Error

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Target

Derivation

1. Split input into 2 regimes

2. if x < -1.3523329577305727e+154 or 439.23543538614194 < x

   1. Initial program 40.8

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

   2. Initial simplification40.8

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

   3. Taylor expanded around -inf 0.0

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

   if -1.3523329577305727e+154 < x < 439.23543538614194

   1. Initial program 0.1

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

   2. Initial simplification0.1

      \[\leadsto \frac{x}{x \cdot x + 1}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt0.1

      \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]

   5. Applied associate-/r*0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3523329577305727 \cdot 10^{+154} \lor \neg \left(x \le 439.23543538614194\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

Details

Time bar (total: 18.9s)Debug log