paweld/fpc
1
0
Fork 0
mirror of https://gitlab.com/freepascal.org/fpc/source.git synced 2025-11-03 04:49:42 +01:00
Code Issues Packages Projects Releases Wiki Activity

+ software implementation of exp(<double>)

Browse Source 
git-svn-id: trunk@5065 -
This commit is contained in:
florian 2006-10-29 18:02:10 +00:00
parent 67851f6aba
commit 02a553668f
1 changed files with 201 additions and 1 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

202
rtl/inc/genmath.inc
View File

@ -597,6 +597,204 @@ type
{$ifndef FPC_SYSTEM_HAS_EXP}
{$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(x: Double):Double;compilerproc;
const
one = 1.0,
halF : array[0..1] of double = (0.5,-0.5);
huge = 1.0e+300;
twom1000 = 9.33263618503218878990e-302; { 2**-1000=0x01700000,0}
o_threshold = 7.09782712893383973096e+02; { 0x40862E42, 0xFEFA39EF }
u_threshold = -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 = 1.44269504088896338700e+00; { 0x3ff71547, 0x652b82fe }
P1 = 1.66666666666666019037e-01; { 0x3FC55555, 0x5555553E }
P2 = -2.77777777770155933842e-03; { 0xBF66C16C, 0x16BEBD93 }
P3 = 6.61375632143793436117e-05; { 0x3F11566A, 0xAF25DE2C }
P4 = -1.65339022054652515390e-06; { 0xBEBBBD41, 0xC5D26BF1 }
P5 = 4.13813679705723846039e-08; { 0x3E663769, 0x72BEA4D0 }
var
double hi : double = 0.0;
double lo = 0.0;
c : double;
t : double;
int32_t k=0;
xsb : longint;
hx,hy,lx : dword;
begin
hx:=float64(x).high;
xsb := (hx shr 31) and 1; { sign bit of x }
hx := hx and $7fffffff; { high word of |x| }
{ filter out non-finite argument }
if hx >= $40862E42 then
begin { if |x|>=709.78... }
if hx >= $7ff00000
begin
lx:=float64(x).low;
if ((hx and $fffff) or lx)<>0 then
begin
result:=x+x; { NaN }
exit;
else
else
begin
if xsb=0 then
result:=x
else
result:=0.0; { exp(+-inf)=begininf,0end }
exit;
end;
end;
if x > o_threshold then
begin
result:=huge*huge; { overflow }
exit;
end;
if x < u_threshold then
begin
result:=twom1000*twom1000; { underflow }
exit;
end;
end;
{ argument reduction }
if hx > $3fd62e42 then
begin { if |x| > 0.5 ln2 }
if hx < $3FF0A2B2 then { and |x| < 1.5 ln2 }
begin
hi := x-ln2HI[xsb];
lo:=ln2LO[xsb];
k := 1-xsb-xsb;
end
else
begin
k := invln2*x+halF[xsb];
t := k;
hi := x - t*ln2HI[0]; { t*ln2HI is exact here }
lo := t*ln2LO[0];
end;
x := hi - lo;
end
else if hx < $3e300000 then
begin { when |x|<2**-28 }
if huge+x>one then
begin
result:=one+x;{ trigger inexact }
exit;
end;
end
else
k := 0;
{ x is now in primary range }
t:=x*x;
c:=x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
if k=0 then
begin
result:=one-((x*c)/(c-2.0)-x);
exit;
end
else
y := one-((lo-(x*c)/(2.0-c))-hi);
if k >= -1021
begin
hy:=float64(y).high;
float64(y).high:=hy+(k shl 20); { add k to y's exponent }
result:=y;
end
else
begin
hy:=float64(y).high;
float64(y).high:=hy+((k+1000) shl 20); { add k to y's exponent }
result:=y*twom1000;
end;
end;
{$else SUPPORT_DOUBLE}
function fpc_exp_real(d: ValReal):ValReal;compilerproc;
{*****************************************************************}
{ Exponential Function }
@ -668,6 +866,8 @@ type
result := d;
end;
end;
{$endif SUPPORT_DOUBLE}
{$endif}
@ -695,7 +895,7 @@ type
else
result:=tr;
end;
{$endif}
{$endif FPC_SYSTEM_HAS_EXP}
{$ifdef FPC_CURRENCY_IS_INT64}