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Target

Derivation

1. Split input into 2 regimes

2. if x < -114.64678370673498 or 104.33422414858215 < 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 add-sqr-sqrt48.3

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

   4. Applied associate-/r*50.2

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

   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

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

   8. 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 -114.64678370673498 < x < 104.33422414858215

   1. Initial program 0.0

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

   3. Applied add-sqr-sqrt0.2

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

   4. Applied associate-/r*0.2

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

   6. Applied *-un-lft-identity0.2

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

   7. Applied *-un-lft-identity0.2

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

   8. Applied distribute-lft-out0.2

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

   9. Simplified0.0

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -114.64678370673498 \lor \neg \left(x \le 104.33422414858215\right):\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} - \left(\frac{-1}{-1 + x} - \frac{1}{1 + x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +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))))

Details

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