xref: /illumos-gate/usr/src/lib/libm/common/m9x/tgamma.c (revision ddc0e0b5)

/*
 * 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
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 */

/*
 * 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 */
	+2.46573585271