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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.116080113353332 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 5.927052566239743 \cdot 10^{-26}\right):\\ \;\;\;\;(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \end{array}\]

Original	37.2
Target	15.1
Herbie	14.1

Derivation

Split input into 2 regimes
if eps < -3.116080113353332e-18 or 5.927052566239743e-26 < eps
1. Initial program 29.9
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-sum1.2
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm
5. Applied div-inv1.2
  \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
6. Applied fma-neg1.2
  \[\leadsto \color{blue}{(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right))_*}\]
if -3.116080113353332e-18 < eps < 5.927052566239743e-26
1. Initial program 45.3
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Taylor expanded around 0 29.8
  \[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)}\]
3. Applied simplify28.5
  \[\leadsto \color{blue}{(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*}\]
Recombined 2 regimes into one program.
Applied simplify14.1
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.116080113353332 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 5.927052566239743 \cdot 10^{-26}\right):\\ \;\;\;\;(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \end{array}}\]

Runtime

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' +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 < -3.116080113353332e-18 or 5.927052566239743e-26 < eps`

`if -3.116080113353332e-18 < eps < 5.927052566239743e-26`

Runtime