freepascal.compiler/rtl/inc/genmath.inc

{
    $Id$
    This file is part of the Free Pascal run time library.
    Copyright (c) 1999-2001 by Several contributors

    Generic mathemtical 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                               }
{-------------------------------------------------------------------------}

{$goto on}

type
    TabCoef = array[0..6] of Real;


const
      PIO2   =  1.57079632679489661923;       {  pi/2        }
      PIO4   =  7.85398163397448309616E-1;    {  pi/4        }
      SQRT2  =  1.41421356237309504880;       {  sqrt(2)     }
      SQRTH  =  7.07106781186547524401E-1;    {  sqrt(2)/2   }
      LOG2E  =  1.4426950408889634073599;     {  1/log(2)    }
      SQ2OPI =  7.9788456080286535587989E-1;  {  sqrt( 2/pi )}
      LOGE2  =  6.93147180559945309417E-1;    {  log(2)      }
      LOGSQ2 =  3.46573590279972654709E-1;    {  log(2)/2    }
      THPIO4 =  2.35619449019234492885;       {  3*pi/4      }
      TWOOPI =  6.36619772367581343075535E-1; {  2/pi        }
      lossth =  1.073741824e9;
      MAXLOG =  8.8029691931113054295988E1;    { log(2**127)  }
      MINLOG = -8.872283911167299960540E1;     { log(2**-128) }

      DP1 =   7.85398125648498535156E-1;
      DP2 =   3.77489470793079817668E-8;
      DP3 =   2.69515142907905952645E-15;

const sincof : TabCoef = (
                1.58962301576546568060E-10,
               -2.50507477628578072866E-8,
                2.75573136213857245213E-6,
               -1.98412698295895385996E-4,
                8.33333333332211858878E-3,
               -1.66666666666666307295E-1, 0);
      coscof : TabCoef = (
               -1.13585365213876817300E-11,
                2.08757008419747316778E-9,
               -2.75573141792967388112E-7,
                2.48015872888517045348E-5,
               -1.38888888888730564116E-3,
                4.16666666666665929218E-2, 0);



{ also necessary for Int() on systems with 64bit floats (JM) }
type
{$ifdef ENDIAN_LITTLE}
  float64 = packed record
    low: longint;
    high: longint;
  end;
{$else}
  float64 = packed record
    high: longint;
    low: longint;
  end;
{$endif}

{$ifndef FPC_SYSTEM_HAS_TRUNC}
type
  float32 = longint;
  flag = byte;

  Function extractFloat64Frac0(a: float64): longint;
  Begin
    extractFloat64Frac0 := a.high and $000FFFFF;
  End;

  Function extractFloat64Frac1(a: float64): longint;
  Begin
    extractFloat64Frac1 := a.low;
  End;

 Function extractFloat64Exp(a: float64): smallint;
 Begin
    extractFloat64Exp:= ( a.high shr 20 ) AND $7FF;
 End;

 Function extractFloat64Sign(a: float64) : flag;
 Begin
    extractFloat64Sign := a.high shr 31;
 End;

Procedure
 shortShift64Left(
     a0:longint; a1:longint; count:smallint; VAR z0Ptr:longint; VAR z1Ptr:longint );
Begin
    z1Ptr := a1 shl count;
    if count = 0 then
      z0Ptr := a0
    else
      z0Ptr := ( a0 shl count ) OR ( a1 shr ( ( - count ) AND 31 ) );
End;

   function float64_to_int32_round_to_zero(a: float64 ): longint;
    Var
      aSign: flag;
      aExp, shiftCount: smallint;
      aSig0, aSig1, absZ, aSigExtra: longint;
      z: smallint;
      label invalid;
   Begin
     aSig1 := extractFloat64Frac1( a );
     aSig0 := extractFloat64Frac0( a );
     aExp := extractFloat64Exp( a );
     aSign := extractFloat64Sign( a );
     shiftCount := aExp - $413;
     if ( 0 <= shiftCount ) then
     Begin
         if ( $41E < aExp ) then
         Begin
             if ( ( aExp = $7FF ) AND  (( aSig0 OR  aSig1 )<>0) ) then
                aSign := 0;
             goto invalid;
         End;
         shortShift64Left(
             aSig0 OR  $00100000, aSig1, shiftCount, absZ, aSigExtra );
     End
     else
     Begin
         if ( aExp < $3FF ) then
         Begin
             float64_to_int32_round_to_zero := 0;
             exit;
         End;
         aSig0 := aSig0 or $00100000;
         aSigExtra := ( aSig0 shl ( shiftCount and 31 ) ) OR  aSig1;
         absZ := aSig0 shr ( - shiftCount );
     End;
     if aSign <> 0 then
       z := - absZ
     else
       z := absZ;
     if ( (( aSign xor flag( z < 0 )) <> 0) AND  (z<>0) ) then
     Begin
  invalid:
        HandleError(207);
        exit;
    End;
    float64_to_int32_round_to_zero := z;
   End;


 Function ExtractFloat32Frac(a : Float32) : longint;
 Begin
    ExtractFloat32Frac := A AND $007FFFFF;
 End;


 Function extractFloat32Exp( a: float32 ): smallint;
  Begin
    extractFloat32Exp := (a shr 23) AND $FF;
  End;

 Function extractFloat32Sign( a: float32 ): Flag;
  Begin
    extractFloat32Sign := a shr 31;
  End;




Function float32_to_int32_round_to_zero( a: Float32 ): longint;
 Var
    aSign : flag;
    aExp, shiftCount : smallint;
    aSig : longint;
    z : longint;
 Begin
    aSig := extractFloat32Frac( a );
    aExp := extractFloat32Exp( a );
    aSign := extractFloat32Sign( a );
    shiftCount := aExp - $9E;
    if ( 0 <= shiftCount ) then
      Begin
        if ( a <> $CF000000 ) then
          Begin
            if ( (aSign=0) or ( ( aExp = $FF ) and (aSig<>0) ) ) then
              Begin
                HandleError(207);
                exit;
              end;
          End;
        HandleError(207);
        exit;
      End
    else
      if ( aExp <= $7E ) then
      Begin
        float32_to_int32_round_to_zero := 0;
        exit;
      End;
    aSig := ( aSig or $00800000 ) shl 8;
    z := aSig shr ( - shiftCount );
    if ( aSign<>0 ) then z := - z;
    float32_to_int32_round_to_zero := z;
 End;


  {$warning FIX ME !! }
  function trunc(d : real) : int64;[internconst:in_const_trunc];
    var
     l: longint;
     f32 : float32;
     f64 : float64;
    Begin
     { in emulation mode the real is equal to a single }
     { otherwise in fpu mode, it is equal to a double  }
     { extended is not supported yet. }
     if sizeof(D) > 8 then
        HandleError(255);
     if sizeof(D)=8 then
       begin
         move(d,f64,sizeof(f64));
         trunc:=float64_to_int32_round_to_zero(f64);
       end
     else
       begin
         move(d,f32,sizeof(f32));
         trunc:=float32_to_int32_round_to_zero(f32);
       end;
    end;
{$endif}




{$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 int(d: double): double;[internconst:in_const_int];
      var
        i0, j0: longint;
        i1: cardinal;
        sx: longint;
      begin
        i0 := float64(d).high;
        i1 := cardinal(float64(d).low);
        sx := i0 and $80000000;
        j0 := ((i0 shr 20) and $7ff) - $3ff;
        if (j0 < 20) then
          begin
            if (j0 < 0) then
              begin
                { the magnitude of the number is < 1 so the result is +-0. }
                float64(d).high := sx;
                float64(d).low := 0;
              end
            else
              begin
                float64(d).high := sx or (i0 and not($fffff shr j0));
                float64(d).low := 0;
              end
          end
        else if (j0 > 51) then
          begin
            if (j0 = $400) then
              { d is inf or NaN }
              exit(d + d); { don't know why they do this (JM) }
          end
        else
          begin
            float64(d).high := i0;
            float64(d).low := longint(i1 and not(cardinal($ffffffff) shr (j0 - 20)));
          end;
        result := d;
      end;

{$else SUPPORT_DOUBLE}


    function int(d : real) : real;[internconst:in_const_int];
      begin
        { this will be correct since real = single in the case of }
        { the motorola version of the compiler...                 }
        int:=real(trunc(d));
      end;
{$endif SUPPORT_DOUBLE}

{$endif}


{$ifndef FPC_SYSTEM_HAS_ABS}

{$ifdef SUPPORT_DOUBLE}

    function abs(d : Double) : Double;[public,alias:'FPC_ABS_REAL'];
    begin
       if (d<0.0) then
         abs := -d
      else
         abs := d ;
    end;

{$else}

    function abs(d : Real) : Real;[public,alias:'FPC_ABS_REAL'];
    begin
       if (d<0.0) then
         abs := -d
      else
         abs := d ;
    end;

{$endif}
    
{$ifdef hascompilerproc}
    function fpc_abs_real(d:Real):Real;compilerproc; external name 'FPC_ABS_REAL';
{$endif hascompilerproc}

{$endif not FPC_SYSTEM_HAS_ABS}


    function frexp(x:Real; var e:Integer ):Real;
    {*  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
      e :=0;
      if (abs(x)<0.5) then
       While (abs(x)<0.5) do
       begin
         x := x*2;
         Dec(e);
       end
      else
       While (abs(x)>1) do
       begin
         x := x/2;
         Inc(e);
       end;
      frexp := x;
    end;


    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;


    function polevl(var x:Real; var Coef:TabCoef; N:Integer):Real;
    {*****************************************************************}
    { Evaluate polynomial                                             }
    {*****************************************************************}
    {                                                                 }
    { SYNOPSIS:                                                       }
    {                                                                 }
    {  int N;                                                         }
    {  double x, y, coef[N+1], polevl[];                              }
    {                                                                 }
    {  y = polevl( x, coef, N );                                      }
    {                                                                 }
    {  DESCRIPTION:                                                   }
    {                                                                 }
    {     Evaluates polynomial of degree N:                           }
    {                                                                 }
    {                       2          N                              }
    {   y  =  C  + C x + C x  +...+ C x                               }
    {          0    1     2          N                                }
    {                                                                 }
    {   Coefficients are stored in reverse order:                     }
    {                                                                 }
    {   coef[0] = C  , ..., coef[N] = C  .                            }
    {              N                   0                              }
    {                                                                 }
    {   The function p1evl() assumes that coef[N] = 1.0 and is        }
    {   omitted from the array.  Its calling arguments are            }
    {   otherwise the same as polevl().                               }
    {                                                                 }
    {  SPEED:                                                         }
    {                                                                 }
    {   In the interest of speed, there are no checks for out         }
    {   of bounds arithmetic.  This routine is used by most of        }
    {   the functions in the library.  Depending on available         }
    {   equipment features, the user may wish to rewrite the          }
    {   program in microcode or assembly language.                    }
    {*****************************************************************}
    var ans : Real;
        i   : Integer;

    begin
      ans := Coef[0];
      for i:=1 to N do
        ans := ans * x + Coef[i];
      polevl:=ans;
    end;


    function p1evl(var x:Real; var Coef:TabCoef; N:Integer):Real;
    {                                                           }
    { Evaluate polynomial when coefficient of x  is 1.0.        }
    { Otherwise same as polevl.                                 }
    {                                                           }
    var
        ans : Real;
        i   : Integer;
    begin
      ans := x + Coef[0];
      for i:=1 to N-1 do
        ans := ans * x + Coef[i];
      p1evl := ans;
    end;





{$ifndef FPC_SYSTEM_HAS_SQR}
    function sqr(d : Real) : Real;[internconst:in_const_sqr];
    begin
      sqr := d*d;
    end;
{$endif}

{$ifndef FPC_SYSTEM_HAS_PI}
    function pi : Real;[internconst:in_const_pi];
    begin
      pi := 3.1415926535897932385;
    end;
{$endif}


{$ifndef FPC_SYSTEM_HAS_SQRT}
    function sqrt(d:Real):Real;[internconst:in_const_sqrt]; [public, alias: 'FPC_SQRT_REAL'];
    {*****************************************************************}
    { 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   : Integer;
        w,z : Real;
    begin
       if( d <= 0.0 ) then
       begin
           if( d < 0.0 ) then
               HandleError(207);
           sqrt := 0.0;
       end
     else
       begin
          w := d;
          { separate exponent and significand }
           z := frexp( d, 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);
          sqrt := d;
       end;
    end;

{$ifdef hascompilerproc}
    function fpc_sqrt_real(d:Real):Real;compilerproc; external name 'FPC_SQRT_REAL';
{$endif hascompilerproc}

{$endif}




{$ifndef FPC_SYSTEM_HAS_EXP}
    function Exp(d:Real):Real;[internconst:in_const_exp];
    {*****************************************************************}
    { 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 : TabCoef = (
           1.26183092834458542160E-4,
           3.02996887658430129200E-2,
           1.00000000000000000000E0, 0, 0, 0, 0);
           Q : TabCoef = (
           3.00227947279887615146E-6,
           2.52453653553222894311E-3,
           2.27266044198352679519E-1,
           2.00000000000000000005E0, 0 ,0 ,0);

           C1 = 6.9335937500000000000E-1;
            C2 = 2.1219444005469058277E-4;
    var n : Integer;
        px, qx, xx : Real;
    begin
      if( d > MAXLOG) then
          HandleError(205)
      else
      if( d < MINLOG ) then
      begin
        HandleError(205);
      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 * polevl( xx, P, 2 );
        d  :=  px/( polevl( xx, Q, 3 ) - px );
        d  := ldexp( d, 1 );
        d  :=  d + 1.0;
        d  := ldexp( d, n );
        Exp := d;
      end;
    end;
{$endif}


{$ifndef FPC_SYSTEM_HAS_ROUND}
{$ifdef hascompilerproc}
    function round(d : Real) : int64;[internconst:in_const_round, external name 'FPC_ROUND'];

    function fpc_round(d : Real) : int64;[public, alias:'FPC_ROUND'];{$ifdef hascompilerproc}compilerproc;{$endif hascompilerproc}
{$else}
    function round(d : Real) : int64;[internconst:in_const_round];
{$endif hascompilerproc}
     var
      fr: Real;
      tr: Real;
    Begin
       fr := abs(Frac(d));
       tr := Trunc(d);
       if fr > 0.5 then
         if d >= 0 then
           result:=Trunc(d)+1
         else
           result:=Trunc(d)-1
       else
       if fr < 0.5 then
          result:=Trunc(d)
       else { fr = 0.5 }
          { check sign to decide ... }
          { as in Turbo Pascal...    }
          if d >= 0.0 then
            result:=Trunc(d)+1
          else
            result:=Trunc(d);
    end;
{$endif}


{$ifdef FPC_CURRENCY_IS_INT64}

    function trunc(c : currency) : int64;
      type
        tmyrec = record
          i: int64;
        end;
      begin
        result := int64(tmyrec(c)) div 10000
      end;


    function trunc(c : comp) : int64;
      begin
        result := c
      end;


    function round(c : currency) : int64;
      type
        tmyrec = record
          i: int64;
        end;
      var
        rem, absrem: longint;
      begin
        { (int64(tmyrec(c))(+/-)5000) div 10000 can overflow }
        result := int64(tmyrec(c)) div 10000;
        rem := int64(tmyrec(c)) - result * 10000;
        absrem := abs(rem);
        if (absrem > 5000) or
           ((absrem = 5000) and
            (rem > 0)) then
          if (rem > 0) then
            inc(result)
          else
            dec(result);
      end;


    function round(c : comp) : int64;
      begin
        result := c
      end;

{$endif FPC_CURRENCY_IS_INT64}



{$ifndef FPC_SYSTEM_HAS_LN}
    function Ln(d:Real):Real;[internconst:in_const_ln];
    {*****************************************************************}
    { Natural Logarithm                                               }
    {*****************************************************************}
    {                                                                 }
    { SYNOPSIS:                                                       }
    {                                                                 }
    { double x, y, log();                                             }
    {                                                                 }
    { y = ln( x );                                                    }
    {                                                                 }
    { DESCRIPTION:                                                    }
    {                                                                 }
    { Returns the base e (2.718...) logarithm of x.                   }
    {                                                                 }
    { The argument is separated into its exponent and fractional      }
    { parts.  If the exponent is between -1 and +1, the logarithm     }
    { of the fraction is approximated by                              }
    {                                                                 }
    {     log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).                   }
    {                                                                 }
    { Otherwise, setting  z = 2(x-1)/x+1),                            }
    {                                                                 }
    {     log(x) = z + z**3 P(z)/Q(z).                                }
    {                                                                 }
    {*****************************************************************}
    const  P : TabCoef = (
     {  Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
         1/sqrt(2) <= x < sqrt(2) }

           4.58482948458143443514E-5,
           4.98531067254050724270E-1,
           6.56312093769992875930E0,
           2.97877425097986925891E1,
           6.06127134467767258030E1,
           5.67349287391754285487E1,
           1.98892446572874072159E1);
       Q : TabCoef = (
           1.50314182634250003249E1,
           8.27410449222435217021E1,
           2.20664384982121929218E2,
           3.07254189979530058263E2,
           2.14955586696422947765E2,
           5.96677339718622216300E1, 0);

     { Coefficients for log(x) = z + z**3 P(z)/Q(z),
        where z = 2(x-1)/(x+1)
        1/sqrt(2) <= x < sqrt(2)  }

       R : TabCoef = (
           -7.89580278884799154124E-1,
            1.63866645699558079767E1,
           -6.41409952958715622951E1, 0, 0, 0, 0);
       S : TabCoef = (
           -3.56722798256324312549E1,
            3.12093766372244180303E2,
           -7.69691943550460008604E2, 0, 0, 0, 0);

    var e : Integer;
       z, y : Real;

    Label Ldone;
    begin
       if( d <= 0.0 ) then
          HandleError(207);
       d := frexp( d, e );

    { logarithm using log(x) = z + z**3 P(z)/Q(z),
      where z = 2(x-1)/x+1) }

       if( (e > 2) or (e < -2) ) then
       begin
         if( d < SQRTH ) then
         begin
           {  2( 2x-1 )/( 2x+1 ) }
          Dec(e, 1);
          z := d - 0.5;
          y := 0.5 * z + 0.5;
         end
         else
         begin
         {   2 (x-1)/(x+1)   }
           z := d - 0.5;
         z := z - 0.5;
         y := 0.5 * d  + 0.5;
         end;
         d := z / y;
         { /* rational form */ }
         z := d*d;
         z := d + d * ( z * polevl( z, R, 2 ) / p1evl( z, S, 3 ) );
         goto ldone;
       end;

    { logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) }

       if( d < SQRTH ) then
       begin
         Dec(e, 1);
         d := ldexp( d, 1 ) - 1.0; {  2x - 1  }
       end
       else
         d := d - 1.0;

       { rational form  }
       z := d*d;
       y := d * ( z * polevl( d, P, 6 ) / p1evl( d, Q, 6 ) );
       y := y - ldexp( z, -1 );   {  y - 0.5 * z  }
       z := d + y;

    ldone:
       { recombine with exponent term }
       if( e <> 0 ) then
       begin
         y := e;
         z := z - y * 2.121944400546905827679e-4;
         z := z + y * 0.693359375;
       end;

       Ln:= z;
    end;
{$endif}


{$ifndef FPC_SYSTEM_HAS_SIN}
    function Sin(d:Real):Real;[internconst:in_const_sin];
    {*****************************************************************}
    { 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  y, z, zz : Real;
         j, sign : Integer;

    begin
      { make argument positive but save the sign }
      sign := 1;
      if( d < 0 ) then
      begin
         d := -d;
         sign := -1;
      end;

      { above this value, approximate towards 0 }
      if( d > lossth ) then
      begin
        sin := 0.0;
        exit;
      end;

      y := Trunc( d/PIO4 ); { integer part of x/PIO4 }

      { strip high bits of integer part to prevent integer overflow }
      z := ldexp( y, -4 );
      z := Trunc(z);           { integer part of y/8 }
      z := y - ldexp( z, 4 );  { y - 16 * (y/16) }

      j := Trunc(z); { convert to integer for tests on the phase angle }
      { map zeros to origin }
      { typecast is to avoid "can't determine which overloaded function }
      { to call"                                                        }
      if odd( longint(j) ) then
      begin
         inc(j);
         y := y + 1.0;
      end;
      j := j and 7; { octant modulo 360 degrees }
      { reflect in x axis }
      if( j > 3) then
      begin
        sign := -sign;
        dec(j, 4);
      end;

      { Extended precision modular arithmetic }
      z := ((d - y * DP1) - y * DP2) - y * DP3;

      zz := z * z;

      if( (j=1) or (j=2) ) then
        y := 1.0 - ldexp(zz,-1) + zz * zz * polevl( zz, coscof, 5 )
      else
      { y = z  +  z * (zz * polevl( zz, sincof, 5 )); }
        y := z  +  z * z * z * polevl( zz, sincof, 5 );

      if(sign < 0) then
      y := -y;
      sin := y;
    end;
{$endif}



{$ifndef FPC_SYSTEM_HAS_COS}
    function Cos(d:Real):Real;[internconst:in_const_cos];
    {*****************************************************************}
    { 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, sign : Integer;
         i : LongInt;
    begin
    { make argument positive }
      sign := 1;
      if( d < 0 ) then
        d := -d;

      { above this value, round towards zero }
      if( d > lossth ) then
      begin
        cos := 0.0;
        exit;
      end;

      y := Trunc( d/PIO4 );
      z := ldexp( y, -4 );
      z := Trunc(z);  { integer part of y/8 }
      z := y - ldexp( z, 4 );  { y - 16 * (y/16) }

      { integer and fractional part modulo one octant }
      i := Trunc(z);
      if odd( i ) then { map zeros to origin }
      begin
        inc(i);
        y := y + 1.0;
      end;
      j := i and 07;
      if( j > 3) then
      begin
        dec(j,4);
        sign := -sign;
      end;
      if( j > 1 ) then
        sign := -sign;

      { Extended precision modular arithmetic  }
      z := ((d - y * DP1) - y * DP2) - y * DP3;

      zz := z * z;

      if( (j=1) or (j=2) ) then
      { y = z  +  z * (zz * polevl( zz, sincof, 5 )); }
        y := z  +  z * z * z * polevl( zz, sincof, 5 )
      else
        y := 1.0 - ldexp(zz,-1) + zz * zz * polevl( zz, coscof, 5 );

      if(sign < 0) then
        y := -y;

      cos := y ;
    end;
{$endif}



{$ifndef FPC_SYSTEM_HAS_ARCTAN}
    function ArcTan(d:Real):Real;[internconst:in_const_arctan];
    {*****************************************************************}
    { 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.                                                           }
    {                                                                 }
    { Range reduction is from four intervals into the interval        }
    { from zero to  tan( pi/8 ).  The approximant uses a rational     }
    { function of degree 3/4 of the form x + x**3 P(x)/Q(x).          }
    {*****************************************************************}
    const P : TabCoef = (
          -8.40980878064499716001E-1,
          -8.83860837023772394279E0,
          -2.18476213081316705724E1,
          -1.48307050340438946993E1, 0, 0, 0);
          Q : TabCoef = (
          1.54974124675307267552E1,
          6.27906555762653017263E1,
          9.22381329856214406485E1,
          4.44921151021319438465E1, 0, 0, 0);

    { tan( 3*pi/8 ) }
    T3P8 = 2.41421356237309504880;
    { tan( pi/8 )   }
    TP8 = 0.41421356237309504880;

    var y,z  : Real;
        Sign : Integer;

    begin
      { make argument positive and save the sign }
      sign := 1;
      if( d < 0.0 ) then
      begin
       sign := -1;
       d := -d;
      end;

      { range reduction }
      if( d > T3P8 ) then
      begin
        y := PIO2;
        d := -( 1.0/d );
      end
      else if( d > TP8 ) then
      begin
        y := PIO4;
        d := (d-1.0)/(d+1.0);
      end
      else
       y := 0.0;

      { rational form in x**2 }

      z := d * d;
      y := y + ( polevl( z, P, 3 ) / p1evl( z, Q, 4 ) ) * z * d + d;

      if( sign < 0 ) then
        y := -y;
      Arctan := y;
    end;
{$endif}


{$ifndef FPC_SYSTEM_HAS_FRAC}
    function frac(d : Real) : Real;[internconst:in_const_frac];
    begin
       frac := d - Int(d);
    end;
{$endif}

{$ifndef FPC_SYSTEM_HAS_POWER}
    function power(bas,expo : real) : real;
     begin
        if bas=0.0 then
          begin
            if expo<>0.0 then
              power:=0.0
            else
              HandleError(207);
          end
        else if expo=0.0 then
         power:=1
        else
        { bas < 0 is not allowed }
         if bas<0.0 then
          handleerror(207)
         else
          power:=exp(ln(bas)*expo);
     end;
{$endif}


{$ifndef FPC_SYSTEM_HAS_POWER_INT64}
   function power(bas,expo : int64) : int64;
     begin
        if bas=0 then
          begin
            if expo<>0 then
              power:=0
            else
              HandleError(207);
          end
        else if expo=0 then
         power:=1
        else
         begin
           if bas<0 then
            begin
              if odd(expo) then
               power:=-round(exp(ln(-bas)*expo))
              else
               power:=round(exp(ln(-bas)*expo));
            end
           else
            power:=round(exp(ln(bas)*expo));
         end;
     end;
{$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 $ffffffff)+dword(q shr 32)*4294967296.0;
  end;
{$endif FPC_SYSTEM_HAS_INT64_TO_DOUBLE}


{$ifndef FPC_SYSTEM_HAS_INT64_TO_DOUBLE}
function fpc_int64_to_double(i : int64): double; compilerproc;
  begin
    if i<0 then
      result:=-double(qword(-i))
    else
      result:=qword(i);
  end;
{$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
         { 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}

{
  $Log$
  Revision 1.18  2004-01-06 21:34:07  peter
    * abs(double) added
    * abs() alias

  Revision 1.17  2004/01/02 17:19:04  jonas
    * if currency = int64, FPC_CURRENCY_IS_INT64 is defined
    + round and trunc for currency and comp if FPC_CURRENCY_IS_INT64 is
      defined
    * if currency = orddef, prefer currency -> int64/qword conversion over
      currency -> float conversions
    * optimized currency/currency if currency = orddef
    * TODO: write FPC_DIV_CURRENCY and FPC_MUL_CURRENCY routines to prevent
        precision loss if currency=int64 and bestreal = double

  Revision 1.16  2003/12/08 19:44:11  jonas
    * use HandleError instead of RunError so exception catching works

  Revision 1.15  2003/09/03 14:09:37  florian
    * arm fixes to the common rtl code
    * some generic math code fixed
    * ...

  Revision 1.14  2003/05/24 13:39:32  jonas
    * fsqrt is an optional instruction in the ppc architecture and isn't
      implemented by any current ppc afaik, so use the generic sqrt routine
      instead (adapted so it works with compilerproc)

  Revision 1.13  2003/05/23 22:58:31  jonas
    * added longint typecase to odd(smallint_var) call to avoid overload
      problem

  Revision 1.12  2003/05/02 15:12:19  jonas
    - removed empty ppc-specific frac()
    + added correct generic frac() implementation for doubles (translated
      from glibc code)

  Revision 1.11  2003/04/23 21:28:21  peter
    * fpc_round added, needed for int64 currency

  Revision 1.10  2003/01/15 00:45:17  peter
    * use generic int64 power

  Revision 1.9  2002/10/12 20:28:49  carl
    * round returns int64

  Revision 1.8  2002/10/07 15:15:02  florian
    * fixed wrong commit

  Revision 1.7  2002/10/07 15:10:45  florian
    + variant wrappers for cmp operators added

  Revision 1.6  2002/09/07 15:07:45  peter
    * old logs removed and tabs fixed

  Revision 1.5  2002/07/28 21:39:29  florian
    * made abs a compiler proc if it is generic

  Revision 1.4  2002/07/28 20:43:48  florian
    * several fixes for linux/powerpc
    * several fixes to MT

}