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

public static double pow(final double a, final double b) {
final int x = (int) (Double.doubleToLongBits(a) >> 32);
final int y = (int) (b * (x - 1072632447) + 1072632447);
return Double.longBitsToDouble(((long) y) << 32);
}

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).

public static double pow(final double a, final double b) {
final long tmp = Double.doubleToLongBits(a);
final long tmp2 = (long)(b * (tmp - 4606921280493453312L)) + 4606921280493453312L;
return Double.longBitsToDouble(tmp2);
}

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.

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;
}

Compiled on my Pentium-M with gcc 4.1.2:

gcc -O3 -march=pentium-m -fomit-frame-pointer

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#:

public static double PowerA(double a, double b) {
int tmp = (int)(BitConverter.DoubleToInt64Bits(a) >> 32);
int tmp2 = (int)(b * (tmp - 1072632447) + 1072632447);
return BitConverter.Int64BitsToDouble(((long)tmp2) << 32);
}

# 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:

public static double exp(double val) {
final long tmp = (long) (1512775 * val + (1072693248 - 60801));
return Double.longBitsToDouble(tmp << 32);
}

## Use Exponential Functions for a^b

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

- Take exponential
a^b = e^(ln(a^b))

- Extract b
a^b = e^(ln(a)*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:

final double tmp = (Double.doubleToLongBits(val) >> 32);

Now solve the equation

tmp = (1512775 * val + (1072693248 - 60801))

for val:

- The original formula
tmp = (1512775 * val + (1072693248 - 60801))

- Perform subtraction
tmp = 1512775 * val + 1072632447

- Bring value to other side
tmp - 1072632447 = 1512775 * val

- Divide by factor
(tmp - 1072632447) / 1512775 = val

- Finally, val on the left side
val = (tmp - 1072632447) / 1512775

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

public double ln(double val) {
final double x = (Double.doubleToLongBits(val) >> 32);
return (x - 1072632447) / 1512775;
}

## Combine Both Approximations

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

public static double pow1(final double a, final double b) {
// calculate ln(a)
final double x = (Double.doubleToLongBits(a) >> 32);
final double ln_a = (x - 1072632447) / 1512775;
// ln(a) * b
final double tmp1 = ln_a * b;
// e^(ln(a) * b)
final long tmp2 = (long) (1512775 * tmp1 + (1072693248 - 60801));
return Double.longBitsToDouble(tmp2 << 32);
}

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

public static double pow2(final double a, final double b) {
final double x = (Double.doubleToLongBits(a) >> 32);
final long tmp2 = (long) (1512775 * (x - 1072632447) / 1512775 * b + (1072693248 - 60801));
return Double.longBitsToDouble(tmp2 << 32);
}

Now simplify `tmp2` calculation:

- The original formula
tmp2 = (1512775 * (x - 1072632447) / 1512775 * b + (1072693248 - 60801))

- We can drop the factor
`1512775`
tmp2 = (x - 1072632447) * b + (1072693248 - 60801)

- And finally, calculate the substraction
tmp2 = b * (x - 1072632447) + 1072632447

## The Result

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

public static double pow(final double a, final double b) {
final int tmp = (int) (Double.doubleToLongBits(a) >> 32);
final int tmp2 = (int) (b * (tmp - 1072632447) + 1072632447);
return Double.longBitsToDouble(((long) tmp2) << 32);
}

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!