Average Error: 0.5 → 0.4
Time: 13.3s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x + \frac{-1}{2}\right) + \frac{\frac{-1}{8}}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x + \frac{-1}{2}\right) + \frac{\frac{-1}{8}}{x}
double f(double x) {
        double r158334 = x;
        double r158335 = 1.0;
        double r158336 = r158334 - r158335;
        double r158337 = sqrt(r158336);
        double r158338 = sqrt(r158334);
        double r158339 = r158337 * r158338;
        return r158339;
}
↓
double f(double x) {
        double r158340 = x;
        double r158341 = -0.5;
        double r158342 = r158340 + r158341;
        double r158343 = -0.125;
        double r158344 = r158343 / r158340;
        double r158345 = r158342 + r158344;
        return r158345;
}

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

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))