Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -1.773639946594851e-15

   1. Initial program 30.8

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

   3. Applied tan-sum0.7

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

   5. Applied flip--0.7

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

   6. Applied associate-/r/0.7

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \tan x\]
   7. Using strategy rm

   8. Applied associate-*r*0.7

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

   if -1.773639946594851e-15 < eps < 1.1218129455385525e-17

   1. Initial program 44.8

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

   2. Taylor expanded around 0 30.9

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

   3. Simplified30.8

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

   if 1.1218129455385525e-17 < eps

   1. Initial program 28.4

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   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. Taylor expanded around inf 0.9

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

   6. Applied tan-quot1.0

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

   7. Applied frac-sub1.0

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

4. Final simplification15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.773639946594851 \cdot 10^{-15}:\\ \;\;\;\;\left(\tan x \cdot \tan \varepsilon + 1\right) \cdot \frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \left(\tan x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le 1.1218129455385525 \cdot 10^{-17}:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot \left(x + \varepsilon\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \sin x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))