Average Error: 29.3 → 0.1
Time: 47.4s
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13097.868846386831:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{-3}{x} - \frac{1}{x \cdot x}\\ \mathbf{elif}\;x \le 10302.787562889038:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(1 + x\right)}{\sqrt[3]{\left(\left(1 + x\right) \cdot \left(x - 1\right)\right) \cdot \left(\left(\left(1 + x\right) \cdot \left(x - 1\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{-3}{x} - \frac{1}{x \cdot x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -13097.868846386831 or 10302.787562889038 < x

    1. Initial program 59.4

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

    3. Applied add-cbrt-cube59.4

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

    4. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

    5. Simplified0.0

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

    if -13097.868846386831 < x < 10302.787562889038

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    5. Applied frac-sub0.1

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

    6. Applied frac-sub0.1

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

    7. Applied frac-sub0.1

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

    8. Applied frac-times0.1

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

    9. Applied frac-times0.1

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

    10. Applied cbrt-div0.1

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

    11. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13097.868846386831:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{-3}{x} - \frac{1}{x \cdot x}\\ \mathbf{elif}\;x \le 10302.787562889038:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(1 + x\right)}{\sqrt[3]{\left(\left(1 + x\right) \cdot \left(x - 1\right)\right) \cdot \left(\left(\left(1 + x\right) \cdot \left(x - 1\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{-3}{x} - \frac{1}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))