Derivation

Split input into 2 regimes
if (* x x) < 0.012073634685286628
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-sqr-sqrt0.0
  \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]
5. Applied associate-/r*0.0
  \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]
if 0.012073634685286628 < (* x x)
1. Initial program 29.8
  \[\frac{x}{x \cdot x + 1}\]
2. Initial simplification29.8
  \[\leadsto \frac{x}{x \cdot x + 1}\]
3. Using strategy rm
4. Applied add-sqr-sqrt29.8
  \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]
5. Applied associate-/r*29.7
  \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]
6. Taylor expanded around inf 0.4
  \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
Recombined 2 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \le 0.012073634685286628:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \end{array}\]

Runtime

herbie shell --seed 2018277 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

In
Out