Error

Try it out

Target

Derivation

1. Split input into 3 regimes

2. if eps < -2.2900358492817126e-29

   1. Initial program 29.9

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

   2. Initial simplification29.9

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

   4. Applied tan-sum2.3

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

   6. Applied flip3--2.4

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

   7. Applied associate-/r/2.4

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

   8. Simplified2.4

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

   10. Applied add-cbrt-cube2.5

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

   11. Applied add-cbrt-cube2.5

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

   12. Applied cbrt-unprod2.4

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

   13. Applied rem-cube-cbrt2.4

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

   14. Simplified2.4

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

   16. Applied add-cbrt-cube2.4

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

   if -2.2900358492817126e-29 < eps < 5.316516443760049e-33

   1. Initial program 45.2

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

   2. Initial simplification45.2

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

   3. Taylor expanded around 0 27.4

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

   4. Simplified27.4

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

   if 5.316516443760049e-33 < eps

   1. Initial program 29.4

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

   2. Initial simplification29.4

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

   4. Applied tan-sum2.5

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

   6. Applied flip3--2.6

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

   7. Applied associate-/r/2.6

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

   8. Simplified2.6

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

   10. Applied add-cbrt-cube2.7

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

   11. Applied add-cbrt-cube2.7

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

   12. Applied cbrt-unprod2.7

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

   13. Applied rem-cube-cbrt2.6

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

   14. Simplified2.6

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

   16. Applied tan-quot2.6

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

   17. Applied tan-quot2.6

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

   18. Applied associate-*r/2.6

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

   19. Applied associate-*l/2.6

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

   20. Applied flip-+2.6

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

   21. Applied frac-add2.7

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

   22. Applied associate-*r/2.7

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

   23. Applied frac-sub2.7

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

   24. Simplified2.7

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

4. Final simplification13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.2900358492817126 \cdot 10^{-29}:\\ \;\;\;\;\left(\left(1 + \tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{{\left(\tan \varepsilon\right)}^{3} \cdot \left({\left(\tan \varepsilon\right)}^{3} \cdot {\left(\tan \varepsilon\right)}^{3}\right)} \cdot \left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le 5.316516443760049 \cdot 10^{-33}:\\ \;\;\;\;\left(\varepsilon \cdot \frac{1}{3} + x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan x \cdot \sin \varepsilon\right)\right) + \cos \varepsilon \cdot \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right) \cdot \frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right)}{1 - {\left(\tan x\right)}^{3} \cdot {\left(\tan \varepsilon\right)}^{3}} - \left(\cos \varepsilon \cdot \sin x\right) \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}{\left(\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018278 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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