freepascal.compiler/packages/numlib/src/spe.pas

{
    This file is part of the Numlib package.
    Copyright (c) 1986-2000 by
     Kees van Ginneken, Wil Kortsmit and Loek van Reij of the
     Computational centre of the Eindhoven University of Technology

    FPC port Code          by Marco van de Voort ([email protected])
             documentation by Michael van Canneyt ([email protected])

    Special functions. (Currently, most of these functions only work for
            ArbFloat=REAL/DOUBLE, not for Extended(at least not at full
            precision, implement the tables in the diverse functions
            for extended)

    See the file COPYING.FPC, included in this distribution,
    for details about the copyright.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

 **********************************************************************}
{$mode objfpc}{$H+}
{$modeswitch nestedprocvars}

{$IFNDEF FPC_DOTTEDUNITS}
unit spe;
{$ENDIF FPC_DOTTEDUNITS}
{$I DIRECT.INC}

interface

{$IFDEF FPC_DOTTEDUNITS}
uses NumLib.Typ;
{$ELSE FPC_DOTTEDUNITS}
uses typ;
{$ENDIF FPC_DOTTEDUNITS}

{  Calculate modified Besselfunction "of the first kind" I0(x) }
function spebi0(x: ArbFloat): ArbFloat;

{  Calculate modified Besselfunction "of the first kind" I1(x) }
function spebi1(x: ArbFloat): ArbFloat;

{  Calculate Besselfunction "of the first kind" J0(x) }
function spebj0(x: ArbFloat): ArbFloat;

{  Calculate Besselfunction "of the first kind" J1(x) }
function spebj1(x: ArbFloat): ArbFloat;

{  Calculate modified Besselfunction "of the second kind" K0(x) }
function spebk0(x: ArbFloat): ArbFloat;

{  Calculate modified Besselfunction "of the second kind" K1(x) }
function spebk1(x: ArbFloat): ArbFloat;

{  Calculate Besselfunction "of the second kind" Y0(x) }
function speby0(x: ArbFloat): ArbFloat;

{  Calculate Besselfunction "of the second kind" Y1(x) }
function speby1(x: ArbFloat): ArbFloat;

{  Entier function, calculates first integer greater or equal than X}
function speent(x: ArbFloat): longint;

{  Errorfunction ( 2/sqrt(pi)* Int(t,0,pi,exp(sqr(t)) )}
function speerf(x: ArbFloat): ArbFloat;

{  Errorfunction's complement ( 2/sqrt(pi)* Int(t,pi,inf,exp(sqr(t))) )}
function speefc(x: ArbFloat): ArbFloat;

{  Calculates the cumulative normal distribution
   N(x) = 1/sqrt(2*pi) * Int(t, -INF, x, exp(t^2/2) ) }
function normaldist(x: ArbFloat): ArbFloat;

{  Inverse of cumulative normal distribution:
   Returns x such that y = normaldist(x) }
function invnormaldist(y: ArbFloat): ArbFloat;

{  Function to calculate the Gamma function ( int(t,0,inf,t^(x-1)*exp(-t)) }
function spegam(x: ArbFloat): ArbFloat;

{  Function to calculate the natural logaritm of the Gamma function}
function spelga(x: ArbFloat): ArbFloat;

{  Function to calculate the lower incomplete Gamma function
     int(t,0,x,exp(-t)t^(s-1)) / spegam(s)  (s > 0) }
function gammap(s, x: ArbFloat): ArbFloat;

{  Function to calculate the upper incomplete Gamma function
     int(t,x,inf,exp(-t)t^(s-1)) / spegam(s)  (s > 0)
     gammaq(s,x) = 1 - gammap(s,x) }
function gammaq(s, x: ArbFloat): ArbFloat;
function invgammaq(s, y: ArbFloat): ArbFloat;

{  Function to calculate the (complete) beta function
      beta(a, b) = int(t, 0, 1, t^(a-1) * (1-t)^(b-1) with a > 0, b > 0
      beta(a, b) = spegam(a) * spegam(b) / spegam(a + b) }
function beta(a, b: ArbFloat): ArbFloat;

{  Function to calculate the (regularized) incomplete beta function
      betai(a, b, x) = int(t, 0, x, t^(x-1) * (1-t)^(y-1) ) / beta(a,b) }
function betai(a, b, x: ArbFloat): ArbFloat;
function invbetai(a, b, y: ArbFloat; eps: ArbFloat = 0.0): ArbFloat;

{ Function to calculate the cumulative chi2 distribution with n degrees of
  freedom (upper tail) }
function chi2dist(x: ArbFloat; n: ArbInt): ArbFloat;
function invchi2dist(y: Arbfloat; n: ArbInt): ArbFloat;

{  Function to calculate Student's t distribution with n degrees of freedom
   (cumulative, upper tail if Tails = 1, else both tails }
type
  TNumTails = 1..2;

function tdist(t: ArbFloat; n: ArbInt; Tails: TNumTails): ArbFloat;
function invtdist(y: ArbFloat; n: ArbInt; Tails: TNumTails; eps: ArbFloat = 0.0): ArbFloat;

{  Function to calculate the cumulative F distribution function of value F
   with n1 and n2 degrees of freedom }
function Fdist(F: ArbFloat; n1, n2: ArbInt): ArbFloat;
function invFdist(p: ArbFloat; n1, n2: ArbInt; eps: ArbFloat = 0.0): ArbFloat;

{  "Calculates" the maximum of two ArbFloat values     }
function spemax(a, b: ArbFloat): ArbFloat; deprecated 'Use max(a,b) in unit math.';

{  Calculates the function value of a polynomial of degree n for variable x.
   The polynomial coefficients a are ordered from lowest to highest degree term.
     y = a0 + a1 x + a2 x^2 + ... + an x^n }
function spepol(x: ArbFloat; var a: ArbFloat; n: ArbInt): ArbFloat;

{ Calc a^b with a and b real numbers}
function spepow(a, b: ArbFloat): ArbFloat; deprecated 'Use power(a,b) in unit math.';

{ Returns sign of x (-1 for x<0, 0 for x=0 and 1 for x>0)  }
function spesgn(x: ArbFloat): ArbInt;  deprecated 'Use sign(x) in unit math.';

{  ArcSin(x) }
function spears(x: ArbFloat): ArbFloat; deprecated 'Use arcsin(x) in unit math.';

{  ArcCos(x) }
function spearc(x: ArbFloat): ArbFloat; deprecated 'Use arccos(x) in unit math.';

{  Sinh(x) }
function spesih(x: ArbFloat): ArbFloat; deprecated 'Use sinh(x) in unit math.';

{  Cosh(x) }
function specoh(x: ArbFloat): ArbFloat; deprecated 'Use cosh(x) in unit math.';

{  Tanh(x) }
function spetah(x: ArbFloat): ArbFloat; deprecated 'Use tanh(x) in unit math.';

{  ArcSinH(x) }
function speash(x: ArbFloat): ArbFloat; deprecated 'Use arcsinh(x) in unit math.';

{  ArcCosh(x) }
function speach(x: ArbFloat): ArbFloat; deprecated 'Use arccosh(x) in unit math';

{  ArcTanH(x) }
function speath(x: ArbFloat): ArbFloat; deprecated 'Use arctanh(x) in unit math';

{ Error numbers used within this unit:

  401 - function spebk0(x) is not defined for x <= 0.
  402 - function spebk1(x) is not defined for x <= 0.
  403 - function speby0(x) is not defined for x <= 0.
  404 - function speby1(x) is not defined for x <= 0.
  405 - function speach(x) is not defined for x < 1
  406 - function speath(x) is not defined for x <= -1 or x >= 1
  407 - function spgam(x): x is too small or too large.
  408 - function splga(x) cannot be calculated for x <= 0, or x is too large.
  409 - function spears(s, x) is not defined for x < -1 or x > 1
  410 - function gammap(s, x) is not defined for s <= 0 or x < 0
  411 - function gammaq(s, x) is not defined for s <= 0 or x < 0
  412 - function beta(a, b) is not defined for a <= 0 or b <= 0
  413 - function betai(a, b, x) is not defined for a <= 0 or b <= 0
  414 - function betai(a, b, x) is not defined for x < 0 or x > 0
  415 - function invtdist(t, n) is not defined for t <= 0 or t >= 1 or n <= 0.
}

implementation

{$IFDEF FPC_DOTTEDUNITS}
uses
  System.Math, NumLib.Roo;
{$ELSE FPC_DOTTEDUNITS}
uses
  math, roo;
{$ENDIF FPC_DOTTEDUNITS}

const
  SQRT2 = 1.4142135623730950488016887242097;   // sqrt(2)
  SQRT2PI = 2.506628274631000502415765284811;  // sqrt(2*pi)
  EXP_2 = 0.13533528323661269189399949497248;  // exp(-2)

function spebi0(x: ArbFloat): ArbFloat;

const

     xvsmall = 3.2e-9;
          a1 : array[0..23] of ArbFloat =
               (  3.08508322553671039e-1, -1.86478066609466760e-1,
                  1.57686843969995904e-1, -1.28895621330524993e-1,
                  9.41616340200868389e-2, -6.04316795007737183e-2,
                  3.41505388391452157e-2, -1.71317947935716536e-2,
                  7.70061052263382555e-3, -3.12923286656374358e-3,
                  1.15888319775791686e-3, -3.93934532072526720e-4,
                  1.23682594989692688e-4, -3.60645571444886286e-5,
                  9.81395862769787105e-6, -2.50298975966588680e-6,
                  6.00566861079330132e-7, -1.36042013507151017e-7,
                  2.92096163521178835e-8, -5.94856273204259507e-9,
                  1.13415934215369209e-9, -2.10071360134551962e-10,
                  4.44484446637868974e-11,-7.48150165756234957e-12);
          a2 : array[0..26] of ArbFloat =
               (  1.43431781856850311e-1, -3.71571542566085323e-2,
                  1.44861237337359455e-2, -6.30121694459896307e-3,
                  2.89362046530968701e-3, -1.37638906941232170e-3,
                  6.72508592273773611e-4, -3.35833513200679384e-4,
                  1.70524543267970595e-4, -8.74354291104467762e-5,
                  4.48739019580173804e-5, -2.28278155280668483e-5,
                  1.14032404021741277e-5, -5.54917762110482949e-6,
                  2.61457634142262604e-6, -1.18752840689765504e-6,
                  5.18632519069546106e-7, -2.17653548816447667e-7,
                  8.75291839187305722e-8, -3.34900221934314738e-8,
                  1.24131668344616429e-8, -4.66215489983794905e-9,
                  1.58599776268172290e-9, -3.80370174256271589e-10,
                  1.23188158175419302e-10,-8.46900307934754898e-11,
                  2.45185252963941089e-11);
           a3: array[0..19] of ArbFloat =
               (  4.01071065066847416e-1,  2.18216817211694382e-3,
                  5.59848253337377763e-5,  2.79770701849785597e-6,
                  2.17160501061222148e-7,  2.36884434055843528e-8,
                  3.44345025431425567e-9,  6.47994117793472057e-10,
                  1.56147127476528831e-10, 4.82726630988879388e-11,
                  1.89599322920800794e-11, 1.05863621425699789e-11,
                  8.27719401266046976e-12, 2.82807056475555021e-12,
                 -4.34624739357691085e-12,-4.29417106720584499e-12,
                  4.30812577328136192e-13, 1.44572313799118029e-12,
                  4.73229306831831040e-14,-1.95679809047625728e-13);


var t : ArbFloat;

begin
  t:=abs(x);
  if t <=xvsmall
  then
    spebi0:=1
  else
  if t <= 4
  then
    spebi0 := exp(t)*spepol(t/2-1, a1[0], SizeOf(a1) div SizeOf(ArbFloat) -1)
  else
  if t <= 12
  then
    spebi0:=exp(t)*spepol(t/4-2, a2[0], SizeOf(a2) div SizeOf(ArbFloat) -1)
  else { t > 12}
    spebi0:=(exp(t)/sqrt(t))*
            spepol(24/t-1, a3[0], SizeOf(a3) div SizeOf(ArbFloat) -1)
end; {spebi0}

function spebi1(x: ArbFloat): ArbFloat;


const xvsmall = 3.2e-9;
      a1: array[0..11] of ArbFloat =
      ( 1.19741654963670236e+0, 9.28758890114609554e-1,
        2.68657659522092832e-1, 4.09286371827770484e-2,
        3.84763940423809498e-3, 2.45224314039278904e-4,
        1.12849795779951847e-5, 3.92368710996392755e-7,
        1.06662712314503955e-8, 2.32856921884663846e-10,
        4.17372709788222413e-12,6.24387910353848320e-14);

      a2: array[0..26] of ArbFloat =
      ( 1.34142493292698178e-1, -2.99140923897405570e-2,
        9.76021102528646704e-3, -3.40759647928956354e-3,
        1.17313412855965374e-3, -3.67626180992174570e-4,
        8.47999438119288094e-5,  5.21557319070236939e-6,
       -2.62051678511418163e-5,  2.47493270133518925e-5,
       -1.79026222757948636e-5,  1.13818992442463952e-5,
       -6.63144162982509821e-6,  3.60186151617732531e-6,
       -1.83910206626348772e-6,  8.86951515545183908e-7,
       -4.05456611578551130e-7,  1.76305222240064495e-7,
       -7.28978293484163628e-8,  2.84961041291017650e-8,
       -1.07563514207617768e-8,  4.11321223904934809e-9,
       -1.41575617446629553e-9,  3.38883570696523350e-10,
       -1.10970391104678003e-10, 7.79929176497056645e-11,
       -2.27061376122617856e-11);

       a3: array[0..19] of ArbFloat =
       ( 3.92624494204116555e-1, -6.40545360348237412e-3,
        -9.12475535508497109e-5, -3.82795135453556215e-6,
        -2.72684545741400871e-7, -2.82537120880041703e-8,
        -3.96757162863209348e-9, -7.28107961041827952e-10,
        -1.72060490748583241e-10,-5.23524129533553498e-11,
        -2.02947854602758139e-11,-1.11795516742222899e-11,
        -8.69631766630563635e-12,-3.05957293450420224e-12,
         4.42966462319664333e-12, 4.47735589657057690e-12,
        -3.95353303949377536e-13,-1.48765082315961139e-12,
        -5.77176811730370560e-14, 1.99448557598015488e-13);

var t : ArbFloat;

begin
  t:=abs(x);
  if t <= xvsmall
  then
    spebi1:=x/2
  else
  if t <= 4
  then
    spebi1:=x*spepol(sqr(t)/8-1, a1[0], sizeof(a1) div sizeof(ArbFloat)-1)
  else
  if t <= 12
  then
    spebi1:=
      exp(t)*spepol(t/4-2, a2[0], sizeof(a2) div sizeof(ArbFloat)-1)*spesgn(x)
  else { t > 12}
    spebi1:=
      (exp(t)/sqrt(t))*
      spepol(24/t-1, a3[0], sizeof(a3) div sizeof(ArbFloat)-1)*spesgn(x)
end; {spebi1}

function spebj0(x: ArbFloat): ArbFloat;
const

       xvsmall = 3.2e-9;
          tbpi = 6.36619772367581343e-1;
           a1 : array[0..5] of ArbFloat =
           ( 1.22200000000000000e-17, -1.94383469000000000e-12,
             7.60816359241900000e-8,  -4.60626166206275050e-4,
             1.58067102332097261e-1,  -8.72344235285222129e-3);

            b1 : array[0..5] of ArbFloat =
            ( - 7.58850000000000000e-16, 7.84869631400000000e-11,
              - 1.76194690776215000e-6,  4.81918006946760450e-3,
              - 3.70094993872649779e-1,  1.57727971474890120e-1);

            c1 : array[0..4] of ArbFloat =
            ( 4.12532100000000000e-14, - 2.67925353056000000e-9,
              3.24603288210050800e-5,  - 3.48937694114088852e-2,
              2.65178613203336810e-1);

            d1 : array[0..13] of ArbFloat =
            ( 9.99457275788251954e-1, -5.36367319213004570e-4,
              6.13741608010926000e-6, -2.05274481565160000e-7,
              1.28037614434400000e-8, -1.21211819632000000e-9,
              1.55005642880000000e-10,-2.48827276800000000e-11,
              4.78702080000000000e-12,-1.06365696000000000e-12,
              2.45294080000000000e-13,-6.41843200000000000e-14,
              3.34028800000000000e-14,-1.17964800000000000e-14);

             d2 : array[0..16] of ArbFloat =
             ( -1.55551138795135187e-2,  6.83314909934390000e-5,
               -1.47713883264594000e-6,  7.10621485930000000e-8,
               -5.66871613024000000e-9,  6.43278173280000000e-10,
               -9.47034774400000000e-11, 1.70330918400000000e-11,
               -3.59094272000000000e-12, 8.59855360000000000e-13,
               -2.28807680000000000e-13, 6.95193600000000000e-14,
               -2.27942400000000000e-14, 4.75136000000000000e-15,
               -1.14688000000000000e-15, 2.12992000000000000e-15,
               -9.83040000000000000e-16);

var t, g, y, t2, a, b, c, cx, sx : ArbFloat;
    i, bov                       : ArbInt;

begin
  t:=abs(x);
  if t<=xvsmall
  then
    spebj0:=1
  else
  if t<=8
  then
    begin
      t:=0.03125*sqr(t)-1; t2:=2*t;
      b:=0; c:=0;
      bov:=sizeof(a1) div sizeof(ArbFloat) - 1;
      for i:=0 to bov do
        begin
          a:=t2*c-b+a1[i];
          if i<5
          then
            b:=t2*a-c+b1[i]
          else
            spebj0:=t*a-c+b1[i];
          if i<bov
          then
            c:=t2*b-a+c1[i]
          else
            if i<5
            then
              spebj0:=t*b-a+c1[i]
        end {i}
    end {abs(x)<=8}
  else
    begin
      g:=t-1/(2*tbpi); y:=sqrt(tbpi/t);
      cx:=cos(g)*y; sx:=-sin(g)*y*8/t;
      t:=128/sqr(t)-1;
      spebj0:=cx*spepol(t, d1[0], sizeof(d1) div sizeof(ArbFloat) - 1)
              + sx*spepol(t, d2[0], sizeof(d2) div sizeof(ArbFloat) - 1)
    end {abs(x)>8}

end {spebj0};

function spebj1(x: ArbFloat): ArbFloat;
const

       xvsmall = 3.2e-9;
          tbpi = 6.36619772367581343e-1;
      a1 : array[0..5] of ArbFloat =
      ( 2.95000000000000000e-18, -5.77740420000000000e-13,
        2.94970700727800000e-8,  -2.60444389348580680e-4,
        1.77709117239728283e-1,  -1.19180116054121687e+0);

      b1 : array[0..5] of ArbFloat =
      ( -1.95540000000000000e-16, 2.52812366400000000e-11,
        -7.61758780540030000e-7,  3.24027018268385747e-3,
        -6.61443934134543253e-1,  6.48358770605264921e-1);

      c1 : array[0..4] of ArbFloat =
      ( 1.13857200000000000e-14, -9.42421298160000000e-10,
        1.58870192399321300e-5,  -2.91755248061542077e-2,
        1.28799409885767762e+0);

       d1 : array[0..13] of ArbFloat =
       ( 1.00090702627808217e+0,  8.98804941670557880e-4,
        -7.95969469843846000e-6,  2.45367662227560000e-7,
        -1.47085129889600000e-8,  1.36030580128000000e-9,
        -1.71310758400000000e-10, 2.72040729600000000e-11,
        -5.19113984000000000e-12, 1.14622464000000000e-12,
        -2.63372800000000000e-13, 6.86387200000000000e-14,
        -3.54508800000000000e-14, 1.24928000000000000e-14);

       d2 : array[0..15] of ArbFloat =
       ( 4.67768740274489776e-2,  -9.62145882205441600e-5,
         1.82120185123076000e-6,  -8.29196070929200000e-8,
         6.42013250344000000e-9,  -7.15110504800000000e-10,
         1.03950931840000000e-10, -1.85248000000000000e-11,
         3.87554432000000000e-12, -9.23228160000000000e-13,
         2.50224640000000000e-13, -7.43936000000000000e-14,
         1.75718400000000000e-14, -4.83328000000000000e-15,
         5.32480000000000000e-15, -2.29376000000000000e-15);

var t, g, y, t2, a, b, c, cx, sx : ArbFloat;
    i, bov                       : ArbInt;

begin
  t:=abs(x);
  if t<xvsmall
  then
    spebj1:=x/2
  else
  if t<=8
  then
    begin
      t:=0.03125*sqr(t)-1; t2:=2*t;
      b:=0; c:=0;
      bov:=sizeof(a1) div sizeof(ArbFloat) - 1;
      for i:=0 to bov do
        begin
          a:=t2*c-b+a1[i];
          if i<bov
          then
            begin
              b:=t2*a-c+b1[i];
              c:=t2*b-a+c1[i]
            end
          else
            spebj1:=(x/8)*(t*a-c+b1[i])
        end {i}
    end {abs(x)<=8}
  else
    begin
      g:=t-1.5/tbpi; y:=sqrt(tbpi/t)*spesgn(x);
      cx:=cos(g)*y; sx:=-sin(g)*y*8/t;
      t:=128/sqr(t)-1;
      spebj1:=cx*spepol(t, d1[0], sizeof(d1) div sizeof(ArbFloat) - 1)
              + sx*spepol(t, d2[0], sizeof(d2) div sizeof(ArbFloat) - 1)
    end {abs(x)>8}
end {spebj1};

function spebk0(x: ArbFloat): ArbFloat;

const

     egam = 0.57721566490153286;
     xvsmall = 3.2e-9;
     highexp = 745;

      a0: array[0..7] of ArbFloat =
      ( 1.12896092945412762e+0,  1.32976966478338191e-1,
        4.07157485171389048e-3,  5.59702338227915383e-5,
        4.34562671546158210e-7,  2.16382411824721532e-9,
        7.49110736894134794e-12, 1.90674197514561280e-14);

      a1: array[0..8] of ArbFloat =
      ( 2.61841879258687055e-1,  1.52436921799395196e-1,
        6.63513979313943827e-3,  1.09534292632401542e-4,
        9.57878493265929443e-7,  5.19906865800665633e-9,
        1.92405264219706684e-11, 5.16867886946332160e-14,
        1.05407718191360000e-16);

      a2: array[0..22] of ArbFloat =
      ( 9.58210053294896496e-1, -1.42477910128828254e-1,
        3.23582010649653009e-2, -8.27780350351692662e-3,
        2.24709729617770471e-3, -6.32678357460594866e-4,
        1.82652460089342789e-4, -5.37101208898441760e-5,
        1.60185974149720562e-5, -4.83134250336922161e-6,
        1.47055796078231691e-6, -4.51017292375200017e-7,
        1.39217270224614153e-7, -4.32185089841834127e-8,
        1.34790467361340101e-8, -4.20597329258249948e-9,
        1.32069362385968867e-9, -4.33326665618780914e-10,
        1.37999268074442719e-10, -3.19241059198852137e-11,
        9.74410152270679245e-12, -7.83738609108569293e-12,
        2.57466288575820595e-12);

      a3: array[0..22] of ArbFloat =
     ( 6.97761598043851776e-1, -1.08801882084935132e-1,
       2.56253646031960321e-2, -6.74459607940169198e-3,
       1.87292939725962385e-3, -5.37145622971910027e-4,
       1.57451516235860573e-4, -4.68936653814896712e-5,
       1.41376509343622727e-5, -4.30373871727268511e-6,
       1.32052261058932425e-6, -4.07851207862189007e-7,
       1.26672629417567360e-7, -3.95403255713518420e-8,
       1.23923137898346852e-8, -3.88349705250555658e-9,
       1.22424982779432970e-9, -4.03424607871960089e-10,
       1.28905587479980147e-10,-2.97787564633235128e-11,
       9.11109430833001267e-12,-7.39672783987933184e-12,
       2.43538242247537459e-12);
      a4: array[0..16] of ArbFloat =
      ( 1.23688664769425422e+0,  -1.72683652385321641e-2,
       -9.25551464765637133e-4,  -9.02553345187404564e-5,
       -6.31692398333746470e-6,  -7.69177622529272933e-7,
       -4.16044811174114579e-8,  -9.41555321137176073e-9,
        1.75359321273580603e-10, -2.22829582288833265e-10,
        3.49564293256545992e-11, -1.11391758572647639e-11,
        2.85481235167705907e-12, -7.31344482663931904e-13,
        2.06328892562554880e-13, -1.28108310826991616e-13,
        4.43741979886551040e-14);


var t: ArbFloat;

begin
  if x<=0
  then
    RunError(401);
  if x<=xvsmall
  then
    spebk0:=-(ln(x/2)+egam)
  else
  if x<=1
  then
    begin
      t:=2*sqr(x)-1;
      spebk0:=-ln(x)*spepol(t, a0[0], sizeof(a0) div sizeof(ArbFloat) - 1)
              + spepol(t, a1[0], sizeof(a1) div sizeof(ArbFloat) - 1)
    end
  else
  if x<=2
  then
    spebk0:=exp(-x)*spepol(2*x-3, a2[0], sizeof(a2) div sizeof(ArbFloat) - 1)
  else
  if x<=4
  then
    spebk0:=exp(-x)*spepol(x-3, a3[0], sizeof(a3) div sizeof(ArbFloat) - 1)
  else
  if x <= highexp
  then
    spebk0:=exp(-x)*
            spepol(10/(1+x)-1, a4[0], sizeof(a4) div sizeof(ArbFloat) - 1)/sqrt(x)
  else
    spebk0:=0
end; {spebk0}

function spebk1(x: ArbFloat): ArbFloat;

const

   xsmall = 7.9e-10;
  highexp = 745;
   a0: array[0..7] of ArbFloat =
   ( 5.31907865913352762e-1,  3.25725988137110495e-2,
     6.71642805873498653e-4,  6.95300274548206237e-6,
     4.32764823642997753e-8,  1.79784792380155752e-10,
     5.33888268665658944e-13, 1.18964962439910400e-15);

   a1: array[0..7] of ArbFloat =
   ( 3.51825828289325536e-1,  4.50490442966943726e-2,
     1.20333585658219028e-3,  1.44612432533006139e-5,
     9.96686689273781531e-8,  4.46828628435618679e-10,
     1.40917103024514301e-12, 3.29881058019865600e-15);

   a2: array[0..23] of ArbFloat =
   ( 1.24316587355255299e+0, -2.71910714388689413e-1,
     8.20250220860693888e-2, -2.62545818729427417e-2,
     8.57388087067410089e-3, -2.82450787841655951e-3,
     9.34594154387642940e-4, -3.10007681013626626e-4,
     1.02982746700060730e-4, -3.42424912211942134e-5,
     1.13930169202553526e-5, -3.79227698821142908e-6,
     1.26265578331941923e-6, -4.20507152338934956e-7,
     1.40138351985185509e-7, -4.66928912168020101e-8,
     1.54456653909012693e-8, -5.13783508140332214e-9,
     1.82808381381205361e-9, -6.15211416898895086e-10,
     1.28044023949946257e-10, -4.02591066627023831e-11,
     4.27404330568767242e-11, -1.46639291782948454e-11);

   a3: array[0..23] of ArbFloat =
   ( 8.06563480128786903e-1,  -1.60052611291327173e-1,
     4.58591528414023064e-2,  -1.42363136684423646e-2,
     4.55865751206724687e-3,  -1.48185472032688523e-3,
     4.85707174778663652e-4,  -1.59994873621599146e-4,
     5.28712919123131781e-5,  -1.75089594354079944e-5,
     5.80692311842296724e-6,  -1.92794586996432593e-6,
     6.40581814037398274e-7,  -2.12969229346310343e-7,
     7.08723366696569880e-8,  -2.35855618461025265e-8,
     7.79421651144832709e-9,  -2.59039399308009059e-9,
     9.20781685906110546e-10, -3.09667392343245062e-10,
     6.44913423545894175e-11, -2.02680401514735862e-11,
     2.14736751065133220e-11, -7.36478297050421658e-12);

    a4: array[0..16] of ArbFloat =
    ( 1.30387573604230402e+0,   5.44845254318931612e-2,
      4.31639434283445364e-3,   4.29973970898766831e-4,
      4.04720631528495020e-5,   4.32776409784235211e-6,
      4.07563856931843484e-7,   4.86651420008153956e-8,
      3.82717692121438315e-9,   6.77688943857588882e-10,
      6.97075379117731379e-12,  1.72026097285930936e-11,
     -2.60774502020271104e-12,  8.58211523713560576e-13,
     -2.19287104441802752e-13,  1.39321122940600320e-13,
     -4.77850238111580160e-14);

var t: ArbFloat;

begin
  if x<=0
  then
    RunError(402);
  if x<=xsmall
  then
    spebk1:=1/x
  else
  if x<=1
  then
    begin
      t:=2*sqr(x)-1;
      spebk1:=( ln(x)*spepol(t, a0[0], sizeof(a0) div sizeof(ArbFloat) - 1)
              -spepol(t, a1[0], sizeof(a1) div sizeof(ArbFloat) -1) )*x + 1/x
    end
  else
  if x<=2
  then
    spebk1:=exp(-x)*spepol(2*x-3, a2[0], sizeof(a2) div sizeof(ArbFloat) - 1)
  else
  if x<=4
  then
    spebk1:=exp(-x)*spepol(x-3, a3[0], sizeof(a3) div sizeof(ArbFloat) - 1)
  else
  if x <= highexp
  then
    spebk1:=exp(-x)*spepol(10/(1+x)-1, a4[0],
            sizeof(a4) div sizeof(ArbFloat) - 1)/sqrt(x)
  else
    spebk1:=0
end; {spebk1}

function speby0(x: ArbFloat): ArbFloat;

const

      tbpi = 6.36619772367581343e-1;
      egam = 5.77215664901532861e-1;
   xvsmall = 3.2e-9;
   a1 : array[0..5] of ArbFloat =
   ( 3.90000000000000000e-19, -8.74734100000000000e-14,
     5.24879478733000000e-9,  -5.63207914105698700e-5,
     4.71966895957633869e-2,   1.79034314077182663e-1);

   b1 : array[0..5] of ArbFloat =
   ( -2.69800000000000000e-17, 4.02633082000000000e-12,
     -1.44072332740190000e-7,  7.53113593257774230e-4,
     -1.77302012781143582e-1, -2.74474305529745265e-1);

   c1 : array[0..5] of ArbFloat =
   ( 1.64349000000000000e-15, -1.58375525420000000e-10,
     3.20653253765480000e-6,  -7.28796247955207918e-3,
     2.61567346255046637e-1,  -3.31461132032849417e-2);

    d1 : array[0..13] of ArbFloat =
    ( 9.99457275788251954e-1, -5.36367319213004570e-4,
      6.13741608010926000e-6, -2.05274481565160000e-7,
      1.28037614434400000e-8, -1.21211819632000000e-9,
      1.55005642880000000e-10,-2.48827276800000000e-11,
      4.78702080000000000e-12,-1.06365696000000000e-12,
      2.45294080000000000e-13,-6.41843200000000000e-14,
      3.34028800000000000e-14,-1.17964800000000000e-14);

    d2 : array[0..16] of ArbFloat =
    (-1.55551138795135187e-2,  6.83314909934390000e-5,
     -1.47713883264594000e-6,  7.10621485930000000e-8,
     -5.66871613024000000e-9,  6.43278173280000000e-10,
     -9.47034774400000000e-11, 1.70330918400000000e-11,
     -3.59094272000000000e-12, 8.59855360000000000e-13,
     -2.28807680000000000e-13, 6.95193600000000000e-14,
     -2.27942400000000000e-14, 4.75136000000000000e-15,
     -1.14688000000000000e-15, 2.12992000000000000e-15,
     -9.83040000000000000e-16);

var t, g, y, t2, a, b, c, cx, sx : ArbFloat;
    i, bov                       : ArbInt;

begin
  if x<=0
  then
    RunError(403);
  if x<=xvsmall
  then
    speby0:=(ln(x/2)+egam)*tbpi
  else
  if x<=8
  then
    begin
      t:=0.03125*sqr(x)-1; t2:=2*t;
      b:=0; c:=0;
      bov:=sizeof(a1) div sizeof(ArbFloat) - 1;
      for i:=0 to bov do
        begin
          a:=t2*c-b+a1[i];
          b:=t2*a-c+b1[i];
          if i<bov
          then
            c:=t2*b-a+c1[i]
          else
            speby0:=t*b-a+c1[i]+tbpi*spebj0(x)*ln(x)
        end {i}
    end {x<=8}
  else
    begin
      g:=x-1/(2*tbpi); y:=sqrt(tbpi/x);
      cx:=cos(g)*y*8/x; sx:=sin(g)*y;
      t:=128/sqr(x)-1;
      speby0:=sx*spepol(t, d1[0], sizeof(d1) div sizeof(ArbFloat) - 1)
            + cx*spepol(t, d2[0], sizeof(d2) div sizeof(ArbFloat) - 1)
    end {x>8}
end {speby0};

function speby1(x: ArbFloat): ArbFloat;

const
    tbpi = 6.36619772367581343e-1;
    xsmall = 7.9e-10;
   a1 : array[0..5] of ArbFloat =
   (-6.58000000000000000e-18, 1.21143321000000000e-12,
    -5.68844003991900000e-8,  4.40478629867099510e-4,
    -2.26624991556754924e-1, -1.28697384381350001e-1);

   b1 : array[0..5] of ArbFloat =
   ( 4.27730000000000000e-16,-5.17212147300000000e-11,
     1.41662436449235000e-6, -5.13164116106108479e-3,
     6.75615780772187667e-1,  2.03041058859342538e-2);

   c1 : array[0..4] of ArbFloat =
   (-2.44094900000000000e-14, 1.87547032473000000e-9,
    -2.83046401495148000e-5,  4.23191803533369041e-2,
    -7.67296362886645940e-1);

   d1 : array[0..13] of ArbFloat =
   ( 1.00090702627808217e+0,  8.98804941670557880e-4,
    -7.95969469843846000e-6,  2.45367662227560000e-7,
    -1.47085129889600000e-8,  1.36030580128000000e-9,
    -1.71310758400000000e-10, 2.72040729600000000e-11,
    -5.19113984000000000e-12, 1.14622464000000000e-12,
    -2.63372800000000000e-13, 6.86387200000000000e-14,
    -3.54508800000000000e-14, 1.24928000000000000e-14);

    d2 : array[0..15] of ArbFloat =
    ( 4.67768740274489776e-2, -9.62145882205441600e-5,
      1.82120185123076000e-6, -8.29196070929200000e-8,
      6.42013250344000000e-9, -7.15110504800000000e-10,
      1.03950931840000000e-10,-1.85248000000000000e-11,
      3.87554432000000000e-12,-9.23228160000000000e-13,
      2.50224640000000000e-13,-7.43936000000000000e-14,
      1.75718400000000000e-14,-4.83328000000000000e-15,
      5.32480000000000000e-15,-2.29376000000000000e-15);

var t, g, y, t2, a, b, c, cx, sx : ArbFloat;
    i, bov                       : ArbInt;

begin
  if x<=0
  then
    RunError(404);
  if x<=xsmall
  then
    speby1:=-tbpi/x
  else
  if x<=8
  then
    begin
      t:=0.03125*sqr(x)-1; t2:=2*t;
      b:=0; c:=0;
      bov:=sizeof(a1) div sizeof(ArbFloat) - 1;
      for i:=0 to bov do
        begin
          a:=t2*c-b+a1[i];
          if i<bov
          then
            begin
              b:=t2*a-c+b1[i];
              c:=t2*b-a+c1[i]
            end
          else
          if bov=3   {single}
          then
            begin
              b:=t2*a-c+b1[i];
              speby1:=(t*b-a+c1[i])*x/8 + spebj1(x)*ln(x)*tbpi - tbpi/x
            end
          else
            speby1:=(t*a-c+b1[i])*x/8 + spebj1(x)*ln(x)*tbpi - tbpi/x
        end {i}
    end {x<=8}
  else
    begin
      g:=x-3/(2*tbpi); y:=sqrt(tbpi/x);
      cx:=cos(g)*y*8/x; sx:=sin(g)*y;
      t:=128/sqr(x)-1;
      speby1:=sx*spepol(t, d1[0], sizeof(d1) div sizeof(ArbFloat) - 1)
            + cx*spepol(t, d2[0], sizeof(d2) div sizeof(ArbFloat) - 1)
    end {x>8}
end {speby1};

function speent(x : ArbFloat): longint;

var tx : longint;

begin
  tx:=trunc(x);
  if x>=0
  then
    speent:=tx
  else
    if x=tx
    then
      speent:=tx
    else
      speent:=tx-1
end; {speent}

function speerf(x : ArbFloat) : ArbFloat;
const

        xup = 6.25;
     sqrtpi = 1.7724538509055160;
     c : array[1..18] of ArbFloat =
     ( 1.9449071068178803e0,  4.20186582324414e-2, -1.86866103976769e-2,
       5.1281061839107e-3,   -1.0683107461726e-3,   1.744737872522e-4,
      -2.15642065714e-5,      1.7282657974e-6,     -2.00479241e-8,
      -1.64782105e-8,         2.0008475e-9,         2.57716e-11,
      -3.06343e-11,           1.9158e-12,           3.703e-13,
      -5.43e-14,             -4.0e-15,              1.2e-15);

     d : array[1..17] of ArbFloat =
     ( 1.4831105640848036e0, -3.010710733865950e-1, 6.89948306898316e-2,
      -1.39162712647222e-2,   2.4207995224335e-3,  -3.658639685849e-4,
       4.86209844323e-5,     -5.7492565580e-6,      6.113243578e-7,
      -5.89910153e-8,         5.2070091e-9,        -4.232976e-10,
       3.18811e-11,          -2.2361e-12,           1.467e-13,
      -9.0e-15,               5.0e-16);

  var t, s, s1, s2, x2: ArbFloat;
         bovc, bovd, j: ArbInt;
begin
  bovc:=sizeof(c) div sizeof(ArbFloat);
  bovd:=sizeof(d) div sizeof(ArbFloat);
  t:=abs(x);
  if t <= 2
  then
    begin
      x2:=sqr(x)-2;
      s1:=d[bovd]; s2:=0; j:=bovd-1;
      s:=x2*s1-s2+d[j];
      while j > 1 do
        begin
          s2:=s1; s1:=s; j:=j-1;
          s:=x2*s1-s2+d[j]
        end;
      speerf:=(s-s2)*x/2
    end
  else
    if t < xup
    then
      begin
        x2:=2-20/(t+3);
        s1:=c[bovc]; s2:=0; j:=bovc-1;
        s:=x2*s1-s2+c[j];
        while j > 1 do
          begin
            s2:=s1; s1:=s; j:=j-1;
            s:=x2*s1-s2+c[j]
          end;
        x2:=((s-s2)/(2*t))*exp(-sqr(x))/sqrtpi;
        speerf:=(1-x2)*spesgn(x)
      end
    else
      speerf:=spesgn(x)
end;  {speerf}

function spemax(a, b: ArbFloat): ArbFloat;
begin
  if a>b
  then
    spemax:=a
  else
    spemax:=b
end; {spemax}

function speefc(x : ArbFloat) : ArbFloat;
const

   xlow = -6.25;
  xhigh = 27.28;
      c : array[0..22] of ArbFloat =
      ( 1.455897212750385e-1, -2.734219314954260e-1,
        2.260080669166197e-1, -1.635718955239687e-1,
        1.026043120322792e-1, -5.480232669380236e-2,
        2.414322397093253e-2, -8.220621168415435e-3,
        1.802962431316418e-3, -2.553523453642242e-5,
       -1.524627476123466e-4,  4.789838226695987e-5,
        3.584014089915968e-6, -6.182369348098529e-6,
        7.478317101785790e-7,  6.575825478226343e-7,
       -1.822565715362025e-7, -7.043998994397452e-8,
        3.026547320064576e-8,  7.532536116142436e-9,
       -4.066088879757269e-9, -5.718639670776992e-10,
        3.328130055126039e-10);

  var t, s : ArbFloat;
begin
  if x <= xlow
  then
    speefc:=2
  else
  if x >= xhigh
  then
    speefc:=0
  else
    begin
      t:=1-7.5/(abs(x)+3.75);
      s:=exp(-x*x)*spepol(t, c[0], sizeof(c) div sizeof(ArbFloat) - 1);
      if x < 0
      then
        speefc:=2-s
      else
        speefc:=s
    end
end {speefc};

{ N(x) = 1/sqrt(2 pi) int(-INF, x, exp(t^2/2) = (1 + erf(x/sqrt(2))) / 2 }
function normaldist(x: ArbFloat): ArbFloat;
begin
  Result := 0.5 * (1.0 + speerf(x / SQRT2));
end;

function invnormaldist(y: ArbFloat): ArbFloat;
{ Ref.: Moshier, "Methods and programs for mathematical function" }
const
  P0: array[0..4] of ArbFloat = (
    -1.23916583867381258016,
    13.9312609387279679503,
    -56.6762857469070293439,
    98.0010754185999661536,
    -59.9633501014107895267);
  Q0: array[0..8] of ArbFloat = (
    -1.18331621121330003142,
    15.9056225126211695515,
    -82.0372256168333339912,
    200.260212380060660359,
    -225.462687854119370527,
    86.3602421390890590575,
    4.67627912898881538453,
    1.95448858338141759834,
    1.0);
  P1: array[0..8] of ArbFloat = (
    -8.57456785154685413611E-4,
    -3.50424626827848203418E-2,
    -1.40256079171354495875E-1,
    2.18663306850790267539,
    14.6849561928858024014,
    44.0805073893200834700,
    57.1628192246421288162,
    31.5251094599893866154,
    4.05544892305962419923);
  Q1: array[0..8] of Arbfloat = (
    -9.33259480895457427372E-4,
    -3.80806407691578277194E-2,
    -1.42182922854787788574E-1,
    2.50464946208309415979,
    15.0425385692907503408,
    41.3172038254672030440,
    45.3907635128879210584,
    15.7799883256466749731,
    1.0);
  P2: array[0..8] of ArbFloat = (
    6.23974539184983293730E-9,
    2.65806974686737550832E-6,
    3.01581553508235416007E-4,
    1.23716634817820021358E-2,
    2.01485389549179081538E-1,
    1.33303460815807542389,
    3.93881025292474443415,
    6.91522889068984211695,
    3.23774891776946035970);
  Q2: array[0..8] of ArbFloat = (
    6.79019408009981274425E-9,
    2.89247864745380683936E-6,
    3.28014464682127739104E-4,
    1.34204006088543189037E-2,
    2.16236993594496635890E-1,
    1.37702099489081330271,
    3.67983563856160859403,
    6.02427039364742014255,
    1.0);
var
  x, x0, x1: ArbFloat;
  yy, y2: ArbFloat;
  z: ArbFloat;
  code: Integer;
begin
  if y <= 0.0 then
  begin
    Result := -giant;
    exit;
  end;

  if y >= 1.0 then
  begin
    Result := +giant;
    exit;
  end;

  code := 1;
  yy := y;
  if yy > 1.0 - EXP_2 then begin    // EXP_2 = exp(-2)
    yy := 1.0 - yy;
    code := 0;
  end;

  if yy > EXP_2 then begin
    yy := yy - 0.5;
    y2 := yy * yy;
    x := y2 * spepol(y2, P0[0], 4) / spepol(y2, Q0[0], 8);
    x := (yy + yy * x) * SQRT2PI;   // SQRT2PI = sqrt(2*pi);
    Result := x;
    exit;
  end;

  x := sqrt(-2.0 * ln(yy));
  x0 := x - ln(x) / x;
  z := 1.0 / x;

  if x < 8.0 then
    x1 := z * spepol(z, P1[0], 8) / spepol(z, Q1[0], 8)
  else
    x1 := z * spepol(z, P2[0], 8) / spepol(z, Q2[0], 8);

  x := x0 - x1;
  if code <> 0 then
    x := -x;
  Result := x;
end;

function spegam(x: ArbFloat): ArbFloat;
const

    tmax = 170;
    a: array[0..23] of ArbFloat =
    ( 8.86226925452758013e-1,  1.61691987244425092e-2,
      1.03703363422075456e-1, -1.34118505705967765e-2,
      9.04033494028101968e-3, -2.42259538436268176e-3,
      9.15785997288933120e-4, -2.96890121633200000e-4,
      1.00928148823365120e-4, -3.36375833240268800e-5,
      1.12524642975590400e-5, -3.75499034136576000e-6,
      1.25281466396672000e-6, -4.17808776355840000e-7,
      1.39383522590720000e-7, -4.64774927155200000e-8,
      1.53835215257600000e-8, -5.11961333760000000e-9,
      1.82243164160000000e-9, -6.13513953280000000e-10,
      1.27679856640000000e-10,-4.01499750400000000e-11,
      4.26560716800000000e-11,-1.46381209600000000e-11);

var tvsmall, t, g: ArbFloat;
             m, i: ArbInt;
begin
  if sizeof(ArbFloat) = 6
  then
    tvsmall:=2*midget
  else
    tvsmall:=midget;
  t:=abs(x);
  if t > tmax
  then
    RunError(407);
  if t < macheps
  then
    begin
      if t < tvsmall
      then
        RunError(407);
      spegam:=1/x
    end
  else  { abs(x) >= macheps }
    begin
      m:=trunc(x);
      if x > 0
      then
        begin
          t:=x-m; m:=m-1; g:=1;
          if m<0
          then
            g:=g/x
          else
            if m>0
            then
              for i:=1 to m do
                g:=(x-i)*g
        end
      else { x < 0 }
        begin
          t:=x-m+1;
          if t=1
          then
            RunError(407);
          m:=1-m;
          g:=x;
          for i:=1 to m do
            g:=(i+x)*g;
          g:=1/g
        end;
      spegam:=spepol(2*t-1, a[0], sizeof(a) div sizeof(ArbFloat) - 1)*g
    end { abs(x) >= macheps }
end; {spegam}

function spelga(x: ArbFloat): ArbFloat;

const

    xbig = 7.7e7;
    xmax = 2.559e305;
  lnr2pi = 9.18938533204672742e-1;
    a: array[0..23] of ArbFloat =
    ( 8.86226925452758013e-1,  1.61691987244425092e-2,
      1.03703363422075456e-1, -1.34118505705967765e-2,
      9.04033494028101968e-3, -2.42259538436268176e-3,
      9.15785997288933120e-4, -2.96890121633200000e-4,
      1.00928148823365120e-4, -3.36375833240268800e-5,
      1.12524642975590400e-5, -3.75499034136576000e-6,
      1.25281466396672000e-6, -4.17808776355840000e-7,
      1.39383522590720000e-7, -4.64774927155200000e-8,
      1.53835215257600000e-8, -5.11961333760000000e-9,
      1.82243164160000000e-9, -6.13513953280000000e-10,
      1.27679856640000000e-10,-4.01499750400000000e-11,
      4.26560716800000000e-11,-1.46381209600000000e-11);
    b: array[0..5] of ArbFloat =
    ( 8.33271644065786580e-2,  -6.16502049453716986e-6,
      3.89978899876484712e-9,  -6.45101975779653651e-12,
      2.00201927337982364e-14, -9.94561064728159347e-17);


var t, g : ArbFloat;
    m, i : ArbInt;

begin
  if x <= 0
  then
    RunError(408);
  if x <= macheps
  then
    spelga:=-ln(x)
  else
  if x <= 15
  then
    begin
      m:=trunc(x); t:=x-m; m:=m-1; g:=1;
      if m < 0
      then
        g:=g/x
      else
      if m > 0
      then
        for i:=1 to m do
          g:=(x-i)*g;
      spelga:=ln(g*spepol(2*t-1, a[0], sizeof(a) div sizeof(ArbFloat) - 1))
    end
  else { x > 15 }
  if x <= xbig
  then
    spelga:=(x-0.5)*ln(x) - x + lnr2pi
            + spepol(450/sqr(x)-1, b[0], sizeof(b) div sizeof(ArbFloat) - 1)/x
  else { x > xbig }
  if x <= xmax
  then
    spelga:=(x-0.5)*ln(x) - x + lnr2pi
  else  { x > xmax => x*ln(x) > giant }
    RunError(408)
end; {spelga}

{ Implements power series expansion for lower incomplete gamma function
  according to
  https://en.wikipedia.org/wiki/Incomplete_gamma_function#Evaluation_formulae
    gamma(s, x) = sum {k = 0 to inf, x^s exp(-x) x^k / (s (s+1) ... (s+k) ) }
  Converges rapidly for x < s + 1 }
function gammaser(s, x: ArbFloat): ArbFloat;
const
  MAX_IT = 100;
  EPS = 1E-7;
var
  delta: Arbfloat;
  sum: ArbFloat;
  k: Integer;
  lngamma: ArbFloat;
begin
  delta := 1 / s;
  sum := delta;
  for k := 1 to MAX_IT do begin
    delta := delta * x / (s + k);
    sum := sum + delta;
    if delta < EPS then break;
  end;
  lngamma := spelga(s);  // log of complete gamma(s)
  Result := exp(s * ln(x) - x + ln(sum) - lngamma);
end;

type
  TCFFunc = function(n: Integer): ArbFloat is nested;

{ Calculates the continued fraction a0 + (b1 / (a1 + b2 / (a2 + b3 / (a3 + b4 /...))))
  using convergents.
  Ref.: http://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=8639&context=rtd
    nth convergent: wn = P(n)/Q(n).
    P(n) = a(n) P(n-1) + b(n) P(n-2)
    Q(n) = a(n) Q(n-1) + b(n) Q(n-2)
    P(-1) = 1, P(0) = a(0), Q(-1) = 0, Q(0) = 1 }
function CalcCF(funca, funcb: TCfFunc; MaxIt: Integer; Eps: ArbFloat): ArbFloat;
var
  Pn, Pn1, Pn2: ArbFloat;
  Qn, Qn1, Qn2: ArbFloat;
  it: Integer;
  prev: ArbFloat;
  a, b: ArbFloat;
begin
  Pn2 := 1.0;
  Pn1 := funca(0);
  Qn2 := 0.0;
  Qn1 := 1.0;
  prev := Giant;
  for it := 1 to MaxIt do begin
    a := funca(it);
    b := funcb(it);
    Pn := a * Pn1 + b * Pn2;
    Qn := a * Qn1 + b * Qn2;
    Result := Pn/Qn;
    if abs(Result - prev) < EPS * abs(Result) then
      exit;
    prev := Result;
    Pn2 := Pn1;
    Pn1 := Pn;
    Qn2 := Qn1;
    Qn1 := Qn;
  end;
end;


{ calculates the upper incomplete gamma function using its continued
  fraction expansion
  https://en.wikipedia.org/wiki/Incomplete_gamma_function#Connection_with_Kummer.27s_confluent_hypergeometric_function }
function gammacf(s, x: ArbFloat): ArbFloat;

  function funca(i: Integer): ArbFloat;
  begin
    if i = 0 then
      Result := 0
    else
    if odd(i) then
      Result := x
    else
      Result := 1;
  end;

  function funcb(i: Integer): ArbFloat;
  begin
    if i = 1 then
      Result := 1
    else
    if odd(i) then
      Result := (i-1) div 2
    else
      Result := i div 2 - s;
  end;

const
  MAX_IT = 100;
  EPS = 1E-7;
begin
  Result := exp(-x + s*ln(x) - spelga(s)) * CalcCF(@funca, @funcb, MAX_IT, EPS);
end;

function gammap(s, x: ArbFloat): ArbFloat;
begin
  if (x < 0.0) or (s <= 0.0) then
    RunError(410);                  // Invalid argument of gammap
  if x = 0.0 then
    Result := 0.0
  else if x < s + 1 then
    Result := gammaser(s, x)        // Use series expansion
  else
    Result := 1.0 - gammacf(s, x);  // Use continued fraction
end;

function gammaq(s, x: ArbFloat): ArbFloat;
begin
  if (x < 0.0) or (s <= 0.0) then
    RunError(411);                  // Invalid argument of gammaq
  if x = 0.0 then
    Result := 1.0
  else if x < s + 1 then
    Result := 1.0 - gammaser(s, x)  // Use series expansion
  else
    Result := gammacf(s, x);        // Use continued fraction
end;

{ Ref.: Moshier, "Methods and programs for mathematical functions" }
function invgammaq(s, y: ArbFloat): ArbFloat;
const
  NUM_IT = 30;
var
  d, y0, x0, xinit, lgm: ArbFloat;
  it: Integer;
  eps: ArbFloat;
begin
  d := 1.0 / (9 * s);
  y0 := invnormaldist(y);
  if y0 = giant then
    exit(0.0);

  y0 := 1.0 - d - y0 * sqrt(d);
  x0 := s * y0 * y0 * y0;
  xinit := x0;
  lgm := spelga(s);
  eps := 2.0 * MachEps;

  for it := 1 to NUM_IT do
  begin
    if (x0 <= 0.0) then   // underflow
      exit(0.0);
    y0 := gammaq(s, x0);
    d := (s - 1.0) * ln(x0) - x0 - lgm;
    if d < -lnGiant then  // underflow
      break;
    d := -exp(d);
    if d = 0.0 then
      break;
    d := (y0 - y) / d;
    x0 := x0 - d;
    if it <= 3 then
      continue;
    if abs(d / x0) < eps then
      break;
  end;
  Result := x0;
end;

{ Calculates the complete beta function based on its property that
    beta(a, b) = gamma(a) * gamma(b) / gamma(a+b)
  https://en.wikipedia.org/wiki/Beta_function }
function beta(a, b: ArbFloat): ArbFloat;
begin
  if (a <= 0) or (b <= 0) then
    RunError(412);
  Result := exp(spelga(a) + spelga(b) - spelga(a + b));
end;

{ Calculates the continued fraction of the incomplete beta function.
  Ref: https://www.encyclopediaofmath.org/index.php/Incomplete_beta-function }
function betaicf(a, b, x: ArbFloat): Arbfloat;

  function funca(i: Integer): ArbFloat;
  begin
    if i = 0 then Result := 0.0 else Result := 1.0;
  end;

  function funcb(i: Integer): ArbFloat;
  var
    am: ArbFloat;
    amm: ArbFloat;
    m: Integer;
  begin
    if i = 1 then
      Result := 1.0
    else begin
      m := (i-1) div 2;
      am := a + m;
      amm := am + m;
      if odd(i) then
        Result := m * (b - m) * x / ((amm - 1) * amm)
      else
        Result := -am * (am + b) * x / (amm * (amm + 1));
    end;
  end;

const
  MAX_IT = 100;
  EPS = 1E-7;
begin
  Result := CalcCF(@funca, @funcb, MAX_IT, EPS);
end;

function betai(a, b, x: ArbFloat): ArbFloat;
var
  factor: ArbFloat;
begin
  // Check for invalid arguments
  if (a <= 0) or (b <= 0) then
    RunError(413);
  if (x < 0) or (x > 1) then
    RunError(414);

  if (x = 0) or (x = 1) then
    factor := 0
  else
    factor := exp(a * ln(x) + b * ln(1.0 - x) + spelga(a + b) - spelga(a) - spelga(b));

  // The continued fraction expansion converges quickly only for
  //    x < (a + 1) / (a + b + 2)
  // For the other case, we apply the relation
  //    beta(a, b, x) = 1 - beta(b, a, 1-x)
  if x < (a + 1) / (a + b + 2) then
    Result := factor * betaicf(a, b, x) / a
  else
    Result := 1.0 - factor * betaicf(b, a, 1.0 - x) / b;
end;

{ Inverse of the incomplete beta function }
function invbetai(a, b, y: ArbFloat; eps: ArbFloat = 0.0): ArbFloat;

  function _betai(x: ArbFloat): ArbFloat;
  begin
    Result := betai(a, b, x) - y;
  end;

var
  term: ArbInt = 0;
begin
  if eps = 0.0 then
    eps := MachEps;
  roof1rn(@_betai, 0, 1, eps, eps, Result, term);
  if term = 3 then
    Result := NaN;
end;

function chi2dist(x: ArbFloat; n: ArbInt): ArbFloat;
begin
  Result := gammaQ(0.5*n, 0.5*x);
end;

function invchi2dist(y: Arbfloat; n: ArbInt): ArbFloat;
begin
  Result := 2.0 * invgammaQ(n/2, y);
//  Result := 2.0 * invgammaQ_alglib(n/2, y);
end;

function tdist(t: ArbFloat; n: ArbInt; Tails: TNumTails): ArbFloat;
begin
  Result := betai(0.5*n, 0.5, n/(n+t*t));
  if Tails = 1 then Result := Result * 0.5;
end;

function invtdist(y: ArbFloat; n: ArbInt; Tails: TNumTails;
  eps: ArbFloat = 0.0): ArbFloat;
var
  w: ArbFloat;
begin
  if (n <= 0) or (y <= 0) or (y >= 1) then
    RunError(415);

  if Tails = 2 then y := y * 0.5;
  w := invbetai(0.5*n, 0.5, 2*y, eps);
  Result := sqrt(n/w - n);
end;

// Calculates the F distribution with n1 and n2 degrees of freedom in the
// numerator and denominator, respectively
function Fdist(F: ArbFloat; n1, n2: ArbInt): ArbFloat;
begin
  Result := betai(n2*0.5, n1*0.5, n2 / (n2 + n1*F));
end;

// Calculates the inverse of the F distribution
// Ref. Moshier, "Methods and programs for mathematical functions"
function invFdist(p: ArbFloat; n1, n2: ArbInt; eps: ArbFloat = 0.0): ArbFloat;
var
  s: ArbFloat;
begin
  if eps = 0.0 then eps := machEps;
  s := invbetai(n2*0.5, n1*0.5, p, eps);
  Result := n2 * (1-s) / (n1 * s);
end;

function spepol(x: ArbFloat; var a: ArbFloat; n: ArbInt): ArbFloat;
var   pa : ^arfloat0;
       i : ArbInt;
    polx : ArbFloat;
begin
  pa:=@a;
  polx:=0;
  for i:=n downto 0 do
    polx:=polx*x+pa^[i];
  spepol:=polx
end {spepol};

function spepow(a, b: ArbFloat): ArbFloat;

   function PowInt(a: double; n: longint): double;
   var a1 : double;
   begin
     if n=0 then PowInt := 1 else
     begin
        a1 := 1;
        if n<0 then begin a := 1/a; n := -n end;
        while n>0
        do if Odd(n)
           then begin Dec(n); a1 := a1*a end
           else begin n := n div 2; a := sqr(a) end;
        PowInt := a1
     end
   end;

var tb : longint;
    fb : double;
begin

  { (a < 0, b niet geheel) of (a = 0, b <= 0), dan afbreken}
  if (a=0) then if (b<=0) then RunError(400) else begin SpePow :=0; exit end;
  tb := Trunc(b); fb := b-tb;
  if (a<0) and (fb<>0) then RunError(400);

  if a>0
  then if fb=0 then SpePow := PowInt(a, tb)
               else SpePow := PowInt(a, tb)*exp(fb*ln(a))
  else if odd(tb) then Spepow := -PowInt(-a, tb)
                  else SpePow := PowInt(-a, tb)

end; {spepow}

function spesgn(x: ArbFloat): ArbInt;

begin
  if x<0
  then
    spesgn:=-1
  else
    if x=0
    then
      spesgn:=0
    else
      spesgn:=1
end; {spesgn}

function spears(x: ArbFloat): ArbFloat;
const

    pi2 = 1.570796326794897;
    a : array[0..17] of ArbFloat =
    (  1.047197551196598e+0, 5.375149359132719e-2, 7.798902238957732e-3,
       1.519668539582420e-3, 3.408637238430600e-4, 8.302317819598986e-5,
       2.134554822576075e-5, 5.701781046148566e-6, 1.566985123962741e-6,
       4.402076871418002e-7, 1.257811162594110e-7, 3.646577948300129e-8,
       1.081021746966715e-8, 3.212744286269388e-9, 8.515014302985799e-10,
       2.513296398553196e-10, 1.342121568282535e-10, 4.210346761190271e-11);

var    y, u, t, s : ArbFloat;
    uprang        : boolean;
begin
  if abs(x) > 1
  then
    RunError(411);
  u:=sqr(x); uprang:= u > 0.5;
  if uprang
  then
    u:=1-u;
  t:=4*u-1; y:=spepol(t, a[0], sizeof(a) div sizeof(ArbFloat) - 1);
  if uprang
  then
    begin
      s:=pi2-sqrt(u)*y;
      if x < 0
      then
        s:=-s;
      spears:=s
    end
  else
    spears:=x*y
end;  {spears}

function spearc(x: ArbFloat): ArbFloat;
const

    pi2 = 1.570796326794897;
    a : array[0..17] of ArbFloat =
    ( 1.047197551196598e+0,  5.375149359132719e-2,  7.798902238957732e-3,
      1.519668539582420e-3,  3.408637238430600e-4,  8.302317819598986e-5,
      2.134554822576075e-5,  5.701781046148566e-6,  1.566985123962741e-6,
      4.402076871418002e-7,  1.257811162594110e-7,  3.646577948300129e-8,
      1.081021746966715e-8,  3.212744286269388e-9,  8.515014302985799e-10,
      2.513296398553196e-10, 1.342121568282535e-10, 4.210346761190271e-11);

var u, t, y, s    : ArbFloat;
    uprang        : boolean;
begin
  if abs(x)>1.0
  then
    RunError(402);
  u:=sqr(x); uprang:=u>0.5;
  if uprang
  then
    u:=1-u;
  t:=4*u-1; y:=spepol(t, a[0], sizeof(a) div sizeof(ArbFloat) - 1);
  if uprang
  then
    begin
      s:=sqrt(u)*y;
      if x<0
      then
        s:=2*pi2-s;
      spearc:=s
    end
  else
    spearc:=pi2-x*y
end;  {spearc}

function spesih(x: ArbFloat): ArbFloat;
const

    a : array[0..6] of ArbFloat =
    ( 1.085441641272607e+0,  8.757509762437522e-2,  2.158779361257021e-3,
      2.549839945498292e-5,  1.761854853281383e-7,  7.980704288665359e-10,
      2.551377137317034e-12);

var u : ArbFloat;
begin
  if abs(x)<=1.0
  then
    begin
      u:=2*sqr(x)-1;
      spesih:=x*spepol(u, a[0], sizeof(a) div sizeof(ArbFloat) - 1)
    end
  else
  begin
    u:=exp(x); spesih:=(u-1/u)/2
  end
end; {spesih}

function specoh(x: ArbFloat): ArbFloat;
var u: ArbFloat;
begin
  u:=exp(x); specoh:=(u+1/u)/2
end; {specoh}

function spetah(x: ArbFloat): ArbFloat;
const
    xhi = 18.50;
    a : array[0..15] of ArbFloat =
    ( 8.610571715805476e-1, -1.158834489728470e-1,  1.918072383973950e-2,
     -3.225255180728459e-3,  5.433071386922689e-4, -9.154289983175165e-5,
      1.542469328074432e-5, -2.599022539340038e-6,  4.379282308765732e-7,
     -7.378980192173815e-8,  1.243517352745986e-8, -2.095373768837420e-9,
      3.509758916273561e-10,-5.908745181531817e-11, 1.124199312776748e-11,
     -1.907888434471600e-12);

var t, y: ArbFloat;

begin
  t:=abs(x);
  if t <= 1
  then
    begin
      y:=2*sqr(x)-1;
      spetah:=x*spepol(y, a[0], sizeof(a) div sizeof(ArbFloat) - 1)
    end
  else
  if t < xhi
  then
    begin
      y:=exp(2*x); spetah:=(y-1)/(y+1)
    end
  else
    spetah:=spesgn(x)
end; {spetah}

function speash(x: ArbFloat): ArbFloat;
const

    xhi = 1e9;
    c : array[0..18] of ArbFloat =
    (  9.312298594527122e-1,  -5.736663926249348e-2,
       9.004288574881897e-3,  -1.833458667045431e-3,
       4.230023450529706e-4,  -1.050715136470630e-4,
       2.740790473603819e-5,  -7.402952157663977e-6,
       2.052474396638805e-6,  -5.807433412373489e-7,
       1.670117348345774e-7,  -4.863477336087045e-8,
       1.432753532351304e-8,  -4.319978113584910e-9,
       1.299779213740398e-9,  -3.394726871170490e-10,
       1.008344962167889e-10, -5.731943029121004e-11,
       1.810792296549804e-11);


var t : ArbFloat;

begin
  t:=abs(x);
  if t <= 1 then
    speash:=x*spepol(2*sqr(x)-1, c[0], sizeof(c) div sizeof(ArbFloat) - 1)
  else
  if t < xhi then
    speash:=ln(sqrt(sqr(x)+1)+t)*spesgn(x)
  else
    speash:=ln(2*t)*spesgn(x)
end; {speash}

function speach(x: ArbFloat): ArbFloat;
const

    xhi = 1e9;

begin
  if x<1 then
    RunError(405);
  if x=1 then
    speach:=0
  else
  if x<=xhi then
    speach:=ln(x+sqrt(sqr(x)-1))
  else
    speach:=ln(2*x)
end; {speach}

function speath(x: ArbFloat): ArbFloat;
const

    c : array[0..19] of ArbFloat =
    ( 1.098612288668110e+0,  1.173605223326117e-1,  2.309071936165689e-2,
      5.449091889986991e-3,  1.404884102286929e-3,  3.816948426588058e-4,
      1.073604335435426e-4,  3.095027782918129e-5,  9.088050814470148e-6,
      2.706881064641104e-6,  8.155200644023077e-7,  2.479830612463254e-7,
      7.588067811453948e-8,  2.339295963220429e-8,  7.408486568719348e-9,
      2.319454882064018e-9,  5.960921368486746e-10, 1.820410351379402e-10,
      1.184977617320312e-10, 3.856235316559190e-11);

var t, u: ArbFloat;
begin
  t:=abs(x);
  if t >= 1 then
    RunError(406);
  u:=sqr(x);
  if u < 0.5 then
    speath:=x*spepol(4*u-1, c[0], sizeof(c) div sizeof(ArbFloat) - 1)
  else { 0.5 < x*x < 1 }
    speath:=ln((1+t)/(1-t))/2*spesgn(x)
end; {speath}

var exitsave : pointer;

procedure MyExit;
{
const ErrorS : array[400..408,1..6] of AnsiChar =
     ('spepow',
      'spebk0',
      'spebk1',
      'speby0',
      'speby1',
      'speach',
      'speath',
      'spegam',
      'spelga'); }

//var ErrFil : text;

begin
     ExitProc := ExitSave;
//     Assign(ErrFil, 'CON');
//    ReWrite(ErrFil);
     if (ExitCode>=400) AND (ExitCode<=415) then
       begin
//         write(ErrFil, 'critical error in ', ErrorS[ExitCode]);
         ExitCode := 201
       end;
//     Close(ErrFil)
end;

begin
   ExitSave := ExitProc;
   ExitProc := @MyExit;
end.