Average Error: 0.5 → 0.4
Time: 15.2s
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 r244849 = x;
        double r244850 = 1.0;
        double r244851 = r244849 - r244850;
        double r244852 = sqrt(r244851);
        double r244853 = sqrt(r244849);
        double r244854 = r244852 * r244853;
        return r244854;
}
↓
double f(double x) {
        double r244855 = x;
        double r244856 = -0.5;
        double r244857 = r244855 + r244856;
        double r244858 = -0.125;
        double r244859 = r244858 / r244855;
        double r244860 = r244857 + r244859;
        return r244860;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))