In a previous post, I discussed representations used for polymorphic parameters. These special data representations are a contract between the language and the runtime which enables the runtime to traverse data structures to collect garbage.

I had neglected to mention this interesting variant of tagged representations that I had come across at one point, but which I believe is not in widespread use. This representation uses full 64-bit floating point representations for all polymorphic values, and encodes non-floating point data in the unused bits of NaN values. IEEE 754 floating point numbers reserve 12 of the 64 bits for flagging NaN values, so the remaining bits are free to use for encoding integers, pointers, and so on.

I haven't found very much empirical data on this approach, but it looks promising. The polymorphic representation is doubled in size, but the structure is unboxed. Thus, the polymorphic function call overhead may be slightly higher, but garbage collector pressure is reduced since less boxing is needed. Numerical code is likely to see the biggest speedups.

Also, this representation can be used to ensure that all values are properly 8-byte aligned, which is often an efficiency gain in modern memory systems. This also has the advantage of permitting unboxed floating point values, which is a significant slowdown for the other tagged representations.

A new representation I recently came across is being used in the SML# language. Consider a typical tagged representation where 1-bit is co-opted to distinguish integers from pointers. Each polymorphic parameter must carry this 1 bit.

But there is no reason for this 1 bit to be contained within the parameter itself. Consider lifting this 1 bit to an extra function parameter, so:

becomes:

The Bool is the flag indicating whether the parameter is a pointer or an integer. But using a full function parameter to hold a single bit is a waste, so let's coalesce all the Bool parameters into a single Bitmask:

Bitmask can be an arbitrarily long sequence of bits, and each bit in the mask corresponds to the bit that would normally be in the pointer/integer representation of the corresponding parameter in a typical tagged representation. By lifting the tag bit to an external parameter, we can now use full natural representations for all parameters.

Calling a polymorphic function now consists of masking and shifting to extract the bit for the given parameter, or assigning one if calling from a monomorphic function.

Bitmasks must also accompany polymorphic data structures. The presence of any polymorphic type variable implies the requirement for a bitmask. It's also not clear how expensive this shifting is in relation to inline tag bits. SML# does not yet have a native code compiler, so any comparisons to other SML compilers aren't representative.

However, the bitmask representation does not unbox floating point numbers, but a hybrid scheme with the above 64-bit representation is possible.

Consider the lifted 1-bit to indicate whether the parameter is word-sized (1), or larger (0). Larger parameters require a full 64-bit representation, while word-sized parameters do not. The garbage collector thus skips any parameters of bitmask tag value 1 (since they are unboxed integers), and analyzes any values of bitmask tag value 0 as a 64-bit value. If the 64-bit is a NaN, the value is further analyzed to extract a single distinguished bit indicating whether it is a pointer. If it is not a pointer, it is skipped (since it is an unboxed floating point value). If it is a pointer, the structure pointed to is traversed.

This last scheme provides natural representations for integers and floating point numbers, and expands the representation of pointers in polymorphic functions by one word. Other tradeoffs are certainly possible.

#### 64-Bit Polymorphic Representation

I had neglected to mention this interesting variant of tagged representations that I had come across at one point, but which I believe is not in widespread use. This representation uses full 64-bit floating point representations for all polymorphic values, and encodes non-floating point data in the unused bits of NaN values. IEEE 754 floating point numbers reserve 12 of the 64 bits for flagging NaN values, so the remaining bits are free to use for encoding integers, pointers, and so on.

I haven't found very much empirical data on this approach, but it looks promising. The polymorphic representation is doubled in size, but the structure is unboxed. Thus, the polymorphic function call overhead may be slightly higher, but garbage collector pressure is reduced since less boxing is needed. Numerical code is likely to see the biggest speedups.

Also, this representation can be used to ensure that all values are properly 8-byte aligned, which is often an efficiency gain in modern memory systems. This also has the advantage of permitting unboxed floating point values, which is a significant slowdown for the other tagged representations.

#### Natural Representations with Bitmasks

A new representation I recently came across is being used in the SML# language. Consider a typical tagged representation where 1-bit is co-opted to distinguish integers from pointers. Each polymorphic parameter must carry this 1 bit.

But there is no reason for this 1 bit to be contained within the parameter itself. Consider lifting this 1 bit to an extra function parameter, so:

val f: 'a → 'b → 'c

becomes:

val f: 'a * Bool → 'b * Bool → 'c * Bool

The Bool is the flag indicating whether the parameter is a pointer or an integer. But using a full function parameter to hold a single bit is a waste, so let's coalesce all the Bool parameters into a single Bitmask:

val f: Bitmask → 'a → 'b → 'c

Bitmask can be an arbitrarily long sequence of bits, and each bit in the mask corresponds to the bit that would normally be in the pointer/integer representation of the corresponding parameter in a typical tagged representation. By lifting the tag bit to an external parameter, we can now use full natural representations for all parameters.

Calling a polymorphic function now consists of masking and shifting to extract the bit for the given parameter, or assigning one if calling from a monomorphic function.

Bitmasks must also accompany polymorphic data structures. The presence of any polymorphic type variable implies the requirement for a bitmask. It's also not clear how expensive this shifting is in relation to inline tag bits. SML# does not yet have a native code compiler, so any comparisons to other SML compilers aren't representative.

However, the bitmask representation does not unbox floating point numbers, but a hybrid scheme with the above 64-bit representation is possible.

Consider the lifted 1-bit to indicate whether the parameter is word-sized (1), or larger (0). Larger parameters require a full 64-bit representation, while word-sized parameters do not. The garbage collector thus skips any parameters of bitmask tag value 1 (since they are unboxed integers), and analyzes any values of bitmask tag value 0 as a 64-bit value. If the 64-bit is a NaN, the value is further analyzed to extract a single distinguished bit indicating whether it is a pointer. If it is not a pointer, it is skipped (since it is an unboxed floating point value). If it is a pointer, the structure pointed to is traversed.

This last scheme provides natural representations for integers and floating point numbers, and expands the representation of pointers in polymorphic functions by one word. Other tradeoffs are certainly possible.

## Comments

I use a whole program analysis which computes all possible polymorphic instantiationsOf course, these representations are only strictly necessary in the presence of separate compilation where the sizes cannot be known in advance.

E.g. a polymorphic function that allocates an array may need to know the size of the elements.I was considering the array case over lunch. Two preliminary solutions:

1. All polymorphic types, including arrays, must use the uniform polymorphic representation (64-bits in the first case, 32-bits in the second); anything larger is boxed.

2. All packed types, like arrays, tuples, vectors, must be handled specially by the language (so it can insert and manage these special headers for them); in this case, it's probably best to try and unify all such packed types.

Of course, there are other options as well, like record polymorphism as used in SML#, but I think I covered those in my last post (though you're right that it has bearing on the second encoding of this post).

There are other designs as well, such as using 2 bits to provide more precision, such as 00=int, 01=pointer, 10=structure of ints, 11=structure containing pointers.