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Target

Derivation

1. Split input into 3 regimes

2. if eps < -1.0632206414980774e-82

   1. Initial program 30.7

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

   3. Applied tan-quot30.6

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

   4. Applied tan-sum6.7

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

   5. Applied frac-sub6.7

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

   if -1.0632206414980774e-82 < eps < 1.496652297597833e-63

   1. Initial program 47.7

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

   3. Applied tan-sum47.7

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

   4. Taylor expanded around 0 30.4

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

   5. Simplified30.4

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

   if 1.496652297597833e-63 < eps

   1. Initial program 29.4

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

   3. Applied tan-sum5.1

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

   5. Applied *-un-lft-identity5.1

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

   6. Applied *-un-lft-identity5.1

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

   7. Applied distribute-lft-out5.1

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

   8. Applied associate-/l*5.2

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

   10. Applied tan-quot5.2

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

   11. Applied associate-*r/5.2

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

4. Final simplification15.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.0632206414980774 \cdot 10^{-82}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.496652297597833 \cdot 10^{-63}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 39.9s)Debug logProfile

herbie shell --seed 2018351 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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