Error

Target

Derivation

1. Split input into 3 regimes

2. if x < -113.67105325092781

   1. Initial program 19.8

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

   2. Taylor expanded around inf 0.4

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

   3. Simplified0.1

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

   5. Applied associate-/r*0.1

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

   if -113.67105325092781 < x < 131.68996983809097

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt1.3

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

   4. Applied add-cube-cbrt1.3

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

   5. Applied prod-diff1.3

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

   6. Applied associate-+l+1.3

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

   7. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \left((\left(-\sqrt[3]{\frac{2}{x}}\right) \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) + \left(\sqrt[3]{\frac{2}{x}} \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right)\right))_* + \frac{1}{x - 1}\right)\]
   8. Using strategy rm

   9. Applied flip-+27.7

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

   10. Applied frac-sub27.7

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

   11. Applied frac-add27.9

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

   12. Simplified59.7

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

   13. Simplified0.1

       \[\leadsto \frac{(\left((x \cdot x + x)_*\right) \cdot \left(\left(\left(\frac{2}{x} - \frac{2}{x}\right) + \frac{1}{x - 1}\right) \cdot \left(\left(\frac{2}{x} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)\right) + \left(\left(\left(\frac{2}{x} - \frac{2}{x}\right) - \frac{1}{x - 1}\right) \cdot \left(x - (x \cdot 2 + 2)_*\right)\right))_*}{\color{blue}{(\left((x \cdot x + x)_*\right) \cdot \left(\frac{-1}{x - 1}\right) + \left((x \cdot x + x)_* \cdot 0\right))_*}}\]

   if 131.68996983809097 < x

   1. Initial program 19.6

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

   2. Taylor expanded around inf 0.6

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

   3. Simplified0.1

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

   5. Applied associate-/l/0.7

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -113.67105325092781:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 131.68996983809097:\\ \;\;\;\;\frac{(\left((x \cdot x + x)_*\right) \cdot \left(\left(\left(\frac{2}{x} - \frac{2}{x}\right) - \frac{1}{x - 1}\right) \cdot \left(\frac{1}{x - 1} + \left(\frac{2}{x} - \frac{2}{x}\right)\right)\right) + \left(\left(\left(\frac{2}{x} - \frac{2}{x}\right) - \frac{1}{x - 1}\right) \cdot \left(x - (x \cdot 2 + 2)_*\right)\right))_*}{(\left((x \cdot x + x)_*\right) \cdot \left(\frac{-1}{x - 1}\right) + \left((x \cdot x + x)_* \cdot 0\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{2}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]

Runtime

Time bar (total: 44.8s)Debug logProfile

herbie shell --seed 2018219 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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