Average Error: 29.2 → 0.1
Time: 34.2s
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10868.378678689824:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{-3}{x} - \frac{1}{x \cdot x}\\ \mathbf{elif}\;x \le 10029.784737411197:\\ \;\;\;\;\frac{{\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}}{\frac{x}{1 + x} \cdot \frac{x}{1 + x} + \left(\frac{x}{1 + x} \cdot \frac{1 + x}{x - 1} + \frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{-3}{x} - \frac{1}{x \cdot x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -10868.378678689824 or 10029.784737411197 < x

    1. Initial program 59.0

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

    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

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

    if -10868.378678689824 < x < 10029.784737411197

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10868.378678689824:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{-3}{x} - \frac{1}{x \cdot x}\\ \mathbf{elif}\;x \le 10029.784737411197:\\ \;\;\;\;\frac{{\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}}{\frac{x}{1 + x} \cdot \frac{x}{1 + x} + \left(\frac{x}{1 + x} \cdot \frac{1 + x}{x - 1} + \frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{-3}{x} - \frac{1}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))