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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1520936995108179.5 \lor \neg \left(x \le 421.41039591667726\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1} \cdot \left(x \cdot x - 1\right)\\ \end{array}\]

Original	14.6
Target	0.1
Herbie	0.0

Derivation

Split input into 2 regimes
if x < -1520936995108179.5 or 421.41039591667726 < x
1. Initial program 30.4
  \[\frac{x}{x \cdot x + 1}\]
2. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
if -1520936995108179.5 < x < 421.41039591667726
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
  \[\leadsto \color{blue}{\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \left(x \cdot x - 1\right)}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1520936995108179.5 \lor \neg \left(x \le 421.41039591667726\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1} \cdot \left(x \cdot x - 1\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018255 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

Error

Try it out

Target

Derivation

`if x < -1520936995108179.5 or 421.41039591667726 < x`

`if -1520936995108179.5 < x < 421.41039591667726`

Runtime

In
Out