Average Error: 0.5 → 0.3
Time: 9.9s
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 r15582 = x;
        double r15583 = 1.0;
        double r15584 = r15582 - r15583;
        double r15585 = sqrt(r15584);
        double r15586 = sqrt(r15582);
        double r15587 = r15585 * r15586;
        return r15587;
}
↓
double f(double x) {
        double r15588 = x;
        double r15589 = 0.5;
        double r15590 = 0.125;
        double r15591 = r15590 / r15588;
        double r15592 = r15589 + r15591;
        double r15593 = r15588 - r15592;
        return r15593;
}

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 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019212 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))