Average Error: 0.5 → 0.3
Time: 17.5s
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 r468906 = x;
        double r468907 = 1.0;
        double r468908 = r468906 - r468907;
        double r468909 = sqrt(r468908);
        double r468910 = sqrt(r468906);
        double r468911 = r468909 * r468910;
        return r468911;
}
↓
double f(double x) {
        double r468912 = x;
        double r468913 = -0.125;
        double r468914 = r468913 / r468912;
        double r468915 = r468912 + r468914;
        double r468916 = 0.5;
        double r468917 = r468915 - r468916;
        return r468917;
}

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

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))