Printing double without losing precision

It’s not correct to say “floating point is inaccurate”, although I admit that’s a useful simplification. If we used base 8 or 16 in real life then people around here would be saying “base 10 decimal fraction packages are inaccurate, why did anyone ever cook those up?”.

The problem is that integral values translate exactly from one base into another, but fractional values do not, because they represent fractions of the integral step and only a few of them are used.

Floating point arithmetic is technically perfectly accurate. Every calculation has one and only one possible result. There is a problem, and it is that most decimal fractions have base-2 representations that repeat. In fact, in the sequence 0.01, 0.02, … 0.99, only a mere 3 values have exact binary representations. (0.25, 0.50, and 0.75.) There are 96 values that repeat and therefore are obviously not represented exactly.

Now, there are a number of ways to write and read back floating point numbers without losing a single bit. The idea is to avoid trying to express the binary number with a base 10 fraction.

  • Write them as binary. These days, everyone implements the IEEE-754 format so as long as you choose a byte order and write or read only that byte order, then the numbers will be portable.
  • Write them as 64-bit integer values. Here you can use the usual base 10. (Because you are representing the 64-bit aliased integer, not the 52-bit fraction.)

You can also just write more decimal fraction digits. Whether this is bit-for-bit accurate will depend on the quality of the conversion libraries and I’m not sure I would count on perfect accuracy (from the software) here. But any errors will be exceedingly small and your original data certainly has no information in the low bits. (None of the constants of physics and chemistry are known to 52 bits, nor has any distance on earth ever been measured to 52 bits of precision.) But for a backup or restore where bit-for-bit accuracy might be compared automatically, this obviously isn’t ideal.

Leave a Comment