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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.3943876692052302 \cdot 10^{-41}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{if}\;\varepsilon \le 4.288585038601844 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Original	36.8
Target	15.1
Herbie	14.3

Derivation

Split input into 3 regimes
if eps < -1.3943876692052302e-41
1. Initial program 29.7
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-quot29.5
  \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
4. Applied tan-sum3.4
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \frac{\sin x}{\cos x}\]
5. Applied frac-sub3.4
  \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
if -1.3943876692052302e-41 < eps < 4.288585038601844e-27
1. Initial program 45.9
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Taylor expanded around 0 29.1
  \[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)}\]
if 4.288585038601844e-27 < eps
1. Initial program 29.6
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-quot29.5
  \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
4. Applied tan-sum1.8
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \frac{\sin x}{\cos x}\]
5. Applied frac-sub1.9
  \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
Recombined 3 regimes into one program.
Removed slow pow expressions.

Runtime

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :herbie-expected 28

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))

Error

Target

Derivation

`if eps < -1.3943876692052302e-41`

`if -1.3943876692052302e-41 < eps < 4.288585038601844e-27`

`if 4.288585038601844e-27 < eps`

Runtime