Error

Derivation

1. Split input into 2 regimes

2. if x < -9757.690344182685 or 12585.124025570638 < x

   1. Initial program 59.3

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

   3. Applied *-un-lft-identity59.3

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

   4. Applied associate-/l*59.3

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{1}{\frac{x - 1}{x + 1}}}\]

   5. Taylor expanded around -inf 0.3

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

   6. 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 -9757.690344182685 < x < 12585.124025570638

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt0.1

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

   4. Applied flip3-+0.1

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

   5. Applied associate-/r/0.1

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

   6. Applied prod-diff0.1

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

   7. Simplified0.1

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

   8. Simplified0.1

      \[\leadsto (\left(\frac{x}{(x \cdot \left(x \cdot x\right) + 1)_*}\right) \cdot \left((x \cdot x + 1)_* - x\right) + \left(\frac{-1 - x}{-1 + x}\right))_* + \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9757.690344182685 \lor \neg \left(x \le 12585.124025570638\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{x}{(x \cdot \left(x \cdot x\right) + 1)_*}\right) \cdot \left((x \cdot x + 1)_* - x\right) + \left(\frac{-1 - x}{-1 + x}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018340 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))