Average Error: 0.5 → 0.4
Time: 11.0s
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 r11074 = x;
        double r11075 = 1.0;
        double r11076 = r11074 - r11075;
        double r11077 = sqrt(r11076);
        double r11078 = sqrt(r11074);
        double r11079 = r11077 * r11078;
        return r11079;
}
↓
double f(double x) {
        double r11080 = x;
        double r11081 = 0.5;
        double r11082 = 0.125;
        double r11083 = r11082 / r11080;
        double r11084 = r11081 + r11083;
        double r11085 = r11080 - r11084;
        return r11085;
}

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 2019326 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))