Error

Derivation

1. Split input into 2 regimes

2. if x < -11443.841380881175 or 14257.434820408032 < x

   1. Initial program 59.3

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

   3. Applied div-inv59.4

      \[\leadsto \color{blue}{x \cdot \frac{1}{x + 1}} - \frac{x + 1}{x - 1}\]

   4. Taylor expanded around inf 0.3

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

   5. Simplified0.3

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

   6. Taylor expanded around -inf 0.3

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

   7. Simplified0.0

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

   if -11443.841380881175 < x < 14257.434820408032

   1. Initial program 0.1

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

   3. Applied div-inv0.1

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11443.841380881175:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 14257.434820408032:\\ \;\;\;\;x \cdot \frac{1}{1 + x} - \frac{1 + x}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))