97 Things Every Programmer Should Know

Chapter 33. Floating-Point Numbers Aren’t Real

Chuck Allison

FLOATING-POINT NUMBERS ARE NOT “REAL NUMBERS” in the mathematical sense, even though they are called real in some programming languages, such as Pascal and Fortran. Real numbers have infinite precision and are therefore continuous and nonlossy; floating-point numbers have limited precision, so they are finite, and they resemble “badly behaved” integers, because they’re not evenly spaced throughout their range.

To illustrate, assign 2147483647 (the largest signed 32-bit integer) to a 32-bit float variable (x, say), and print it. You’ll see 2147483648. Now print x-64. Still 2147483648. Now print x-65, and you’ll get 2147483520! Why? Because the spacing between adjacent floats in that range is 128, and floating-point operations round to the nearest floating-point number.

IEEE floating-point numbers are fixed-precision numbers based on base-two scientific notation: 1.d₁d₂…d_{p 1} x 2^e, where p is the precision (24 for float, 53 for double). The spacing between two consecutive numbers is 2^1-p+e, which can be safely approximated by ε|x|, where ε is the machine epsilon (2^1-p).

Knowing the spacing in the neighborhood of a floating-point number can help you avoid classic numerical blunders. For example, if you’re performing an iterative calculation, such as searching for the root of an equation, there’s no sense in asking for greater precision ...