Average Error: 29.1 → 0.2
Time: 41.9s
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13124.679046151381:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) \cdot 3\\ \mathbf{elif}\;x \le 14964.993179788482:\\ \;\;\;\;x \cdot \frac{1}{1 + x} - \frac{1 + x}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) \cdot 3\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -13124.679046151381 or 14964.993179788482 < x

    1. Initial program 59.1

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

    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. Simplified0.3

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

    if -13124.679046151381 < x < 14964.993179788482

    1. Initial program 0.1

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

    3. Applied div-inv0.1

      \[\leadsto \color{blue}{x \cdot \frac{1}{x + 1}} - \frac{x + 1}{x - 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13124.679046151381:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) \cdot 3\\ \mathbf{elif}\;x \le 14964.993179788482:\\ \;\;\;\;x \cdot \frac{1}{1 + x} - \frac{1 + x}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) \cdot 3\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))