Error

Try it out

Target

Derivation

1. Split input into 2 regimes

2. if eps < -6.820762576371901e-09 or 1.5103998066481984e-21 < eps

   1. Initial program 29.4

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

   3. Applied sin-sum0.9

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

   if -6.820762576371901e-09 < eps < 1.5103998066481984e-21

   1. Initial program 45.2

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

   3. Applied diff-sin45.2

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

   4. Simplified0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
   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-log-exp0.5

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

4. Final simplification0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.820762576371901 \cdot 10^{-09} \lor \neg \left(\varepsilon \le 1.5103998066481984 \cdot 10^{-21}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\log \left(e^{\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)}\right) \cdot \left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)}\right) \cdot 2\\ \end{array}\]

Runtime

Time bar (total: 23.6s)Debug logProfile

herbie shell --seed 2018220 +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)))