Muldis::D::Core::Rational - Muldis D rational numeric operators
This document is Muldis::D::Core::Rational version 0.148.0.
This document is part of the Muldis D language specification, whose root document is Muldis::D; you should read that root document before you read this one, which provides subservient details. Moreover, you should read the Muldis::D::Core document before this current document, as that forms its own tree beneath a root document branch.
This document describes essentially all of the core Muldis D operators that are specific to the core data type Rat, essentially all the generic ones that a typical programming language should have.
Rat
This documentation is pending.
Following are all the data types described in this document, arranged in a type graph according to their proper sub|supertype relationships:
sys.std.Core.Type.Universal sys.std.Core.Type.Scalar sys.std.Core.Type.DHScalar sys.std.Core.Type.Cat.DHScalarWP # The following are all regular ordered scalar types. sys.std.Core.Type.Rat sys.std.Core.Type.Rational.BRat sys.std.Core.Type.Rational.DRat
BRat (binary rational) is a proper subtype of Rat where the radix is 2; it is the best option to exactly represent rational numbers that are conceptually binary or octal or hexadecimal, such as most IEEE-754 floating point numbers.
BRat
radix
DRat (decimal rational) is a proper subtype of Rat where the radix is 10 (or if it could be without the float possrep normalization constraint); it is the best option to exactly represent rational numbers that are conceptually the decimal numbers that humans typically work with.
DRat
float
function order (Order <-- topic : Rat, other : Rat, misc_args? : Tuple, is_reverse_order? : Bool) implements sys.std.Core.Ordered.order {...}
This is a (total) order-determination function specific to Rat. Its only valid misc_args argument is Tuple:D0.
order-determination
misc_args
Tuple:D0
function abs (NNRat <-- topic : Rat) implements sys.std.Core.Numeric.abs {...}
This function results in the absolute value of its argument.
function sum (Rat <-- topic? : bag_of.Rat) implements sys.std.Core.Numeric.sum {...}
This function results in the sum of the N element values of its argument; it is a reduction operator that recursively takes each pair of input values and adds (which is both commutative and associative) them together until just one is left, which is the result. If topic has zero values, then sum results in the rational zero, which is the identity value for addition.
topic
sum
function diff (Rat <-- minuend : Rat, subtrahend : Rat) implements sys.std.Core.Numeric.diff {...}
This function results in the difference when its subtrahend argument is subtracted from its minuend argument.
subtrahend
minuend
function abs_diff (Rat <-- topic : Rat, other : Rat) implements sys.std.Core.Numeric.abs_diff {...}
This symmetric function results in the absolute difference between its 2 arguments.
function product (Rat <-- topic? : bag_of.Rat) implements sys.std.Core.Numeric.product {...}
This function results in the product of the N element values of its argument; it is a reduction operator that recursively takes each pair of input values and multiplies (which is both commutative and associative) them together until just one is left, which is the result. If topic has zero values, then product results in the rational 1, which is the identity value for multiplication.
product
function frac_quotient (Rat <-- dividend : Rat, divisor : Rat) implements sys.std.Core.Numeric.frac_quotient {...}
This function results in the rational quotient when its dividend argument is divided by its divisor argument using the semantics of real number division. This function will fail if divisor is zero.
dividend
divisor
function whole_quotient (Int <-- dividend : Rat, divisor : Rat, round_meth : RoundMeth) implements sys.std.Core.Numeric.whole_quotient {...}
This function results in the integer quotient when its dividend argument is divided by its divisor argument using the semantics of real number division, and then the latter's result is rounded to the same or nearest integer, where the nearest is determined by the rounding method specified by the round_meth argument. This function will fail if divisor is zero. TODO: Consider making the result a whole-number Rat instead.
round_meth
function remainder (Rat <-- dividend : Rat, divisor : Rat, round_meth : RoundMeth) implements sys.std.Core.Numeric.remainder {...}
This function results in the rational remainder when its dividend argument is divided by its divisor argument using the semantics of real number division, and then the latter's result is rounded to the same or nearest integer. The semantics of this function preserve the identity x mod y = x - y * (x div y) (read x as dividend and y as divisor) where the division has the same semantics as sys.std.Core.Rational.whole_quotient (rounding guided by round_meth); the sign of this function's result always matches the sign of the dividend or the divisor if round_meth is ToZero (aka truncate) or Down (aka floor), respectively. This function will fail if divisor is zero.
x mod y = x - y * (x div y)
x
y
sys.std.Core.Rational.whole_quotient
ToZero
Down
function quot_and_rem (Tuple <-- dividend : Rat, divisor : Rat, round_meth : RoundMeth) implements sys.std.Core.Numeric.quot_and_rem {...}
This function results in a binary tuple whose attribute names are quotient and remainder and whose respective attribute values are what sys.std.Core.Rational.whole_quotient and sys.std.Core.Rational.remainder would result in when given the same arguments. This function will fail if divisor is zero.
quotient
remainder
sys.std.Core.Rational.remainder
function range (Rat <-- topic : set_of.Rat) implements sys.std.Core.Numeric.range {...}
This function results in the difference between the lowest and highest element values of its argument. If topic has zero values, then this function will fail.
function frac_mean (Rat <-- topic : bag_of.Rat) implements sys.std.Core.Numeric.frac_mean {...}
This function results in the rational mean or arithmetic average of the N element values of its argument. It is equivalent to first taking the sum of the input values, and dividing that sum by the count of the input values using the semantics of real number division. If topic has zero values, then this function will fail.
function median (set_of.Rat <-- topic : bag_of.Rat) implements sys.std.Core.Numeric.median {...}
This function results in the 1 or 2 median values of the N element values of its argument; they are returned as a set. It is equivalent to first arranging the input values from least to greatest, and then taking the single middle value, if the count of input values is odd, or taking the 2 middle values, if the count of input values is even (but if the 2 middle values are the same value, the output has one element). If topic has zero values, then the result set is empty.
function frac_mean_of_median (Rat <-- topic : bag_of.Rat) implements sys.std.Core.Numeric.frac_mean_of_median {...}
This function is a wrapper over sys.std.Core.Rational.median that will result in the rational mean of its result elements; it will fail if there are zero elements.
sys.std.Core.Rational.median
function mode (set_of.Rat <-- topic : bag_of.Rat) implements sys.std.Core.Numeric.mode {...}
This function results in the mode of the N element values of its argument; it is the set of values that appear the most often as input elements, and all have the same count of occurrances. As a trivial case, if all input elements have the same count of occurrances, then they will all be in the output. If topic has zero values, then the result set is empty.
function power_with_whole_exp (Rat <-- radix : Rat, exponent : Int) implements sys.std.Core.Numeric.power_with_whole_exp {...}
This function results in a rational number that is the result of its radix argument taken to the power of its integer exponent argument. This function will result in 1 if radix and exponent are both zero (rather than failing).
exponent
These functions implement commonly used rational numeric operations.
function round (Rat <-- topic : Rat, round_rule : RatRoundRule) {...}
This function results in the rational that is equal to or otherwise nearest to its topic argument, where the nearest is determined by the rational rounding rule specified by the round_rule argument.
round_rule
function power (PRat <-- radix : PRat, exponent : Rat, round_rule : RatRoundRule) {...}
This function results in its (positive rational) radix argument taken to the power of its exponent argument. Since the result would be an irrational number in the general case, the round_rule argument specifies how to coerce the conceptual result into a rational number that is the actual result. Note that, while this function might conceptually have multiple real number results for some fractional exponent, it will always only result in the one that is positive. Note that this operation is also known as exponentiation or **.
**
function log (Rat <-- topic : PRat, radix : PRat, round_rule : RatRoundRule) {...}
This function results in the logarithm of its topic argument to the base given in its (positive rational) radix argument. The round_rule parameter is as per power.
power
function natural_power (PRat <-- exponent : Rat, round_rule : RatRoundRule) {...}
This function results in the special mathematical constant e (which is the base of the natural logarithm) taken to the power of its exponent argument. The round_rule parameter is as per power. Note that this operation is also known as e**.
e**
function natural_log (Rat <-- topic : PRat, round_rule : RatRoundRule) {...}
This function results in the natural logarithm of its topic argument. The round_rule parameter is as per power. Note that this operation is also known as log-e.
log-e
These system-service routines provide ways to get random numbers from the system. Where the results are in the range between truly random and pseudo-random is, for the moment, an implementation detail, but the details of these functions is subject to become more formalized later.
system-service fetch_random (&target : Rat, radix : PInt2_N, max_denom : PInt, interval : sp_interval_of.Rat) [...]
This system-service routine will update the variable supplied as its target argument so that it holds a randomly generated rational value that is included within the interval defined by its interval argument. The denominator attribute of the generated value will be a non-negative power of radix that is not larger than max_denom. This function will fail if interval represents an empty interval.
target
interval
max_denom
Go to Muldis::D for the majority of distribution-internal references, and Muldis::D::SeeAlso for the majority of distribution-external references.
Darren Duncan (darren@DarrenDuncan.net)
darren@DarrenDuncan.net
This file is part of the formal specification of the Muldis D language.
Muldis D is Copyright © 2002-2011, Muldis Data Systems, Inc.
See the LICENSE AND COPYRIGHT of Muldis::D for details.
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