Error

Target

Derivation

1. Initial program 37.4

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

2. Initial simplification37.4

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

4. Applied sin-sum21.8

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

5. Applied associate--l+0.4

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

7. Applied *-un-lft-identity0.4

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

8. Applied distribute-rgt-out--0.4

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

9. Taylor expanded around inf 21.8

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

10. Simplified0.4

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

11. Final simplification0.4

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

Runtime

Time bar (total: 20.7s)Debug logProfile

herbie shell --seed 2018353 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))