Error

Target

Derivation

1. Split input into 2 regimes

2. if (fma (* x eps) (fma (* x eps) eps eps) eps) < -5.629571522578517e-10 or 5.7595172635421755e-71 < (fma (* x eps) (fma (* x eps) eps eps) eps)

   1. Initial program 33.5

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

   3. Applied tan-sum9.4

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

   5. Applied flip3--9.4

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

   6. Applied associate-/r/9.4

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

   7. Applied fma-neg9.4

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

   if -5.629571522578517e-10 < (fma (* x eps) (fma (* x eps) eps eps) eps) < 5.7595172635421755e-71

   1. Initial program 43.4

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

   2. Taylor expanded around 0 23.9

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

   3. Applied simplify22.6

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

4. Applied simplify14.3

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le -5.629571522578517 \cdot 10^{-10} \lor \neg \left((\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le 5.7595172635421755 \cdot 10^{-71}\right):\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \tan \varepsilon \cdot \tan x\right) + 1\right) + \left(-\tan x\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \end{array}}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' +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)))