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Derivation

1. Split input into 2 regimes

2. if x < -11997.32666981334 or 11909.34130082208 < x

   1. Initial program 59.2

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

   3. Applied frac-sub60.9

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

   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. Simplified0.0

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

   if -11997.32666981334 < x < 11909.34130082208

   1. Initial program 0.1

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

   3. Applied frac-sub0.1

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

   5. Applied distribute-rgt-in0.1

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

   6. Applied associate--r+0.1

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

   7. Simplified0.1

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

   9. Applied flip-+0.1

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

   10. Applied associate-*l/0.1

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

   11. Applied associate-/r/0.1

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11997.32666981334 \lor \neg \left(x \le 11909.34130082208\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) \cdot x - \left(1 + x\right) \cdot x\right) - \left(1 + x\right)}{\left(x - 1\right) \cdot \left(x \cdot x - 1\right)} \cdot \left(x - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

Details

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