Average Error: 0.5 → 0.3
Time: 13.4s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - 0.5\right) - \frac{0.125}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - 0.5\right) - \frac{0.125}{x}
double f(double x) {
        double r334815 = x;
        double r334816 = 1.0;
        double r334817 = r334815 - r334816;
        double r334818 = sqrt(r334817);
        double r334819 = sqrt(r334815);
        double r334820 = r334818 * r334819;
        return r334820;
}
↓
double f(double x) {
        double r334821 = x;
        double r334822 = 0.5;
        double r334823 = r334821 - r334822;
        double r334824 = 0.125;
        double r334825 = r334824 / r334821;
        double r334826 = r334823 - r334825;
        return r334826;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  2. Taylor expanded around inf 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \left(x - 0.5\right) - \frac{0.125}{x}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))