A function derives from the `function_class`

class. This
class is an interface to the name of the function, its associated real
operator, and six theorems.

struct function_class { function_class(real_op_type t, int mask); virtual interval round (interval const &, std::string &) const; virtual interval enforce (interval const &, std::string &) const; virtual interval absolute_error (std::string &) const; virtual interval relative_error (std::string &) const; virtual interval absolute_error_from_exact_bnd (interval const &, std::string &) const; virtual interval absolute_error_from_exact_abs (interval const &, std::string &) const; virtual interval absolute_error_from_approx_bnd(interval const &, std::string &) const; virtual interval absolute_error_from_approx_abs(interval const &, std::string &) const; virtual interval relative_error_from_exact_bnd (interval const &, std::string &) const; virtual interval relative_error_from_exact_abs (interval const &, std::string &) const; virtual interval relative_error_from_approx_bnd(interval const &, std::string &) const; virtual interval relative_error_from_approx_abs(interval const &, std::string &) const; virtual std::string description() const = 0; virtual std::string pretty_name() const = 0; virtual ~function_class(); };

The `description`

function should return the internal
name of the rounding operator. It will be used when generating the
notations of the proof. When the generated notation cannot be reduced to
a simple name, comma-separated additional parameters can be appended. The
back-end will take care of formatting the final string. This remark also
applies to names returned by the theorem methods (see below). The
`pretty_name`

function returns a name that can be used in
messages displayed to the user. Ideally, this string can be reused in an
input script.

The `real_op_type`

value is the associated real operator. This will be
`UOP_ID`

(the unary identity function) for standard rounding
operators. But it can be more complex if needed:

enum real_op_type { UOP_ID, UOP_NEG, UOP_ABS, BOP_ADD, BOP_SUB, BOP_MUL, BOP_DIV, ... };

The type will indicate to the parser the number of arguments the function
requires. For example, if the `BOP_DIAM`

type is associated to the
function `f`

, then `f`

will be parsed as a binary
function. But the type is also used by the rewriting engines in order to derive
default rules for this function. These rules involve the associated real
operator (the diamond in this example).

For these rules and the following theorems to be useful, the expressions [math] and [math] have to be close to each other. Bounding their distance is the purpose of the last ten theorems. The first two theorems compute the range of [math] itself.

It is better for the proof engine not to consider theorems that never
return a useful range. The `mask`

argument of the `function_class`

constructor is a combination of the following flags. They indicate which
theorems are known. The corresponding methods should therefore have been overloaded.

struct function_class { static const int TH_RND, TH_ENF, TH_ABS, TH_REL, TH_ABS_EXA_BND, TH_ABS_EXA_ABS, TH_ABS_APX_BND, TH_ABS_APX_ABS, TH_REL_EXA_BND, TH_REL_EXA_ABS, TH_REL_APX_BND, TH_REL_APX_ABS; };

All the virtual methods for theorems have a similar specification.
If the result is the undefined interval `interval()`

, the
theorem does not apply. Otherwise, the last parameter is updated with the
name of the theorem that was used for computing the returned interval.
The proof generator will then generate an internal node from the two
intervals and the name. When defining a new rounding operator,
overloading does not have to be comprehensive; some functions may be
ignored and the engine will work around the missing theorems.

`round`

Given the range of [math], compute the range of [math].

`enforce`

Given the range of [math], compute a stricter range of it.

`absolute_error`

Given no range, compute the range of [math].

`relative_error`

Given no range, compute the range of [math].

`absolute_error_from_exact_bnd`

Given the range of [math], compute the range of [math].

`absolute_error_from_exact_abs`

Given the range of [math], compute the range of [math].

`absolute_error_from_approx_bnd`

Given the range of [math], compute the range of [math].

`absolute_error_from_approx_abs`

Given the range of [math], compute the range of [math].

`relative_error_from_exact_bnd`

Given the range of [math], compute the range of [math].

`relative_error_from_exact_abs`

Given the range of [math], compute the range of [math].

`relative_error_from_approx_bnd`

Given the range of [math], compute the range of [math].

`relative_error_from_approx_abs`

Given the range of [math], compute the range of [math].

The `enforce`

theorem is meant to trim the bounds of a range.
For example, if this expression is an integer between 1.7 and 3.5, then it is
also a real number between 2 and 3. This property is especially useful when doing a
dichotomy resolution, since some of the smaller intervals may be reduced to a
single exact value through this theorem.

Since the undefined interval is used when a theorem does not apply, it
cannot be used by `enforce`

to flag an empty interval in case of a
contradiction. The method should instead return an interval that does not
intersect the initial interval. Due to formal certification considerations, it
should however be in the rounded outward version of the initial interval. For
example, if the expression is an integer between 1.3 and 1.7, then the method
should return an interval contained in [1,1.3[ or ]1.7,2]. For practical
reasons, [1,1] and [2,2] are the most interesting answers.

Because functions can be templated by parameters. They have to be
generated by the parser on the fly. This is done by invoking the functional
method of an object derived from the `function_generator`

class. For identical parameters, the same `function_class`

object should be returned, which means that they have to be
cached.

struct function_generator { function_generator(const char *); virtual function_class const *operator()(function_params const &) const = 0; virtual ~function_generator() {} };

The constructor of this class requires the name of the function template,
so that it gets registered by the parser. `operator()`

is called
with a vector of encoded parameters.

If a function has no template parameters, the `default_function_generator`

class can be used instead to register it. The first parameter of the constructor
is the function name. The second one is the address of the
`function_class`

object.

default_function_generator::default_function_generator(const char *, function_class const *);