Code Like It's 198x

Divide a Word by a Byte on NES

A quotient without its remainder throws useful information away.

A quotient without its remainder throws useful information away. Keep both when converting counters, coordinates or scores.

NES shows the working diagnostic row 8E 00 06 01 E8 03 07 00.
A completed native caller displays 8E 00 06 01 E8 03 07 00. The public field order below identifies these bytes; the capture is a state diagnostic, not a recording of a finished game.

Use it in a project

Divide the unsigned low-byte-first 16-bit numerator by DIVISOR 1–255. Return a 16-bit quotient, a remainder below the divisor and VALID=1. Division by zero returns zero outputs and VALID=0. Inputs are preserved.

Include component.inc once (MOS callers include the shared 6502 implementation through it). Reserve the public fields in this exact order: QUOTIENT_LO, QUOTIENT_HI, REMAINDER, VALID, NUMERATOR_LO, NUMERATOR_HI, DIVISOR. They occupy 7 contiguous bytes. Also reserve the private workspace: T0, T1, T2 (3 bytes). Array names in the explanation refer to the numbered consecutive bindings, such as C0, C1 and so on.

Call word_division_init once at the intended reset boundary. It clears inputs as well as outputs and workspace. Set your starting state and configuration afterwards, then call word_division_step whenever the described event occurs. The helper reads no device, waits for no frame and installs no interrupt handler.

Ownership Fields
Retain between calls none
Supply before service NUMERATOR_LO, NUMERATOR_HI and DIVISOR
Read after service QUOTIENT_LO, QUOTIENT_HI, REMAINDER and VALID

Long division consumes sixteen numerator bits. The remainder temporarily needs a ninth bit when shifted; keeping that carry separately prevents a large divisor from being compared against a wrapped remainder.

Reusable initialisation and service

6502.inc
; SPDX-License-Identifier: MIT
; Divide a Word by a Byte. See README.md for the byte interface and ownership.
; init clears the public fields and private workspace; set inputs afterwards.
word_division_init:
    lda #0
    sta QUOTIENT_LO
    sta QUOTIENT_HI
    sta REMAINDER
    sta VALID
    sta NUMERATOR_LO
    sta NUMERATOR_HI
    sta DIVISOR
    sta T0
    sta T1
    sta T2
    rts
word_division_step:
; QUOTIENT_LO = 0
    lda #0
    sta QUOTIENT_LO
; QUOTIENT_HI = 0
    lda #0
    sta QUOTIENT_HI
; REMAINDER = 0
    lda #0
    sta REMAINDER
; VALID = 0
    lda #0
    sta VALID
; if DIVISOR == 0:
    lda DIVISOR
    cmp #0
    beq word_division_L4
word_division_L5:
    jmp word_division_L2
word_division_L4:
    jmp word_division_L1
word_division_L1:
; return
    rts
    jmp word_division_L3
word_division_L2:
word_division_L3:
; VALID = 1
    lda #1
    sta VALID
; QUOTIENT_LO = NUMERATOR_LO
    lda NUMERATOR_LO
    sta QUOTIENT_LO
; QUOTIENT_HI = NUMERATOR_HI
    lda NUMERATOR_HI
    sta QUOTIENT_HI
; T0 = 16
    lda #16
    sta T0
; while T0 != 0:
word_division_L6:
    lda T0
    cmp #0
    bne word_division_L9
word_division_L10:
    jmp word_division_L8
word_division_L9:
    jmp word_division_L7
word_division_L7:
; T1 = QUOTIENT_HI >> 7
    lda QUOTIENT_HI
    lsr
    lsr
    lsr
    lsr
    lsr
    lsr
    lsr
    sta T1
; T2 = QUOTIENT_LO >> 7
    lda QUOTIENT_LO
    lsr
    lsr
    lsr
    lsr
    lsr
    lsr
    lsr
    sta T2
; QUOTIENT_HI = QUOTIENT_HI << 1
    lda QUOTIENT_HI
    asl
    sta QUOTIENT_HI
; QUOTIENT_HI = QUOTIENT_HI | T2
    lda QUOTIENT_HI
    ora T2
    sta QUOTIENT_HI
; QUOTIENT_LO = QUOTIENT_LO << 1
    lda QUOTIENT_LO
    asl
    sta QUOTIENT_LO
; T2 = REMAINDER >> 7
    lda REMAINDER
    lsr
    lsr
    lsr
    lsr
    lsr
    lsr
    lsr
    sta T2
; REMAINDER = REMAINDER << 1
    lda REMAINDER
    asl
    sta REMAINDER
; REMAINDER = REMAINDER | T1
    lda REMAINDER
    ora T1
    sta REMAINDER
; if T2 != 0:
    lda T2
    cmp #0
    bne word_division_L14
word_division_L15:
    jmp word_division_L12
word_division_L14:
    jmp word_division_L11
word_division_L11:
; REMAINDER = REMAINDER - DIVISOR
    lda REMAINDER
    sec
    sbc DIVISOR
    sta REMAINDER
; QUOTIENT_LO = QUOTIENT_LO | 1
    lda QUOTIENT_LO
    ora #1
    sta QUOTIENT_LO
    jmp word_division_L13
word_division_L12:
; if REMAINDER >= DIVISOR:
    lda REMAINDER
    cmp DIVISOR
    bcs word_division_L19
word_division_L20:
    jmp word_division_L17
word_division_L19:
    jmp word_division_L16
word_division_L16:
; REMAINDER = REMAINDER - DIVISOR
    lda REMAINDER
    sec
    sbc DIVISOR
    sta REMAINDER
; QUOTIENT_LO = QUOTIENT_LO | 1
    lda QUOTIENT_LO
    ora #1
    sta QUOTIENT_LO
    jmp word_division_L18
word_division_L17:
word_division_L18:
word_division_L13:
; T0 = T0 - 1
    lda T0
    sec
    sbc #1
    sta T0
    jmp word_division_L6
word_division_L8:
    rts
component_end:

Machine contract

A/X/status may change; Y is preserved. The 2A03 has no decimal arithmetic mode. A direct JSR needs two return-address stack bytes. The helper makes no nested calls and adds no other stack use. Its named scratch is private workspace, not a pointer, a ROM variable or shared zero-page space that the game may borrow concurrently. The samples use fixed symbol bindings. Scope or rename every label and binding, including the component_end measurement marker, and allocate a separate block for each independent instance. Different helpers can also use common field names, so combining them needs separate namespaces or explicitly renamed bindings. Give each block one mutation owner.

The NROM caller owns PPU setup and background display, with NMI and DMC disabled. Table and packet sizes are bounded to fit the cartridge caller and internal RAM; the helper owns no PPU registers.

Adapt the caller

demo.asm supplies the bindings and include order. exercise.inc first runs table cases, then a separate client retaining state between calls. A game can call the public step entry directly and remove the diagnostic packet and display. Do not copy the diagnostic’s hardware takeover into an existing game loop.

Build with make ASM198X=/path/to/asm198x. The Makefile produces normal, custom, deliberately wrong and repaired programs. Tools must already be installed; the sample downloads nothing.

Bindings and include order

demo.asm
.segment "HEADER"
.byte "NES",$1a,2,1,0,0,0,0,0,0,0,0,0,0
QUOTIENT_LO=$80
QUOTIENT_HI=$81
REMAINDER=$82
VALID=$83
NUMERATOR_LO=$84
NUMERATOR_HI=$85
DIVISOR=$86
T0=$87
T1=$88
T2=$89
STATE=$80
input_ptr=$70
cases_left=$72
ptr=$74
saved_y=$76
OUTPUT=$0200
preview=OUTPUT+1400
ready=OUTPUT+1408
CUSTOM=0
BROKEN=0
COYOTE_STEPS=4
BUFFER_STEPS=4
TEXT_CAP=8
PROPERTY_COUNT=16
STREAM_BYTES=13
CRC_POLY=7
.include "caller.inc"
.include "exercise.inc"
component_start:
.include "component.inc"
.include "data.inc"
.include "cases.inc"
code_end:
.segment "VECTORS"
.word nmi,reset,irq
.segment "CHARS"
.include "../../input/controller-reading/edges-font.inc"
.res 8192-272,0

A client retaining state between calls

exercise.inc, excerpt
preview_rules:
.if CUSTOM
; NUMERATOR_LO = 255
    lda #255
    sta NUMERATOR_LO
; NUMERATOR_HI = 255
    lda #255
    sta NUMERATOR_HI
; DIVISOR = 255
    lda #255
    sta DIVISOR
.if BROKEN
; NUMERATOR_HI = 0
    lda #0
    sta NUMERATOR_HI
.endif
    jsr checked_step
    jsr emit_state
    rts
.else
; NUMERATOR_LO = 232
    lda #232
    sta NUMERATOR_LO
; NUMERATOR_HI = 3
    lda #3
    sta NUMERATOR_HI
; DIVISOR = 7
    lda #7
    sta DIVISOR
.if BROKEN
; NUMERATOR_HI = 0
    lda #0
    sta NUMERATOR_HI
.endif
    jsr checked_step
    jsr emit_state
.endif
    rts

Customise it

Divide the largest word by the largest byte. The maintained custom caller changes its inputs or client sequence rather than hiding a different implementation behind the same label. Compare every state of the two clients, not only their final display.

Custom client selection

custom.asm
.segment "HEADER"
.byte "NES",$1a,2,1,0,0,0,0,0,0,0,0,0,0
QUOTIENT_LO=$80
QUOTIENT_HI=$81
REMAINDER=$82
VALID=$83
NUMERATOR_LO=$84
NUMERATOR_HI=$85
DIVISOR=$86
T0=$87
T1=$88
T2=$89
STATE=$80
input_ptr=$70
cases_left=$72
ptr=$74
saved_y=$76
OUTPUT=$0200
preview=OUTPUT+1400
ready=OUTPUT+1408
CUSTOM=1
BROKEN=0
COYOTE_STEPS=2
BUFFER_STEPS=2
TEXT_CAP=4
PROPERTY_COUNT=16
STREAM_BYTES=13
CRC_POLY=49
.include "caller.inc"
.include "exercise.inc"
component_start:
.include "component.inc"
.include "data.inc"
.include "cases.inc"
code_end:
.segment "VECTORS"
.word nmi,reset,irq
.segment "CHARS"
.include "../../input/controller-reading/edges-font.inc"
.res 8192-272,0

The custom diagnostic row is 01 01 00 01 FF FF FF 00. The full client trace in the source explains how it gets there.

Mistake and repair

The wrong caller drops the numerator’s high byte. The division remains internally consistent but answers a different question. Preserve both bytes when passing the numerator.

The BROKEN branch lives in the sequential client, not the reusable helper. wrong.asm selects it; repaired.asm restores the normal call sequence. The repaired executable matches the normal program (for Spectrum, compare loaded CODE rather than the tape filename).

Select the deliberately wrong client

wrong.asm
.segment "HEADER"
.byte "NES",$1a,2,1,0,0,0,0,0,0,0,0,0,0
QUOTIENT_LO=$80
QUOTIENT_HI=$81
REMAINDER=$82
VALID=$83
NUMERATOR_LO=$84
NUMERATOR_HI=$85
DIVISOR=$86
T0=$87
T1=$88
T2=$89
STATE=$80
input_ptr=$70
cases_left=$72
ptr=$74
saved_y=$76
OUTPUT=$0200
preview=OUTPUT+1400
ready=OUTPUT+1408
CUSTOM=0
BROKEN=1
COYOTE_STEPS=4
BUFFER_STEPS=4
TEXT_CAP=8
PROPERTY_COUNT=16
STREAM_BYTES=13
CRC_POLY=7
.include "caller.inc"
.include "exercise.inc"
component_start:
.include "component.inc"
.include "data.inc"
.include "cases.inc"
code_end:
.segment "VECTORS"
.word nmi,reset,irq
.segment "CHARS"
.include "../../input/controller-reading/edges-font.inc"
.res 8192-272,0
NES shows the deliberate error row 21 00 01 01 E8 00 07 00.
The wrong client completes with 21 00 01 01 E8 00 07 00; its changed inputs and state explain the result.

Cost and alternatives

The measured helper extent is 210 code bytes; public state is 7 bytes and private scratch is 3 bytes. There are no immutable helper tables. Caller data, the test cases, return-address stack, alignment and display buffers cost additional space. These are assembled sizes, not cycle measurements.

Sixteen stages independent of quotient size. Repeated subtraction is shorter for a tightly bounded small quotient but can take 65,535 subtractions for division by one. A 68000 DIVU path can exploit native word arithmetic after validating its quotient range.

Unsigned division rounds down. A signed quotient needs a stated rounding rule. The helper does not turn an arbitrary random byte modulo a range into an unbiased random choice.

Execution evidence

Four variants run on NTSC NROM, background only, DMC/NMI disabled, using Asm198x 0.0.72, Emu198x 0.44.2. Each executes 64 boundary/varied/error table cases, dirty initialisation and the separate retained-state client. An independent mathematical/container model checks every public byte after every call. This is bounded example coverage, not exhaustive correctness for every possible state.

The records retain loaded immutable code, completion, register sentinels, full packets, source/tool/firmware hashes and captures. The public gate validates these retained observations and rejects altered evidence; it does not run restricted firmware. No firmware is distributed. The eight displayed bytes are the first eight public fields, zero-padded when shorter; the packet includes every field.

Where it fits

Decimal formatting: Repeated quotient/remainder extraction is the arithmetic behind base-ten digits.

The Craft · Maths for Games · Source and native evidence.