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Derivation

1. Split input into 2 regimes

2. if eps < -63538.036661244936 or 0.11700375221913029 < eps

   1. Initial program 30.3

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

   2. Initial simplification30.3

      \[\leadsto \cos \left(\varepsilon + x\right) - \cos x\]
   3. Using strategy rm

   4. Applied cos-sum0.8

      \[\leadsto \color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} - \cos x\]

   if -63538.036661244936 < eps < 0.11700375221913029

   1. Initial program 48.5

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

   2. Initial simplification48.5

      \[\leadsto \cos \left(\varepsilon + x\right) - \cos x\]
   3. Using strategy rm

   4. Applied diff-cos37.2

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

   5. Simplified0.8

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
   6. Using strategy rm

   7. Applied associate-*r*0.8

      \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -63538.036661244936 \lor \neg \left(\varepsilon \le 0.11700375221913029\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 17.6s)Debug logProfile

herbie shell --seed 2018227 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))