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Derivation

1. Split input into 2 regimes

2. if x < -13507.472987014315 or 12154.943809828786 < x

   1. Initial program 29.3

      \[\frac{1}{x + 1} - \frac{1}{x}\]

   2. Taylor expanded around inf 0.9

      \[\leadsto \color{blue}{\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)}\]
   3. Using strategy rm

   4. Applied pow-flip0.0

      \[\leadsto \frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \color{blue}{{x}^{\left(-2\right)}}\right)\]

   5. Simplified0.0

      \[\leadsto \frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + {x}^{\color{blue}{-2}}\right)\]

   if -13507.472987014315 < x < 12154.943809828786

   1. Initial program 0.1

      \[\frac{1}{x + 1} - \frac{1}{x}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13507.472987014315 \lor \neg \left(x \le 12154.943809828786\right):\\ \;\;\;\;\frac{1}{{x}^{3}} - \left({x}^{-2} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} - \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Details

Time bar (total: 9.2s)Debug log