Average Error: 0.5 → 0.4
Time: 10.7s
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 r329424 = x;
        double r329425 = 1.0;
        double r329426 = r329424 - r329425;
        double r329427 = sqrt(r329426);
        double r329428 = sqrt(r329424);
        double r329429 = r329427 * r329428;
        return r329429;
}
↓
double f(double x) {
        double r329430 = x;
        double r329431 = 0.5;
        double r329432 = 0.125;
        double r329433 = r329432 / r329430;
        double r329434 = r329431 + r329433;
        double r329435 = r329430 - r329434;
        return r329435;
}

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 2019138 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))