Average Error: 0.5 → 0.4
Time: 7.9s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}
double f(double x) {
        double r132521 = x;
        double r132522 = 1.0;
        double r132523 = r132521 - r132522;
        double r132524 = sqrt(r132523);
        double r132525 = sqrt(r132521);
        double r132526 = r132524 * r132525;
        return r132526;
}
↓
double f(double x) {
        double r132527 = x;
        double r132528 = 0.125;
        double r132529 = r132528 / r132527;
        double r132530 = r132527 - r132529;
        double r132531 = 0.5;
        double r132532 = r132530 - r132531;
        return r132532;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))