Average Error: 0.5 → 0.4
Time: 9.7s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(\frac{-1}{2} + x\right) - \frac{\frac{1}{8}}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(\frac{-1}{2} + x\right) - \frac{\frac{1}{8}}{x}
double f(double x) {
        double r179294 = x;
        double r179295 = 1.0;
        double r179296 = r179294 - r179295;
        double r179297 = sqrt(r179296);
        double r179298 = sqrt(r179294);
        double r179299 = r179297 * r179298;
        return r179299;
}
↓
double f(double x) {
        double r179300 = -0.5;
        double r179301 = x;
        double r179302 = r179300 + r179301;
        double r179303 = 0.125;
        double r179304 = r179303 / r179301;
        double r179305 = r179302 - r179304;
        return r179305;
}

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

    \[\leadsto \color{blue}{\left(x + \frac{-1}{2}\right) - \frac{\frac{1}{8}}{x}}\]

  4. Final simplification0.4

    \[\leadsto \left(\frac{-1}{2} + x\right) - \frac{\frac{1}{8}}{x}\]

Reproduce

herbie shell --seed 2019155 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))