Average Error: 0.5 → 0.4
Time: 13.8s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\frac{-1}{2} + \left(x - \frac{\frac{1}{8}}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\frac{-1}{2} + \left(x - \frac{\frac{1}{8}}{x}\right)
double f(double x) {
        double r208186 = x;
        double r208187 = 1.0;
        double r208188 = r208186 - r208187;
        double r208189 = sqrt(r208188);
        double r208190 = sqrt(r208186);
        double r208191 = r208189 * r208190;
        return r208191;
}
↓
double f(double x) {
        double r208192 = -0.5;
        double r208193 = x;
        double r208194 = 0.125;
        double r208195 = r208194 / r208193;
        double r208196 = r208193 - r208195;
        double r208197 = r208192 + r208196;
        return r208197;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))