Average Error: 0.5 → 0.4
Time: 6.4s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(\frac{1}{2} + \frac{\frac{1}{8}}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(\frac{1}{2} + \frac{\frac{1}{8}}{x}\right)
double f(double x) {
        double r79363 = x;
        double r79364 = 1.0;
        double r79365 = r79363 - r79364;
        double r79366 = sqrt(r79365);
        double r79367 = sqrt(r79363);
        double r79368 = r79366 * r79367;
        return r79368;
}
↓
double f(double x) {
        double r79369 = x;
        double r79370 = 0.5;
        double r79371 = 0.125;
        double r79372 = r79371 / r79369;
        double r79373 = r79370 + r79372;
        double r79374 = r79369 - r79373;
        return r79374;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))