Average Error: 29.7 → 0.1
Time: 13.8s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9516.22025087149:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 11259.474582447798:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9516.22025087149:\\
\;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\

\mathbf{elif}\;x \le 11259.474582447798:\\
\;\;\;\;\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\

\end{array}
double f(double x) {
        double r2250780 = x;
        double r2250781 = 1.0;
        double r2250782 = r2250780 + r2250781;
        double r2250783 = r2250780 / r2250782;
        double r2250784 = r2250780 - r2250781;
        double r2250785 = r2250782 / r2250784;
        double r2250786 = r2250783 - r2250785;
        return r2250786;
}
↓
double f(double x) {
        double r2250787 = x;
        double r2250788 = -9516.22025087149;
        bool r2250789 = r2250787 <= r2250788;
        double r2250790 = -3.0;
        double r2250791 = r2250787 * r2250787;
        double r2250792 = r2250791 * r2250787;
        double r2250793 = r2250790 / r2250792;
        double r2250794 = 1.0;
        double r2250795 = r2250794 / r2250791;
        double r2250796 = 3.0;
        double r2250797 = r2250796 / r2250787;
        double r2250798 = r2250795 + r2250797;
        double r2250799 = r2250793 - r2250798;
        double r2250800 = 11259.474582447798;
        bool r2250801 = r2250787 <= r2250800;
        double r2250802 = r2250787 + r2250794;
        double r2250803 = r2250787 / r2250802;
        double r2250804 = r2250787 - r2250794;
        double r2250805 = r2250802 / r2250804;
        double r2250806 = r2250803 - r2250805;
        double r2250807 = exp(r2250806);
        double r2250808 = log(r2250807);
        double r2250809 = r2250801 ? r2250808 : r2250799;
        double r2250810 = r2250789 ? r2250799 : r2250809;
        return r2250810;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -9516.22025087149 or 11259.474582447798 < x

    1. Initial program 59.3

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

    2. Taylor expanded around -inf 0.3

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.0

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

    if -9516.22025087149 < x < 11259.474582447798

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied add-log-exp0.1

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

    5. Applied diff-log0.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9516.22025087149:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 11259.474582447798:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))