Question

I am interested to learn about the binary format for a single or a double type used by C++ on Intel based systems.

I have avoided the use of floating point numbers in cases where the data needs to potentially be read or written by another system (i.e. files or networking). I do realise that I could use fixed point numbers instead, and that fixed point is more accurate, but I am interested to learn about the floating point format.

Answer 1

Floating-point format is determined by the processor, not the language or compiler. These days almost all processors (including all Intel desktop machines) either have no floating-point unit or have one that complies with IEEE 754. You get two or three different sizes (Intel with SSE offers 32, 64, and 80 bits) and each one has a sign bit, an exponent, and a significand. The number represented is usually given by this formula:

sign * (2**(E-k)) * (1 + S / (2**k'))

where k’ is the number of bits in the significand and k is a constant around the middle range of exponents. There are special representations for zero (plus and minus zero) as well as infinities and other “not a number” (NaN) values.

There are definite quirks; for example, the fraction 1/10 cannot be represented exactly as a binary IEEE standard floating-point number. For this reason the IEEE standard also provides for a decimal representation, but this is used primarily by handheld calculators and not by general-purpose computers.

Recommended reading: David Golberg’s What Every Computer Scientist Should Know About Floating-Point Arithmetic