Mostly thanks to this reddit discussion, I have updated my pow() approximation for C / C++. I have now two different versions:

inline double fastPow(double a, double b) { union { double d; int x[2]; } u = { a }; u.x[1] = (int)(b * (u.x[1] - 1072632447) + 1072632447); u.x[0] = 0; return u.d; }

This new code uses the union trick, instead of the weird casting trick I’ve used before. This means that `-fno-strict-aliasing` is no more required any more when compiling, and it is also a bit faster because one less temporary variables is needed. When you have a little endian machine, you have to exchange u.x[0] and u.x[1]. On my PC, this version is 4.2 times faster than the much more precise pow().

Besides that, I also have now a slower approximation that has much less error when the exponent is larger than 1. It makes use exponentiation by squaring, which is exact for the integer part of the exponent, and uses only the exponent’s fraction for the approximation:

// should be much more precise with large b inline double fastPrecisePow(double a, double b) { // calculate approximation with fraction of the exponent int e = (int) b; union { double d; int x[2]; } u = { a }; u.x[1] = (int)((b - e) * (u.x[1] - 1072632447) + 1072632447); u.x[0] = 0; // exponentiation by squaring with the exponent's integer part // double r = u.d makes everything much slower, not sure why double r = 1.0; while (e) { if (e & 1) { r *= a; } a *= a; e >>= 1; } return r * u.d; }

This code is 3.3 times faster than pow(). Writing a microbenchmark is not easy, so I have posted mine here. Here is also a Java version of the more accurate pow approximation.

Any ideas how this could be improved? Please post them!

The new approximation does not work with negative exponents because the loop will never end. Is the first approximation valid for negative exponents?

It does not work with negative constants, but you can use this trick:

a^(-b) = 1/(a^b)

so when b is negative, just calculate

1.0/pow(a, abs(b))