Average Error: 0.5 → 0.4
Time: 8.1s
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 r10618 = x;
        double r10619 = 1.0;
        double r10620 = r10618 - r10619;
        double r10621 = sqrt(r10620);
        double r10622 = sqrt(r10618);
        double r10623 = r10621 * r10622;
        return r10623;
}
↓
double f(double x) {
        double r10624 = x;
        double r10625 = 0.5;
        double r10626 = 0.125;
        double r10627 = r10626 / r10624;
        double r10628 = r10625 + r10627;
        double r10629 = r10624 - r10628;
        return r10629;
}

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

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

  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 2019235 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))