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!
Javadoc Search Engine Updated
The Javadoc Search Engine now searches JDK 7 too:
- Go here: javadoc.ankerl.com
It’s still a draft, so the documentation is surely subject to change. You can also use the search directly from here:
happy hacking!
Java Developer Kit (JDK) Search Engine
Thanks to Google Co-op I have just created a new search engine that searches the JDK documentation. It is quite convenient because you can use the labels to choose which version you like to see:
Happy googling.
Exponential Functions: Benchmarks, 8 Times Faster Math.pow()
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:
Read more
Statistical Unit Tests with ensure4j
As part of another project I am developing ensure4j. The syntax (see the examples here) is working quite nicely, ensure4j is already very useful for internal use.
Lately I was busy adding tests that are able to verify if some code (e.g. an optimizer that uses random, like genetic algorithm, simulated annealing, …) produces the desired in e.g. 95% of the cases (Wikipedia has a nice practical example for confidence intervals).
Example Usage
Here is an example that tests the nonsense code Math.random() * 2.
ensure(new Experiment() {
public double measure() {
return Math.random() * 2;
}
}).between(0.9, 1.1, 0.95).sample(10, 100);
The code most likely does not make much sense out of context like this, so here is an explanation of what it does: