Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -5.315332958531357e+47 or 536.2819961347985 < x

   1. Initial program 33.4

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

   2. Simplified33.4

      \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
   3. Using strategy rm

   4. Applied clear-num33.3

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

   5. Taylor expanded around -inf 0.0

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

   if -5.315332958531357e+47 < x < 536.2819961347985

   1. Initial program 0.0

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

   2. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
   3. Using strategy rm

   4. Applied clear-num0.2

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.315332958531357 \cdot 10^{+47} \lor \neg \left(x \le 536.2819961347985\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{(x \cdot x + 1)_*}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

Details

Time bar (total: 13.5s)Debug log