Error

Derivation

1. Split input into 2 regimes

2. if x < -13564.282494661315 or 1.0092784462051199 < x

   1. Initial program 58.9

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

   3. Applied flip--58.9

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

   5. Applied div-inv59.1

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

   6. Applied associate-*l*59.1

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

   8. Applied associate-*l/59.1

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

   9. Applied associate-*r/58.9

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

   10. Applied associate-*l/58.9

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

   11. Applied frac-sub60.6

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

   12. Simplified60.7

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

   13. Taylor expanded around inf 0.6

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

   14. Simplified0.3

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

   if -13564.282494661315 < x < 1.0092784462051199

   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 div-inv0.1

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

   6. Applied associate-*l*0.1

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

   8. Applied associate-*l/0.1

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

   9. Applied associate-*r/0.1

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

   10. Applied associate-*l/0.1

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

   11. Applied frac-sub0.1

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

   12. Simplified0.1

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

   14. Applied expm1-log1p-u0.1

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13564.282494661315 \lor \neg \left(x \le 1.0092784462051199\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{(e^{\log_* (1 + \left(\frac{x \cdot x}{\frac{1 + x}{-1 + x}} - \frac{1 + x}{-1 + x} \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)\right))} - 1)^*}{\left(x - 1\right) \cdot \left(1 + x\right)}}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

Details

Time bar (total: 3.2m)Debug log