Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -5.454647893991481e-78

   1. Initial program 30.5

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

   3. Applied tan-sum6.4

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

   5. Applied div-inv6.4

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

   6. Applied fma-neg6.4

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

   8. Applied add-cube-cbrt6.6

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

   9. Applied fma-def6.6

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

   if -5.454647893991481e-78 < eps < 4.606203760411138e-25

   1. Initial program 45.8

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

   2. Taylor expanded around 0 31.3

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

   3. Simplified31.2

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

   if 4.606203760411138e-25 < eps

   1. Initial program 29.7

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

   3. Applied tan-sum1.5

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

   5. Applied div-inv1.5

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

   6. Applied fma-neg1.5

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

   8. Applied flip--1.6

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

4. Final simplification15.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.454647893991481 \cdot 10^{-78}:\\ \;\;\;\;(\left((\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \left(\sqrt[3]{\tan x}\right) + \left(\tan \varepsilon\right))_*\right) \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 4.606203760411138 \cdot 10^{-25}:\\ \;\;\;\;(\left(\varepsilon \cdot \left(x + \varepsilon\right)\right) \cdot x + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\tan \varepsilon + \tan x\right) \cdot \left(\frac{1}{\frac{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}{\tan \varepsilon \cdot \tan x + 1}}\right) + \left(-\tan x\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +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)))