Average Error: 0.5 → 0.4
Time: 2.4s
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 r2512 = x;
        double r2513 = 1.0;
        double r2514 = r2512 - r2513;
        double r2515 = sqrt(r2514);
        double r2516 = sqrt(r2512);
        double r2517 = r2515 * r2516;
        return r2517;
}
↓
double f(double x) {
        double r2518 = x;
        double r2519 = 0.125;
        double r2520 = 1.0;
        double r2521 = r2520 / r2518;
        double r2522 = 0.5;
        double r2523 = fma(r2519, r2521, r2522);
        double r2524 = r2518 - r2523;
        return r2524;
}

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 2020083 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))