Average Error: 0.5 → 0.3
Time: 15.1s
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 r398327 = x;
        double r398328 = 1.0;
        double r398329 = r398327 - r398328;
        double r398330 = sqrt(r398329);
        double r398331 = sqrt(r398327);
        double r398332 = r398330 * r398331;
        return r398332;
}
↓
double f(double x) {
        double r398333 = x;
        double r398334 = 0.5;
        double r398335 = r398333 - r398334;
        double r398336 = 0.125;
        double r398337 = r398336 / r398333;
        double r398338 = r398335 - r398337;
        return r398338;
}

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 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019200 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))