Error

Derivation

1. Split input into 3 regimes

2. if (- (* (cos x) (cos eps)) (+ (* (sin x) (sin eps)) (cos x))) < -0.023980124914600554

   1. Initial program 22.0

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

   3. Applied cos-sum0.5

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

   4. Applied associate--l-0.5

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

   if -0.023980124914600554 < (- (* (cos x) (cos eps)) (+ (* (sin x) (sin eps)) (cos x))) < 1.6445526954123046e-08

   1. Initial program 47.8

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

   3. Applied diff-cos36.8

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

   4. Applied simplify1.1

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

   if 1.6445526954123046e-08 < (- (* (cos x) (cos eps)) (+ (* (sin x) (sin eps)) (cos x)))

   1. Initial program 58.7

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

   3. Applied cos-sum1.0

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

Runtime

Time bar (total: 40.8s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))