Average Error: 0.5 → 0.4
Time: 11.3s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + \frac{0.125}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \frac{0.125}{x}\right)
double f(double x) {
        double r11055 = x;
        double r11056 = 1.0;
        double r11057 = r11055 - r11056;
        double r11058 = sqrt(r11057);
        double r11059 = sqrt(r11055);
        double r11060 = r11058 * r11059;
        return r11060;
}
↓
double f(double x) {
        double r11061 = x;
        double r11062 = 0.5;
        double r11063 = 0.125;
        double r11064 = r11063 / r11061;
        double r11065 = r11062 + r11064;
        double r11066 = r11061 - r11065;
        return r11066;
}

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. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))