Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -1.6209673985070623e-48

   1. Initial program 29.7

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

   3. Applied tan-sum3.6

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

   5. Applied flip3--3.7

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

   6. Applied associate-/r/3.7

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

   7. Applied fma-neg3.6

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

   9. Applied tan-quot3.7

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

   10. Applied associate-*r/3.7

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

   11. Applied cube-div3.7

       \[\leadsto (\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - \color{blue}{\frac{{\left(\tan x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3}}}}\right) \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) + \left(-\tan x\right))_*\]
   12. Using strategy rm

   13. Applied add-cube-cbrt3.9

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

   14. Applied unpow-prod-down3.9

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

   15. Simplified3.7

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

   if -1.6209673985070623e-48 < eps < 5.179340602069793e-69

   1. Initial program 47.7

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

   3. Applied tan-sum47.7

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

   4. Taylor expanded around 0 31.0

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

   5. Simplified31.0

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

   if 5.179340602069793e-69 < eps

   1. Initial program 31.0

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

   3. Applied tan-sum5.9

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

   5. Applied flip3--5.9

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

   6. Applied associate-/r/5.9

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

   7. Applied fma-neg5.9

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

   9. Applied tan-quot5.9

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

   10. Applied associate-*r/5.9

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

   11. Applied cube-div5.9

       \[\leadsto (\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - \color{blue}{\frac{{\left(\tan x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3}}}}\right) \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) + \left(-\tan x\right))_*\]
   12. Using strategy rm

   13. Applied expm1-log1p-u5.9

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

4. Final simplification15.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.6209673985070623 \cdot 10^{-48}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \frac{{\left(\tan x \cdot \sin \varepsilon\right)}^{3}}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot {\left(\sqrt[3]{\cos \varepsilon}\right)}^{3}}}\right) \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 5.179340602069793 \cdot 10^{-69}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \frac{{\left(\tan x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3}}}\right) \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + (e^{\log_* (1 + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right))} - 1)^*\right)\right) + \left(-\tan x\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018349 +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)))