Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -1.1336421505985684e-48

   1. Initial program 29.5

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

   3. Applied tan-sum3.1

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

   5. Applied flip--3.1

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

   6. Applied associate-/r/3.1

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

   8. Applied tan-quot3.1

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

   9. Applied associate-*r/3.1

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

   10. Applied tan-quot3.1

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

   11. Applied associate-*r/3.2

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

   12. Applied frac-times3.2

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

   14. Applied tan-quot3.1

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

   15. Applied associate-*r/3.1

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

   if -1.1336421505985684e-48 < eps < 1.7863185171509975e-61

   1. Initial program 47.1

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

   3. Applied tan-sum47.1

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

   5. Applied flip--47.1

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

   6. Applied associate-/r/47.1

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

   8. Applied tan-quot47.1

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

   9. Applied associate-*r/47.1

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

   10. Applied tan-quot47.1

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

   11. Applied associate-*r/47.1

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

   12. Applied frac-times47.1

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

   13. Taylor expanded around 0 30.9

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

   14. Simplified30.9

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

   if 1.7863185171509975e-61 < eps

   1. Initial program 30.2

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

   3. Applied tan-sum5.8

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

   5. Applied flip--5.9

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

4. Final simplification15.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1336421505985684 \cdot 10^{-48}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \frac{\left(\tan x \cdot \sin \varepsilon\right) \cdot \left(\tan x \cdot \sin \varepsilon\right)}{\cos \varepsilon \cdot \cos \varepsilon}} \cdot \left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} + 1\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 1.7863185171509975 \cdot 10^{-61}:\\ \;\;\;\;\left(\left(x + \varepsilon\right) \cdot \varepsilon\right) \cdot x + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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