Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -508.7686280028254 or 449.3527534388389 < x

   1. Initial program 30.3

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

   2. Taylor expanded around inf 0.0

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

   if -508.7686280028254 < x < 449.3527534388389

   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

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

4. Applied simplify0.0

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -508.7686280028254 \lor \neg \left(x \le 449.3527534388389\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - x \cdot x\right) + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{(\left({x}^{3}\right) \cdot \left({x}^{3}\right) + 1)_*}\\ \end{array}}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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