Error

Derivation

1. Split input into 2 regimes

2. if x < -11997.32666981334 or 11909.34130082208 < 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 -11997.32666981334 < x < 11909.34130082208

   1. Initial program 0.1

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

   3. Applied frac-sub0.1

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

   5. Applied distribute-rgt-in0.1

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

   6. Applied associate--r+0.1

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

   7. Simplified0.1

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

   9. Applied flip3-+0.1

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

   10. Applied associate-*l/0.1

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

   11. Applied associate-/r/0.1

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

   12. Simplified0.1

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11997.32666981334 \lor \neg \left(x \le 11909.34130082208\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(x \cdot x + \left(1 - x\right))_* \cdot \frac{\left(\left(x - 1\right) \cdot x - \left(1 + x\right) \cdot x\right) - \left(1 + x\right)}{\left(x - 1\right) \cdot \left(1 + {x}^{3}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

Details

Time bar (total: 4.6m)Debug log