Average Error: 0.5 → 0.4
Time: 7.4s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + \frac{0.125}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \frac{0.125}{x}\right)
double f(double x) {
        double r9358 = x;
        double r9359 = 1.0;
        double r9360 = r9358 - r9359;
        double r9361 = sqrt(r9360);
        double r9362 = sqrt(r9358);
        double r9363 = r9361 * r9362;
        return r9363;
}
↓
double f(double x) {
        double r9364 = x;
        double r9365 = 0.5;
        double r9366 = 0.125;
        double r9367 = r9366 / r9364;
        double r9368 = r9365 + r9367;
        double r9369 = r9364 - r9368;
        return r9369;
}

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

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

  4. Final simplification0.4

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

Reproduce

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