Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -2.166459689224054e+31 or 436.9147618452836 < x

   1. Initial program 31.8

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

   2. Simplified31.8

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

   4. Applied *-un-lft-identity31.8

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

   5. Applied associate-/r*31.8

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

   6. Taylor expanded around inf 0.0

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

   7. Simplified0.0

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

   if -2.166459689224054e+31 < x < 436.9147618452836

   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 *-un-lft-identity0.0

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

   5. Applied associate-/r*0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.166459689224054 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \le 436.9147618452836:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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