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Target

Derivation

1. Split input into 2 regimes

2. if eps < -3.0289062418692434e-15 or 1.8235088267822702e-58 < eps

   1. Initial program 30.1

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

   2. Initial simplification30.1

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

   4. Applied tan-sum2.7

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

   6. Applied tan-quot2.7

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

   7. Applied associate-*l/2.7

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

   9. Applied *-un-lft-identity2.7

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

   10. Applied associate-/l*2.8

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

   if -3.0289062418692434e-15 < eps < 1.8235088267822702e-58

   1. Initial program 45.6

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

   2. Initial simplification45.6

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

   4. Applied tan-sum45.6

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

   5. Taylor expanded around 0 27.2

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

   6. Simplified27.2

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

4. Final simplification13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.0289062418692434 \cdot 10^{-15} \lor \neg \left(\varepsilon \le 1.8235088267822702 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{1}{\frac{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \frac{1}{3} + x\right) + \varepsilon\\ \end{array}\]

Runtime

Time bar (total: 36.2s)Debug logProfile

herbie shell --seed 2018339 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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