Average Error: 0.5 → 0.4
Time: 4.3s
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 r20872 = x;
        double r20873 = 1.0;
        double r20874 = r20872 - r20873;
        double r20875 = sqrt(r20874);
        double r20876 = sqrt(r20872);
        double r20877 = r20875 * r20876;
        return r20877;
}
↓
double f(double x) {
        double r20878 = x;
        double r20879 = 0.125;
        double r20880 = 1.0;
        double r20881 = r20880 / r20878;
        double r20882 = 0.5;
        double r20883 = fma(r20879, r20881, r20882);
        double r20884 = r20878 - r20883;
        return r20884;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.5

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

  2. Taylor expanded around inf 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))