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Target

Derivation

1. Split input into 3 regimes

2. if (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x)) < -4.037825951534793e-07

   1. Initial program 30.2

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

   3. Applied sin-sum0.5

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

   if -4.037825951534793e-07 < (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x)) < 1.960167980894928e-13

   1. Initial program 44.8

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

   3. Applied diff-sin44.8

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

   4. Simplified0.3

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

   if 1.960167980894928e-13 < (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x))

   1. Initial program 29.4

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

   3. Applied sin-sum0.7

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

   4. Applied associate--l+0.7

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

4. Final simplification0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x \le -4.037825951534793 \cdot 10^{-07}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x\\ \mathbf{elif}\;\left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x \le 1.960167980894928 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \sin x + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Runtime

Time bar (total: 25.0s)Debug logProfile

herbie shell --seed 2018217 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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