Error

Target

Derivation

1. Split input into 3 regimes

2. if (fma (* x eps) (fma (* x eps) eps eps) eps) < -8.53710463573885e-09

   1. Initial program 33.9

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

   3. Applied tan-sum7.8

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

   5. Applied flip--7.9

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

   6. Applied associate-/r/7.9

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

   7. Applied fma-neg7.9

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

   if -8.53710463573885e-09 < (fma (* x eps) (fma (* x eps) eps eps) eps) < 1.1108439280395142e-47

   1. Initial program 43.3

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

   2. Taylor expanded around 0 23.9

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

   3. Applied simplify22.6

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

   if 1.1108439280395142e-47 < (fma (* x eps) (fma (* x eps) eps eps) eps)

   1. Initial program 33.4

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

   3. Applied tan-sum9.2

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

   5. Applied add-cube-cbrt9.3

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

   7. Applied tan-quot9.4

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

   8. Applied frac-sub9.4

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

   9. Applied simplify9.4

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

   10. Applied simplify9.2

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

4. Applied simplify14.0

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le -8.53710463573885 \cdot 10^{-09}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}\right) \cdot \left(\tan \varepsilon \cdot \tan x + 1\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le 1.1108439280395142 \cdot 10^{-47}:\\ \;\;\;\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\tan \varepsilon + \tan x\right) \cdot \left(\cos x\right) + \left((\left(\tan x \cdot \sin x\right) \cdot \left(\tan \varepsilon\right) + \left(-\sin x\right))_*\right))_*}{(\left(-\cos x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\cos x\right))_*}\\ \end{array}}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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