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Target

Derivation

1. Initial program 36.7

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

3. Applied diff-sin37.0

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

4. Simplified15.1

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

5. Taylor expanded around inf 15.1

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

6. Simplified15.1

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

8. Applied fma-udef15.1

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

9. Applied cos-sum0.3

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

11. Applied sub-neg0.3

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

12. Applied distribute-rgt-in0.3

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

13. Final simplification0.3

    \[\leadsto 2 \cdot \left(\left(\left(-\sin x\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\]

Runtime

Time bar (total: 22.0s)Debug logProfile

herbie shell --seed 2018351 +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)))