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
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.
I use Linux, Java 1.6.0-b105 with the server VM, and execute the benchmark with this command:
sudo nice -n -20 java -cp . -server PowTest
The approximation is 27 times faster than Math.pow() on my Pentium-M. On a Pentium 4 it is 41 times faster. Unfortunately, microbenchmarks are difficult to do in Java, so your mileage may vary. You can download the benchmark PowTest.java and have a look, I have tried to prevent overoptimization while still having a low overhead.
Approximation of pow() in C and C++
double pow(double a, double b) {
int tmp = (*(1 + (int *)&a));
int tmp2 = (int)(b * (tmp - 1072632447) + 1072632447);
double p = 0.0;
*(1 + (int * )&p) = tmp2;
return p;
}
Compiled on my Pentium-M with gcc 4.1.2:
gcc -O3 -march=pentium-m -fomit-frame-pointer -fno-strict-aliasing
This version is 7.8 times faster than pow() from the standard library.
WARNING! you HAVE to use the -fno-strict-aliasing option, or this does not work!
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
Voíla, 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!
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