Average Error: 0.5 → 0.4
Time: 8.7s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + \frac{0.125}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \frac{0.125}{x}\right)
double f(double x) {
        double r357608 = x;
        double r357609 = 1.0;
        double r357610 = r357608 - r357609;
        double r357611 = sqrt(r357610);
        double r357612 = sqrt(r357608);
        double r357613 = r357611 * r357612;
        return r357613;
}
↓
double f(double x) {
        double r357614 = x;
        double r357615 = 0.5;
        double r357616 = 0.125;
        double r357617 = r357616 / r357614;
        double r357618 = r357615 + r357617;
        double r357619 = r357614 - r357618;
        return r357619;
}

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

    \[\leadsto \color{blue}{x - \left(0.125 \cdot \frac{1}{x} + 0.5\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{x - \left(0.5 + \frac{0.125}{x}\right)}\]

  4. Final simplification0.4

    \[\leadsto x - \left(0.5 + \frac{0.125}{x}\right)\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))