Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -1.5128669689931483e-47

   1. Initial program 29.8

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

   3. Applied tan-sum3.5

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

   5. Applied flip--3.6

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

   6. Applied associate-/r/3.5

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

   8. Applied tan-quot3.6

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

   9. Applied associate-*r/3.6

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

   11. Applied tan-quot3.6

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

   12. Applied associate-*r/3.6

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

   if -1.5128669689931483e-47 < eps < 5.068526259562942e-24

   1. Initial program 45.5

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

   3. Applied tan-sum45.5

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

   5. Applied flip--45.5

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

   6. Applied associate-/r/45.5

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

   8. Applied tan-quot45.5

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

   9. Applied associate-*r/45.5

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

   10. Taylor expanded around 0 30.2

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

   11. Simplified30.2

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

   if 5.068526259562942e-24 < eps

   1. Initial program 29.9

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   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 add-cbrt-cube1.6

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

   6. Applied add-cbrt-cube1.7

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

   7. Applied cbrt-unprod1.6

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

   8. Simplified1.6

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

4. Final simplification14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5128669689931483 \cdot 10^{-47}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} \cdot \left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} + 1\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 5.068526259562942 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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