Error

Target

Derivation

1. Split input into 3 regimes

2. if (+ (fma (pow eps 3) (* x x) (* x (* eps eps))) eps) < -2.084504492194263e-45

   1. Initial program 35.2

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

   3. Applied tan-quot35.1

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

   4. Applied tan-sum9.6

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \frac{\sin x}{\cos x}\]

   5. Applied frac-sub9.7

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

   if -2.084504492194263e-45 < (+ (fma (pow eps 3) (* x x) (* x (* eps eps))) eps) < 2.1892169813184298e-07

   1. Initial program 39.8

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

   2. Taylor expanded around 0 16.2

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

   3. Simplified16.2

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

   if 2.1892169813184298e-07 < (+ (fma (pow eps 3) (* x x) (* x (* eps eps))) eps)

   1. Initial program 36.5

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

   3. Applied tan-sum14.8

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

   5. Applied add-cbrt-cube14.9

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

   6. Applied add-cbrt-cube14.9

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

   7. Applied cbrt-unprod14.9

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

   8. Simplified14.9

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

4. Final simplification14.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon + (\left({\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right))_* \le -2.084504492194263 \cdot 10^{-45}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon + (\left({\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right))_* \le 2.1892169813184298 \cdot 10^{-07}:\\ \;\;\;\;\varepsilon + (\left({\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 5.3m)Debug logProfile

herbie shell --seed 2018214 +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)))