Derivation

Split input into 2 regimes
if x < -641.2554917333738 or 416.7099922766305 < x
1. Initial program 30.1
  \[\frac{x}{x \cdot x + 1}\]
2. Initial simplification30.1
  \[\leadsto \frac{x}{(x \cdot x + 1)_*}\]
3. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
if -641.2554917333738 < x < 416.7099922766305
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 add-cube-cbrt0.0
  \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{(x \cdot x + 1)_*} \cdot \sqrt[3]{(x \cdot x + 1)_*}\right) \cdot \sqrt[3]{(x \cdot x + 1)_*}}}\]
5. Applied associate-/r*0.0
  \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt[3]{(x \cdot x + 1)_*} \cdot \sqrt[3]{(x \cdot x + 1)_*}}}{\sqrt[3]{(x \cdot x + 1)_*}}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -641.2554917333738 \lor \neg \left(x \le 416.7099922766305\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{(x \cdot x + 1)_*} \cdot \sqrt[3]{(x \cdot x + 1)_*}}}{\sqrt[3]{(x \cdot x + 1)_*}}\\ \end{array}\]

Runtime

herbie shell --seed 2018234 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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