Average Error: 0.5 → 0.4
Time: 8.5s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(\frac{0.125}{x} + 0.5\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(\frac{0.125}{x} + 0.5\right)
double f(double x) {
        double r11713 = x;
        double r11714 = 1.0;
        double r11715 = r11713 - r11714;
        double r11716 = sqrt(r11715);
        double r11717 = sqrt(r11713);
        double r11718 = r11716 * r11717;
        return r11718;
}
↓
double f(double x) {
        double r11719 = x;
        double r11720 = 0.125;
        double r11721 = r11720 / r11719;
        double r11722 = 0.5;
        double r11723 = r11721 + r11722;
        double r11724 = r11719 - r11723;
        return r11724;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))