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Derivation

1. Split input into 2 regimes

2. if x < -0.01262787248960153 or 1.853550105999067e-15 < x

   1. Initial program 2.0

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

   3. Applied flip--2.2

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

   4. Applied associate-/l/2.2

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

   5. Simplified1.1

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

   7. Applied times-frac1.1

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

   8. Simplified0.9

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

   if -0.01262787248960153 < x < 1.853550105999067e-15

   1. Initial program 61.7

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

   3. Applied flip--61.7

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

   4. Applied associate-/l/61.7

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

   5. Simplified30.7

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

   6. 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 simplification0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.01262787248960153 \lor \neg \left(x \le 1.853550105999067 \cdot 10^{-15}\right):\\ \;\;\;\;\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

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