Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -1.0208141978551163e+48 or 423.1303199100469 < x

   1. Initial program 32.1

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

   2. Simplified32.1

      \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]

   3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]

   if -1.0208141978551163e+48 < x < 423.1303199100469

   1. Initial program 0.0

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

   2. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0208141978551163 \cdot 10^{+48} \lor \neg \left(x \le 423.1303199100469\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

Details

Time bar (total: 13.0s)Debug log