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Derivation

1. Split input into 2 regimes

2. if x < -10509.011418440743 or 14170.303121934652 < 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

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

   if -10509.011418440743 < x < 14170.303121934652

   1. Initial program 0.1

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

   3. Applied flip3--0.1

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

   5. Applied cube-div0.1

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10509.011418440743 \lor \neg \left(x \le 14170.303121934652\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{1 + x}\right)}^{3} - \frac{{\left(1 + x\right)}^{3}}{{\left(x - 1\right)}^{3}}}{\frac{x}{1 + x} \cdot \frac{x}{1 + x} + \left(\frac{x}{1 + x} \cdot \frac{1 + x}{x - 1} + \frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018286 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))