Sometimes the useful answer is the whole-number distance, with no floating point in sight.
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
; 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:
ld a,0
ld (ROOT),a
ld (REMAINDER_LO),a
ld (REMAINDER_HI),a
ld (NUMBER_LO),a
ld (NUMBER_HI),a
ld (T0),a
ld (T1),a
ld (T2),a
ld (T3),a
ret
integer_square_root_step:
; ROOT = 0
ld a,0
ld (ROOT),a
; REMAINDER_LO = NUMBER_LO
ld a,(NUMBER_LO)
ld (REMAINDER_LO),a
; REMAINDER_HI = NUMBER_HI
ld a,(NUMBER_HI)
ld (REMAINDER_HI),a
; T0 = 1
ld a,1
ld (T0),a
; T1 = 0
ld a,0
ld (T1),a
; T2 = 1
ld a,1
ld (T2),a
; while T2 == 1:
integer_square_root_L1:
ld a,(T2)
cp 1
jp z,integer_square_root_L2
jp integer_square_root_L3
integer_square_root_L2:
; if REMAINDER_HI < T1:
ld a,(REMAINDER_HI)
ld b,a
ld a,(T1)
ld c,a
ld a,b
cp c
jp c,integer_square_root_L4
jp integer_square_root_L5
integer_square_root_L4:
; return
ret
jp integer_square_root_L6
integer_square_root_L5:
integer_square_root_L6:
; if REMAINDER_HI == T1:
ld a,(REMAINDER_HI)
ld b,a
ld a,(T1)
ld c,a
ld a,b
cp c
jp z,integer_square_root_L7
jp integer_square_root_L8
integer_square_root_L7:
; if REMAINDER_LO < T0:
ld a,(REMAINDER_LO)
ld b,a
ld a,(T0)
ld c,a
ld a,b
cp c
jp c,integer_square_root_L10
jp integer_square_root_L11
integer_square_root_L10:
; return
ret
jp integer_square_root_L12
integer_square_root_L11:
integer_square_root_L12:
jp integer_square_root_L9
integer_square_root_L8:
integer_square_root_L9:
; T3 = 0
ld a,0
ld (T3),a
; if REMAINDER_LO < T0:
ld a,(REMAINDER_LO)
ld b,a
ld a,(T0)
ld c,a
ld a,b
cp c
jp c,integer_square_root_L13
jp integer_square_root_L14
integer_square_root_L13:
; T3 = 1
ld a,1
ld (T3),a
jp integer_square_root_L15
integer_square_root_L14:
integer_square_root_L15:
; REMAINDER_LO = REMAINDER_LO - T0
ld a,(REMAINDER_LO)
ld b,a
ld a,(T0)
ld c,a
ld a,b
sub c
ld (REMAINDER_LO),a
; REMAINDER_HI = REMAINDER_HI - T1
ld a,(REMAINDER_HI)
ld b,a
ld a,(T1)
ld c,a
ld a,b
sub c
ld (REMAINDER_HI),a
; REMAINDER_HI = REMAINDER_HI - T3
ld a,(REMAINDER_HI)
ld b,a
ld a,(T3)
ld c,a
ld a,b
sub c
ld (REMAINDER_HI),a
; ROOT = ROOT + 1
ld a,(ROOT)
add a,1
ld (ROOT),a
; T3 = T0
ld a,(T0)
ld (T3),a
; T0 = T0 + 2
ld a,(T0)
add a,2
ld (T0),a
; if T0 < T3:
ld a,(T0)
ld b,a
ld a,(T3)
ld c,a
ld a,b
cp c
jp c,integer_square_root_L16
jp integer_square_root_L17
integer_square_root_L16:
; T1 = T1 + 1
ld a,(T1)
add a,1
ld (T1),a
jp integer_square_root_L18
integer_square_root_L17:
integer_square_root_L18:
jp integer_square_root_L1
integer_square_root_L3:
ret
component_end:Machine contract
AF/BC/DE/HL may change; IX/IY are preserved. A direct CALL 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 48K diagnostic boots through ROM tape loading and USR, disables interrupts and owns the screen. The helper performs no port or screen writes.
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
org $8000
CUSTOM equ 0
BROKEN equ 0
COYOTE_STEPS equ 4
BUFFER_STEPS equ 4
TEXT_CAP equ 8
PROPERTY_COUNT equ 16
STREAM_BYTES equ 13
CRC_POLY equ 7
include "caller.inc"
include "exercise.inc"
component_start:
include "component.inc"
include "data.inc"
include "cases.inc"
include "../../physics/rectangle-overlap/display.inc"
code_end:
STATE:
ROOT: defb 0
REMAINDER_LO: defb 0
REMAINDER_HI: defb 0
NUMBER_LO: defb 0
NUMBER_HI: defb 0
T0: defb 0
T1: defb 0
T2: defb 0
T3: defb 0
input_ptr: defw 0
output_ptr: defw 0
cases_left: defb 0
ready: defb 0
preview: defs 8,0
OUTPUT: defs 1300,0
end startA client retaining state between calls
preview_rules:
if CUSTOM
; NUMBER_LO = 255
ld a,255
ld (NUMBER_LO),a
; NUMBER_HI = 255
ld a,255
ld (NUMBER_HI),a
if BROKEN
; NUMBER_HI = 0
ld a,0
ld (NUMBER_HI),a
endif
call checked_step
call emit_state
ret
else
; NUMBER_LO = 232
ld a,232
ld (NUMBER_LO),a
; NUMBER_HI = 3
ld a,3
ld (NUMBER_HI),a
if BROKEN
; NUMBER_HI = 0
ld a,0
ld (NUMBER_HI),a
endif
call checked_step
call emit_state
endif
retCustomise 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
org $8000
CUSTOM equ 1
BROKEN equ 0
COYOTE_STEPS equ 2
BUFFER_STEPS equ 2
TEXT_CAP equ 4
PROPERTY_COUNT equ 16
STREAM_BYTES equ 13
CRC_POLY equ 49
include "caller.inc"
include "exercise.inc"
component_start:
include "component.inc"
include "data.inc"
include "cases.inc"
include "../../physics/rectangle-overlap/display.inc"
code_end:
STATE:
ROOT: defb 0
REMAINDER_LO: defb 0
REMAINDER_HI: defb 0
NUMBER_LO: defb 0
NUMBER_HI: defb 0
T0: defb 0
T1: defb 0
T2: defb 0
T3: defb 0
input_ptr: defw 0
output_ptr: defw 0
cases_left: defb 0
ready: defb 0
preview: defs 8,0
OUTPUT: defs 1300,0
end startThe 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
org $8000
CUSTOM equ 0
BROKEN equ 1
COYOTE_STEPS equ 4
BUFFER_STEPS equ 4
TEXT_CAP equ 8
PROPERTY_COUNT equ 16
STREAM_BYTES equ 13
CRC_POLY equ 7
include "caller.inc"
include "exercise.inc"
component_start:
include "component.inc"
include "data.inc"
include "cases.inc"
include "../../physics/rectangle-overlap/display.inc"
code_end:
STATE:
ROOT: defb 0
REMAINDER_LO: defb 0
REMAINDER_HI: defb 0
NUMBER_LO: defb 0
NUMBER_HI: defb 0
T0: defb 0
T1: defb 0
T2: defb 0
T3: defb 0
input_ptr: defw 0
output_ptr: defw 0
cases_left: defb 0
ready: defb 0
preview: defs 8,0
OUTPUT: defs 1300,0
end start
Cost and alternatives
The measured helper extent is 253 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 48K PAL ROM tape load and USR, 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.