Average Error: 14.8 → 0.0
Time: 5.3s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -57137068010548903280640:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 8124.998191315608892182353883981704711914:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -57137068010548903280640:\\
\;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\

\mathbf{elif}\;x \le 8124.998191315608892182353883981704711914:\\
\;\;\;\;\frac{x}{1 + x \cdot x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\

\end{array}
double f(double x) {
        double r2435916 = x;
        double r2435917 = r2435916 * r2435916;
        double r2435918 = 1.0;
        double r2435919 = r2435917 + r2435918;
        double r2435920 = r2435916 / r2435919;
        return r2435920;
}
↓
double f(double x) {
        double r2435921 = x;
        double r2435922 = -5.71370680105489e+22;
        bool r2435923 = r2435921 <= r2435922;
        double r2435924 = 1.0;
        double r2435925 = r2435924 / r2435921;
        double r2435926 = 1.0;
        double r2435927 = 5.0;
        double r2435928 = pow(r2435921, r2435927);
        double r2435929 = r2435926 / r2435928;
        double r2435930 = r2435921 * r2435921;
        double r2435931 = r2435930 * r2435921;
        double r2435932 = r2435926 / r2435931;
        double r2435933 = r2435929 - r2435932;
        double r2435934 = r2435925 + r2435933;
        double r2435935 = 8124.998191315609;
        bool r2435936 = r2435921 <= r2435935;
        double r2435937 = r2435926 + r2435930;
        double r2435938 = r2435921 / r2435937;
        double r2435939 = r2435936 ? r2435938 : r2435934;
        double r2435940 = r2435923 ? r2435934 : r2435939;
        return r2435940;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.8
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.71370680105489e+22 or 8124.998191315609 < x

    1. Initial program 30.6

      \[\frac{x}{x \cdot x + 1}\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{{x}^{5}} + \frac{1}{x}\right) - 1 \cdot \frac{1}{{x}^{3}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right) + \frac{1}{x}}\]

    if -5.71370680105489e+22 < x < 8124.998191315609

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -57137068010548903280640:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 8124.998191315608892182353883981704711914:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))