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Target

Derivation

1. Split input into 2 regimes

2. if eps < -1.1997626042950266e-08 or 7.806571631081903e-09 < eps

   1. Initial program 29.4

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

   3. Applied sin-sum0.5

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

   4. Applied associate--l+0.5

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

   if -1.1997626042950266e-08 < eps < 7.806571631081903e-09

   1. Initial program 44.0

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

   3. Applied diff-sin44.1

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

   4. Simplified0.3

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

   6. Applied add-cbrt-cube0.5

      \[\leadsto 2 \cdot \left(\color{blue}{\sqrt[3]{\left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)}} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
   7. Using strategy rm

   8. Applied add-cbrt-cube0.4

      \[\leadsto 2 \cdot \left(\sqrt[3]{\left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)}}} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1997626042950266 \cdot 10^{-08} \lor \neg \left(\varepsilon \le 7.806571631081903 \cdot 10^{-09}\right):\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)} \cdot \left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)}\right) \cdot 2\\ \end{array}\]

Runtime

Time bar (total: 22.0s)Debug logProfile

herbie shell --seed 2018290 +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)))