Error

Try it out

Target

Derivation

1. Split input into 2 regimes

2. if x < -32889.63765713185 or 425.99930081705185 < x

   1. Initial program 30.0

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

   3. Applied flip-+47.6

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

   4. Applied associate-/r/47.7

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

   5. Taylor expanded around inf 0.0

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

   if -32889.63765713185 < x < 425.99930081705185

   1. Initial program 0.0

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

   3. Applied flip-+0.0

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

   4. Applied associate-/r/0.0

      \[\leadsto \color{blue}{\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \left(x \cdot x - 1\right)}\]
   5. Using strategy rm

   6. Applied pow10.0

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

   7. Applied pow10.0

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

   8. Applied pow-prod-down0.0

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

   9. Simplified0.0

      \[\leadsto {\color{blue}{\left(\frac{\left(-x\right) + {x}^{3}}{-1 + {x}^{4}}\right)}}^{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -32889.63765713185 \lor \neg \left(x \le 425.99930081705185\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-x\right) + {x}^{3}}{-1 + {x}^{4}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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

Details

Time bar (total: 15.2s)Debug log