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Target

Derivation

1. Split input into 2 regimes

2. if x < -93901838.54092976 or 1037.5541891955904 < x

   1. Initial program 19.1

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

   3. Applied frac-sub52.2

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

   4. Applied frac-add50.7

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

   6. Applied flip-+50.7

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

   7. Applied associate-*l/50.7

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

   8. Applied associate-*l/50.7

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

   9. Applied associate-/r/50.7

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

   10. Taylor expanded around inf 0.6

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

   11. Simplified0.1

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

   if -93901838.54092976 < x < 1037.5541891955904

   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

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

   6. Applied flip-+0.0

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

   7. Applied associate-*l/0.0

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

   8. Applied associate-*l/0.0

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

   9. Applied associate-/r/0.0

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

   11. Applied associate-*l/0.0

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

   12. Simplified0.0

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -93901838.54092976 \lor \neg \left(x \le 1037.5541891955904\right):\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(1 + x\right) + \left(x + -1\right)\right) \cdot x - \left(2 + 2 \cdot x\right) \cdot \left(x + -1\right)\right) \cdot \left(x + -1\right)}{\left(\left(x \cdot x - 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018249 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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