Average Error: 0.5 → 0.4
Time: 2.2s
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 r1760 = x;
        double r1761 = 1.0;
        double r1762 = r1760 - r1761;
        double r1763 = sqrt(r1762);
        double r1764 = sqrt(r1760);
        double r1765 = r1763 * r1764;
        return r1765;
}
↓
double f(double x) {
        double r1766 = x;
        double r1767 = 0.125;
        double r1768 = 1.0;
        double r1769 = r1768 / r1766;
        double r1770 = 0.5;
        double r1771 = fma(r1767, r1769, r1770);
        double r1772 = r1766 - r1771;
        return r1772;
}

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 2019354 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))