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

⬇

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.786347343699343 \cdot 10^{-16}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{if}\;\varepsilon \le 3.94735216048035 \cdot 10^{-08}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \end{array}\]

Derivation

Split input into 2 regimes.
if eps < -8.786347343699343e-16 or 3.94735216048035e-08 < eps
1. Initial program 31.2
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum 1.7
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Applied associate--l- 1.7
  \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
if -8.786347343699343e-16 < eps < 3.94735216048035e-08
1. Initial program 49.6
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied diff-cos 38.0
  \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Applied simplify 0.2
  \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie shell --seed '#(644180380 3784176976 401987740 22459203 1940947670 3323606534)'
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Error

Derivation

`if eps < -8.786347343699343e-16 or 3.94735216048035e-08 < eps`

`if -8.786347343699343e-16 < eps < 3.94735216048035e-08`

Runtime