Error

Try it out

Target

Derivation

1. Split input into 2 regimes

2. if (+ (* x (pow eps 2)) (+ eps (* (pow x 2) (pow eps 3)))) < -1.059607752271381e-24 or 5.431768626105551e-126 < (+ (* x (pow eps 2)) (+ eps (* (pow x 2) (pow eps 3))))

   1. Initial program 35.0

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

   3. Applied tan-sum14.1

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

   5. Applied add-cube-cbrt14.2

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

   6. Applied associate-*r*14.2

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\tan x \cdot \left(\sqrt[3]{\tan \varepsilon} \cdot \sqrt[3]{\tan \varepsilon}\right)\right) \cdot \sqrt[3]{\tan \varepsilon}}} - \tan x\]
   7. Using strategy rm

   8. Applied tan-quot14.3

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

   9. Applied frac-sub14.3

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

   10. Simplified13.2

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

   11. Simplified13.0

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

   if -1.059607752271381e-24 < (+ (* x (pow eps 2)) (+ eps (* (pow x 2) (pow eps 3)))) < 5.431768626105551e-126

   1. Initial program 42.3

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

   2. Taylor expanded around 0 16.9

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

4. Final simplification14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\varepsilon}^{2} \cdot x + \left(\varepsilon + {\varepsilon}^{3} \cdot {x}^{2}\right) \le -1.059607752271381 \cdot 10^{-24}:\\ \;\;\;\;\frac{\tan x \cdot \left(\tan \varepsilon \cdot \sin x\right) + \left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right)}{\cos x - \left(\tan \varepsilon \cdot \cos x\right) \cdot \tan x}\\ \mathbf{elif}\;{\varepsilon}^{2} \cdot x + \left(\varepsilon + {\varepsilon}^{3} \cdot {x}^{2}\right) \le 5.431768626105551 \cdot 10^{-126}:\\ \;\;\;\;{\varepsilon}^{2} \cdot x + \left(\varepsilon + {\varepsilon}^{3} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x \cdot \left(\tan \varepsilon \cdot \sin x\right) + \left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right)}{\cos x - \left(\tan \varepsilon \cdot \cos x\right) \cdot \tan x}\\ \end{array}\]

Runtime

Time bar (total: 56.2s)Debug logProfile

herbie shell --seed 2018216 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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