Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -4.4697484735015806e-82

   1. Initial program 30.7

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

   3. Applied tan-sum6.6

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

   5. Applied add-sqr-sqrt35.2

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

   6. Applied flip3--35.2

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

   7. Applied associate-/r/35.2

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

   8. Applied prod-diff35.2

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

   9. Simplified35.2

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

   10. Simplified6.7

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

   12. Applied *-un-lft-identity6.7

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

   13. Applied times-frac6.6

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

   14. Applied fma-neg6.6

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

   15. Simplified6.6

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

   if -4.4697484735015806e-82 < eps < 9.01472419115293e-53

   1. Initial program 47.2

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

   3. Applied tan-sum47.2

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

   5. Applied add-sqr-sqrt55.4

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

   6. Applied flip3--55.4

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

   7. Applied associate-/r/55.4

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

   8. Applied prod-diff55.7

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

   9. Simplified55.7

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

   10. Simplified47.2

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

   11. Taylor expanded around 0 26.7

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

   12. Simplified26.7

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

   if 9.01472419115293e-53 < eps

   1. Initial program 29.3

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

   3. Applied tan-sum4.1

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

   5. Applied add-sqr-sqrt33.5

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

   6. Applied flip3--33.6

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

   7. Applied associate-/r/33.6

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

   8. Applied prod-diff33.5

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

   9. Simplified33.5

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

   10. Simplified4.1

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

   12. Applied flip--4.2

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

   13. Applied associate-/r/4.2

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

   14. Applied fma-neg4.2

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

4. Final simplification13.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.4697484735015806 \cdot 10^{-82}:\\ \;\;\;\;(\left((\left((\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + 1)_*\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1)_*\right) \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 9.01472419115293 \cdot 10^{-53}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\left(\tan \varepsilon + \tan x\right) \cdot (\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_*}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3} \cdot {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right) \cdot \left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right) + \left(-\tan x\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018351 +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)))