Derivation

Split input into 2 regimes
if x < -426.2099919485139 or 451.36740085518375 < x
1. Initial program 29.4
  \[\frac{x}{x \cdot x + 1}\]
2. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
if -426.2099919485139 < x < 451.36740085518375
1. Initial program 0.0
  \[\frac{x}{x \cdot x + 1}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.0
  \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]
4. Applied associate-/r*0.0
  \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]
5. Using strategy rm
6. Applied add-cbrt-cube0.0
  \[\leadsto \frac{\frac{x}{\color{blue}{\sqrt[3]{\left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}\right) \cdot \sqrt{x \cdot x + 1}}}}}{\sqrt{x \cdot x + 1}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -426.2099919485139 \lor \neg \left(x \le 451.36740085518375\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{\left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}\right) \cdot \sqrt{x \cdot x + 1}}}}{\sqrt{x \cdot x + 1}}\\ \end{array}\]

Runtime

herbie shell --seed 2018296 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

In
Out