Average Error: 0.5 → 0.3
Time: 18.5s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x + \frac{-1}{2}\right) + \frac{\frac{-1}{8}}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x + \frac{-1}{2}\right) + \frac{\frac{-1}{8}}{x}
double f(double x) {
        double r364572 = x;
        double r364573 = 1.0;
        double r364574 = r364572 - r364573;
        double r364575 = sqrt(r364574);
        double r364576 = sqrt(r364572);
        double r364577 = r364575 * r364576;
        return r364577;
}
↓
double f(double x) {
        double r364578 = x;
        double r364579 = -0.5;
        double r364580 = r364578 + r364579;
        double r364581 = -0.125;
        double r364582 = r364581 / r364578;
        double r364583 = r364580 + r364582;
        return r364583;
}

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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))