freepascal.compiler/rtl/inc/genmath.inc

{
    This file is part of the Free Pascal run time library.
    Copyright (c) 1999-2007 by Several contributors

    Generic mathematical routines (on type real)

    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.

 **********************************************************************}
{*************************************************************************}
{  Credits                                                                }
{*************************************************************************}
{       Copyright Abandoned, 1987, Fred Fish                              }
{                                                                         }
{       This previously copyrighted work has been placed into the         }
{       public domain by the author (Fred Fish) and may be freely used    }
{       for any purpose, private or commercial.  I would appreciate       }
{       it, as a courtesy, if this notice is left in all copies and       }
{       derivative works.  Thank you, and enjoy...                        }
{                                                                         }
{       The author makes no warranty of any kind with respect to this     }
{       product and explicitly disclaims any implied warranties of        }
{       merchantability or fitness for any particular purpose.            }
{-------------------------------------------------------------------------}
{       Copyright (c) 1992 Odent Jean Philippe                            }
{                                                                         }
{       The source can be modified as long as my name appears and some    }
{       notes explaining the modifications done are included in the file. }
{-------------------------------------------------------------------------}
{       Copyright (c) 1997 Carl Eric Codere                               }
{-------------------------------------------------------------------------}
{-------------------------------------------------------------------------
 Using functions from AMath/DAMath libraries, which are covered by the
 following license:

 (C) Copyright 2009-2013 Wolfgang Ehrhardt

 This software is provided 'as-is', without any express or implied warranty.
 In no event will the authors be held liable for any damages arising from
 the use of this software.

 Permission is granted to anyone to use this software for any purpose,
 including commercial applications, and to alter it and redistribute it
 freely, subject to the following restrictions:

 1. The origin of this software must not be misrepresented; you must not
    claim that you wrote the original software. If you use this software in
    a product, an acknowledgment in the product documentation would be
    appreciated but is not required.

 2. Altered source versions must be plainly marked as such, and must not be
    misrepresented as being the original software.

 3. This notice may not be removed or altered from any source distribution.
----------------------------------------------------------------------------}

type
  PReal = ^Real;
{ float64 definition is now in genmathh.inc,
  to ensure that float64 will always be in
  the system interface symbol table. }

const
      PIO4   =  7.85398163397448309616E-1;    {  pi/4        }
      SQRT2  =  1.41421356237309504880;       {  sqrt(2)     }
      LOG2E  =  1.4426950408889634073599;     {  1/log(2)    }
      lossth =  1.073741824e9;
      MAXLOG =  8.8029691931113054295988E1;    { log(2**127)  }
      MINLOG = -8.872283911167299960540E1;     { log(2**-128) }
      H2_54: double = 18014398509481984.0;    {2^54}
      huge: double = 1e300;
      one:  double = 1.0;
      zero: double = 0;

{$if not defined(FPC_SYSTEM_HAS_SIN) or not defined(FPC_SYSTEM_HAS_COS)}
const sincof : array[0..5] of Real = (
                1.58962301576546568060E-10,
               -2.50507477628578072866E-8,
                2.75573136213857245213E-6,
               -1.98412698295895385996E-4,
                8.33333333332211858878E-3,
               -1.66666666666666307295E-1);
      coscof : array[0..5] of Real = (
               -1.13585365213876817300E-11,
                2.08757008419747316778E-9,
               -2.75573141792967388112E-7,
                2.48015872888517045348E-5,
               -1.38888888888730564116E-3,
                4.16666666666665929218E-2);
{$endif}

{*
-------------------------------------------------------------------------------
Raises the exceptions specified by `flags'.  Floating-point traps can be
defined here if desired.  It is currently not possible for such a trap
to substitute a result value.  If traps are not implemented, this routine
should be simply `softfloat_exception_flags |= flags;'.
-------------------------------------------------------------------------------
*}
procedure float_raise(i: TFPUException);
begin
  float_raise([i]);
end;

procedure float_raise(i: TFPUExceptionMask);
var
  pflags: ^TFPUExceptionMask;
  unmasked_flags: TFPUExceptionMask;
Begin
  { taking address of threadvar produces somewhat more compact code }
  pflags := @softfloat_exception_flags;
  pflags^:=pflags^ + i;
  unmasked_flags := pflags^ - softfloat_exception_mask;
  { before we raise the exception, we mark it as handled,
    this behaviour is similiar to the hardware handler in SignalToRunerror }
  SysResetFPU;
  if (float_flag_invalid in unmasked_flags) then
     HandleError(207)
  else if (float_flag_divbyzero in unmasked_flags) then
     HandleError(208)
  else if (float_flag_overflow in unmasked_flags) then
     HandleError(205)
  else if (float_flag_underflow in unmasked_flags) then
     HandleError(206)
  else if (float_flag_inexact in unmasked_flags) then
     HandleError(207)
  else if (float_flag_denormal in unmasked_flags) then
     HandleError(216);
end;


{ This function does nothing, but its argument is expected to be an expression
  which causes FPE when calculated. If exception is masked, it just returns true,
  allowing to use it in expressions. }
function fpe_helper(x: valreal): boolean;
begin
  result:=true;
end;


{$ifdef SUPPORT_DOUBLE}

{$ifndef FPC_HAS_FLOAT64HIGH}
{$define FPC_HAS_FLOAT64HIGH}
function float64high(d: double): longint; inline;
begin
  result:=float64(d).high;
end;


procedure float64sethigh(var d: double; l: longint); inline;
begin
  float64(d).high:=l;
end;

{$endif FPC_HAS_FLOAT64HIGH}

{$ifndef FPC_HAS_FLOAT64LOW}
{$define FPC_HAS_FLOAT64LOW}
function float64low(d: double): longint; inline;
begin
  result:=float64(d).low;
end;


procedure float64setlow(var d: double; l: longint); inline;
begin
  float64(d).low:=l;
end;

{$endif FPC_HAS_FLOAT64LOW}

{$endif SUPPORT_DOUBLE}


{$ifndef FPC_SYSTEM_HAS_TRUNC}

{$ifdef SUPPORT_DOUBLE}
   { based on softfloat float64_to_int64_round_to_zero }
   function fpc_trunc_real(d : valreal) : int64; compilerproc;
     var
       aExp, shiftCount : smallint;
       aSig : int64;
       z : int64;
       a: float64;
     begin
       a:=float64(d);
       aSig:=(int64(a.high and $000fffff) shl 32) or longword(a.low);
       aExp:=(a.high shr 20) and $7FF;
       if aExp<>0 then
         aSig:=aSig or $0010000000000000;
       shiftCount:= aExp-$433;
       if 0<=shiftCount then
         begin
           if aExp>=$43e then
             begin
               if (a.high<>longint($C3E00000)) or (a.low<>0) then
                 begin
                   fpe_helper(zero/zero);
                   if (longint(a.high)>=0) or ((aExp=$7FF) and
                      (aSig<>$0010000000000000 )) then
                     begin
                       result:=$7FFFFFFFFFFFFFFF;
                       exit;
                     end;
                 end;
               result:=$8000000000000000;
               exit;
             end;
           z:=aSig shl shiftCount;
         end
       else
         begin
           if aExp<$3fe then
             begin
               result:=0;
               exit;
             end;
           z:=aSig shr -shiftCount;
           {
           if (aSig shl (shiftCount and 63))<>0 then
             float_exception_flags |= float_flag_inexact;
           }
         end;
       if longint(a.high)<0 then
         z:=-z;
       result:=z;
     end;

{$else SUPPORT_DOUBLE}
  { based on softfloat float32_to_int64_round_to_zero }
  Function fpc_trunc_real( d: valreal ): int64; compilerproc;
    Var
       a : float32;
       aExp, shiftCount : smallint;
       aSig : longint;
       aSig64, z : int64;
    Begin
       a := float32(d);
       aSig := a and $007FFFFF;
       aExp := (a shr 23) and $FF;
       shiftCount := aExp - $BE;
       if ( 0 <= shiftCount ) then
         Begin
           if ( a <> Float32($DF000000) ) then
             Begin
               fpe_helper( zero/zero );
               if ( (longint(a)>=0) or ( ( aExp = $FF ) and (aSig<>0) ) ) then
                 Begin
                   result:=$7fffffffffffffff;
                   exit;
                 end;
             End;
           result:=$8000000000000000;
           exit;
         End
       else
         if ( aExp <= $7E ) then
         Begin
           result := 0;
           exit;
         End;
       aSig64 := int64( aSig or $00800000 ) shl 40;
       z := aSig64 shr ( - shiftCount );
       if ( longint(a)<0 ) then z := - z;
       result := z;
    End;
{$endif SUPPORT_DOUBLE}

{$endif not FPC_SYSTEM_HAS_TRUNC}


{$ifndef FPC_SYSTEM_HAS_INT}

{$ifdef SUPPORT_DOUBLE}

    { straight Pascal translation of the code for __trunc() in }
    { the file sysdeps/libm-ieee754/s_trunc.c of glibc (JM)    }
    function fpc_int_real(d: ValReal): ValReal;compilerproc;
      var
        i0, j0: longint;
        i1: cardinal;
        sx: longint;
        f64 : float64;
      begin
        f64:=float64(d);
        i0 := f64.high;
        i1 := cardinal(f64.low);
        sx := i0 and $80000000;
        j0 := ((i0 shr 20) and $7ff) - $3ff;
        if (j0 < 20) then
          begin
            if (j0 < 0) then
              begin
                { the magnitude of the number is < 1 so the result is +-0. }
                f64.high := sx;
                f64.low := 0;
              end
            else
              begin
                f64.high := sx or (i0 and not($fffff shr j0));
                f64.low := 0;
              end
          end
        else if (j0 > 51) then
          begin
            if (j0 = $400) then
              { d is inf or NaN }
              exit(d + d); { don't know why they do this (JM) }
          end
        else
          begin
            f64.high := i0;
            f64.low := longint(i1 and not(cardinal($ffffffff) shr (j0 - 20)));
          end;
        result:=double(f64);
      end;

{$else SUPPORT_DOUBLE}

    function fpc_int_real(d : ValReal) : ValReal;compilerproc;
      begin
        { this will be correct since real = single in the case of }
        { the motorola version of the compiler...                 }
        result:=ValReal(trunc(d));
      end;
{$endif SUPPORT_DOUBLE}

{$endif not FPC_SYSTEM_HAS_INT}


{$ifndef FPC_SYSTEM_HAS_ABS}

    function fpc_abs_real(d : ValReal) : ValReal;compilerproc;
    begin
       if (d<0.0) then
         result := -d
      else
         result := d ;
    end;

{$endif not FPC_SYSTEM_HAS_ABS}


{$ifndef SYSTEM_HAS_FREXP}
    procedure frexp(X: Real; out Mantissa: Real; out Exponent: longint);
    {*  frexp() extracts the exponent from x.  It returns an integer     *}
    {*  power of two to expnt and the significand between 0.5 and 1      *}
    {*  to y.  Thus  x = y * 2**expn.                                    *}
    begin
      exponent:=0;
      if (abs(x)<0.5) then
       While (abs(x)<0.5) do
       begin
         x := x*2;
         Dec(exponent);
       end
      else
       While (abs(x)>1) do
       begin
         x := x/2;
         Inc(exponent);
       end;
      Mantissa := x;
    end;
{$endif not SYSTEM_HAS_FREXP}


{$ifndef SYSTEM_HAS_LDEXP}
{$ifdef SUPPORT_DOUBLE}
{ ldexpd function adapted from DAMath library (C) Copyright 2013 Wolfgang Ehrhardt }
    function ldexp( x: Real; N: Integer):Real;
    {* ldexp() multiplies x by 2**n.                                    *}
    var
      i: integer;
    begin
      i := (float64high(x) and $7ff00000) shr 20;
      {if +-INF, NaN, 0 or if e=0 return d}
      if (i=$7FF) or (N=0) or (x=0.0) then
        ldexp := x
      else if i=0 then    {Denormal: result = d*2^54*2^(e-54)}
        ldexp := ldexp(x*H2_54, N-54)
      else
      begin
        N := N+i;
        if N>$7FE then  { overflow }
        begin
          if x>0.0 then
            ldexp := 2.0*huge
          else
            ldexp := (-2.0)*huge;
        end
        else if N<1 then
        begin
          {underflow or denormal}
          if N<-53 then
            ldexp := 0.0
          else
          begin
            {Denormal: result = d*2^(e+54)/2^54}
            inc(N,54);
            float64sethigh(x,(float64high(x) and $800FFFFF) or (longint(N) shl 20));
            ldexp := x/H2_54;
          end;
        end
        else
        begin
          float64sethigh(x,(float64high(x) and $800FFFFF) or (longint(N) shl 20));
          ldexp := x;
        end;
      end;
    end;
{$else SUPPORT_DOUBLE}
    function ldexp( x: Real; N: Integer):Real;
    {* ldexp() multiplies x by 2**n.                                    *}
    var r : Real;
    begin
      R := 1;
      if N>0 then
         while N>0 do
         begin
            R:=R*2;
            Dec(N);
         end
      else
        while N<0 do
        begin
           R:=R/2;
           Inc(N);
        end;
      ldexp := x * R;
    end;
{$endif SUPPORT_DOUBLE}
{$endif not SYSTEM_HAS_LDEXP}


    function floord(x: double): double; inline;
    begin
      result := int(x);
      if result>x then
        result := result - 1.0;
    end;

   {*
    * ====================================================
    * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
    *
    * Developed at SunPro, a Sun Microsystems, Inc. business.
    * Permission to use, copy, modify, and distribute this
    * software is freely granted, provided that this notice
    * is preserved.
    * ====================================================
    *
    * Pascal port of this routine comes from DAMath library
    * (C) Copyright 2013 Wolfgang Ehrhardt
    *
    * k_rem_pio2 return the last three bits of N with y = x - N*pi/2
    * so that |y| < pi/2.
    *
    * The method is to compute the integer (mod 8) and fraction parts of
    * (2/pi)*x without doing the full multiplication. In general we
    * skip the part of the product that are known to be a huge integer
    * (more accurately, = 0 mod 8 ). Thus the number of operations are
    * independent of the exponent of the input.
    *
    * (2/pi) is represented by an array of 24-bit integers in ipio2[].
    *
    * Input parameters:
    *      x[]     The input value (must be positive) is broken into nx
    *              pieces of 24-bit integers in double precision format.
    *              x[i] will be the i-th 24 bit of x. The scaled exponent
    *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
    *              match x's up to 24 bits.
    *
    *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
    *                      e0 = ilogb(z)-23
    *                      z  = scalbn(z,-e0)
    *              for i = 0,1,2
    *                      x[i] = floor(z)
    *                      z    = (z-x[i])*2**24
    *
    *
    *      y[]     output result in an array of double precision numbers.
    *              The dimension of y[] is:
    *                      24-bit  precision       1
    *                      53-bit  precision       2
    *                      64-bit  precision       2
    *                      113-bit precision       3
    *              The actual value is the sum of them. Thus for 113-bit
    *              precison, one may have to do something like:
    *
    *              long double t,w,r_head, r_tail;
    *              t = (long double)y[2] + (long double)y[1];
    *              w = (long double)y[0];
    *              r_head = t+w;
    *              r_tail = w - (r_head - t);
    *
    *      e0      The exponent of x[0]. Must be <= 16360 or you need to
    *              expand the ipio2 table.
    *
    *      nx      dimension of x[]
    *
    *      prec    an integer indicating the precision:
    *                      0       24  bits (single)
    *                      1       53  bits (double)
    *                      2       64  bits (extended)
    *                      3       113 bits (quad)
    *
    * Here is the description of some local variables:
    *
    *      jk      jk+1 is the initial number of terms of ipio2[] needed
    *              in the computation. The recommended value is 2,3,4,
    *              6 for single, double, extended,and quad.
    *
    *      jz      local integer variable indicating the number of
    *              terms of ipio2[] used.
    *
    *      jx      nx - 1
    *
    *      jv      index for pointing to the suitable ipio2[] for the
    *              computation. In general, we want
    *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
    *              is an integer. Thus
    *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
    *              Hence jv = max(0,(e0-3)/24).
    *
    *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
    *
    *      q[]     double array with integral value, representing the
    *              24-bits chunk of the product of x and 2/pi.
    *
    *      q0      the corresponding exponent of q[0]. Note that the
    *              exponent for q[i] would be q0-24*i.
    *
    *      PIo2[]  double precision array, obtained by cutting pi/2
    *              into 24 bits chunks.
    *
    *      f[]     ipio2[] in floating point
    *
    *      iq[]    integer array by breaking up q[] in 24-bits chunk.
    *
    *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
    *
    *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
    *              it also indicates the *sign* of the result.
    *}

    {PIo2[] double array, obtained by cutting pi/2 into 24 bits chunks.}
    const
      PIo2chunked: array[0..7] of double = (
        1.57079625129699707031e+00, { 0x3FF921FB, 0x40000000 }
        7.54978941586159635335e-08, { 0x3E74442D, 0x00000000 }
        5.39030252995776476554e-15, { 0x3CF84698, 0x80000000 }
        3.28200341580791294123e-22, { 0x3B78CC51, 0x60000000 }
        1.27065575308067607349e-29, { 0x39F01B83, 0x80000000 }
        1.22933308981111328932e-36, { 0x387A2520, 0x40000000 }
        2.73370053816464559624e-44, { 0x36E38222, 0x80000000 }
        2.16741683877804819444e-51  { 0x3569F31D, 0x00000000 }
      );

    {Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi }
      ipio2: array[0..65] of longint = (
        $A2F983, $6E4E44, $1529FC, $2757D1, $F534DD, $C0DB62,
        $95993C, $439041, $FE5163, $ABDEBB, $C561B7, $246E3A,
        $424DD2, $E00649, $2EEA09, $D1921C, $FE1DEB, $1CB129,
        $A73EE8, $8235F5, $2EBB44, $84E99C, $7026B4, $5F7E41,
        $3991D6, $398353, $39F49C, $845F8B, $BDF928, $3B1FF8,
        $97FFDE, $05980F, $EF2F11, $8B5A0A, $6D1F6D, $367ECF,
        $27CB09, $B74F46, $3F669E, $5FEA2D, $7527BA, $C7EBE5,
        $F17B3D, $0739F7, $8A5292, $EA6BFB, $5FB11F, $8D5D08,
        $560330, $46FC7B, $6BABF0, $CFBC20, $9AF436, $1DA9E3,
        $91615E, $E61B08, $659985, $5F14A0, $68408D, $FFD880,
        $4D7327, $310606, $1556CA, $73A8C9, $60E27B, $C08C6B);

      init_jk: array[0..3] of integer = (2,3,4,6); {initial value for jk}
      two24:  double = 16777216.0;                 {2^24}
      twon24: double = 5.9604644775390625e-08;     {1/2^24}

    type
      TDA02 = array[0..2] of double; { 3 elements is enough for float128 }

    { inline to make use of the fact prec is always 2. }
    function k_rem_pio2(const x: TDA02; out y: TDA02; e0, nx, prec: integer): sizeint; inline;
    var
      i,ih,j,jz,jx,jv,jp,jk,carry,k,n,q0: longint;
      t: longint;
      iq: array[0..19] of longint;
      f,fq,q: array[0..19] of double;
      z,fw: double;
    begin
      {initialize jk}
      jk := init_jk[prec];
      jp := jk;

      {determine jx,jv,q0, note that 3>q0}
      jx := nx-1;
      jv := (e0-3) div 24; if jv<0 then jv := 0;
      q0 := e0-24*(jv+1);

      {set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk]}
      j := jv-jx;
      for i:=0 to jx+jk do
      begin
        if j<0 then f[i] := 0.0 else f[i] := ipio2[j];
        inc(j);
      end;

      {compute q[0],q[1],...q[jk]}
      for i:=0 to jk do
      begin
        fw := 0.0;
        for j:=0 to jx do
          fw := fw + x[j]*f[jx+i-j];
        q[i] := fw;
      end;
      jz := jk;

      repeat
        {distill q[] into iq[] reversingly}
        i := 0;
        z := q[jz];
        for j:=jz downto 1 do
        begin
          fw    := trunc(twon24*z);
          iq[i] := trunc(z-two24*fw);
          z     := q[j-1]+fw;
          inc(i);
        end;

        {compute n}
        z  := ldexp(z,q0);              {actual value of z}
        z  := z - 8.0*floord(z*0.125);  {trim off integer >= 8}
        n  := trunc(z);
        z  := z - n;
        ih := 0;
        if q0>0 then
        begin
          {need iq[jz-1] to determine n}
          t  := (iq[jz-1] shr (24-q0));
          inc(n,t);
          dec(iq[jz-1], t shl (24-q0));
          ih := iq[jz-1] shr (23-q0);
        end
        else if q0=0 then
          ih := iq[jz-1] shr 23
        else if z>=0.5 then
          ih := 2;

        if ih>0 then    {q > 0.5}
        begin
          inc(n);
          carry := 0;
          for i:=0 to jz-1 do
          begin
            {compute 1-q}
            t := iq[i];
            if carry=0 then
            begin
              if t<>0 then
              begin
                carry := 1;
                iq[i] := $1000000 - t;
              end
            end
            else
              iq[i] := $ffffff - t;
          end;
          if q0>0 then
          begin
            {rare case: chance is 1 in 12}
            case q0 of
              1: iq[jz-1] := iq[jz-1] and $7fffff;
              2: iq[jz-1] := iq[jz-1] and $3fffff;
            end;
          end;
          if ih=2 then
          begin
            z := 1.0 - z;
            if carry<>0 then
              z := z - ldexp(1.0,q0);
          end;
        end;

        {check if recomputation is needed}
        if z<>0.0 then
          break;
        t := 0;
        for i:=jz-1 downto jk do
          t := t or iq[i];
        if t<>0 then
          break;

        {need recomputation}
        k := 1;
        while iq[jk-k]=0 do    {k = no. of terms needed}
          inc(k);
        for i:=jz+1 to jz+k do
        begin
          {add q[jz+1] to q[jz+k]}
          f[jx+i] := ipio2[jv+i];
          fw := 0.0;
          for j:=0 to jx do
            fw := fw + x[j]*f[jx+i-j];
          q[i] := fw;
        end;
        inc(jz,k);
      until False;

      {chop off zero terms}
      if z=0.0 then
      begin
        repeat
          dec(jz);
          dec(q0,24);
        until iq[jz]<>0;
      end
      else
      begin
        {break z into 24-bit if necessary}
        z := ldexp(z,-q0);
        if z>=two24 then
        begin
          fw := trunc(twon24*z);
          iq[jz] := trunc(z-two24*fw);
          inc(jz);
          inc(q0,24);
          iq[jz] := trunc(fw);
        end
        else
          iq[jz] := trunc(z);
      end;

      {convert integer "bit" chunk to floating-point value}
      fw := ldexp(1.0,q0);
      for i:=jz downto 0 do
      begin
        q[i] := fw*iq[i];
        fw := fw*twon24;
      end;

      {compute PIo2[0,...,jp]*q[jz,...,0]}
      for i:=jz downto 0 do
      begin
        fw :=0.0;
        k := 0;
        while (k<=jp) and (k<=jz-i) do
        begin
          fw := fw + double(PIo2chunked[k])*(q[i+k]);
          inc(k);
        end;
        fq[jz-i] := fw;
      end;

      {compress fq[] into y[]}
      case prec of
        0:
        begin
          fw := 0.0;
          for i:=jz downto 0 do
            fw := fw + fq[i];
          if ih=0 then
            y[0] := fw
          else
            y[0] := -fw;
        end;

        1, 2:
        begin
          fw := 0.0;
          for i:=jz downto 0 do
            fw := fw + fq[i];
          if ih=0 then
            y[0] := fw
          else
            y[0] := -fw;
          fw := fq[0]-fw;
          for i:=1 to jz do
            fw := fw + fq[i];
          if ih=0 then
            y[1] := fw
          else
            y[1] := -fw;
        end;

        3:
        begin
          {painful}
          for i:=jz downto 1 do
          begin
            fw     := fq[i-1]+fq[i];
            fq[i]  := fq[i]+(fq[i-1]-fw);
            fq[i-1]:= fw;
          end;
          for i:=jz downto 2 do
          begin
            fw     := fq[i-1]+fq[i];
            fq[i]  := fq[i]+(fq[i-1]-fw);
            fq[i-1]:= fw;
          end;
          fw := 0.0;
          for i:=jz downto 2 do
            fw := fw + fq[i];
          if ih=0 then
          begin
            y[0] := fq[0];
            y[1] := fq[1];
            y[2] := fw;
          end
          else
          begin
            y[0] := -fq[0];
            y[1] := -fq[1];
            y[2] := -fw;
          end;
        end;
      end;
      k_rem_pio2 := n and 7;
    end;


    { Argument reduction of x:  z = x - n*Pi/2, |z| <= Pi/4, result = n mod 8.}
    { Uses Payne/Hanek if |x| >= lossth, Cody/Waite otherwise}
    function rem_pio2_unlikely(x: double; out z: double): sizeint;
    var
      e0,nx: longint;
      tx,ty: TDA02;
    begin
      z := abs(x);
      e0 := (float64high(z) shr 20)-1046;
      if (e0 = ($7ff-1046)) then  { z is Inf or NaN }
      begin
{$push} {$optimization nofastmath}
        z := x - x;
{$pop}
        exit(0);
      end;
      float64sethigh(z,float64high(z) - (e0 shl 20));
      tx[0] := trunc(z);
      z := (z-tx[0])*two24;
      tx[1] := trunc(z);
      tx[2] := (z-tx[1])*two24;
      nx := 3;
      while (tx[nx-1]=0.0) do dec(nx);  { skip zero terms }
      result := k_rem_pio2(tx,ty,e0,nx,2);
      z := ty[0] + ty[1];
      if x<0 then
      begin
        result := (-result) and 7;
        z := -z;
      end;
    end;

    function rem_pio2(x: double; out z: double): sizeint;
    const
      tol = double(2.384185791015625E-7);  {lossth*eps_d}
      DP1 = double(7.85398125648498535156E-1);
      DP2 = double(3.77489470793079817668E-8);
      DP3 = double(2.69515142907905952645E-15);
    var
      i: longint;
      y: double;
    begin
      y := abs(x);
      if (y < PIO4) then
      begin
        z := x;
        exit(0);
      end
      else if (y < lossth) then
      begin
        y := floord(x/PIO4);
        i := trunc(y - 16.0*floord(y*0.0625));
        if odd(i) then
        begin
          inc(i);
          y := y + 1.0;
        end;
        z := ((x - y * DP1) - y * DP2) - y * DP3;

        {If x is near a multiple of Pi/2, the C/W relative error may be large.}
        {In this case redo the calculation with the Payne/Hanek algorithm.    }
        if abs(z) > tol then
          exit(i shr 1 and 7);
      end;
      result := rem_pio2_unlikely(x, z);
    end;


{$ifndef FPC_SYSTEM_HAS_SQR}
    function fpc_sqr_real(d : ValReal) : ValReal;compilerproc;{$ifdef MATHINLINE}inline;{$endif}
    begin
      result := d*d;
    end;
{$endif}



{$ifndef FPC_SYSTEM_HAS_SQRT}
    function fpc_sqrt_real(d:ValReal):ValReal;compilerproc;
    {*****************************************************************}
    { Square root                                                     }
    {*****************************************************************}
    {                                                                 }
    { SYNOPSIS:                                                       }
    {                                                                 }
    { double x, y, sqrt();                                            }
    {                                                                 }
    { y = sqrt( x );                                                  }
    {                                                                 }
    { DESCRIPTION:                                                    }
    {                                                                 }
    { Returns the square root of x.                                   }
    {                                                                 }
    { Range reduction involves isolating the power of two of the      }
    { argument and using a polynomial approximation to obtain         }
    { a rough value for the square root.  Then Heron's iteration      }
    { is used three times to converge to an accurate value.           }
    {*****************************************************************}
    var e   : Longint;
        w,z : Real;
    begin
       if( d <= 0.0 ) then
       begin
           if d < 0.0 then
             result:=zero/zero
           else
             result := 0.0;
       end
     else
       begin
          w := d;
          { separate exponent and significand }
          frexp( d, z, e );

          {  approximate square root of number between 0.5 and 1  }
          {  relative error of approximation = 7.47e-3            }
          d := 4.173075996388649989089E-1 + 5.9016206709064458299663E-1 * z;

          { adjust for odd powers of 2 }
          if odd(e) then
             d := d*SQRT2;

          { re-insert exponent }
          d := ldexp( d, (e div 2) );

          { Newton iterations: }
          d := 0.5*(d + w/d);
          d := 0.5*(d + w/d);
          d := 0.5*(d + w/d);
          d := 0.5*(d + w/d);
          d := 0.5*(d + w/d);
          d := 0.5*(d + w/d);
          result := d;
       end;
    end;

{$endif}


{$ifndef FPC_SYSTEM_HAS_EXP}
{$ifdef SUPPORT_DOUBLE}
    {
     This code was translated from uclib code, the original code
     had the following copyright notice:

     *
     * ====================================================
     * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
     *
     * Developed at SunPro, a Sun Microsystems, Inc. business.
     * Permission to use, copy, modify, and distribute this
     * software is freely granted, provided that this notice
     * is preserved.
     * ====================================================
     *}

    {*
     * Returns the exponential of x.
     *
     * Method
     *   1. Argument reduction:
     *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
     *  Given x, find r and integer k such that
     *
     *               x = k*ln2 + r,  |r| <= 0.5*ln2.
     *
     *      Here r will be represented as r = hi-lo for better
     *  accuracy.
     *
     *   2. Approximation of exp(r) by a special rational function on
     *  the interval [0,0.34658]:
     *  Write
     *      R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
     *      We use a special Reme algorithm on [0,0.34658] to generate
     *  a polynomial of degree 5 to approximate R. The maximum error
     *  of this polynomial approximation is bounded by 2**-59. In
     *  other words,
     *      R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
     *      (where z=r*r, and the values of P1 to P5 are listed below)
     *  and
     *      |                  5          |     -59
     *      | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
     *      |                             |
     *  The computation of exp(r) thus becomes
     *                     2*r
     *      exp(r) = 1 + -------
     *                    R - r
     *                         r*R1(r)
     *             = 1 + r + ----------- (for better accuracy)
     *                        2 - R1(r)
     *  where
                         2       4             10
     *     R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
     *
     *   3. Scale back to obtain exp(x):
     *  From step 1, we have
     *     exp(x) = 2^k * exp(r)
     *
     * Special cases:
     *  exp(INF) is INF, exp(NaN) is NaN;
     *  exp(-INF) is 0, and
     *  for finite argument, only exp(0)=1 is exact.
     *
     * Accuracy:
     *  according to an error analysis, the error is always less than
     *  1 ulp (unit in the last place).
     *
     * Misc. info.
     *  For IEEE double
     *      if x >  7.09782712893383973096e+02 then exp(x) overflow
     *      if x < -7.45133219101941108420e+02 then exp(x) underflow
     *
     * Constants:
     * The hexadecimal values are the intended ones for the following
     * constants. The decimal values may be used, provided that the
     * compiler will convert from decimal to binary accurately enough
     * to produce the hexadecimal values shown.
     *
    }
    function fpc_exp_real(d: ValReal):ValReal;compilerproc;
      const
        halF : array[0..1] of double = (0.5,-0.5);
        twom1000: double = 9.33263618503218878990e-302;     { 2**-1000=0x01700000,0}
        o_threshold: double =  7.09782712893383973096e+02;  { 0x40862E42, 0xFEFA39EF }
        u_threshold: double = -7.45133219101941108420e+02;  { 0xc0874910, 0xD52D3051 }
        ln2HI  : array[0..1] of double = ( 6.93147180369123816490e-01,  { 0x3fe62e42, 0xfee00000 }
             -6.93147180369123816490e-01); { 0xbfe62e42, 0xfee00000 }
        ln2LO : array[0..1] of double = (1.90821492927058770002e-10,  { 0x3dea39ef, 0x35793c76 }
             -1.90821492927058770002e-10); { 0xbdea39ef, 0x35793c76 }
        invln2: double =  1.44269504088896338700e+00; { 0x3ff71547, 0x652b82fe }
        P1: double   =  1.66666666666666019037e-01; { 0x3FC55555, 0x5555553E }
        P2: double   = -2.77777777770155933842e-03; { 0xBF66C16C, 0x16BEBD93 }
        P3: double   =  6.61375632143793436117e-05; { 0x3F11566A, 0xAF25DE2C }
        P4: double   = -1.65339022054652515390e-06; { 0xBEBBBD41, 0xC5D26BF1 }
        P5: double   =  4.13813679705723846039e-08; { 0x3E663769, 0x72BEA4D0 }
      var
        c,hi,lo,t,y : double;
        k,xsb : longint;
        hx,hy,lx : dword;
      begin
        hi:=0.0;
        lo:=0.0;
        k:=0;
        hx:=float64high(d);
        xsb := (hx shr 31) and 1;               { sign bit of d }
        hx := hx and $7fffffff;                 { high word of |d| }

        { filter out non-finite argument }
        if hx >= $40862E42 then
          begin                  { if |d|>=709.78... }
            if hx >= $7ff00000 then
              begin
                lx:=float64low(d);
                if ((hx and $fffff) or lx)<>0 then
                  begin
                    result:=d+d; { NaN }
                    exit;
                  end
                else
                  begin
                    if xsb=0 then
                      result:=d
                    else
                      result:=0.0;    { exp(+-inf)=(inf,0) }
                    exit;
                  end;
              end;
            if d > o_threshold then begin
              result:=huge*huge;  { overflow }
              exit;
            end;
            if d < u_threshold then begin
              result:=twom1000*twom1000; { underflow }
              exit;
            end;
          end;
        { argument reduction }
        if hx > $3fd62e42 then
          begin           { if  |d| > 0.5 ln2 }
            if hx < $3FF0A2B2 then { and |d| < 1.5 ln2 }
              begin
                hi := d-ln2HI[xsb];
                lo:=ln2LO[xsb];
                k := 1-xsb-xsb;
              end
            else
              begin
               k  := trunc(invln2*d+halF[xsb]);
               t  := k;
               hi := d - t*ln2HI[0];    { t*ln2HI is exact here }
               lo := t*ln2LO[0];
              end;
            d  := hi - lo;
          end
        else if hx < $3e300000 then
          begin     { when |d|<2**-28 }
            if huge+d>one then
              begin
                result:=one+d;{ trigger inexact }
                exit;
              end;
          end
        else
          k := 0;

        { d is now in primary range }
        t:=d*d;
        c:=d - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
        if k=0 then
          begin
            result:=one-((d*c)/(c-2.0)-d);
            exit;
          end
        else
          y := one-((lo-(d*c)/(2.0-c))-hi);

        if k >= -1021 then
          begin
            hy:=float64high(y);
            float64sethigh(y,longint(hy)+(k shl 20));        { add k to y's exponent }
            result:=y;
          end
        else
          begin
            hy:=float64high(y);
            float64sethigh(y,longint(hy)+((k+1000) shl 20)); { add k to y's exponent }
            result:=y*twom1000;
          end;
      end;

{$else SUPPORT_DOUBLE}

    function fpc_exp_real(d: ValReal):ValReal;compilerproc;
    {*****************************************************************}
    { Exponential Function                                            }
    {*****************************************************************}
    {                                                                 }
    { SYNOPSIS:                                                       }
    {                                                                 }
    { double x, y, exp();                                             }
    {                                                                 }
    { y = exp( x );                                                   }
    {                                                                 }
    { DESCRIPTION:                                                    }
    {                                                                 }
    { Returns e (2.71828...) raised to the x power.                   }
    {                                                                 }
    { Range reduction is accomplished by separating the argument      }
    { into an integer k and fraction f such that                      }
    {                                                                 }
    {     x    k  f                                                   }
    {    e  = 2  e.                                                   }
    {                                                                 }
    { A Pade' form of degree 2/3 is used to approximate exp(f)- 1     }
    { in the basic range [-0.5 ln 2, 0.5 ln 2].                       }
    {*****************************************************************}
    const  P : array[0..2] of Real = (
           1.26183092834458542160E-4,
           3.02996887658430129200E-2,
           1.00000000000000000000E0);
           Q : array[0..3] of Real = (
           3.00227947279887615146E-6,
           2.52453653553222894311E-3,
           2.27266044198352679519E-1,
           2.00000000000000000005E0);

           C1 = 6.9335937500000000000E-1;
            C2 = 2.1219444005469058277E-4;
    var n : Integer;
        px, qx, xx : Real;
    begin
      if( d > MAXLOG) then
          float_raise(float_flag_overflow)
      else
      if( d < MINLOG ) then
      begin
        float_raise(float_flag_underflow);
        result:=0; { Result if underflow masked }
      end
      else
      begin

     { Express e**x = e**g 2**n }
     {   = e**g e**( n loge(2) ) }
     {   = e**( g + n loge(2) )  }

        px := d * LOG2E;
        qx := Trunc( px + 0.5 ); { Trunc() truncates toward -infinity. }
        n  := Trunc(qx);
        d  := d - qx * C1;
        d  := d + qx * C2;

      { rational approximation for exponential  }
      { of the fractional part: }
      { e**x - 1  =  2x P(x**2)/( Q(x**2) - P(x**2) )  }
        xx := d * d;
        px := d * ((P[0] * xx + P[1]) * xx + P[2]);
        d  :=  px/( (((Q[0] * xx + Q[1]) * xx + Q[2]) * xx + Q[3]) - px );
        d  := 2 * d + 1.0;
        d  := ldexp( d, n );
        result := d;
      end;
    end;
{$endif SUPPORT_DOUBLE}

{$endif}


{$ifndef FPC_SYSTEM_HAS_ROUND}
    function fpc_round_real(d : ValReal) : int64;compilerproc;
     var
      tmp: double;
      j0: longint;
      hx: longword;
      sx: longint;
    const
      H2_52: array[0..1] of double = (
        4.50359962737049600000e+15,
       -4.50359962737049600000e+15
      );
    Begin
      { This basically calculates trunc((d+2**52)-2**52) }
      hx:=float64high(d);
      j0:=((hx shr 20) and $7ff) - $3ff;
      sx:=hx shr 31;
      hx:=(hx and $fffff) or $100000;

      if j0>=52 then         { No fraction bits, already integer }
        begin
          if j0>=63 then     { Overflow, let trunc() raise an exception }
            exit(trunc(d))   { and/or return +/-MaxInt64 if it's masked }
          else
            result:=((int64(hx) shl 32) or dword(float64low(d))) shl (j0-52);
        end
      else
        begin
          { Rounding happens here. It is important that the expression is not
            optimized by selecting a larger type to store 'tmp'. }
          tmp:=H2_52[sx]+d;
          d:=tmp-H2_52[sx];
          hx:=float64high(d);
          j0:=((hx shr 20) and $7ff)-$3ff;
          hx:=(hx and $fffff) or $100000;
          if j0<=20 then
            begin
              if j0<0 then
                exit(0)
              else           { more than 32 fraction bits, low dword discarded }
                result:=hx shr (20-j0);
            end
          else
            result:=(int64(hx) shl (j0-20)) or (float64low(d) shr (52-j0));
        end;
      if sx<>0 then
        result:=-result;
    end;
{$endif FPC_SYSTEM_HAS_ROUND}


{$ifndef FPC_SYSTEM_HAS_LN}
    function fpc_ln_real(d:ValReal):ValReal;compilerproc;
    {
     This code was translated from uclib code, the original code
     had the following copyright notice:

     *
     * ====================================================
     * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
     *
     * Developed at SunPro, a Sun Microsystems, Inc. business.
     * Permission to use, copy, modify, and distribute this
     * software is freely granted, provided that this notice
     * is preserved.
     * ====================================================
     *}

    {*****************************************************************}
    { Natural Logarithm                                               }
    {*****************************************************************}
    {*
     * SYNOPSIS:
     *
     * double x, y, log();
     *
     * y = ln( x );
     *
     * DESCRIPTION:
     *
     * Returns the base e (2.718...) logarithm of x.
     *
     * Method :
     *   1. Argument Reduction: find k and f such that
     *                   x = 2^k * (1+f),
     *         where  sqrt(2)/2 < 1+f < sqrt(2) .
     *
     *   2. Approximation of log(1+f).
     *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
     *            = 2s + 2/3 s**3 + 2/5 s**5 + .....,
     *            = 2s + s*R
     *      We use a special Reme algorithm on [0,0.1716] to generate
     *   a polynomial of degree 14 to approximate R The maximum error
     *   of this polynomial approximation is bounded by 2**-58.45. In
     *   other words,
     *                      2      4      6      8      10      12      14
     *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
     *      (the values of Lg1 to Lg7 are listed in the program)
     *      and
     *          |      2          14          |     -58.45
     *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
     *          |                             |
     *   Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
     *   In order to guarantee error in log below 1ulp, we compute log
     *   by
     *           log(1+f) = f - s*(f - R)         (if f is not too large)
     *           log(1+f) = f - (hfsq - s*(hfsq+R)).    (better accuracy)
     *
     *   3. Finally,  log(x) = k*ln2 + log(1+f).
     *                       = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
     *      Here ln2 is split into two floating point number:
     *                   ln2_hi + ln2_lo,
     *      where n*ln2_hi is always exact for |n| < 2000.
     *
     * Special cases:
     *   log(x) is NaN with signal if x < 0 (including -INF) ;
     *   log(+INF) is +INF; log(0) is -INF with signal;
     *   log(NaN) is that NaN with no signal.
     *
     * Accuracy:
     *   according to an error analysis, the error is always less than
     *   1 ulp (unit in the last place).
     *}
    const
      ln2_hi: double = 6.93147180369123816490e-01;    { 3fe62e42 fee00000 }
      ln2_lo: double = 1.90821492927058770002e-10;    { 3dea39ef 35793c76 }
      two54:  double = 1.80143985094819840000e+16;    { 43500000 00000000 }
      Lg1: double = 6.666666666666735130e-01;         { 3FE55555 55555593 }
      Lg2: double = 3.999999999940941908e-01;         { 3FD99999 9997FA04 }
      Lg3: double = 2.857142874366239149e-01;         { 3FD24924 94229359 }
      Lg4: double = 2.222219843214978396e-01;         { 3FCC71C5 1D8E78AF }
      Lg5: double = 1.818357216161805012e-01;         { 3FC74664 96CB03DE }
      Lg6: double = 1.531383769920937332e-01;         { 3FC39A09 D078C69F }
      Lg7: double = 1.479819860511658591e-01;         { 3FC2F112 DF3E5244 }

    var
      hfsq,f,s,z,R,w,t1,t2,dk: double;
      k,hx,i,j: longint;
      lx: longword;
{$push}
{ if we have to check manually fpu exceptions, then force the exit statements here to
  throw one }
{$CHECKFPUEXCEPTIONS+}
{ turn off fastmath as it converts zero/zero into 1 and thus not raising an exception }
{$OPTIMIZATION NOFASTMATH}
    begin
      hx := float64high(d);
      lx := float64low(d);

      k := 0;
      if (hx < $00100000) then              { x < 2**-1022  }
      begin
        if (((hx and $7fffffff) or longint(lx))=0) then
          exit(-two54/zero);                { log(+-0)=-inf }
        if (hx<0) then
          exit(zero/zero);                 { log(-#) = NaN }
        dec(k, 54); d := d * two54;         { subnormal number, scale up x }
        hx := float64high(d);
      end;
      if (hx >= $7ff00000) then
        exit(d+d);
{$pop}
      inc(k, (hx shr 20)-1023);
      hx := hx and $000fffff;
      i := (hx + $95f64) and $100000;
      float64sethigh(d,hx or (i xor $3ff00000));   { normalize x or x/2 }
      inc(k, (i shr 20));
      f := d-1.0;
      if (($000fffff and (2+hx))<3) then    { |f| < 2**-20 }
      begin
        if (f=zero) then
        begin
          if (k=0) then
            exit(zero)
          else
          begin
            dk := k;
            exit(dk*ln2_hi+dk*ln2_lo);
          end;
        end;
        R := f*f*(0.5-0.33333333333333333*f);
        if (k=0) then
          exit(f-R)
        else
        begin
          dk := k;
          exit(dk*ln2_hi-((R-dk*ln2_lo)-f));
        end;
      end;
      s := f/(2.0+f);
      dk := k;
      z := s*s;
      i := hx-$6147a;
      w := z*z;
      j := $6b851-hx;
      t1 := w*(Lg2+w*(Lg4+w*Lg6));
      t2 := z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
      i := i or j;
      R := t2+t1;
      if (i>0) then
      begin
        hfsq := 0.5*f*f;
        if (k=0) then
          result := f-(hfsq-s*(hfsq+R))
        else
          result := dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
      end
      else
      begin
        if (k=0) then
          result := f-s*(f-R)
        else
          result := dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
      end;
    end;
{$endif}


{$ifndef FPC_SYSTEM_HAS_SIN}
    function fpc_Sin_real(d:ValReal):ValReal;compilerproc;
    {*****************************************************************}
    { Circular Sine                                                   }
    {*****************************************************************}
    {                                                                 }
    { SYNOPSIS:                                                       }
    {                                                                 }
    { double x, y, sin();                                             }
    {                                                                 }
    { y = sin( x );                                                   }
    {                                                                 }
    { DESCRIPTION:                                                    }
    {                                                                 }
    { Range reduction is into intervals of pi/4.  The reduction       }
    { error is nearly eliminated by contriving an extended            }
    { precision modular arithmetic.                                   }
    {                                                                 }
    { Two polynomial approximating functions are employed.            }
    { Between 0 and pi/4 the sine is approximated by                  }
    {      x  +  x**3 P(x**2).                                        }
    { Between pi/4 and pi/2 the cosine is represented as              }
    {      1  -  x**2 Q(x**2).                                        }
    {*****************************************************************}
    var  z, zz : Real;
         j : sizeint;

    begin
      { This seemingly useless condition ensures that sin(-0.0)=-0.0 }
      if (d=0.0) then
        exit(d);
      j := rem_pio2(d,z);

      zz := z * z;
      if j and 1<>0 then { j and 3 = 1 or j and 3 = 3 }
        result := 1.0 - zz * 0.5 + zz * zz * (((((coscof[0] * zz + coscof[1]) * zz + coscof[2]) * zz + coscof[3]) * zz + coscof[4]) * zz + coscof[5])
      else
        result := z + zz * z * (((((sincof[0] * zz + sincof[1]) * zz + sincof[2]) * zz + sincof[3]) * zz + sincof[4]) * zz + sincof[5]);

      if j and 2<>0 then { j and 3 = 2 or j and 3 = 3 }
        result := -result;
    end;
{$endif}



{$ifndef FPC_SYSTEM_HAS_COS}
    function fpc_Cos_real(d:ValReal):ValReal;compilerproc;
    {*****************************************************************}
    { Circular cosine                                                 }
    {*****************************************************************}
    {                                                                 }
    { Circular cosine                                                 }
    {                                                                 }
    { SYNOPSIS:                                                       }
    {                                                                 }
    { double x, y, cos();                                             }
    {                                                                 }
    { y = cos( x );                                                   }
    {                                                                 }
    { DESCRIPTION:                                                    }
    {                                                                 }
    { Range reduction is into intervals of pi/4.  The reduction       }
    { error is nearly eliminated by contriving an extended            }
    { precision modular arithmetic.                                   }
    {                                                                 }
    { Two polynomial approximating functions are employed.            }
    { Between 0 and pi/4 the cosine is approximated by                }
    {      1  -  x**2 Q(x**2).                                        }
    { Between pi/4 and pi/2 the sine is represented as                }
    {      x  +  x**3 P(x**2).                                        }
    {*****************************************************************}
    var  y, z, zz : Real;
         j : sizeint;
    begin
      j := rem_pio2(d,z);

      zz := z * z;
      if j and 1<>0 then { j and 3 = 1 or j and 3 = 3 }
        result := z + zz * z * (((((sincof[0] * zz + sincof[1]) * zz + sincof[2]) * zz + sincof[3]) * zz + sincof[4]) * zz + sincof[5])
      else
        result := 1.0 - zz * 0.5 + zz * zz * (((((coscof[0] * zz + coscof[1]) * zz + coscof[2]) * zz + coscof[3]) * zz + coscof[4]) * zz + coscof[5]);

      if (j+1) and 2<>0 then { j and 3 = 1 or j and 3 = 2 }
        result := -result;
    end;
{$endif}



{$ifndef FPC_SYSTEM_HAS_ARCTAN}
    function fpc_ArcTan_real(d:ValReal):ValReal;compilerproc;
    {
     This code was translated from uclibc code, the original code
     had the following copyright notice:

     *
     * ====================================================
     * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
     *
     * Developed at SunPro, a Sun Microsystems, Inc. business.
     * Permission to use, copy, modify, and distribute this
     * software is freely granted, provided that this notice
     * is preserved.
     * ====================================================
     *}

    {********************************************************************}
    { Inverse circular tangent (arctangent)                              }
    {********************************************************************}
    {                                                                    }
    { SYNOPSIS:                                                          }
    {                                                                    }
    { double x, y, atan();                                               }
    {                                                                    }
    { y = atan( x );                                                     }
    {                                                                    }
    { DESCRIPTION:                                                       }
    {                                                                    }
    { Returns radian angle between -pi/2 and +pi/2 whose tangent         }
    { is x.                                                              }
    {                                                                    }
    { Method                                                             }
    {   1. Reduce x to positive by atan(x) = -atan(-x).                  }
    {   2. According to the integer k=4t+0.25 chopped, t=x, the argument }
    {      is further reduced to one of the following intervals and the  }
    {      arctangent of t is evaluated by the corresponding formula:    }
    {                                                                    }
    {   [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)   }
    {   [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )      }
    {   [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )          }
    {   [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )     }
    {   [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )                 }
    {********************************************************************}
    const
      atanhi: array [0..3] of double = (
        4.63647609000806093515e-01, { atan(0.5)hi 0x3FDDAC67, 0x0561BB4F }
        7.85398163397448278999e-01, { atan(1.0)hi 0x3FE921FB, 0x54442D18 }
        9.82793723247329054082e-01, { atan(1.5)hi 0x3FEF730B, 0xD281F69B }
        1.57079632679489655800e+00  { atan(inf)hi 0x3FF921FB, 0x54442D18 }
      );

      atanlo: array [0..3] of double = (
        2.26987774529616870924e-17, { atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 }
        3.06161699786838301793e-17, { atan(1.0)lo 0x3C81A626, 0x33145C07 }
        1.39033110312309984516e-17, { atan(1.5)lo 0x3C700788, 0x7AF0CBBD }
        6.12323399573676603587e-17  { atan(inf)lo 0x3C91A626, 0x33145C07 }
      );

      aT: array[0..10] of double = (
         3.33333333333329318027e-01,  { 0x3FD55555, 0x5555550D }
        -1.99999999998764832476e-01,  { 0xBFC99999, 0x9998EBC4 }
         1.42857142725034663711e-01,  { 0x3FC24924, 0x920083FF }
        -1.11111104054623557880e-01,  { 0xBFBC71C6, 0xFE231671 }
         9.09088713343650656196e-02,  { 0x3FB745CD, 0xC54C206E }
        -7.69187620504482999495e-02,  { 0xBFB3B0F2, 0xAF749A6D }
         6.66107313738753120669e-02,  { 0x3FB10D66, 0xA0D03D51 }
        -5.83357013379057348645e-02,  { 0xBFADDE2D, 0x52DEFD9A }
         4.97687799461593236017e-02,  { 0x3FA97B4B, 0x24760DEB }
        -3.65315727442169155270e-02,  { 0xBFA2B444, 0x2C6A6C2F }
         1.62858201153657823623e-02   { 0x3F90AD3A, 0xE322DA11 }
      );

    var
      w,s1,s2,z: double;
      ix,hx,id: longint;
      low: longword;
    begin
      hx:=float64high(d);
      ix := hx and $7fffffff;
      if (ix>=$44100000) then   { if |x| >= 2^66 }
      begin
        low:=float64low(d);
        if (ix > $7ff00000) or ((ix = $7ff00000) and (low<>0)) then
          exit(d+d);          { NaN }
        if (hx>0) then
          exit(atanhi[3]+atanlo[3])
        else
          exit(-atanhi[3]-atanlo[3]);
      end;
      if (ix < $3fdc0000) then  { |x| < 0.4375 }
      begin
        if (ix < $3e200000) then  { |x| < 2^-29 }
        begin
          if (huge+d>one) then exit(d);    { raise inexact }
        end;
        id := -1;
      end
      else
      begin
        d := abs(d);
        if (ix < $3ff30000) then    { |x| < 1.1875 }
        begin
          if (ix < $3fe60000) then  { 7/16 <=|x|<11/16 }
          begin
            id := 0; d := (2.0*d-one)/(2.0+d);
          end
          else                      { 11/16<=|x|< 19/16 }
          begin
            id := 1; d := (d-one)/(d+one);
          end
        end
        else
        begin
          if (ix < $40038000) then   { |x| < 2.4375 }
          begin
            id := 2; d := (d-1.5)/(one+1.5*d);
          end
          else                       { 2.4375 <= |x| < 2^66 }
          begin
            id := 3; d := -1.0/d;
          end;
        end;
      end;
    { end of argument reduction }
      z := d*d;
      w := z*z;
      { break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly }
      s1 := z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
      s2 := w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
      if (id<0) then
        result := d - d*(s1+s2)
      else
      begin
        z := atanhi[id] - ((d*(s1+s2) - atanlo[id]) - d);
        if hx<0 then
          result := -z
        else
          result := z;
      end;
    end;
{$endif}


{$ifndef FPC_SYSTEM_HAS_FRAC}
{$push}
{$ifndef VER3_2}
{ if we have to check manually fpu exceptions, then force the result assignment statement here to
  throw one }
{$CHECKFPUEXCEPTIONS+}
{ turn off fastmath as it converts zero/zero into 0 and thus not raising an exception }
{$OPTIMIZATION NOFASTMATH}
{$endif VER3_2}
function fpc_frac_real(d : ValReal) : ValReal;compilerproc;
  begin
    { Nan or +/-Inf }
    if (float64high(d) and $7ff00000)=$7ff00000 then
      { return NaN }
{$ifdef VER3_2}
      { fix bootstrapping with 3.2 on arm-linux }
      result := (d-d)/0.0
{$else VER3_2}
      result := zero/zero
{$endif VER3_2}
    else
      result := d - Int(d);
  end;
{$pop}
{$endif}


{$ifdef FPC_INCLUDE_SOFTWARE_INT64_TO_DOUBLE}

{$ifndef FPC_SYSTEM_HAS_QWORD_TO_DOUBLE}
function fpc_qword_to_double(q : qword): double; compilerproc;
  begin
     result:=dword(q and $ffffffff)+dword(q shr 32)*double(4294967296.0);
  end;
{$endif FPC_SYSTEM_HAS_INT64_TO_DOUBLE}


{$ifndef FPC_SYSTEM_HAS_INT64_TO_DOUBLE}
function fpc_int64_to_double(i : int64): double; compilerproc;
  begin
    result:=dword(i and $ffffffff)+longint(i shr 32)*double(4294967296.0);
  end;
{$endif FPC_SYSTEM_HAS_INT64_TO_DOUBLE}

{$endif FPC_INCLUDE_SOFTWARE_INT64_TO_DOUBLE}


{$ifdef SUPPORT_DOUBLE}
{****************************************************************************
                    Helper routines to support old TP styled reals
 ****************************************************************************}

{$ifndef FPC_SYSTEM_HAS_REAL2DOUBLE}
    function real2double(r : real48) : double;

      var
         res : array[0..7] of byte;
         exponent : word;

      begin
         { check for zero }
         if r[0]=0 then
           begin
             real2double:=0.0;
             exit;
           end;

         { copy mantissa }
         res[0]:=0;
         res[1]:=r[1] shl 5;
         res[2]:=(r[1] shr 3) or (r[2] shl 5);
         res[3]:=(r[2] shr 3) or (r[3] shl 5);
         res[4]:=(r[3] shr 3) or (r[4] shl 5);
         res[5]:=(r[4] shr 3) or (r[5] and $7f) shl 5;
         res[6]:=(r[5] and $7f) shr 3;

         { copy exponent }
         { correct exponent: }
         exponent:=(word(r[0])+(1023-129));
         res[6]:=res[6] or ((exponent and $f) shl 4);
         res[7]:=exponent shr 4;

         { set sign }
         res[7]:=res[7] or (r[5] and $80);
         real2double:=double(res);
      end;
{$endif FPC_SYSTEM_HAS_REAL2DOUBLE}
{$endif SUPPORT_DOUBLE}

{$ifdef SUPPORT_EXTENDED}
{ fast 10^n routine }
function FPower10(val: Extended; Power: Longint): Extended;
  const
    pow32 : array[0..31] of extended =
      (
        1e0,1e1,1e2,1e3,1e4,1e5,1e6,1e7,1e8,1e9,1e10,
        1e11,1e12,1e13,1e14,1e15,1e16,1e17,1e18,1e19,1e20,
        1e21,1e22,1e23,1e24,1e25,1e26,1e27,1e28,1e29,1e30,
        1e31
      );
    pow512 : array[0..15] of extended =
      (
        1,1e32,1e64,1e96,1e128,1e160,1e192,1e224,
        1e256,1e288,1e320,1e352,1e384,1e416,1e448,
        1e480
      );
    pow4096 : array[0..9] of extended =
      (1,1e512,1e1024,1e1536,
       1e2048,1e2560,1e3072,1e3584,
       1e4096,1e4608
      );

    negpow32 : array[0..31] of extended =
      (
        1e-0,1e-1,1e-2,1e-3,1e-4,1e-5,1e-6,1e-7,1e-8,1e-9,1e-10,
        1e-11,1e-12,1e-13,1e-14,1e-15,1e-16,1e-17,1e-18,1e-19,1e-20,
        1e-21,1e-22,1e-23,1e-24,1e-25,1e-26,1e-27,1e-28,1e-29,1e-30,
        1e-31
      );
    negpow512 : array[0..15] of extended =
      (
        0,1e-32,1e-64,1e-96,1e-128,1e-160,1e-192,1e-224,
        1e-256,1e-288,1e-320,1e-352,1e-384,1e-416,1e-448,
        1e-480
      );
    negpow4096 : array[0..9] of extended =
      (
        0,1e-512,1e-1024,1e-1536,
        1e-2048,1e-2560,1e-3072,1e-3584,
        1e-4096,1e-4608
      );

  begin
    if Power<0 then
      begin
        Power:=-Power;
        result:=val*negpow32[Power and $1f];
        power:=power shr 5;
        if power<>0 then
          begin
            result:=result*negpow512[Power and $f];
            power:=power shr 4;
            if power<>0 then
              begin
                if power<=9 then
                  result:=result*negpow4096[Power]
                else
                  result:=1.0/0.0;
              end;
          end;
      end
    else
      begin
        result:=val*pow32[Power and $1f];
        power:=power shr 5;
        if power<>0 then
          begin
            result:=result*pow512[Power and $f];
            power:=power shr 4;
            if power<>0 then
              begin
                if power<=9 then
                  result:=result*pow4096[Power]
                else
                  result:=1.0/0.0;
              end;
          end;
      end;
  end;
{$endif SUPPORT_EXTENDED}

{$if defined(SUPPORT_EXTENDED) or defined(FPC_SOFT_FPUX80)}
function TExtended80Rec.Mantissa(IncludeHiddenBit: Boolean = False) : QWord;
  begin
    if IncludeHiddenbit then
      Result:=Frac
    else
      Result:=Frac and $7fffffffffffffff;
  end;

function TExtended80Rec.Fraction : Extended;
  begin
{$ifdef SUPPORT_EXTENDED}
    Result:=system.frac(Value);
{$else}
    Result:=Frac / Double (1 shl 63) / 2.0;
{$endif}
  end;


function TExtended80Rec.Exponent : Longint;
  var
    E: QWord;
  begin
    Result := 0;
    E := GetExp;
    if (0<E) and (E<2*Bias+1) then
      Result:=Exp-Bias
    else if (Exp=0) and (Frac<>0) then
      Result:=-(Bias-1);
  end;


function TExtended80Rec.GetExp : QWord;
  begin
    Result:=_Exp and $7fff;
  end;


procedure TExtended80Rec.SetExp(e : QWord);
  begin
    _Exp:=(_Exp and $8000) or (e and $7fff);
  end;


function TExtended80Rec.GetSign : Boolean;
  begin
    Result:=(_Exp and $8000)<>0;
  end;


procedure TExtended80Rec.SetSign(s : Boolean);
  begin
    _Exp:=(_Exp and $7ffff) or (ord(s) shl 15);
  end;

{
  Based on information taken from http://en.wikipedia.org/wiki/Extended_precision#x86_Extended_Precision_Format
}
function TExtended80Rec.SpecialType : TFloatSpecial;
  const
    Denormal : array[boolean] of TFloatSpecial = (fsDenormal,fsNDenormal);
  begin
    case Exp of
      0:
        begin
          if Mantissa=0 then
            begin
              if Sign then
                Result:=fsNZero
              else
                Result:=fsZero
            end
          else
            Result:=Denormal[Sign];
        end;
      $7fff:
        case (Frac shr 62) and 3 of
          0,1:
            Result:=fsInvalidOp;
          2:
            begin
              if (Frac and $3fffffffffffffff)=0 then
                begin
                  if Sign then
                    Result:=fsNInf
                  else
                    Result:=fsInf;
                end
              else
                Result:=fsNaN;
            end;
          3:
            Result:=fsNaN;
        end
      else
        begin
          if (Frac and $8000000000000000)=0 then
            Result:=fsInvalidOp
          else
            begin
              if Sign then
                Result:=fsNegative
              else
                Result:=fsPositive;
            end;
        end;
      end;
  end;

procedure TExtended80Rec.BuildUp(const _Sign: Boolean; const _Mantissa: QWord; const _Exponent: Longint);
begin
{$ifdef SUPPORT_EXTENDED}
  Value := 0.0;
{$else SUPPORT_EXTENDED}
  FillChar(Value, SizeOf(Value),0);
{$endif SUPPORT_EXTENDED}
  if (_Mantissa=0) and (_Exponent=0) then
    SetExp(0)
  else
    SetExp(_Exponent + Bias);
  SetSign(_Sign);
  Frac := _Mantissa;
end;
{$endif SUPPORT_EXTENDED or FPC_SOFT_FPUX80}


{$ifdef SUPPORT_DOUBLE}
function TDoubleRec.Mantissa(IncludeHiddenBit: Boolean = False) : QWord;
  begin
    Result:=(Data and $fffffffffffff);
    if (Result=0) and (GetExp=0) then Exit;
    if IncludeHiddenBit then Result := Result or $10000000000000; //add the hidden bit
  end;


function TDoubleRec.Fraction : ValReal;
  begin
    Result:=system.frac(Value);
  end;


function TDoubleRec.Exponent : Longint;
  var
    E: QWord;
  begin
    Result := 0;
    E := GetExp;
    if (0<E) and (E<2*Bias+1) then
      Result:=Exp-Bias
    else if (Exp=0) and (Frac<>0) then
      Result:=-(Bias-1);
  end;


function TDoubleRec.GetExp : QWord;
  begin
    Result:=(Data and $7ff0000000000000) shr 52;
  end;


procedure TDoubleRec.SetExp(e : QWord);
  begin
    Data:=(Data and $800fffffffffffff) or ((e and $7ff) shl 52);
  end;


function TDoubleRec.GetSign : Boolean;
  begin
    Result:=(Data and $8000000000000000)<>0;
  end;


procedure TDoubleRec.SetSign(s : Boolean);
  begin
    Data:=(Data and $7fffffffffffffff) or (QWord(ord(s)) shl 63);
  end;


function TDoubleRec.GetFrac : QWord;
  begin
    Result := Data and $fffffffffffff;
  end;


procedure TDoubleRec.SetFrac(e : QWord);
  begin
    Data:=(Data and $fff0000000000000) or (e and $fffffffffffff);
  end;

{
  Based on information taken from http://en.wikipedia.org/wiki/Double_precision#x86_Extended_Precision_Format
}
function TDoubleRec.SpecialType : TFloatSpecial;
  const
    Denormal : array[boolean] of TFloatSpecial = (fsDenormal,fsNDenormal);
  begin
    case Exp of
      0:
        begin
          if Mantissa=0 then
            begin
              if Sign then
                Result:=fsNZero
              else
                Result:=fsZero
            end
          else
            Result:=Denormal[Sign];
        end;
      $7ff:
        if Mantissa=0 then
          begin
            if Sign then
              Result:=fsNInf
            else
              Result:=fsInf;
          end
        else
          Result:=fsNaN;
      else
        begin
          if Sign then
            Result:=fsNegative
          else
            Result:=fsPositive;
        end;
      end;
  end;

procedure TDoubleRec.BuildUp(const _Sign: Boolean; const _Mantissa: QWord; const _Exponent: Longint);
  begin
    Value := 0.0;
    SetSign(_Sign);
    if (_Mantissa=0) and (_Exponent=0) then
      Exit //SetExp(0)
    else
      SetExp(_Exponent + Bias);
    SetFrac(_Mantissa and $fffffffffffff); //clear top bit
  end;
{$endif SUPPORT_DOUBLE}


{$ifdef SUPPORT_SINGLE}
function TSingleRec.Mantissa(IncludeHiddenBit: Boolean = False) : QWord;
  begin
    Result:=(Data and $7fffff);
    if (Result=0) and (GetExp=0) then Exit;
    if IncludeHiddenBit then Result:=Result or $800000; //add the hidden bit
  end;


function TSingleRec.Fraction : ValReal;
  begin
    Result:=system.frac(Value);
  end;


function TSingleRec.Exponent : Longint;
  var
    E: QWord;
  begin
    Result := 0;
    E := GetExp;
    if (0<E) and (E<2*Bias+1) then
      Result:=Exp-Bias
    else if (Exp=0) and (Frac<>0) then
      Result:=-(Bias-1);
  end;


function TSingleRec.GetExp : QWord;
  begin
    Result:=(Data and $7f800000) shr 23;
  end;


procedure TSingleRec.SetExp(e : QWord);
  begin
    Data:=(Data and $807fffff) or ((e and $ff) shl 23);
  end;


function TSingleRec.GetSign : Boolean;
  begin
    Result:=(Data and $80000000)<>0;
  end;


procedure TSingleRec.SetSign(s : Boolean);
  begin
    Data:=(Data and $7fffffff) or (DWord(ord(s)) shl 31);
  end;


function TSingleRec.GetFrac : QWord;
  begin
    Result:=Data and $7fffff;
  end;


procedure TSingleRec.SetFrac(e : QWord);
  begin
    Data:=(Data and $ff800000) or (e and $7fffff);
  end;

{
  Based on information taken from http://en.wikipedia.org/wiki/Single_precision#x86_Extended_Precision_Format
}
function TSingleRec.SpecialType : TFloatSpecial;
  const
    Denormal : array[boolean] of TFloatSpecial = (fsDenormal,fsNDenormal);
  begin
    case Exp of
      0:
        begin
          if Mantissa=0 then
            begin
              if Sign then
                Result:=fsNZero
              else
                Result:=fsZero
            end
          else
            Result:=Denormal[Sign];
        end;
      $ff:
        if Mantissa=0 then
          begin
            if Sign then
              Result:=fsNInf
            else
              Result:=fsInf;
          end
        else
          Result:=fsNaN;
      else
        begin
          if Sign then
            Result:=fsNegative
          else
            Result:=fsPositive;
        end;
      end;
  end;

procedure TSingleRec.BuildUp(const _Sign: Boolean; const _Mantissa: QWord; const _Exponent: Longint);
  begin
    Value := 0.0;
    SetSign(_Sign);
    if (_Mantissa=0) and (_Exponent=0) then
      Exit //SetExp(0)
    else
      SetExp(_Exponent + Bias);
    SetFrac(_Mantissa and $7fffff); //clear top bit
  end;
{$endif SUPPORT_SINGLE}