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25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Use is subject to license terms.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT OFF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * True gamma function
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * double tgamma(double x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Less that one ulp for both positive and negative arguments.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * A: For negative argument
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (1) gamma(-n or -inf) is NaN
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (2) Underflow Threshold
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (3) Reduction to gamma(1+x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * B: For x between 1 and 2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * C: For x between 0 and 1
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * D: For x between 2 and 8
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * E: Overflow thresold {see over.c}
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * F: For overflow_threshold >= x >= 8
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Implementation details
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -----------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (A) For negative argument, use gamma(-x) = ------------------------.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (sin(pi*x)*gamma(1+x))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (1) gamma(-n or -inf) is NaN with invalid signal by SUSv3 spec.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (Ideally, gamma(-n) = 1/sinpi(n) = (-1)**(n+1) * inf.)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (2) Underflow Threshold. For each precision, there is a value T
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * such that when x>T and when x is not an integer, gamma(-x) will
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * always underflow. A table of the underflow threshold value is given
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * below. For proof, see file "under.c".
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Precision underflow threshold T =
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ----------------------------------------------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * single 41.000041962 = 41 + 11 ULP
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (machine format) 4224000B
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * double 183.000000000000312639 = 183 + 11 ULP
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (machine format) 4066E000 0000000B
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * quad 1774.0000000000000000000000000000017749370 = 1774 + 9 ULP
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (machine format) 4009BB80000000000000000000000009
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ----------------------------------------------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (3) Reduction to gamma(1+x).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Because of (1) and (2), we need only consider non-integral x
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * such that 0<x<T. Let k = [x] and z = x-[x]. Define
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * sin(x*pi) cos(x*pi)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpsin(x) = --------- and kpcos(x) = --------- . Then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(-x) = --------------------.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -kpsin(x)*gamma(1+x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Since x = k+z,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -sin(x*pi) = -sin(k*pi+z*pi) = (-1) *sin(z*pi),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * we have -kpsin(x) = (-1) * kpsin(z). We can further
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * reduce z to t by
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (I) t = z when 0.00000 <= z < 0.31830...
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (II) t = 0.5-z when 0.31830... <= z < 0.681690...
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (III) t = 1-z when 0.681690... <= z < 1.00000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * and correspondingly
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (I) kpsin(z) = kpsin(t) ... 0<= z < 0.3184
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (II) kpsin(z) = kpcos(t) ... |t| < 0.182
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (III) kpsin(z) = kpsin(t) ... 0<= t < 0.3184
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Using a special Remez algorithm, we obtain the following polynomial
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * approximation for kpsin(t) for 0<=t<0.3184:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Computation note: in simulating higher precision arithmetic, kcpsin
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * return head = t and tail = ks[0]*t^3 + (...) to maintain extra bits.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Quad precision, remez error <= 2**(-129.74)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpsin(t) = t + ks[0] * t + ks[1] * t + ... + ks[12] * t
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[ 0] = -1.64493406684822643647241516664602518705158902870e+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[ 1] = 8.11742425283353643637002772405874238094995726160e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[ 2] = -1.90751824122084213696472111835337366232282723933e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[ 3] = 2.61478478176548005046532613563241288115395517084e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[ 4] = -2.34608103545582363750893072647117829448016479971e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[ 5] = 1.48428793031071003684606647212534027556262040158e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[ 6] = -6.97587366165638046518462722252768122615952898698e-0006
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[ 7] = 2.53121740413702536928659271747187500934840057929e-0007
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[ 8] = -7.30471182221385990397683641695766121301933621956e-0009
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[ 9] = 1.71653847451163495739958249695549313987973589884e-0010
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[10] = -3.34813314714560776122245796929054813458341420565e-0012
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[11] = 5.50724992262622033449487808306969135431411753047e-0014
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[12] = -7.67678132753577998601234393215802221104236979928e-0016
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Double precision, Remez error <= 2**(-62.9)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpsin(t) = t + ks[0] * t + ks[1] * t + ... + ks[6] * t
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[0] = -1.644934066848226406065691 (0x3ffa51a6 625307d3)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[1] = 8.11742425283341655883668741874008920850698590621e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[2] = -1.90751824120862873825597279118304943994042258291e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[3] = 2.61478477632554278317289628332654539353521911570e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[4] = -2.34607978510202710377617190278735525354347705866e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[5] = 1.48413292290051695897242899977121846763824221705e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[6] = -6.87730769637543488108688726777687262485357072242e-0006
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Single precision, Remez error <= 2**(-34.09)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpsin(t) = t + ks[0] * t + ks[1] * t + ... + ks[3] * t
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[0] = -1.64493404985645811354476665052005342839447790544e+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[1] = 8.11740794458351064092797249069438269367389272270e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[2] = -1.90703144603551216933075809162889536878854055202e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ks[3] = 2.55742333994264563281155312271481108635575331201e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Computation note: in simulating higher precision arithmetic, kcpsin
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * return head = t and tail = kc[0]*t^3 + (...) to maintain extra bits
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * And for kpcos(t) for |t|< 0.183:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Quad precision, remez <= 2**(-122.48)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpcos(t) = 1/pi + pi/2 * t + kc[2] * t + ... + kc[11] * t
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[2] = 1.29192819501249250731151312779548918765320728489e+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[3] = -4.25027339979557573976029596929319207009444090366e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[4] = 7.49080661650990096109672954618317623888421628613e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[5] = -8.21458866111282287985539464173976555436050215120e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[6] = 6.14202578809529228503205255165761204750211603402e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[7] = -3.33073432691149607007217330302595267179545908740e-0005
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[8] = 1.36970959047832085796809745461530865597993680204e-0006
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[9] = -4.41780774262583514450246512727201806217271097336e-0008
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[10]= 1.14741409212381858820016567664488123478660705759e-0009
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[11]= -2.44261236114707374558437500654381006300502749632e-0011
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Double precision, remez < 2**(61.91)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpcos(t) = 1/pi + pi/2 *t + kc[2] * t + ... + kc[6] * t
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[2] = 1.29192819501230224953283586722575766189551966008e+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[3] = -4.25027339940149518500158850753393173519732149213e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[4] = 7.49080625187015312373925142219429422375556727752e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[5] = -8.21442040906099210866977352284054849051348692715e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[6] = 6.10411356829515414575566564733632532333904115968e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Single precision, remez < 2**(-30.13)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpcos(t) = kc[0] + kc[1] * t + ... + kc[3] * t
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[0] = 3.18309886183790671537767526745028724068919291480e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[1] = -1.57079581447762568199467875065854538626594937791e+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[2] = 1.29183528092558692844073004029568674027807393862e+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kc[3] = -4.20232949771307685981015914425195471602739075537e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Computation note: in simulating higher precision arithmetic, kcpcos
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * return head = 1/pi chopped, and tail = pi/2 *t^2 + (tail part of 1/pi
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * + ...) to maintain extra bits precision. In particular, pi/2 * t^2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * is calculated with great care.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Thus, the computation of gamma(-x), x>0, is:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Let k = int(x), z = x-k.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * For z in (I)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(-x) = ------------------- ;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpsin(z)*gamma(1+x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * otherwise, for z in (II),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(-x) = ----------------------- ;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpcos(0.5-z)*gamma(1+x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * otherwise, for z in (III),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(-x) = --------------------- .
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpsin(1-z)*gamma(1+x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Thus, the computation of gamma(-x) reduced to the computation of
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(1+x) and kpsin(), kpcos().
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (B) For x between 1 and 2. We break [1,2] into three parts:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GT1 = [1.0000, 1.2845]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GT2 = [1.2844, 1.6374]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GT3 = [1.6373, 2.0000]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * For x in GTi, i=1,2,3, let
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * z1 = 1.134861805732790769689793935774652917006
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gz1 = gamma(z1) = 0.9382046279096824494097535615803269576988
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * tz1 = gamma'(z1) = -0.3517214357852935791015625000000000000000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * z2 = 1.461632144968362341262659542325721328468e+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gz2 = gamma(z2) = 0.8856031944108887002788159005825887332080
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * tz2 = gamma'(z2) = 0.00
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * z3 = 1.819773101100500601787868704921606996312e+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gz3 = gamma(z3) = 0.9367814114636523216188468970808378497426
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * tz3 = gamma'(z3) = 0.2805306315422058105468750000000000000000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * y = x-zi ... for extra precision, write y = y.h + y.l
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(x) = gzi + tzi*(y.h+y.l) + y*y*Ri(y),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = gzi.h + (tzi*y.h + ((tzi*y.l+gzi.l) + y*y*Ri(y)))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (I) For double precision
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Ri(y) = Pi(y)/Qi(y), i=1,2,3;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P1(y) = p1[0] + p1[1]*y + ... + p1[4]*y^4
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Q1(y) = q1[0] + q1[1]*y + ... + q1[5]*y^5
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P2(y) = p2[0] + p2[1]*y + ... + p2[3]*y^3
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Q2(y) = q2[0] + q2[1]*y + ... + q2[6]*y^6
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P3(y) = p3[0] + p3[1]*y + ... + p3[4]*y^4
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Q3(y) = q3[0] + q3[1]*y + ... + q3[5]*y^5
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Remez precision of Ri(y):
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |gamma(x)-(gzi+tzi*y) - y*y*Ri(y)| <= 2**-62.3 ... for i = 1
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * <= 2**-59.4 ... for i = 2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * <= 2**-62.1 ... for i = 3
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (II) For quad precision
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Ri(y) = Pi(y)/Qi(y), i=1,2,3;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P1(y) = p1[0] + p1[1]*y + ... + p1[9]*y^9
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Q1(y) = q1[0] + q1[1]*y + ... + q1[8]*y^8
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P2(y) = p2[0] + p2[1]*y + ... + p2[9]*y^9
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Q2(y) = q2[0] + q2[1]*y + ... + q2[9]*y^9
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P3(y) = p3[0] + p3[1]*y + ... + p3[9]*y^9
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Q3(y) = q3[0] + q3[1]*y + ... + q3[9]*y^9
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Remez precision of Ri(y):
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |gamma(x)-(gzi+tzi*y) - y*y*Ri(y)| <= 2**-118.2 ... for i = 1
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * <= 2**-126.8 ... for i = 2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * <= 2**-119.5 ... for i = 3
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (III) For single precision
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Ri(y) = Pi(y), i=1,2,3;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P1(y) = p1[0] + p1[1]*y + ... + p1[5]*y^5
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P2(y) = p2[0] + p2[1]*y + ... + p2[5]*y^5
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P3(y) = p3[0] + p3[1]*y + ... + p3[4]*y^4
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Remez precision of Ri(y):
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |gamma(x)-(gzi+tzi*y) - y*y*Ri(y)| <= 2**-30.8 ... for i = 1
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * <= 2**-31.6 ... for i = 2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * <= 2**-29.5 ... for i = 3
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Notes. (1) GTi and zi are choosen to balance the interval width and
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * minimize the distant between gamma(x) and the tangent line at
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * zi. In particular, we have
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |gamma(x)-(gzi+tzi*(x-zi))| <= 0.01436... for x in [1,z2]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * <= 0.01265... for x in [z2,2]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (2) zi are slightly adjusted so that tzi=gamma'(zi) is very
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * close to a single precision value.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Coefficents: Single precision
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P1[0] = 7.09087253435088360271451613398019280077561279443e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P1[1] = -5.17229560788652108545141978238701790105241761089e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P1[2] = 5.23403394528150789405825222323770647162337764327e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P1[3] = -4.54586308717075010784041566069480411732634814899e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P1[4] = 4.20596490915239085459964590559256913498190955233e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * P1[5] = -3.57307589712377520978332185838241458642142185789e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[0] = 4.28486983980295198166056119223984284434264344578e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[1] = -1.30704539487709138528680121627899735386650103914e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[2] = 1.60856285038051955072861219352655851542955430871e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[3] = -9.22285161346010583774458802067371182158937943507e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[4] = 7.19240511767225260740890292605070595560626179357e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[5] = -4.88158265593355093703112238534484636193260459574e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[0] = 3.82409531118807759081121479786092134814808872880e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[1] = 2.65309888180188647956400403013495759365167853426e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[2] = 8.06815109775079171923561169415370309376296739835e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[3] = -1.54821591666137613928840890835174351674007764799e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[4] = 1.76308239242717268530498313416899188157165183405e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Coefficents: Double precision
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[0] = 0.70908683619977797008004927192814648151397705078125000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[1] = 1.71987061393048558089579513384356441668351720061e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[2] = -3.19273345791990970293320316122813960527705450671e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[3] = 8.36172645419110036267169600390549973563534476989e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[4] = 1.13745336648572838333152213474277971244629758101e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[1] = 9.71980217826032937526460731778472389791321968082e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[2] = -7.43576743326756176594084137256042653497087666030e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[3] = -1.19345944932265559769719470515102012246995255372e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[4] = 1.59913445751425002620935120470781382215050284762e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[5] = 1.12601136853374984566572691306402321911547550783e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[0] = 0.42848681585558601181418225678498856723308563232421875
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[1] = 6.53596762668970816023718845105667418483122103629e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[2] = -6.97280829631212931321050770925128264272768936731e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[3] = 6.46342359021981718947208605674813260166116632899e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[1] = 4.57572620560506047062553957454062012327519313936e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[2] = -2.52182594886075452859655003407796103083422572036e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[3] = -1.82970945407778594681348166040103197178711552827e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[4] = 2.43574726993169566475227642128830141304953840502e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[5] = -5.20390406466942525358645957564897411258667085501e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[6] = 4.79520251383279837635552431988023256031951133885e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[0] = 0.382409479734567459008331979930517263710498809814453125
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[1] = 1.42876048697668161599069814043449301572928034140e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[2] = 3.42157571052250536817923866013561760785748899071e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[3] = -5.01542621710067521405087887856991700987709272937e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[4] = 8.89285814866740910123834688163838287618332122670e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[1] = 3.04253086629444201002215640948957897906299633168e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[2] = -2.23162407379999477282555672834881213873185520006e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[3] = -1.05060867741952065921809811933670131427552903636e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[4] = 1.70511763916186982473301861980856352005926669320e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[5] = -2.12950201683609187927899416700094630764182477464e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Note that all pi0 are exact in double, which is obtained by a
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * special Remez Algorithm.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Coefficents: Quad precision
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[0] = 0.709086836199777919037185741507610124611513720557
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[1] = 4.45754781206489035827915969367354835667391606951e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[2] = 3.21049298735832382311662273882632210062918153852e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[3] = -5.71296796342106617651765245858289197369688864350e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[4] = 6.04666892891998977081619174969855831606965352773e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[5] = 8.99106186996888711939627812174765258822658645168e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[6] = -6.96496846144407741431207008527018441810175568949e-0005
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[7] = 1.52597046118984020814225409300131445070213882429e-0005
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[8] = 5.68521076168495673844711465407432189190681541547e-0007
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p1[9] = 3.30749673519634895220582062520286565610418952979e-0008
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[0] = 1.0+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[1] = 1.35806511721671070408570853537257079579490650668e+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[2] = 2.97567810153429553405327140096063086994072952961e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[3] = -1.52956835982588571502954372821681851681118097870e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[4] = -2.88248519561420109768781615289082053597954521218e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[5] = 1.03475311719937405219789948456313936302378395955e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[6] = 4.12310203243891222368965360124391297374822742313e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[7] = -3.12653708152290867248931925120380729518332507388e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q1[8] = 2.36672170850409745237358105667757760527014332458e-0005
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[0] = 0.428486815855585429730209907810650616737756697477
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[1] = 2.63622124067885222919192651151581541943362617352e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[2] = 3.85520683670028865731877276741390421744971446855e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[3] = 3.05065978278128549958897133190295325258023525862e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[4] = 2.48232934951723128892080415054084339152450445081e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[5] = 3.67092777065632360693313762221411547741550105407e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[6] = 3.81228045616085789674530902563145250532194518946e-0006
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[7] = 4.61677225867087554059531455133839175822537617677e-0006
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[8] = 2.18209052385703200438239200991201916609364872993e-0007
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p2[9] = 1.00490538985245846460006244065624754421022542454e-0008
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[1] = 9.20276350207639290567783725273128544224570775056e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[2] = -4.79533683654165107448020515733883781138947771495e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[3] = -1.24538337585899300494444600248687901947684291683e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[4] = 4.49866050763472358547524708431719114204535491412e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[5] = 7.20715455697920560621638325356292640604078591907e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[6] = -8.68513169029126780280798337091982780598228096116e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[7] = -1.25104431629401181525027098222745544809974229874e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[8] = 3.10558344839000038489191304550998047521253437464e-0005
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q2[9] = -1.76829227852852176018537139573609433652506765712e-0006
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[0] = 0.3824094797345675048502747661075355640070439388902
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[1] = 3.42198093076618495415854906335908427159833377774e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[2] = 9.63828189500585568303961406863153237440702754858e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[3] = 8.76069421042696384852462044188520252156846768667e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[4] = 1.86477890389161491224872014149309015261897537488e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[5] = 8.16871354540309895879974742853701311541286944191e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[6] = 6.83783483674600322518695090864659381650125625216e-0005
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[7] = -1.10168269719261574708565935172719209272190828456e-0006
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[8] = 9.66243228508380420159234853278906717065629721016e-0007
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p3[9] = 2.31858885579177250541163820671121664974334728142e-0008
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[1] = 8.25479821168813634632437430090376252512793067339e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[2] = -1.62251363073937769739639623669295110346015576320e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[3] = -1.10621286905916732758745130629426559691187579852e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[4] = 3.48309693970985612644446415789230015515365291459e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[5] = 6.73553737487488333032431261131289672347043401328e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[6] = -7.63222008393372630162743587811004613050245128051e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[7] = -1.35792670669190631476784768961953711773073251336e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[8] = 3.19610150954223587006220730065608156460205690618e-0005
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q3[9] = -1.82096553862822346610109522015129585693354348322e-0006
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (C) For x between 0 and 1.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Let P stand for the number of significant bits in the working precision.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (1)For 0 <= x <= 2 , gamma(x) is computed by --- rounded to nearest.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The error is bound by 0.739 ulp(gamma(x)) in IEEE double precision.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Since -------- ~ x + 0.577...*x - ..., we have, for small x,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ----------- < gamma(x) < --- and
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * x(1+0.578x) x
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 0 < --- - gamma(x) <= --- - ----------- < 0.578
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * x x x(1+0.578x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The error is thus bounded by --- ulp(---) + 0.578. Since x <= 2 ,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * --- >= 2 , ulp(---) >= ulp(2 ) >= 2. Thus 0.578=0.289*2<=0.289ulp(-)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * | gamma(x) - [---] rounded | <= (0.5+0.289)*ulp(---).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Note that for x<= 2 , it is easy to see that ulp(---)=ulp(gamma(x))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * except only when x = 2 , (n<= -53). In such cases, --- is exact
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * and therefore the error is bounded by
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 0.298*ulp(---) = 0.298*2*ulp(gamma(x)) = 0.578ulp(gamma(x)).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Thus we conclude that the error in gamma is less than 0.739 ulp.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (2)Otherwise, for x in GTi-1 (see B), let y = x-(zi-1). From (B) we obtain
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(1+x) = gy.h + gy.l, then compute gamma(x) by -----------.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Implementaion note. Write x = x.h+x.l, and Let th = ----- chopped to
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 20 bits, then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(x) = th + (---------- - th )
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = th + ---*(gy.h-th*x.h+gy.l-th*x.l)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (D) For x between 2 and 8. Let n = 1+x chopped to an integer. Then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(x)=(x-1)*(x-2)*...*(x-n)*gamma(x-n)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Since x-n is between 1 and 2, we can apply (B) to compute gamma(x).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Implementation detail. The computation of (x-1)(x-2)...(x-n) in simulated
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * higher precision arithmetic can be somewhat optimized. For example, in
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * computing (x-1)*(x-2)*(x-3)*(x-4), if we compute (x-1)*(x-4) = z.h+z.l,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * then (x-2)(x-3) = z.h+2+z.l readily. In below, we list the expression
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * of the formula to compute gamma(x).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Assume x-n is in GTi (i=1,2, or 3, see B for detail). Let y = x - n - zi.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * By (B) we have gamma(x-n) = gy.h+gy.l. If x = x.h+x.l, then we have
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * n=1 (x in [2,3]):
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(x) = (x-1)*gamma(x-1) = (x-1)*(gy.h+gy.l)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = [(x.h-1)+x.l]*(gy.h+gy.l)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * n=2 (x in [3,4]):
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(x) = (x-1)(x-2)*gamma(x-2) = (x-1)*(x-2)*(gy.h+gy.l)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = ((x.h-2)+x.l)*((x.h-1)+x.l)*(gy.h+gy.l)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = [x.h*(x.h-3)+2+x.l*(x+(x.h-3))]*(gy.h+gy.l)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * n=3 (x in [4,5])
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(x) = (x-1)(x-2)(x-3)*(gy.h+gy.l)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = (x.h*(x.h-3)+2+x.l*(x+(x.h-3)))*[((x.h-3)+x.l)(gy.h+gy.l)]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * n=4 (x in [5,6])
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(x) = [(x-1)(x-4)]*[(x-2)(x-3)]*(gy.h+gy.l)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = [(x.h*(x.h-5)+4+x.l(x+(x.h-5)))]*[(x-2)*(x-3)]*(gy.h+gy.l)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = (y.h+y.l)*(y.h+1+y.l)*(gy.h+gy.l)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * n=5 (x in [6,7])
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(x) = [(x-1)(x-4)]*[(x-2)(x-3)]*[(x-5)*(gy.h+gy.l)]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * n=6 (x in [7,8])
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(x) = [(x-1)(x-6)]*[(x-2)(x-5)]*[(x-3)(x-4)]*(gy.h+gy.l)]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = [(y.h+y.l)(y.h+4+y.l)][(y.h+6+y.l)(gy.h+gy.l)]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (E)Overflow Thresold. For x > Overflow thresold of gamma,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * return huge*huge (overflow).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * By checking whether lgamma(x) >= 2**{128,1024,16384}, one can
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * determine the overflow threshold for x in single, double, and
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * quad precision. See over.c for details.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The overflow threshold of gamma(x) are
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * single: x = 3.5040096283e+01
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = 0x420C290F (IEEE single)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * double: x = 1.71624376956302711505e+02
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = 0x406573FAE561F647 (IEEE double)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * quad: x = 1.7555483429044629170038892160702032034177e+03
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = 0x4009B6E3180CD66A5C4206F128BA77F4 (quad)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (F)For overflow_threshold >= x >= 8, we use asymptotic approximation.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (1) Stirling's formula
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = L1 + L2 + L3,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * L1(x) = (x-.5)*(log(x)-1),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * L2 = .5(log(2pi)-1) = 0.41893853....,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * L3(x) = (1/x)P(1/(x*x)),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The range of L1,L2, and L3 are as follows:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ------------------------------------------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Range(L1) = (single) [8.09..,88.30..] =[2** 3.01..,2** 6.46..]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (double) [8.09..,709.3..] =[2** 3.01..,2** 9.47..]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (quad) [8.09..,11356.10..]=[2** 3.01..,2** 13.47..]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Range(L2) = 0.41893853.....
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Range(L3) = [0.0104...., 0.00048....] =[2**-6.58..,2**-11.02..]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ------------------------------------------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Gamma(x) is then computed by exp(L1+L2+L3).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (2) Error analysis of (F):
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * --------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The error in Gamma(x) depends on the error inherited in the computation
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * of L= L1+L2+L3. Let L' be the computed value of L. The absolute error
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * in L' is t = L-L'. Since exp(L') = exp(L-t) = exp(L)*exp(t) ~
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (1+t)*exp(L), the relative error in exp(L') is approximately t.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * To guarantee the relatively accuracy in exp(L'), we would like
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |t| < 2**(-P-5) where P denotes for the number of significant bits
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * of the working precision. Consequently, each of the L1,L2, and L3
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * must be computed with absolute error bounded by 2**(-P-5) in absolute
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Since L2 is a constant, it can be pre-computed to the desired accuracy.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Also |L3| < 2**-6; therefore, it suffices to compute L3 with the
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * working precision. That is,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * L3(x) approxmiate log(G(x))-(x-.5)(log(x)-1)-.5(log(2pi)-1)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * to a precision bounded by 2**(-P-5).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * _________V___________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * L1(x): |_________|___________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * __ ________________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * L2: |__|________________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * __________________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * + L3(x): |__________________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -------------------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * [leading] + [Trailing]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * For L1(x)=(x-0.5)*(log(x)-1), we need ilogb(L1(x))+5 extra bits for
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * both multiplicants to guarantee L1(x)'s absolute error is bounded by
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2**(-P-5) in absolute value. Here ilogb(y) is defined to be the unbias
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * binary exponent of y in IEEE format. We can get x-0.5 to the desire
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * accuracy easily. It remains to compute log(x)-1 with ilogb(L1(x))+5
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * extra bits accracy. Note that the range of L1 is 88.30.., 709.3.., and
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 11356.10... for single, double, and quadruple precision, we have
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * single double quadruple
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ------------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ilogb(L1(x))+5 <= 11 14 18
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ------------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (3) Table Driven Method for log(x)-1:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * --------------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Let x = 2**n * y, where 1 <= y < 2. Let Z={z(i),i=1,...,m}
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * be a set of predetermined evenly distributed floating point numbers
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * in [1, 2]. Let z(j) be the closest one to y, then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * log(x)-1 = n*log(2)-1 + log(y)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = n*log(2)-1 + log(z(j)*y/z(j))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = n*log(2)-1 + log(z(j)) + log(y/z(j))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = T1(n) + T2(j) + T3,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * where T1(n) = n*log(2)-1 and T2(j) = log(z(j)). Both T1 and T2 can be
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * pre-calculated and be looked-up in a table. Note that 8 <= x < 1756
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * implies 3<=n<=10 implies 1.079.. < T1(n) < 6.931.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * y-z(i) y 1+s
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * For T3, let s = --------; then ----- = ----- and
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * y+z(i) z(i) 1-s
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T3 = log(-----) = 2s + --- s + --- s + ....
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Suppose the first term 2s is compute in extra precision. The
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * dominating error in T3 would then be the rounding error of the
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * second term 2/3*s**3. To force the rounding bounded by
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * the required accuracy, we have
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * single: |2/3*s**3| < 2**-11 == > |s|<0.09014...
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * double: |2/3*s**3| < 2**-14 == > |s|<0.04507...
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * quad : |2/3*s**3| < 2**-18 == > |s|<0.01788... = 2**(-5.80..)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Base on this analysis, we choose Z = {z(i)|z(i)=1+i/64+1/128, 0<=i<=63}.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * For any y in [1,2), let j = [64*y] chopped to integer, then z(j) is
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * the closest to y, and it is not difficult to see that |s| < 2**(-8).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Please note that the polynomial approximation of T3 must be accurate
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -24-11 -35 -53-14 -67 -113-18 -131
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * to 2 =2 , 2 = 2 , and 2 =2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * for single, double, and quadruple precision respectively.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Inplementation notes.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (1) Table look-up entries for T1(n) and T2(j), as well as the calculation
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * of the leading term 2s in T3, are broken up into leading and trailing
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * part such that (leading part)* 2**24 will always be an integer. That
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * will guarantee the addition of the leading parts will be exact.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * _________V___________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T1(n): |_________|___________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * _______ ______________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T2(j): |_______|______________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ____ _______________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2s: |____|_______________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * __________________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * + T3(s)-2s: |__________________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -------------------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * [leading] + [Trailing]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (2) How to compute 2s accurately.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (A) Compute v = 2s to the working precision. If |v| < 2**(-18),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (B) chopped v to 2**(-24): v = ((int)(v*2**24))/2**24
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (C) 2s = v + (2s - v), where
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2s - v = --- * (2(y-z) - v*(y+z) )
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = --- * ( [2(y-z) - v*(y+z)_h ] - v*(y+z)_l )
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * where (y+z)_h = (y+z) rounded to 24 bits by (double)(float),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * and (y+z)_l = ((z+z)-(y+z)_h)+(y-z). Note the the quantity
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * in [] is exact.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (3) Remez approximation for (T3(s)-2s)/s = T3[0]*s + T3[1]*s + ...:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Single precision: 1 term (compute in double precision arithmetic)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T3(s) = 2s + S1*s^3, S1 = 0.6666717231848518054693623697539230
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Remez error: |T3(s)/s - (2s+S1*s^3)| < 2**(-35.87)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Double precision: 3 terms, Remez error is bounded by 2**(-72.40),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * see "tgamma_log"
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Quad precision: 7 terms, Remez error is bounded by 2**(-136.54),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * see "tgammal_log"
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The computation of 0.5*(ln(2pi)-1):
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 0.5*(ln(2pi)-1) = 0.4189385332046727417803297364056176398614...
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * split 0.5*(ln(2pi)-1) to hln2pi_h + hln2pi_l, where hln2pi_h is the
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * leading 21 bits of the constant.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * hln2pi_h= 0.4189383983612060546875
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * hln2pi_l= 1.348434666870928297364056176398612173648e-07
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The computation of 1/x*P(1/x^2) = log(G(x))-(x-.5)(ln(x)-1)-(.5ln(2pi)-1):
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Let s = 1/x <= 1/8 < 0.125. We have
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * quad precision
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |GP(s) - s*P(s^2)| <= 2**(-120.6), where
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP(s) = GP0*s+GP1*s +GP2*s +... +GP19*s ,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP0 = 0.083333333333333333333333333333333172839171301
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * hex 0x3ffe5555 55555555 55555555 55555548
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP1 = -2.77777777777777777777777777492501211999399424104e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP2 = 7.93650793650793650793635650541638236350020883243e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP3 = -5.95238095238095238057299772679324503339241961704e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP4 = 8.41750841750841696138422987977683524926142600321e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP5 = -1.91752691752686682825032547823699662178842123308e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP6 = 6.41025641022403480921891559356473451161279359322e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP7 = -2.95506535798414019189819587455577003732808185071e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP8 = 1.79644367229970031486079180060923073476568732136e-0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP9 = -1.39243086487274662174562872567057200255649290646e+0000
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP10 = 1.34025874044417962188677816477842265259608269775e+0001
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP11 = -1.56803713480127469414495545399982508700748274318e+0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP12 = 2.18739841656201561694927630335099313968924493891e+0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP13 = -3.55249848644100338419187038090925410976237921269e+0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP14 = 6.43464880437835286216768959439484376449179576452e+0005
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP15 = -1.20459154385577014992600342782821389605893904624e+0007
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP16 = 2.09263249637351298563934942349749718491071093210e+0008
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP17 = -2.96247483183169219343745316433899599834685703457e+0009
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP18 = 2.88984933605896033154727626086506756972327292981e+0010
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP19 = -1.40960434146030007732838382416230610302678063984e+0011
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * double precision
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |GP(s) - s*P(s^2)| <= 2**(-63.5), where
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 3 5 7 9 11 13 15
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP(s) = GP0*s+GP1*s +GP2*s +GP3*s +GP4*s +GP5*s +GP6*s +GP7*s ,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP0= 0.0833333333333333287074040640618477 (3FB55555 55555555)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP1= -2.77777777776649355200565611114627670089130772843e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP2= 7.93650787486083724805476194170211775784158551509e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP3= -5.95236628558314928757811419580281294593903582971e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP4= 8.41566473999853451983137162780427812781178932540e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP5= -1.90424776670441373564512942038926168175921303212e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP6= 5.84933161530949666312333949534482303007354299178e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP7= -1.59453228931082030262124832506144392496561694550e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * single precision
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |GP(s) - s*P(s^2)| <= 2**(-37.78), where
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP(s) = GP0*s+GP1*s +GP2*s
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP0 = 8.33333330959694065245736888749042811909994573178e-0002
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP1 = -2.77765545601667179767706600890361535225507762168e-0003
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * GP2 = 7.77830853479775281781085278324621033523037489883e-0004
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Implementation note:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * z = (1/x), z2 = z*z, z4 = z2*z2;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * p = z*(GP0+z2*(GP1+....+z2*GP7))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = z*(GP0+(z4*(GP2+z4*(GP4+z4*GP6))+z2*(GP1+z4*(GP3+z4*(GP5+z4*GP7)))))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Adding everything up:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * t = rr.h*ww.h+hln2pi_h ... exact
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * w = (hln2pi_l + ((x-0.5)*ww.l+rr.l*ww.h)) + p
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Computing exp(t+w):
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * s = t+w; write s = (n+j/32)*ln2+r, |r|<=(1/64)*ln2, then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * exp(s) = 2**n * (2**(j/32) + 2**(j/32)*expm1(r)), where
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * expm1(r) = r + Et1*r^2 + Et2*r^3 + ... + Et5*r^6, and
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2**(j/32) is obtained by table look-up S[j]+S_trail[j].
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Remez error bound:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |exp(r) - (1+r+Et1*r^2+...+Et5*r^6)| <= 2^(-63).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* Hex value of GP0 shoule be 3FB55555 55555555 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisstatic const double c[] = {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.66666666666666740682e-01, /* A1=T3[0] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.99999999955626478023093908674902212920e-01, /* A2=T3[1] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.85720221533145659809237398709372330980e-01, /* A3=T3[2] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +0.0833333333333333287074040640618477, /* GP[0] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -2.77777777776649355200565611114627670089130772843e-03,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +7.93650787486083724805476194170211775784158551509e-04,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -5.95236628558314928757811419580281294593903582971e-04,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.41566473999853451983137162780427812781178932540e-04,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.90424776670441373564512942038926168175921303212e-03,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.84933161530949666312333949534482303007354299178e-03,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.59453228931082030262124832506144392496561694550e-02,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.18937683105468750000e-01, /* hln2pi_h */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.50099203991780279640e-07, /* hln2pi_l */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.18938533204672741744150788368695779923320328369e-01, /* hln2pi */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.16608493865351192653e-02, /* ln2_32hi */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.96317165397058656257e-12, /* ln2_32lo */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.61662413084468283841e+01, /* invln2_32 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.66666666665223585560605991943703896196054020060e-01, /* Et2 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.16666666665895103520154073534275286743788421687e-02, /* Et3 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.33336844093536520775865096538773197505523826029e-03, /* Et4 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.38889201930843436040204096950052984793587640227e-03, /* Et5 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * double precision coefficients for computing log(x)-1 in tgamma.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * See "algorithm" for details
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T1(n) = T1[2n,2n+1] = n*log(2)-1,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T2(j) = T2[2j,2j+1] = log(z[j]),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T3(s) = 2s + T3[0]s^3 + T3[1]s^5 + T3[2]s^7
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = 2s + A1*s^3 + A2*s^5 + A3*s^7 (see const A1,A2,A3)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (1) the leading entries are truncated to 24 binary point.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (2) Remez error for T3(s) is bounded by 2**(-72.4)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.00000000000000000000e+00, /* 0xBFF00000 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +0.00000000000000000000e+00, /* 0x00000000 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -3.06852817535400390625e-01, /* 0xBFD3A37A 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.90465429995776763166e-09, /* 0xBE205C61 0x0CA86C38 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.86294305324554443359e-01, /* 0x3FD8B90B 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.57953361754750897367e-08, /* 0x3E6DF473 0xDE6AF279 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.07944148778915405273e+00, /* 0x3FF14564 0x70000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.38906818755173187963e-08, /* 0x3E6CEEAD 0xCDA06BB5 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.77258867025375366211e+00, /* 0x3FFC5C85 0xF0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.19860275755595544734e-08, /* 0x3E6BE8E7 0xBCD5E4F2 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.46573585271835327148e+00, /* 0x4003B9D3 0xB8000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.00813732756017835330e-08, /* 0x3E6AE321 0xAC0B5E2E */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.15888303518295288086e+00, /* 0x40094564 0x78000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.81767189756440192100e-08, /* 0x3E69DD5B 0x9B40D76B */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.85203021764755249023e+00, /* 0x400ED0F5 0x38000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.62720646756862482697e-08, /* 0x3E68D795 0x8A7650A7 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.54517740011215209961e+00, /* 0x40122E42 0xFC000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.43674103757284839467e-08, /* 0x3E67D1CF 0x79ABC9E4 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.23832458257675170898e+00, /* 0x4014F40B 0x5C000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.24627560757707130063e-08, /* 0x3E66CC09 0x68E14320 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.93147176504135131836e+00, /* 0x4017B9D3 0xBC000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.05581017758129486834e-08, /* 0x3E65C643 0x5816BC5D */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +7.78210163116455078125e-03, /* 0x3F7FE020 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.88108903981662140884e-08, /* 0x3E64D620 0xCF11F86F */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.31670141220092773438e-02, /* 0x3F97B918 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.51595251008850513740e-08, /* 0x3E683EAD 0x88D54940 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.83188128471374511719e-02, /* 0x3FA39E86 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.14549991480218823411e-08, /* 0x3E6B9FEB 0xD5FA9016 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.32444715499877929688e-02, /* 0x3FAB42DC 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.29688244898971182165e-08, /* 0x3E671197 0x1BEC28D1 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.79506063461303710938e-02, /* 0x3FB16536 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.55623773783008185114e-08, /* 0x3E6DD46F 0x5C1D0C4C */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.24436545372009277344e-02, /* 0x3FB51B07 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.46738736635337847313e-08, /* 0x3E4F830C 0x1FB493C7 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +9.67295765876770019531e-02, /* 0x3FB8C345 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.98708741103424492282e-08, /* 0x3E6AC633 0x641EB597 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.10814332962036132812e-01, /* 0x3FBC5E54 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.33782539813823062226e-08, /* 0x3E61EB78 0xE862BAC3 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.24703466892242431641e-01, /* 0x3FBFEC91 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.16087148042227818450e-08, /* 0x3E48EDF5 0x5D551729 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.38402283191680908203e-01, /* 0x3FC1B72A 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.96674382274822001957e-08, /* 0x3E654BD9 0xE80A4181 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.51916027069091796875e-01, /* 0x3FC371FC 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.49567501781968021494e-08, /* 0x3E500F47 0xBA1DE6CB */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.65249526500701904297e-01, /* 0x3FC526E5 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.63946052585787334062e-08, /* 0x3E68E86D 0x0DE8B900 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.78407609462738037109e-01, /* 0x3FC6D60F 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.80100802600100279538e-08, /* 0x3E69C674 0x8723551E */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.91394805908203125000e-01, /* 0x3FC87FA0 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.70914263296092971436e-08, /* 0x3E694832 0x44240802 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.04215526580810546875e-01, /* 0x3FCA23BC 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.48478803446288209001e-08, /* 0x3E4FE2B5 0x63193712 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.16873884201049804688e-01, /* 0x3FCBC286 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.40995645549315919488e-08, /* 0x3E6D0B63 0x358A7E74 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.29374051094055175781e-01, /* 0x3FCD5C21 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.99707906542102284117e-08, /* 0x3E6AD3EE 0xE456E443 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.41719901561737060547e-01, /* 0x3FCEF0AD 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.53254081075974352804e-08, /* 0x3E62F716 0x4D948638 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.53915190696716308594e-01, /* 0x3FD04025 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.92842471355435739091e-08, /* 0x3E54B4D0 0x40DAE27C */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.65963494777679443359e-01, /* 0x3FD1058B 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.37194584979797487125e-08, /* 0x3E6CD725 0x6A8C4FD0 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.77868449687957763672e-01, /* 0x3FD1C898 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.31549854251447496506e-09, /* 0x3E16999F 0xAFBC68E7 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.89633274078369140625e-01, /* 0x3FD2895A 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.85046735362538929911e-08, /* 0x3E53DE86 0xA35EB493 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.01261305809020996094e-01, /* 0x3FD347DD 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.47691407849191245052e-08, /* 0x3E5A987D 0x54D64567 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.12755703926086425781e-01, /* 0x3FD40430 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.07781046260499658610e-09, /* 0x3E3A1A9F 0x8EF4304A */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.24119448661804199219e-01, /* 0x3FD4BE5F 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.99924077768719198045e-08, /* 0x3E557778 0xA0DB4C99 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.35355520248413085938e-01, /* 0x3FD57677 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.16727247443196802771e-08, /* 0x3E57455A 0x6C549AB7 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.46466720104217529297e-01, /* 0x3FD62C82 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.72419910516215900493e-08, /* 0x3E695CE3 0xCA97B7B0 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.57455849647521972656e-01, /* 0x3FD6E08E 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.92742818015697624778e-08, /* 0x3E6515D0 0xF1C609CA */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.68325531482696533203e-01, /* 0x3FD792A5 0x40000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.96760111198451042238e-08, /* 0x3E5FDD47 0xA27C15DA */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.79078328609466552734e-01, /* 0x3FD842D1 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.43255029056564770289e-08, /* 0x3E5A1E8B 0x17493B14 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.89716744422912597656e-01, /* 0x3FD8F11E 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.71711261571421332726e-09, /* 0x3E3CD98B 0x1DF85DA7 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.00243163108825683594e-01, /* 0x3FD99D95 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.01818702333557515008e-09, /* 0x3E117E08 0xACBA92EF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.10659909248352050781e-01, /* 0x3FDA4840 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.57369163351530571459e-08, /* 0x3E50E5BB 0x0A2BFCA7 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.20969247817993164062e-01, /* 0x3FDAF129 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.68261364720663662040e-08, /* 0x3E6923BC 0x358899C2 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.31173443794250488281e-01, /* 0x3FDB9858 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.10241208525779214510e-08, /* 0x3E569310 0xFB598FB1 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.41274523735046386719e-01, /* 0x3FDC3DD7 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.70698288427707487748e-08, /* 0x3E63E6D6 0xA6B9D9E1 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.51274633407592773438e-01, /* 0x3FDCE1AF 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.07318658117071930723e-08, /* 0x3E470BE7 0xD6F6FA58 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.61175680160522460938e-01, /* 0x3FDD83E7 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.49616477054305011286e-08, /* 0x3E62C517 0x9F2828AE */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.70979690551757812500e-01, /* 0x3FDE2488 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.46670332000468969567e-08, /* 0x3E5A7C6C 0x261CBD8F */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.80688512325286865234e-01, /* 0x3FDEC399 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.70204650424422423704e-08, /* 0x3E52468C 0xC0175CEE */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.90303933620452880859e-01, /* 0x3FDF6123 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.44247409572909703749e-08, /* 0x3E6D3814 0x5630A2B6 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.99827861785888671875e-01, /* 0x3FDFFD2E 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +7.77056065794633071345e-09, /* 0x3E40AFE9 0x30AB2FA0 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.09261846542358398438e-01, /* 0x3FE04BDF 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.52474495483665749052e-08, /* 0x3E6DA926 0xD265FCC1 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.18607735633850097656e-01, /* 0x3FE0986F 0x40000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.85741955344967264536e-08, /* 0x3E5EAE6A 0x41723FB5 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.27867078781127929688e-01, /* 0x3FE0E449 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.08397144554263914271e-08, /* 0x3E474732 0x2FDBAB97 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.37041425704956054688e-01, /* 0x3FE12F71 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.01919275998792285777e-08, /* 0x3E6593EF 0xBC530123 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.46132385730743408203e-01, /* 0x3FE179EA 0xA0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.18673922421792693237e-08, /* 0x3E6BD899 0xA0BFC60E */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.55141448974609375000e-01, /* 0x3FE1C3B8 0x00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.85658922177154808539e-08, /* 0x3E6F713C 0x24BC94F9 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.64070105552673339844e-01, /* 0x3FE20CDC 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.27321296262276338905e-08, /* 0x3E6192AB 0x6D93503D */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.72919726371765136719e-01, /* 0x3FE2555B 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.71900203723740076878e-08, /* 0x3E5D31EF 0x96780876 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.81691682338714599609e-01, /* 0x3FE29D37 0xE0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.72959078829112371070e-08, /* 0x3E6EC2B0 0x8AC85CD7 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.90387403964996337891e-01, /* 0x3FE2E474 0x20000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.26371800367512948470e-08, /* 0x3E66E402 0x68405422 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.99008142948150634766e-01, /* 0x3FE32B13 0x20000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.66979327646159769249e-08, /* 0x3E69121D 0x71320557 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.07555210590362548828e-01, /* 0x3FE37117 0xA0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.96341792466729582847e-08, /* 0x3E654747 0xB5C5DD02 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.16029858589172363281e-01, /* 0x3FE3B684 0x40000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.86263416563663175432e-08, /* 0x3E53FFF8 0x455F1DBE */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.24433279037475585938e-01, /* 0x3FE3FB5B 0x80000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.97441791510503832111e-09, /* 0x3E4345BD 0x096D3A75 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.32766664028167724609e-01, /* 0x3FE43F9F 0xE0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.54287010493641158796e-09, /* 0x3E37CE73 0x3BD393DD */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.41031146049499511719e-01, /* 0x3FE48353 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.33714317793368531132e-08, /* 0x3E61EA88 0xDF73D5E9 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.49227917194366455078e-01, /* 0x3FE4C679 0xA0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.94307433638127158696e-08, /* 0x3E5F99DC 0x7362D1DA */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.57358050346374511719e-01, /* 0x3FE50913 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.23619855184231409785e-08, /* 0x3E5802D0 0xD6979675 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.65422618389129638672e-01, /* 0x3FE54B24 0x60000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.41559608102782173188e-08, /* 0x3E4E6652 0x5EA4550A */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.73422634601593017578e-01, /* 0x3FE58CAD 0xA0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.06105737027198329700e-08, /* 0x3E65CD79 0x893092F2 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.81359171867370605469e-01, /* 0x3FE5CDB1 0xC0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.29405324634793230630e-08, /* 0x3E6C6C17 0x648CF6E4 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.89233243465423583984e-01, /* 0x3FE60E32 0xE0000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.77733853963405370102e-08, /* 0x3E644788 0xD8CA7C89 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* S[j],S_trail[j] = 2**(j/32.) for the final computation of exp(t+w) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisstatic const double S[] = {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.00000000000000000000e+00, /* 3FF0000000000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.02189714865411662714e+00, /* 3FF059B0D3158574 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.04427378242741375480e+00, /* 3FF0B5586CF9890F */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.06714040067682369717e+00, /* 3FF11301D0125B51 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.09050773266525768967e+00, /* 3FF172B83C7D517B */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.11438674259589243221e+00, /* 3FF1D4873168B9AA */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.13878863475669156458e+00, /* 3FF2387A6E756238 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.16372485877757747552e+00, /* 3FF29E9DF51FDEE1 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.18920711500272102690e+00, /* 3FF306FE0A31B715 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.21524735998046895524e+00, /* 3FF371A7373AA9CB */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.24185781207348400201e+00, /* 3FF3DEA64C123422 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.26905095719173321989e+00, /* 3FF44E086061892D */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.29683955465100964055e+00, /* 3FF4BFDAD5362A27 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.32523664315974132322e+00, /* 3FF5342B569D4F82 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.35425554693689265129e+00, /* 3FF5AB07DD485429 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.38390988196383202258e+00, /* 3FF6247EB03A5585 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.41421356237309514547e+00, /* 3FF6A09E667F3BCD */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.44518080697704665027e+00, /* 3FF71F75E8EC5F74 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.47682614593949934623e+00, /* 3FF7A11473EB0187 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.50916442759342284141e+00, /* 3FF82589994CCE13 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.54221082540794074411e+00, /* 3FF8ACE5422AA0DB */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.57598084510788649659e+00, /* 3FF93737B0CDC5E5 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.61049033194925428347e+00, /* 3FF9C49182A3F090 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.64575547815396494578e+00, /* 3FFA5503B23E255D */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.68179283050742900407e+00, /* 3FFAE89F995AD3AD */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.71861929812247793414e+00, /* 3FFB7F76F2FB5E47 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.75625216037329945351e+00, /* 3FFC199BDD85529C */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.79470907500310716820e+00, /* 3FFCB720DCEF9069 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.83400808640934243066e+00, /* 3FFD5818DCFBA487 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.87416763411029996256e+00, /* 3FFDFC97337B9B5F */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.91520656139714740007e+00, /* 3FFEA4AFA2A490DA */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.95714412417540017941e+00, /* 3FFF50765B6E4540 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +0.00000000000000000000e+00,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +5.10922502897344389359e-17, /* 3C8D73E2A475B465 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.55188970553796365958e-17, /* 3C98A62E4ADC610A */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -7.89985396684158212226e-17, /* BC96C51039449B3A */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -3.04678207981247114697e-17, /* BC819041B9D78A76 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.04102784568455709549e-16, /* 3C9E016E00A2643C */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.91281267602540777782e-17, /* 3C99B07EB6C70573 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.82920483692409349872e-17, /* 3C8612E8AFAD1255 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.98201523146564611098e-17, /* 3C86F46AD23182E4 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -7.71263069268148813091e-17, /* BC963AEABF42EAE2 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.65802759183693679123e-17, /* 3C8ADA0911F09EBC */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.66793213134218609523e-18, /* 3C489B7A04EF80D0 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.53825027948883149593e-17, /* 3C7D4397AFEC42E2 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -2.85873121003886075697e-17, /* BC807ABE1DB13CAC */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +7.70094837980298946162e-17, /* 3C96324C054647AD */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -6.77051165879478628716e-17, /* BC9383C17E40B497 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -9.66729331345291345105e-17, /* BC9BDD3413B26456 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -3.02375813499398731940e-17, /* BC816E4786887A99 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -3.48399455689279579579e-17, /* BC841577EE04992F */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.01645532775429503911e-16, /* BC9D4C1DD41532D8 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +7.94983480969762085616e-17, /* 3C96E9F156864B27 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.01369164712783039808e-17, /* BC675FC781B57EBC */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.47071925697978878522e-17, /* 3C7C7C46B071F2BE */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.01256799136747726038e-16, /* BC9D2F6EDB8D41E1 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.19901002058149652013e-17, /* 3C97A1CD345DCC81 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.85138041826311098821e-17, /* BC75584F7E54AC3B */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.96014069544887330703e-17, /* 3C811065895048DD */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.82274584279120867698e-17, /* 3C7503CBD1E949DB */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.28310722424562658722e-17, /* 3C82ED02D75B3706 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -6.12276341300414256164e-17, /* BC91A5CD4F184B5C */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.06199460561959626376e-16, /* BC9E9C23179C2893 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.96076779103666776760e-17, /* 3C99D3E12DD8A18B */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* Primary interval GTi() */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +0.70908683619977797008004927192814648151397705078125000,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.71987061393048558089579513384356441668351720061e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -3.19273345791990970293320316122813960527705450671e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.36172645419110036267169600390549973563534476989e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.13745336648572838333152213474277971244629758101e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +9.71980217826032937526460731778472389791321968082e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -7.43576743326756176594084137256042653497087666030e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.19345944932265559769719470515102012246995255372e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.59913445751425002620935120470781382215050284762e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.12601136853374984566572691306402321911547550783e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +0.42848681585558601181418225678498856723308563232421875,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.53596762668970816023718845105667418483122103629e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -6.97280829631212931321050770925128264272768936731e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.46342359021981718947208605674813260166116632899e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.57572620560506047062553957454062012327519313936e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -2.52182594886075452859655003407796103083422572036e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.82970945407778594681348166040103197178711552827e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.43574726993169566475227642128830141304953840502e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -5.20390406466942525358645957564897411258667085501e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +4.79520251383279837635552431988023256031951133885e-0004,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +0.382409479734567459008331979930517263710498809814453125,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.42876048697668161599069814043449301572928034140e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.42157571052250536817923866013561760785748899071e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -5.01542621710067521405087887856991700987709272937e-0004,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.89285814866740910123834688163838287618332122670e-0004,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +3.04253086629444201002215640948957897906299633168e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -2.23162407379999477282555672834881213873185520006e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.05060867741952065921809811933670131427552903636e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.70511763916186982473301861980856352005926669320e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -2.12950201683609187927899416700094630764182477464e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisstatic const double
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT ON */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* compute gamma(y=yh+yl) for y in GT1 = [1.0000, 1.2845] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* assume yh got 20 significant bits */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t3 = (z * (P10 + y * ((P11 + y * P12) + z * (P13 + y * P14)))) /
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis (Q10 + y * ((Q11 + y * Q12) + z * ((Q13 + Q14 * y) + z * Q15)));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis r.h = (double) ((float) (t4 + GZ1_h + t3));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* compute gamma(y=yh+yl) for y in GT2 = [1.2844, 1.6374] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* assume yh got 20 significant bits */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t3 = (z * (P20 + y * P21 + z * (P22 + y * P23))) /
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis (z * z) * ((Q24 + Q25 * y) + z * Q26))) + GZ2_l;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* compute gamma(y=yh+yl) for y in GT3 = [1.6373, 2.0000] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* assume yh got 20 significant bits */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t3 = (z * (P30 + y * ((P31 + y * P32) + z * (P33 + y * P34)))) /
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis (Q30 + y * ((Q31 + y * Q32) + z * ((Q33 + Q34 * y) + z * Q35)));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis r.h = (double) ((float) (t4 + GZ3_h + t3));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT OFF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * return tgamma(x) scaled by 2**-m for 8<x<=171.62... using Stirling's formula
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = L1 + L2 + L3,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT ON */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis double z, t1, t2, t3, z2, t5, w, y, u, r, z4, v, t24 = 16777216.0,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT OFF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * compute ss = ss.h+ss.l = log(x)-1 (see tgamma_log.h for details)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T1(n) = T1[2n,2n+1] = n*log(2)-1,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T2(j) = T2[2j,2j+1] = log(z[j]),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T3(s) = 2s + A1[0]s^3 + A2[1]s^5 + A3[2]s^7
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (1) the leading entries are truncated to 24 binary point.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (2) Remez error for T3(s) is bounded by 2**(-72.4)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * _________V___________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T1(n): |_________|___________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * _______ ______________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * T2(j): |_______|______________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ____ _______________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2s: |____|_______________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * __________________________
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * + T3(s)-2s: |__________________________|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -------------------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * [leading] + [Trailing]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT ON */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis n2 = (ix >> 20) - 0x3ff; /* exponent of x, range:3-7 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis ix = (ix & 0x000fffff) | 0x3ff00000; /* y = scale x to [1,2] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis __HI(z) = (ix & 0xffffc000) | 0x2000; /* z[j]=1+j/64+1/128 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t2 = t4 + u * ((two + z2 * A1) + (z2 * z2) * (A2 + z2 * A3));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t5 = t4 + u * (z2 * A1 + (z2 * z2) * (A2 + z2 * A3));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t2 = t5 + r * ((two * t2 - v * t1) - v * (y - (t1 - z)));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * compute ww = (x-.5)*(log(x)-1) + .5*(log(2pi)-1) + 1/x*(P(1/x^2)))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * where ss = log(x) - 1 in already in extra precision
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis r_h = (double) ((float) r);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t1 = z2 * (GP1 + z4 * (GP3 + z4 * (GP5 + z4 * GP7)));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis k = (int) ((w_h + w_l) * invln2_32 + half);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* compute the exponential of w_h+w_l */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis *m = (k >> 5);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t3 = (double) k;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* perform w - k*ln2_32 (represent as w_h - w_l) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* compute exp(w_h+w_l) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis zz.l = S_trail[j] * (one + t3) + S[j] * t3;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT OFF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpsin(x)= sin(pi*x)/pi
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 3 5 7 9 11 13 15
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = x+ks[0]*x +ks[1]*x +ks[2]*x +ks[3]*x +ks[4]*x +ks[5]*x +ks[6]*x
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.64493406684822640606569,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +8.11742425283341655883668741874008920850698590621e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.90751824120862873825597279118304943994042258291e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +2.61478477632554278317289628332654539353521911570e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -2.34607978510202710377617190278735525354347705866e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.48413292290051695897242899977121846763824221705e-0004,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -6.87730769637543488108688726777687262485357072242e-0006,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT ON */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* assume x is not tiny and positive */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t3 = (t1 * z) * ((ks[1] + z * ks[2] + t2 * ks[3]) + (z * t2) *
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT OFF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kpcos(x)= cos(pi*x)/pi
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2 4 6 8 10 12
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = 1/pi +kc[0]*x +kc[1]*x +kc[2]*x +kc[3]*x +kc[4]*x +kc[5]*x
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisstatic const double one_pi_h = 0.318309886183790635705292970,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis one_pi_l = 3.583247455607534006714276420e-17;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis npi_2_l = -0.00829632679489661923132169163975055099555883223;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -1.57079632679489661923132169163975055099555883223e+0000,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +1.29192819501230224953283586722575766189551966008e+0000,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -4.25027339940149518500158850753393173519732149213e-0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +7.49080625187015312373925142219429422375556727752e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis -8.21442040906099210866977352284054849051348692715e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis +6.10411356829515414575566564733632532333904115968e-0004,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT ON */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* assume x is not tiny and positive */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t1 = (double) ((float) x);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t2 = npi_2_l * z + npi_2_h * (x + t1) * (x - t1);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t3 = one_pi_l + x4 * ((kc[1] + z * kc[2]) + x4 * (kc[3] + z *
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t4 *= npi_2_h; /* npi_2_h is 5 bits const. The product is exact */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xx.l = x8 + t4; /* that will minimized the rounding error in xx.l */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT OFF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisstatic const double
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* 0.134861805732790769689793935774652917006 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* 0.461632144968362341262659542325721328468 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* 0.819773101100500601787868704921606996312 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* 1.134861805732790769689793935774652917006 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT ON */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* gamma(x+i) for 0 <= x < 1 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis double r1, r2, t2, z, xh, xl, yh, yl, zh, z1, z2, zl, x5, wh, wl;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* compute yy = gamma(x+1) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (x > 0.2845) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (x > 0.6374) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* compute gamma(x+i) = (x+i-1)*...*(x+1)*yy, 0<i<8 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis case 0: /* yy/x */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xh = (double) ((float) x); /* x is not tiny */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis rr.h = (double) ((float) ((yy.h + yy.l) * r1));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis zh = (double) ((float) z);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis rr.l = z * yy.l + (x - (zh - one)) * yy.h;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xh = (double) ((float) z);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xl = (x - (zh - one)) * (z2 + zh) - (xh - zh * (zh + one));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis __HI(zh) &= 0xfffffff8; /* zh 18 bits mantissa */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xh = (double) ((float) z);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xl = zl * (z2 + zh * (z1 + zh)) - (xh - zh * (zh * zh - one));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis case 5: /* ((x+1)*(x+4)*(x+2)*(x+3))*yy */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis yh = (double) ((float) z);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis yl = (x - (zh - 2.0)) * (z2 + zh) - (yh - zh * (zh + one));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xh = (double) ((float) z);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis case 6: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5))*yy */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis yh = (double) ((float) z);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis yl = z1 * (z2 + zh) - (yh - zh * (zh + one));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xh = (double) ((float) z);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* xh+xl=(x+1)*...*(x+4) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* wh+wl=(x+5)*yy */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis wh = (double) ((float) (x5 * (yy.h + yy.l)));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis wl = (z1 * yy.h + x5 * yy.l) - (wh - zh * yy.h);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis case 7: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6))*yy */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis yh = (double) ((float) z); /* yh+yl = (x+3)(x+4) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis yl = (x - (zh - 3.0)) * (z2 + zh) - (yh - (zh * (zh + one)));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xh = (double) ((float) z);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* xh+xl=(x+2)*...*(x+5) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* wh+wl=(x+1)(x+6)*yy */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis wh = (double) ((float) (z2 * (yy.h + yy.l)));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis wl = (z2 * yy.l + yl * yy.h) - (wh - (yh - 6.0) * yy.h);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis double t, t1, t2, t3, t4, t5, w, y, z, z1, z2, z3, z5;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* +Inf -> +Inf, -Inf or NaN -> NaN */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (hx > 0x406573fa || /* x > 171.62... overflow to +inf */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (z * z);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* negative x */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* INDENT OFF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * compute: xk =
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -2 ... x is an even int (-inf is even)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -1 ... x is an odd int
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * +0 ... x is not an int but chopped to an even int
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * +1 ... x is not an int but chopped to an odd int
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* INDENT ON */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* ideally gamma(-n)= (-1)**(n+1) * inf, but c99 expect NaN */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return ((x - x) / (x - x)); /* 0/0 = NaN */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* negative underflow thresold */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (ix > 0x4066e000 || (ix == 0x4066e000 && lx > 11)) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* x < -183.0 - 11ulp */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* now compute gamma(x) by -1/((sin(pi*y)/pi)*gamma(1+y)), y = -x */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * First compute ss = -sin(pi*y)/pi , so that
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * gamma(x) = 1/(ss*gamma(1+y))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis z = y - (double) j;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (z > 0.3183098861837906715377675)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (z > 0.6816901138162093284622325)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* Then compute ww = gamma(1+y), note that result scale to 2**m */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if ((lx & 1) == 0) { /* y+1 exact (note that y<184) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (t == y) { /* y+one exact */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis } else { /* use y*gamma(y) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t1 = (double) ((float) y);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* t4 will not be too large */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis ww.l = y * (ww.l - (t2 - ww.h)) + (y - t1) * t2;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* compute 1/(ss*ww) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis z1 = ss.l - (t1 - ss.h); /* (t1,z1) = ss */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis z2 = ww.l - (t2 - ww.h); /* (t2,z2) = ww */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis z5 = z1 * t4 + t1 * z2; /* (t5,z5) = ss*ww */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t1 = (double) ((float) t3); /* (t1,z1) = ss*ww */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t2 = (double) ((float) z3); /* leading 1/(ss*ww) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* check whether z*2**-m underflow */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis } else if ((m - j) > 52) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* underflow */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* subnormal */