Error

Try it out

Target

Derivation

1. Split input into 3 regimes

2. if eps < -5.44910382112671e-54

   1. Initial program 30.8

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

   2. Initial simplification30.8

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

   4. Applied tan-sum4.5

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

   6. Applied *-un-lft-identity4.5

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

   7. Applied *-un-lft-identity4.5

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

   8. Applied distribute-lft-out4.5

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

   9. Applied associate-/l*4.6

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

   11. Applied tan-quot4.6

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

   12. Applied frac-sub4.7

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

   13. Simplified4.7

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

   if -5.44910382112671e-54 < eps < 8.057210057566332e-125

   1. Initial program 48.4

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

   2. Initial simplification48.4

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

   4. Applied tan-sum48.4

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

   5. Taylor expanded around 0 32.5

      \[\leadsto \color{blue}{\varepsilon + \left({x}^{3} \cdot {\varepsilon}^{2} + \left({x}^{2} \cdot \varepsilon + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\frac{4}{3} \cdot \left({x}^{2} \cdot {\varepsilon}^{3}\right) + {x}^{4} \cdot {\varepsilon}^{3}\right)\right)\right)\right)}\]

   6. Simplified32.5

      \[\leadsto \color{blue}{\left(\left(\varepsilon + {x}^{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \left(\left(x \cdot x\right) \cdot \varepsilon + \left(\frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + \left(\left(x \cdot \frac{4}{3}\right) \cdot x + {x}^{4}\right) \cdot {\varepsilon}^{3}}\]

   if 8.057210057566332e-125 < eps

   1. Initial program 32.1

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

   2. Initial simplification32.1

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

   4. Applied tan-sum9.6

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

   5. Taylor expanded around inf 9.6

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

   7. Applied tan-quot9.6

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

   8. Applied frac-sub9.6

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

4. Final simplification16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.44910382112671 \cdot 10^{-54}:\\ \;\;\;\;\frac{\cos x - \frac{1 - \tan \varepsilon \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{\sin x}}}{\cos x \cdot \frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}}\\ \mathbf{elif}\;\varepsilon \le 8.057210057566332 \cdot 10^{-125}:\\ \;\;\;\;\left(x \cdot \left(\frac{4}{3} \cdot x\right) + {x}^{4}\right) \cdot {\varepsilon}^{3} + \left(\left(\left(x \cdot x\right) \cdot \varepsilon + \left(\frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {x}^{3} + \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \sin x \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon}\right)}{\left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}\\ \end{array}\]

Runtime

Time bar (total: 46.1s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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