Error

Derivation

1. Initial program 39.9

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

3. Applied diff-cos34.6

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

4. Simplified15.2

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

5. Taylor expanded around inf 15.2

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

6. Simplified15.1

   \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left((\frac{1}{2} \cdot \varepsilon + x)_*\right)\right)}\]
7. Using strategy rm

8. Applied fma-udef15.1

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

9. Applied sin-sum0.4

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

10. Taylor expanded around inf 0.4

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

11. Simplified0.4

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

12. Final simplification0.4

    \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot (\left(\sin x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right))_*\right) \cdot -2\]

Runtime

Time bar (total: 20.4s)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))