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

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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).

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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.

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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:

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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#:

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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:

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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
1a^b = e^(ln(a^b))
- Extract b
1a^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:

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

Now solve the equation

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

for val:

- The original formula
1tmp = (1512775 * val + (1072693248 - 60801))
- Perform subtraction
1tmp = 1512775 * val + 1072632447
- Bring value to other side
1tmp - 1072632447 = 1512775 * val
- Divide by factor
1(tmp - 1072632447) / 1512775 = val
- Finally, val on the left side
1val = (tmp - 1072632447) / 1512775

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

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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)`:

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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

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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
1tmp2 = (1512775 * (x - 1072632447) / 1512775 * b + (1072693248 - 60801))
- We can drop the factor
`1512775`1tmp2 = (x - 1072632447) * b + (1072693248 - 60801) - And finally, calculate the substraction
1tmp2 = b * (x - 1072632447) + 1072632447

## The Result

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

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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!

Would it be possible to use this with float value instead of doubles?