Average Error: 0.5 → 0.4
Time: 13.1s
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 r158349 = x;
        double r158350 = 1.0;
        double r158351 = r158349 - r158350;
        double r158352 = sqrt(r158351);
        double r158353 = sqrt(r158349);
        double r158354 = r158352 * r158353;
        return r158354;
}
↓
double f(double x) {
        double r158355 = x;
        double r158356 = -0.5;
        double r158357 = r158355 + r158356;
        double r158358 = -0.125;
        double r158359 = r158358 / r158355;
        double r158360 = r158357 + r158359;
        return r158360;
}

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 2019149 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))