Average Error: 0.5 → 0.4
Time: 6.8s
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 r66587 = x;
        double r66588 = 1.0;
        double r66589 = r66587 - r66588;
        double r66590 = sqrt(r66589);
        double r66591 = sqrt(r66587);
        double r66592 = r66590 * r66591;
        return r66592;
}
↓
double f(double x) {
        double r66593 = x;
        double r66594 = 0.5;
        double r66595 = -0.125;
        double r66596 = r66595 / r66593;
        double r66597 = r66594 - r66596;
        double r66598 = r66593 - r66597;
        return r66598;
}

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 2019121 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))