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Target

Derivation

1. Initial program 36.8

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

2. Initial simplification36.8

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

4. Applied sin-sum21.6

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

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 add-log-exp14.3

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

8. Applied add-log-exp0.5

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

9. Applied diff-log0.5

   \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\log \left(\frac{e^{\cos \varepsilon \cdot \sin x}}{e^{\sin x}}\right)}\]

10. Simplified0.5

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

11. Final simplification0.5

    \[\leadsto \log \left(e^{\sin x \cdot \cos \varepsilon - \sin x}\right) + \cos x \cdot \sin \varepsilon\]

Runtime

Time bar (total: 23.9s)Debug logProfile

herbie shell --seed 2018285 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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