14 Tiny Zuse

14 : Tiny Zuse

Design render
  • Author: Florian Stolz
  • Description: Minimal Implementation of a Zuse Z3-style FPU for Addition/Subtraction
  • GitHub repository
  • Clock: 10000000 Hz

Quick Overview

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.

Number Representation

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:

  1. The number is positive, so the sign bit is 1. s = 1
  2. Convert the integer part to binary. 42 = 101010
  3. The highest bit is in position 5 (counting from 0). Thus, e = 5
  4. Now convert the fractional part to binary:

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

  1. Take the number above and remove the dot. Now you got your mantissa

  2. Thus in general the Z3 number will be 1 (sign) | 0000101 (exponent) | 10101000111101 (mantissa)

  3. 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.

How to test

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

External hardware

Use the on board RPi 2400 for uart connections. It uses the default uart ports suggested by tiny tapeout.

IO

#InputOutputBidirectional
0
1
2
3rx
4tx
5
6
7

Chip location

Controller Mux Mux Mux Mux Mux Mux Mux Mux Mux Mux Mux Analog Mux Mux Mux Mux Mux Mux Mux Mux Mux Mux Mux Analog Mux Mux Mux Mux Mux Mux Mux Mux Mux Mux tt_um_chip_rom (Chip ROM) tt_um_factory_test (TinyTapeout 06 Factory Test) tt_um_analog_factory_test (TT06 Analog Factory Test) tt_um_analog_factory_test (TT06 Analog Factory Test) tt_um_urish_charge_pump (Dickson Charge Pump) tt_um_psychogenic_wowa (WoWA) tt_um_oscillating_bones (Oscillating Bones) tt_um_kevinwguan (Crossbar Array) tt_um_coloquinte_moosic (Moosic logic-locked design) tt_um_alexsegura_pong (Pong) tt_um_iron_violet_simon (Iron Violet) tt_um_tomkeddie_a (VGA Experiments in Tennis) tt_um_MichaelBell_tinyQV (TinyQV Risc-V SoC) tt_um_andychip1_sn74169 (sn74169) tt_um_mattvenn_r2r_dac (Analog 8bit R2R DAC) tt_um_thorkn_audiochip_v2 (AudioChip_V2) tt_um_faramire_gate_guesser (Gate Guesser) tt_um_urish_simon (Simon Says game) tt_um_TT06_SAR_wulffern (TT06 8-bit SAR ADC) tt_um_soundgen (soundgen) tt_um_ledcontroller_Gatsch (ledcontroller) tt_um_digitaler_filter_rathmayr (Digitaler Filter) tt_um_histefan_top (Snake Game) tt_um_mayrmichael_wave_generator (Wave Generator) tt_um_advanced_counter (jku-tt06-advanced-counter) tt_um_FanCTRL_DomnikBrandstetter (PI-Based Fan Controller) tt_um_ps2_morse_encoder_top (PS/2 Keyboard to Morse Code Encoder) tt_um_calculator_muehlbb (16-bit calculator) tt_um_hpretl_tt06_tempsens (Temperature Sensor NG) tt_um_haeuslermarkus_fir_filter (FIR Filter with adaptable coefficients) tt_um_mattvenn_rgb_mixer (RGB Mixer demo) tt_um_analog_loopback (Analog loopback) tt_um_entwurf_integrierter_schaltungen_hadner (Projekt KEIS Hadner Thomas) tt_um_seven_segment_fun1 (7-segment-FUN) tt_um_moving_average_master (Moving average filter) tt_um_rgbled_decoder (SPI to RGBLED Decoder/Driver) tt_um_4bit_cpu_with_fsm (4-Bit CPU mit FSM) tt_um_flappy_bird (Flappy Bird) tt_um_drops (drops) tt_um_enieman (UART-Programmable RISC-V 32I Core) tt_um_gabejessil_timer (2 Player Game) tt_um_wokwi_384804985843168257 (playwithnumbers) tt_um_wokwi_384711264596377601 (luckyCube) tt_um_hpretl_tt06_tdc (Synthesized Time-to-Digital Converter (TDC)) tt_um_wokwi_384437973887503361 (Asynchronous Down Counter) tt_um_spi_pwm_djuara (spi_pwm) tt_um_SteffenReith_PiMACTop (PiMAC) tt_um_mattvenn_relax_osc (Relaxation oscillator) tt_um_jv_sigdel (1st passive Sigma Delta ADC) tt_um_wokwi_392873974467527681 (PILIPINAS) tt_um_scorbetta_goa (GOA - grogu on ASIC) tt_um_sanojn_ttrpg_dice (TTRPG Dice + simple I2C peripheral) tt_um_urish_dffram (DFFRAM Example (128 bytes)) tt_um_lucaz97_monobit (Monobit Test) tt_um_noritsuna_i4004 (i4004 for TinyTapeout) tt_um_hpretl_tt06_tdc_v2 (Synthesized Time-to-Digital Converter (TDC) v2) tt_um_vaf_555_timer (A 555-Timer Clone for Tiny Tapeout 6) tt_um_obriensp_be8 (8-bit CPU with Debugger (Lite)) tt_um_toivoh_retro_console (Retro Console) tt_um_mattvenn_inverter (Double Inverter) tt_um_SteffenReith_ASGTop (ASG) tt_um_lucaz97_rng_tests (rng Test) tt_um_dieroller_nathangross1 (Die Roller) tt_um_kwilke_cdc_fifo (Clock Domain Crossing FIFO) tt_um_spiff42_exp_led_pwm (LED PWM controller) tt_um_devinatkin_fastreadout (Fast Readout Image Sensor Prototype) tt_um_ja1tye_tiny_cpu (Tiny 8-bit CPU) tt_um_7seg_animated (Animated 7-segment character display) tt_um_neurocore (Neurocore) tt_um_zhwa_rgb_mixer (RGB Mixer) tt_um_wokwi_394704587372210177 (Cambio de giro de motor CD) tt_um_ian_keypad_controller (Keypad controller) tt_um_urish_spell (SPELL) tt_um_vks_pll (PLL blocks) tt_um_fountaincoder_top (multimac) tt_um_dsatizabal_opamp (Simple FET OpAmp with Sky130.) tt_um_obriensp_be8_nomacro (8-bit CPU with Debugger) tt_um_LFSR_shivam (10-bit Linear feedback shift register) tt_um_shivam (Pulse Width Modulation) tt_um_algofoogle_tt06_grab_bag (TT06 Grab Bag) tt_um_meiniKi_tt06_fazyrv_exotiny (FazyRV-ExoTiny) tt_um_wokwi_394888799427677185 (4-bit stochastic multiplier traditional) tt_um_QIF_8bit (8 Bit Digital QIF) tt_um_MATTHIAS_M_PAL_TOP_WRAPPER (easy PAL) tt_um_andrewtron3000 (Rule 30 Engine!) tt_um_tommythorn_4b_cpu_v2 (Silly 4b CPU v2) tt_um_aerox2_jrb8_computer (The James Retro Byte 8 computer) tt_um_wokwi_394898807123828737 (4-bit Stochastic Multiplier Compact with Stochastic Resonator) tt_um_argunda_tiny_opamp (Tiny Opamp) tt_um_fdc_chip (Frequency to digital converters (asynchronous and synchronous)) tt_um_8bit_cpu (8-Bit CPU In a Week) tt_um_mitssdd (co processor for precision farming) tt_um_fstolzcode (Tiny Zuse) tt_um_liu3hao_rv32e_min_mcu (tt06-RV32E_MinMCU) tt_um_kianV_rv32ima_uLinux_SoC (KianV uLinux SoC) tt_um_wokwi_395444977868278785 (*NOT WORKING* HP 5082-7500 Decoder) tt_um_wokwi_394618582085551105 (Keypad Decoder) tt_um_wokwi_395054820631340033 (Workshop Hackaday Juli) tt_um_wokwi_395055035944909825 (Some_LEDs) tt_um_wokwi_395055351144787969 (Hack a day Tiny Tapeout project) tt_um_wokwi_395054823569451009 (First TT Project) tt_um_wokwi_395054823837887489 (Dice) tt_um_wokwi_395055341723330561 (Workshop_chip) tt_um_jduchniewicz_prng (8-bit PRNG) tt_um_wokwi_395054564978002945 (Bestagon LED matrix driver) tt_um_wokwi_395054466384583681 (1-Bit ALU 2) tt_um_wokwi_395058308283408385 (test for tiny tapeout hackaday) tt_um_s1pu11i_simple_nco (Simple NCO) tt_um_wokwi_395055359324730369 (Tiny_Tapeout_6_Frank) tt_um_disp1 (Display test 1) tt_um_pckys_game (PCKY´s Successive Approximation Game) tt_um_tiny_shader_mole99 (Tiny Shader) tt_um_wokwi_393815624518031361 (My Chip) tt_um_minibyte (Minibyte CPU) tt_um_emilian_rf_playground (IDAC8 based on divide current by 2) tt_um_triple_watchdog (Triple Watchdog) tt_um_wokwi_395142547244224513 (EFAB Demo 2) tt_um_chisel_hello_schoeberl (Chisel Hello World) tt_um_aiju_8080 (8080 CPU) tt_um_wokwi_395134712676183041 (Inverters) tt_um_nubcore_default_tape (DEFAULT) tt_um_wuehr1999_servotester (Servotester) tt_um_wokwi_395055722430895105 (Servo Signal Tester) tt_um_exai_izhikevich_neuron (Izhikevich Neuron) tt_um_lisa (LISA 8-Bit Microcontroller) tt_um_wokwi_394707429798790145 (32-Bit Galois Linear Feedback Shift Register) tt_um_CKPope_top (X/Y Controller) tt_um_MNSLab_BLDC (Universal Motor and Actuator Controller) tt_um_couchand_dual_deque (Dual Deque) tt_um_JamesTimothyMeech_inverter (Programmable Thing) tt_um_signed_unsigned_4x4_bit_multiplier (Signed Unsigned multiplyer) tt_um_lipsi_schoeberl (Lipsi: Probably the Smallest Processor in the World) tt_um_i_tree_batzolislefteris (Anomaly Detection using Isolation trees) tt_um_wokwi_394830069681034241 (Cyclic Redundancy Check 8 bit) tt_um_rng_3_lucaz97 (RNG3) tt_um_wokwi_395263962779770881 (Bivium-B Non-Linear Feedback Shift Register) tt_um_dvxf_dj8 (DJ8 8-bit CPU) tt_um_silicon_tinytapeout_lm07 (Digital Temperature Monitor) tt_um_htfab_flash_adc (Flash ADC) tt_um_chisel_pong (Chisel Pong) tt_um_wokwi_395414987024660481 (HELP for tinyTapeout) tt_um_jorgenkraghjakobsen_toi2s (SPDIF to I2S decoder) tt_um_cmerrill_pdm (Parallel / SPI modulation tester) tt_um_csit_luks (CSIT-Luks) tt_um_wokwi_395357890431011841 (Trivium Non-Linear Feedback Shift Register) tt_um_drburke3_top (SADdiff_v1) tt_um_cejmu_riscv (TinyRV1 CPU) tt_um_rejunity_current_cmp (Analog Current Comparator) tt_um_loco_choco (BF Processor) tt_um_qubitbytes_alive (It's Alive) tt_um_wokwi_395061443288867841 (BCD to single 7 segment display Converter) tt_um_SJ (SiliconJackets_Systolic_Array) tt_um_ejfogleman_smsdac (8-bit DEM R2R DAC) tt_um_wokwi_395055455727667201 (Hardware Trojan Part II) tt_um_ericsmi_weste_problem_4_11 (Measurement of CMOS VLSI Design Problem 4.11) tt_um_wokwi_395034561853515777 (2 bit Binary Calculator) tt_um_mw73_pmic (Power Management IC) tt_um_Counter_1_shivam (8-bit Binary Counter) tt_um_wokwi_395054508867644417 (SynchMux) tt_um_otp_encryptor (TT06 OTP Encryptor) tt_um_wokwi_395514572866576385 (Parity Generator) tt_um_ADPCM_COMPRESSOR (ADPCM Encoder Audio Compressor) tt_um_3515_sequenceDetector (Sequence detector using 7-segment) tt_um_faramire_stopwatch (Simple Stopwatch) tt_um_ks_pyamnihc (Karplus-Strong String Synthesis) tt_um_dlmiles_muldiv8 (MULDIV unit (8-bit signed/unsigned)) tt_um_dlmiles_muldiv8_sky130faha (MULDIV unit (8-bit signed/unsigned) with sky130 HA/FA cells) tt_um_tommythorn_ncl_lfsr (NCL LFSR) tt_um_lk_ans_top (ANS Encoder/Decoder) tt_um_MichaelBell_latch_mem (Latch RAM (64 bytes)) tt_um_wokwi_395179352683141121 (Combination Lock) tt_um_Uart_Transciver (UART Transceiver) tt_um_dgkaminski (4-Bit ALU) tt_um_DigitalClockTop (TDM Digital Clock) tt_um_wokwi_394640918790880257 (IFSC Keypad Locker) tt_um_wokwi_395355133883896833 (BIT COMPARATOR) tt_um_alu (SumLatchUART_System) tt_um_alfiero88_VCII (VCII) tt_um_ALU (3-bit ALU) tt_um_topTDC (Convertidor de Tiempo a Digital (TDC)) tt_um_UABCReloj (24 H Clock) tt_um_CDMA_Santiago (CDMA_2024) tt_um_dr_skyler_clock (Clock) tt_um_motor (motor a pasos) tt_um_mult_2b (mult_2b) tt_um_CodHex7seg (Decodificador binario a display 7 segmentos hexadecimal) tt_um_S2P (Serial to Parallel Register) tt_um_PWM (PWM) tt_um_ss_register (serie_serie_register) tt_um_stepper (Stepper) tt_um_g3f (Generador digital trifásico) tt_um_ALU_DECODERS (ALU with a Gray and Octal decoders) tt_um_ram (4 bit RAM) tt_um_sap_1 (SAP-1 Computer) tt_um_guitar_pedal (Integrated Distorion Pedal) tt_um_mbalestrini_usb_cdc_devices (Two ports USB CDC device) tt_um_adammaj (Tiny ALU) tt_um_wokwi_395567106413190145 (4-Bit Full Adder and Subtractor with Hardware Trojan) tt_um_gak25_8bit_cpu_ext (Most minimal extension of friend's 'CPU In a Week' in a day) tt_um_hsc_tdc (UCSC HW Systems Collective, TDC) tt_um_BoothMulti_hhrb98 (UACJ-MIE-Booth 4) tt_um_dlmiles_poc_fskmodem_hdlctrx (FSK Modem +HDLC +UART (PoC)) tt_um_simplez_rcoeurjoly (tt6-simplez) tt_um_nurirfansyah_alits01 (Analog Test Circuit ITS: VCO) tt_um_ppca (drEEm tEEm PPCA) tt_um_wokwi_395522292785089537 (Displays CIt) tt_um_fpu (Dgrid_FPU) tt_um_duk_lif (Leaky Integrate and fire neuron(LIF)) tt_um_bomba1 (Latin_bomba) tt_um_chatgpt_rsnn_paolaunisa (ChatGPT designed Recurrent Spiking Neural Network) tt_um_bit_ctrl (Bit Control) tt_um_array_multiplier_hhrb98 (Array Multiplier) tt_um_wallace_hhrb98 (UACJ-Wallace multiplier) tt_um_I2C_to_SPI (TinyTapeout SPI Master) tt_um_rng (Random number generator) tt_um_wokwi_395599496098067457 (EVEN AND ODD COUNTERS) tt_um_8bitALU (8bit ALU) tt_um_aleena (Analog Sigmoid) tt_um_rejunity_1_58bit (Ternary 1.58-bit x 8-bit matrix multiplier) tt_um_rejunity_fp4_mul_i8 (FP4 x 8-bit matrix multiplier) tt_um_PWM_Controller (PWM Controller) tt_um_couchand_cora16 (CORA-16) tt_um_frq_divider (clk frequency divider controled by rom) tt_um_wokwi_390913889347409921 (Notre Dame Dorms LED) tt_um_timer_counter_UGM (4-Digit Scanning Digital Timer Counter) tt_um_koconnor_kstep (kstep) tt_um_lancemitrex (DIP Switch to HEX 7-segment Display) tt_um_PWM_Sine_UART (PWM_Sinewave_UART) tt_um_nicklausthompson_twi_monitor (TWI Monitor) tt_um_wokwi_395615790979120129 (Cambio de giro de motor CD) tt_um_ancho (Circuito PWM con ciclo de trabajo configurable) tt_um_wokwi_395618714068432897 (32b Fibonacci Original) tt_um_voting_thingey (Voting thingey) tt_um_hsc_tdc_buf (UCSC HW Systems Collective, TDC - BUF2x1) tt_um_hsc_tdc_mux (UCSC HW Systems Collective, TDC - MUX2x1) tt_um_petersn_micro1 (14 Hour Simple Computer) tt_um_sanojn_tlv2556_interface (UART interface to ADC TLV2556 (VHDL Test)) tt_um_gray_sobel (Gray scale and Sobel filter) tt_um_wokwi_395614106833794049 (Universal gates) Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available Available