To write a computer program to print the first 1000 nonnegative powers of two, do you think you’d need to use arbitrary precision arithmetic? After all, 21000 is a 302-digit number. How about printing the first 1000 negative powers of two? 2-1000 weighs in at a whopping 1000 decimal places. It turns out all you need is standard double-precision floating-point arithmetic — and the right compiler!
Recently I showed that programming languages vary in how much precision they allow in printed floating-point fractions. Not only do they vary, but most don’t meet my standard — printing, to full precision, decimal values that have exact floating-point representations. Here I’ll present a similar study for floating-point integers, which had similar results.
Interestingly, programming languages vary in how much precision they allow in printed floating-point fractions. You would think they’d all be the same, allowing you to print as many decimal places as you ask for. After all, a floating-point fraction is a dyadic fraction; it has as many decimal places as it has bits in its fractional representation.
Consider the dyadic fraction 5,404,319,552,844,595/253. Its decimal expansion is 0.59999999999999997779553950749686919152736663818359375, and its binary expansion is 0.10011001100110011001100110011001100110011001100110011. Both are 53 digits long. The ideal programming language lets you print all 53 decimal places, because all are meaningful. Unfortunately, many languages won’t let you do that; they typically cap the number of decimal places at between 15 and 17, which for our example might be 0.59999999999999998.
To write a program to check if an integer is a power of two, you could follow two basic strategies: check the number based on its decimal value, or check it based on its binary representation. The former approach is more human-friendly but generally less efficient; the latter approach is more machine-friendly but generally more efficient. We will explore both approaches, comparing ten different but equivalent C functions.
A power of two, when expressed as a binary number, is easy to spot: it has one, and only one, 1 bit. For example, 1000, 10, and 0.001 are powers of two. Inside a computer, however, numbers are more generally represented in binary code, not as “pure” binary numbers. As a result, you may not be able to look at the binary representation of a number and tell at a glance whether it’s a power of two or not; it depends on how it’s encoded.