Derivation

Split input into 2 regimes
if x < -9330.22904458982 or 443.5527729523362 < x
1. Initial program 29.5
  \[\frac{x}{x \cdot x + 1}\]
2. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
if -9330.22904458982 < x < 443.5527729523362
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
  \[\leadsto \color{blue}{\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \left(x \cdot x - 1\right)}\]
5. Applied simplify0.0
  \[\leadsto \color{blue}{\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1}} \cdot \left(x \cdot x - 1\right)\]
Recombined 2 regimes into one program.
Applied simplify0.0
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -9330.22904458982 \lor \neg \left(x \le 443.5527729523362\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1}\\ \end{array}}\]

Runtime

herbie shell --seed 2018198 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

In
Out