...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

template <class T> struct eps_tolerance { eps_tolerance(); eps_tolerance(int bits); bool operator()(const T& a, const T& b)const; };

`eps_tolerance`

is the usual
termination condition used with these root finding functions. Its `operator()`

will return true when the relative distance between *a*
and *b* is less than four times the machine epsilon
for T, or 2^{1-bits}, whichever is the larger. In other words, you set *bits*
to the number of bits of precision you want in the result. The minimal
tolerance of *four times the machine epsilon of type T*
is required to ensure that we get back a bracketing interval, since this
must clearly be at greater than one epsilon in size. While in theory a
maximum distance of twice machine epsilon is possible to achieve, in practice
this results in a great deal of "thrashing" given that the function
whose root is being found can only ever be accurate to 1 epsilon at best.

struct equal_floor { equal_floor(); template <class T> bool operator()(const T& a, const T& b)const; };

This termination condition is used when you want to find an integer result
that is the *floor* of the true root. It will terminate
as soon as both ends of the interval have the same *floor*.

struct equal_ceil { equal_ceil(); template <class T> bool operator()(const T& a, const T& b)const; };

This termination condition is used when you want to find an integer result
that is the *ceil* of the true root. It will terminate
as soon as both ends of the interval have the same *ceil*.

struct equal_nearest_integer { equal_nearest_integer(); template <class T> bool operator()(const T& a, const T& b)const; };

This termination condition is used when you want to find an integer result
that is the *closest* to the true root. It will terminate
as soon as both ends of the interval round to the same nearest integer.