Average Error: 0.5 → 0.4
Time: 10.0s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - 0.5\right) - \frac{0.125}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - 0.5\right) - \frac{0.125}{x}
double f(double x) {
        double r10568 = x;
        double r10569 = 1.0;
        double r10570 = r10568 - r10569;
        double r10571 = sqrt(r10570);
        double r10572 = sqrt(r10568);
        double r10573 = r10571 * r10572;
        return r10573;
}
↓
double f(double x) {
        double r10574 = x;
        double r10575 = 0.5;
        double r10576 = r10574 - r10575;
        double r10577 = 0.125;
        double r10578 = r10577 / r10574;
        double r10579 = r10576 - r10578;
        return r10579;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))