How is floating point conversion actually done in C++?(double to float or float to double)

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The cvtsd2ss instruction uses the FPU’s rounding mode to do the conversion. The default rounding mode is round-to-nearest-even.

In order to follow the algorithm, it helps to keep in mind the information at the IEEE 754-1985 Wikipedia page, especially the diagrams representing the layout.

First, the exponent of the target float is computed: the double type has a wider range than float, so the result may be 0.0f (or a denormal) for a very small double, or an infinite value for a very large double.

For the usual case of a normal double being converted to a normal float (roughly, when the unbiased exponent of the double can be represented in the 8 bits of a single-precision representation), the first 23 bits of the destination significand start out the same as the most significant of the original number’s 52-bit significand.

Then there is the problem of rounding:

  • if the left-over bits are below 10..0, then the target significand is left as-is.

  • If the left-over bits are above 10..0, then the target significand is incremented. If incrementing it makes it overflow (because it is already 1..1), then the carry is propagated into the exponent bits. This produces the correct result because of the careful way the IEEE 754 layout has been designed.

  • If the bits left over are exactly 10..0, then the double is exactly midway between two floats. Of these two choices, the one with the last bit 0 (“even”) is chosen.

After this step, the target significand corresponds to the float nearest to the original double.

The directed rounding modes are only simpler. The case where the target float is a denormal is slightly more complicated (one must be careful to avoid “double-rounding”).

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