Average Error: 0.5 → 0.5
Time: 2.9s
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 r9250 = x;
        double r9251 = 1.0;
        double r9252 = r9250 - r9251;
        double r9253 = sqrt(r9252);
        double r9254 = sqrt(r9250);
        double r9255 = r9253 * r9254;
        return r9255;
}
↓
double f(double x) {
        double r9256 = x;
        double r9257 = 0.125;
        double r9258 = 1.0;
        double r9259 = r9258 / r9256;
        double r9260 = 0.5;
        double r9261 = fma(r9257, r9259, r9260);
        double r9262 = r9256 - r9261;
        return r9262;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.5

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

  2. Taylor expanded around inf 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))