Average Error: 15.3 → 0.0
Time: 8.5s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7852.73119197666:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 226367535.9280359:\\ \;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7852.73119197666:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x}\\

\mathbf{elif}\;x \le 226367535.9280359:\\
\;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + -1}\\

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

\end{array}
double f(double x) {
        double r926879 = x;
        double r926880 = r926879 * r926879;
        double r926881 = 1.0;
        double r926882 = r926880 + r926881;
        double r926883 = r926879 / r926882;
        return r926883;
}
↓
double f(double x) {
        double r926884 = x;
        double r926885 = -7852.73119197666;
        bool r926886 = r926884 <= r926885;
        double r926887 = 1.0;
        double r926888 = 5.0;
        double r926889 = pow(r926884, r926888);
        double r926890 = r926887 / r926889;
        double r926891 = r926887 / r926884;
        double r926892 = r926884 * r926884;
        double r926893 = r926891 / r926892;
        double r926894 = r926890 - r926893;
        double r926895 = r926894 + r926891;
        double r926896 = 226367535.9280359;
        bool r926897 = r926884 <= r926896;
        double r926898 = r926892 - r926887;
        double r926899 = r926892 * r926892;
        double r926900 = -1.0;
        double r926901 = r926899 + r926900;
        double r926902 = r926884 / r926901;
        double r926903 = r926898 * r926902;
        double r926904 = r926897 ? r926903 : r926895;
        double r926905 = r926886 ? r926895 : r926904;
        return r926905;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7852.73119197666 or 226367535.9280359 < x

    1. Initial program 31.2

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

    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

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

    if -7852.73119197666 < x < 226367535.9280359

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

    3. Applied flip-+0.0

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

    4. Applied associate-/r/0.0

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + -1}} \cdot \left(x \cdot x - 1\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7852.73119197666:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 226367535.9280359:\\ \;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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