# Distance.bas

This calculates an approximation for the distance formula **r= SQR(x ^{2} + y^{2})**,
based on two parameters,

`x`

and `y`

. The return value is not guaranteed to be accurate -
and indeed can be as high as 10% inaccurate as x and y approach 255 (the upper limit for input).
The return value is an integer - chosen because screen is 256 pixels wide, and the diagonal across the screen
is bigger than 1 byte can hold.If you need accurate results, you should go with `iSqrt`

or `fSqrt`

from this library.

For speed, this can't be beaten, however.

Comparing -
`answer = distance(i, j)`

against
`answer = iSqrt(i * i + j * j)`

shows over a range of `i`

and `j`

`1..250`

:

- distance 8.98 seconds

iSqrt 50.1 seconds

Distance is definitely faster, if you're willing to accept the greater inaccuracy (you probably are).

By the by - standard floating point square root:

`fSqrt`

function: 44 minutes (2625.14 seconds)`SQR`

(ROM) - 122 minutes. (7336.86 seconds)

Shows how awful that ROM SQR routine really is...

Formula is: in a right angle triangle with sides A and B, and hypotenuse H, as an estimate of length of H, it returns (A + B) - (half the smallest of A and B) - (1/4 the smallest of A and B) + (1/16 the smallest of A and B)

```
FUNCTION fastcall distance (a as ubyte, b as ubyte) as uInteger
REM returns a fast approximation of SQRT (a^2 + b^2) - the distance formula, generated from taylor series expansion.
REM This version fundamentally by Alcoholics Anonymous, improving on Britlion's earlier version - which itself
REM was suggested, with thanks, by NA_TH_AN.
asm
POP HL ;' return address
;' First parameter in A
POP BC ;' second parameter -> B
PUSH HL ;' put return back
;' First find out which is bigger - A or B.
cp b
ld c,b
jr nc, distance_AisMAX
ld c,a
distance_AisMAX:
;' c = MIN(a,b)
srl c ;' c = MIN/2
sub c ;' a = A - MIN/2
srl c ;' c = MIN/4
sub c ;' a = A - MIN/2 - MIN/4
srl c
srl c ;' c = MIN/16
add a,c ;' a = A - MIN/2 - MIN/4 + MIN/16
add a,b ;' a = A + B - MIN/2 - MIN/4 + MIN/16
ld l,a
ld h,0 ;' hl = result
ret nc
inc h ;' catch 9th bit
END ASM
END FUNCTION
```