## Defining rounding operators for a new arithmetic

### Function classes

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
{
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 $f(a, b)$ and $a \diamond b$ 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 $f(a, b)$ 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 $a \diamond b$, compute the range of $f(a, b)$.

enforce

Given the range of $f(a, b)$, compute a stricter range of it.

absolute_error

Given no range, compute the range of $f(a, b) - a \diamond b$.

relative_error

Given no range, compute the range of $\frac{f(a, b) - a \diamond b}{a \diamond b}$.

absolute_error_from_exact_bnd

Given the range of $a \diamond b$, compute the range of $f(a, b) - a \diamond b$.

absolute_error_from_exact_abs

Given the range of $|a \diamond b|$, compute the range of $f(a, b) - a \diamond b$.

absolute_error_from_approx_bnd

Given the range of $f(a, b)$, compute the range of $f(a, b) - a \diamond b$.

absolute_error_from_approx_abs

Given the range of $|f(a, b)|$, compute the range of $f(a, b) - a \diamond b$.

relative_error_from_exact_bnd

Given the range of $a \diamond b$, compute the range of $\frac{f(a, b) - a \diamond b}{a \diamond b}$.

relative_error_from_exact_abs

Given the range of $|a \diamond b|$, compute the range of $\frac{f(a, b) - a \diamond b}{a \diamond b}$.

relative_error_from_approx_bnd

Given the range of $f(a, b)$, compute the range of $\frac{f(a, b) - a \diamond b}{a \diamond b}$.

relative_error_from_approx_abs

Given the range of $|f(a, b)|$, compute the range of $\frac{f(a, b) - a \diamond b}{a \diamond b}$.

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.

### Function generators

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 *);