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Derivation

1. Split input into 2 regimes

2. if x < -0.030418058561901353 or 0.033075082181641506 < x

   1. Initial program 1.0

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

   3. Applied *-un-lft-identity1.0

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

   4. Applied times-frac0.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]

   if -0.030418058561901353 < x < 0.033075082181641506

   1. Initial program 61.4

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

   3. Applied flip--61.4

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

   4. Applied associate-/l/61.4

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

   5. Simplified29.3

      \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
   6. Using strategy rm

   7. Applied add-exp-log29.3

      \[\leadsto \frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \color{blue}{e^{\log \left(1 + \cos x\right)}}}\]

   8. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.030418058561901353 \lor \neg \left(x \le 0.033075082181641506\right):\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Runtime

Time bar (total: 42.3s)Debug logProfile

herbie shell --seed 2018263 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))