Average Error: 0.5 → 0.3
Time: 7.6s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x + \left(\frac{-1}{2} - \frac{\frac{1}{8}}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x + \left(\frac{-1}{2} - \frac{\frac{1}{8}}{x}\right)
double f(double x) {
        double r197640 = x;
        double r197641 = 1.0;
        double r197642 = r197640 - r197641;
        double r197643 = sqrt(r197642);
        double r197644 = sqrt(r197640);
        double r197645 = r197643 * r197644;
        return r197645;
}
↓
double f(double x) {
        double r197646 = x;
        double r197647 = -0.5;
        double r197648 = 0.125;
        double r197649 = r197648 / r197646;
        double r197650 = r197647 - r197649;
        double r197651 = r197646 + r197650;
        return r197651;
}

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

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))