Error

Target

Derivation

1. Split input into 2 regimes

2. if (fma eps (fma (* x eps) (* x eps) (* x eps)) eps) < -7.377007838795717e-15 or 5.060445514815462e-26 < (fma eps (fma (* x eps) (* x eps) (* x eps)) eps)

   1. Initial program 33.0

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

   3. Applied tan-sum8.5

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

   5. Applied add-cube-cbrt8.8

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

   6. Applied flip3--8.8

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

   7. Applied associate-/r/8.8

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

   8. Applied prod-diff8.8

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

   9. Applied simplify8.4

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

   10. Applied simplify8.5

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

   12. Applied tan-quot8.5

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

   13. Applied frac-sub8.6

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

   if -7.377007838795717e-15 < (fma eps (fma (* x eps) (* x eps) (* x eps)) eps) < 5.060445514815462e-26

   1. Initial program 41.7

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

   2. Taylor expanded around 0 22.5

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

   3. Applied simplify21.3

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

4. Applied simplify13.6

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\varepsilon \cdot \left((\left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right) + \left(x \cdot \varepsilon\right))_*\right) + \varepsilon)_* \le -7.377007838795717 \cdot 10^{-15}:\\ \;\;\;\;\frac{\cos x \cdot (\left(\tan \varepsilon + \tan x\right) \cdot \left((\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right))_*\right) + \left(\tan \varepsilon + \tan x\right))_* - \sin x \cdot \left(1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}{\cos x \cdot \left(1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}\\ \mathbf{if}\;(\varepsilon \cdot \left((\left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right) + \left(x \cdot \varepsilon\right))_*\right) + \varepsilon)_* \le 5.060445514815462 \cdot 10^{-26}:\\ \;\;\;\;(\varepsilon \cdot \left((\left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right) + \left(x \cdot \varepsilon\right))_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot (\left(\tan \varepsilon + \tan x\right) \cdot \left((\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right))_*\right) + \left(\tan \varepsilon + \tan x\right))_* - \sin x \cdot \left(1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}{\cos x \cdot \left(1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)}\\ \end{array}}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +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)))