Average Error: 29.3 → 0.1
Time: 4.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10207.34946242376827285625040531158447266 \lor \neg \left(x \le 8646.459775303861533757299184799194335938\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1} - \left(x + 1\right) \cdot \frac{1}{x - 1}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10207.34946242376827285625040531158447266 \lor \neg \left(x \le 8646.459775303861533757299184799194335938\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r96147 = x;
        double r96148 = 1.0;
        double r96149 = r96147 + r96148;
        double r96150 = r96147 / r96149;
        double r96151 = r96147 - r96148;
        double r96152 = r96149 / r96151;
        double r96153 = r96150 - r96152;
        return r96153;
}
↓
double f(double x) {
        double r96154 = x;
        double r96155 = -10207.349462423768;
        bool r96156 = r96154 <= r96155;
        double r96157 = 8646.459775303862;
        bool r96158 = r96154 <= r96157;
        double r96159 = !r96158;
        bool r96160 = r96156 || r96159;
        double r96161 = 1.0;
        double r96162 = -r96161;
        double r96163 = 2.0;
        double r96164 = pow(r96154, r96163);
        double r96165 = r96162 / r96164;
        double r96166 = 3.0;
        double r96167 = r96166 / r96154;
        double r96168 = r96165 - r96167;
        double r96169 = 3.0;
        double r96170 = pow(r96154, r96169);
        double r96171 = r96166 / r96170;
        double r96172 = r96168 - r96171;
        double r96173 = r96154 + r96161;
        double r96174 = r96154 / r96173;
        double r96175 = 1.0;
        double r96176 = r96154 - r96161;
        double r96177 = r96175 / r96176;
        double r96178 = r96173 * r96177;
        double r96179 = r96174 - r96178;
        double r96180 = exp(r96179);
        double r96181 = log(r96180);
        double r96182 = r96160 ? r96172 : r96181;
        return r96182;
}

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 < -10207.349462423768 or 8646.459775303862 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -10207.349462423768 < x < 8646.459775303862

    1. Initial program 0.1

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

    3. Applied div-inv0.1

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\left(x + 1\right) \cdot \frac{1}{x - 1}}\]
    4. Using strategy rm

    5. Applied add-log-exp0.1

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

    6. Applied add-log-exp0.1

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

    7. Applied diff-log0.1

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

    8. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10207.34946242376827285625040531158447266 \lor \neg \left(x \le 8646.459775303861533757299184799194335938\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1} - \left(x + 1\right) \cdot \frac{1}{x - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))