Average Error: 0.5 → 0.3
Time: 18.4s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x + \left(\frac{-1}{2} - \frac{\frac{1}{8}}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x + \left(\frac{-1}{2} - \frac{\frac{1}{8}}{x}\right)
double f(double x) {
        double r556837 = x;
        double r556838 = 1.0;
        double r556839 = r556837 - r556838;
        double r556840 = sqrt(r556839);
        double r556841 = sqrt(r556837);
        double r556842 = r556840 * r556841;
        return r556842;
}
↓
double f(double x) {
        double r556843 = x;
        double r556844 = -0.5;
        double r556845 = 0.125;
        double r556846 = r556845 / r556843;
        double r556847 = r556844 - r556846;
        double r556848 = r556843 + r556847;
        return r556848;
}

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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019146 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))