IMPORTANT NOTICE: please make sure you are using GMP 4.1.4 or later (wrong results may be obtained with earlier versions).
Elements of a RingTwinFloat try to act as though they were unlimited
precision floating point values (while using only a finite precision).
RingTwinFloat uses a heuristic to monitor loss of precision during
computation, and will throw a RingTwinFloat::InsufficientPrecision object if
it detects an unacceptable loss of precision. Beware that this is only
a probabilistic heuristic which can underestimate precision loss. A
RingTwinFloat::InsufficientPrecision object may also be caught as an
ErrorInfo object having error code ERR::InsuffPrec (see error).
EXAMPLE:
If epsilon is a non-zero RingTwinFloat value then equality test (1+epsilon == 1)
can give three possible outcomes (depending on the chosen precision and the
relative size of epsilon compared to 1): true if epsilon is very small
compared to 1, false if epsilon is "relatively large", or it may throw
RingTwinFloat::InsufficientPrecision if epsilon is neither very small
nor relatively large.
RingTwinFloat uses a heuristic for guessing when the difference of two
almost equal values should be regarded as zero. While the heuristic is
usually very reliable, it is possible to construct examples where the
heuristic fails: see ex-RingTwinFloat1.C.
The function IsInteger will return false for any value of magnitude
greater than or equal to 2^PrecisionBits(RR). Recognition of integers
is heuristic; failures in either sense are possible but are also
unlikely.
See RingElem RingTwinFloat for operations on its elements.
There are two constructors for RingTwinFloat: one takes a single
argument being a lower bound on the number of bits' of "heuristically
guaranteed" precision desired (in the mantissa); the other expects 3
args being the minimum "heuristically guaranteed" precision in the
value, a "buffer precision" and the number of noise bits to be
appended. A minimum precision of 8 bits is imposed; smaller
precisions are automatically increased to 8.
All arguments are MachineInt
NewRingTwinFloat(AccuracyBits) implies as many buffer bits as accuracy bits
NewRingTwinFloat(AccuracyBits, BufferBits, NoiseBits)
RingTwinFloat(R) -- sort of downcast the ring R to a twin-float ring;
will throw an ErrorInfo object with code ERR::NotRingTwinFloat if needed.
Let S be a ring
IsRingTwinFloat(S) -- true iff S is actually a RingTwinFloat
RingTwinFloatPtr(S) -- pointer to the twin-float impl (for calling mem fns);
will throw an ErrorInfo object with code ERR::NotRingTwinFloat if needed.
In addition to the standard ring operations, a FractionField
may be used in:
PrecisionBits(RR) -- gives the mantissa precision specified in the ctor
Let RR be a RingTwinFloat and S any ring
NewApproxHom(RR, S) -- creates the homomorphism RR --> S
(but see also CanonicalHom)
As usual the class RingTwinFloat is just a reference counting smart pointer
to an object of type RingTwinFloatImpl (which is the one which really does
the work). The implementation of the smart pointer class RingTwinFloat is
altogether straightfoward (just the same as any of the other smart
pointer ring classes).
The implementation is based on Traverso's idea of "paired floats": each value is represented as two almost equal floating point numbers. The difference between the two numbers is intended to give a good indication of how much "noise" there is in the values. Here we shall allow larger tuples of floating point numbers. Arithmetic is performed independently on each component: e.g.
(a[0],a[1]) + (b[0],b[1]) ==> (a[0]+b[0] , a[1]+b[1])
The consistency of the components is checked after every operation.
The main "trick" in the implementation of RingTwinFloatImpl is that its
elements are MultipleFloats (just a C array of mpf_t values). The
number of components in a MultipleFloat value is determined by
RingTwinFloatImpl::myNumCompts -- currently fixed equal to 2 at compile
time. Furthermore the values of these components must all be very close
to each other. Indeed the function RingTwinFloatImpl::myCheckConsistency
checks this condition: two outcomes are possible:
- (1) all the components are very close to each other;
- (2) at least one component is quite far from another.
-
In case (1) nothing more happens. In case (2) it is evident that an
accumulated loss of precision has become unacceptable, and this triggers
an exception of type RingTwinFloat::InsufficientPrecision. The addition and
subtraction functions check explicitly for near cancellation, and force
the result to be zero in such cases.
The bit precision parameter specified when creating a RingTwinFloat is used
in the following way (with the underlying principle being that elements
of RingTwinFloat(N) should have at least roughly N bits of reliable value).
The digits in the mantissa (of each component in a MultipleFloat) are
conceptually divided into three regions:
A A A A...A A A B B B B....B B B B C C C....C C C
<- N bits -> <- sqrt(N) bits -> <- N/2 bits ->
The region A comprises as many bits as the precision requested, and may
be regarded as being correct with high probability.
The region B comprises "guard digits": these digits are NOT regarded
as being correct, but regions A and B of all components must be equal.
Finally, region C is for "noise", and may be different in different
components.
When an integer is converted to a MultipleFloat, the component with
index 0 takes on the closest possible value to the integer while the
other component(s) have about sqrt(N) bits of uniform random "noise"
added to them (the random value may be positive or negative).
Special action is taken if there is large drop in magnitude during an addition
(or subtraction): if the magnitude drops by more than N+sqrt(N) bits
then the answer is forced to be equal to zero. There is a remote chance
of erroneously computing zero when two almost equal values are
subtracted. It does not seem to be possible to avoid this using limited
precision arithmetic.
Special action is taken if a "noisy" component happens to be too close to the value at index 0: in this case more random noise is added. This can happen, for instance, if a value is divided by itself.
Sorry about the long name; it tries to be descriptive. The function is used to detect cancellation in sums/differences and also for comparisons between twin-float values.
This function captures the "heuristic equality" of twin-float values.
If the values are "clearly unequal" is returns false3; if the values
are equal according to the heuristic it returns true3; otherwise it
returns uncertain3.
There is a discussion in redmine issue 859 about possible definitions of
"heuristically equal" and "clearly unequal". In practice there are two
reasonable candidate definitions: one requires the outer intervals to be
disjoint, the other allows some overlap. The code implements both, and uses
the flag myEqTestMode to choose between them at run-time. Currently
this flag is set when the ring is created, and cannot be changed (because
I'm too lazy to write the necessary simple code).
It took me a while to find a satisfactory definition for the member
function myFloor (even though the final code is fairly simple).
myCeil is quite analogous. myNearestInt calls indirectly
either myFloor or myCeil; it is so simple it must be right,
right?
I eventually settled on the following definition for myFloor.
If the argument satisfies the IsInteger predicate then the floor
function must surely give precisely that integer. Otherwise the argument
(call it X) is not an integer, and the floor of X, if it exists,
will be that integer N which satisfies the two-part condition
N <= X and N+1 > X. If there is no such integer N
then the floor cannot be computed, and an InsufficientPrecision
exception must be thrown. In fact, there is an obvious candidate for N,
namely the floor of the first component of the internal representation of X
(it would be trickier to use the floor of the second component). Clearly N
can be no larger than this candidate, since otherwise the first part of the
condition would fail; and if N were any smaller then the second part would fail.
The code is ugly.
The functions perturb, ApproximatelyEqual and myCmp do "wasteful"
alloc/free of temporary mpf_t values. myCmp can be done better.
What about a function which finds a continued fraction approximant to a
RingTwinFloat value? It seems hard to implement such a function "outside"
RingTwinFloatImpl as InsufficientPrecision will be triggered long before
ambiguity is encountered in the continued fraction.
myIsInteger needs to be rewritten more sensibly (using mpf_ceil or
mpf_floor perhaps?)
How to print out floats when they appear as coeffs in a polynomial??? What are the "best" criteria for printing out a float so that it looks like an integer? Should the integer-like printout contain a decimal point to emphasise that the value may not be exact?
Is it really necessary to call myCheckConsistency after multiplication
and division? The accumulated loss of precision must grow quite slowly.
Yes, it is necessary: consider computing 1^1000000 (or any other high power).
What about COMPLEX floats???
When a MultipleFloat is duplicated should its components be perturbed?
AsMPF is an UGLY function: signature reveals too much about the impl!
myNumCompts could be chosen by the user at run-time; in which case it
must become a per-instance data member (instead of static). I'd guess
that 2, 3 or 4 would be the best compromise.
RingTwinFloatImpl::myOutput:
myCheckConsistency (I'm a little uneasy about this invisible link)
Should there be a means of mapping an element of a high precision RingTwinFloat
to a lower precision RingTwinFloat (without having to pass through an external
representation, such as a rational number)?
It seems wasteful to use two mpf_t values to represent a single RingTwinFloat
value. Would it not be better to keep the main value and an "epsilon" (held as
a double and an int exponent? Would it matter that "epsilon" has only
limited precision?