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Target

Derivation

1. Split input into 2 regimes

2. if x < -215629.11924862463 or 20830.631465752547 < x

   1. Initial program 30.6

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

   2. Taylor expanded around inf 0.0

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

   if -215629.11924862463 < x < 20830.631465752547

   1. Initial program 0.0

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

   3. Applied flip3-+0.0

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

   4. Applied associate-/r/0.0

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

   5. Applied simplify0.0

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

4. Applied simplify0.0

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -215629.11924862463 \lor \neg \left(x \le 20830.631465752547\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 - x \cdot x\right)\right) \cdot \frac{x}{{\left(x \cdot x\right)}^{3} + 1}\\ \end{array}}\]

Runtime

Time bar (total: 39.5s)Debug logProfile

herbie shell --seed 2018214 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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