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 r2567 = x;
        double r2568 = 1.0;
        double r2569 = r2567 - r2568;
        double r2570 = sqrt(r2569);
        double r2571 = sqrt(r2567);
        double r2572 = r2570 * r2571;
        return r2572;
}
↓
double f(double x) {
        double r2573 = x;
        double r2574 = 0.125;
        double r2575 = 1.0;
        double r2576 = r2575 / r2573;
        double r2577 = 0.5;
        double r2578 = fma(r2574, r2576, r2577);
        double r2579 = r2573 - r2578;
        return r2579;
}

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 2019352 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))