c - Implementing single-precision division as double-precision multiplication

Question

Question

For a C99 compiler implementing exact IEEE 754 arithmetic, do values of f, divisor of type float exist such that f / divisor != (float)(f * (1.0 / divisor))?

EDIT: By “implementing exact IEEE 754 arithmetic” I mean a compiler that rightfully defines FLT_EVAL_METHOD as 0.

Context

A C compiler that provides IEEE 754-compliant floating-point can only replace a single-precision division by a constant by a single-precision multiplication by the inverse if said inverse is itself representable exactly as a float.

In practice, this only happens for powers of two. So a programmer, Alex, may be confident that f / 2.0f will be compiled as if it had been f * 0.5f, but if it is acceptable for Alex to multiply by 0.10f instead of dividing by 10, Alex should express it by writing the multiplication in the program, or by using a compiler option such as GCC's -ffast-math.

This question is about transforming a single-precision division into a double-precision multiplication. Does it always produce the correctly rounded result? Is there a chance that it could be cheaper, and thus be an optimization that compilers might make (even without -ffast-math)?

I have compared (float)(f * 0.10) and f / 10.0f for all single-precision values of f between 1 and 2, without finding any counter-example. This should cover all divisions of normal floats producing a normal result.

Then I generalized the test to all divisors with the program below:

#include <float.h>
#include <math.h>
#include <stdio.h>

int main(void){
  for (float divisor = 1.0; divisor != 2.0; divisor = nextafterf(divisor, 2.0))
    {
      double factor = 1.0 / divisor; // double-precision inverse
      for (float f = 1.0; f != 2.0; f = nextafterf(f, 2.0))
        {
          float cr = f / divisor;
          float opt = f * factor; // double-precision multiplication
          if (cr != opt)
            printf("For divisor=%a, f=%a, f/divisor=%a but (float)(f*factor)=%a
",
                   divisor, f, cr, opt);
        }
    }
}

The search space is just large enough to make this interesting (2⁴⁶). The program is currently running. Can someone tell me whether it will print something, perhaps with an explanation why or why not, before it has finished?

Answer 1

Your program won't print anything, assuming round-ties-to-even rounding mode. The essence of the argument is as follows:

We're assuming that both f and divisor are between 1.0 and 2.0. So f = a / 2^23 and divisor = b / 2^23 for some integers a and b in the range [2^23, 2^24). The case divisor = 1.0 isn't interesting, so we can further assume that b > 2^23.

The only way that (float)(f * (1.0 / divisor)) could give the wrong result would be for the exact value f / divisor to be so close to a halfway case (i.e., a number exactly halfway between two single-precision floats) that the accumulated errors in the expression f * (1.0 / divisor) push us to the other side of that halfway case from the true value.

But that can't happen. For simplicity, let's first assume that f >= divisor, so that the exact quotient is in [1.0, 2.0). Now any halfway case for single precision in the interval [1.0, 2.0) has the form c / 2^24 for some odd integer c with 2^24 < c < 2^25. The exact value of f / divisor is a / b, so the absolute value of the difference f / divisor - c / 2^24 is bounded below by 1 / (2^24 b), so is at least 1 / 2^48 (since b < 2^24). So we're more than 16 double-precision ulps away from any halfway case, and it should be easy to show that the error in the double precision computation can never exceed 16 ulps. (I haven't done the arithmetic, but I'd guess it's easy to show an upper bound of 3 ulps on the error.)

So f / divisor can't be close enough to a halfway case to create problems. Note that f / divisor can't be an exact halfway case, either: since c is odd, c and 2^24 are relatively prime, so the only way we could have c / 2^24 = a / b is if b is a multiple of 2^24. But b is in the range (2^23, 2^24), so that's not possible.

The case where f < divisor is similar: the halfway cases then have the form c / 2^25 and the analogous argument shows that abs(f / divisor - c / 2^25) is greater than 1 / 2^49, which again gives us a margin of 16 double-precision ulps to play with.