For the Rex.bin file the two tasks of translating from Rex.bin's modified S-record format to actual (physical) Rex memory image and back again (which includes calculating the checksum) are relatively simple programming jobs.

Q: How do the Rex records differ from standard S-Records?
It appears to be a bastardized Motorola S-Record. It follows the format but other than the Sx line headers being in Ascii, the data is in raw binary. (The standard is all data is in ascii hex and each Sx line terminated with CRLF).
The 8-bit checksum is the one's complement sum of the bytes after the Sx header up to the checksum ( = ~(unsigned sum of data) & 0xFF).

The overall format used rex.bin is:

SO, length(byte), addr(2 bytes) = 0000, text data usually title, version etc., chksum
16-bit address:
S1, length(byte), addr(2 bytes), data (length-3 bytes), chksum (byte)
S9, length(byte) = 3, starting execution address in 2 bytes, chksum (byte)
24-bit address:
S2, length(byte), addr(3 bytes), data (length-4 bytes), chksum (byte)
S8, length(byte) = 4, starting execution address in 3 bytes, chksum (byte)
32-bit address:
S3, length(byte), addr(4 bytes), data (length-5 bytes), chksum (byte)
S7, length(byte) = 5, starting execution address in 4 bytes, chksum (byte)
Count of total records of type S1,S2,S3 already sent
S5, length(byte) = 3 , count (2 bytes)"

Q: How are the checksums calculated?
If you want to modify, for example, Rex.bin, to amend the firmware and then upload this you must get the checksums correct or it will be rejected.
If you are doing checksums on less then 128 bytes using C, you don't have to worry about any sign conventions as the sum of 128 bytes will always be 0x7FF or less. So to calculate the checksum:
Add up all ascii hexbytes following the "Sx" part of the S-record line. Bitwise complement it (zero bits become one bits and vice-versa) and mask off the low byte (& 0xFF) (or mask off the low byte and then complement it).

If your checksum does exceed 0x7FF (the max positive number in 16bit integers) then in C (which uses two's complement sign conventions) you need to be careful as numeric intepretation of the sum can screw up the resulting eight-bit checksum result.

Q: What is the standard S-Record format?
An S-record file consists of a sequence of specially formatted ASCII character strings.

The order of S-records within a file is of no significance and no particular order may be assumed.

The general format of an S-record follows:

+-------------------//------------------//-----------------------+
| type | count | address  |            data           | checksum |
+-------------------//------------------//-----------------------+

type -- A char[2] field. These characters describe the type of record (S0, S1, S2, S3, S5, S7, S8, or S9).

count -- A char[2] field. These characters when paired and interpreted as a hexadecimal value, display the count of remaining character pairs in the record. The count field can have values in the range of 0x3 (2 bytes of address + 1 byte checksum = 3, a not very useful record) to 0xff; this is the count of remaining character pairs, including checksum.

address -- A char[4,6, or 8] field. These characters grouped and interpreted as a hexadecimal value, display the address at which the data field is to be loaded into memory. The length of the field depends on the number of bytes necessary to hold the address. A 2-byte address uses 4 characters, a 3-byte address uses 6 characters, and a 4-byte address uses 8 characters.

data -- A char [0-64] field. These characters when paired and interpreted as hexadecimal values represent the memory loadable data or descriptive information.

checksum -- A char[2] field. These characters when paired and interpreted as a hexadecimal value display the least significant byte of the ones complement (see below) of the sum of the byte values represented by the pairs of characters making up the count, the address, and the data fields.

Each record is terminated with a line feed. If any additional or different record terminator(s) or delay characters are needed during transmission to the target system it is the responsibility of the transmitting program to provide them.

S0 Record. The type of record is 'S0' (0x5330). The address field is unused and will be filled with zeros (0x0000). The header information within the data field is divided into the following subfields.

mname is char[20] and is the module name.
ver is char[2] and is the version number.
rev is char[2] and is the revision number.
description is char[0-36] and is a text comment.

Each of the subfields is composed of ASCII bytes whose associated characters, when paired, represent one byte hexadecimal values in the case of the version and revision numbers, or represent the hexadecimal values of the ASCII characters comprising the module name and description.

S1 Record. The type of record field is 'S1' (0x5331). The address field is intrepreted as a 2-byte address. The data field is composed of memory loadable data.

S2 Record. The type of record field is 'S2' (0x5332). The address field is intrepreted as a 3-byte address. The data field is composed of memory loadable data.

S3 Record. The type of record field is 'S3' (0x5333). The address field is intrepreted as a 4-byte address. The data field is composed of memory loadable data.

S5 Record. The type of record field is 'S5' (0x5335). The address field is intrepreted as a 2-byte value and contains the count of S1, S2, and S3 records previously transmitted. There is no data field.

S7 Record. The type of record field is 'S7' (0x5337). The address field contains the starting execution address and is intrepreted as 4-byte address. There is no data field.

S8 Record. The type of record field is 'S8' (0x5338). The address field contains the starting execution address and is intrepreted as 3-byte address. There is no data field.

S9 Record. The type of record field is 'S9' (0x5339). The address field contains the starting execution address and is intrepreted as 2-byte address. There is no data field.

EXAMPLE

Shown below is a typical S-record format file.

S00600004844521B
S1130000285F245F2212226A000424290008237C2A
S11300100002000800082629001853812341001813
S113002041E900084E42234300182342000824A952
S107003000144ED492
S5030004F8
S9030000FC

The file consists of one S0 record, four S1 records, one S5 record and an S9 record.

The S0 record is comprised as follows:

• S0 S-record type S0, indicating it is a header record.
• 06 Hexadecimal 06 (decimal 6), indicating that six character pairs (or ASCII bytes) follow.
• 00 00 Four character 2-byte address field, zeroes in this example.
  
• 48 44 52 ASCII H, D, and R - "HDR".
• 1B The checksum.

The first S1 record is comprised as follows:

• S1 S-record type S1, indicating it is a data record to be loaded at a 2-byte address.
• 13 Hexadecimal 13 (decimal 19), indicating that nineteen character pairs, representing a 2 byte address, 16 bytes of binary data, and a 1 byte checksum, follow.
• 00 00 Four character 2-byte address field; hexidecimal address 0x0000, where the data which follows is to be loaded.
• 28 5F 24 5F 22 12 22 6A 00 04 24 29 00 08 23 7C Sixteen character pairs representing the actual binary data.
• 2A The checksum.

The second and third S1 records each contain 0x13 (19) character pairs and are ended with checksums of 13 and 52, respectively. The fourth S1 record contains 07 character pairs and has a checksum of 92.

The S5 record is comprised as follows:

• S5 S-record type S5, indicating it is a count record indicating the number of S1 records
• 03 Hexadecimal 03 (decimal 3), indicating that three character pairs follow.
• 00 04 Hexadecimal 0004 (decimal 4), indicating that there are four data records previous to this record.
• F8 The checksum.

The S9 record is comprised as follows:

• S9 S-record type S9, indicating it is a termination record.
• 03 Hexadecimal 03 (decimal 3), indicating that three character pairs follow.
• 00 00 The address field, hexadecimal 0 (decimal 0) indicating the starting execution address.
• FC The checksum.

Q: What is "Ones complement"?
A system used in some computers to represent negative numbers. To negate a number, each bit of the number is inverted (zeros are replaced with ones and vice versa). This has the consequence that there are two reperesentations for zero, either all zeros or all ones.

...
000...00011 = +3
000...00010 = +2
000...00001 = +1
000...00000 = +0
111...11111 = -0
111...11110 = -1
111...11101 = -2
111...11100 = -3
...

Naive logic for ones complement addition might easily conclude that -0 + 1 = +0. The twos complement avoids this by using all ones to represent -1.