Proposal for Prolog Standard core update wrt floating point arithmetic

Creative Commons Licence
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Joachim Schimpf
14 Oct 2009
20 Oct 2009


Our aim is to update the standard in such a way that it accommodates Prolog processors that implement: We do not at this point try to

Concrete Suggestions

We suggest the following changes to the Prolog standard 13211-1 (1995). They are motivated and discussed in detail in the subsequent document.
  1. Change 7.9.2 and 5.5.10 to allow implementation defined choice between exception raising and the delivery of continuation values, see here and here
  2. Correct mistakes in exception specifications in section 9, see here
  3. Add float-related flags, see here
  4. Add copysign/2 function, see here
  5. Add nexttoward/2 function, see here
  6. Add syntax definitions for infinity and NaN, see here
  7. Change wording in section 8.7.2 to allow for exact comparison, see here
  8. Amend to accomodate floating point overflow, see here

How the Prolog standard defines floating point operations

The Prolog standard refers to the Standard for Language Independent Arithmetic [LIA], which characterises floating point numbers as a subset F of the real numbers R, characterised by Derived values are An extended set F* is defined, which is used for defining rounding, but not for accommodating infinity or NaN (which are not concepts defined in LIA).

The semantics of a language arithmetic is defined in terms of operations on F, which in turn are defined in terms of the common mathematical operations on R (of which F is a subset), plus a rounding function. Since infinity and NaN are not members of R, they cannot be covered this way, nor can signed zeros be distinguished. This is an intrinsic problem with basing the Prolog semantics on LIA.

Floating point results in LIA can be numbers in F, or 'exceptional values' like float_overflow, undefined, underflow (but these are just conceptual things). LIA section 6 allows different 'notifications' of these 'exceptional values', among them the raising of language exceptions, but also the production of 'continuation values' which are not members of F. This is where infinities and NaNs fit in: they are implementation-defined ways of providing notification of the conceptual exceptions. So, a language implementation is allowed to do this by defining which 'continuation values' are produced for which exception (to do this with IEEE arithmetic is rather straightforward).

The Prolog standard however forbids this by stating in 7.9.2 and 9.1.2 that an 'exceptional value' result E gives rise to an evaluation_error(E). This must be changed.

Once 'continuation values' are introduced, they can occur as arguments of arithmetic operations. The problem is then to define the operations in these cases. LIA is no help here, since it only describes operation on F. We are faced with 3 alternatives:

Suggestion: Explicitly refer to IEEE 754 for the specification of behaviour with non-F values. For operations not in IEEE 754, define it in the Prolog standard.

This means that a Prolog does not have to implement IEEE 754 arithmetic, but if it provides infinities and NaNs, they must behave like IEEE 754 ones for the operations that are relevant for Prolog. This seems a more helpful solution than leaving it implementation-defined.


ISO 31211-1 defines 4 float-related exceptions, based on LIA, with rather obvious correspondence to IEEE 754 exceptions for the floats: Section 7.9.2 and 9.1.2 prescribe that an 'exceptional value' result E gives rise to an evaluation_error(E), which obviously disallows returning infinities or NaNs. Section 5.5.10 says "a processor may support the value of an expression being a value of an additional type instead of an exceptional value". If the word 'type' here is used in the sense defined in 3.186, then this would refer to an additional type like 'complex' (allowing e.g. sqrt(-2) to be computed without exception), rather than allowing an additional value for the existing float type to be returned.

Suggestion: change wording to make evaluation_errors implementation defined.

The implementation shall have the option of raising evaluation_error(E) or continuing with an implementation-defined 'continuation value'. For each exception individually, an implementation can choose between

The choices must be reflected in the values of the corresponding new Prolog flags.

Corrections to Section 9

We review here the exceptions that can be raised by the different operations when using IEEE arithmetic. Mistakes here must be corrected and new exceptions added because we are going to use the exceptions to specify where infinities and NaNs are being created.

An undefined exception can occur with:

as before
change: log(0) now to zero_divisor
as before
to be added
was zero_divisor in 13211-1
to be added
to be added
to be added
These correspond to the cases where IEEE 754 would produce the INVALID exception and produce a NaN. These are the only functions that produce NaNs from defined inputs. IEEE 754 additionally lists remainder(Any,0) and remainder(inf,Any).

A zero_divisor exception can occur with:

change: 0/0 now undefined
was undefined exception in 13211-1
was undefined exception in 13211-1
to be added, if implemented
These correspond to the cases where IEEE 754 would produce the DIVIDEBYZERO exception and produce an infinity.

underflow exceptions can occur in X**Y, add, sub, mul, div.


The first two are required by IEEE 754.

Special Values

Negative zeros

If negative zeros are handled by a system, they must satisfy the following requirements: Because 0.0 and -0.0 are easier to distinguish in Prolog than in other languages, it is advisable to specify precisely where negative zeros may be generated. Otherwise code that matches or unifies against 0.0 will unexpectedly fail.


If a system supports infinities, they shall


If supported, similar requirements as for infinities, i.e. they shall

In my opinion, while infinities are very useful to have in Prolog because they extend the computation domain in a natural way, NaNs fit much less naturally. Binding a variable to a special value that says "there is no solution" isn't the Prolog way, the logical behaviour is to fail when there is no solution (although the logical failure might be replaced by an exception for practical purposes, analogous to type failures/errors). Adding NaNs seems to create far more problems than it solves, e.g. regarding comparisons.


The following is standard conforming behaviour with double floats, and implemented by most systems:
?- 18014398509481985 > 18014398509481984.0.
?- 18014398509481985 < 18014398509481984.0.
?- 18014398509481983 > 18014398509481984.0.
?- 18014398509481983 < 18014398509481984.0.
Infinities are a special case of this:
?- 2^1024 < 1.0Inf.
This is because the integer is inexactly converted to float, then the comparison is done with floats. This semantics is prescribed by the standard (section 8.7.2). It would be possible (and advocated in [OKPL]) to require, or at least allow, exact comparison instead, which means changing section 8.7.2.

Another way to fix it is to raise an exception on inexact (implicit) conversion, i.e. earlier than overflow. The above examples would then raise an exception, which the programmer can circumvent by an explicit float/1 conversion. It would not give a way to get the correct result though.

Suggestion: change section 8.7.2 to allow all of current behaviour, inexact exception, or exact comparison.

Minimum and Maximum

If we, as previously suggested, do not standardise mixed type operation on these primitives, the problems are minimal (when allowing mixed type arguments, the issues from comparison would carry over). Special cases in float-float comparison are
min(0.0,-0.0)           -0.0    see below
min(-0.0,0.0)           -0.0    see below
max(0.0,-0.0)           0.0
max(-0.0,0.0)           0.0
max(1.0Inf,Any)         1.0Inf
min(1.0Inf,Any)         Any
max(-1.0Inf,Any)        Any
min(-1.0Inf,Any)        1.0Inf
min(NaN,Any)            Any     see below
max(NaN,Any)            Any     see below
IEEE 754 (2008) relaxes min(-0.0,0.0), not specifying the sign of the result! This is probably not advisable for Prolog, where the zeros can easily be told apart.

IEEE 754 (2008) specifies minNum() and maxNum() in this way, i.e. preferring the non-NaN argument. [KAH] says there are good reasons to do so, but that they are disputed.

Results with IEEE arithmetic

Special values as arguments

For most operations, the behaviour can be defined by referring to the corresponding definitions in IEEE 754. Only operations with no obvious counterpart there should be explicit in the Prolog standard. The following is a summary.

Unary Functions

                0.0     -0.0    1.0Inf  -1.0Inf NaN     Remark

Standard Unary Functions

+/1             0.0     -0.0    1.0Inf  -1.0Inf NaN
-/1             -0.0    0.0     -1.0Inf 1.0Inf  NaN
abs/1           0.0     0.0     1.0Inf  1.0Inf  NaN
sqrt/1          0.0     -0.0    1.0Inf  Ex/NaN  NaN     (1)
sign/1          0.0     0.0?    1.0     -1.0    NaN     (see below)
float_i_part/1  0.0     -0.0    1.0Inf  -1.0Inf NaN
float_f_part/1  0.0     -0.0    0.0     0.0     NaN
float/1         0.0     -0.0    1.0Inf  -1.0Inf NaN

sin/1           0.0     -0.0    Ex/NaN  Ex/NaN  NaN
cos/1           1.0     1.0     Ex/NaN  Ex/NaN  NaN
tan/1           0.0     -0.0    Ex/NaN  Ex/NaN  NaN
asin/1          0.0     -0.0    Ex/NaN  Ex/NaN  NaN
acos/1          pi/2    pi/2    Ex/NaN  Ex/NaN  NaN
atan/1          0.0     -0.0    pi/2    -pi/2   NaN
exp/1           1.0     1.0     1.0Inf  0.0     NaN
log/1           -1.0Inf -1.0Inf 1.0Inf  Ex/NaN  NaN     (4)

floor/1         0       0       Ex      Ex      Ex      (2),(5)
truncate/1      0       0       Ex      Ex      Ex      (5)
round/1         0       0       Ex      Ex      Ex      (5)
ceiling/1       0       0       Ex      Ex      Ex      (5)

Non-standard unary functions (for illustration)         (5)

ffloor/1        0.0     -0.0    1.0Inf  -1.0Inf NaN     (2)
ftruncate/1     0.0     -0.0    1.0Inf  -1.0Inf NaN     (3)
fround/1        0.0     -0.0    1.0Inf  -1.0Inf NaN     (3)
fceiling/1      0.0     -0.0    1.0Inf  -1.0Inf NaN     (3)
sqrt(-0.0) is defined as -0.0 in [sqrt], i.e. it does not behave like a small negative number (which would be undefined).
floor(-0.0) is defined as returning -0.0 (rather than -1.0) in [floor], i.e. it does not behave like a small negative number.
ceil/trunc/round(-0.0) are all defined as -0.0, not 0.0 in [ceil], [trunc], [round].
log(-0.0) is -1.0Inf (rather than domain error) according to [log], i.e. it does not behave like a small negative number (which would be undefined).
It is regrettable that the standard defines floor, ceiling, truncate and round with an implicit conversion to integer type, because this makes them partial functions. We would have been better off having these return floats (or whatever the argument type was), and have a separate pure type conversion to integer. It's probably too late to change that. I also seem to recall that it was in an earlier standard draft, but was later rejected.

sign/1 and copysign/2

sign/1 is implemented incorrectly by a number of systems:
                13211-1 SP3.11  SWI5.6/Yap5.1/ECL6.0
sign(5)         1       1       1
sign(0)         0       0       0
sign(-5)        -1      -1      -1
sign(5.0)       1.0     1.0     1
sign(0.0)       0.0     0.0     0
sign(-0.0)      unspec  0.0     0
sign(-5.0)      -1.0    -1.0    -1
Unlike earlier drafts, the final 13211-1 specifies 0.0 (rather than 1.0) as the result of sign(0.0). This means we cannot simulate copysign(X,Y) via X*sign(Y), in particular not for copysign(X, -0.0). This was modelled after the signF operation in LIA. LIA changed its mind in a recent draft update (2007) and replaced signF with a function signumF wich returns 1.0 (for positive numbers and 0.0) and -1.0 (for negative numbers and -0.0). We should either change our sign/1 function, or add a copysign/2 function.

How to specify sign(-0.0) is a bit of a dilemma. Returning 0.0 is nice in the sense that there are only 3 discrete sign values, like in the integer case (but why is the result then not integer -1,0,1 in the first place?). On the other hand, simply losing the sign bit of -0.0 looks wrong for a float to float operation. That's probably a reason why IEEE 754 does not define such an operation. As a last resort, it could be left implementation-defined.

[remark added 2011-03-08: Java defines signum() as returning -0.0 for -0.0, which would be a way to subtly improve the current sign/1, but is still no replacement for copysign/2.]


+(-0.0,0.0)             0.0
+(0.0,-0.0)             0.0
+(-0.0,-0.0)            -0.0
+(-0.0,NZ)              NZ		(NZ nonzero)
+(1.0Inf,-1.0Inf)       Ex/NaN
+(1.0Inf,NotInf)        1.0Inf
+(-1.0Inf,1.0Inf)       Ex/NaN
+(-1.0Inf,NotInf)       -1.0Inf
+(NaN,Any)              NaN		(Any is not NaN)


-(0.0,0.0)              0.0
-(-0.0,0.0)             -0.0
-(0.0,-0.0)             0.0
-(-0.0,-0.0)            0.0
-(-0.0,NZ)              -NZ
-(NZ,-0.0)              NZ
-(-1.0Inf,-1.0Inf)      Ex/NaN
-(1.0Inf,1.0Inf)        Ex/NaN
-(1.0Inf,-1.0Inf)       1.0Inf
-(-1.0Inf,1.0Inf)       -1.0Inf
-(1.0Inf,NotInf)        1.0Inf
-(NotInf,1.0Inf)        -1.0Inf		(NotInf is not infinity)
-(-1.0Inf,NotInf)       -1.0Inf
-(NotInf,-1.0Inf)       1.0Inf
-(NaN,Any)              NaN
-(Any,NaN)              NaN


*(-0.0,Pos)             -0.0
*(-0.0,Neg)             0.0
*(1.0Inf,1.0Inf)        1.0Inf
*(1.0Inf,-1.0Inf)       -1.0Inf
*(-1.0Inf,-1.0Inf)      1.0Inf
*(1.0Inf,NotInf)        1.0Inf
*(-1.0Inf,NotInf)       -1.0Inf
*(NaN,Any)              NaN


/(-0.0,0.0)             Ex/NaN
/(-0.0,-0.0)            Ex/NaN
/(-0.0,PNZ)             -0.0		(PNZ positive nonzero)
/(-0.0,NNZ)             0.0		(NNZ negative nonzero)
/(PNZ,-0.0)             -1.0Inf
/(NNZ,-0.0)             1.0Inf
/(-1.0Inf,-1.0Inf)      Ex/NaN
/(1.0Inf,1.0Inf)        Ex/NaN
/(1.0Inf,-1.0Inf)       Ex/NaN
/(-1.0Inf,1.0Inf)       Ex/NaN
/(1.0Inf,PNZF)          1.0Inf		(PNZF pos nonzero finite)
/(PNZF,1.0Inf)          0.0
/(-1.0Inf,PNZF)         -1.0Inf
/(PNZF,-1.0Inf)         -0.0
/(1.0Inf,NNZF)          -1.0Inf		(NNZF neg nonzero finite)
/(NNZF,1.0Inf)          -0.0
/(-1.0Inf,NNZF)         1.0Inf
/(NNZF,-1.0Inf)         0.0
/(NaN,Any)              NaN
/(Any,NaN)              NaN

Special values as results of operations on non-special values

Special values infinity and NaN occur as a result of an 'exceptional value' result from an arithmetic operation (if the corresponding evaluation_error(E) is not raised, see exceptions).

Negative zeros occur as results as specified in IEEE 754, e.g.

-0.0 is 0.0 * -1.0.
-0.0 is 0.0 / -1.0.
and additionally in
float_integer_part(-0.01)	-0.0
float_fractional_part(-2.0)	-0.0
float_integer_part(-0.0)	-0.0
float_fractional_part(-0.0)	-0.0
The latter two result from the requirement that
X is float_integer_part(X) + float_fractional_part(X).


The following proposal for floating point flags is based on

FlagExample (double)Comment
float_radix2Radix of the representation [LIA]
float_precision53Digits in radix base [LIA]
float_emin-1022Smallest exponent in radix base [LIA]
float_emax1023Largest exponent in radix base [LIA]
float_denormtrueWhether denormalised values are used [LIA]
float_iec_559trueFlag required by [LIA]. Indicates full conformity to IEC 559 (IEEE 754)
float_min2.2250738585072014e-308Smallest representable positive number. Can be computed from the above, but tediously. Can be computed as nexttoward(0.0,1.0)
float_max1.7976931348623157e+308Largest representable finite number. Can be computed from the above, but tediously. Can be computed as nexttoward(1.0Inf,0.0) if infinity is supported. Flag needed to raise representation_error(float_max) in term reading.
float_max_integer9007199254740992.0Largest integer that can be represented accurately. Can be computed as radix**precision
float_overflowinfinityBehaviour on float_overflow during computation or reading (error, infinity) [OKE]
float_zero_divinfinityBehaviour on zero_divisor exception (error, infinity)
float_undefinederrorBehaviour on undefined result (error, nan)
float_underflowignoreBehaviour on underflow (ignore, error)
float_roundingto_nearestRounding mode (to_nearest, to_nearest_tta, to_positive, to_negative, to_zero). May be writeable.
FlagExample (double)Comment
float_inexactignoreBehaviour on inexact result (ignore, error). As the standard does not define this exception, this is reserved for implementations that do.
float_min_integer-9007199254740992.0Can be computed as -max_integer
float_min_exponent-307Only useful in relation to decimal I/O. Can be computed as ceiling((emin-1)*log(radix,10))
float_max_exponent308Only useful in relation to decimal I/O. Can be computed as floor((emax+1)*log(radix,10))
float_epsilon2.2204460492503131e-16Likely to be misused. nexttoward/2 is more useful. Can be computed as radix**(1-precision), or nexttoward(1.0,2.0)-1.0
float_digits16Number of significant decimal digits. Can be computed more precisely as precision/log(radix,10)
float_width24Number of characters required to canonically write a float
float_formatieee(double)A term describing the underlying implementation. Difficult to define the possible values, e.g. when it's 'ieee with some features missing'.

New Functions


As suggested above, the nexttoward/2 function (see [nexttoward], or nextAfter() in IEEE 754) has many good uses. It computes the next representable float in a particular direction. This can be be used to perform safe rounding up or down, when following a primitive arithmetic operation that is known to be correct to within one ULP, thus supporting interval arithmetic. It also can be used to compute special values like float_min and float_max, and float_epsilon (see above).

nexttoward(X,Y) evaluates the expressions X and Y with values VX and VY and computes the next representable floating-point value following VX in the direction of VY. Examples:

nexttoward(1.0,2.0)                             1.0000000000000002
nexttoward(1.0,-1.0)                            9.99999999999999889e-01
nexttoward(1.0Inf,0.0)                          1.7976931348623157e+308
nexttoward(0.0,1.0)                             4.94065645841246544e-324
nexttoward(9007199254740992.0,1.0Inf)           9007199254740994.0
nexttoward(1.7976931348623157e+308,1.0Inf)      1.0Inf
nexttoward(0.0,-1.0)                            -4.94065645841246544e-324
nexttoward(-0.0,-1.0)                           -4.94065645841246544e-324
nexttoward(-4.94065645841246544e-324,1.0)	-0.0


Because of the shortcomings of the existing sign/1 function, we recommend the addition of the copysign/2 function with the semantics specified in IEEE 754 (this is the same as SIGN() in Fortran).

copysign(X,Y) evaluates the expressions X and Y with values VX and VY and produces a value with the magnitude of VX and the sign of VY. Examples:

copysign(3.0, 2.0)       3.0
copysign(3.0, 0.0)       3.0
copysign(3.0,-0.0)      -3.0
copysign(3.0,-2.0)      -3.0
copysign(Any,1.0NaN)    Any
copysign(Any,-1.0NaN)   -Any
copysign(Any,PNZ)       Any
copysign(Any,NNZ)       -Any
copysign(Any,0.0)       Any
copysign(Any,-0.0)      -Any
Note the significance of the sign on zero and NaNs (copysign doesn't propagate NaNs like other functions).


The canonical syntax for a negative zero shall be -0.0. On input, any representation of a negative number too small to represent may be rounded to -0.0. Processors that do not implement negative zeros shall produce positive zero instead.

The canonical syntax for positive infinity shall be 1.0Inf, and -1.0Inf for negative infinity. On input, the actual values of the digits are ignored, so 3.456Inf=1.0Inf, and even 0.0Inf=1.0Inf. Note that the syntax is chosen not to conflict with other valid Prolog syntax, in particular, the suffix is required to start with a capital letter, so that 1.0inf can still serve as postfix syntax for inf(1.0). Processors that do not implement an infinity representation shall raise a representation_error(float_overflow) exception during reading.

The canonical syntax for NaNs shall be 1.xxxNaN or -1.xxxNaN. The value before the NaN suffix is obtained by changing the NaN's exponent to 0. While this printed representation does not allow for easy human interpretation of the NaN's bit pattern, it requires the minimum in terms of syntax extension. Processors that support NaNs are free to provide an operation to decode NaNs programmatically, and to support other (non-canonical) output formats. Again, the NaN suffix shall have exactly this capitalisation, to avoid syntax conflict with a possible postfix operator. Processors that do not implement a NaN representation shall raise a representation_error(float_undefined) exception during reading.

Suggestion: modify 6.4.5 to allow the above forms, and add the possible exceptions to the predicates of the read-family.

SICStus and Yap use +inf as syntax for infinities, which conflicts with the parsing of +(inf) in prefix form. Some degree of compatibility can be achieved by making inf/0 a function returning infinity.

Canonical Output

Section 7.10.5(c) specifies that if the write-option quoted(true) is in effect, then a float must be output in such a way that read_term/3 will read it back as exactly the same value (this will normally mean having to print one or even two digits beyond the last significant one). The same must be true for the additional values negative zero, infinity and NaN. Not all current systems get that right currently.

Pitfall: a printed representation that reads back correctly on one system may not do so on another, because of different rounding behaviour of the read routines. The standard can probably interpreted to mean that only the same Prolog processor has to read the value back accurately.

Reading Floats

Standard behaviour on reading overly large integers is defined in as raising the exception representation_error(max_integer). Behaviour on reading overly large floats is not defined, but should be added here: A representation_error(max_float) shall be raised when the float_overflow flag has the value 'error', and an attempt is made to read a number greater than specified in max_float (but smaller than infinity).

Suggestion: add additional error condition to


Off-Topic: Integer operations

Bad decisions in the standard that should ideally be resolved:


  1. ISO 31211-1 is the Prolog standard from 1995
  2. [LIA] ISO 10967-1 is the Language Independent Arithmetic standard from 1994. Part 1, Part 2. See also 2007 draft
  3. IEEE 754 is the standard on floating point numbers, originally from 1985, updated in 2008. IEEE final, Earlier draft.
  4. Discussion on comparison variants
  5. [KAH] W.Kahan, Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic
  6. [OKE] library(environment) by Richard A. O'Keefe.
  7. [OKPL] Richard A. O'Keefe: Am elementary Prolog Library
  8. Discussion on floats in Prolog comp.lang.prolog also summarised here