Average Error: 0.5 → 0.4
Time: 13.3s
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 r157700 = x;
        double r157701 = 1.0;
        double r157702 = r157700 - r157701;
        double r157703 = sqrt(r157702);
        double r157704 = sqrt(r157700);
        double r157705 = r157703 * r157704;
        return r157705;
}
↓
double f(double x) {
        double r157706 = x;
        double r157707 = -0.5;
        double r157708 = r157706 + r157707;
        double r157709 = -0.125;
        double r157710 = r157709 / r157706;
        double r157711 = r157708 + r157710;
        return r157711;
}

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 2019152 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))