Testing x == y

You rarely want to perform tests such as x==y on floating-point variables, except when filtering special cases. The reason is that roundoff errors usually cause differences in mathematically equivalent expressions. Even the tiniest error can foil a test for strict equality. For example, the formula for parallel resistors used in "Arithmetic with infinities" on page 195 is often rewritten in the mathematically equivalent form

With , , and hypothetical four-digit decimal arithmetic, the first formula yields 845.3 and the second 845.2. (The actual value is 845.240479...)

As another example, consider computing given the value of f at x. You might proceed by discrete steps from x to y, but using a test for equality to terminate the process could lead to an infinite loop.

  float stepsize = (y - x) / n;
  do {
      // recompute stepsize if necessary
      x += stepsize;
      f_at_x = (/* estimate f(x) */)
  
} until (x >= y);

  bool ExcessiveRelativeError(double s, double t, double tolerance) {
      return (bool) (Abs((s - t)/t) > tolerance);
  }