Average Error: 0.5 → 0.4
Time: 13.7s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}
double f(double x) {
        double r16831 = x;
        double r16832 = 1.0;
        double r16833 = r16831 - r16832;
        double r16834 = sqrt(r16833);
        double r16835 = sqrt(r16831);
        double r16836 = r16834 * r16835;
        return r16836;
}
↓
double f(double x) {
        double r16837 = x;
        double r16838 = 1.0;
        double r16839 = 8.0;
        double r16840 = r16838 / r16839;
        double r16841 = r16840 / r16837;
        double r16842 = r16837 - r16841;
        double r16843 = 2.0;
        double r16844 = r16838 / r16843;
        double r16845 = r16842 - r16844;
        return r16845;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))