Average Error: 0.5 → 0.4
Time: 6.8s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(\frac{-1}{2} + x\right) - \frac{\frac{1}{8}}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(\frac{-1}{2} + x\right) - \frac{\frac{1}{8}}{x}
double f(double x) {
        double r139785 = x;
        double r139786 = 1.0;
        double r139787 = r139785 - r139786;
        double r139788 = sqrt(r139787);
        double r139789 = sqrt(r139785);
        double r139790 = r139788 * r139789;
        return r139790;
}
↓
double f(double x) {
        double r139791 = -0.5;
        double r139792 = x;
        double r139793 = r139791 + r139792;
        double r139794 = 0.125;
        double r139795 = r139794 / r139792;
        double r139796 = r139793 - r139795;
        return r139796;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))