Optimized pow() approximation for Java, C / C++, and C#

I have already written about approximations of e^x, log(x) and pow(a, b) in my post Optimized Exponential Functions for Java. Now I have more 🙂 In particular, the pow() function is now even faster, simpler, and more accurate. Without further ado, I proudly give you the brand new approximation:

Approximation of pow() in Java

This is really very compact. The calculation only requires 2 shifts, 1 mul, 2 add, and 2 register operations. That’s it! In my tests it usually within an error margin of 5% to 12%, in extreme cases sometimes up to 25%. A careful analysis is left as an exercise for the reader. This is very usable for in e.g. metaheuristics or neural nets.

UPDATE, December 10, 2011

I just managed to make the above code about 30% faster than the one above on my machine. The error is a tiny fraction different (not better or worse).

This new approximation is about 23 times as fast as Math.pow() on my machine (Intel Core2 Quad, Q9550, Java 1.7.0_01-b08, 64-Bit Server VM). Unfortunately, microbenchmarks are difficult to do in Java, so your mileage may vary. You can download the benchmark PowBench.java and have a look, I have tried to prevent overoptimization, and substract the overhead introduced due to this preventation.

Approximation of pow() in C and C++

UPDATE, January 25, 2012

The code below is updated with using union, you do not need -fno-strict-aliasing any more for compiling. Also, here is a more precise version of the approximation.

Compiled on my Pentium-M with gcc 4.1.2:

This version is 7.8 times faster than pow() from the standard library.

Approximation of pow() in C#

Jason Jung has posted a port of the this code to C#:

How the Approximation was Developed

It is quite impossible to understand what is going on in this function, it just magically works. To shine a bit more light on it, here is a detailed description how I have developed this.

Approximation of e^x

As described here, the paper “A Fast, Compact Approximation of the Exponential Function” develops a C macro that does a good job at exploiting the IEEE 754 floating-point representation to calculate e^x. This macro can be transformed into Java code straightforward, which looks like this:

Use Exponential Functions for a^b

Thanks to the power of math, we know that a^b can be transformed like this:

  1. Take exponential
  2. Extract b

Now we have expressed the pow calculation with e^x and ln(x). We already have the e^x approximation, but no good ln(x). The old approximation is very bad, so we need a better one. So what now?

Approximation of ln(x)

Here comes the big trick: Rember that we have the nice e^x approximation? Well, ln(x) is exactly the inverse function! That means we just need to transform the above approximation so that the output of e^x is transformed back into the original input.

That’s not too difficult. Have a look at the above code, we now take the output and move backwards to undo the calculation. First reverse the shift:

Now solve the equation

for val:

  1. The original formula
  2. Perform subtraction
  3. Bring value to other side
  4. Divide by factor
  5. Finally, val on the left side

Voíla, now we have a nice approximation of ln(x):

Combine Both Approximations

Finally we can combine the two approximations into e^(ln(a) * b):

Between the two shifts, we can simply insert the tmp1 calculation into the tmp2 calculation to get

Now simplify tmp2 calculation:

  1. The original formula
  2. We can drop the factor 1512775
  3. And finally, calculate the substraction

The Result

That’s it! Add some casts, and the complete function is the same as above.

This concludes my little tutorial on microoptimization of the pow() function. If you have come this far, I congratulate your presistence 🙂

UPDATE Recently there several other approximative pow calculation methods have been developed, here are some others that I have found through reddit:

  • Fast pow() With Adjustable Accuracy — This looks quite a bit more sophisticated and precise than my approximation. Written in C and for float values. A Java port should not be too difficult.
  • Fast SSE2 pow: tables or polynomials? — Uses SSE operation and seems to be a bit faster than the table approach from the link above with the potential to scale better when due to less cache usage.

Please post what you think about this!