Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -1.5274795464155212e-44

   1. Initial program 30.7

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

   2. Initial simplification30.7

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

   4. Applied tan-sum3.8

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

   6. Applied div-inv3.8

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

   7. Applied fma-neg3.8

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

   9. Applied flip3--3.8

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

   if -1.5274795464155212e-44 < eps < 5.227487673487659e-24

   1. Initial program 46.1

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

   2. Initial simplification46.1

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

   3. Taylor expanded around 0 28.4

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)}\]

   4. Simplified29.0

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

   if 5.227487673487659e-24 < eps

   1. Initial program 30.5

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

   2. Initial simplification30.5

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

   4. Applied tan-sum1.4

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

   6. Applied div-inv1.4

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

   7. Applied fma-neg1.4

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

   9. Applied add-log-exp1.5

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

4. Final simplification14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5274795464155212 \cdot 10^{-44}:\\ \;\;\;\;(\left(\tan \varepsilon + \tan x\right) \cdot \left(\frac{1}{\frac{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}{1 + \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \tan \varepsilon \cdot \tan x\right)}}\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 5.227487673487659 \cdot 10^{-24}:\\ \;\;\;\;(\varepsilon \cdot \left(\varepsilon \cdot (\varepsilon \cdot \frac{1}{3} + x)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\tan \varepsilon + \tan x\right) \cdot \left(\frac{1}{\log \left(e^{1 - \tan \varepsilon \cdot \tan x}\right)}\right) + \left(-\tan x\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018234 +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)))