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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.684447861881018 \cdot 10^{+27} \lor \neg \left(x \le 945.5765595016253\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \end{array}\]

Original	15.4
Target	0.1
Herbie	0.0

Derivation

Split input into 2 regimes
if x < -4.684447861881018e+27 or 945.5765595016253 < x
1. Initial program 31.9
  \[\frac{x}{x \cdot x + 1}\]
2. Applied simplify31.9
  \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
3. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
if -4.684447861881018e+27 < x < 945.5765595016253
1. Initial program 0.0
  \[\frac{x}{x \cdot x + 1}\]
2. Applied simplify0.0
  \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
Recombined 2 regimes into one program.
Applied simplify0.0
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -4.684447861881018 \cdot 10^{+27} \lor \neg \left(x \le 945.5765595016253\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \end{array}}\]

Runtime

herbie shell --seed 2020178 +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 < -4.684447861881018e+27 or 945.5765595016253 < x`

`if -4.684447861881018e+27 < x < 945.5765595016253`

Runtime