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Target

Derivation

1. Split input into 3 regimes

2. if eps < -2.6102094036101758e-58

   1. Initial program 29.5

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

   3. Applied tan-sum4.6

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

   5. Applied *-un-lft-identity4.6

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

   6. Applied *-un-lft-identity4.6

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

   7. Applied distribute-lft-out4.6

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

   8. Applied associate-/l*4.6

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

   10. Applied tan-quot4.7

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

   11. Applied frac-sub4.7

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

   12. Simplified4.7

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

   if -2.6102094036101758e-58 < eps < 2.7120892299828264e-22

   1. Initial program 45.4

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

   3. Applied tan-sum45.4

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

   5. Applied *-un-lft-identity45.4

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

   6. Applied *-un-lft-identity45.4

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

   7. Applied distribute-lft-out45.4

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

   8. Applied associate-/l*45.6

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

   9. Taylor expanded around 0 30.4

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

   10. Simplified30.3

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

   if 2.7120892299828264e-22 < eps

   1. Initial program 29.6

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

   3. Applied tan-sum1.3

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

   5. Applied tan-quot1.3

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

   6. Applied frac-sub1.3

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

4. Final simplification15.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.6102094036101758 \cdot 10^{-58}:\\ \;\;\;\;\frac{\cos x - \frac{1 - \tan x \cdot \tan \varepsilon}{\frac{\tan x + \tan \varepsilon}{\sin x}}}{\cos x \cdot \frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}}\\ \mathbf{elif}\;\varepsilon \le 2.7120892299828264 \cdot 10^{-22}:\\ \;\;\;\;\left(x + \varepsilon\right) \cdot \left(x \cdot \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan x + \tan \varepsilon\right) - \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Runtime

Time bar (total: 40.5s)Debug logProfile

herbie shell --seed 2018290 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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