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Target

Derivation

1. Split input into 2 regimes

2. if x < -1.5716236815582468e+71 or 509.4123991212909 < x

   1. Initial program 33.9

      \[\frac{x}{x \cdot x + 1}\]
   2. Using strategy rm

   3. Applied div-inv34.0

      \[\leadsto \color{blue}{x \cdot \frac{1}{x \cdot x + 1}}\]

   4. Taylor expanded around -inf 0.0

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

   if -1.5716236815582468e+71 < x < 509.4123991212909

   1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
   2. Using strategy rm

   3. Applied flip-+0.0

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

   4. Applied associate-/r/0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5716236815582468 \cdot 10^{+71} \lor \neg \left(x \le 509.4123991212909\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1} \cdot \left(x \cdot x - 1\right)\\ \end{array}\]

Runtime

Time bar (total: 15.4s)Debug logProfile

herbie shell --seed 2018290 
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))