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Derivation

1. Split input into 2 regimes

2. if eps < -6.030207407183357e-05 or 4.096802796851688e-05 < eps

   1. Initial program 30.9

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

   2. Initial simplification30.9

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

   4. Applied cos-sum0.9

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

   5. Applied associate--l-0.9

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

   if -6.030207407183357e-05 < eps < 4.096802796851688e-05

   1. Initial program 49.3

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

   2. Initial simplification49.3

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

   4. Applied diff-cos37.8

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

   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 log1p-expm1-u0.5

      \[\leadsto \color{blue}{\log_* (1 + (e^{-2 \cdot \left(\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)} - 1)^*)}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.030207407183357 \cdot 10^{-05} \lor \neg \left(\varepsilon \le 4.096802796851688 \cdot 10^{-05}\right):\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)} - 1)^*)\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 9.1s)Debug log