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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -5.864261687274507 \cdot 10^{+52} \lor \neg \left(x \le 15389527.932909489\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(x \cdot x + 1)_*} \cdot x\\ \end{array}\]

Original	14.6
Target	0.1
Herbie	0.0

Derivation

Split input into 2 regimes
if x < -5.864261687274507e+52 or 15389527.932909489 < x
1. Initial program 32.1
  \[\frac{x}{x \cdot x + 1}\]
2. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
if -5.864261687274507e+52 < x < 15389527.932909489
1. Initial program 0.0
  \[\frac{x}{x \cdot x + 1}\]
2. Using strategy rm
3. Applied div-inv0.0
  \[\leadsto \color{blue}{x \cdot \frac{1}{x \cdot x + 1}}\]
4. Applied simplify0.0
  \[\leadsto x \cdot \color{blue}{\frac{1}{(x \cdot x + 1)_*}}\]
Recombined 2 regimes into one program.
Applied simplify0.0
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -5.864261687274507 \cdot 10^{+52} \lor \neg \left(x \le 15389527.932909489\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(x \cdot x + 1)_*} \cdot x\\ \end{array}}\]

Runtime

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +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 < -5.864261687274507e+52 or 15389527.932909489 < x`

`if -5.864261687274507e+52 < x < 15389527.932909489`

Runtime