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Derivation

1. Split input into 2 regimes

2. if eps < -3.3020901580082745e-05 or 0.02011113661874671 < eps

   1. Initial program 29.8

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

   3. Applied cos-sum0.9

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

   if -3.3020901580082745e-05 < eps < 0.02011113661874671

   1. Initial program 49.5

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

   3. Applied diff-cos37.9

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

   4. Simplified0.6

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

4. Final simplification0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.3020901580082745 \cdot 10^{-05} \lor \neg \left(\varepsilon \le 0.02011113661874671\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 17.0s)Debug log