Average Error: 28.8 → 0.1
Time: 46.4s
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12193.125583501567:\\ \;\;\;\;\frac{\frac{-3}{x} - \frac{\frac{-3}{x}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{1 - \frac{1}{x \cdot x}} - \frac{1}{x \cdot x}\\ \mathbf{elif}\;x \le 10283.176249851711:\\ \;\;\;\;\frac{\log \left(e^{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}\right)}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{x} - \frac{\frac{-3}{x}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{1 - \frac{1}{x \cdot x}} - \frac{1}{x \cdot x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -12193.125583501567 or 10283.176249851711 < 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

      \[\leadsto \color{blue}{\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{-3}{x} - \frac{1}{x \cdot x}}\]
    4. Using strategy rm

    5. Applied flip-+0.0

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{1}{x \cdot x} \cdot \frac{1}{x \cdot x}}{1 - \frac{1}{x \cdot x}}} \cdot \frac{-3}{x} - \frac{1}{x \cdot x}\]

    6. Applied associate-*l/0.0

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

    7. Simplified0.0

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

    if -12193.125583501567 < x < 10283.176249851711

    1. Initial program 0.1

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

    3. Applied flip--0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]
    4. Using strategy rm

    5. Applied add-log-exp0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12193.125583501567:\\ \;\;\;\;\frac{\frac{-3}{x} - \frac{\frac{-3}{x}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{1 - \frac{1}{x \cdot x}} - \frac{1}{x \cdot x}\\ \mathbf{elif}\;x \le 10283.176249851711:\\ \;\;\;\;\frac{\log \left(e^{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}\right)}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{x} - \frac{\frac{-3}{x}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{1 - \frac{1}{x \cdot x}} - \frac{1}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))