.71653847451163495739958249695549313987973589884e-0010
* ks[10] = -3.34813314714560776122245796929054813458341420565e-0012
* ks[11] = 5.50724992262622033449487808306969135431411753047e-0014
* ks[12] = -7.67678132753577998601234393215802221104236979928e-0016
*
* Double precision, Remez error <= 2**(-62.9)
* 3 5 15
* kpsin(t) = t + ks[0] * t + ks[1] * t + ... + ks[6] * t
*
* ks[0] = -1.644934066848226406065691 (0x3ffa51a6 625307d3)
* ks[1] = 8.11742425283341655883668741874008920850698590621e-0001
* ks[2] = -1.90751824120862873825597279118304943994042258291e-0001
* ks[3] = 2.61478477632554278317289628332654539353521911570e-0002
* ks[4] = -2.34607978510202710377617190278735525354347705866e-0003
* ks[5] = 1.48413292290051695897242899977121846763824221705e-0004
* ks[6] = -6.87730769637543488108688726777687262485357072242e-0006
*
* Single precision, Remez error <= 2**(-34.09)
* 3 5 9
* kpsin(t) = t + ks[0] * t + ks[1] * t + ... + ks[3] * t
*
* ks[0] = -1.64493404985645811354476665052005342839447790544e+0000
* ks[1] = 8.11740794458351064092797249069438269367389272270e-0001
* ks[2] = -1.90703144603551216933075809162889536878854055202e-0001
* ks[3] = 2.55742333994264563281155312271481108635575331201e-0002
*
* Computation note: in simulating higher precision arithmetic, kcpsin
* return head = t and tail = kc[0]*t^3 + (...) to maintain extra bits
* precision.
*
* And for kpcos(t) for |t|< 0.183:
*
* Quad precision, remez <= 2**(-122.48)
* 2 4 22
* kpcos(t) = 1/pi + pi/2 * t + kc[2] * t + ... + kc[11] * t
*
* kc[2] = 1.29192819501249250731151312779548918765320728489e+0000
* kc[3] = -4.25027339979557573976029596929319207009444090366e-0001
* kc[4] = 7.49080661650990096109672954618317623888421628613e-0002
* kc[5] = -8.21458866111282287985539464173976555436050215120e-0003
* kc[6] = 6.14202578809529228503205255165761204750211603402e-0004
* kc[7] = -3.33073432691149607007217330302595267179545908740e-0005
* kc[8] = 1.36970959047832085796809745461530865597993680204e-0006
* kc[9] = -4.41780774262583514450246512727201806217271097336e-0008
* kc[10]= 1.14741409212381858820016567664488123478660705759e-0009
* kc[11]= -2.44261236114707374558437500654381006300502749632e-0011
*
* Double precision, remez < 2**(61.91)
* 2 4 12
* kpcos(t) = 1/pi + pi/2 *t + kc[2] * t + ... + kc[6] * t
*
* kc[2] = 1.29192819501230224953283586722575766189551966008e+0000
* kc[3] = -4.25027339940149518500158850753393173519732149213e-0001
* kc[4] = 7.49080625187015312373925142219429422375556727752e-0002
* kc[5] = -8.21442040906099210866977352284054849051348692715e-0003
* kc[6] = 6.10411356829515414575566564733632532333904115968e-0004
*
* Single precision, remez < 2**(-30.13)
* 2 6
* kpcos(t) = kc[0] + kc[1] * t + ... + kc[3] * t
*
* kc[0] = 3.18309886183790671537767526745028724068919291480e-0001
* kc[1] = -1.57079581447762568199467875065854538626594937791e+0000
* kc[2] = 1.29183528092558692844073004029568674027807393862e+0000
* kc[3] = -4.20232949771307685981015914425195471602739075537e-0001
*
* Computation note: in simulating higher precision arithmetic, kcpcos
* return head = 1/pi chopped, and tail = pi/2 *t^2 + (tail part of 1/pi
* + ...) to maintain extra bits precision. In particular, pi/2 * t^2
* is calculated with great care.
*
* Thus, the computation of gamma(-x), x>0, is:
* Let k = int(x), z = x-k.
* For z in (I)
* k+1
* (-1)
* gamma(-x) = ------------------- ;
* kpsin(z)*gamma(1+x)
*
* otherwise, for z in (II),
* k+1
* (-1)
* gamma(-x) = ----------------------- ;
* kpcos(0.5-z)*gamma(1+x)
*
* otherwise, for z in (III),
* k+1
* (-1)
* gamma(-x) = --------------------- .
* kpsin(1-z)*gamma(1+x)
*
* Thus, the computation of gamma(-x) reduced to the computation of
* gamma(1+x) and kpsin(), kpcos().
*
* (B) For x between 1 and 2. We break [1,2] into three parts:
* GT1 = [1.0000, 1.2845]
* GT2 = [1.2844, 1.6374]
* GT3 = [1.6373, 2.0000]
*
* For x in GTi, i=1,2,3, let
* z1 = 1.134861805732790769689793935774652917006
* gz1 = gamma(z1) = 0.9382046279096824494097535615803269576988
* tz1 = gamma'(z1) = -0.3517214357852935791015625000000000000000
*
* z2 = 1.461632144968362341262659542325721328468e+0000
* gz2 = gamma(z2) = 0.8856031944108887002788159005825887332080
* tz2 = gamma'(z2) = 0.00
*
* z3 = 1.819773101100500601787868704921606996312e+0000
* gz3 = gamma(z3) = 0.9367814114636523216188468970808378497426
* tz3 = gamma'(z3) = 0.2805306315422058105468750000000000000000
*
* and
* y = x-zi ... for extra precision, write y = y.h + y.l
* Then
* gamma(x) = gzi + tzi*(y.h+y.l) + y*y*Ri(y),
* = gzi.h + (tzi*y.h + ((tzi*y.l+gzi.l) + y*y*Ri(y)))
* = gy.h + gy.l
* where
* (I) For double precision
*
* Ri(y) = Pi(y)/Qi(y), i=1,2,3;
*
* P1(y) = p1[0] + p1[1]*y + ... + p1[4]*y^4
* Q1(y) = q1[0] + q1[1]*y + ... + q1[5]*y^5
*
* P2(y) = p2[0] + p2[1]*y + ... + p2[3]*y^3
* Q2(y) = q2[0] + q2[1]*y + ... + q2[6]*y^6
*
* P3(y) = p3[0] + p3[1]*y + ... + p3[4]*y^4
* Q3(y) = q3[0] + q3[1]*y + ... + q3[5]*y^5
*
* Remez precision of Ri(y):
* |gamma(x)-(gzi+tzi*y) - y*y*Ri(y)| <= 2**-62.3 ... for i = 1
* <= 2**-59.4 ... for i = 2
* <= 2**-62.1 ... for i = 3
*
* (II) For quad precision
*
* Ri(y) = Pi(y)/Qi(y), i=1,2,3;
*
* P1(y) = p1[0] + p1[1]*y + ... + p1[9]*y^9
* Q1(y) = q1[0] + q1[1]*y + ... + q1[8]*y^8
*
* P2(y) = p2[0] + p2[1]*y + ... + p2[9]*y^9
* Q2(y) = q2[0] + q2[1]*y + ... + q2[9]*y^9
*
* P3(y) = p3[0] + p3[1]*y + ... + p3[9]*y^9
* Q3(y) = q3[0] + q3[1]*y + ... + q3[9]*y^9
*
* Remez precision of Ri(y):
* |gamma(x)-(gzi+tzi*y) - y*y*Ri(y)| <= 2**-118.2 ... for i = 1
* <= 2**-126.8 ... for i = 2
* <= 2**-119.5 ... for i = 3
*
* (III) For single precision
*
* Ri(y) = Pi(y), i=1,2,3;
*
* P1(y) = p1[0] + p1[1]*y + ... + p1[5]*y^5
*
* P2(y) = p2[0] + p2[1]*y + ... + p2[5]*y^5
*
* P3(y) = p3[0] + p3[1]*y + ... + p3[4]*y^4
*
* Remez precision of Ri(y):
* |gamma(x)-(gzi+tzi*y) - y*y*Ri(y)| <= 2**-30.8 ... for i = 1
* <= 2**-31.6 ... for i = 2
* <= 2**-29.5 ... for i = 3
*
* Notes. (1) GTi and zi are choosen to balance the interval width and
* minimize the distant between gamma(x) and the tangent line at
* zi. In particular, we have
* |gamma(x)-(gzi+tzi*(x-zi))| <= 0.01436... for x in [1,z2]
* <= 0.01265... for x in [z2,2]
*
* (2) zi are slightly adjusted so that tzi=gamma'(zi) is very
* close to a single precision value.
*
* Coefficents: Single precision
* i= 1:
* P1[0] = 7.09087253435088360271451613398019280077561279443e-0001
* P1[1] = -5.17229560788652108545141978238701790105241761089e-0001
* P1[2] = 5.23403394528150789405825222323770647162337764327e-0001
* P1[3] = -4.54586308717075010784041566069480411732634814899e-0001
* P1[4] = 4.20596490915239085459964590559256913498190955233e-0001
* P1[5] = -3.57307589712377520978332185838241458642142185789e-0001
*
* i = 2:
* p2[0] = 4.28486983980295198166056119223984284434264344578e-0001
* p2[1] = -1.30704539487709138528680121627899735386650103914e-0001
* p2[2] = 1.60856285038051955072861219352655851542955430871e-0001
* p2[3] = -9.22285161346010583774458802067371182158937943507e-0002
* p2[4] = 7.19240511767225260740890292605070595560626179357e-0002
* p2[5] = -4.88158265593355093703112238534484636193260459574e-0002
*
* i = 3
* p3[0] = 3.82409531118807759081121479786092134814808872880e-0001
* p3[1] = 2.65309888180188647956400403013495759365167853426e-0002
* p3[2] = 8.06815109775079171923561169415370309376296739835e-0002
* p3[3] = -1.54821591666137613928840890835174351674007764799e-0002
* p3[4] = 1.76308239242717268530498313416899188157165183405e-0002
*
* Coefficents: Double precision
* i = 1:
* p1[0] = 0.70908683619977797008004927192814648151397705078125000
* p1[1] = 1.71987061393048558089579513384356441668351720061e-0001
* p1[2] = -3.19273345791990970293320316122813960527705450671e-0002
* p1[3] = 8.36172645419110036267169600390549973563534476989e-0003
* p1[4] = 1.13745336648572838333152213474277971244629758101e-0003
* q1[0] = 1.0
* q1[1] = 9.71980217826032937526460731778472389791321968082e-0001
* q1[2] = -7.43576743326756176594084137256042653497087666030e-0002
* q1[3] = -1.19345944932265559769719470515102012246995255372e-0001
* q1[4] = 1.59913445751425002620935120470781382215050284762e-0002
* q1[5] = 1.12601136853374984566572691306402321911547550783e-0003
* i = 2:
* p2[0] = 0.42848681585558601181418225678498856723308563232421875
* p2[1] = 6.53596762668970816023718845105667418483122103629e-0002
* p2[2] = -6.97280829631212931321050770925128264272768936731e-0003
* p2[3] = 6.46342359021981718947208605674813260166116632899e-0003
* q2[0] = 1.0
* q2[1] = 4.57572620560506047062553957454062012327519313936e-0001
* q2[2] = -2.52182594886075452859655003407796103083422572036e-0001
* q2[3] = -1.82970945407778594681348166040103197178711552827e-0002
* q2[4] = 2.43574726993169566475227642128830141304953840502e-0002
* q2[5] = -5.20390406466942525358645957564897411258667085501e-0003
* q2[6] = 4.79520251383279837635552431988023256031951133885e-0004
* i = 3:
* p3[0] = 0.382409479734567459008331979930517263710498809814453125
* p3[1] = 1.42876048697668161599069814043449301572928034140e-0001
* p3[2] = 3.42157571052250536817923866013561760785748899071e-0003
* p3[3] = -5.01542621710067521405087887856991700987709272937e-0004
* p3[4] = 8.89285814866740910123834688163838287618332122670e-0004
* q3[0] = 1.0
* q3[1] = 3.04253086629444201002215640948957897906299633168e-0001
* q3[2] = -2.23162407379999477282555672834881213873185520006e-0001
* q3[3] = -1.05060867741952065921809811933670131427552903636e-0002
* q3[4] = 1.70511763916186982473301861980856352005926669320e-0002
* q3[5] = -2.12950201683609187927899416700094630764182477464e-0003
*
* Note that all pi0 are exact in double, which is obtained by a
* special Remez Algorithm.
*
* Coefficents: Quad precision
* i = 1:
* p1[0] = 0.709086836199777919037185741507610124611513720557
* p1[1] = 4.45754781206489035827915969367354835667391606951e-0001
* p1[2] = 3.21049298735832382311662273882632210062918153852e-0002
* p1[3] = -5.71296796342106617651765245858289197369688864350e-0003
* p1[4] = 6.04666892891998977081619174969855831606965352773e-0003
* p1[5] = 8.99106186996888711939627812174765258822658645168e-0004
* p1[6] = -6.96496846144407741431207008527018441810175568949e-0005
* p1[7] = 1.52597046118984020814225409300131445070213882429e-0005
* p1[8] = 5.68521076168495673844711465407432189190681541547e-0007
* p1[9] = 3.30749673519634895220582062520286565610418952979e-0008
* q1[0] = 1.0+0000
* q1[1] = 1.35806511721671070408570853537257079579490650668e+0000
* q1[2] = 2.97567810153429553405327140096063086994072952961e-0001
* q1[3] = -1.52956835982588571502954372821681851681118097870e-0001
* q1[4] = -2.88248519561420109768781615289082053597954521218e-0002
* q1[5] = 1.03475311719937405219789948456313936302378395955e-0002
* q1[6] = 4.12310203243891222368965360124391297374822742313e-0004
* q1[7] = -3.12653708152290867248931925120380729518332507388e-0004
* q1[8] = 2.36672170850409745237358105667757760527014332458e-0005
*
* i = 2:
* p2[0] = 0.428486815855585429730209907810650616737756697477
* p2[1] = 2.63622124067885222919192651151581541943362617352e-0001
* p2[2] = 3.85520683670028865731877276741390421744971446855e-0002
* p2[3] = 3.05065978278128549958897133190295325258023525862e-0003
* p2[4] = 2.48232934951723128892080415054084339152450445081e-0003
* p2[5] = 3.67092777065632360693313762221411547741550105407e-0004
* p2[6] = 3.81228045616085789674530902563145250532194518946e-0006
* p2[7] = 4.61677225867087554059531455133839175822537617677e-0006
* p2[8] = 2.18209052385703200438239200991201916609364872993e-0007
* p2[9] = 1.00490538985245846460006244065624754421022542454e-0008
* q2[0] = 1.0
* q2[1] = 9.20276350207639290567783725273128544224570775056e-0001
* q2[2] = -4.79533683654165107448020515733883781138947771495e-0003
* q2[3] = -1.24538337585899300494444600248687901947684291683e-0001
* q2[4] = 4.49866050763472358547524708431719114204535491412e-0003
* q2[5] = 7.20715455697920560621638325356292640604078591907e-0003
* q2[6] = -8.68513169029126780280798337091982780598228096116e-0004
* q2[7] = -1.25104431629401181525027098222745544809974229874e-0004
* q2[8] = 3.10558344839000038489191304550998047521253437464e-0005
* q2[9] = -1.76829227852852176018537139573609433652506765712e-0006
*
* i = 3
* p3[0] = 0.3824094797345675048502747661075355640070439388902
* p3[1] = 3.42198093076618495415854906335908427159833377774e-0001
* p3[2] = 9.63828189500585568303961406863153237440702754858e-0002
* p3[3] = 8.76069421042696384852462044188520252156846768667e-0003
* p3[4] = 1.86477890389161491224872014149309015261897537488e-0003
* p3[5] = 8.16871354540309895879974742853701311541286944191e-0004
* p3[6] = 6.83783483674600322518695090864659381650125625216e-0005
* p3[7] = -1.10168269719261574708565935172719209272190828456e-0006
* p3[8] = 9.66243228508380420159234853278906717065629721016e-0007
* p3[9] = 2.31858885579177250541163820671121664974334728142e-0008
* q3[0] = 1.0
* q3[1] = 8.25479821168813634632437430090376252512793067339e-0001
* q3[2] = -1.62251363073937769739639623669295110346015576320e-0002
* q3[3] = -1.10621286905916732758745130629426559691187579852e-0001
* q3[4] = 3.48309693970985612644446415789230015515365291459e-0003
* q3[5] = 6.73553737487488333032431261131289672347043401328e-0003
* q3[6] = -7.63222008393372630162743587811004613050245128051e-0004
* q3[7] = -1.35792670669190631476784768961953711773073251336e-0004
* q3[8] = 3.19610150954223587006220730065608156460205690618e-0005
* q3[9] = -1.82096553862822346610109522015129585693354348322e-0006
*
* (C) For x between 0 and 1.
* Let P stand for the number of significant bits in the working precision.
* -P 1
* (1)For 0 <= x <= 2 , gamma(x) is computed by --- rounded to nearest.
* x
* The error is bound by 0.739 ulp(gamma(x)) in IEEE double precision.
* Proof.
* 1 2
* Since -------- ~ x + 0.577...*x - ..., we have, for small x,
* gamma(x)
* 1 1
* ----------- < gamma(x) < --- and
* x(1+0.578x) x
* 1 1 1
* 0 < --- - gamma(x) <= --- - ----------- < 0.578
* x x x(1+0.578x)
* 1 1 -P
* The error is thus bounded by --- ulp(---) + 0.578. Since x <= 2 ,
* 2 x
* 1 P 1 P 1
* --- >= 2 , ulp(---) >= ulp(2 ) >= 2. Thus 0.578=0.289*2<=0.289ulp(-)
* x x x
* Thus
* 1 1
* | gamma(x) - [---] rounded | <= (0.5+0.289)*ulp(---).
* x x
* -P 1
* Note that for x<= 2 , it is easy to see that ulp(---)=ulp(gamma(x))
* x
* n 1
* except only when x = 2 , (n<= -53). In such cases, --- is exact
* x
* and therefore the error is bounded by
* 1
* 0.298*ulp(---) = 0.298*2*ulp(gamma(x)) = 0.578ulp(gamma(x)).
* x
* Thus we conclude that the error in gamma is less than 0.739 ulp.
*
* (2)Otherwise, for x in GTi-1 (see B), let y = x-(zi-1). From (B) we obtain
* gamma(1+x)
* gamma(1+x) = gy.h + gy.l, then compute gamma(x) by -----------.
* x
* gy.h
* Implementaion note. Write x = x.h+x.l, and Let th = ----- chopped to
* x
* 20 bits, then
* gy.h+gy.l
* gamma(x) = th + (---------- - th )
* x
* 1
* = th + ---*(gy.h-th*x.h+gy.l-th*x.l)
* x
*
* (D) For x between 2 and 8. Let n = 1+x chopped to an integer. Then
*
* gamma(x)=(x-1)*(x-2)*...*(x-n)*gamma(x-n)
*
* Since x-n is between 1 and 2, we can apply (B) to compute gamma(x).
*
* Implementation detail. The computation of (x-1)(x-2)...(x-n) in simulated
* higher precision arithmetic can be somewhat optimized. For example, in
* computing (x-1)*(x-2)*(x-3)*(x-4), if we compute (x-1)*(x-4) = z.h+z.l,
* then (x-2)(x-3) = z.h+2+z.l readily. In below, we list the expression
* of the formula to compute gamma(x).
*
* Assume x-n is in GTi (i=1,2, or 3, see B for detail). Let y = x - n - zi.
* By (B) we have gamma(x-n) = gy.h+gy.l. If x = x.h+x.l, then we have
* n=1 (x in [2,3]):
* gamma(x) = (x-1)*gamma(x-1) = (x-1)*(gy.h+gy.l)
* = [(x.h-1)+x.l]*(gy.h+gy.l)
* n=2 (x in [3,4]):
* gamma(x) = (x-1)(x-2)*gamma(x-2) = (x-1)*(x-2)*(gy.h+gy.l)
* = ((x.h-2)+x.l)*((x.h-1)+x.l)*(gy.h+gy.l)
* = [x.h*(x.h-3)+2+x.l*(x+(x.h-3))]*(gy.h+gy.l)
* n=3 (x in [4,5])
* gamma(x) = (x-1)(x-2)(x-3)*(gy.h+gy.l)
* = (x.h*(x.h-3)+2+x.l*(x+(x.h-3)))*[((x.h-3)+x.l)(gy.h+gy.l)]
* n=4 (x in [5,6])
* gamma(x) = [(x-1)(x-4)]*[(x-2)(x-3)]*(gy.h+gy.l)
* = [(x.h*(x.h-5)+4+x.l(x+(x.h-5)))]*[(x-2)*(x-3)]*(gy.h+gy.l)
* = (y.h+y.l)*(y.h+1+y.l)*(gy.h+gy.l)
* n=5 (x in [6,7])
* gamma(x) = [(x-1)(x-4)]*[(x-2)(x-3)]*[(x-5)*(gy.h+gy.l)]
* n=6 (x in [7,8])
* gamma(x) = [(x-1)(x-6)]*[(x-2)(x-5)]*[(x-3)(x-4)]*(gy.h+gy.l)]
* = [(y.h+y.l)(y.h+4+y.l)][(y.h+6+y.l)(gy.h+gy.l)]
*
* (E)Overflow Thresold. For x > Overflow thresold of gamma,
* return huge*huge (overflow).
*
* By checking whether lgamma(x) >= 2**{128,1024,16384}, one can
* determine the overflow threshold for x in single, double, and
* quad precision. See over.c for details.
*
* The overflow threshold of gamma(x) are
*
* single: x = 3.5040096283e+01
* = 0x420C290F (IEEE single)
* double: x = 1.71624376956302711505e+02
* = 0x406573FAE561F647 (IEEE double)
* quad: x = 1.7555483429044629170038892160702032034177e+03
* = 0x4009B6E3180CD66A5C4206F128BA77F4 (quad)
*
* (F)For overflow_threshold >= x >= 8, we use asymptotic approximation.
* (1) Stirling's formula
*
* log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
* = L1 + L2 + L3,
* where
* L1(x) = (x-.5)*(log(x)-1),
* L2 = .5(log(2pi)-1) = 0.41893853....,
* L3(x) = (1/x)P(1/(x*x)),
*
* The range of L1,L2, and L3 are as follows:
*
* ------------------------------------------------------------------
* Range(L1) = (single) [8.09..,88.30..] =[2** 3.01..,2** 6.46..]
* (double) [8.09..,709.3..] =[2** 3.01..,2** 9.47..]
* (quad) [8.09..,11356.10..]=[2** 3.01..,2** 13.47..]
* Range(L2) = 0.41893853.....
* Range(L3) = [0.0104...., 0.00048....] =[2**-6.58..,2**-11.02..]
* ------------------------------------------------------------------
*
* Gamma(x) is then computed by exp(L1+L2+L3).
*
* (2) Error analysis of (F):
* --------------------------
* The error in Gamma(x) depends on the error inherited in the computation
* of L= L1+L2+L3. Let L' be the computed value of L. The absolute error
* in L' is t = L-L'. Since exp(L') = exp(L-t) = exp(L)*exp(t) ~
* (1+t)*exp(L), the relative error in exp(L') is approximately t.
*
* To guarantee the relatively accuracy in exp(L'), we would like
* |t| < 2**(-P-5) where P denotes for the number of significant bits
* of the working precision. Consequently, each of the L1,L2, and L3
* must be computed with absolute error bounded by 2**(-P-5) in absolute
* value.
*
* Since L2 is a constant, it can be pre-computed to the desired accuracy.
* Also |L3| < 2**-6; therefore, it suffices to compute L3 with the
* working precision. That is,
* L3(x) approxmiate log(G(x))-(x-.5)(log(x)-1)-.5(log(2pi)-1)
* to a precision bounded by 2**(-P-5).
*
* 2**(-6)
* _________V___________________
* L1(x): |_________|___________________|
* __ ________________________
* L2: |__|________________________|
* __________________________
* + L3(x): |__________________________|
* -------------------------------------------
* [leading] + [Trailing]
*
* For L1(x)=(x-0.5)*(log(x)-1), we need ilogb(L1(x))+5 extra bits for
* both multiplicants to guarantee L1(x)'s absolute error is bounded by
* 2**(-P-5) in absolute value. Here ilogb(y) is defined to be the unbias
* binary exponent of y in IEEE format. We can get x-0.5 to the desire
* accuracy easily. It remains to compute log(x)-1 with ilogb(L1(x))+5
* extra bits accracy. Note that the range of L1 is 88.30.., 709.3.., and
* 11356.10... for single, double, and quadruple precision, we have
*
* single double quadruple
* ------------------------------------
* ilogb(L1(x))+5 <= 11 14 18
* ------------------------------------
*
* (3) Table Driven Method for log(x)-1:
* --------------------------------------
* Let x = 2**n * y, where 1 <= y < 2. Let Z={z(i),i=1,...,m}
* be a set of predetermined evenly distributed floating point numbers
* in [1, 2]. Let z(j) be the closest one to y, then
* log(x)-1 = n*log(2)-1 + log(y)
* = n*log(2)-1 + log(z(j)*y/z(j))
* = n*log(2)-1 + log(z(j)) + log(y/z(j))
* = T1(n) + T2(j) + T3,
*
* where T1(n) = n*log(2)-1 and T2(j) = log(z(j)). Both T1 and T2 can be
* pre-calculated and be looked-up in a table. Note that 8 <= x < 1756
* implies 3<=n<=10 implies 1.079.. < T1(n) < 6.931.
*
*
* y-z(i) y 1+s
* For T3, let s = --------; then ----- = ----- and
* y+z(i) z(i) 1-s
* 1+s 2 3 2 5
* T3 = log(-----) = 2s + --- s + --- s + ....
* 1-s 3 5
*
* Suppose the first term 2s is compute in extra precision. The
* dominating error in T3 would then be the rounding error of the
* second term 2/3*s**3. To force the rounding bounded by
* the required accuracy, we have
* single: |2/3*s**3| < 2**-11 == > |s|<0.09014...
* double: |2/3*s**3| < 2**-14 == > |s|<0.04507...
* quad : |2/3*s**3| < 2**-18 == > |s|<0.01788... = 2**(-5.80..)
*
* Base on this analysis, we choose Z = {z(i)|z(i)=1+i/64+1/128, 0<=i<=63}.
* For any y in [1,2), let j = [64*y] chopped to integer, then z(j) is
* the closest to y, and it is not difficult to see that |s| < 2**(-8).
* Please note that the polynomial approximation of T3 must be accurate
* -24-11 -35 -53-14 -67 -113-18 -131
* to 2 =2 , 2 = 2 , and 2 =2
* for single, double, and quadruple precision respectively.
*
* Inplementation notes.
* (1) Table look-up entries for T1(n) and T2(j), as well as the calculation
* of the leading term 2s in T3, are broken up into leading and trailing
* part such that (leading part)* 2**24 will always be an integer. That
* will guarantee the addition of the leading parts will be exact.
*
* 2**(-24)
* _________V___________________
* T1(n): |_________|___________________|
* _______ ______________________
* T2(j): |_______|______________________|
* ____ _______________________
* 2s: |____|_______________________|
* __________________________
* + T3(s)-2s: |__________________________|
* -------------------------------------------
* [leading] + [Trailing]
*
* (2) How to compute 2s accurately.
* (A) Compute v = 2s to the working precision. If |v| < 2**(-18),
* stop.
* (B) chopped v to 2**(-24): v = ((int)(v*2**24))/2**24
* (C) 2s = v + (2s - v), where
* 1
* 2s - v = --- * (2(y-z) - v*(y+z) )
* y+z
* 1
* = --- * ( [2(y-z) - v*(y+z)_h ] - v*(y+z)_l )
* y+z
* where (y+z)_h = (y+z) rounded to 24 bits by (double)(float),
* and (y+z)_l = ((z+z)-(y+z)_h)+(y-z). Note the the quantity
* in [] is exact.
* 2 4
* (3) Remez approximation for (T3(s)-2s)/s = T3[0]*s + T3[1]*s + ...:
* Single precision: 1 term (compute in double precision arithmetic)
* T3(s) = 2s + S1*s^3, S1 = 0.6666717231848518054693623697539230
* Remez error: |T3(s)/s - (2s+S1*s^3)| < 2**(-35.87)
* Double precision: 3 terms, Remez error is bounded by 2**(-72.40),
* see "tgamma_log"
* Quad precision: 7 terms, Remez error is bounded by 2**(-136.54),
* see "tgammal_log"
*
* The computation of 0.5*(ln(2pi)-1):
* 0.5*(ln(2pi)-1) = 0.4189385332046727417803297364056176398614...
* split 0.5*(ln(2pi)-1) to hln2pi_h + hln2pi_l, where hln2pi_h is the
* leading 21 bits of the constant.
* hln2pi_h= 0.4189383983612060546875
* hln2pi_l= 1.348434666870928297364056176398612173648e-07
*
* The computation of 1/x*P(1/x^2) = log(G(x))-(x-.5)(ln(x)-1)-(.5ln(2pi)-1):
* Let s = 1/x <= 1/8 < 0.125. We have
* quad precision
* |GP(s) - s*P(s^2)| <= 2**(-120.6), where
* 3 5 39
* GP(s) = GP0*s+GP1*s +GP2*s +... +GP19*s ,
* GP0 = 0.083333333333333333333333333333333172839171301
* hex 0x3ffe5555 55555555 55555555 55555548
* GP1 = -2.77777777777777777777777777492501211999399424104e-0003
* GP2 = 7.93650793650793650793635650541638236350020883243e-0004
* GP3 = -5.95238095238095238057299772679324503339241961704e-0004
* GP4 = 8.41750841750841696138422987977683524926142600321e-0004
* GP5 = -1.91752691752686682825032547823699662178842123308e-0003
* GP6 = 6.41025641022403480921891559356473451161279359322e-0003
* GP7 = -2.95506535798414019189819587455577003732808185071e-0002
* GP8 = 1.79644367229970031486079180060923073476568732136e-0001
* GP9 = -1.39243086487274662174562872567057200255649290646e+0000
* GP10 = 1.34025874044417962188677816477842265259608269775e+0001
* GP11 = -1.56803713480127469414495545399982508700748274318e+0002
* GP12 = 2.18739841656201561694927630335099313968924493891e+0003
* GP13 = -3.55249848644100338419187038090925410976237921269e+0004
* GP14 = 6.43464880437835286216768959439484376449179576452e+0005
* GP15 = -1.20459154385577014992600342782821389605893904624e+0007
* GP16 = 2.09263249637351298563934942349749718491071093210e+0008
* GP17 = -2.96247483183169219343745316433899599834685703457e+0009
* GP18 = 2.88984933605896033154727626086506756972327292981e+0010
* GP19 = -1.40960434146030007732838382416230610302678063984e+0011
*
* double precision
* |GP(s) - s*P(s^2)| <= 2**(-63.5), where
* 3 5 7 9 11 13 15
* GP(s) = GP0*s+GP1*s +GP2*s +GP3*s +GP4*s +GP5*s +GP6*s +GP7*s ,
*
* GP0= 0.0833333333333333287074040640618477 (3FB55555 55555555)
* GP1= -2.77777777776649355200565611114627670089130772843e-0003
* GP2= 7.93650787486083724805476194170211775784158551509e-0004
* GP3= -5.95236628558314928757811419580281294593903582971e-0004
* GP4= 8.41566473999853451983137162780427812781178932540e-0004
* GP5= -1.90424776670441373564512942038926168175921303212e-0003
* GP6= 5.84933161530949666312333949534482303007354299178e-0003
* GP7= -1.59453228931082030262124832506144392496561694550e-0002
* single precision
* |GP(s) - s*P(s^2)| <= 2**(-37.78), where
* 3 5
* GP(s) = GP0*s+GP1*s +GP2*s
* GP0 = 8.33333330959694065245736888749042811909994573178e-0002
* GP1 = -2.77765545601667179767706600890361535225507762168e-0003
* GP2 = 7.77830853479775281781085278324621033523037489883e-0004
*
*
* Implementation note:
* z = (1/x), z2 = z*z, z4 = z2*z2;
* p = z*(GP0+z2*(GP1+....+z2*GP7))
* = z*(GP0+(z4*(GP2+z4*(GP4+z4*GP6))+z2*(GP1+z4*(GP3+z4*(GP5+z4*GP7)))))
*
* Adding everything up:
* t = rr.h*ww.h+hln2pi_h ... exact
* w = (hln2pi_l + ((x-0.5)*ww.l+rr.l*ww.h)) + p
*
* Computing exp(t+w):
* s = t+w; write s = (n+j/32)*ln2+r, |r|<=(1/64)*ln2, then
* exp(s) = 2**n * (2**(j/32) + 2**(j/32)*expm1(r)), where
* expm1(r) = r + Et1*r^2 + Et2*r^3 + ... + Et5*r^6, and
* 2**(j/32) is obtained by table look-up S[j]+S_trail[j].
* Remez error bound:
* |exp(r) - (1+r+Et1*r^2+...+Et5*r^6)| <= 2^(-63).
*/
#include "libm.h"
#define __HI(x) ((int *) &x)[HIWORD]
#define __LO(x) ((unsigned *) &x)[LOWORD]
struct Double {
double h;
double l;
};
/* Hex value of GP0 shoule be 3FB55555 55555555 */
static const double c[] = {
+1.0,
+2.0,
+0.5,
+1.0e-300,
+6.66666666666666740682e-01, /* A1=T3[0] */
+3.99999999955626478023093908674902212920e-01, /* A2=T3[1] */
+2.85720221533145659809237398709372330980e-01, /* A3=T3[2] */
+0.0833333333333333287074040640618477, /* GP[0] */
-2.77777777776649355200565611114627670089130772843e-03,
+7.93650787486083724805476194170211775784158551509e-04,
-5.95236628558314928757811419580281294593903582971e-04,
+8.41566473999853451983137162780427812781178932540e-04,
-1.90424776670441373564512942038926168175921303212e-03,
+5.84933161530949666312333949534482303007354299178e-03,
-1.59453228931082030262124832506144392496561694550e-02,
+4.18937683105468750000e-01, /* hln2pi_h */
+8.50099203991780279640e-07, /* hln2pi_l */
+4.18938533204672741744150788368695779923320328369e-01, /* hln2pi */
+2.16608493865351192653e-02, /* ln2_32hi */
+5.96317165397058656257e-12, /* ln2_32lo */
+4.61662413084468283841e+01, /* invln2_32 */
+5.0000000000000000000e-1, /* Et1 */
+1.66666666665223585560605991943703896196054020060e-01, /* Et2 */
+4.16666666665895103520154073534275286743788421687e-02, /* Et3 */
+8.33336844093536520775865096538773197505523826029e-03, /* Et4 */
+1.38889201930843436040204096950052984793587640227e-03, /* Et5 */
};
#define one c[0]
#define two c[1]
#define half c[2]
#define tiny c[3]
#define A1 c[4]
#define A2 c[5]
#define A3 c[6]
#define GP0 c[7]
#define GP1 c[8]
#define GP2 c[9]
#define GP3 c[10]
#define GP4 c[11]
#define GP5 c[12]
#define GP6 c[13]
#define GP7 c[14]
#define hln2pi_h c[15]
#define hln2pi_l c[16]
#define hln2pi c[17]
#define ln2_32hi c[18]
#define ln2_32lo c[19]
#define invln2_32 c[20]
#define Et1 c[21]
#define Et2 c[22]
#define Et3 c[23]
#define Et4 c[24]
#define Et5 c[25]
/*
* double precision coefficients for computing log(x)-1 in tgamma.
* See "algorithm" for details
*
* log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
* j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
* T1(n) = T1[2n,2n+1] = n*log(2)-1,
* T2(j) = T2[2j,2j+1] = log(z[j]),
* T3(s) = 2s + T3[0]s^3 + T3[1]s^5 + T3[2]s^7
* = 2s + A1*s^3 + A2*s^5 + A3*s^7 (see const A1,A2,A3)
* Note
* (1) the leading entries are truncated to 24 binary point.
* See Remezpak/sun/tgamma_log_64.c
* (2) Remez error for T3(s) is bounded by 2**(-72.4)
* See mpremez/work/Log/tgamma_log_4_outr2
*/
static const double T1[] = {
-1.00000000000000000000e+00, /* 0xBFF00000 0x00000000 */
+0.00000000000000000000e+00, /* 0x00000000 0x00000000 */
-3.06852817535400390625e-01, /* 0xBFD3A37A 0x00000000 */
-1.90465429995776763166e-09, /* 0xBE205C61 0x0CA86C38 */
+3.86294305324554443359e-01, /* 0x3FD8B90B 0xC0000000 */
+5.57953361754750897367e-08, /* 0x3E6DF473 0xDE6AF279 */
+1.07944148778915405273e+00, /* 0x3FF14564 0x70000000 */
+5.38906818755173187963e-08, /* 0x3E6CEEAD 0xCDA06BB5 */
+1.77258867025375366211e+00, /* 0x3FFC5C85 0xF0000000 */
+5.19860275755595544734e-08, /* 0x3E6BE8E7 0xBCD5E4F2 */
+2.46573585271835327148e+00, /* 0x4003B9D3 0xB8000000 */
+5.00813732756017835330e-08, /* 0x3E6AE321 0xAC0B5E2E */
+3.15888303518295288086e+00, /* 0x40094564 0x78000000 */
+4.81767189756440192100e-08, /* 0x3E69DD5B 0x9B40D76B */
+3.85203021764755249023e+00, /* 0x400ED0F5 0x38000000 */
+4.62720646756862482697e-08, /* 0x3E68D795 0x8A7650A7 */
+4.54517740011215209961e+00, /* 0x40122E42 0xFC000000 */
+4.43674103757284839467e-08, /* 0x3E67D1CF 0x79ABC9E4 */
+5.23832458257675170898e+00, /* 0x4014F40B 0x5C000000 */
+4.24627560757707130063e-08, /* 0x3E66CC09 0x68E14320 */
+5.93147176504135131836e+00, /* 0x4017B9D3 0xBC000000 */
+4.05581017758129486834e-08, /* 0x3E65C643 0x5816BC5D */
};
static const double T2[] = {
+7.78210163116455078125e-03, /* 0x3F7FE020 0x00000000 */
+3.88108903981662140884e-08, /* 0x3E64D620 0xCF11F86F */
+2.31670141220092773438e-02, /* 0x3F97B918 0x00000000 */
+4.51595251008850513740e-08, /* 0x3E683EAD 0x88D54940 */
+3.83188128471374511719e-02, /* 0x3FA39E86 0x00000000 */
+5.14549991480218823411e-08, /* 0x3E6B9FEB 0xD5FA9016 */
+5.32444715499877929688e-02, /* 0x3FAB42DC 0x00000000 */
+4.29688244898971182165e-08, /* 0x3E671197 0x1BEC28D1 */
+6.79506063461303710938e-02, /* 0x3FB16536 0x00000000 */
+5.55623773783008185114e-08, /* 0x3E6DD46F 0x5C1D0C4C */
+8.24436545372009277344e-02, /* 0x3FB51B07 0x00000000 */
+1.46738736635337847313e-08, /* 0x3E4F830C 0x1FB493C7 */
+9.67295765876770019531e-02, /* 0x3FB8C345 0x00000000 */
+4.98708741103424492282e-08, /* 0x3E6AC633 0x641EB597 */
+1.10814332962036132812e-01, /* 0x3FBC5E54 0x00000000 */
+3.33782539813823062226e-08, /* 0x3E61EB78 0xE862BAC3 */
+1.24703466892242431641e-01, /* 0x3FBFEC91 0x00000000 */
+1.16087148042227818450e-08, /* 0x3E48EDF5 0x5D551729 */
+1.38402283191680908203e-01, /* 0x3FC1B72A 0x80000000 */
+3.96674382274822001957e-08, /* 0x3E654BD9 0xE80A4181 */
+1.51916027069091796875e-01, /* 0x3FC371FC 0x00000000 */
+1.49567501781968021494e-08, /* 0x3E500F47 0xBA1DE6CB */
+1.65249526500701904297e-01, /* 0x3FC526E5 0x80000000 */
+4.63946052585787334062e-08, /* 0x3E68E86D 0x0DE8B900 */
+1.78407609462738037109e-01, /* 0x3FC6D60F 0x80000000 */
+4.80100802600100279538e-08, /* 0x3E69C674 0x8723551E */
+1.91394805908203125000e-01, /* 0x3FC87FA0 0x00000000 */
+4.70914263296092971436e-08, /* 0x3E694832 0x44240802 */
+2.04215526580810546875e-01, /* 0x3FCA23BC 0x00000000 */
+1.48478803446288209001e-08, /* 0x3E4FE2B5 0x63193712 */
+2.16873884201049804688e-01, /* 0x3FCBC286 0x00000000 */
+5.40995645549315919488e-08, /* 0x3E6D0B63 0x358A7E74 */
+2.29374051094055175781e-01, /* 0x3FCD5C21 0x00000000 */
+4.99707906542102284117e-08, /* 0x3E6AD3EE 0xE456E443 */
+2.41719901561737060547e-01, /* 0x3FCEF0AD 0x80000000 */
+3.53254081075974352804e-08, /* 0x3E62F716 0x4D948638 */
+2.53915190696716308594e-01, /* 0x3FD04025 0x80000000 */
+1.92842471355435739091e-08, /* 0x3E54B4D0 0x40DAE27C */
+2.65963494777679443359e-01, /* 0x3FD1058B 0xC0000000 */
+5.37194584979797487125e-08, /* 0x3E6CD725 0x6A8C4FD0 */
+2.77868449687957763672e-01, /* 0x3FD1C898 0xC0000000 */
+1.31549854251447496506e-09, /* 0x3E16999F 0xAFBC68E7 */
+2.89633274078369140625e-01, /* 0x3FD2895A 0x00000000 */
+1.85046735362538929911e-08, /* 0x3E53DE86 0xA35EB493 */
+3.01261305809020996094e-01, /* 0x3FD347DD 0x80000000 */
+2.47691407849191245052e-08, /* 0x3E5A987D 0x54D64567 */
+3.12755703926086425781e-01, /* 0x3FD40430 0x80000000 */
+6.07781046260499658610e-09, /* 0x3E3A1A9F 0x8EF4304A */
+3.24119448661804199219e-01, /* 0x3FD4BE5F 0x80000000 */
+1.99924077768719198045e-08, /* 0x3E557778 0xA0DB4C99 */
+3.35355520248413085938e-01, /* 0x3FD57677 0x00000000 */
+2.16727247443196802771e-08, /* 0x3E57455A 0x6C549AB7 */
+3.46466720104217529297e-01, /* 0x3FD62C82 0xC0000000 */
+4.72419910516215900493e-08, /* 0x3E695CE3 0xCA97B7B0 */
+3.57455849647521972656e-01, /* 0x3FD6E08E 0x80000000 */
+3.92742818015697624778e-08, /* 0x3E6515D0 0xF1C609CA */
+3.68325531482696533203e-01, /* 0x3FD792A5 0x40000000 */
+2.96760111198451042238e-08, /* 0x3E5FDD47 0xA27C15DA */
+3.79078328609466552734e-01, /* 0x3FD842D1 0xC0000000 */
+2.43255029056564770289e-08, /* 0x3E5A1E8B 0x17493B14 */
+3.89716744422912597656e-01, /* 0x3FD8F11E 0x80000000 */
+6.71711261571421332726e-09, /* 0x3E3CD98B 0x1DF85DA7 */
+4.00243163108825683594e-01, /* 0x3FD99D95 0x80000000 */
+1.01818702333557515008e-09, /* 0x3E117E08 0xACBA92EF */
+4.10659909248352050781e-01, /* 0x3FDA4840 0x80000000 */
+1.57369163351530571459e-08, /* 0x3E50E5BB 0x0A2BFCA7 */
+4.20969247817993164062e-01, /* 0x3FDAF129 0x00000000 */
+4.68261364720663662040e-08, /* 0x3E6923BC 0x358899C2 */
+4.31173443794250488281e-01, /* 0x3FDB9858 0x80000000 */
+2.10241208525779214510e-08, /* 0x3E569310 0xFB598FB1 */
+4.41274523735046386719e-01, /* 0x3FDC3DD7 0x80000000 */
+3.70698288427707487748e-08, /* 0x3E63E6D6 0xA6B9D9E1 */
+4.51274633407592773438e-01, /* 0x3FDCE1AF 0x00000000 */
+1.07318658117071930723e-08, /* 0x3E470BE7 0xD6F6FA58 */
+4.61175680160522460938e-01, /* 0x3FDD83E7 0x00000000 */
+3.49616477054305011286e-08, /* 0x3E62C517 0x9F2828AE */
+4.70979690551757812500e-01, /* 0x3FDE2488 0x00000000 */
+2.46670332000468969567e-08, /* 0x3E5A7C6C 0x261CBD8F */
+4.80688512325286865234e-01, /* 0x3FDEC399 0xC0000000 */
+1.70204650424422423704e-08, /* 0x3E52468C 0xC0175CEE */
+4.90303933620452880859e-01, /* 0x3FDF6123 0xC0000000 */
+5.44247409572909703749e-08, /* 0x3E6D3814 0x5630A2B6 */
+4.99827861785888671875e-01, /* 0x3FDFFD2E 0x00000000 */
+7.77056065794633071345e-09, /* 0x3E40AFE9 0x30AB2FA0 */
+5.09261846542358398438e-01, /* 0x3FE04BDF 0x80000000 */
+5.52474495483665749052e-08, /* 0x3E6DA926 0xD265FCC1 */
+5.18607735633850097656e-01, /* 0x3FE0986F 0x40000000 */
+2.85741955344967264536e-08, /* 0x3E5EAE6A 0x41723FB5 */
+5.27867078781127929688e-01, /* 0x3FE0E449 0x80000000 */
+1.08397144554263914271e-08, /* 0x3E474732 0x2FDBAB97 */
+5.37041425704956054688e-01, /* 0x3FE12F71 0x80000000 */
+4.01919275998792285777e-08, /* 0x3E6593EF 0xBC530123 */
+5.46132385730743408203e-01, /* 0x3FE179EA 0xA0000000 */
+5.18673922421792693237e-08, /* 0x3E6BD899 0xA0BFC60E */
+5.55141448974609375000e-01, /* 0x3FE1C3B8 0x00000000 */
+5.85658922177154808539e-08, /* 0x3E6F713C 0x24BC94F9 */
+5.64070105552673339844e-01, /* 0x3FE20CDC 0xC0000000 */
+3.27321296262276338905e-08, /* 0x3E6192AB 0x6D93503D */
+5.72919726371765136719e-01, /* 0x3FE2555B 0xC0000000 */
+2.71900203723740076878e-08, /* 0x3E5D31EF 0x96780876 */
+5.81691682338714599609e-01, /* 0x3FE29D37 0xE0000000 */
+5.72959078829112371070e-08, /* 0x3E6EC2B0 0x8AC85CD7 */
+5.90387403964996337891e-01, /* 0x3FE2E474 0x20000000 */
+4.26371800367512948470e-08, /* 0x3E66E402 0x68405422 */
+5.99008142948150634766e-01, /* 0x3FE32B13 0x20000000 */
+4.66979327646159769249e-08, /* 0x3E69121D 0x71320557 */
+6.07555210590362548828e-01, /* 0x3FE37117 0xA0000000 */
+3.96341792466729582847e-08, /* 0x3E654747 0xB5C5DD02 */
+6.16029858589172363281e-01, /* 0x3FE3B684 0x40000000 */
+1.86263416563663175432e-08, /* 0x3E53FFF8 0x455F1DBE */
+6.24433279037475585938e-01, /* 0x3FE3FB5B 0x80000000 */
+8.97441791510503832111e-09, /* 0x3E4345BD 0x096D3A75 */
+6.32766664028167724609e-01, /* 0x3FE43F9F 0xE0000000 */
+5.54287010493641158796e-09, /* 0x3E37CE73 0x3BD393DD */
+6.41031146049499511719e-01, /* 0x3FE48353 0xC0000000 */
+3.33714317793368531132e-08, /* 0x3E61EA88 0xDF73D5E9 */
+6.49227917194366455078e-01, /* 0x3FE4C679 0xA0000000 */
+2.94307433638127158696e-08, /* 0x3E5F99DC 0x7362D1DA */
+6.57358050346374511719e-01, /* 0x3FE50913 0xC0000000 */
+2.23619855184231409785e-08, /* 0x3E5802D0 0xD6979675 */
+6.65422618389129638672e-01, /* 0x3FE54B24 0x60000000 */
+1.41559608102782173188e-08, /* 0x3E4E6652 0x5EA4550A */
+6.73422634601593017578e-01, /* 0x3FE58CAD 0xA0000000 */
+4.06105737027198329700e-08, /* 0x3E65CD79 0x893092F2 */
+6.81359171867370605469e-01, /* 0x3FE5CDB1 0xC0000000 */
+5.29405324634793230630e-08, /* 0x3E6C6C17 0x648CF6E4 */
+6.89233243465423583984e-01, /* 0x3FE60E32 0xE0000000 */
+3.77733853963405370102e-08, /* 0x3E644788 0xD8CA7C89 */
};
/* S[j],S_trail[j] = 2**(j/32.) for the final computation of exp(t+w) */
static const double S[] = {
+1.00000000000000000000e+00, /* 3FF0000000000000 */
+1.02189714865411662714e+00, /* 3FF059B0D3158574 */
+1.04427378242741375480e+00, /* 3FF0B5586CF9890F */
+1.06714040067682369717e+00, /* 3FF11301D0125B51 */
+1.09050773266525768967e+00, /* 3FF172B83C7D517B */
+1.11438674259589243221e+00, /* 3FF1D4873168B9AA */
+1.13878863475669156458e+00, /* 3FF2387A6E756238 */
+1.16372485877757747552e+00, /* 3FF29E9DF51FDEE1 */
+1.18920711500272102690e+00, /* 3FF306FE0A31B715 */
+1.21524735998046895524e+00, /* 3FF371A7373AA9CB */
+1.24185781207348400201e+00, /* 3FF3DEA64C123422 */
+1.26905095719173321989e+00, /* 3FF44E086061892D */
+1.29683955465100964055e+00, /* 3FF4BFDAD5362A27 */
+1.32523664315974132322e+00, /* 3FF5342B569D4F82 */
+1.35425554693689265129e+00, /* 3FF5AB07DD485429 */
+1.38390988196383202258e+00, /* 3FF6247EB03A5585 */
+1.41421356237309514547e+00, /* 3FF6A09E667F3BCD */
+1.44518080697704665027e+00, /* 3FF71F75E8EC5F74 */
+1.47682614593949934623e+00, /* 3FF7A11473EB0187 */
+1.50916442759342284141e+00, /* 3FF82589994CCE13 */
+1.54221082540794074411e+00, /* 3FF8ACE5422AA0DB */
+1.57598084510788649659e+00, /* 3FF93737B0CDC5E5 */
+1.61049033194925428347e+00, /* 3FF9C49182A3F090 */
+1.64575547815396494578e+00, /* 3FFA5503B23E255D */
+1.68179283050742900407e+00, /* 3FFAE89F995AD3AD */
+1.71861929812247793414e+00, /* 3FFB7F76F2FB5E47 */
+1.75625216037329945351e+00, /* 3FFC199BDD85529C */
+1.79470907500310716820e+00, /* 3FFCB720DCEF9069 */
+1.83400808640934243066e+00, /* 3FFD5818DCFBA487 */
+1.87416763411029996256e+00, /* 3FFDFC97337B9B5F */
+1.91520656139714740007e+00, /* 3FFEA4AFA2A490DA */
+1.95714412417540017941e+00, /* 3FFF50765B6E4540 */
};
static const double S_trail[] = {
+0.00000000000000000000e+00,
+5.10922502897344389359e-17, /* 3C8D73E2A475B465 */
+8.55188970553796365958e-17, /* 3C98A62E4ADC610A */
-7.89985396684158212226e-17, /* BC96C51039449B3A */
-3.04678207981247114697e-17, /* BC819041B9D78A76 */
+1.04102784568455709549e-16, /* 3C9E016E00A2643C */
+8.91281267602540777782e-17, /* 3C99B07EB6C70573 */
+3.82920483692409349872e-17, /* 3C8612E8AFAD1255 */
+3.98201523146564611098e-17, /* 3C86F46AD23182E4 */
-7.71263069268148813091e-17, /* BC963AEABF42EAE2 */
+4.65802759183693679123e-17, /* 3C8ADA0911F09EBC */
+2.66793213134218609523e-18, /* 3C489B7A04EF80D0 */
+2.53825027948883149593e-17, /* 3C7D4397AFEC42E2 */
-2.85873121003886075697e-17, /* BC807ABE1DB13CAC */
+7.70094837980298946162e-17, /* 3C96324C054647AD */
-6.77051165879478628716e-17, /* BC9383C17E40B497 */
-9.66729331345291345105e-17, /* BC9BDD3413B26456 */
-3.02375813499398731940e-17, /* BC816E4786887A99 */
-3.48399455689279579579e-17, /* BC841577EE04992F */
-1.01645532775429503911e-16, /* BC9D4C1DD41532D8 */
+7.94983480969762085616e-17, /* 3C96E9F156864B27 */
-1.01369164712783039808e-17, /* BC675FC781B57EBC */
+2.47071925697978878522e-17, /* 3C7C7C46B071F2BE */
-1.01256799136747726038e-16, /* BC9D2F6EDB8D41E1 */
+8.19901002058149652013e-17, /* 3C97A1CD345DCC81 */
-1.85138041826311098821e-17, /* BC75584F7E54AC3B */
+2.96014069544887330703e-17, /* 3C811065895048DD */
+1.82274584279120867698e-17, /* 3C7503CBD1E949DB */
+3.28310722424562658722e-17, /* 3C82ED02D75B3706 */
-6.12276341300414256164e-17, /* BC91A5CD4F184B5C */
-1.06199460561959626376e-16, /* BC9E9C23179C2893 */
+8.96076779103666776760e-17, /* 3C99D3E12DD8A18B */
};
/* Primary interval GTi() */
static const double cr[] = {
/* p1, q1 */
+0.70908683619977797008004927192814648151397705078125000,
+1.71987061393048558089579513384356441668351720061e-0001,
-3.19273345791990970293320316122813960527705450671e-0002,
+8.36172645419110036267169600390549973563534476989e-0003,
+1.13745336648572838333152213474277971244629758101e-0003,
+1.0,
+9.71980217826032937526460731778472389791321968082e-0001,
-7.43576743326756176594084137256042653497087666030e-0002,
-1.19345944932265559769719470515102012246995255372e-0001,
+1.59913445751425002620935120470781382215050284762e-0002,
+1.12601136853374984566572691306402321911547550783e-0003,
/* p2, q2 */
+0.42848681585558601181418225678498856723308563232421875,
+6.53596762668970816023718845105667418483122103629e-0002,
-6.97280829631212931321050770925128264272768936731e-0003,
+6.46342359021981718947208605674813260166116632899e-0003,
+1.0,
+4.57572620560506047062553957454062012327519313936e-0001,
-2.52182594886075452859655003407796103083422572036e-0001,
-1.82970945407778594681348166040103197178711552827e-0002,
+2.43574726993169566475227642128830141304953840502e-0002,
-5.20390406466942525358645957564897411258667085501e-0003,
+4.79520251383279837635552431988023256031951133885e-0004,
/* p3, q3 */
+0.382409479734567459008331979930517263710498809814453125,
+1.42876048697668161599069814043449301572928034140e-0001,
+3.42157571052250536817923866013561760785748899071e-0003,
-5.01542621710067521405087887856991700987709272937e-0004,
+8.89285814866740910123834688163838287618332122670e-0004,
+1.0,
+3.04253086629444201002215640948957897906299633168e-0001,
-2.23162407379999477282555672834881213873185520006e-0001,
-1.05060867741952065921809811933670131427552903636e-0002,
+1.70511763916186982473301861980856352005926669320e-0002,
-2.12950201683609187927899416700094630764182477464e-0003,
};
#define P10 cr[0]
#define P11 cr[1]
#define P12 cr[2]
#define P13 cr[3]
#define P14 cr[4]
#define Q10 cr[5]
#define Q11 cr[6]
#define Q12 cr[7]
#define Q13 cr[8]
#define Q14 cr[9]
#define Q15 cr[10]
#define P20 cr[11]
#define P21 cr[12]
#define P22 cr[13]
#define P23 cr[14]
#define Q20 cr[15]
#define Q21 cr[16]
#define Q22 cr[17]
#define Q23 cr[18]
#define Q24 cr[19]
#define Q25 cr[20]
#define Q26 cr[21]
#define P30 cr[22]
#define P31 cr[23]
#define P32 cr[24]
#define P33 cr[25]
#define P34 cr[26]
#define Q30 cr[27]
#define Q31 cr[28]
#define Q32 cr[29]
#define Q33 cr[30]
#define Q34 cr[31]
#define Q35 cr[32]
static const double
GZ1_h = +0.938204627909682398190,
GZ1_l = +5.121952600248205157935e-17,
GZ2_h = +0.885603194410888749921,
GZ2_l = -4.964236872556339810692e-17,
GZ3_h = +0.936781411463652347038,
GZ3_l = -2.541923110834479415023e-17,
TZ1 = -0.3517214357852935791015625,
TZ3 = +0.280530631542205810546875;
/* INDENT ON */
/* compute gamma(y=yh+yl) for y in GT1 = [1.0000, 1.2845] */
/* assume yh got 20 significant bits */
static struct Double
GT1(double yh, double yl) {
double t3, t4, y, z;
struct Double r;
y = yh + yl;
z = y * y;
t3 = (z * (P10 + y * ((P11 + y * P12) + z * (P13 + y * P14)))) /
(Q10 + y * ((Q11 + y * Q12) + z * ((Q13 + Q14 * y) + z * Q15)));
t3 += (TZ1 * yl + GZ1_l);
t4 = TZ1 * yh;
r.h = (double) ((float) (t4 + GZ1_h + t3));
t3 += (t4 - (r.h - GZ1_h));
r.l = t3;
return (r);
}
/* compute gamma(y=yh+yl) for y in GT2 = [1.2844, 1.6374] */
/* assume yh got 20 significant bits */
static struct Double
GT2(double yh, double yl) {
double t3, y, z;
struct Double r;
y = yh + yl;
z = y * y;
t3 = (z * (P20 + y * P21 + z * (P22 + y * P23))) /
(Q20 + (y * ((Q21 + Q22 * y) + z * Q23) +
(z * z) * ((Q24 + Q25 * y) + z * Q26))) + GZ2_l;
r.h = (double) ((float) (GZ2_h + t3));
r.l = t3 - (r.h - GZ2_h);
return (r);
}
/* compute gamma(y=yh+yl) for y in GT3 = [1.6373, 2.0000] */
/* assume yh got 20 significant bits */
static struct Double
GT3(double yh, double yl) {
double t3, t4, y, z;
struct Double r;
y = yh + yl;
z = y * y;
t3 = (z * (P30 + y * ((P31 + y * P32) + z * (P33 + y * P34)))) /
(Q30 + y * ((Q31 + y * Q32) + z * ((Q33 + Q34 * y) + z * Q35)));
t3 += (TZ3 * yl + GZ3_l);
t4 = TZ3 * yh;
r.h = (double) ((float) (t4 + GZ3_h + t3));
t3 += (t4 - (r.h - GZ3_h));
r.l = t3;
return (r);
}
/* INDENT OFF */
/*
* return tgamma(x) scaled by 2**-m for 8<x<=171.62... using Stirling's formula
* log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
* = L1 + L2 + L3,
*/
/* INDENT ON */
static struct Double
large_gam(double x, int *m) {
double z, t1, t2, t3, z2, t5, w, y, u, r, z4, v, t24 = 16777216.0,
p24 = 1.0 / 16777216.0;
int n2, j2, k, ix, j;
unsigned lx;
struct Double zz;
double u2, ss_h, ss_l, r_h, w_h, w_l, t4;
/* INDENT OFF */
/*
* compute ss = ss.h+ss.l = log(x)-1 (see tgamma_log.h for details)
*
* log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
* j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
* T1(n) = T1[2n,2n+1] = n*log(2)-1,
* T2(j) = T2[2j,2j+1] = log(z[j]),
* T3(s) = 2s + A1[0]s^3 + A2[1]s^5 + A3[2]s^7
* Note
* (1) the leading entries are truncated to 24 binary point.
* (2) Remez error for T3(s) is bounded by 2**(-72.4)
* 2**(-24)
* _________V___________________
* T1(n): |_________|___________________|
* _______ ______________________
* T2(j): |_______|______________________|
* ____ _______________________
* 2s: |____|_______________________|
* __________________________
* + T3(s)-2s: |__________________________|
* -------------------------------------------
* [leading] + [Trailing]
*/
/* INDENT ON */
ix = __HI(x);
lx = __LO(x);
n2 = (ix >> 20) - 0x3ff; /* exponent of x, range:3-7 */
n2 += n2; /* 2n */
ix = (ix & 0x000fffff) | 0x3ff00000; /* y = scale x to [1,2] */
__HI(y) = ix;
__LO(y) = lx;
__HI(z) = (ix & 0xffffc000) | 0x2000; /* z[j]=1+j/64+1/128 */
__LO(z) = 0;
j2 = (ix >> 13) & 0x7e; /* 2j */
t1 = y + z;
t2 = y - z;
r = one / t1;
t1 = (double) ((float) t1);
u = r * t2; /* u = (y-z)/(y+z) */
t4 = T2[j2 + 1] + T1[n2 + 1];
z2 = u * u;
k = __HI(u) & 0x7fffffff;
t3 = T2[j2] + T1[n2];
if ((k >> 20) < 0x3ec) { /* |u|<2**-19 */
t2 = t4 + u * ((two + z2 * A1) + (z2 * z2) * (A2 + z2 * A3));
} else {
t5 = t4 + u * (z2 * A1 + (z2 * z2) * (A2 + z2 * A3));
u2 = u + u;
v = (double) ((int) (u2 * t24)) * p24;
t2 = t5 + r * ((two * t2 - v * t1) - v * (y - (t1 - z)));
t3 += v;
}
ss_h = (double) ((float) (t2 + t3));
ss_l = t2 - (ss_h - t3);
/*
* compute ww = (x-.5)*(log(x)-1) + .5*(log(2pi)-1) + 1/x*(P(1/x^2)))
* where ss = log(x) - 1 in already in extra precision
*/
z = one / x;
r = x - half;
r_h = (double) ((float) r);
w_h = r_h * ss_h + hln2pi_h;
z2 = z * z;
w = (r - r_h) * ss_h + r * ss_l;
z4 = z2 * z2;
t1 = z2 * (GP1 + z4 * (GP3 + z4 * (GP5 + z4 * GP7)));
t2 = z4 * (GP2 + z4 * (GP4 + z4 * GP6));
t1 += t2;
w += hln2pi_l;
w_l = z * (GP0 + t1) + w;
k = (int) ((w_h + w_l) * invln2_32 + half);
/* compute the exponential of w_h+w_l */
j = k & 0x1f;
*m = (k >> 5);
t3 = (double) k;
/* perform w - k*ln2_32 (represent as w_h - w_l) */
t1 = w_h - t3 * ln2_32hi;
t2 = t3 * ln2_32lo;
w = w_l - t2;
w_h = t1 + w_l;
w_l = t2 - (w_l - (w_h - t1));
/* compute exp(w_h+w_l) */
z = w_h - w_l;
z2 = z * z;
t1 = z2 * (Et1 + z2 * (Et3 + z2 * Et5));
t2 = z2 * (Et2 + z2 * Et4);
t3 = w_h - (w_l - (t1 + z * t2));
zz.l = S_trail[j] * (one + t3) + S[j] * t3;
zz.h = S[j];
return (zz);
}
/* INDENT OFF */
/*
* kpsin(x)= sin(pi*x)/pi
* 3 5 7 9 11 13 15
* = x+ks[0]*x +ks[1]*x +ks[2]*x +ks[3]*x +ks[4]*x +ks[5]*x +ks[6]*x
*/
static const double ks[] = {
-1.64493406684822640606569,
+8.11742425283341655883668741874008920850698590621e-0001,
-1.90751824120862873825597279118304943994042258291e-0001,
+2.61478477632554278317289628332654539353521911570e-0002,
-2.34607978510202710377617190278735525354347705866e-0003,
+1.48413292290051695897242899977121846763824221705e-0004,
-6.87730769637543488108688726777687262485357072242e-0006,
};
/* INDENT ON */
/* assume x is not tiny and positive */
static struct Double
kpsin(double x) {
double z, t1, t2, t3, t4;
struct Double xx;
z = x * x;
xx.h = x;
t1 = z * x;
t2 = z * z;
t4 = t1 * ks[0];
t3 = (t1 * z) * ((ks[1] + z * ks[2] + t2 * ks[3]) + (z * t2) *
(ks[4] + z * ks[5] + t2 * ks[6]));
xx.l = t4 + t3;
return (xx);
}
/* INDENT OFF */
/*
* kpcos(x)= cos(pi*x)/pi
* 2 4 6 8 10 12
* = 1/pi +kc[0]*x +kc[1]*x +kc[2]*x +kc[3]*x +kc[4]*x +kc[5]*x
*/
static const double one_pi_h = 0.318309886183790635705292970,
one_pi_l = 3.583247455607534006714276420e-17;
static const double npi_2_h = -1.5625,
npi_2_l = -0.00829632679489661923132169163975055099555883223;
static const double kc[] = {
-1.57079632679489661923132169163975055099555883223e+0000,
+1.29192819501230224953283586722575766189551966008e+0000,
-4.25027339940149518500158850753393173519732149213e-0001,
+7.49080625187015312373925142219429422375556727752e-0002,
-8.21442040906099210866977352284054849051348692715e-0003,
+6.10411356829515414575566564733632532333904115968e-0004,
};
/* INDENT ON */
/* assume x is not tiny and positive */
static struct Double
kpcos(double x) {
double z, t1, t2, t3, t4, x4, x8;
struct Double xx;
z = x * x;
xx.h = one_pi_h;
t1 = (double) ((float) x);
x4 = z * z;
t2 = npi_2_l * z + npi_2_h * (x + t1) * (x - t1);
t3 = one_pi_l + x4 * ((kc[1] + z * kc[2]) + x4 * (kc[3] + z *
kc[4] + x4 * kc[5]));
t4 = t1 * t1; /* 48 bits mantissa */
x8 = t2 + t3;
t4 *= npi_2_h; /* npi_2_h is 5 bits const. The product is exact */
xx.l = x8 + t4; /* that will minimized the rounding error in xx.l */
return (xx);
}
/* INDENT OFF */
static const double
/* 0.134861805732790769689793935774652917006 */
t0z1 = 0.1348618057327907737708,
t0z1_l = -4.0810077708578299022531e-18,
/* 0.461632144968362341262659542325721328468 */
t0z2 = 0.4616321449683623567850,
t0z2_l = -1.5522348162858676890521e-17,
/* 0.819773101100500601787868704921606996312 */
t0z3 = 0.8197731011005006118708,
t0z3_l = -1.0082945122487103498325e-17;
/* 1.134861805732790769689793935774652917006 */
/* INDENT ON */
/* gamma(x+i) for 0 <= x < 1 */
static struct Double
gam_n(int i, double x) {
struct Double rr = {0.0L, 0.0L}, yy;
double r1, r2, t2, z, xh, xl, yh, yl, zh, z1, z2, zl, x5, wh, wl;
/* compute yy = gamma(x+1) */
if (x > 0.2845) {
if (x > 0.6374) {
r1 = x - t0z3;
r2 = (double) ((float) (r1 - t0z3_l));
t2 = r1 - r2;
yy = GT3(r2, t2 - t0z3_l);
} else {
r1 = x - t0z2;
r2 = (double) ((float) (r1 - t0z2_l));
t2 = r1 - r2;
yy = GT2(r2, t2 - t0z2_l);
}
} else {
r1 = x - t0z1;
r2 = (double) ((float) (r1 - t0z1_l));
t2 = r1 - r2;
yy = GT1(r2, t2 - t0z1_l);
}
/* compute gamma(x+i) = (x+i-1)*...*(x+1)*yy, 0<i<8 */
switch (i) {
case 0: /* yy/x */
r1 = one / x;
xh = (double) ((float) x); /* x is not tiny */
rr.h = (double) ((float) ((yy.h + yy.l) * r1));
rr.l = r1 * (yy.h - rr.h * xh) -
((r1 * rr.h) * (x - xh) - r1 * yy.l);
break;
case 1: /* yy */
rr.h = yy.h;
rr.l = yy.l;
break;
case 2: /* (x+1)*yy */
z = x + one; /* may not be exact */
zh = (double) ((float) z);
rr.h = zh * yy.h;
rr.l = z * yy.l + (x - (zh - one)) * yy.h;
break;
case 3: /* (x+2)*(x+1)*yy */
z1 = x + one;
z2 = x + 2.0;
z = z1 * z2;
xh = (double) ((float) z);
zh = (double) ((float) z1);
xl = (x - (zh - one)) * (z2 + zh) - (xh - zh * (zh + one));
rr.h = xh * yy.h;
rr.l = z * yy.l + xl * yy.h;
break;
case 4: /* (x+1)*(x+3)*(x+2)*yy */
z1 = x + 2.0;
z2 = (x + one) * (x + 3.0);
zh = z1;
__LO(zh) = 0;
__HI(zh) &= 0xfffffff8; /* zh 18 bits mantissa */
zl = x - (zh - 2.0);
z = z1 * z2;
xh = (double) ((float) z);
xl = zl * (z2 + zh * (z1 + zh)) - (xh - zh * (zh * zh - one));
rr.h = xh * yy.h;
rr.l = z * yy.l + xl * yy.h;
break;
case 5: /* ((x+1)*(x+4)*(x+2)*(x+3))*yy */
z1 = x + 2.0;
z2 = x + 3.0;
z = z1 * z2;
zh = (double) ((float) z1);
yh = (double) ((float) z);
yl = (x - (zh - 2.0)) * (z2 + zh) - (yh - zh * (zh + one));
z2 = z - 2.0;
z *= z2;
xh = (double) ((float) z);
xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0));
rr.h = xh * yy.h;
rr.l = z * yy.l + xl * yy.h;
break;
case 6: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5))*yy */
z1 = x + 2.0;
z2 = x + 3.0;
z = z1 * z2;
zh = (double) ((float) z1);
yh = (double) ((float) z);
z1 = x - (zh - 2.0);
yl = z1 * (z2 + zh) - (yh - zh * (zh + one));
z2 = z - 2.0;
x5 = x + 5.0;
z *= z2;
xh = (double) ((float) z);
zh += 3.0;
xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0));
/* xh+xl=(x+1)*...*(x+4) */
/* wh+wl=(x+5)*yy */
wh = (double) ((float) (x5 * (yy.h + yy.l)));
wl = (z1 * yy.h + x5 * yy.l) - (wh - zh * yy.h);
rr.h = wh * xh;
rr.l = z * wl + xl * wh;
break;
case 7: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6))*yy */
z1 = x + 3.0;
z2 = x + 4.0;
z = z2 * z1;
zh = (double) ((float) z1);
yh = (double) ((float) z); /* yh+yl = (x+3)(x+4) */
yl = (x - (zh - 3.0)) * (z2 + zh) - (yh - (zh * (zh + one)));
z1 = x + 6.0;
z2 = z - 2.0; /* z2 = (x+2)*(x+5) */
z *= z2;
xh = (double) ((float) z);
xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0));
/* xh+xl=(x+2)*...*(x+5) */
/* wh+wl=(x+1)(x+6)*yy */
z2 -= 4.0; /* z2 = (x+1)(x+6) */
wh = (double) ((float) (z2 * (yy.h + yy.l)));
wl = (z2 * yy.l + yl * yy.h) - (wh - (yh - 6.0) * yy.h);
rr.h = wh * xh;
rr.l = z * wl + xl * wh;
}
return (rr);
}
double
tgamma(double x) {
struct Double ss, ww;
double t, t1, t2, t3, t4, t5, w, y, z, z1, z2, z3, z5;
int i, j, k, m, ix, hx, xk;
unsigned lx;
hx = __HI(x);
lx = __LO(x);
ix = hx & 0x7fffffff;
y = x;
if (ix < 0x3ca00000)
return (one / x); /* |x| < 2**-53 */
if (ix >= 0x7ff00000)
/* +Inf -> +Inf, -Inf or NaN -> NaN */
return (x * ((hx < 0)? 0.0 : x));
if (hx > 0x406573fa || /* x > 171.62... overflow to +inf */
(hx == 0x406573fa && lx > 0xE561F647)) {
z = x / tiny;
return (z * z);
}
if (hx >= 0x40200000) { /* x >= 8 */
ww = large_gam(x, &m);
w = ww.h + ww.l;
__HI(w) += m << 20;
return (w);
}
if (hx > 0) { /* 0 < x < 8 */
i = (int) x;
ww = gam_n(i, x - (double) i);
return (ww.h + ww.l);
}
/* negative x */
/* INDENT OFF */
/*
* compute: xk =
* -2 ... x is an even int (-inf is even)
* -1 ... x is an odd int
* +0 ... x is not an int but chopped to an even int
* +1 ... x is not an int but chopped to an odd int
*/
/* INDENT ON */
xk = 0;
if (ix >= 0x43300000) {
if (ix >= 0x43400000)
xk = -2;
else
xk = -2 + (lx & 1);
} else if (ix >= 0x3ff00000) {
k = (ix >> 20) - 0x3ff;
if (k > 20) {
j = lx >> (52 - k);
if ((j << (52 - k)) == lx)
xk = -2 + (j & 1);
else
xk = j & 1;
} else {
j = ix >> (20 - k);
if ((j << (20 - k)) == ix && lx == 0)
xk = -2 + (j & 1);
else
xk = j & 1;
}
}
if (xk < 0)
/* ideally gamma(-n)= (-1)**(n+1) * inf, but c99 expect NaN */
return ((x - x) / (x - x)); /* 0/0 = NaN */
/* negative underflow thresold */
if (ix > 0x4066e000 || (ix == 0x4066e000 && lx > 11)) {
/* x < -183.0 - 11ulp */
z = tiny / x;
if (xk == 1)
z = -z;
return (z * tiny);
}
/* now compute gamma(x) by -1/((sin(pi*y)/pi)*gamma(1+y)), y = -x */
/*
* First compute ss = -sin(pi*y)/pi , so that
* gamma(x) = 1/(ss*gamma(1+y))
*/
y = -x;
j = (int) y;
z = y - (double) j;
if (z > 0.3183098861837906715377675)
if (z > 0.6816901138162093284622325)
ss = kpsin(one - z);
else
ss = kpcos(0.5 - z);
else
ss = kpsin(z);
if (xk == 0) {
ss.h = -ss.h;
ss.l = -ss.l;
}
/* Then compute ww = gamma(1+y), note that result scale to 2**m */
m = 0;
if (j < 7) {
ww = gam_n(j + 1, z);
} else {
w = y + one;
if ((lx & 1) == 0) { /* y+1 exact (note that y<184) */
ww = large_gam(w, &m);
} else {
t = w - one;
if (t == y) { /* y+one exact */
ww = large_gam(w, &m);
} else { /* use y*gamma(y) */
if (j == 7)
ww = gam_n(j, z);
else
ww = large_gam(y, &m);
t4 = ww.h + ww.l;
t1 = (double) ((float) y);
t2 = (double) ((float) t4);
/* t4 will not be too large */
ww.l = y * (ww.l - (t2 - ww.h)) + (y - t1) * t2;
ww.h = t1 * t2;
}
}
}
/* compute 1/(ss*ww) */
t3 = ss.h + ss.l;
t4 = ww.h + ww.l;
t1 = (double) ((float) t3);
t2 = (double) ((float) t4);
z1 = ss.l - (t1 - ss.h); /* (t1,z1) = ss */
z2 = ww.l - (t2 - ww.h); /* (t2,z2) = ww */
t3 = t3 * t4; /* t3 = ss*ww */
z3 = one / t3; /* z3 = 1/(ss*ww) */
t5 = t1 * t2;
z5 = z1 * t4 + t1 * z2; /* (t5,z5) = ss*ww */
t1 = (double) ((float) t3); /* (t1,z1) = ss*ww */
z1 = z5 - (t1 - t5);
t2 = (double) ((float) z3); /* leading 1/(ss*ww) */
z2 = z3 * (t2 * z1 - (one - t2 * t1));
z = t2 - z2;
/* check whether z*2**-m underflow */
if (m != 0) {
hx = __HI(z);
i = hx & 0x80000000;
ix = hx ^ i;
j = ix >> 20;
if (j > m) {
ix -= m << 20;
__HI(z) = ix ^ i;
} else if ((m - j) > 52) {
/* underflow */
if (xk == 0)
z = -tiny * tiny;
else
z = tiny * tiny;
} else {
/* subnormal */
m -= 60;
t = one;
__HI(t) -= 60 << 20;
ix -= m << 20;
__HI(z) = ix ^ i;
z *= t;
}
}
return (z);
}