Average Error: 0.5 → 0.4
Time: 6.2s
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 r67628 = x;
        double r67629 = 1.0;
        double r67630 = r67628 - r67629;
        double r67631 = sqrt(r67630);
        double r67632 = sqrt(r67628);
        double r67633 = r67631 * r67632;
        return r67633;
}
↓
double f(double x) {
        double r67634 = x;
        double r67635 = 0.5;
        double r67636 = -0.125;
        double r67637 = r67636 / r67634;
        double r67638 = r67635 - r67637;
        double r67639 = r67634 - r67638;
        return r67639;
}

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

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))