Average Error: 0.5 → 0.4
Time: 12.0s
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 r11760 = x;
        double r11761 = 1.0;
        double r11762 = r11760 - r11761;
        double r11763 = sqrt(r11762);
        double r11764 = sqrt(r11760);
        double r11765 = r11763 * r11764;
        return r11765;
}
↓
double f(double x) {
        double r11766 = x;
        double r11767 = 0.125;
        double r11768 = r11767 / r11766;
        double r11769 = r11766 - r11768;
        double r11770 = 0.5;
        double r11771 = r11769 - r11770;
        return r11771;
}

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 0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019294 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))