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Target

Derivation

1. Initial program 37.0

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

3. Applied tan-sum21.8

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

4. Taylor expanded around inf 21.9

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

6. Applied div-inv22.0

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

7. Applied *-un-lft-identity22.0

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

8. Applied *-un-lft-identity22.0

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

9. Applied distribute-lft-out22.0

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

10. Applied prod-diff22.2

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

11. Simplified15.7

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

12. Simplified13.0

    \[\leadsto \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\right) + \color{blue}{0}\]
13. Using strategy rm

14. Applied associate-*r/13.0

    \[\leadsto \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \color{blue}{\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\right) + 0\]

15. Final simplification13.0

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

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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

Details

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