Error

Try it out

Target

Derivation

1. Split input into 3 regimes

2. if eps < -1.347520604115508e-19

   1. Initial program 29.0

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

   2. Initial simplification29.0

      \[\leadsto \tan \left(\varepsilon + x\right) - \tan x\]
   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 *-un-lft-identity1.0

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

   7. Applied associate-/l*1.1

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

   if -1.347520604115508e-19 < eps < 4.1131778345912215e-18

   1. Initial program 45.0

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

   2. Initial simplification45.0

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

   4. Applied tan-sum45.0

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

   6. Applied flip3--45.0

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

   7. Applied associate-/r/45.0

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

   8. Simplified45.0

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

   9. Taylor expanded around 0 27.9

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

   10. Simplified27.9

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

   if 4.1131778345912215e-18 < eps

   1. Initial program 30.0

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

   2. Initial simplification30.0

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

   4. Applied tan-sum1.2

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

   6. Applied flip3--1.2

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

   7. Applied associate-/r/1.2

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

   8. Simplified1.2

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

   10. Applied add-cbrt-cube1.2

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

   12. Applied tan-quot1.2

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

   13. Applied tan-quot1.3

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

   14. Applied frac-times1.2

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

   15. Applied cube-div1.2

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

   16. Applied tan-quot1.3

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

   17. Applied associate-*r/1.3

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

   18. Applied cube-div1.3

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

   19. Applied tan-quot1.3

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

   20. Applied associate-*r/1.3

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

   21. Applied cube-div1.3

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

   22. Applied frac-times1.3

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

   23. Applied frac-times1.3

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

   24. Applied cbrt-div1.3

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

4. Final simplification13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.347520604115508 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 4.1131778345912215 \cdot 10^{-18}:\\ \;\;\;\;\varepsilon + \left(x + \varepsilon \cdot \frac{1}{3}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \left(1 + \tan x \cdot \tan \varepsilon\right)\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - \frac{\sqrt[3]{{\left(\sin \varepsilon \cdot \sin x\right)}^{3} \cdot \left({\left(\sin x \cdot \tan \varepsilon\right)}^{3} \cdot {\left(\sin x \cdot \tan \varepsilon\right)}^{3}\right)}}{\sqrt[3]{{\left(\cos \varepsilon \cdot \cos x\right)}^{3} \cdot \left({\left(\cos x\right)}^{3} \cdot {\left(\cos x\right)}^{3}\right)}}} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018354 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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