ARTICLE: "math-interval-properties""Properties of interval arithmetic"
"For some operations, interval arithmetic yields inaccurate results, either because the result of lifting some operations to intervals does not result in intervals (bitwise operations, for example) or for the sake of simplicity of implementation."
$nl
"However, one important property holds for all operations. Suppose " { $emphasis "I, J" } " are intervals and " { $emphasis "op" } " is an operation. If " { $emphasis "x" } " is an element of " { $emphasis "I" } " and " { $emphasis "y" } " is an element of " { $emphasis "J" } ", then " { $emphasis "x op y" } " is an element of " { $emphasis "I op J" } "."
$nl
"In other words, the resulting interval might be an overestimate, but it is never an underestimate.";
"Interval arithmetic is performed on ranges of real numbers, rather than exact values. It is used by the Factor compiler to convert arbitrary-precision arithmetic to machine arithmetic, by inferring bounds for integer calculations."
{ $class-description "An interval represents a set of real numbers between two endpoints; the endpoints can either be included or excluded from the interval."
"The " { $snippet "from" } " and " { $snippet "to" } " slots store endpoints, represented as arrays of the shape " { $snippet "{ number included? }" } "."
{ $description "Creates a new interval that excludes the lower endpoint and includes the upper endpoint." } ;
HELP:[a,a]
{ $values { "a"real } { "interval" interval } }
{ $description "Creates a new interval consisting of a single point." } ;
HELP:[-inf,a]
{ $values { "a"real } { "interval" interval } }
{ $description "Creates a new interval containing all real numbers less than or equal to " { $snippet "a" } ", together with negative infinity." } ;
HELP:[-inf,a)
{ $values { "a"real } { "interval" interval } }
{ $description "Creates a new interval containing all real numbers less than " { $snippet "a" } ", together with negative infinity." } ;
HELP:[a,inf]
{ $values { "a"real } { "interval" interval } }
{ $description "Creates a new interval containing all real numbers greater than or equal to " { $snippet "a" } ", together with positive infinity." } ;
HELP:(a,inf]
{ $values { "a"real } { "interval" interval } }
{ $description "Creates a new interval containing all real numbers greater than " { $snippet "a" } ", together with positive infinity." } ;
{ $description "Shifts " { $snippet "i1" } " to the left by " { $snippet "i2" } " bits. Outputs " { $link f } " if the endpoints of either " { $snippet "i1" } " or " { $snippet "i2" } " are not integers." } ;
{ $description "Outputs an interval containing all possible values obtained by aplying " { $link mod } " to elements of " { $snippet "i1" } " and " { $snippet "i2" } "." } ;
{ $description "Outputs an interval containing all possible values obtained by aplying " { $link rem } " to elements of " { $snippet "i1" } " and " { $snippet "i2" } "." } ;
{ $description "Outputs an interval containing all possible values obtained by aplying " { $link bitand } " to elements of " { $snippet "i1" } " and " { $snippet "i2" } "." } ;
{ $description "Outputs an interval containing all possible values obtained by aplying " { $link bitor } " to elements of " { $snippet "i1" } " and " { $snippet "i2" } "." } ;
{ $description "Outputs an interval containing all possible values obtained by aplying " { $link bitxor } " to elements of " { $snippet "i1" } " and " { $snippet "i2" } "." } ;
{ $description "Outputs the smallest interval containing the set-theoretic union of " { $snippet "i1" } " and " { $snippet "i2" } " (the union itself may not be an interval)." } ;
{ $description "Outputs the reciprocal of an interval. Outputs " { $link f } " if " { $snippet "i1" } " contains points arbitrarily close to zero." } ;
HELP:interval-2/
{ $values { "i1" interval } { "i2" interval } }
{ $description "Shifts an interval to the right by one bit." } ;
HELP:interval-bitnot
{ $values { "i1" interval } { "i2" interval } }
{ $description "Computes the bitwise complement of the interval." } ;
{ $description "Shifts " { $snippet "i1" } " to the left by " { $snippet "i2" } " bits. Outputs " { $link f } " if the endpoints of either " { $snippet "i1" } " or " { $snippet "i2" } " are not integers, or if the endpoints of " { $snippet "i2" } " are so large that the resulting interval will consume too much memory." } ;
HELP:incomparable
{ $description "Output value from " { $link interval<= } ", " { $link interval< } ", " { $link interval>= } " and " { $link interval> } " in the case where the result of the comparison is ambiguous." } ;
{ $description "Outputs the interval consisting of points from " { $snippet "i1" } " which are less than all points in " { $snippet "i2" } ". If the resulting interval is empty, outputs " { $link f } "." } ;
{ $description "Outputs the interval consisting of points from " { $snippet "i1" } " which are less or equal to all points in " { $snippet "i2" } ". If the resulting interval is empty, outputs " { $link f } "." } ;
{ $description "Outputs the interval consisting of points from " { $snippet "i1" } " which are greater than all points in " { $snippet "i2" } ". If the resulting interval is empty, outputs " { $link f } "." } ;
{ $description "Outputs the interval consisting of points from " { $snippet "i1" } " which are greater than or equal to all points in " { $snippet "i2" } ". If the resulting interval is empty, outputs " { $link f } "." } ;
{ $description "Outputs a closed interval which is equal as a set to " { $snippet "i1" } ", when " { $snippet "i1" } " is viewed as an interval over in integers (that is, a discrete set)." } ;
