The Secret Life of NaN

The floating point standard defines a special value called Not-a-Number (NaN) which is used to represent, well, values that aren’t numbers. Double precision NaNs come with a payload of 51 bits which can be used for whatever you want– one especially fun hack is using the payload to represent all other non-floating point values and their types at runtime in dynamically typed languages.

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I gave a lightning talk about the secret life of NaN at !!con West 2019– it has fewer details, but more jokes; you can find a recording here.

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the non-secret life of NaN

When I say “NaN” and also “floating point”, I specifically mean the representations defined in IEEE 754-2008, the ubiquitous floating point standard. This standard was born in 1985 (with much drama!) out of a need for a canonical representation which would allow code to be portable by quelling the anarchy induced by the menagerie of inconsistent floating point representations used by different processors.

Floating point values are a discrete logarithmic-ish approximation to real numbers; below is a visualization of the points defined by a toy floating point like representation with 3 bits of exponent, 3 bits of mantissa (the image is from the paper “How do you compute the midpoint of an interval?”, which points out arithmetic artifacts that commonly show up in midpoint computations).

Since the NaN I’m talking about doesn’t exist outside of IEEE 754-2008, let’s briefly take a look at the spec.

An extremely brief overview of IEEE 754-2008

The standard defines these logarithmic-ish distributions of values with base-2 and base-10. For base-2, the standard defines representations for bit-widths for all powers of two between 16 bits wide and 256 bits wide; for base-10 it defines representations for bit-widths for all powers of two between 32 bits wide and 128 bits wide. (Well, almost. For the exact spec check out page 13 spec). These are the only standardized bitwidths, meaning, if a processor supports 32 bit floating point values, then it’s highly likely it will support it in the standard compliant representation.

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