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Target

Derivation

1. Split input into 3 regimes

2. if x < -107.3467989013554

   1. Initial program 19.1

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

   3. Applied flip-+50.8

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

   4. Applied associate-/r/53.1

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

   5. Taylor expanded around -inf 0.5

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

   6. Simplified0.1

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

   7. Taylor expanded around inf 0.5

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

   if -107.3467989013554 < x < 104.27982465385276

   1. Initial program 0.0

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

   3. Applied flip-+0.0

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

   4. Applied associate-/r/0.1

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

   if 104.27982465385276 < x

   1. Initial program 18.7

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

   3. Applied flip-+49.9

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

   4. Applied associate-/r/52.7

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

   5. Taylor expanded around -inf 0.5

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

   6. Simplified0.1

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

   8. Applied div-inv0.1

      \[\leadsto \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \color{blue}{\frac{2}{x} \cdot \frac{1}{x \cdot x}}\]
   9. Using strategy rm

   10. Applied pow20.1

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

   11. Applied pow-flip0.1

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

   12. Simplified0.1

       \[\leadsto \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{2}{x} \cdot {x}^{\color{blue}{-2}}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -107.3467989013554:\\ \;\;\;\;\frac{2}{{x}^{3}} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 104.27982465385276:\\ \;\;\;\;\left(\frac{1}{x \cdot x - 1} \cdot \left(x - 1\right) - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + {x}^{-2} \cdot \frac{2}{x}\\ \end{array}\]

Runtime

Time bar (total: 47.2s)Debug logProfile

herbie shell --seed 2018273 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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