Average Error: 0.5 → 0.4
Time: 16.8s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\frac{\frac{-1}{8}}{x} + \left(x + \frac{-1}{2}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\frac{\frac{-1}{8}}{x} + \left(x + \frac{-1}{2}\right)
double f(double x) {
        double r711615 = x;
        double r711616 = 1.0;
        double r711617 = r711615 - r711616;
        double r711618 = sqrt(r711617);
        double r711619 = sqrt(r711615);
        double r711620 = r711618 * r711619;
        return r711620;
}
↓
double f(double x) {
        double r711621 = -0.125;
        double r711622 = x;
        double r711623 = r711621 / r711622;
        double r711624 = -0.5;
        double r711625 = r711622 + r711624;
        double r711626 = r711623 + r711625;
        return r711626;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))