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Target

Derivation

1. Initial program 36.4

   \[\tan \left(x + \varepsilon\right) - \tan x\]

2. Initial simplification36.4

   \[\leadsto \tan \left(\varepsilon + x\right) - \tan x\]
3. Using strategy rm

4. Applied tan-sum21.1

   \[\leadsto \color{blue}{\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x}} - \tan x\]
5. Using strategy rm

6. Applied tan-quot21.2

   \[\leadsto \frac{\tan \varepsilon + \color{blue}{\frac{\sin x}{\cos x}}}{1 - \tan \varepsilon \cdot \tan x} - \tan x\]

7. Applied tan-quot21.3

   \[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} + \frac{\sin x}{\cos x}}{1 - \tan \varepsilon \cdot \tan x} - \tan x\]

8. Applied frac-add21.3

   \[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}}}{1 - \tan \varepsilon \cdot \tan x} - \tan x\]

9. Applied associate-/l/21.3

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

11. Applied tan-quot21.2

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

12. Applied frac-sub21.2

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

13. Taylor expanded around -inf 0.4

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

14. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sin \varepsilon}{\cos x}}{\cos \varepsilon \cdot \cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}}\]

15. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{\sin \varepsilon}{\cos x}}{\cos x \cdot \cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}\]

Runtime

Time bar (total: 44.7s)Debug logProfile

herbie shell --seed 2018295 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

  (- (tan (+ x eps)) (tan x)))