Average Error: 0.5 → 0.4
Time: 8.4s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - 0.5\right) - \frac{0.125}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - 0.5\right) - \frac{0.125}{x}
double f(double x) {
        double r9402 = x;
        double r9403 = 1.0;
        double r9404 = r9402 - r9403;
        double r9405 = sqrt(r9404);
        double r9406 = sqrt(r9402);
        double r9407 = r9405 * r9406;
        return r9407;
}
↓
double f(double x) {
        double r9408 = x;
        double r9409 = 0.5;
        double r9410 = r9408 - r9409;
        double r9411 = 0.125;
        double r9412 = r9411 / r9408;
        double r9413 = r9410 - r9412;
        return r9413;
}

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

    \[\leadsto \color{blue}{x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\left(x - 0.5\right) - \frac{0.125}{x}}\]

  4. Final simplification0.4

    \[\leadsto \left(x - 0.5\right) - \frac{0.125}{x}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))