Average Error: 0.5 → 0.4
Time: 6.9s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(\frac{-1}{2} + x\right) + \frac{\frac{-1}{8}}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(\frac{-1}{2} + x\right) + \frac{\frac{-1}{8}}{x}
double f(double x) {
        double r164583 = x;
        double r164584 = 1.0;
        double r164585 = r164583 - r164584;
        double r164586 = sqrt(r164585);
        double r164587 = sqrt(r164583);
        double r164588 = r164586 * r164587;
        return r164588;
}
↓
double f(double x) {
        double r164589 = -0.5;
        double r164590 = x;
        double r164591 = r164589 + r164590;
        double r164592 = -0.125;
        double r164593 = r164592 / r164590;
        double r164594 = r164591 + r164593;
        return r164594;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))