Companion to hypot()

Question

The hypot function, introduced into C in the 1999 revision of the language, calculates the hypotenuse of a right triangle given the other sides as arguments, but with care taken to avoid the over/underflow which would result from the naive implementation as

double hypot(double a, double b)
{
  return sqrt(a*a + b*b);
}

I find myself with the need for companion functionality: given a side and the hypotenuse of a triangle, find the third side (avoiding under/overflow). I can think of a few ways to do this, but wondered if there was an existing "best practice"?

My target is Python, but really I'm looking for algorithm pointers.

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Thanks for the replies. In case anyone is interested in the result, my C99 implementation can be found here and a Python version here, part of the Hypothesis project.

Answer 1

The first thing to do is factorize:

b = sqrt(h*h - a*a) = sqrt((h-a)*(h+a))

We have not only avoided some overflow, but also gained accuracy.

If any factor is close to 1E+154 = sqrt(1E+308) (max with IEEE 754 64 bits float) then we must also avoid overflow:

sqrt((h-a)*(h+a)) = sqrt(h-a) * sqrt(h+a)

This case is very unlikely, so the two sqrt's are justified, even if its slower than just a sqrt.

Notice that if h ~ 5E+7 * a then h ~ b which means that there are not enough digits to represent b as different from h.