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Derivation

1. Split input into 2 regimes

2. if x < -9805.251439711301 or 9393.816243652562 < x

   1. Initial program 59.3

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

   3. Applied add-cbrt-cube59.3

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

   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. Simplified0.0

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

   7. Applied associate-+l-0.0

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

   if -9805.251439711301 < x < 9393.816243652562

   1. Initial program 0.1

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

   3. Applied add-cbrt-cube0.1

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9805.251439711301 \lor \neg \left(x \le 9393.816243652562\right):\\ \;\;\;\;\frac{-3}{x} - \left(\frac{1}{x \cdot x} - \frac{\frac{-3}{x}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \sqrt[3]{\frac{1 + x}{x - 1} \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}\\ \end{array}\]

Runtime

Time bar (total: 38.6s)Debug logProfile

herbie shell --seed 2018250 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))