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