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Target

Derivation

1. Split input into 3 regimes

2. if eps < -3.4079781252857947e-40

   1. Initial program 30.4

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

   2. Initial simplification30.4

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

   4. Applied tan-sum3.0

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

   6. Applied flip3--3.1

      \[\leadsto \frac{\tan \varepsilon + \tan x}{\color{blue}{\frac{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}{1 \cdot 1 + \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)}}} - \tan x\]

   7. Applied associate-/r/3.1

      \[\leadsto \color{blue}{\frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} - \tan x\]

   8. Simplified3.1

      \[\leadsto \frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \color{blue}{\left(\left(1 + \tan x \cdot \tan \varepsilon\right) + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)} - \tan x\]
   9. Using strategy rm

   10. Applied unpow33.1

       \[\leadsto \frac{\tan \varepsilon + \tan x}{{1}^{3} - \color{blue}{\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}} \cdot \left(\left(1 + \tan x \cdot \tan \varepsilon\right) + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) - \tan x\]
   11. Using strategy rm

   12. Applied tan-quot3.1

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

   13. Applied tan-quot3.1

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

   14. Applied associate-*l/3.1

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

   15. Applied tan-quot3.1

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

   16. Applied associate-*l/3.1

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

   17. Applied frac-times3.2

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

   18. Applied flip-+3.2

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

   19. Applied frac-add3.2

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

   20. Applied frac-times3.1

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

   21. Applied frac-sub3.2

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

   22. Simplified3.2

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

   23. Simplified3.2

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

   if -3.4079781252857947e-40 < eps < 7.036047147535559e-18

   1. Initial program 45.9

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

   2. Initial simplification45.9

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

   4. Applied tan-sum45.9

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

   5. Taylor expanded around 0 28.4

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

   6. Simplified28.4

      \[\leadsto \color{blue}{\varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \frac{1}{3} \cdot \varepsilon\right)}\]

   if 7.036047147535559e-18 < eps

   1. Initial program 30.3

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

   2. Initial simplification30.3

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

   4. Applied tan-sum1.0

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

   6. Applied flip3--1.0

      \[\leadsto \frac{\tan \varepsilon + \tan x}{\color{blue}{\frac{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}{1 \cdot 1 + \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)}}} - \tan x\]

   7. Applied associate-/r/1.0

      \[\leadsto \color{blue}{\frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} - \tan x\]

   8. Simplified1.0

      \[\leadsto \frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \color{blue}{\left(\left(1 + \tan x \cdot \tan \varepsilon\right) + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)} - \tan x\]
   9. Using strategy rm

   10. Applied unpow31.0

       \[\leadsto \frac{\tan \varepsilon + \tan x}{{1}^{3} - \color{blue}{\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}} \cdot \left(\left(1 + \tan x \cdot \tan \varepsilon\right) + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) - \tan x\]
3. Recombined 3 regimes into one program.

4. Final simplification14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.4079781252857947 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\cos x \cdot \left(\left(\tan x + \tan \varepsilon\right) \cdot \left(\cos x \cdot \cos x\right)\right)\right) + \left(\cos x \cdot \left(\left(\tan \varepsilon \cdot \sin x\right) \cdot \left(\tan x + \tan \varepsilon\right)\right)\right) \cdot \left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan \varepsilon \cdot \sin x\right)\right)\right) - \left(1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right) \cdot \left(\left(\left(\cos x \cdot \cos x\right) \cdot \sin x\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right)}{\left(1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right) \cdot \left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right)\right)}\\ \mathbf{elif}\;\varepsilon \le 7.036047147535559 \cdot 10^{-18}:\\ \;\;\;\;\varepsilon + \left(x + \frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(\left(\tan x \cdot \tan \varepsilon + 1\right) + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) - \tan x\\ \end{array}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018346 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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