\[\cos \left(x + \varepsilon\right) - \cos x\]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -843.3970976391065:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\varepsilon \le 0.7431075649161416:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Derivation

Split input into 2 regimes
if eps < -843.3970976391065 or 0.7431075649161416 < eps
1. Initial program 29.9
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum0.8
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
if -843.3970976391065 < eps < 0.7431075649161416
1. Initial program 49.3
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied diff-cos38.5
  \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Applied simplify1.0
  \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)}\]
5. Using strategy rm
6. Applied associate-*r*1.0
  \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}\]
Recombined 2 regimes into one program.
Removed slow pow expressions.

Runtime

herbie shell --seed '#(1062930989 876886121 3990119081 3032829768 3060892583 1929069376)' 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Error

Derivation

`if eps < -843.3970976391065 or 0.7431075649161416 < eps`

`if -843.3970976391065 < eps < 0.7431075649161416`

Runtime