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Derivation

1. Split input into 2 regimes

2. if x < -292.47442415369994 or 484.8501943030043 < x

   1. Initial program 29.3

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

   2. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{4}} + \left(2 \cdot \frac{1}{{x}^{6}} + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

   3. Simplified0.8

      \[\leadsto \color{blue}{\frac{-2}{{x}^{6}} + \left(\frac{-2}{{x}^{4}} + \frac{-2}{x \cdot x}\right)}\]
   4. Using strategy rm

   5. Applied associate-/r*0.1

      \[\leadsto \frac{-2}{{x}^{6}} + \left(\frac{-2}{{x}^{4}} + \color{blue}{\frac{\frac{-2}{x}}{x}}\right)\]

   if -292.47442415369994 < x < 484.8501943030043

   1. Initial program 0.0

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -292.47442415369994 \lor \neg \left(x \le 484.8501943030043\right):\\ \;\;\;\;\left(\frac{\frac{-2}{x}}{x} + \frac{-2}{{x}^{4}}\right) + \frac{-2}{{x}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} - \frac{1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))

Details

Time bar (total: 31.8s)Debug log