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Target

Derivation

1. Split input into 3 regimes

2. if eps < -2.404824026878926e-47

   1. Initial program 30.2

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

   2. Initial simplification30.2

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

   4. Applied tan-sum4.2

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

   6. Applied flip3--4.2

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

   7. Applied associate-/r/4.2

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

   8. Simplified4.3

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

   10. Applied tan-quot4.3

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

   11. Applied tan-quot4.3

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

   12. Applied associate-*l/4.3

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

   13. Applied associate-*l/4.3

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

   14. Applied flip-+4.3

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

   15. Applied frac-add4.3

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

   16. Applied associate-*r/4.3

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

   17. Applied frac-sub4.3

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

   18. Simplified4.3

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

   if -2.404824026878926e-47 < eps < 5.139869012831936e-33

   1. Initial program 46.2

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

   2. Initial simplification46.2

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

   3. Taylor expanded around 0 27.1

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

   4. Simplified27.1

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

   if 5.139869012831936e-33 < 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-sum2.7

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

   5. Taylor expanded around inf 2.7

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

   7. Applied tan-quot2.8

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

   8. Applied frac-sub2.8

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

4. Final simplification14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.404824026878926 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(\cos x \cdot \frac{\tan \varepsilon + \tan x}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \left(\left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \cos x + \left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \left(\left(\tan \varepsilon \cdot \sin x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right) - \left(\cos x \cdot \sin x\right) \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}{\left(\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 5.139869012831936 \cdot 10^{-33}:\\ \;\;\;\;\left(\varepsilon \cdot \frac{1}{3} + x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan \varepsilon + \tan x\right) \cdot \cos x - \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right) \cdot \sin x}{\cos x \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018352 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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