Average Error: 0.5 → 0.4
Time: 8.6s
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 r174463 = x;
        double r174464 = 1.0;
        double r174465 = r174463 - r174464;
        double r174466 = sqrt(r174465);
        double r174467 = sqrt(r174463);
        double r174468 = r174466 * r174467;
        return r174468;
}
↓
double f(double x) {
        double r174469 = x;
        double r174470 = 0.5;
        double r174471 = -0.125;
        double r174472 = r174471 / r174469;
        double r174473 = r174470 - r174472;
        double r174474 = r174469 - r174473;
        return r174474;
}

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 2019129 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))