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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.004513004279039 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 416.49421652298525:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Original	14.4
Target	0.1
Herbie	0.0

Derivation

Split input into 2 regimes
if x < -3.004513004279039e+22 or 416.49421652298525 < x
1. Initial program 30.7
  \[\frac{x}{x \cdot x + 1}\]
2. Simplified30.7
  \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
3. Taylor expanded around -inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
4. Simplified0.0
  \[\leadsto \color{blue}{\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}}\]
if -3.004513004279039e+22 < x < 416.49421652298525
1. Initial program 0.0
  \[\frac{x}{x \cdot x + 1}\]
2. Simplified0.0
  \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.004513004279039 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 416.49421652298525:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 +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 < -3.004513004279039e+22 or 416.49421652298525 < x`

`if -3.004513004279039e+22 < x < 416.49421652298525`

Reproduce