Error

Derivation

1. Split input into 2 regimes

2. if x < -2.1484138251760253e+33 or 4921536.861747979 < x

   1. Initial program 60.1

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

   3. Applied frac-sub61.9

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

   4. Simplified55.8

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

   5. Taylor expanded around -inf 32.8

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

   6. Simplified32.8

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

   8. Applied flip3-+45.8

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

   9. Applied associate-*l/50.9

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

   10. Applied associate-/r/50.9

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

   11. Simplified50.9

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

   12. Taylor expanded around -inf 0.3

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

   13. Simplified0.0

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

   if -2.1484138251760253e+33 < x < 4921536.861747979

   1. Initial program 2.2

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

   3. Applied frac-sub2.2

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

   4. Simplified1.3

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

   5. Taylor expanded around -inf 0.0

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

   6. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.1484138251760253 \cdot 10^{+33} \lor \neg \left(x \le 4921536.861747979\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x} \cdot \left(1 + \frac{3}{x}\right) - \frac{3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{(-3 \cdot x + -1)_*}{\left(1 + x\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

Details

Time bar (total: 1.7m)Debug log