Error

Target

Derivation

1. Split input into 3 regimes

2. if (tan eps) < -1.0838418945046034e-13

   1. Initial program 29.4

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

   3. Applied tan-sum0.8

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

   5. Applied tan-quot0.8

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

   6. Applied tan-quot0.8

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

   7. Applied frac-times0.8

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

   if -1.0838418945046034e-13 < (tan eps) < 6.014310192619639e-17

   1. Initial program 45.0

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

   2. Taylor expanded around 0 29.7

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

   3. Simplified28.6

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

   if 6.014310192619639e-17 < (tan eps)

   1. Initial program 30.4

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

   3. Applied tan-sum0.9

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

   5. Applied add-cube-cbrt1.3

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

   6. Applied flip3--1.3

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

   7. Applied associate-/r/1.3

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

   8. Applied prod-diff1.3

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

   9. Simplified0.9

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

   10. Simplified0.9

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

4. Final simplification13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \varepsilon \le -1.0838418945046034 \cdot 10^{-13}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\tan \varepsilon \le 6.014310192619639 \cdot 10^{-17}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left((\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \left(\tan x \cdot \tan \varepsilon\right))_*\right) \cdot \left(\tan \varepsilon + \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018217 +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)))