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lustrec / doc / integer_division.org @ a6e85cdc

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Integer division in LustreC
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* Issue
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Integer division / and associated modulo mod
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a = (a / b) * b + (a mod b)
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Division between two integers can be interpreted in different ways
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- a C like division where sign(a mod b) = sign a
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- a euclidean division where 0 <= a mod b < |b|
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In both cases they satisfy the above equation.
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Kind model-checker or Horn encoding assumes the mathematical definition, ie. the
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euclidean division, while lustreC or the Verimag compiler rely on the C
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definition.
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In the following we deonote by div_C/mod_C and div_M/mod_M the functions in C
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and math, respectively.
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As an example -4 div_C 3 = -1 while -4 div_M 3 = 2
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Some properties:
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- we have a div_M b = a div_C b when a = b * k 
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- we have a mod_C b = 0 \equiv a mod_M b = 0. 
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* From C to Euclidian
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a mod_M b = (a mod_C b) + (a < 0 ? abs(b) : 0) 
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a div_M b = (a - (a mod_M b)) div_C b
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          = (a - ((a mod_C b) + (a < 0 ? abs(b) : 0))) div_C b
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* From Euclidian to C
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a mod_C b = (a >= 0 ? a mod_M b : - ((-a) mod_M b)) 
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            (using math def to ensure positiveness of remainder))
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          = (a mod_M b) - (a < 0 ? abs(b) : 0)
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            (using the def of mod_M above)
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a div_C b = (a - (a mod_C b)) div_M b
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          = (a - ((a mod_M b) - (a < 0 ? abs(b) : 0))) div_M b
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Let's chosse the second, simpler, def of mod_C
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