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Target

Derivation

1. Split input into 2 regimes

2. if x < -578.1237044981676 or 402.7494417162069 < x

   1. Initial program 30.5

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

   2. Taylor expanded around inf 0.0

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

   if -578.1237044981676 < x < 402.7494417162069

   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. Applied simplify0.0

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

4. Applied simplify0.0

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -578.1237044981676 \lor \neg \left(x \le 402.7494417162069\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1^2 + x^2}^*}}{\sqrt{1 + x \cdot x}}\\ \end{array}}\]

Runtime

Time bar (total: 41.7s)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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