I have updated the code for the Math.pow() approximation, now it is 11 times faster on my Pentium IV. Read Optimized Exponential Functions for Java for more information. Now I can also give you some benchmarks:

Benchmarks

Math.log() — 11.7 times faster

• 6.233 sec, Math.log(x)
• 0.531 sec, 6*(x-1)/ (x + 1 + 4*(Math.sqrt(x)))

Math.exp() — 5.3 times faster

• 5.920 sec, Math.exp(x)
• 1.108 sec, exp optimized with IEEE 754 trick

Math.pow() — 8.7 times faster

• 15.967 sec, Math.pow(a, b)
• 11.014 sec, e^(b * log(a))
• 7.607 sec, e^(b * log(a)) + IEEE 754 trick
• 2.109 sec, e^(b * log(a)) + IEEE 754 trick + LOG approximation
• 1.827 sec, simplified everything, see Optimized Exponential Functions for Java

For accurate measurements I have performed each calculation 20 million times and used a random number generator to prevent optimization. I have measured the overhead of iterating and random number generation (3.969 sec) and substracted this from each measurement so that only the pure functional code is measured.