Error

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Target

Derivation

1. Split input into 3 regimes

2. if eps < -1.890229421761351e-33

   1. Initial program 30.6

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

   3. Applied tan-sum2.5

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

   5. Applied flip3--2.6

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

   6. Applied associate-/r/2.5

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

   7. Simplified2.5

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

   9. Applied add-cbrt-cube2.6

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

   10. Applied add-cbrt-cube2.7

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

   11. Applied cbrt-unprod2.6

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

   12. Applied rem-cube-cbrt2.6

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

   13. Simplified2.5

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

   15. Applied tan-quot2.6

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

   16. Applied tan-quot2.6

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

   17. Applied associate-*l/2.6

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

   18. Applied tan-quot2.6

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

   19. Applied associate-*l/2.6

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

   20. Applied frac-times2.6

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

   21. Applied flip-+2.6

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

   22. Applied frac-add2.7

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

   23. Applied associate-*r/2.7

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

   24. Applied frac-sub2.7

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

   25. Simplified2.7

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

   if -1.890229421761351e-33 < eps < 1.374570900499627e-25

   1. Initial program 46.0

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

   3. Applied tan-sum46.0

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

   5. Applied flip3--46.0

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

   6. Applied associate-/r/46.0

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

   7. Simplified46.0

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

   9. Applied add-cbrt-cube46.0

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

   10. Applied add-cbrt-cube46.0

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

   11. Applied cbrt-unprod46.0

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

   12. Applied rem-cube-cbrt46.0

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

   13. Simplified46.0

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

   14. Taylor expanded around 0 30.5

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

   15. Simplified30.4

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

   if 1.374570900499627e-25 < eps

   1. Initial program 29.8

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

   3. Applied tan-sum1.6

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

   5. Applied flip3--1.6

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

   6. Applied associate-/r/1.6

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

   7. Simplified1.7

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

   9. Applied add-cbrt-cube1.7

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

   10. Applied add-cbrt-cube1.8

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

   11. Applied cbrt-unprod1.7

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

   12. Applied rem-cube-cbrt1.7

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

   13. Simplified1.6

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

   15. Applied tan-quot1.7

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

   16. Applied tan-quot1.7

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

   17. Applied associate-*l/1.7

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

   18. Applied tan-quot1.7

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

   19. Applied associate-*r/1.7

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

   20. Applied frac-times1.7

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

   21. Applied flip-+1.7

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

   22. Applied frac-add1.7

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

   23. Applied associate-*r/1.7

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

   24. Applied frac-sub1.8

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

   25. Simplified1.8

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

4. Final simplification15.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.890229421761351 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\cos x \cdot \left(\tan x + \tan \varepsilon\right)}{1 - {\left(\tan \varepsilon\right)}^{3} \cdot {\left(\tan x\right)}^{3}} \cdot \left(\left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\cos \varepsilon \cdot \cos \varepsilon\right) + \left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \left(\left(\sin \varepsilon \cdot \tan x\right) \cdot \left(\sin \varepsilon \cdot \tan x\right)\right)\right) - \left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \left(\sin x \cdot \left(\cos \varepsilon \cdot \cos \varepsilon\right)\right)}{\cos x \cdot \left(\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)\right)}\\ \mathbf{elif}\;\varepsilon \le 1.374570900499627 \cdot 10^{-25}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\tan x + \tan \varepsilon\right)\right) + \left(\left(\sin x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\left(\tan x + \tan \varepsilon\right) \cdot \sin \varepsilon\right)}{\frac{1 - {\left(\tan \varepsilon\right)}^{3} \cdot {\left(\tan x\right)}^{3}}{\cos x}} - \left(\left(\sin x \cdot \cos x\right) \cdot \cos \varepsilon\right) \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}{\cos x \cdot \left(\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018355 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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