Error

Derivation

1. Split input into 3 regimes

2. if eps < -1.4194207403981303e-05

   1. Initial program 30.9

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

   3. Applied cos-sum0.9

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

   4. Applied associate--l-0.9

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

   5. Applied simplify0.9

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

   if -1.4194207403981303e-05 < eps < 2.6713101620310196e-06

   1. Initial program 48.8

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

   3. Applied diff-cos37.1

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

   4. Applied simplify0.6

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

   6. Applied log1p-expm1-u0.6

      \[\leadsto -2 \cdot \color{blue}{\log_* (1 + (e^{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)}\]

   if 2.6713101620310196e-06 < eps

   1. Initial program 30.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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: 47.9s)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))