Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -5.9223589551452404e-18

   1. Initial program 30.3

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

   3. Applied tan-sum1.0

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

   5. Applied tan-quot1.0

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

   6. Applied associate-*l/1.0

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

   8. Applied div-inv1.1

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

   9. Applied fma-neg1.0

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

   if -5.9223589551452404e-18 < eps < 9.534906536733132e-38

   1. Initial program 45.4

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

   3. Applied tan-sum45.4

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

   5. Applied tan-quot45.4

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

   6. Applied associate-*l/45.4

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

   7. Taylor expanded around 0 30.8

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

   8. Simplified30.8

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

   if 9.534906536733132e-38 < eps

   1. Initial program 28.4

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

   3. Applied tan-sum2.9

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

   5. Applied tan-quot3.0

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

   6. Applied associate-*l/3.0

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

   8. Applied add-cbrt-cube3.0

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

4. Final simplification15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.9223589551452404 \cdot 10^{-18}:\\ \;\;\;\;(\left(\tan \varepsilon + \tan x\right) \cdot \left(\frac{1}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 9.534906536733132 \cdot 10^{-38}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{\left(\frac{\sin x \cdot \tan \varepsilon}{\cos x} \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018348 +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)))