Abstraction: binratintvl
Version: 1
Date: 29 November 2003; revised 29 November 2003
Author: Mike O'Donnell

Derived from ratintvl-4

Values in the binratintvl abstraction are closed intervals
[(m-1)*2^n, (m+1)*2^n], where m and n are integers. They are intended
to represent approximate information about real numbers, but all
procedures are defined in terms of their effects on the intervals.

Constructors
------------

(make_binratintvl r1 r2) takes integers m and n, and returns
    a representation of the closed interval [(m-1)*2^n, (m+1)*2^n].

(rational->ratintvl r) returns the smallest closed binary rational
    interval containing the given rational r.

Predicates
----------

(binratintvl? expr) returns #t if expr is a ratintvl, #f otherwise.

(member_binratintvl? r bri) returns #t if the rational r is a member of
    the ratintvl bri, #f otherwise.

(contained_binratintvl? bri1 bri2) returns #t if bri1 is a subinterval
    (possibly equal) of bri2, #f otherwise.

(equal_binratintvl? bri1 bri2) returns #t if bri1=bri2, #f otherwise.

(precedes_binratintvl? bri1 bri2) returns #t if all reals in ri1
    strictly precede all reals in bri2.

(overlaps_binratintvl? bri1 bri2) returns #t if there is at least one
    real number that is a member of both bri1 and bri2 (equivalently,
    if the intersection of bri1 and bri2 is not empty), #f otherwise.

(lbounds_binratintvl? bri1, bri2) returns #t if bri1 contains at least
    one real less than all reals in bri2, #f otherwise.

(ubounded_binratintvl? bri1, bri2) returns #t if bri2 contains at
    least one real greater than all reals in bri1, #f otherwise.

Operations
----------

(union_binratintvl bri1 bri2) returns the smallest binratintvl
    containing all reals in bri1 and all reals in bri2.

(intersect_binratintvl bri1 bri2) returns the smallest binratintvl
    containing all reals that are in both of bri1 and bri2.

(add_binratintvl bri1 bri2) returns the smallest binratintvl containing
    all sums of one real in bri1 plus one real in bri2.

(sub_binratintvl bri1 bri2) returns the smallest binratintvl
    containing all differences of one real in bri1 minus one real in
    bri2.

(mult_binratintvl bri1 bri2) returns the binratintvl containing all
    products of one real in bri1 times one real in bri2.

(div_binratintvl bri1 bri2) returns the binratintvl containing all
    ratios of one real in bri1 divided by one real in bri2.

(power_binratintvl bri int) returns the binratintvl containing all
    values of one real in bri to the integer power int.

Selectors
---------

(min_binratintvl->rational bri) returns the left rational endpoint
    of bri. 

(max_binratintvl->rational bri) returns the right rational endpoint
    of bri.

(abs_tolerance_binratintvl bri) returns the width r2-r1 of bri=[r1,r2].

(rel_tolerance_binratintvl bri) returns the relative width
    (r2-r1)/max(|r1|,|r2|) of bri=[r1,r2].

(mantissa_binratintvl bri) returns m, where bri=[(m-1)*2^n, (m+1)*2^n].

(exponent_binratintvl bri) returns n, where bri=[(m-1)*2^n, (m+1)*2^n].

Type conversions
----------------

(mid_binratintvl->rational bri) returns the rational midpoint of bri.

(binratintvl->ratintvl bri) returns the interval bri represented as a
    ratintvl instead of a binratintvl.

(ratintvl->binratintvl ri) returns the smallest binratintvl containing
    the ratintvl ri.

String representations
----------------------

(std_binratintvl->string bri) returns a character string representing
    the binratintvl bri. Use the following format:

        [<r1>,<r2>]

    <r1> and <r2> are rationals, in the standard <em>Scheme</em>
    notation.

(mid_err_binratintvl->string bri) returns a character string
    representing the binratintvl bri. Use the following format:

        <dec1>+-<dec2>

    <dec1> and <dec2> are exact decimal representations of the
    midpoint and half width of bri.

(decimal_sn_ratintvl->string bri) returns a character string displaying
    the most accurate decimal representation of bri in scientific
    notation. Use the following format:

        <msgn><mdigit>.<mdigits>E<esgn><edigits>

    Each of <msgn> and <esgn> is either "-" or "". <mdigit> is one
    decimal digit from 0-9. <mdigits> is a sequence of 0 or more
    decimal digits. <edigits> is a sequence of 1 or more decimal
    digits. The first <mdigit>, and every digit in <mdigits>, must be
    significant (i.e., correct for every real in bri). <edigits> must
    have no leading 0s, except for a single 0 if the exponent value
    is 0.

(decimal_ratintvl->string bri) returns a character string displaying

    the most accurate representation of bri in normal decimal
    notation. Use the following format:

        <sgn><idigits>.<ddigits>

    <sgn> is either "-" or "". <idigits> is a sequence of 1 or more
    decimal digits. <ddigits> is a sequence of 0 or more decimal
    digits. Every digit in <idigits> and <ddigits> must be significant
    (i.e., correct for every real in bri). <idigits> must have no
    leading 0s, except for a single 0 if the integer part
    is 0. <ddigits> must have trailing 0s if they are significant.