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Derivation

1. Split input into 2 regimes

2. if x < -11360.435045225768 or 11442.78162855777 < x

   1. Initial program 59.2

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

   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

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

   if -11360.435045225768 < x < 11442.78162855777

   1. Initial program 0.1

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

   3. Applied flip3--0.1

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

   4. Applied associate-/r/0.1

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

   5. Simplified0.1

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

   7. Applied associate-*l/0.1

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

   8. Applied frac-sub0.1

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

   9. Simplified0.1

      \[\leadsto \frac{\color{blue}{\left({x}^{4} + \left(-x\right)\right) - \left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(1 + x\right) + x \cdot x\right)}}{\left(x + 1\right) \cdot \left({x}^{3} - {1}^{3}\right)}\]
   10. Using strategy rm

   11. Applied flip3-+0.1

       \[\leadsto \frac{\left({x}^{4} + \left(-x\right)\right) - \left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \color{blue}{\frac{{\left(1 + x\right)}^{3} + {\left(x \cdot x\right)}^{3}}{\left(1 + x\right) \cdot \left(1 + x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(1 + x\right) \cdot \left(x \cdot x\right)\right)}}}{\left(x + 1\right) \cdot \left({x}^{3} - {1}^{3}\right)}\]

   12. Applied flip-+0.1

       \[\leadsto \frac{\left({x}^{4} + \left(-x\right)\right) - \left(\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}} \cdot \left(1 + x\right)\right) \cdot \frac{{\left(1 + x\right)}^{3} + {\left(x \cdot x\right)}^{3}}{\left(1 + x\right) \cdot \left(1 + x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(1 + x\right) \cdot \left(x \cdot x\right)\right)}}{\left(x + 1\right) \cdot \left({x}^{3} - {1}^{3}\right)}\]

   13. Applied associate-*l/0.1

       \[\leadsto \frac{\left({x}^{4} + \left(-x\right)\right) - \color{blue}{\frac{\left(1 \cdot 1 - x \cdot x\right) \cdot \left(1 + x\right)}{1 - x}} \cdot \frac{{\left(1 + x\right)}^{3} + {\left(x \cdot x\right)}^{3}}{\left(1 + x\right) \cdot \left(1 + x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(1 + x\right) \cdot \left(x \cdot x\right)\right)}}{\left(x + 1\right) \cdot \left({x}^{3} - {1}^{3}\right)}\]

   14. Applied frac-times0.1

       \[\leadsto \frac{\left({x}^{4} + \left(-x\right)\right) - \color{blue}{\frac{\left(\left(1 \cdot 1 - x \cdot x\right) \cdot \left(1 + x\right)\right) \cdot \left({\left(1 + x\right)}^{3} + {\left(x \cdot x\right)}^{3}\right)}{\left(1 - x\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(1 + x\right) \cdot \left(x \cdot x\right)\right)\right)}}}{\left(x + 1\right) \cdot \left({x}^{3} - {1}^{3}\right)}\]

   15. Simplified0.1

       \[\leadsto \frac{\left({x}^{4} + \left(-x\right)\right) - \frac{\color{blue}{\left({\left(1 + x\right)}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \left(\left(1 - x \cdot x\right) \cdot \left(1 + x\right)\right)}}{\left(1 - x\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(1 + x\right) \cdot \left(x \cdot x\right)\right)\right)}}{\left(x + 1\right) \cdot \left({x}^{3} - {1}^{3}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11360.435045225768 \lor \neg \left(x \le 11442.78162855777\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({x}^{4} + \left(-x\right)\right) - \frac{\left(\left(1 + x\right) \cdot \left(1 - x \cdot x\right)\right) \cdot \left({\left(x \cdot x\right)}^{3} + {\left(1 + x\right)}^{3}\right)}{\left(1 - x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(x \cdot x\right) \cdot \left(1 + x\right)\right) + \left(1 + x\right) \cdot \left(1 + x\right)\right)}}{\left({x}^{3} - 1\right) \cdot \left(1 + x\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018355 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))