Average Error: 0.5 → 0.4
Time: 8.2s
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 r134319 = x;
        double r134320 = 1.0;
        double r134321 = r134319 - r134320;
        double r134322 = sqrt(r134321);
        double r134323 = sqrt(r134319);
        double r134324 = r134322 * r134323;
        return r134324;
}
↓
double f(double x) {
        double r134325 = x;
        double r134326 = 0.5;
        double r134327 = -0.125;
        double r134328 = r134327 / r134325;
        double r134329 = r134326 - r134328;
        double r134330 = r134325 - r134329;
        return r134330;
}

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 2019120 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))