Average Error: 0.5 → 0.3
Time: 8.4s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}
double f(double x) {
        double r172298 = x;
        double r172299 = 1.0;
        double r172300 = r172298 - r172299;
        double r172301 = sqrt(r172300);
        double r172302 = sqrt(r172298);
        double r172303 = r172301 * r172302;
        return r172303;
}
↓
double f(double x) {
        double r172304 = x;
        double r172305 = 0.125;
        double r172306 = r172305 / r172304;
        double r172307 = r172304 - r172306;
        double r172308 = 0.5;
        double r172309 = r172307 - r172308;
        return r172309;
}

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 0.3

    \[\leadsto \color{blue}{x - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}}\]

  4. Final simplification0.3

    \[\leadsto \left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))