Error

Derivation

1. Split input into 2 regimes

2. if eps < -0.15281236229617784 or 0.00023034535512454643 < eps

   1. Initial program 30.4

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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 -0.15281236229617784 < eps < 0.00023034535512454643

   1. Initial program 49.0

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

   3. Applied diff-cos37.5

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

   4. Applied simplify0.7

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

4. Applied simplify0.7

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.15281236229617784 \lor \neg \left(\varepsilon \le 0.00023034535512454643\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}}\]

Runtime

Time bar (total: 34.7s)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))