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Target

Derivation

1. Split input into 3 regimes

2. if eps < -3.848860041120697e-74

   1. Initial program 31.0

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

   2. Initial simplification31.0

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

   4. Applied tan-quot30.8

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

   5. Applied tan-sum5.9

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

   6. Applied frac-sub6.0

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

   if -3.848860041120697e-74 < eps < 1.0275046713596955e-89

   1. Initial program 48.0

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

   2. Initial simplification48.0

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

   4. Applied tan-sum48.0

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

   6. Applied tan-quot48.0

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

   7. Applied associate-*l/48.0

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

   9. Applied add-cbrt-cube48.0

      \[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \color{blue}{\sqrt[3]{\left(\frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}}}} - \tan x\]

   10. Taylor expanded around 0 27.8

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

   11. Simplified27.8

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

   if 1.0275046713596955e-89 < eps

   1. Initial program 30.3

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

   2. Initial simplification30.3

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

   4. Applied tan-sum7.3

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

   6. Applied *-un-lft-identity7.3

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

   7. Applied associate-/l*7.3

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

4. Final simplification14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.848860041120697 \cdot 10^{-74}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \sin x}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.0275046713596955 \cdot 10^{-89}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon \cdot \frac{1}{3}\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 38.5s)Debug logProfile

herbie shell --seed 2018297 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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