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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -13084.334255292539 \lor \neg \left(x \le 10285.963445549292\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1}{\frac{x - 1}{1 + x}}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < -13084.334255292539 or 10285.963445549292 < x
1. Initial program 59.3
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
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 -13084.334255292539 < x < 10285.963445549292
1. Initial program 0.1
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied clear-num0.1
  \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{1}{\frac{x - 1}{x + 1}}}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13084.334255292539 \lor \neg \left(x \le 10285.963445549292\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1}{\frac{x - 1}{1 + x}}\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	31.3	0.1	0.0	31.3	99.9%

herbie shell --seed 2018351 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

Error

Derivation

`if x < -13084.334255292539 or 10285.963445549292 < x`

`if -13084.334255292539 < x < 10285.963445549292`

Runtime