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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.435762579177024 \cdot 10^{-49}:\\ \;\;\;\;(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;\varepsilon \le 1.4084902029441485 \cdot 10^{-34}:\\ \;\;\;\;(\varepsilon \cdot \left((\left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right) + \left(x \cdot \varepsilon\right))_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}\right) + \left(-\tan x\right))_*\\ \end{array}\]

Original	37.3
Target	15.1
Herbie	14.4

Derivation

Split input into 2 regimes
if eps < -2.435762579177024e-49 or 1.4084902029441485e-34 < eps
1. Initial program 30.3
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-sum3.1
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm
5. Applied div-inv3.1
  \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
6. Applied fma-neg3.1
  \[\leadsto \color{blue}{(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right))_*}\]
7. Using strategy rm
8. Applied tan-quot3.1
  \[\leadsto (\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}}\right) + \left(-\tan x\right))_*\]
9. Applied tan-quot3.1
  \[\leadsto (\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \color{blue}{\frac{\sin x}{\cos x}} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right) + \left(-\tan x\right))_*\]
10. Applied frac-times3.1
  \[\leadsto (\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \color{blue}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}}\right) + \left(-\tan x\right))_*\]
if -2.435762579177024e-49 < eps < 1.4084902029441485e-34
1. Initial program 46.2
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Taylor expanded around 0 29.9
  \[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)}\]
3. Applied simplify28.7
  \[\leadsto \color{blue}{(\varepsilon \cdot \left((\left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right) + \left(x \cdot \varepsilon\right))_*\right) + \varepsilon)_*}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' +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)))

Error

Target

Derivation

`if eps < -2.435762579177024e-49 or 1.4084902029441485e-34 < eps`

`if -2.435762579177024e-49 < eps < 1.4084902029441485e-34`

Runtime