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Target

Derivation

1. Split input into 3 regimes

2. if eps < -2.388080272206088e-71

   1. Initial program 30.5

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

   3. Applied tan-sum5.6

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

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

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

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

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

   7. Applied distribute-lft-out5.6

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

   8. Applied associate-/l*5.6

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

   if -2.388080272206088e-71 < eps < 8.03573431146359e-62

   1. Initial program 47.2

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

   2. Taylor expanded around 0 31.5

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

   3. Simplified31.4

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

   if 8.03573431146359e-62 < eps

   1. Initial program 31.4

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

   3. Applied tan-sum5.5

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

   4. Taylor expanded around -inf 5.5

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

   6. Applied associate-/l*5.5

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

   8. Applied tan-quot5.7

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

   9. Applied tan-quot5.7

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

   10. Applied frac-add5.7

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

4. Final simplification16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.388080272206088 \cdot 10^{-71}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 8.03573431146359 \cdot 10^{-62}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x \cdot \cos \varepsilon + \sin \varepsilon \cdot \cos x}{\cos x \cdot \cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos x \cdot \cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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

Details

Time bar (total: 33.1s)Debug log