Error

Target

Derivation

1. Split input into 2 regimes

2. if (fma (* x eps) (fma (* x eps) eps eps) eps) < -1.6525039405208001e-12 or 7.18388088886727e-56 < (fma (* x eps) (fma (* x eps) eps eps) eps)

   1. Initial program 32.9

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

   3. Applied tan-sum8.3

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

   5. Applied div-inv8.4

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

   6. Applied fma-neg8.3

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

   if -1.6525039405208001e-12 < (fma (* x eps) (fma (* x eps) eps eps) eps) < 7.18388088886727e-56

   1. Initial program 43.1

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

   2. Taylor expanded around 0 22.9

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

   3. Simplified21.9

      \[\leadsto \color{blue}{(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*}\]
3. Recombined 2 regimes into one program.

4. Final simplification13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le -1.6525039405208001 \cdot 10^{-12}:\\ \;\;\;\;(\left(\tan \varepsilon + \tan x\right) \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le 7.18388088886727 \cdot 10^{-56}:\\ \;\;\;\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\tan \varepsilon + \tan x\right) \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018215 +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)))