Hmm.. It looks complex.

$Using\quad a\quad corollary\quad of\quad Cauchy's\quad integral\quad formula:\\ { f }^{ (n) }(a)=\frac { n! }{ 2\pi i } \oint _{ C }{ \frac { f(z) }{ (z-a)^{ n+1 } } dz } \quad for\quad an\quad analytic\quad f\quad in\quad a\\ closed\quad contour\quad C,\quad and\quad setting\quad n=3;\quad a=0\quad \\ and\quad f(z)=sinh(2z),\quad we\quad have,\quad \\ f^{ ''' }(z)=8cosh(2\times 0)\times \frac { 2\pi i }{ 3! } =\quad I\\ \Rightarrow \quad I\quad =\quad i8\pi /3\quad \Rightarrow \quad exp(I)=-1/2+i\sqrt { 3 } /2\\ \Rightarrow \quad Re(e^{ I })\quad =\quad \boxed { -0.5 }$