Average Error: 0.5 → 0.4
Time: 1.9s
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 r1749 = x;
        double r1750 = 1.0;
        double r1751 = r1749 - r1750;
        double r1752 = sqrt(r1751);
        double r1753 = sqrt(r1749);
        double r1754 = r1752 * r1753;
        return r1754;
}
↓
double f(double x) {
        double r1755 = x;
        double r1756 = 0.125;
        double r1757 = 1.0;
        double r1758 = r1757 / r1755;
        double r1759 = 0.5;
        double r1760 = fma(r1756, r1758, r1759);
        double r1761 = r1755 - r1760;
        return r1761;
}

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 2020057 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))