\[\sqrt{x - 1} \cdot \sqrt{x}\]

↓

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

double f(double x) {
        double r7177341 = x;
        double r7177342 = 1.0;
        double r7177343 = r7177341 - r7177342;
        double r7177344 = sqrt(r7177343);
        double r7177345 = sqrt(r7177341);
        double r7177346 = r7177344 * r7177345;
        return r7177346;
}

↓

double f(double x) {
        double r7177347 = x;
        double r7177348 = 0.5;
        double r7177349 = -0.125;
        double r7177350 = r7177349 / r7177347;
        double r7177351 = r7177348 - r7177350;
        double r7177352 = r7177347 - r7177351;
        return r7177352;
}

\sqrt{x - 1} \cdot \sqrt{x}

↓

x - \left(\frac{1}{2} - \frac{\frac{-1}{8}}{x}\right)

Derivation

Initial program 0.5
\[\sqrt{x - 1} \cdot \sqrt{x}\]
Taylor expanded around inf 0.4
\[\leadsto \color{blue}{x - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)}\]
Simplified0.4
\[\leadsto \color{blue}{x - \left(\frac{1}{2} - \frac{\frac{-1}{8}}{x}\right)}\]
Final simplification0.4
\[\leadsto x - \left(\frac{1}{2} - \frac{\frac{-1}{8}}{x}\right)\]

Reproduce

herbie shell --seed 2019101 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))

Error

Derivation

Reproduce