Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -3.004513004279039e+22 or 416.49421652298525 < x

   1. Initial program 30.7

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

   3. Applied add-cube-cbrt31.2

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

   4. Applied *-un-lft-identity31.2

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

   5. Applied times-frac31.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{x \cdot x + 1} \cdot \sqrt[3]{x \cdot x + 1}} \cdot \frac{x}{\sqrt[3]{x \cdot x + 1}}}\]

   6. Taylor expanded around inf 0.0

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

   7. Simplified0.0

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

   if -3.004513004279039e+22 < x < 416.49421652298525

   1. Initial program 0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.004513004279039 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 416.49421652298525:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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