Derivation

Split input into 2 regimes
if x < -0.03509091473179428 or 0.027662060737258805 < 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 -0.03509091473179428 < x < 0.027662060737258805
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. Simplified0.0
  \[\leadsto \color{blue}{(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - x \cdot \left(x \cdot \frac{1}{24}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03509091473179428 \lor \neg \left(x \le 0.027662060737258805\right):\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - \left(\frac{1}{24} \cdot x\right) \cdot x\\ \end{array}\]

Runtime

herbie shell --seed 2018214 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))