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| This should be evaluated before using any of the following operations.
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| :: ufb | this parameter is not used except for type checking
| | -> ra | | | -> (b, b) | | | Convert from the associated interval type to the base type.
(The types are determined by the given example function.)
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| :: ufb | this parameter is not used except for type checking
| | -> (b, b) | | | -> ra | | | Convert from the base type to the associated interval type.
(The types are determined by the given example function.)
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| A linear ordering on basic functions, which can be syntactic and rather arbitrary.
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| :: Int | number of decimal digits to show
| | -> Bool | whether to show granularity
| | -> Bool | whether to show internal structure
| | -> ufb | | | -> String | |
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| Check internal consistency of the basic function, typically absence of NaN.
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| :: String | indentification of caller location for easier debugging
| | -> ufb | | | -> ufb | | | Check internal consistency of the basic function and report problem if any.
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| Get the granularity of the coefficients inside this basic function.
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Get the degree of this basic function.
If the function is a polynomial, this function should
return its degree.
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| Decrease the degree of a basic function, rounding pointwise upwards.
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Get the term size of this basic function.
If the function is a polynomial, this function should
return the number of terms in the polynomial.
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| Decrease the size of this basic function, rounding pointwise upwards.
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| getVariables :: ufb -> [varid] | Source |
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| Get a list of all variables featured in this basic function.
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| Construct a constant basic function.
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| constEncl :: (b, b) -> (ufb, ufb) | Source |
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| Construct a constant basic enclosure (negated lower bound, upper bound).
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| :: b | value at 0
| | -> Map varid b | ascent of each base vector
| | -> ufb | | | Construct an affine basic function.
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| Find an upper bound of a basic function over [-1,1]^n.
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> ufb | f1 | | -> ufb | f2 | | -> ufb | | | Approximate the function max(f1,f2) from above.
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> ufb | f1 | | -> ufb | f2 | | -> ufb | | | Approximate the function min(f1,f2) from above.
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| Pointwise exact negation of a basic function
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| scaleUp :: b -> ufb -> ufb | Source |
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| Multiply a basic function by a scalar, rounding upwards.
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| :: Int | maximum polynomial degree
| | -> Int | maximum term count
| | -> ra | | | -> ufb | | | -> ufb | | | Multiply a basic function by an approximation of a scalar,
rounding upwards.
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| (+^) :: ufb -> ufb -> ufb | Source |
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| Pointwise upwards rounded addition
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| (-^) :: ufb -> ufb -> ufb | Source |
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| Pointwise upwards rounded subtraction
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| (*^) :: ufb -> ufb -> ufb | Source |
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| Pointwise upwards rounded multiplication
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| :: Int | maximum polynomial degree
| | -> Int | maximum term count
| | -> (ufb, ufb) | | | -> (ufb, ufb) | | | -> (ufb, ufb) | | Enclosure multiplication
IMPORTANT: enclosure = (negated lower bound, upper bound)
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| Approximate the function 1/f from above, assuming
f does not hit zero in the unit domain.
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> EffortIndex | | | -> (ufb, ufb) | enclosure of f
| | -> (ufb, ufb) | | Approximate the reciprocal of an enclosure, assuming
f does not hit zero in the unit domain.
IMPORTANT: enclosure = (negated lower bound, upper bound)
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| evalUp :: boxb -> ufb -> b | Source |
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| Evaluate a basic function at a point rounding upwards
using a basic number for both the point and the result.
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| evalApprox :: boxra -> ufb -> ra | Source |
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| Safely evaluate a basic function at a point using a real number approximation
for both the point and the result.
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| partialEvalApproxUp :: boxra -> ufb -> ufb | Source |
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| Partially evaluate a basic function at a lower-dimensional point
given using a real number approximation.
Approximate the resulting function from above.
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> ufb | function f
| | -> varid | variable v to substitute in f
| | -> ufb | function f_v to substitute for v
that maps [-1,1] into [-1,1]
| | -> ufb | pointwise upper bound of f[v |-> f_v]
| | Compose two basic functions, rounding downwards and upwards,
assuming f_v ranges within the domain [-1,1].
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> ufb | function f
| | -> varid | variable v to substitute in f
| | -> (ufb, ufb) | enclosure of a function f_v to substitute for v
that maps [-1,1] into [-1,1]
| | -> (ufb, ufb) | enclosure of f[v |-> f_v]
| | Compose two basic functions, rounding downwards and upwards,
assuming f_v ranges within the domain [-1,1].
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> ufb | function f
| | -> Map varid ufb | variables to substitute and for each variable v,
function f_v to substitute for v
that maps [-1,1] into [-1,1]
| | -> ufb | pointwise upper bound of f[v |-> f_v]
| | Substitute several variables in a basic function with other basic functions,
rounding downwards and upwards, assuming each f_v ranges
within the domain [-1,1].
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> ufb | function f
| | -> Map varid (ufb, ufb) | variables to substitute and for each variable v,
enclosure of a function f_v to substitute for v
that maps [-1,1] into [-1,1]
| | -> (ufb, ufb) | enclosure of f[v |-> f_v]
| | Substitute several variables in a basic function with other basic functions,
rounding downwards and upwards, assuming each f_v ranges
within the domain [-1,1].
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> EffortIndex | how hard to try when approximating exp as a polynomial
| | -> (ufb, ufb) | f | | -> (ufb, ufb) | | | Approximate sqrt(f) for enclosures.
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> EffortIndex | how hard to try when approximating exp as a polynomial
| | -> (ufb, ufb) | f | | -> (ufb, ufb) | | | Approximate exp(f) for enclosures.
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> EffortIndex | how hard to try when approximating log as a polynomial
| | -> (ufb, ufb) | f | | -> (ufb, ufb) | | | Approximate log(f) for enclosures.
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> EffortIndex | how hard to try when approximating sin as a polynomial
| | -> (ufb, ufb) | f | | -> (ufb, ufb) | | | Approximate sin(f) for enclosures,
assuming the range of f is within [-pi2,pi2].
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> EffortIndex | how hard to try when approximating cos as a polynomial
| | -> (ufb, ufb) | f | | -> (ufb, ufb) | | | Approximate cos(f) for enclosures,
assuming the range of f is within [-pi2,pi2].
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| :: Int | max degree for result
| | -> Int | max approx size for result
| | -> EffortIndex | how hard to try when approximating cos as a polynomial
| | -> (ufb, ufb) | f | | -> (ufb, ufb) | | | Approximate atan(f) for enclosures.
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| :: varid | variable to integrate by
| | -> ufb | f | | -> (ufb, ufb) | | | Approximate the primitive function of f from below and from above.
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| :: [varid] | dimensions to include in the measuring domain;
have to include all those present in f
| | -> ufb | f | | -> b | | | Measure the volume between a function
and the zero hyperplane on the domain [-1,1]^n.
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Instances