Interval arithmetic

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Interval arithmetic

is a mathematical technique used to put bounds on rounding errors
and measurement errors in mathematical computation.

Numerical methods using interval arithmetic can guarantee reliable,
mathematically correct results.

Wiki

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https://mauriciopoppe.github.io/interval-arithmetic/

Interval arithmetic

Instead of using a single floating point number as an approximation of a real number,
interval arithmetic represents the approximated value as a set of possible values
(considering the numbers that floating point cannot represent),
let's say we want to represent the number 1 / 3, as a single floating point number
it's approximated as 0.333333..., in the end there will be some 333... decimals
that will be lost due to the nature of floating point,
instead we can represent this number with the interval [0.2, 0.4],
with this interval we're completely sure that 1 / 3 is within the interval
(although the interval is also representing many more numbers),
to improve the scope of the interval we have to understand that numbers
in JavaScript are represented with 64 bits,
therefore to get the next floating point number of a single precision number
the last bit needs to be incremented to get the upper bound,
and the last bit also needs to be decremented to get the lower point

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Description

An interval is a pair of numbers which represents all the numbers between them,
closed means that the bounds are also included in the representation,
extended real because the real number system is extended with two elements:
-∞ and +∞ representing negative infinity and positive infinity respectively.

The implementation is a modified port of the Boost's interval arithmetic library,
the modifications are based on some guidelines from the following papers/presentations:

Interval Arithmetic: from Principles to Implementation - T. Hickey, Q. Ju, M.H. van Emden

Interval Arithmetic: Python Implementation and Applications - Stefano Taschini

The Boost interval arithmetic library - Hervé Brönnimann, Guillaume Melquiond,
Sylvain Pion

Graphing equations with generalized interval arithmetic - Jeffrey Allen Tupper

Floating point operations

Floating point is a way to represent a real number in an approximate way
(due to the finite space existing on a computer),
most calculations with real numbers will produce quantities that
cannot be exactly represented with the space allocated and
therefore this operation needs to be rounded in order to
fit back into its finite representation, such errors are described in more detail here