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Target

Derivation

1. Split input into 2 regimes

2. if x < -6351279.368385906 or 471.8837632120897 < x

   1. Initial program 30.8

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

   3. Applied add-sqr-sqrt30.8

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

   4. Applied associate-/r*30.7

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

   5. Taylor expanded around -inf 0.0

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

   if -6351279.368385906 < x < 471.8837632120897

   1. Initial program 0.0

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

   3. Applied add-sqr-sqrt0.0

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

   4. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity0.0

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

   7. Applied *-un-lft-identity0.0

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

   8. Applied times-frac0.0

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

   9. Simplified0.0

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

   10. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6351279.368385906 \lor \neg \left(x \le 471.8837632120897\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Runtime

Time bar (total: 12.9s)Debug logProfile

herbie shell --seed 2018286 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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