power : Java Glossary

Introduction

To raise a number to a power, you can use Math.pow( double x, double p).

This also works for non integer p and negative p. So you can compute roots with it too, e.g. the twelfth root of 2 is Math. pow( 2, 1.0d/12.0d ). (This number was the secret of Bach’s equal tempering that permitted changes of key without “wolves“.)

Math. pow is a very expensive operation. It has to calculate the natural logarithm of x using polynomial interpolation (lots of multiplies), then it multiplies by p, then it calculates e to that power, again using polynomial interpolation. It also has to ensure if both operands are precise integers represented as doubles and the result is an integer that can be precisely represented as a double, then the result must be bang on. Avoid Math.pow if you possibly can. If you look at the source code for  Math.pow you may think I am all wet. Much of this complexity is hidden inside the floating point hardware. Like all floating point routines, Math.pow’s results are approximate.

If you want perfection, use long, BigInteger or BigDecimal. For squared x² and cubed x² you can use x*x or x*x*x. For 2ⁿ you can use 1<<n. For other integral powers you can use Patricia Shanahan’s method, which it turns out is almost identical to the method Knuth gives on page 462 of The Art of Computer Programming Volume 2 Seminumerical Algorithms. The method dates back to 200 BC in India.

Fast log₂ Algorithm

Powers of Two

Magic Round Binary Numbers

Icons come in 16, 32, 24, 64, 128 and 256 sizes. These are magic numbers when you express them in binary, much like 10, 100, 100 are in decimal notation. SHA digests come in 32, 64 and 128 byte lengths, again all powers of two. A megabyte is 1,048,576 bytes, a power of two, 2²⁰. A gigabyte is 1,073,741,824 bytes, again power of two, 2³⁰.