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package interval_intel
Interval library in OCaml. ONLY FOR INTEL PROCESSORS.
All operations use correct rounding.
It is recommended to open this module. It will put into scope the interval type and a module I (see Interval.I) containing interval operations:
open Interval_intel
let x = I.(v 0.5 1. + sin(v 3. 3.125))When the module I is open, the integer operators (+, -,...) and the floating point ones (+., -.,...) are redefined. If, in the middle of an expression, you need to use the usual operators, locally open the module I.U as in I.(x /. U.(float(n + 1))).
The names symbols for infix operators have been chosen to try to make the standard cases short and the overall expression readable. The rationale is as follows.
- The integer operators
+,-,*,/,~-and~+act on intervals; - The float operators
+.,-.,/.take an interval to the left and a float to the right, but*.does the opposite. This is to match the standard presentation of polynomials. Example:I.(3. *. x**2 + 2. *. x +. 4.). WARNING: you should only use these functions when the float argument represent the exact mathematical value (taking into account that the compiler will round literals). For example,42.is fine while0.1is not (because it has an infinite binary representation). - Operators
+:,-:,*:and/:are as in the previous point but with the interval and float swapped. Note that+:and*:are not really needed because the operations are commutative but are present for consistency. - For exponentiation,
**has been chosen for integer exponent because they are the more frequent,**.when the exponent is a float,**:when the base is a float and***for exponentiation of two intervals.
You do not have to worry about remembering these rules. The type system will enforce them.
It is not mandatory, but still wise, to read the documentation of the Fpu module.
This library has been mainly designed to be used in a branch and bound optimization algorithm. So, some choices have been made:
- NaN is never used. We either extend functions by pseudo continuity or raise exceptions. For example,
{low=2.;high=3.} / {low=0.;high=2.}returns{low=1.;high=Inf}, while{low=2.;high=3.} / {low=0.;high=0.}or{low=0.;high=0.} / {low=0.;high=0.}raisesInterval_base.Division_by_zero. - Intervals [+Inf,+Inf] or [-Inf,-Inf] are never used and never returned.
- When using a float in the following operations, it must never be equal to +Inf or -Inf or Nan.
- Functions such as
log,sqrt,acosorasinare restricted to their definition domain but raise an exception rather than returning an empty interval: for examplesqrt {low=-4;high=4}returns{low=0;high=2}whilesqrt {low=-4;high=-2}will raise an exception.
Another design choice was to have non mutable elements in interval structure, and to maintain an "ordinary" syntax for operations, such as let a = b + c in ... thus mapping interval computation formula on airthmetic formula. We could have instead chosen to have mutable elements, and to write for example (Interval.add a b c) to perform “a ← b + c”. The first choice is, to our point of view, more elegant and easier to use. The second is more efficient, especially when computing functions with many temporary results, which force the GC to create and destroy lot of intervals when using the implementation we chose. Nothing's perfect.
The older deprecated interface is still available.
The library is implemented in x87 assembly mode and is quite efficient (see below).
Intervals (for Intel processors)
The interval type. Be careful however when creating intervals. For example, the following code: let a = {low=1./.3.; high=1./.3.} creates an interval which does NOT contain the mathematical object 1/3.
If you want to create an interval representing 1/3, you have to write let a = I.(inv(v 3. 3.)) because rounding will then be properly handled by I.inv and the resulting interval will indeed contain the exact value of 1/3.
module I : sig ... endInterval operations. Locally open this module — using e.g. I.(...) — to redefine classical arithmetic operators for interval arithmetic.
module Fpu : sig ... endAccess to low level floating point functions. THIS LIBRARY ONLY WORKS FOR INTEL PROCESSORS.
module RoundDown = Fpu.RoundDownmodule RoundUp = Fpu.RoundUpmodule Low = RoundDownmodule High = RoundUpOld interface (deprecated)
The functions below are the ones of the older versions of Interval. They will soon be removed.
type t = intervalval zero_I : intervalNeutral element for addition
val one_I : intervalNeutral element for multiplication
val pi_I : intervalpi with bounds properly rounded
val e_I : intervale with bounds properly rounded
Prints an interval with the same format applied to both endpoints. Formats follow the same specification than the one used for the regular printf function
val fprintf_I :
out_channel ->
(float -> string, unit, string) format ->
interval ->
unitPrints an interval into an out_channel with the same format applied to both endpoints
Returns a string holding the interval printed with the same format applied to both endpoints
val float_i : int -> intervalReturns the interval containing the float conversion of an integer
val compare_I_f : interval -> float -> intcompare_I_f a x returns 1 if a.high<x, 0 if a.low<=x<=a.high and -1 if x<a.low
val size_I : interval -> floatsize_I a returns a.high-a.low
sgn a returns {low=float (compare a.low 0.);high=float
(compare a.high 0.)}
abs_I a returns {low=a.low;high=a.high} if a.low>=0., {low=-a.high;high=-a.low} if a.high<=0., and {low=0.;high=max -a.low a.high} otherwise
union_I_I a b returns {low=min a.low b.low; high=max a.high b.high}
max_I_I a b returns {low=max a.low b.low; high=max a.high b.high}
min_I_I a b returns {low=min a.low b.low; high=min a.high b.high}
a *$. x multiplies a by x according to interval arithmetic and returns the proper result. If x=0. then zero_I is returned
x *$. a multiplies a by x according to interval arithmetic and returns the proper result. If x=0. then zero_I is returned.
a *$ b multiplies a by b according to interval arithmetic and returns the proper result. If a=zero_I or b=zero_I then zero_I is returned
a /$. x divides a by x according to interval arithmetic and returns the proper result. Raise Failure "/$." if x=0.
x /.$ a divides x by a according to interval arithmetic and returns the result. Raise Failure "/.$" if a=zero_I
a /$ b divides the first interval by the second according to interval arithmetic and returns the proper result. Raise Failure "/$" if b=zero_I
mod_I_f a f returns a mod f according to interval arithmetic et OCaml mod_float definition. Raise Failure
"mod_I_f" if f=0.
sqrt_I a returns {low=sqrt a;high=sqrt b} if a>=0., {low=0.;high=sqrt b} if a<0.<=b. Raise Failure "sqrt_I" if b<0.
Pow_I_i a n with n integer returns interval a raised to nth power according to interval arithmetic. If n=0 then {low=1.;high=1.} is returned. Raise Failure "pow_I_f" if n<=0 and a=zero_I. Computed with exp-log in base2.
a **$. f returns interval a raised to f power according to interval arithmetic. If f=0. then {low=1.;high=1.} is returned. Raise Failure "**$." if f<=0. and a=zero_I or if f is not an integer value and a.high<0.. Computed with exp-log in base2.
a **$ b returns interval a raised to b power according to interval arithmetic, considering the restriction of x power y to x >= 0. Raise Failure "**$" if a.high < 0 or (a.high=0. and
b.high<=0.)
x **.$ a returns float x raised to interval a power according to interval arithmetic, considering the restiction of x power y to x >= 0. Raise Failure "**.$" if x < 0 and a.high <= 0
log_I a returns {low=log a.low; high=log a.high} if a.low>0., {low=neg_infinity; high=log a.high} if a.low<0<=a.high. Raise Failure "log_I" if a.high<=0.
cos_I a returns the proper extension of cos to arithmetic interval Returns [-1,1] if one of the bounds is greater or lower than +/-2**53
sin_I a returns the proper extension of sin to arithmetic interval Returns [-1,1] if one of the bounds is greater or lower than +/-2**53
tan_I a returns the proper extension of tan to arithmetic interval Returns [-Inf,Inf] if one of the bounds is greater or lower than +/-2**53
acos_I a raise Failure "acos_I" if a.low>1. or a.high<-1., else returns {low=if a.high<1. then acos a.high else 0;
high=if a.low>-1. then acos a.low else pi}. All values are in [0,pi].
asin_I a raise Failure "asin_I" if a.low>1. or a.high<-1. else returns {low=if a.low>-1. then asin a.low else -pi/2;
high=if a.low<1. then asin a.high else pi/2}. All values are in [-pi/2,pi/2].
atan2mod_I_I y x returns the proper extension of interval arithmetic to atan2 but with values in [-pi,2 pi] instead of [-pi,pi]. This can happen when y.low<0 and y.high>0 and x.high<0: then the returned interval is {low=atan2 y.high
x.high;high=(atan2 y.low x.high)+2 pi}. This preserves the best inclusion function possible but is not compatible with the standard definition of atan2
Same function as above but when y.low<0 and y.high>0 and x.high<0 the returned interval is [-pi,pi]. This does not preserve the best inclusion function but is compatible with the atan2 regular definition
val size_max_X : interval array -> floatComputes the size of the largest interval of the interval vector
val size_mean_X : interval array -> floatComputes the mean of the size of intervals of the interval vector
Prints an interval vector with the same format applied to all endpoints.
val fprintf_X :
out_channel ->
(float -> string, unit, string) format ->
interval array ->
unitPrints an interval vector into an out_channel with the same format applied to all endpoints
Returns a string holding the interval vector printed with the same format applied to all endpoints
val print_X : interval array -> unitDeprecated
val print_I : interval -> unitDeprecated
val size_X : interval array -> floatDeprecated
val size2_X : interval array -> floatDeprecated
val (<$.) : interval -> float -> intDeprecated
Performance
Intel Atom 230 Linux 32 bits:
ftanspeed (10000000 calls): 2.528158fcosspeed (10000000 calls): 2.076129fsinspeed (10000000 calls): 1.972123I.tanspeed (10000000 calls): 4.416276I.cosspeed (10000000 calls): 4.936308I.sinspeed (10000000 calls): 5.396338faddspeed (10000000 calls): 0.980062fsubspeed (10000000 calls): 0.980061fmulspeed (10000000 calls): 0.980061fdivspeed (10000000 calls): 1.424089I.( + )speed (10000000 calls): 1.656103I.( - )speed (10000000 calls): 1.636103I.( * )speed (10000000 calls): 4.568285I.( / )speed (10000000 calls): 4.552285
Intel 980X Linux 64 bits:
ftanspeed (10000000 calls): 0.472029fcosspeed (10000000 calls): 0.400025fsinspeed (10000000 calls): 0.400025I.tanspeed (10000000 calls): 0.752047I.cosspeed (10000000 calls): 1.036065I.sinspeed (10000000 calls): 1.104069faddspeed (10000000 calls): 0.124008fsubspeed (10000000 calls): 0.120008fmulspeed (10000000 calls): 0.128008fdivspeed (10000000 calls): 0.156010I.( + )speed (10000000 calls): 0.340021I.( - )speed (10000000 calls): 0.332021I.( * )speed (10000000 calls): 0.556035I.( / )speed (10000000 calls): 0.468029