{
$Id: genmath.inc,v 1.18 2004/01/06 21:34:07 peter Exp $
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: genmath.inc,v $
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
}
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