Binary number representation

Question

First, this isn't a question about precision or anything like that.

My question is, how does the compiler decide how to represent a number?

Let's take C for example. I write

double d = 4.5632;

How does it pick its binary representation? I know it's not represented exactly, so how does it choose the closest representable number? Is it done at compile time? Is it done by the CPU or the OS?

Please only answer if you know how this happens, answers like "don't worry about it" are not helpful. Also, "it depends on the platform" isn't helpful also, you can pick a platform and explain for that.

Answer 1

The compiler doesn't decide (typically). The CPU (typically) has a floating-point unit, which requires floating-point values to be represented in a particular format (it's typically IEEE-754). Of course, it's possible to emulate an entirely different architecture, in which case the compiler/emulator author is free to pick an entirely different representation. But this isn't typical.

As to how the specific lexical representation 4.5632 is converted to the underlying representation, that's specified by the C standard. So from section 6.4.4.2 of the C99 standard (I've highlighted the most relevant part):

The significand part is interpreted as a (decimal or hexadecimal) rational number; the digit sequence in the exponent part is interpreted as a decimal integer. For decimal floating constants, the exponent indicates the power of 10 by which the significand part is to be scaled. For hexadecimal floating constants, the exponent indicates the power of 2 by which the significand part is to be scaled. For decimal floating constants, and also for hexadecimal floating constants when FLT_RADIX is not a power of 2, the result is either the nearest representable value, or the larger or smaller representable value immediately adjacent to the nearest representable value, chosen in an implementation-defined manner. For hexadecimal floating constants when FLT_RADIX is a power of 2, the result is correctly rounded.

This will be done at compile-time (although the standard doesn't mandate that).