  This paper proves explicit bilipschitz bounds on the change in metric between
the thick part of a cusped hyperbolic 3-manifold N and the thick part of any of
its long Dehn fillings. Given a bilipschitz constant J > 1 and a thickness
constant epsilon > 0, we quantify how long a Dehn filling suffices to guarantee
a J-bilipschitz map on epsilon-thick parts. A similar theorem without
quantitative control was previously proved by Brock and Bromberg, applying
Hodgson and Kerckhoff's theory of cone deformations. We achieve quantitative
control by bounding the analytic quantities that control the infinitesimal
change in metric during the cone deformation.
  Our quantitative results have two immediate applications. First, we relate
the Margulis number of N to the Margulis numbers of its Dehn fillings. In
particular, we give a lower bound on the systole of any closed 3-manifold M
whose Margulis number is less than 0.29. Combined with Shalen's upper bound on
the volume of such a manifold, this gives a procedure to compute the finite
list of 3-manifolds whose Margulis numbers are below 0.29.
  Our second application is to the cosmetic surgery conjecture. Given the
systole of a one-cusped hyperbolic manifold N, we produce an explicit upper
bound on the length of a slope involved in a cosmetic surgery on N. This
reduces the cosmetic surgery conjecture on N to an explicit finite search.
