Average Error: 0.5 → 0.3
Time: 9.0s
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 r164843 = x;
        double r164844 = 1.0;
        double r164845 = r164843 - r164844;
        double r164846 = sqrt(r164845);
        double r164847 = sqrt(r164843);
        double r164848 = r164846 * r164847;
        return r164848;
}
↓
double f(double x) {
        double r164849 = x;
        double r164850 = 0.5;
        double r164851 = -0.125;
        double r164852 = r164851 / r164849;
        double r164853 = r164850 - r164852;
        double r164854 = r164849 - r164853;
        return r164854;
}

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 2019119 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))