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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -697137081738736.5 \lor \neg \left(x \le 471.71787806799045\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \end{array}\]

Original	14.7
Target	0.1
Herbie	0.0

Derivation

Split input into 2 regimes
if x < -697137081738736.5 or 471.71787806799045 < x
1. Initial program 30.5
  \[\frac{x}{x \cdot x + 1}\]
2. Initial simplification30.5
  \[\leadsto \frac{x}{(x \cdot x + 1)_*}\]
3. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
if -697137081738736.5 < x < 471.71787806799045
1. Initial program 0.0
  \[\frac{x}{x \cdot x + 1}\]
2. Initial simplification0.0
  \[\leadsto \frac{x}{(x \cdot x + 1)_*}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -697137081738736.5 \lor \neg \left(x \le 471.71787806799045\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \end{array}\]

Runtime

herbie shell --seed 2018242 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

Error

Target

Derivation

`if x < -697137081738736.5 or 471.71787806799045 < x`

`if -697137081738736.5 < x < 471.71787806799045`

Runtime