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Derivation

1. Split input into 2 regimes

2. if x < -0.03247091185753463 or 0.03064339360211378 < x

   1. Initial program 1.1

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

   3. Applied associate-/r*0.5

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

   if -0.03247091185753463 < x < 0.03064339360211378

   1. Initial program 61.3

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

   3. Applied flip--61.3

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

   4. Applied associate-/l/61.3

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

   5. Simplified29.9

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

   7. Applied flip3-+29.9

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

   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 simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03247091185753463 \lor \neg \left(x \le 0.03064339360211378\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \frac{1}{720} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

Time bar (total: 19.6s)Debug log