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Target

Derivation

1. Split input into 3 regimes

2. if x < -21541054.182414927

   1. Initial program 18.8

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied flip-+18.8

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

   4. Taylor expanded around -inf 0.6

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]

   5. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}}\]

   if -21541054.182414927 < x < 1992.9042173635328

   1. Initial program 0.4

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied frac-sub0.4

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

   4. Applied frac-add0.0

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

   if 1992.9042173635328 < x

   1. Initial program 19.8

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied flip-+19.8

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1} \cdot \frac{1}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\]
   4. Using strategy rm

   5. Applied associate-*l/22.3

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

   6. Applied frac-sub24.5

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

   7. Applied associate-*l/24.5

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

   8. Applied frac-sub52.0

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

   9. Applied associate-/l/52.0

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

   10. Simplified19.8

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

   11. Taylor expanded around -inf 0.4

       \[\leadsto \frac{\color{blue}{-\left(4 \cdot \frac{1}{{x}^{4}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}\right)\right)}}{\left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right) \cdot \left(\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)\right)}\]

   12. Simplified0.4

       \[\leadsto \frac{\color{blue}{\frac{-4}{{x}^{4}} - \left(\frac{\frac{4}{x}}{x} + \frac{4}{x}\right)}}{\left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right) \cdot \left(\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -21541054.182414927:\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 1992.9042173635328:\\ \;\;\;\;\frac{\left(x - 2 \cdot \left(1 + x\right)\right) \cdot \left(x - 1\right) + x \cdot \left(1 + x\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4}{{x}^{4}} - \left(\frac{4}{x} + \frac{\frac{4}{x}}{x}\right)}{\left(\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}\right) \cdot \left(\left(x \cdot \left(1 + x\right)\right) \cdot \left(x - 1\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 8.5m)Debug logProfile

herbie shell --seed 2018278 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))