Error

Derivation

1. Split input into 2 regimes

2. if x < -11810.647953296477 or 9392.348407456098 < x

   1. Initial program 59.4

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

   3. Applied add-log-exp59.4

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

   4. Taylor expanded around inf 0.3

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

   5. 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 -11810.647953296477 < x < 9392.348407456098

   1. Initial program 0.1

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

   3. Applied add-log-exp0.1

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

   5. Applied log1p-expm1-u0.1

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11810.647953296477 \lor \neg \left(x \le 9392.348407456098\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{\log \left(e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\right)} - 1)^*)\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

Details

Time bar (total: 3.5m)Debug log