Average Error: 29.9 → 0.1
Time: 10.6s
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11843.2408407861 \lor \neg \left(x \le 12015.839524391253\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -11843.2408407861 or 12015.839524391253 < x

    1. Initial program 59.4

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

    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

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

    if -11843.2408407861 < x < 12015.839524391253

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11843.2408407861 \lor \neg \left(x \le 12015.839524391253\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))