Error

Derivation

1. Split input into 2 regimes

2. if x < -15731.583965646452 or 9865.742309267576 < 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

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

   if -15731.583965646452 < x < 9865.742309267576

   1. Initial program 0.1

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

   3. Applied add-cbrt-cube0.1

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15731.583965646452 \lor \neg \left(x \le 9865.742309267576\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)}\\ \end{array}\]

Reproduce

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

Details

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