The decimal data sub-type in XL-VBA

In VBA, the most straightforward way to add some extra precision to calculations with doubles is to try to harness the sub-type decimal of the data type variant. According to the VBA language specification provided by MS quite recently, a variable of sub-type decimal comprises a numerator of 96 bits (or 12 octets) of precision that encode an unsigned integer in binary format, and a denominator comprising 8 bits (or 1 octet) that encode the power of ten by which the numerator has to be divided in order to yield the mathematical value that is represented by the variable. MS further state that there also is a sign bit, and that a decimal is "A rational number represented in a 14 byte data structure". I started to doubt this statement when reading this information on the decimal sub-type (in German). There, a decimal is said to be a data structure of 16 bytes, and moreover, a rather peculiar order of the 12 octets of the numerator is
mentioned. Consequently, I started to check out the details by myself.

For that, in my VBA7.1 I declared a call to the kernel function RtlMoveMemory that can be accessed via the system library kernel32.dll as follows:
Private Declare PtrSafe Sub MoveMem Lib "kernel32" Alias "RtlMoveMemory" (PointerToMemory As Any, PointerToBuffer As Any, ByVal AmountOfOctets As Long)

In order to call this function properly from within a VBA UDF, an array of Bytes of desired length has to be declared. Finally, it will contain an exact copy of the block of computer memory that starts at address PointerToMemory and comprises AmountOfOctets octets. For that, the VBA UDF called HexDump, which is of return type String, is defined like this:
Function HexDump$(ByVal PointerToBufferStart As LongLong, ByVal myOctets As Long)
In HexDump, the amount of octets to be returned is rounded up to the next equal or larger multiple of 16 and
assigned to the variable NumOctets. Then an array of type Byte and of length NumOctets is declared:
ReDim BytArr(1 To NumOctets) As Byte .
When calling MoveMem, the pointer to the start of BytArr in memory is obtained implicitly by simply submitting the reference to its first element:
Call MoveMem(BytArr(1), ByVal PointerToBufferStart, ByVal NumOctets).
The rest of MoveMem is only concerned with creating a proper string from all the elements of BytArr in a form that is easily perceivable by humans.

I then created some decimals according to the following scheme:
Function some_decimals$()

Dim …, d18, d19, d20, d21, d22
d18 = CDec("1234567890123456789012345")
d19 = CDec("123456789012345678901234567")
d20 = CDec("12345678901234567890123456789")
d21 = CDec("1.2345678901234567890123456789")
d22 = CDec("-1.2345678901234567890123456789")
some_decimals = HexDump(VarPtr(d22), 528&)

Quite surprisingly for me at least, the position of the variables in memory is found to be in reversed order of declaration, which means the pointer to the variable declared least has to be submitted to HexDump in order to get back a block of memory that contains all the variables declared. The relevant part of the string returned by HexDump, which encodes the values of all octets in hexadecimal notation, reads as follows:

0E 00 1C 80 32 1B E4 27 15 81 39 6E B1 C9 BE 46
00 00 00 00 00 00 00 00 0E 00 1C 00 32 1B E4 27
15 81 39 6E B1 C9 BE 46 00 00 00 00 00 00 00 00
0E 00 00 00 32 1B E4 27 15 81 39 6E B1 C9 BE 46
00 00 00 00 00 00 00 00 0E 00 00 00 FD 1E 66 00
87 4B 9F 2C A8 F2 58 F1 00 00 00 00 00 00 00 00
0E 00 00 00 6E 05 01 00 79 DF E2 3D 44 A6 36 0F

As I shall show in detail, the first 16 octets indeed encode the value of d22 as a decimal correctly. Yet, the effective amount of memory claimed by each of consecutively declared decimals is neither 14 nor 16 octets, but equals 24 octets, at least on my Win7x64 XL15x64 PC. Consequently, in order to catch the entire block of 22 decimals, an amount of at least 21*24+16 = 520 octets is needed, which is then internally rounded up to 33*16 = 528 octets.

In the German resource mentioned above, it is claimed that the 16 octets of a decimal start with an identifier octet having a value of 14 dec = 0E hex, and that the following octet remains unused. The 3rd octet is said to encode the power of 10 of the denominator (here: 28 dec = 1C hex or 0 dec = 0 hex), and the 4th octet is claimed to encode a positive sign by a value of 0, and a negative one by 128 dec = 80 hex. As can easily be seen from the list given above, all of these statements hold true. Therefore, the trailing 12 octets are needed to encode the 96 bits of the numerator. In order to understand the values returned by HexDump, I harnessed xNumbers for converting “12345678901234567890123456789” into a binary string. This string is found to comprise 94 bits, so that it is presented here on three lines each of which gives 4 octets, starting with the most significant one. We have:

OOIO-OIII IIIO-OIOO OOOI-IOII OOII-OOIO
2-7 E-4 1-B 3-2
OIOO-OIIO IOII-IIIO IIOO-IOOI IOII-OOOI
4-6 B-E C-9 B-1
OIIO-IIIO OOII-IOOI IOOO-OOOI OOOI-OIOI
6-E 3-9 8-1 1-5

so that the 94 bits in big endian byte order and in hexadecimal notation read:
27 E4 1B 32 46 BE C9 B1 6E 39 81 15
which in little endian byte order reads:
15 81 39 6E B1 C9 BE 46 32 1B E4 27 .
When this sequence is compared to that returned by HexDump, the peculiar byte order by which variable values of sub-type decimal store the numerator in computer memory becomes obvious:
32 1B E4 27 15 81 39 6E B1 C9 BE 46
The first block in memory comprises the 4 most significant consecutive octets (green), and the second one the 8 least significant (blue) of them, which is just the opposite of what is said in the mentioned German resource.

When comparing the 16 net octets of the values assigned to d20, d21, and d22, we have:

d22 = CDec("-1.2345678901234567890123456789")
d22 = 0E 00 1C 80 32 1B E4 27 15 81 39 6E B1 C9 BE 46
d21 = CDec("+1.2345678901234567890123456789")
d21 = 0E 00 1C 00 32 1B E4 27 15 81 39 6E B1 C9 BE 46
d20 = CDec( "+12345678901234567890123456789")
d20 = 0E 00 00 00 32 1B E4 27 15 81 39 6E B1 C9 BE 46

This means the power of 10 of the denominator, which equals 28 dec = 1C hex for both d22 and d21, and 0 dec/hex for d20, and that is encoded in the 3rd octet, can also be interpreted as the position of where to insert the decimal separator when counting positions of the unsigned numerator from right to left. Finally, there is a sign bit, but effectively, a decimal uses a sign octet.

Summarising, variables of sub-type decimal encode their values very in-efficiently. On the other hand, they allow for exact calculations with decimal fractions, which makes them the optimal data type in the area of finance. Due to the fact that only the basic algebraic operators support decimals, any function that is to be evaluated needs to be implemented from scratch using only the four basic operators +,-,·, and /, or calling other functions supporting decimals that have already been implemented. Not every double can be converted directly into a decimal due to either the limited range of the latter of at maximum ±(2^96-1) = ±7.9228162514264337593543950335E28 or their minimum resolution of 1E-28 or a combination of both effects. Moreover, only doubles having no more than 29 significant decimal figures can be exactly casted into decimals. When scaling doubles in order to make them fit the limits of decimals, only factors equalling integral powers of 2 should be used in order to leave the floating-point mantissa unchanged. This will be demonstrated in a forthcoming post.

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