Error

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Target

Derivation

1. Split input into 2 regimes

2. if x < -7.345940124114164e+23 or 937904.7083747658 < x

   1. Initial program 31.6

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

   3. Applied *-un-lft-identity31.6

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

   4. Applied associate-/r*31.6

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

   5. Taylor expanded around inf 0.0

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

   if -7.345940124114164e+23 < x < 937904.7083747658

   1. Initial program 0.0

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

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

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

   4. Applied associate-/r*0.0

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

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

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

   7. Applied div-inv0.0

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

   8. Applied times-frac0.0

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

   9. Simplified0.0

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

   10. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.345940124114164 \cdot 10^{+23} \lor \neg \left(x \le 937904.7083747658\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

Details

Time bar (total: 7.0s)Debug log