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Target

Derivation

1. Split input into 3 regimes

2. if eps < -25.400968012366302

   1. Initial program 29.8

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

   3. Applied sin-sum0.4

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

   4. Applied associate--l+0.4

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

   if -25.400968012366302 < eps < 9.406785545346194e-09

   1. Initial program 44.2

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

   3. Applied diff-sin44.2

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

   4. Simplified0.7

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

   if 9.406785545346194e-09 < eps

   1. Initial program 28.7

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

   3. Applied sin-sum0.5

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

4. Final simplification0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -25.400968012366302:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 9.406785545346194 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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

Details

Time bar (total: 14.7s)Debug log