Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -6.533455973501571e-37

   1. Initial program 29.6

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

   3. Applied tan-sum2.9

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

   5. Applied add-sqr-sqrt31.9

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

   6. Applied flip3--31.9

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

   7. Applied associate-/r/31.9

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

   8. Applied prod-diff31.9

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

   9. Simplified31.8

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

   10. Simplified2.9

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right) + \color{blue}{0}\]
   11. Using strategy rm

   12. Applied tan-quot2.9

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{1 - {\left(\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \tan x\right)}^{3}} - \tan x\right) + 0\]

   13. Applied associate-*l/2.9

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{1 - {\color{blue}{\left(\frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}\right)}}^{3}} - \tan x\right) + 0\]

   14. Applied cube-div2.9

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{1 - \color{blue}{\frac{{\left(\sin \varepsilon \cdot \tan x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3}}}} - \tan x\right) + 0\]

   if -6.533455973501571e-37 < eps < 1.0018624295577554e-47

   1. Initial program 46.2

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

   3. Applied tan-sum46.2

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

   5. Applied add-sqr-sqrt54.9

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

   6. Applied flip3--54.9

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

   7. Applied associate-/r/54.9

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

   8. Applied prod-diff55.1

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

   9. Simplified55.1

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

   10. Simplified46.2

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right) + \color{blue}{0}\]
   11. Using strategy rm

   12. Applied flip3--46.2

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{\color{blue}{\frac{{1}^{3} - {\left({\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}^{3}}{1 \cdot 1 + \left({\left(\tan \varepsilon \cdot \tan x\right)}^{3} \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3} + 1 \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}}} - \tan x\right) + 0\]

   13. Applied associate-/r/46.2

       \[\leadsto \left(\color{blue}{\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{{1}^{3} - {\left({\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}^{3}} \cdot \left(1 \cdot 1 + \left({\left(\tan \varepsilon \cdot \tan x\right)}^{3} \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3} + 1 \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)\right)} - \tan x\right) + 0\]

   14. Applied fma-neg46.2

       \[\leadsto \color{blue}{(\left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{{1}^{3} - {\left({\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}^{3}}\right) \cdot \left(1 \cdot 1 + \left({\left(\tan \varepsilon \cdot \tan x\right)}^{3} \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3} + 1 \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)\right) + \left(-\tan x\right))_*} + 0\]

   15. Taylor expanded around 0 28.1

       \[\leadsto \color{blue}{\left(x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)\right)} + 0\]

   16. Simplified28.1

       \[\leadsto \color{blue}{(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*} + 0\]

   if 1.0018624295577554e-47 < eps

   1. Initial program 29.9

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   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 add-sqr-sqrt34.1

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

   6. Applied flip3--34.2

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

   7. Applied associate-/r/34.2

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

   8. Applied prod-diff34.1

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

   9. Simplified34.1

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

   10. Simplified3.6

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right) + \color{blue}{0}\]
   11. Using strategy rm

   12. Applied flip3--3.6

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{\color{blue}{\frac{{1}^{3} - {\left({\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}^{3}}{1 \cdot 1 + \left({\left(\tan \varepsilon \cdot \tan x\right)}^{3} \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3} + 1 \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}}} - \tan x\right) + 0\]

   13. Applied associate-/r/3.6

       \[\leadsto \left(\color{blue}{\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{{1}^{3} - {\left({\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}^{3}} \cdot \left(1 \cdot 1 + \left({\left(\tan \varepsilon \cdot \tan x\right)}^{3} \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3} + 1 \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)\right)} - \tan x\right) + 0\]

   14. Applied fma-neg3.6

       \[\leadsto \color{blue}{(\left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{{1}^{3} - {\left({\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}^{3}}\right) \cdot \left(1 \cdot 1 + \left({\left(\tan \varepsilon \cdot \tan x\right)}^{3} \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3} + 1 \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)\right) + \left(-\tan x\right))_*} + 0\]
3. Recombined 3 regimes into one program.

4. Final simplification13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.533455973501571 \cdot 10^{-37}:\\ \;\;\;\;\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan \varepsilon + \tan x\right)}{1 - \frac{{\left(\sin \varepsilon \cdot \tan x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3}}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 1.0018624295577554 \cdot 10^{-47}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan \varepsilon + \tan x\right)}{1 - {\left({\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}^{3}}\right) \cdot \left(\left({\left(\tan \varepsilon \cdot \tan x\right)}^{3} + {\left(\tan \varepsilon \cdot \tan x\right)}^{3} \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right) + 1\right) + \left(-\tan x\right))_*\\ \end{array}\]

Runtime

Time bar (total: 57.0s)Debug logProfile

herbie shell --seed 2018285 +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)))