Average Error: 0.5 → 0.4
Time: 7.7s
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 r199773 = x;
        double r199774 = 1.0;
        double r199775 = r199773 - r199774;
        double r199776 = sqrt(r199775);
        double r199777 = sqrt(r199773);
        double r199778 = r199776 * r199777;
        return r199778;
}
↓
double f(double x) {
        double r199779 = x;
        double r199780 = -0.5;
        double r199781 = 0.125;
        double r199782 = r199781 / r199779;
        double r199783 = r199780 - r199782;
        double r199784 = r199779 + r199783;
        return r199784;
}

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 2019130 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))