# Arithmetic Operations

A project log for Novasaur Retrocomputer

Retrocomputer with serial and video built from only 1978-era TTL logic

The hardware is complete and now collects dust as the firmware development grinds on. The virtual CPU is taking shape and lots of interesting problems are getting solved. One of the major hurdles is building out a functional ALU. The current hardware really doesn't have one, but can utilize lookup tables in the ROM to perform arithmetic operations.

One example is addition. The ROM can be used to add two 8-bits value to produce an 8-bit sum. The easiest way to do this is with a 256x256 table to store the 65,536 possible values (64k bytes). This is a lot of memory though, so the problem is broken down in to two steps using two 256x16 tables (4k bytes each). The first table takes a full 8-bit value and adds the lower nibble. The second table takes the result of the first step and adds the upper nibble.

```example: 19 + 28
19
8 +
---
27
2  +
---
47```

Both methods have a major flaw though. The ROM produces an 8-bit result, but adding two 8-bit values results in a 9-bit value; there is no carry from the ROM lookup table. The carry is an important component of any ALU, not only a carry out, but also a carry in to allow the chaining of arithmetic operations. Other flags are also useful, such as an overflow to indicate a carry from a signed addition, a zero flag to indicate if the result was zero, etc. The only flag the ROM table can produce is the negative flag, since it is the most-significant bit of the result. This is therefore the only flag available for performing conditional logic.

The negative flag can be used to determine the carry by looking at the before and after state of the addition. This takes a lot of cycles and complicates things. The solution is to add another pair of tables to the ROM to create the Arithmetic Flag (AF) operation. This is a little more complex than it sounds due to the way the nibble cycles work. The first cycle returns the flags of adding the two lower nibbles: These include the carry and borrow from adding/subtracting the first 4 bits and a zero and parity flag for the first 4 bits of the result. The second table completes the flag operation using the initial 4 flags and processing the upper nibbles.

To find the flags, the AF operation proceeds an ADD operation to generate the following:

```Flag bits - CNZPHOBL
C - Carry (from bit 7)
N - Negative (sign of result)
Z - Zero (high if result is 0)
P - Parity (parity of result)
H - Half carry (carry from bit 3)
O - Overflow (carry from bit 6)
B - Borrow (from bit 7 if subtraction)
L - Low borrow (from bit 3 if subtraction)
```

The original SUB operation was removed to make room for AF leaving ADD as the only byte-wide arithmetic operation in the ROM. Subtractions can still be done by using the 2COM function (in the default set of unary operations) to negate the value being added. An additional pair of flags indicate if there was a borrow from the equivalent subtraction.

The final problem is the carry in. There's no easy solution to this, so the entire ADD/AF sequence is needed to first conditionally add/subtract 1 to the accumulator if the carry/borrow bit is set. Then the normal arithmetic operation is performed and the flags from both operations combined. This makes add with carry, and subtract with borrow, some of the most expensive operations on this type of restricted hardware.

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