This is a partial recreation of the original Zuse Z3 ALU. In Germany the Zuse Z3 is generally regarded as the first computer, however, unlike the ENIAC it is not turing-complete. Even though it may not be the first turing-complete computer, to the best of my knowledge, it and its predecessor the Zuse Z1 contains the first implementation of a floating point unit. It works purely on floating point numbers and only understands a few commands: loading/storing, reading in data and performing addition/subtraction, multiplication/division and lastly computing the square root. It only employs two registers and 64 memory locations.
This is not a faithful recreation, because I did not want to convert the relay-based logic 1:1 to verilog. The memory is missing as well. However, it retains the original floating point format as well as algorithms.
This project uses the Zuse Z3 floating point format, but without using hidden digits. All floats must be normalized, meaning the mantissa must be within 1.0 to 1.99999. The mantissa is 15 bits long, the msb must always be 1 to comply with the previously mentioned normalization (normally, this digit is hidden and used implicitly, but not in this design).
A number is represented via: +/- x * 2 ^ e. The sign is represented by a single bit (1 = positive, 0 = negative). X is the mantissa. E is the exponent: A signed 7 bit number!
In order to convert a decimal number, for example, 42.24 to the Z3 format perform the following steps:
0.24 * 2 = 0.48 (0)
0.48 * 2 = 0.96 (0)
0.96 * 2 = 1.92 (1)
0.92 * 2 = 1.84 (1)
0.84 * 2 = 1.64 (1)
0.64 * 2 = 1.28 (1)
0.28 * 2 = 0.56 (0)
0.56 * 2 = 1.12 (1)
(this would continue, but we have enough digits to fill our mantissa)
Our number is thus: 101010.00111101
Take the number above and remove the dot. Now you got your mantissa
Thus in general the Z3 number will be 1 (sign) | 0000101 (exponent) | 10101000111101 (mantissa)
You can verify this by computing:
2^5 *( 2^0 + 2^-2 + 2^-4 + 2^-8 + 2^-9 + 2^-10 + 2^-11 + 2^-13) = 42.23828125
This is not exactly 42.24, which is to be expected, because some decimal numbers are not representible in binary, thus inducing a rounding error.
Here are some example bit strings in Python format, which you can send to the FPU:
b'\x85\xab\x00' = 42.75
b'\x82\xe0\x00' = 7.0
The number 0 is represented by any value, which has the exponent -64.
Infinity is represented by any value, which has the exponent 63.
The design was created for a 10 MHz clock and uses a 9600 baud UART connection for communication. It lets you load values into the FPU and perform addition or subtraction.
The commands require a single byte and are defined as follows:
0x82: Set the R1 register
0x83: Set the R2 register
0x84: Set the status register (e.g., overflow, underflow)
0x85: Read the R1 register
0x86: Read the R2 register
0x87: Read the result register
0x88: Perform R1 + R2
0x89: Perform R1 - R2
0x8A: Perform R1 * R2
0x8B: Perform R1 / R2
0x8C: Perform sqrt(R1)
After sending the command to write a register you need to send 3 additional bytes where the first byte contains the sign bit (7) and the exponent (6:0), the following byte defines the mantissa bits 15 to 7. Remember that bit 15 must be 1. The last byte defines the lower mantissa bits. Notice how we transmit 16 bits but only use 15 bits of information. The lowest bit of the last byte is thus ignored. You do not get an ack from the board! Simply read the register back if you are unsure if the transmission worked.
If you send a read command, you will receive 3 bytes in the exact same format as above. First the sign and exponent in the first byte followed by the mantissa bytes.
If you send a command to compute a result, you will receive no answer. You will have to manually read the result register. Do not worry, the FSM should wait until the FPU is done, so no reading of undefined data will happen!
Using the example values from above, here is a complete command sequence:
b'\x82\x85\xab\x00' sends the number 42.75
b'\x83\x82\xe0\x00' send the number 7.0
b'\x88' sends the ADD command
b'\x87' reads the result register
The result should be b'\x85\xc7\x00'
The status register can signify the following events:
The result was zero bit position 0
The computation overflowed bit position 1
The computation underflowed bit position 2
Use the on board RPi 2400 for uart connections. It uses the default uart ports suggested by tiny tapeout.
| # | Input | Output | Bidirectional |
|---|---|---|---|
| 0 | |||
| 1 | |||
| 2 | |||
| 3 | rx | ||
| 4 | tx | ||
| 5 | |||
| 6 | |||
| 7 |