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2016 Update

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Tina Müller 2016-12-05 22:15:40 +01:00
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Task/Safe-addition/00DESCRIPTION
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Implementation of [[wp:Interval_arithmetic|interval arithmetic]] and more generally fuzzy number arithmetic require operations that yield safe upper and lower bounds of the exact result. For example, for an addition, it is the operations +↑ and +↓ defined as: ''a'' +↓ ''b'' ≤ ''a'' + ''b'' ≤ ''a'' +↑ ''b''. Additionally it is desired that the width of the interval (''a'' +↑ ''b'') - (''a'' +↓ ''b'') would be about the machine epsilon after removing the exponent part.
Implementation of   [[wp:Interval_arithmetic|interval arithmetic]]   and more generally fuzzy number arithmetic require operations that yield safe upper and lower bounds of the exact result.
Differently to the standard floating-point arithmetic, safe interval arithmetic is '''accurate''' (but still imprecise). I.e. the result of each defined operation contains (though does not identify) the exact mathematical outcome.
For example, for an addition, it is the operations &nbsp; <big> +&uarr; </big> &nbsp; and &nbsp; <big> +&darr; </big> &nbsp; defined as: &nbsp; <big> ''a'' +&darr; ''b'' &le; ''a'' + ''b'' &le; ''a'' +&uarr; ''b''. </big>
Usually a [[wp:Floating_Point_Unit|FPU's]] have machine +,-,*,/ operations accurate within the machine precision. To illustrate it, let us consider a machine with decimal floating-point arithmetic that has the precision is 3 decimal points. If the result of the machine addition is 1.23, then the exact mathematical result is within the interval ]1.22, 1.24[. When the machine rounds towards zero, then the exact result is within [1.23,1.24[. This is the basis for an implementation of safe addition.
Additionally it is desired that the width of the interval &nbsp; <big> (''a'' +&uarr; ''b'') - (''a'' +&darr; ''b'') </big> &nbsp; would be about the machine epsilon after removing the exponent part.
===Task===
Show how +&darr; and +&uarr; can be implemented in your language using the standard floating-point type. Define an interval type based on the standard floating-point one, and implement an interval-valued addition of two floating-point numbers considering them exact, in short an operation that yields the interval [''a'' +&darr; ''b'', ''a'' +&uarr; ''b''].
Differently to the standard floating-point arithmetic, safe interval arithmetic is '''accurate''' (but still imprecise).
I.E.: &nbsp; the result of each defined operation contains (though does not identify) the exact mathematical outcome.
Usually a &nbsp; [[wp:Floating_Point_Unit|FPU's]] &nbsp; have machine &nbsp; <big> +,-,*,/ </big> &nbsp; operations accurate within the machine precision.
To illustrate it, let us consider a machine with decimal floating-point arithmetic that has the precision is '''3''' decimal points.
If the result of the machine addition is &nbsp; <big> 1.23, </big> &nbsp; then the exact mathematical result is within the interval &nbsp; <big> ]1.22, 1.24[. </big>
When the machine rounds towards zero, then the exact result is within &nbsp; <big> [1.23,1.24[. </big> &nbsp; This is the basis for an implementation of safe addition.
;Task;
Show how &nbsp; <big> +&darr; </big> &nbsp; and &nbsp; <big> +&uarr; </big> &nbsp; can be implemented in your language using the standard floating-point type.
Define an interval type based on the standard floating-point one, &nbsp; and implement an interval-valued addition of two floating-point numbers considering them exact, in short an operation that yields the interval &nbsp; <big> [''a'' +&darr; ''b'', ''a'' +&uarr; ''b'']. </big>
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