Error

Derivation

1. Split input into 2 regimes

2. if x < -11486.711370277431 or 11208.13028654597 < x

   1. Initial program 59.3

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

   3. Applied div-inv59.5

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

   4. Applied fma-neg60.2

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

   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 -11486.711370277431 < x < 11208.13028654597

   1. Initial program 0.1

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

   3. Applied div-inv0.1

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

   4. Applied fma-neg0.1

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11486.711370277431 \lor \neg \left(x \le 11208.13028654597\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(x \cdot \left(\frac{1}{1 + x}\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 48.2s)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))