Derivation

Split input into 2 regimes
if x < -0.026062455830960528 or 0.03184327862110824 < x
1. Initial program 1.0
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm
3. Applied add-sqr-sqrt1.1
  \[\leadsto \frac{\color{blue}{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}}{x \cdot x}\]
4. Applied times-frac0.6
  \[\leadsto \color{blue}{\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}}\]
if -0.026062455830960528 < x < 0.03184327862110824
1. Initial program 61.3
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}}\]
3. Applied simplify0.0
  \[\leadsto \color{blue}{(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - x \cdot \left(\frac{1}{24} \cdot x\right)}\]
Recombined 2 regimes into one program.
Applied simplify0.3
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.026062455830960528 \lor \neg \left(x \le 0.03184327862110824\right):\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \mathbf{else}:\\ \;\;\;\;(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - \left(\frac{1}{24} \cdot x\right) \cdot x\\ \end{array}}\]

Runtime

herbie shell --seed 2018167 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))