Average Error: 15.1 → 0.0
Time: 10.1s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2678435.1094305497:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 11907527.731338572:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2678435.1094305497:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 11907527.731338572:\\
\;\;\;\;\frac{x}{1 + x \cdot x}\\

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

\end{array}
double f(double x) {
        double r1125825 = x;
        double r1125826 = r1125825 * r1125825;
        double r1125827 = 1.0;
        double r1125828 = r1125826 + r1125827;
        double r1125829 = r1125825 / r1125828;
        return r1125829;
}
↓
double f(double x) {
        double r1125830 = x;
        double r1125831 = -2678435.1094305497;
        bool r1125832 = r1125830 <= r1125831;
        double r1125833 = 1.0;
        double r1125834 = r1125833 / r1125830;
        double r1125835 = r1125830 * r1125830;
        double r1125836 = r1125834 / r1125835;
        double r1125837 = r1125834 - r1125836;
        double r1125838 = 5.0;
        double r1125839 = pow(r1125830, r1125838);
        double r1125840 = r1125833 / r1125839;
        double r1125841 = r1125837 + r1125840;
        double r1125842 = 11907527.731338572;
        bool r1125843 = r1125830 <= r1125842;
        double r1125844 = r1125833 + r1125835;
        double r1125845 = r1125830 / r1125844;
        double r1125846 = r1125843 ? r1125845 : r1125841;
        double r1125847 = r1125832 ? r1125841 : r1125846;
        return r1125847;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.1
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2678435.1094305497 or 11907527.731338572 < x

    1. Initial program 30.4

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -2678435.1094305497 < x < 11907527.731338572

    1. Initial program 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2678435.1094305497:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 11907527.731338572:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))