Error

Derivation

1. Split input into 2 regimes

2. if x < -110399846259660.97 or 106500.10447990824 < x

   1. Initial program 60.0

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

   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 -110399846259660.97 < x < 106500.10447990824

   1. Initial program 0.5

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

   3. Applied clear-num0.5

      \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x}}} - \frac{x + 1}{x - 1}\]
   4. Using strategy rm

   5. Applied frac-sub0.3

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

   6. Simplified0.0

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

   7. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -110399846259660.97 \lor \neg \left(x \le 106500.10447990824\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 - x\right) + x\right) - \frac{1 + x}{x}}{\frac{(x \cdot x + -1)_*}{x}}\\ \end{array}\]

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed 2018304 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))