Error

Try it out

Target

Derivation

1. Initial program 37.6

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

3. Applied diff-sin37.9

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

4. Applied simplify15.2

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

5. Taylor expanded around 0 15.1

   \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(x + \frac{1}{2} \cdot \varepsilon\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
6. Using strategy rm

7. Applied cos-sum0.3

   \[\leadsto 2 \cdot \left(\color{blue}{\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
8. Using strategy rm

9. Applied add-cbrt-cube0.4

   \[\leadsto 2 \cdot \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sin x \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

10. Applied add-cbrt-cube0.4

    \[\leadsto 2 \cdot \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}} \cdot \sqrt[3]{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

11. Applied cbrt-unprod0.4

    \[\leadsto 2 \cdot \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right) \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

12. Applied simplify0.4

    \[\leadsto 2 \cdot \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{\color{blue}{{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}^{3}}}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Runtime

Time bar (total: 43.4s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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