Find an Integer Square Root on NES | Pattern Library

Sometimes the useful answer is the whole-number distance, with no floating point in sight.

NES shows the working diagnostic row 1F 27 00 E8 03 00 00 00. — A completed native caller displays 1F 27 00 E8 03 00 00 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

NUMBER_LO/NUMBER_HI hold an unsigned 16-bit number. Return ROOT=floor(sqrt(number)) and the remainder number−ROOT² as a low-byte-first word. Inputs are preserved. The remainder can reach 510, so it is not a single byte.

Include component.inc once (MOS callers include the shared 6502 implementation through it). Reserve the public fields in this exact order: ROOT, REMAINDER_LO, REMAINDER_HI, NUMBER_LO, NUMBER_HI. They occupy 5 contiguous bytes. Also reserve the private workspace: T0, T1, T2, T3 (4 bytes). Array names in the explanation refer to the numbered consecutive bindings, such as C0, C1 and so on.

Call integer_square_root_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 integer_square_root_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	NUMBER_LO and NUMBER_HI
Read after service	ROOT, REMAINDER_LO and REMAINDER_HI

The first n odd numbers sum to n². Subtract 1, 3, 5 and so on until the next subtraction would go negative. The number of accepted subtractions is the root.

Reusable initialisation and service

6502.inc

; SPDX-License-Identifier: MIT
; Find an Integer Square Root. See README.md for the byte interface and ownership.
; init clears the public fields and private workspace; set inputs afterwards.
integer_square_root_init:
    lda #0
    sta ROOT
    sta REMAINDER_LO
    sta REMAINDER_HI
    sta NUMBER_LO
    sta NUMBER_HI
    sta T0
    sta T1
    sta T2
    sta T3
    rts
integer_square_root_step:
; ROOT = 0
    lda #0
    sta ROOT
; REMAINDER_LO = NUMBER_LO
    lda NUMBER_LO
    sta REMAINDER_LO
; REMAINDER_HI = NUMBER_HI
    lda NUMBER_HI
    sta REMAINDER_HI
; T0 = 1
    lda #1
    sta T0
; T1 = 0
    lda #0
    sta T1
; T2 = 1
    lda #1
    sta T2
; while T2 == 1:
integer_square_root_L1:
    lda T2
    cmp #1
    beq integer_square_root_L4
integer_square_root_L5:
    jmp integer_square_root_L3
integer_square_root_L4:
    jmp integer_square_root_L2
integer_square_root_L2:
; if REMAINDER_HI < T1:
    lda REMAINDER_HI
    cmp T1
    bcc integer_square_root_L9
integer_square_root_L10:
    jmp integer_square_root_L7
integer_square_root_L9:
    jmp integer_square_root_L6
integer_square_root_L6:
; return
    rts
    jmp integer_square_root_L8
integer_square_root_L7:
integer_square_root_L8:
; if REMAINDER_HI == T1:
    lda REMAINDER_HI
    cmp T1
    beq integer_square_root_L14
integer_square_root_L15:
    jmp integer_square_root_L12
integer_square_root_L14:
    jmp integer_square_root_L11
integer_square_root_L11:
; if REMAINDER_LO < T0:
    lda REMAINDER_LO
    cmp T0
    bcc integer_square_root_L19
integer_square_root_L20:
    jmp integer_square_root_L17
integer_square_root_L19:
    jmp integer_square_root_L16
integer_square_root_L16:
; return
    rts
    jmp integer_square_root_L18
integer_square_root_L17:
integer_square_root_L18:
    jmp integer_square_root_L13
integer_square_root_L12:
integer_square_root_L13:
; T3 = 0
    lda #0
    sta T3
; if REMAINDER_LO < T0:
    lda REMAINDER_LO
    cmp T0
    bcc integer_square_root_L24
integer_square_root_L25:
    jmp integer_square_root_L22
integer_square_root_L24:
    jmp integer_square_root_L21
integer_square_root_L21:
; T3 = 1
    lda #1
    sta T3
    jmp integer_square_root_L23
integer_square_root_L22:
integer_square_root_L23:
; REMAINDER_LO = REMAINDER_LO - T0
    lda REMAINDER_LO
    sec
    sbc T0
    sta REMAINDER_LO
; REMAINDER_HI = REMAINDER_HI - T1
    lda REMAINDER_HI
    sec
    sbc T1
    sta REMAINDER_HI
; REMAINDER_HI = REMAINDER_HI - T3
    lda REMAINDER_HI
    sec
    sbc T3
    sta REMAINDER_HI
; ROOT = ROOT + 1
    lda ROOT
    clc
    adc #1
    sta ROOT
; T3 = T0
    lda T0
    sta T3
; T0 = T0 + 2
    lda T0
    clc
    adc #2
    sta T0
; if T0 < T3:
    lda T0
    cmp T3
    bcc integer_square_root_L29
integer_square_root_L30:
    jmp integer_square_root_L27
integer_square_root_L29:
    jmp integer_square_root_L26
integer_square_root_L26:
; T1 = T1 + 1
    lda T1
    clc
    adc #1
    sta T1
    jmp integer_square_root_L28
integer_square_root_L27:
integer_square_root_L28:
    jmp integer_square_root_L1
integer_square_root_L3:
    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
ROOT=$80
REMAINDER_LO=$81
REMAINDER_HI=$82
NUMBER_LO=$83
NUMBER_HI=$84
T0=$85
T1=$86
T2=$87
T3=$88
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
; NUMBER_LO = 255
    lda #255
    sta NUMBER_LO
; NUMBER_HI = 255
    lda #255
    sta NUMBER_HI
.if BROKEN
; NUMBER_HI = 0
    lda #0
    sta NUMBER_HI
.endif
    jsr checked_step
    jsr emit_state
    rts
.else
; NUMBER_LO = 232
    lda #232
    sta NUMBER_LO
; NUMBER_HI = 3
    lda #3
    sta NUMBER_HI
.if BROKEN
; NUMBER_HI = 0
    lda #0
    sta NUMBER_HI
.endif
    jsr checked_step
    jsr emit_state
.endif
    rts

Customise it

Exercise the maximum input and its two-byte remainder. 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
ROOT=$80
REMAINDER_LO=$81
REMAINDER_HI=$82
NUMBER_LO=$83
NUMBER_HI=$84
T0=$85
T1=$86
T2=$87
T3=$88
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 FF FE 01 FF FF 00 00 00. The full client trace in the source explains how it gets there.

Mistake and repair

The wrong caller keeps only the low input byte. Large distances collapse into small unrelated values. Supply the complete squared value, including its high byte.

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
ROOT=$80
REMAINDER_LO=$81
REMAINDER_HI=$82
NUMBER_LO=$83
NUMBER_HI=$84
T0=$85
T1=$86
T2=$87
T3=$88
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 0F 07 00 E8 00 00 00 00. — The wrong client completes with 0F 07 00 E8 00 00 00 00; its changed inputs and state explain the result.

Cost and alternatives

The measured helper extent is 192 code bytes; public state is 5 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.

Zero to 255 subtraction stages. A bitwise restoring square root has a fixed number of bit-pair stages but is less direct to read. Use a table only when the input range and memory cost justify it.

This readable version subtracts consecutive odd numbers and can take 255 iterations. Collision tests usually compare squared distances instead, avoiding the square root entirely.

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

Byte products: Squaring a byte is a direct way to produce a full-width input for this routine.

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