Decode Bounded Byte Deltas on NES

A path or table whose values change slowly can be stored as a starting value followed by small signed differences.

Use it in a project

COUNT 0–8 deltas are two’s-complement bytes. Starting at BASE, add each delta modulo 256 and write successive values to R0..R7. Return WRITTEN=COUNT and VALID=1. Invalid count leaves all output bytes unchanged and returns WRITTEN=0, VALID=0. BASE is preserved.

Include component.inc once (MOS callers include the shared 6502 implementation through it). Reserve the public fields in this exact order: WRITTEN, VALID, COUNT, BASE, Q0, Q1, Q2, Q3, Q4, Q5, Q6, Q7, R0, R1, R2, R3, R4, R5, R6, R7. They occupy 20 contiguous bytes. Also reserve the private workspace: T0, T1, T2, EXPR_SAVE (4 bytes). Array names in the explanation refer to the numbered consecutive bindings, such as C0, C1 and so on.

Call bounded_delta_decoder_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 bounded_delta_decoder_step whenever the described event occurs. The helper reads no device, waits for no frame and installs no interrupt handler.

Keep a running byte total. Two’s-complement negative deltas work through modulo addition, so no sign branch is needed.

Reusable initialisation and service

; SPDX-License-Identifier: MIT
; Decode Bounded Byte Deltas. See README.md for the byte interface and ownership.
; init clears the public fields and private workspace; set inputs afterwards.
bounded_delta_decoder_init:
    lda #0
    sta WRITTEN
    sta VALID
    sta COUNT
    sta BASE
    sta Q0
    sta Q1
    sta Q2
    sta Q3
    sta Q4
    sta Q5
    sta Q6
    sta Q7
    sta R0
    sta R1
    sta R2
    sta R3
    sta R4
    sta R5
    sta R6
    sta R7
    sta T0
    sta T1
    sta T2
    sta EXPR_SAVE
    rts
bounded_delta_decoder_step:
; WRITTEN = 0
    lda #0
    sta WRITTEN
; VALID = 0
    lda #0
    sta VALID
; if COUNT > 8:
    lda COUNT
    cmp #8
    beq bounded_delta_decoder_L5
    bcs bounded_delta_decoder_L4
bounded_delta_decoder_L5:
    jmp bounded_delta_decoder_L2
bounded_delta_decoder_L4:
    jmp bounded_delta_decoder_L1
bounded_delta_decoder_L1:
; return
    rts
    jmp bounded_delta_decoder_L3
bounded_delta_decoder_L2:
bounded_delta_decoder_L3:
; VALID = 1
    lda #1
    sta VALID
; T0 = BASE
    lda BASE
    sta T0
; T1 = 0
    lda #0
    sta T1
; while T1 < COUNT:
bounded_delta_decoder_L6:
    lda T1
    cmp COUNT
    bcc bounded_delta_decoder_L9
bounded_delta_decoder_L10:
    jmp bounded_delta_decoder_L8
bounded_delta_decoder_L9:
    jmp bounded_delta_decoder_L7
bounded_delta_decoder_L7:
; T2 = Q[T1]
    ldx T1
    lda Q0,x
    sta T2
; T0 = T0 + T2
    lda T0
    clc
    adc T2
    sta T0
; R[T1] = T0
    lda T0
    sta EXPR_SAVE
    ldx T1
    lda EXPR_SAVE
    sta R0,x
; T1 = T1 + 1
    lda T1
    clc
    adc #1
    sta T1
    jmp bounded_delta_decoder_L6
bounded_delta_decoder_L8:
; WRITTEN = COUNT
    lda COUNT
    sta WRITTEN
    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

.segment "HEADER"
.byte "NES",$1a,2,1,0,0,0,0,0,0,0,0,0,0
WRITTEN=$80
VALID=$81
COUNT=$82
BASE=$83
Q0=$84
Q1=$85
Q2=$86
Q3=$87
Q4=$88
Q5=$89
Q6=$8a
Q7=$8b
R0=$8c
R1=$8d
R2=$8e
R3=$8f
R4=$90
R5=$91
R6=$92
R7=$93
T0=$94
T1=$95
T2=$96
EXPR_SAVE=$97
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

preview_rules:
.if CUSTOM
; COUNT = 3
    lda #3
    sta COUNT
; BASE = 255
    lda #255
    sta BASE
; Q0 = 1
    lda #1
    sta Q0
; Q1 = 128
    lda #128
    sta Q1
; Q2 = 127
    lda #127
    sta Q2
.if BROKEN
; BASE = 0
    lda #0
    sta BASE
.endif
    jsr checked_step
    jsr emit_state
    rts
.else
; COUNT = 4
    lda #4
    sta COUNT
; BASE = 10
    lda #10
    sta BASE
; Q0 = 1
    lda #1
    sta Q0
; Q1 = 255
    lda #255
    sta Q1
; Q2 = 2
    lda #2
    sta Q2
; Q3 = 254
    lda #254
    sta Q3
.if BROKEN
; BASE = 0
    lda #0
    sta BASE
.endif
    jsr checked_step
    jsr emit_state
.endif
    rts

Customise it

Exercise wrapping and the −128 signed delta. 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

.segment "HEADER"
.byte "NES",$1a,2,1,0,0,0,0,0,0,0,0,0,0
WRITTEN=$80
VALID=$81
COUNT=$82
BASE=$83
Q0=$84
Q1=$85
Q2=$86
Q3=$87
Q4=$88
Q5=$89
Q6=$8a
Q7=$8b
R0=$8c
R1=$8d
R2=$8e
R3=$8f
R4=$90
R5=$91
R6=$92
R7=$93
T0=$94
T1=$95
T2=$96
EXPR_SAVE=$97
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 03 01 03 FF 01 80 7F 00. The full client trace in the source explains how it gets there.

Mistake and repair

The wrong caller discards the starting value. Every decoded coordinate is shifted. Preserve BASE alongside the delta stream.

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

.segment "HEADER"
.byte "NES",$1a,2,1,0,0,0,0,0,0,0,0,0,0
WRITTEN=$80
VALID=$81
COUNT=$82
BASE=$83
Q0=$84
Q1=$85
Q2=$86
Q3=$87
Q4=$88
Q5=$89
Q6=$8a
Q7=$8b
R0=$8c
R1=$8d
R2=$8e
R3=$8f
R4=$90
R5=$91
R6=$92
R7=$93
T0=$94
T1=$95
T2=$96
EXPR_SAVE=$97
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

Cost and alternatives

The measured helper extent is 139 code bytes; public state is 20 bytes and private scratch is 4 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.

Linear in at most eight deltas. Random access needs a checkpoint or decoding the preceding deltas; absolute values cost the same number of bytes unless deltas are further packed.

This format is lossless modulo 256, with exactly one output per delta. It does not itself compress bytes; a narrower packed-delta format needs escape codes for large changes.

Execution evidence

Four variants run on NTSC NROM, background only, DMC/NMI disabled, using Asm198x 0.0.72, Emu198x 0.44.2. Each executes 50 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

Bounded RLE decoding: Choose an encoding that matches repetition or small changes in the actual data.

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