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Target

Derivation

1. Split input into 2 regimes

2. if eps < -4.381173797888801e-25 or 1.3072855954539968e-17 < eps

   1. Initial program 30.5

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

   2. Initial simplification30.5

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

   4. Applied tan-sum1.5

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

   6. Applied flip3--1.5

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

   7. Applied associate-/r/1.5

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

   8. Simplified1.5

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

   10. Applied add-cbrt-cube1.6

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

   11. Applied add-cbrt-cube1.6

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

   12. Applied cbrt-unprod1.6

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

   13. Applied rem-cube-cbrt1.5

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

   14. Simplified1.5

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

   if -4.381173797888801e-25 < eps < 1.3072855954539968e-17

   1. Initial program 45.4

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

   2. Initial simplification45.4

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

   4. Applied tan-sum45.4

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

   6. Applied tan-quot45.4

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

   7. Applied associate-*r/45.4

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

   8. Taylor expanded around 0 27.7

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

   9. Simplified27.7

      \[\leadsto \color{blue}{\varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \frac{1}{3} \cdot \varepsilon\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.381173797888801 \cdot 10^{-25} \lor \neg \left(\varepsilon \le 1.3072855954539968 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \left(\tan x \cdot \tan x\right)\right) \cdot {\left(\tan \varepsilon\right)}^{3}} \cdot \left(\left(1 + \tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \frac{1}{3} + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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

Details

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