Average Error: 0.5 → 0.4
Time: 13.0s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - 0.5\right) - \frac{0.125}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - 0.5\right) - \frac{0.125}{x}
double f(double x) {
        double r11075 = x;
        double r11076 = 1.0;
        double r11077 = r11075 - r11076;
        double r11078 = sqrt(r11077);
        double r11079 = sqrt(r11075);
        double r11080 = r11078 * r11079;
        return r11080;
}
↓
double f(double x) {
        double r11081 = x;
        double r11082 = 0.5;
        double r11083 = r11081 - r11082;
        double r11084 = 0.125;
        double r11085 = r11084 / r11081;
        double r11086 = r11083 - r11085;
        return r11086;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))