Average Error: 0.5 → 0.4
Time: 2.5s
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 r2125 = x;
        double r2126 = 1.0;
        double r2127 = r2125 - r2126;
        double r2128 = sqrt(r2127);
        double r2129 = sqrt(r2125);
        double r2130 = r2128 * r2129;
        return r2130;
}
↓
double f(double x) {
        double r2131 = x;
        double r2132 = 0.125;
        double r2133 = 1.0;
        double r2134 = r2133 / r2131;
        double r2135 = 0.5;
        double r2136 = fma(r2132, r2134, r2135);
        double r2137 = r2131 - r2136;
        return r2137;
}

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 2020089 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))