Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -1.16838641534245e-26

   1. Initial program 28.2

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

   3. Applied tan-sum1.9

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

   5. Applied div-inv1.9

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

   6. Applied fma-neg1.9

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

   8. Applied flip-+2.0

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

   if -1.16838641534245e-26 < eps < 7.604015002971267e-18

   1. Initial program 45.7

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

   2. Taylor expanded around 0 30.9

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

   3. Simplified30.9

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

   if 7.604015002971267e-18 < eps

   1. Initial program 30.0

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   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 div-inv0.9

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

   6. Applied fma-neg0.8

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

   7. Taylor expanded around inf 0.9

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

   9. Applied add-cbrt-cube0.9

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

   10. Applied add-cbrt-cube0.9

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

   11. Applied cbrt-unprod0.9

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

   12. Simplified0.9

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

4. Final simplification15.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.16838641534245 \cdot 10^{-26}:\\ \;\;\;\;(\left(\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\tan x - \tan \varepsilon}\right) \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 7.604015002971267 \cdot 10^{-18}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\tan \varepsilon + \tan x\right) \cdot \left(\frac{1}{1 - \frac{\sqrt[3]{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}}{\cos \varepsilon \cdot \cos x}}\right) + \left(-\tan x\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 +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)))