Average Error: 0.5 → 0.4
Time: 7.8s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - \frac{0.125}{x}\right) - 0.5\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - \frac{0.125}{x}\right) - 0.5
double f(double x) {
        double r11708 = x;
        double r11709 = 1.0;
        double r11710 = r11708 - r11709;
        double r11711 = sqrt(r11710);
        double r11712 = sqrt(r11708);
        double r11713 = r11711 * r11712;
        return r11713;
}
↓
double f(double x) {
        double r11714 = x;
        double r11715 = 0.125;
        double r11716 = r11715 / r11714;
        double r11717 = r11714 - r11716;
        double r11718 = 0.5;
        double r11719 = r11717 - r11718;
        return r11719;
}

Error

Bits error versus x

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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 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))