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Derivation

1. Split input into 2 regimes

2. if eps < -0.00027952165095120534 or 2.1823998195395567e-05 < eps

   1. Initial program 30.1

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

   2. Initial simplification30.1

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

   4. Applied cos-sum0.9

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

   if -0.00027952165095120534 < eps < 2.1823998195395567e-05

   1. Initial program 49.2

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

   2. Initial simplification49.2

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

   4. Applied diff-cos37.6

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

   5. Simplified0.5

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

   7. Applied add-cube-cbrt1.6

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

4. Final simplification1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.00027952165095120534 \lor \neg \left(\varepsilon \le 2.1823998195395567 \cdot 10^{-05}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 19.7s)Debug logProfile

herbie shell --seed 2018340 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))