Average Error: 0.5 → 0.4
Time: 10.2s
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 r147839 = x;
        double r147840 = 1.0;
        double r147841 = r147839 - r147840;
        double r147842 = sqrt(r147841);
        double r147843 = sqrt(r147839);
        double r147844 = r147842 * r147843;
        return r147844;
}
↓
double f(double x) {
        double r147845 = x;
        double r147846 = -0.5;
        double r147847 = 0.125;
        double r147848 = r147847 / r147845;
        double r147849 = r147846 - r147848;
        double r147850 = r147845 + r147849;
        return r147850;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019144 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))