\[\frac{1 - \cos x}{{x}^2}\]

⬇

\[\begin{array}{l} \mathbf{if}\;x \le -5.458497096453193 \cdot 10^{-06}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \le 85.36504497937187:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array}\]

Derivation

Split input into 2 regimes.
if x < -5.458497096453193e-06 or 85.36504497937187 < x
1. Initial program 1.2
  \[\frac{1 - \cos x}{{x}^2}\]
2. Using strategy rm
3. Applied square-mult 1.2
  \[\leadsto \frac{1 - \cos x}{\color{blue}{x \cdot x}}\]
4. Applied *-un-lft-identity 1.2
  \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{x \cdot x}\]
5. Applied times-frac 0.6
  \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
if -5.458497096453193e-06 < x < 85.36504497937187
1. Initial program 61.4
  \[\frac{1 - \cos x}{{x}^2}\]
2. Applied taylor 0.0
  \[\leadsto \left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^2\]
3. Taylor expanded around 0 0.0
  
  \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^2}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie --seed '#(2450642636 2863344379 4098708838 1734408276 15201110 2842944231)'
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (sqr x)))

Error

Derivation

`if x < -5.458497096453193e-06 or 85.36504497937187 < x`

`if -5.458497096453193e-06 < x < 85.36504497937187`

Runtime