Integer Square Roots in Z80 machine code
========================================
A simple, but inefficient way of calculating integer square roots is to
count how many times increasing odd numbers can be subtracted from the
starting value. For example:

32-1=31 -> 31-3=28 -> 28-5=23 -> 23-7=16 -> 16-9=7 -> 7-11<0
        1          2          3          4          5

-> SQR(30)=5, remainder 7

This can be done in Z80 machine code as follows. To calculate roots of
numbers larger than 16 bits different code is needed, as the square root
of &10000 (a 17-bit number) is &100 (a 9-bit number).

; Calculate 16-bit square root
; ----------------------------
; On entry  HL=input value
; On exit   DE=corrupted
;            A=root
;           HL=remainder
; Size      14 bytes
;
.sqr
LD   DE,1         ; Initialise to first subtrand
LD   A,D          ; Initial root=0
.sqr_loop         ; Repeatedly subtract increasing
AND  A            ;   odd numbers until HL<0
SBC  HL,DE        ; HL=HL-subtrand
JR   C,sqr_done   ; HL<0, all done
INC  A            ; Increase root
INC  DE
INC  DE           ; step subtrand +2 to next odd number
JR   sqr_loop     ; Loop to subtract next odd number
.sqr_done
ADD  HL,DE        ; A=root, HL=remainder

You can test this in BBC BASIC by terminating the code with:

LD H,L:LD L,A:EXX ; HL'=remainder*256+root
RET

and running:

FOR L%=1 TO 65535:H%=L%DIV256:R%=USR sqr:PRINT L%,~R%:NEXT


The following code calculates roots of 8-bit numbers by using an 8-bit
counter and root. As it calculates roots of 8-bit numbers the maximum it
will calculate is the integer square root of 255 which is 15.

; Calculate 8-bit square root
; ---------------------------
; On entry   A=input value
; On exit    A=corrupted
;            C=corrupted
;            B=root
; Size       11 bytes
;
.sqr8
LD   BC,1         ; Initialise C=first subtrand, B=SQR(0)
.sqr8_loop        ; Repeatedly subtract increasing
SUB  A,C          ; Subtract current odd number
JR   C,sqr8_done  ; A<0, all done
INC  B            ; Increase root
INC  C
INC  C            ; Step to next odd number
JR   sqr8_loop    ; Loop to subtract next odd number
.sqr8_done
                  ; B=root