Re: gamma approximation in 48/49/50? Message #20 Posted by hugh steers on 25 Sept 2009, 8:06 p.m., in response to message #19 by Eric Smith
maybe it really does use this formula and there is no mistake here. except there is an outer "driver" routine that conditions the input.
say we want 12 accurate digits. I have the (alleged) 48 logic as:
double _gamma48gx(double x)
{
const double k0 = 49.5920119017593;
const double k1 = 3.03479606073615;
const double k2 = 7.65594818140208;
const double k3 = 0.933584905660377;
const double k4 = 0.121142857142857;
const double k5 = 0.4;
const double k6 = 0.918938533204673;
double tx2 = 12.0 * x * x; // 12 x squared
double ln_gamma = x/(k4/(k2/((k0/(tx2+60))+tx2+k1)+tx2+k3)+tx2+k5)x+k6+(x0.5)*log(x);
return exp (ln_gamma);
}
this barely gets about 8 digits for numbers starting at 6. But, if you bump all inputs up to around 100, things are ok.
eg
double gamma48gx(double x)
{
double g;
double p = 1.0;
while (x < 100)
{
p *= x;
x = x + 1;
}
return _gamma48gx(x)/p;
}
ok, its not very good because you have to perform all those multiplies, but it does get the right answer.
so i had a look at what also might be done with the same amount of work (ie 4 divides, some adds plus log/exp), i am finding the Stieltjes forms quite good. for example, i get this:
double gammaStieltjes4(double x)
{
// get 12 digits x > 8
const double a = 1.0/12;
const double b = 1.0/30;
const double c = 53.0/210;
const double d = 129.0/250; // adapted
// would be a constant
double e = sqrt(2*PI);
double s;
s = a/(x + b/(x + c/(x + d/x)));
return e*exp((x.5)*log(x)  x + s);
}
this starts getting 12 correct digits somewhere between 7 and 8 and would be easy to bump up in the same was suggested above, but with only a few extra multiples (maybe just one). Either that or use the next best version of the approx (ie one more term).
in this example, i got surprisingly accurate results from this 4 divide logic. actually this was a mistake, i had a typo in the last coefficient that turned out to make it *more* accurate!
After some experimentation, it turns out to be a benefit to adjust the last coefficient term to take account of the fact that you are stopping there (can compute the adjustment from the next term ignored). I did this and arrived at my "adapted" term here.
i still don't really know what the 48 actually does.
