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Target

Derivation

1. Split input into 2 regimes

2. if x < -7293977845865.593 or 1675.9184660645035 < x

   1. Initial program 30.4

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

   2. Initial simplification30.4

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

   3. Taylor expanded around -inf 0.0

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

   if -7293977845865.593 < x < 1675.9184660645035

   1. Initial program 0.0

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

   2. Initial simplification0.0

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

   4. Applied *-un-lft-identity0.0

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

   5. Applied associate-/r*0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7293977845865.593 \lor \neg \left(x \le 1675.9184660645035\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Runtime

Time bar (total: 6.6s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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