Average Error: 0.5 → 0.4
Time: 2.3s
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 r2523 = x;
        double r2524 = 1.0;
        double r2525 = r2523 - r2524;
        double r2526 = sqrt(r2525);
        double r2527 = sqrt(r2523);
        double r2528 = r2526 * r2527;
        return r2528;
}
↓
double f(double x) {
        double r2529 = x;
        double r2530 = 0.125;
        double r2531 = 1.0;
        double r2532 = r2531 / r2529;
        double r2533 = 0.5;
        double r2534 = fma(r2530, r2532, r2533);
        double r2535 = r2529 - r2534;
        return r2535;
}

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 2019346 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))