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Target

Derivation

1. Split input into 2 regimes

2. if x < -7200.610100540406 or 2038012.1201151453 < x

   1. Initial program 30.7

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

   3. Applied add-sqr-sqrt30.7

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

   4. Applied associate-/r*30.6

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

   5. Taylor expanded around -inf 0.0

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

   if -7200.610100540406 < x < 2038012.1201151453

   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. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7200.610100540406 \lor \neg \left(x \le 2038012.1201151453\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + {\left(x \cdot x\right)}^{3}} \cdot \left(1 - \left(x \cdot x - {x}^{4}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 28.2s)Debug logProfile

herbie shell --seed 2018346 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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