Average Error: 0.5 → 0.5
Time: 3.6s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \mathsf{fma}\left(0.125, \frac{1}{x}, 0.5\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \mathsf{fma}\left(0.125, \frac{1}{x}, 0.5\right)
double f(double x) {
        double r19823 = x;
        double r19824 = 1.0;
        double r19825 = r19823 - r19824;
        double r19826 = sqrt(r19825);
        double r19827 = sqrt(r19823);
        double r19828 = r19826 * r19827;
        return r19828;
}
↓
double f(double x) {
        double r19829 = x;
        double r19830 = 0.125;
        double r19831 = 1.0;
        double r19832 = r19831 / r19829;
        double r19833 = 0.5;
        double r19834 = fma(r19830, r19832, r19833);
        double r19835 = r19829 - r19834;
        return r19835;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.5

    \[\sqrt{x - 1} \cdot \sqrt{x}\]

  2. Taylor expanded around inf 0.5

    \[\leadsto \color{blue}{x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)}\]

  3. Simplified0.5

    \[\leadsto \color{blue}{x - \mathsf{fma}\left(0.125, \frac{1}{x}, 0.5\right)}\]

  4. Final simplification0.5

    \[\leadsto x - \mathsf{fma}\left(0.125, \frac{1}{x}, 0.5\right)\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))