Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -59936.72473869073 or 51139.341490639854 < x

   1. Initial program 30.1

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

   2. Simplified30.1

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

   3. Taylor expanded around -inf 0.0

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

   if -59936.72473869073 < x < 51139.341490639854

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -59936.72473869073 \lor \neg \left(x \le 51139.341490639854\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

Details

Time bar (total: 12.6s)Debug log