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Target

Derivation

1. Split input into 3 regimes

2. if eps < -5.381143162538014e-10

   1. Initial program 29.9

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

   3. Applied tan-sum0.5

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

   5. Applied *-un-lft-identity0.5

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

   6. Applied associate-/l*0.6

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

   8. Applied tan-quot0.6

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

   9. Applied associate-*l/0.6

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

   11. Applied div-sub0.7

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

   if -5.381143162538014e-10 < eps < 2.0047394400257063e-27

   1. Initial program 45.6

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

   3. Applied tan-sum45.5

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

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

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

   6. Applied associate-/l*45.7

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

   8. Applied tan-quot45.7

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

   9. Applied associate-*l/45.7

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

   10. Taylor expanded around 0 30.9

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

   11. Simplified30.8

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

   if 2.0047394400257063e-27 < eps

   1. Initial program 29.1

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

   3. Applied tan-sum1.9

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

   5. Applied *-un-lft-identity1.9

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

   6. Applied associate-/l*2.0

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

   8. Applied tan-quot2.0

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

   9. Applied associate-*l/2.0

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

   11. Applied flip--2.2

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

4. Final simplification15.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.381143162538014 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{1}{\tan x + \tan \varepsilon} - \frac{\frac{\sin x \cdot \tan \varepsilon}{\cos x}}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.0047394400257063 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}{\tan x + \tan \varepsilon}} \cdot \frac{1}{\frac{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}{\tan x + \tan \varepsilon}} - \tan x \cdot \tan x}{\frac{1}{\frac{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}{\tan x + \tan \varepsilon}} + \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(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: 38.7s)Debug log