Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -2.7102452299401797e-21

   1. Initial program 30.0

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

   3. Applied tan-quot29.9

      \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]

   4. Applied tan-sum1.6

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

   5. Applied frac-sub1.6

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

   6. Simplified1.6

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

   if -2.7102452299401797e-21 < eps < 1.9799621094739053e-19

   1. Initial program 45.2

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

   2. Taylor expanded around 0 29.4

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

   3. Simplified28.4

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

   if 1.9799621094739053e-19 < eps

   1. Initial program 30.5

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

   3. Applied tan-sum1.2

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

   5. Applied tan-quot1.2

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

   6. Applied tan-quot1.2

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

   7. Applied frac-times1.2

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}} - \tan x\]
3. Recombined 3 regimes into one program.

4. Final simplification14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.7102452299401797 \cdot 10^{-21}:\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan x + \tan \varepsilon\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + \left(-1\right))_* \cdot \sin x\right))_*}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 1.9799621094739053 \cdot 10^{-19}:\\ \;\;\;\;(\varepsilon \cdot \left(\left(x \cdot \varepsilon\right) \cdot (x \cdot \varepsilon + 1)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018221 +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)))