An Integer square root is the nearest whole number smaller than the full square root answer. So the integer square root of 10 is 3 instead of 3.162277. You'd get the same answer as INT(SQR(x)) with an integer square root function.
For things like games programming, this is often near enough - for example, the distance formula, based on pythagoras' equation A^2=B^2+C^2 only works if you square root the answer. If you need to find the distances between your items, then you're going to be doing a lot of square roots, and you're going to need to do them FAST. (That said the even faster solution might be this one: distance.bas or if you don't need the actual distance just the answer to the question "which is further away?" then not square rooting, and comparing distance1^2 with distance2^2 still tells you which is nearer. Berksman  written in ZX Basic, does this trick of never doing the square root part, for example.
Anyway, this function returns integer square roots. For numbers less than 65536, it's about 100 times faster, because it can do 16 bit calculation. For longer numbers, it has to do 32 bit calculations, which are less than optimal on an 8 bit processor! It's still about 50 times faster than the ROM routine, however.
If you want completely accurate results, you should use the floating point fast routine over at fSqrt.bas.
FUNCTION FASTCALL iSqrt (num AS ULONG) AS UINTEGER REM incoming is DEHL REM output is HL ASM LD A,D OR E JP Z, sqrtLF16bit ; we're inside a 16 bit number. We can use the faster version. LD b,16 ; b times round EXX ; Out TO root AND REM - we're doing most of this in alternate registers. LD DE,0 LD HL,0 ; DEHL = REMainder LD BC,0 ; BC = root EXX ;back TO num AND LOOP sqrtLFasmloop: EXX ; out TO root AND REM SLA C ; root <<= 1 RL B ; SLA L ; REM=rem<<1 RL H ; RL E ; RL D ; SLA L ; REM=rem<<1 RL H ; RL E ; RL D ; EXX ; back TO Num AND LOOP LD a,d ; A = inputnum>>30 AND 192 RLCA RLCA SLA L ; num <<= 1 RL H RL E RL D SLA L ; num <<= 1 RL H RL E RL D EXX ; out TO root AND REM ADD A,L ; a=a+L ; REM=REM+num>>30 LD L,A ; a-> L ; JR NC, sqrtLFasmloophop1 ; INC H JR NC, sqrtLFasmloophop1 INC DE ; ; sqrtLFasmloophop1: INC BC ; root=root+1 sqrtLFasmloophop2: ; DEHL = REMainder ; BC = root ; IF REM >= root then LD A,D OR E JR NZ, sqrtLFasmthen ; IF REM > 65535 then rem is definitely > root and we go to true LD A, H CP B JR C, sqrtLFasmelse ; H<B - that is REM<root so rem>=root is false and we go to else JR NZ, sqrtLFasmthen ; H isn't zero though, so we could do a carry from it, so we're good to say HL is larger. ; IF h is out, THEN it's down to L and C LD A,L CP C JR C, sqrtLFasmelse ; L<C - that is REM<root so rem>=root is false and we go to else ; must be true - GO TO true. sqrtLFasmthen: ;REMainder=remainder-root AND A ; clear carry flag SBC HL,BC ; take root away from the lower half of REM. JP NC, sqrtLFasmhop3 ; we didn't take away too much, so we're okay to loop round. ; IF we're here, we did take away too much. We need to borrow from DE DEC DE ; borrow off DE sqrtLFasmhop3: INC BC ;root=root+1 JP sqrtLFasmloopend ;else sqrtLFasmelse: DEC BC ;root=root-1 ;end IF sqrtLFasmloopend: EXX ; back TO num DJNZ sqrtLFasmloop EXX ; out TO root AND REM PUSH BC EXX ; back TO normal POP HL SRA H RES 7,H RR L ; Hl=HL/2 - root/2 is the answer. jr sqrtLFexitFunction sqrtLF16bit: ld a,l ld l,h ld de,0040h ; 40h appends "01" TO D ld h,d ld b,7 sqrtLFsqrt16loop: sbc hl,de ; IF speed is critical, AND you don't mind spending the extra bytes, ; you could unroll this LOOP 7 times instead of DJNZ. ; deprecated because of issues - jr nc,$+3 (note that IF you unroll this LOOP, you'll need 7 labels for the jumps the other way!) jr nc,sqrtLFsqrthop1 add hl,de sqrtLFsqrthop1: ccf rl d rla adc hl,hl rla adc hl,hl DJNZ sqrtLFsqrt16loop sbc hl,de ; optimised last iteration ccf rl d ld h,0 ld l,d ld de,0 sqrtLFexitFunction: END ASM END FUNCTION