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Target

Derivation

1. Split input into 2 regimes

2. if x < -1.0159564017437261 or 104.16183730286129 < x

   1. Initial program 19.0

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

   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. Simplified0.3

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

   5. Applied associate-/r*0.3

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

   if -1.0159564017437261 < x < 104.16183730286129

   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-sqrt0.0

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

   4. Applied associate-/r*0.0

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0159564017437261 \lor \neg \left(x \le 104.16183730286129\right):\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x - 1} + \left(\frac{\frac{1}{\sqrt{1 + x}}}{\sqrt{1 + x}} - \frac{2}{x}\right)\\ \end{array}\]

Runtime

Time bar (total: 34.2s)Debug logProfile

herbie shell --seed 2018254 +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))))