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Target

Derivation

1. Initial program 36.6

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

2. Initial simplification36.6

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

4. Applied tan-sum21.3

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

6. Applied flip--21.3

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

7. Applied associate-/r/21.3

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

8. Taylor expanded around -inf 21.4

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

9. Simplified0.6

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

11. Applied frac-sub2.4

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

12. Simplified0.5

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

13. Final simplification0.5

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018230 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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