musicdsp.org source code archive - WSInf

fast exp() approximations (click this to go back to the index)
Type : Taylor series approximation
References : Posted by scoofy[AT]inf[DOT]elte[DOT]hu
Notes :
I needed a fast exp() approximation in the -3.14..3.14 range, so I made some approximations based on the tanh() code posted in the archive by
Fuzzpilz. Should be pretty straightforward, but someone may find this useful.
The increasing numbers in the name of the function means increasing precision. Maximum error in the -1..1 range:
fastexp3: 0.05 (1.8%)
fastexp4: 0.01 (0.36%)
fastexp5: 0.0016152 (0.59%)
fastexp6: 0.0002263 (0.0083%)
fastexp7: 0.0000279 (0.001%)
fastexp8: 0.0000031 (0.00011%)
fastexp9: 0.0000003 (0.000011%)
Maximum error in the -3.14..3.14 range:
fastexp3: 8.8742 (38.4%)
fastexp4: 4.8237 (20.8%)
fastexp5: 2.28 (9.8%)
fastexp6: 0.9488 (4.1%)
fastexp7: 0.3516 (1.5%)
fastexp8: 0.1172 (0.5%)
fastexp9: 0.0355 (0.15%)
These were done using the Taylor series, for example I got fastexp4 by using:
exp(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + ...
= (24 + 24x + x^2*12 + x^3*4 + x^4) / 24
(using Horner-scheme:)
= (24 + x * (24 + x * (12 + x * (4 + x)))) * 0.041666666f
Code :
inline float fastexp3(float x) {
return (6+x*(6+x*(3+x)))*0.16666666f;
}
inline float fastexp4(float x) {
return (24+x*(24+x*(12+x*(4+x))))*0.041666666f;
}
inline float fastexp5(float x) {
return (120+x*(120+x*(60+x*(20+x*(5+x)))))*0.0083333333f;
}
inline float fastexp6(float x) {
return 720+x*(720+x*(360+x*(120+x*(30+x*(6+x))))))*0.0013888888f;
}
inline float fastexp7(float x) {
return (5040+x*(5040+x*(2520+x*(840+x*(210+x*(42+x*(7+x)))))))*0.00019841269f;
}
inline float fastexp8(float x) {
return (40320+x*(40320+x*(20160+x*(6720+x*(1680+x*(336+x*(56+x*(8+x))))))))*2.4801587301e-5;
}
inline float fastexp9(float x) {
return (362880+x*(362880+x*(181440+x*(60480+x*(15120+x*(3024+x*(504+x*(72+x*(9+x)))))))))*2.75573192e-6;
}
Comments
from : scoofy@inf.elte.hu
comment : These series converge fast only near zero. But there is an identity:
exp(x) = exp(a) * exp(x-a)
So, if you want a relatively fast polynomial approximation for exp(x) for 0 to ~7.5, you can use:
// max error in the 0 .. 7.5 range: ~0.45%
inline float fastexp(float const &x)
{
if (x