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Derivation

1. Split input into 2 regimes

2. if x < -0.033303858259176554 or 0.033056562733425277 < x

   1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]

   2. Initial simplification1.0

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

   4. Applied associate-/r*0.5

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

   if -0.033303858259176554 < x < 0.033056562733425277

   1. Initial program 61.2

      \[\frac{1 - \cos x}{x \cdot x}\]

   2. Initial simplification61.2

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

   4. Applied flip--61.2

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

   5. Applied associate-/l/61.2

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

   6. Simplified29.8

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

   8. Applied times-frac30.7

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

   9. Simplified30.7

      \[\leadsto \frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}\]

   10. 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 simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.033303858259176554 \lor \neg \left(x \le 0.033056562733425277\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \frac{1}{720} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Runtime

Time bar (total: 23.1s)Debug logProfile

herbie shell --seed 2018256 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))