Average Error: 0.5 → 0.4
Time: 2.0s
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 r2408 = x;
        double r2409 = 1.0;
        double r2410 = r2408 - r2409;
        double r2411 = sqrt(r2410);
        double r2412 = sqrt(r2408);
        double r2413 = r2411 * r2412;
        return r2413;
}
↓
double f(double x) {
        double r2414 = x;
        double r2415 = 0.125;
        double r2416 = 1.0;
        double r2417 = r2416 / r2414;
        double r2418 = 0.5;
        double r2419 = fma(r2415, r2417, r2418);
        double r2420 = r2414 - r2419;
        return r2420;
}

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 2020027 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))