Average Error: 0.5 → 0.4
Time: 9.8s
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 r335894 = x;
        double r335895 = 1.0;
        double r335896 = r335894 - r335895;
        double r335897 = sqrt(r335896);
        double r335898 = sqrt(r335894);
        double r335899 = r335897 * r335898;
        return r335899;
}
↓
double f(double x) {
        double r335900 = x;
        double r335901 = 0.5;
        double r335902 = 0.125;
        double r335903 = r335902 / r335900;
        double r335904 = r335901 + r335903;
        double r335905 = r335900 - r335904;
        return r335905;
}

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 2019135 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))