z80:Math Routines
From Learn @ Cemetech
Jump to navigationJump to searchIf you want to make games, you will likely need math routines, if you are making a math program, you need math, and if you are making a utility, you will need math. You will need math in a good number of programs, so here are some routines that might prove useful.
Contents
- 1 Multiplication
- 1.1 DE_Times_A, 16-bit output
- 1.2 DE_Times_A, 24-bit output
- 1.3 DE_Times_A, Unrolled, 16-bit output
- 1.4 DE_Times_A, unrolled, 24-bit output
- 1.5 B_Times_DE, 16-bit output
- 1.6 B_Times_DE, 24-bit output
- 1.7 DE_Times_BC, 32-bit result
- 1.8 DE_Times_BC, pseudo-unrolled result
- 1.9 BC_Times_DE, fully unrolled
- 1.10 C_Times_D
- 1.11 mul32, 64-bit output
- 1.12 BCDE_Times_A
- 1.13 H_Times_E
- 1.14 L_Squared (fast)
- 2 Absolute Value
- 3 Division
- 4 Square Root
- 5 ConvFloat (or ConvOP1)
- 6 ConvStr16
- 7 gcdHL_DE
- 8 Modulus
- 9 Random Numbers
- 10 Fixed Point Math
Multiplication
DE_Times_A, 16-bit output
At 13 bytes, this code is a pretty decent balance of speed and size. It multiplies DE by A and returns a 16-bit result in HL.
DE_Times_A:
;Inputs: DE,A
;Outputs: HL is product, B=0, A,C,DE preserved
;342cc~390cc, avg= 366cc
;size: 13 bytes
ld b,8
ld hl,0
_:
add hl,hl \ rlca \ jr nc,$+3 \ add hl,de \ djnz -_
ret
DE_Times_A, 24-bit output
This version takes only a minor tweak to return the full 24-bit result.
DE_Times_A:
;Inputs: DE,A
;Outputs: A:HL is product, BC=0,DE preserved
;343cc~423cc, avg= 383cc
;size: 14 bytes
ld bc,0800h
ld h,c
ld l,c
_:
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
djnz -_
ret
DE_Times_A, Unrolled, 16-bit output
Unrolled routines are larger in most cases, but they can really save on speed. This is 25% faster at its slowest, 40% faster at its fastest:
DE_Times_A: ;Inputs: DE,A ;Outputs: A:HL is product, BC,DE preserved ;min: 203cc ;max: 268cc ;avg: 235cc ;size: 43 bytes
ld hl,0 \ rlca \ jr nc,$+5 \ ld h,d \ ld e,l
add hl,hl \ rlca \ jr nc,$+3 \ add hl,de
add hl,hl \ rlca \ jr nc,$+3 \ add hl,de
add hl,hl \ rlca \ jr nc,$+3 \ add hl,de
add hl,hl \ rlca \ jr nc,$+3 \ add hl,de
add hl,hl \ rlca \ jr nc,$+3 \ add hl,de
add hl,hl \ rlca \ jr nc,$+3 \ add hl,de
add hl,hl \ rlca \ ret nc \ add hl,de \ ret
DE_Times_A, unrolled, 24-bit output
This version takes only a minor tweak to return the full 24-bit result. This is roughly 34% faster, being unrolled.
DE_Times_A:
;Inputs: DE,A
;Outputs: A:HL is product, C=0, B,DE preserved
;207cc~300cc, avg= 253.5cc
;size: 14 bytes
ld hl,0
ld c,h
;or a Uncomment to allow early exit if A=0
;ret z
add a,a \ jr nc,$+5 \ ld h,d \ ld l,e
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ ret nc \ add hl,de \ adc a,c
ret
B_Times_DE, 16-bit output
This routine removes leading zeroes before finishing the multiplication, which take a little more code, but results in a faster average speed.
B_Times_DE: ;Inputs: A,DE ;Outputs: HL=product, B=0, A=input B, C,DE unafected ;22 bytes ;B=0: 26cc ;B=1: 201cc ;B>1: 219cc~339cc ;avg=319.552734375cc (319+283/512)
ld hl,0
or b
ret z
ld b,8
rlca
dec b
jr nc,$-2
ld h,d
ld l,e
ret z
_:
add hl,hl
rlca
jr nc,$+3
add hl,de
djnz -_
ret
B_Times_DE, 24-bit output
This routine removes leading zeroes before finishing the multiplication, which take a little more code, but This routine uses another clever way of optimizing for speed without unrolling. The result is slightly larger and a bit faster. The idea here is to remove leading zeros before multiplying.
B_Times_DE:
;Inputs: A,DE
;Outputs: HL=product, B=0, A=input B, C,DE unafected
;22 bytes
ld hl,0
or b
ret z
ld bc,800h
rla
dec b
jr nc,$-2
ld h,d
ld l,e
ret z
_:
add hl,hl
rla
jr nc,$+3
add hl,de
adc a,c
djnz -_
ret
DE_Times_BC, 32-bit result
DE_Times_BC:
;Inputs:
; DE and BC are factors
;Outputs:
; A is 0
; BC is not changed
; DE:HL is the product
;902cc~1206cc, avg=1050cc
;20 bytes
ld hl,0
ld a,16
Mul_Loop_1:
add hl,hl
rl e \ rl d
jr nc,$+6
add hl,bc
jr nc,$+3
inc de
dec a
jr nz,Mul_Loop_1
ret
DE_Times_BC, pseudo-unrolled result
For 5 bytes more, you can cut out 93cc from the average. This method somehat unrolls the code by making the iterated code into a subroutine, then place a call to that subroutine that falls through to the same subroutine, essentially iterating it twice. Now do the same to this subroutine, and you have iterated 4 times, do this twice more to get the 16 iterations we need.
DE_Times_BC:
;Inputs:
; DE and BC are factors
;Outputs:
; A,BC are not changed
; DE:HL is the product
;25 bytes
;873cc~1289cc, avg=957cc
ld hl,0
call +_
_:
call +_
_:
call +_
_:
call +_
_:
;38cc or 54cc or 64cc, avg=43.25
add hl,hl
rl e \ rl d
ret nc
add hl,bc
ret nc
inc de
ret
BC_Times_DE, fully unrolled
This is very fast, averaging less than 600cc. 38% and 43% faster than the pseudo-unrolled and regular routine, but 123 bytes.
BC_Times_DE:
;BC*DE->BCHL
;out: E=0, A,D are destroyed
;Assuming B==0,C==0 128cc
;Assuming B==0,C!=0 317cc~414cc, avg 365.5c
; B!=0,C==0 317cc~422cc, avg 367.5c
;Assuming B!=0,C!=0 527cc~695cc, avg 598.5
;Overall average: 78209011/131072=596.68740081787109375
;123 bytes
ld a,b
ld hl,0
ld b,h
or a
jr z,+_
add a,a \ jr nc,$+5 \ ld h,d \ ld l,e
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
_:
push hl
ld h,b
ld l,b
ld b,a
ld a,c
ld c,b
or a
jr z,+_
add a,a \ jr nc,$+5 \ ld h,d \ ld l,e
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,c
_:
pop de
ld c,d
ld d,e
ld e,0
add hl,de
ret nc
adc a,c
ld c,a
ret nc
inc b
ret
source: Zeda's Pastebin/BC_Times_DE
C_Times_D
C_Time_D:
;Outputs:
; A is the result
; B is 0
ld b,8 ;7 7
xor a ;4 4
rlca ;4*8 32
rlc c ;8*8 64
jr nc,$+3 ;(12|11) 96|88
add a,d ;--
djnz $-6 ;13*7+8 99
ret ;10 10
;304+b (b is number of bits)
;308 is average speed.
;12 bytes
mul32, 64-bit output
With these sizes, we need to use RAM to hold intermediate values. mul16 needs to perform DE*BC => DEHL
mul32:
;;uses karatsuba multiplication
;;var_x * var_y
;;z0 holds the 64-bit result
;;708cc+6a+13b+42c +3mul
;;Avg: 2464.110153
;;Max:2839cc, 92cc faster
;;Min:2178cc (early can make it faster, though), 167cc faster
ld de,(var_x) ;\
ld bc,(var_y) ; |compute z0,z2
push bc ; | var_y
call mul16 ; |
ld (var_z0),hl ; |
ld bc,(var_y+2) ; |
ld (var_z0+2),de; |
ld de,(var_x+2) ; |
push bc ; | var_y+2
call mul16 ; |
ld (var_z2),hl ; |
ld (var_z2+2),de;/ 208cc
xor a ;\
ld hl,(var_x) ; |
ld de,(var_x+2) ; |
add hl,de ; |
rra ; |
pop de ; |
ex (sp),hl ; |
add hl,de ; |
pop bc ; |
ex de,hl ; | 109cc
push de ; |if bit0=1, add DE<<16 to result
push bc ; |
push af ; |c flag means add BC<<16 to result
call mul16 ; |
ex de,hl ; |
pop af ; |
pop bc ; |
jr nc,$+3 ; | 86+6a
add hl,bc ; |
pop bc ; |
rla ; |
jr nc,$+4 ; | 26+13b
add hl,bc ; |
adc a,0 ; |z1 = AHLDE-z2-z1
ex de,hl \ ld bc,(var_z0) \ sbc hl,bc
ex de,hl \ ld bc,(var_z0+2) \ sbc hl,bc
sbc a,0
ex de,hl \ ld bc,(var_z2) \ sbc hl,bc
ex de,hl \ ld bc,(var_z2+2) \ sbc hl,bc
sbc a,0 ; |z1 = AHLDE
ld b,h \ ld c,l ;/ z1 = ABCDE
ld hl,(var_z0+2);\
add hl,de ; |Add:
ld (var_z0+2),hl; |z2z0
ld hl,(var_z2) ; | z1
adc hl,bc ; |----
ld (var_z2),hl ; |
ret nc ; | 279+42c
ld hl,(var_z2+2); |
inc hl ; |
ld (var_z2+2),hl; |
ret ;/
source Zeda's Pastebin/mul32
BCDE_Times_A
BCDE_Times_A:
;Inputs: BC:DE,A
;Outputs: A:HL:IX is the 40-bit product, BC,DE unaffected
;503cc~831cc
;667cc average
;29 bytes
ld ix,0
ld hl,0
call +_
_:
call +_
_:
call +_
_:
add ix,ix \ adc hl,hl \ rla \ ret nc
add ix,de \ adc hl,bc \ adc a,0
ret
source: Zeda's Pastebin/BCDE_Times_A
H_Times_E
This is the fastest and smallest rolled 8-bit multiplication routine here, and it returns the full 16-bit result.
H_Times_E:
;Inputs:
; H,E
;Outputs:
; HL is the product
; D,B are 0
; A,E,C are preserved
;Size: 12 bytes
;Speed: 311+6b, b is the number of bits set in the input HL
; average is 335 cycles
; max required is 359 cycles
ld d,0 ;1600 7 7
ld l,d ;6A 4 4
ld b,8 ;0608 7 7
;
add hl,hl ;29 11*8 88
jr nc,$+3 ;3001 12*8-5b 96-5b
add hl,de ;19 11*b 11b
djnz $-4 ;10FA 13*8-5 99
;
ret ;C9 10 10
== H_Times_E == (Unrolled)
H_Times_E:
;Inputs:
; H,E
;Outputs:
; HL is the product
; D,B are 0
; A,E,C are preserved
;Size: 38 bytes
;Speed: 198+6b+9p-7s, b is the number of bits set in the input H, p is if it is odd, s is the upper bit of h
; average is 226.5 cycles (108.5 cycles saved)
; max required is 255 cycles (104 cycles saved)
ld d,0 ;1600 7 7
ld l,d ;6A 4 4
ld b,8 ;0608 7 7
;
sla h ; 8
jr nc,$+3 ;3001 12-b
ld l,e ;6B --
add hl,hl ;29 11
jr nc,$+3 ;3001 12+6b
add hl,de ;19 --
add hl,hl ;29 11
jr nc,$+3 ;3001 12+6b
add hl,de ;19 --
add hl,hl ;29 11
jr nc,$+3 ;3001 12+6b
add hl,de ;19 --
add hl,hl ;29 11
jr nc,$+3 ;3001 12+6b
add hl,de ;19 --
add hl,hl ;29 11
jr nc,$+3 ;3001 12+6b
add hl,de ;19 --
add hl,hl ;29 11
jr nc,$+3 ;3001 12+6b
add hl,de ;19 --
add hl,hl ;29 11
ret nc ;D0 11+15p
add hl,de ;19 --
ret ;C9 --
L_Squared (fast)
The following provides an optimized algorithm to square an 8-bit number, but it only returns the lower 8 bits.
L_sqrd: ;Input: L ;Output: L*L->A ;151 t-states ;37 bytes ld h,l ;First iteration, get the lowest 3 bits sla l rr h sbc a,a or l ;second iteration, get the next 2 bits ld c,a rr h sbc a,a xor l and $F8 add a,c ;third iteration, get the next 2 bits ld c,a sla l rr h sbc a,a xor l and $E0 add a,c ;fourth iteration, get the last bit ld c,a ld a,l add a,a rrc h xor h and $80 xor c neg ret
Absolute Value
Here are a handful of optimised routines for the absolute value of a number:
absHL
absHL:
bit 7,h
ret z
xor a \ sub l \ ld l,a
sbc a,a \ sub h \ ld h,a
ret
absDE
absDE:
bit 7,d
ret z
xor a \ sub e \ ld e,a
sbc a,a \ sub d \ ld d,a
ret
absBC
absBC:
bit 7,b
ret z
xor a \ sub c \ ld c,a
sbc a,a \ sub b \ ld b,a
ret
absA
absA:
or a
ret p
neg
ret
abs[reg8]
abs[reg8]:
xor a
sub [reg8]
ret m
ld [reg8],a
ret
Division
C_Div_D
This is a simple 8-bit division routine:
C_Div_D:
;Inputs:
; C is the numerator
; D is the denominator
;Outputs:
; A is the remainder
; B is 0
; C is the result of C/D
; D,E,H,L are not changed
;
ld b,8
xor a
sla c
rla
cp d
jr c,$+4
inc c
sub d
djnz $-8
ret
BC_Div_DE
This divides BC by DE, storing the result in AC, remainder in HL
BC_Div_DE:
;Inputs: BC,DE
;Outputs: DE unaffected, HL is remainder, AC is quotient, B=0
;20 bytes
;1098cc~1258cc, avg=1178cc
ld hl,0
ld a,b
ld b,16
_:
sll c \ rla \ adc hl,hl \ sbc hl,de \ jr nc,$+4 \ add hl,de \ dec c
djnz -_
ret
DEHL_Div_C
This divides the 32-bit value in DEHL by C:
DEHL_Div_C:
;Inputs:
; DEHL is a 32 bit value where DE is the upper 16 bits
; C is the value to divide DEHL by
;Outputs:
; A is the remainder
; B is 0
; C is not changed
; DEHL is the result of the division
;
ld b,32
xor a
add hl,hl
rl e \ rl d
rla
cp c
jr c,$+4
inc l
sub c
djnz $-11
ret
DEHLIX_Div_C
DEHLIX_Div_C:
;Inputs:
; DEHLIX is a 48 bit value where DE is the upper 16 bits
; C is the value to divide DEHL by
;Outputs:
; A is the remainder
; B is 0
; C is not changed
; DEHLIX is the result of the division
;
ld b,48
xor a
add ix,ix
adc hl,hl
rl e \ rl d
rla
cp c
jr c,$+5
inc ixl
sub c
djnz $-15
ret
HL_Div_C
HL_Div_C:
;Inputs:
; HL is the numerator
; C is the denominator
;Outputs:
; A is the remainder
; B is 0
; C is not changed
; DE is not changed
; HL is the quotient
;
ld b,16
xor a
add hl,hl
rla
cp c
jr c,$+4
inc l
sub c
djnz $-7
ret
HLDE_Div_C
HLDE_Div_C:
;Inputs:
; HLDE is a 32 bit value where HL is the upper 16 bits
; C is the value to divide HLDE by
;Outputs:
; A is the remainder
; B is 0
; C is not changed
; HLDE is the result of the division
;
ld b,32
xor a
sll e \ rl d
adc hl,hl
rla
cp c
jr c,$+4
inc e
sub c
djnz $-12
ret
RoundHL_Div_C
Returns the result of the division rounded to the nearest integer.
RoundHL_Div_C:
;Inputs:
; HL is the numerator
; C is the denominator
;Outputs:
; A is twice the remainder of the unrounded value
; B is 0
; C is not changed
; DE is not changed
; HL is the rounded quotient
; c flag set means no rounding was performed
; reset means the value was rounded
;
ld b,16
xor a
add hl,hl
rla
cp c
jr c,$+4
inc l
sub c
djnz $-7
add a,a
cp c
jr c,$+3
inc hl
ret
Speed Optimised HL_div_10
By adding 9 bytes to the code, we save 87 cycles: (min speed = 636 t-states)
DivHLby10:
;Inputs:
; HL
;Outputs:
; HL is the quotient
; A is the remainder
; DE is not changed
; BC is 10
ld bc,$0D0A
xor a
add hl,hl \ rla
add hl,hl \ rla
add hl,hl \ rla
add hl,hl \ rla
cp c
jr c,$+4
sub c
inc l
djnz $-7
ret
Speed Optimised EHL_Div_10
By adding 20 bytes to the routine, we actually save 301 t-states. The speed is quite fast at a minimum of 966 t-states and a max of 1002:
DivEHLby10:
;Inputs:
; EHL
;Outputs:
; EHL is the quotient
; A is the remainder
; D is not changed
; BC is 10
ld bc,$050a
xor a
sla e \ rla
sla e \ rla
sla e \ rla
sla e \ rla
cp c
jr c,$+4
sub c
inc e
djnz $-8
ld b,16
add hl,hl
rla
cp c
jr c,$+4
sub c
inc l
djnz $-7
ret
Speed Optimised DEHL_Div_10
The minimum speed is now 1350 t-states. The cost was 15 bytes, the savings were 589 t-states
DivDEHLby10:
;Inputs:
; DEHL
;Outputs:
; DEHL is the quotient
; A is the remainder
; BC is 10
ld bc,$0D0A
xor a
ex de,hl
add hl,hl \ rla
add hl,hl \ rla
add hl,hl \ rla
add hl,hl \ rla
cp c
jr c,$+4
sub c
inc l
djnz $-7
ex de,hl
ld b,16
add hl,hl
rla
cp c
jr c,$+4
sub c
inc l
djnz $-7
ret
A_Div_C (small)
This routine should only be used when C is expected to be greater than 16. In this case, the naive way is actually the fastest and smallest way:
ld b,-1
sub c
inc b
jr nc,$-2
add a,c
Now B is the quotient, A is the remainder. It takes at least 26 t-states and at most 346 if you ensure that c>16
E_div_10 (tiny+fast)
This is how it would appear inline, since it is so small at 10 bytes (and 81 t-states). It divides E by 10, returning the result in H :
e_div_10:
;returns result in H
ld d,0
ld h,d \ ld l,e
add hl,hl
add hl,de
add hl,hl
add hl,hl
add hl,de
add hl,hl
Square Root
RoundSqrtE
Returns the square root of E, rounded to the nearest integer:
;===============================================================
sqrtE:
;===============================================================
;Input:
; E is the value to find the square root of
;Outputs:
; A is E-D^2
; B is 0
; D is the rounded result
; E is not changed
; HL is not changed
;Destroys:
; C
;
xor a ;1 4 4
ld d,a ;1 4 4
ld c,a ;1 4 4
ld b,4 ;2 7 7
sqrtELoop:
rlc d ;2 8 32
ld c,d ;1 4 16
scf ;1 4 16
rl c ;2 8 32
rlc e ;2 8 32
rla ;1 4 16
rlc e ;2 8 32
rla ;1 4 16
cp c ;1 4 16
jr c,$+4 ;4 12|15 48+3x
inc d ;-- -- --
sub c ;-- -- --
djnz sqrtELoop ;2 13|8 47
cp d ;1 4 4
jr c,$+3 ;3 12|11 12|11
inc d ;-- -- --
ret ;1 10 10
;===============================================================
;Size : 29 bytes
;Speed : 347+3x cycles plus 1 if rounded down
; x is the number of set bits in the result.
;===============================================================
SqrtE
This returns the square root of E (rounded down).
;===============================================================
sqrtE:
;===============================================================
;Input:
; E is the value to find the square root of
;Outputs:
; A is E-D^2
; B is 0
; D is the result
; E is not changed
; HL is not changed
;Destroys:
; C=2D+1 if D is even, 2D-1 if D is odd
xor a ;1 4 4
ld d,a ;1 4 4
ld c,a ;1 4 4
ld b,4 ;2 7 7
sqrtELoop:
rlc d ;2 8 32
ld c,d ;1 4 16
scf ;1 4 16
rl c ;2 8 32
rlc e ;2 8 32
rla ;1 4 16
rlc e ;2 8 32
rla ;1 4 16
cp c ;1 4 16
jr c,$+4 ;4 12|15 48+3x
inc d ;-- -- --
sub c ;-- -- --
djnz sqrtELoop ;2 13|8 47
ret ;1 10 10
;===============================================================
;Size : 25 bytes
;Speed : 332+3x cycles
; x is the number of set bits in the result. This will not
; exceed 4, so the range for cycles is 332 to 344. To put this
; into perspective, under the slowest conditions (4 set bits
; in the result at 6MHz), this can execute over 18000 times
; in a second.
;===============================================================
SqrtHL
This returns the square root of HL (rounded down). It is faster than division, interestingly:
SqrtHL4:
;39 bytes
;Inputs:
; HL
;Outputs:
; BC is the remainder
; D is not changed
; E is the square root
; H is 0
;Destroys:
; A
; L is a value of either {0,1,4,5}
; every bit except 0 and 2 are always zero
ld bc,0800h ;3 10 ;10
ld e,c ;1 4 ;4
xor a ;1 4 ;4
SHL4Loop: ; ;
add hl,hl ;1 11 ;88
rl c ;2 8 ;64
adc hl,hl ;2 15 ;120
rl c ;2 8 ;64
jr nc,$+4 ;2 7|12 ;96+3y ;y is the number of overflows. max is 2
set 0,l ;2 8 ;--
ld a,e ;1 4 ;32
add a,a ;1 4 ;32
ld e,a ;1 4 ;32
add a,a ;1 4 ;32
bit 0,l ;2 8 ;64
jr nz,$+5 ;2 7|12 ;144-6y
sub c ;1 4 ;32
jr nc,$+7 ;2 7|12 ;96+15x ;number of bits in the result
ld a,c ;1 4 ;
sub e ;1 4 ;
inc e ;1 4 ;
sub e ;1 4 ;
ld c,a ;1 4 ;
djnz SHL4Loop ;2 13|8 ;99
bit 0,l ;2 8 ;8
ret z ;1 11|19 ;11+8z
inc b ;1 ;
ret ;1 ;
;1036+15x-3y+8z
;x is the number of set bits in the result
;y is the number of overflows (max is 2)
;z is 1 if 'b' is returned as 1
;max is 1154 cycles
;min is 1032 cycles
SqrtL
This returns the square root of L, rounded down:
SqrtL:
;Inputs:
; L is the value to find the square root of
;Outputs:
; C is the result
; B,L are 0
; DE is not changed
; H is how far away it is from the next smallest perfect square
; L is 0
; z flag set if it was a perfect square
;Destroyed:
; A
ld bc,400h ; 10 10
ld h,c ; 4 4
sqrt8Loop: ;
add hl,hl ;11 44
add hl,hl ;11 44
rl c ; 8 32
ld a,c ; 4 16
rla ; 4 16
sub a,h ; 4 16
jr nc,$+5 ;12|19 48+7x
inc c
cpl
ld h,a
djnz sqrt8Loop ;13|8 47
ret ;10 10
;287+7x
;19 bytes
ConvFloat (or ConvOP1)
This converts a floating point number pointed to by DE to a 16 bit-value. This is like bcall(_ConvOP1) without the limit of 9999 and a bit more flexible (since the number doesn't need to be at OP1):
ConvOP1:
;;Output: HL is the 16-bit result.
ld de,OP1
ConvFloat:
;;Input: DE points to the float.
;;Output: HL is the 16-bit result.
;;Errors: DataType if the float is negative or complex
;; Domain if the integer exceeds 16 bits.
;;Timings: Assume no errors were called.
;; Input is on:
;; (0,1) => 59cc Average=59
;; 0 or [1,10) => 120cc or 129cc =124.5
;; [10,100) => 176cc or 177cc =176.5
;; [100,1000) => 309cc, 310cc, 318cc, or 319cc. =314
;; [1000,10000) => 376cc to 378cc =377
;; [10000,65536) => 514cc to 516cc, or 523cc to 525cc =519.5
;;Average case: 496.577178955078125cc
;;vs 959.656982421875cc
;;87 bytes
ld a,(de)
or a
jr nz,ErrDataType
inc de
ld hl,0
ld a,(de)
inc de
sub 80h
ret c
jr z,final
cp 5
jp c,enterloop
ErrDomain:
;Throws a domain error.
bcall(_ErrDomain)
ErrDataType:
;Throws a data type error.
bcall(_ErrDataType)
loop:
ld a,b
ld b,h
ld c,l
add hl,hl
add hl,bc
add hl,hl
add hl,hl
add hl,hl
add hl,bc
add hl,hl
add hl,hl
enterloop:
ld b,a
ex de,hl
ld a,(hl) \ and $F0 \ rra \ ld c,a \ rra \ rra \ sub c \ add a,(hl)
inc hl
ex de,hl
add a,l
ld l,a
jr nc,$+3
inc h
dec b
ret z
djnz loop
ld b,h
ld c,l
xor a
;check overflow in this mul by 10!
add hl,hl \ adc a,a
add hl,hl \ adc a,a
add hl,bc \ adc a,0
add hl,hl \ adc a,a
jr nz,ErrDomain
final:
ld a,(de)
rrca
rrca
rrca
rrca
and 15
add a,l
ld l,a
ret nc
inc h
ret nz
jr ErrDomain
source: Cemetech/Useful Routines
ConvStr16
This will convert a string of base-10 digits to a 16-bit value. Useful for parsing numbers in a string:
;===============================================================
ConvRStr16:
;===============================================================
;Input:
; DE points to the base 10 number string in RAM.
;Outputs:
; HL is the 16-bit value of the number
; DE points to the byte after the number
; BC is HL/10
; z flag reset (nz)
; c flag reset (nc)
;Destroys:
; A (actually, add 30h and you get the ending token)
;Size: 23 bytes
;Speed: 104n+42+11c
; n is the number of digits
; c is at most n-2
; at most 595 cycles for any 16-bit decimal value
;===============================================================
ld hl,0 ; 10 : 210000
ConvLoop: ;
ld a,(de) ; 7 : 1A
sub 30h ; 7 : D630
cp 10 ; 7 : FE0A
ret nc ;5|11 : D0
inc de ; 6 : 13
;
ld b,h ; 4 : 44
ld c,l ; 4 : 4D
add hl,hl ; 11 : 29
add hl,hl ; 11 : 29
add hl,bc ; 11 : 09
add hl,hl ; 11 : 29
;
add a,l ; 4 : 85
ld l,a ; 4 : 6F
jr nc,ConvLoop ;12|23: 30EE
inc h ; --- : 24
jr ConvLoop ; --- : 18EB
gcdHL_DE
This computes the Greatest Common Divisor of HL and DE, using the binary gcd algorithm.
gcdHL_DE:
;gcd(HL,DE)->HL
;binary GCD algorithm
ld a,h \ or l \ ret z
ex de,hl
ld a,h \ or l \ ret z
sbc hl,de
add hl,de
ret z
ld b,1
ld a,e \ or l \ rra \ jr c,+_
inc b
rr h \ rr l
rr d \ rr e
ld a,e \ or l \ rra \ jr nc,$-12
_:
srl h \ rr l \ jr nc,$-4 \ adc hl,hl
ex de,hl
_:
srl h \ rr l \ jr nc,$-4 \ adc hl,hl
xor a \ sbc hl,de
jr z,+_
jr nc,-_ \ sub l \ ld l,a \ sbc a,a \ sub h \ ld h,a
jp -_-1
_:
ex de,hl
dec b
ret z
add hl,hl
djnz $-1
ret
Modulus
DEHL_mod_3
See below.
HLDE_mod_3
See below.
HL_mod_3
See below.
A_mod_3
Computes A mod 3 (essentially, the remainder of A after division by 3):
DEHL_mod_3:
HLDE_mod_3:
;Inputs:
; HL:DE unsigned integer
;Outputs:
; A = HLDE mod 3
; Z flag is set if divisible by 3
;Destroys:
; C
; 27 bytes, 103cc,118cc,104cc,119cc, avg=114.75cc
add hl,de
jr nc,$+3
dec hl
HL_mod_3:
;Inputs:
; HL unsigned integer
;Outputs:
; A = HL mod 3
; Z flag is set if divisible by 3
;Destroys:
; C
; 23 bytes, 80cc or 95cc, avg 91.25cc
ld a,h
add a,l
sbc a,0 ;conditional decrement
A_mod_3:
;Inputs:
; A unsigned integer
;Outputs:
; A = A mod 3
; Z flag is set if divisible by 3
;Destroys:
; C
; 19 bytes, 65cc or 80cc, avg=76.25cc
ld c,a ;add nibbles
rrca / rrca / rrca / rrca
add a,c
adc a,0 ;n mod 15 (+1) in both nibbles
ld c,a ;add half nibbles
rrca / rrca
add a,c
adc a,1
ret z
and 3
dec a
ret
source: Cemetech/Fast 8-bit mod 3
A_mod_10:
This is not a typical method used, but it is small and fast at 196 to 201 t-states, 12 bytes
ld bc,05A0h
Loop:
sub c
jr nc,$+3
add a,c
srl c
djnz Loop
ret
Random Numbers
rand8_LCG
This is one of many variations of PRNGs. This routine is not particularly useful for many games, but is fairly useful for shuffling a deck of cards. It uses SMC, but that can be fixed by defining randSeed elsewhere and using ld a,(randSeed) at the beginning.
rand8_LCG:
;f(n+1)=13f(n)+83
;97 cycles
randSeed=$+1
ld a,3
ld c,a
add a,a
add a,c
add a,a
add a,a
add a,c
add a,83
ld (randSeed),a
ret
rand16_LCG
Similar to the rand8_LCG, this generates a a sequence of pseudo-random values that has a cycle of 65536 (so it will hit every single 16-bit integer):
rand16_LCG:
;f(n+1)=241f(n)+257 ;65536
;181 cycles, add 17 if called
;Outputs:
; BC was the previous pseudorandom value
; HL is the next pseudorandom value
;Notes:
; You can also use B,C,H,L as pseudorandom 8-bit values
; this will generate all 8-bit values
randSeed=$+1
ld hl,235
ld c,l
ld b,h
add hl,hl
add hl,bc
add hl,hl
add hl,bc
add hl,hl
add hl,bc
add hl,hl
add hl,hl
add hl,hl
add hl,hl
add hl,bc
inc h
inc hl
ld (randSeed),hl
ret
rand; best quality:speed
This routine uses a combined LFSR and LCG to offer an extremely fast and proven high quality pseudo random numbers.
rand:
;;Tested and passes all CAcert tests
;;Uses a very simple 32-bit LCG and 32-bit LFSR
;;it has a period of 18,446,744,069,414,584,320
;;roughly 18.4 quintillion.
;;291cc
;;58 bytes
seed1_0=$+1
ld hl,12345
seed1_1=$+1
ld de,6789
ld b,h
ld c,l
add hl,hl \ rl e \ rl d
add hl,hl \ rl e \ rl d
inc l
add hl,bc
ld (seed1_0),hl
ld hl,(seed1_1)
adc hl,de
ld (seed1_1),hl
ex de,hl
seed2_0=$+1
ld hl,9876
seed2_1=$+1
ld bc,54321
add hl,hl \ rl c \ rl b
ld (seed2_1),bc
sbc a,a
and %11000101
xor l
ld l,a
ld (seed2_0),hl
ex de,hl
add hl,bc
ret
source: Cemetech/Useful Routines
rand; very fast
The cycle for this is more limited, but is still quite large and well suited to games.
rand:
;collab with Runer112
;;Output:
;; HL is a pseudo-random int
;; A and BC are also, but much weaker and smaller cycles
;; Preserves DE
;;148cc, super fast
;;26 bytes
;;period length: 4,294,901,760
seed1=$+1
ld hl,9999
ld b,h
ld c,l
add hl,hl
add hl,hl
inc l
add hl,bc
ld (seed1),hl
seed2=$+1
ld hl,987
add hl,hl
sbc a,a
and %00101101
xor l
ld l,a
ld (seed2),hl
add hl,bc
ret
source: Zeda's Pastebin/rand
randInt
This returns a random integer on [0,A-1].
rand:
;;Input: A is the range.
;;Output: Returns in A a random number from 0 to B-1.
;; B=0
;; DE is not changed
;;Destroys:
;; HL
;;Speed:
;; 322cc to 373cc, 347.5cc average
push af
call rand
ex de,hl
pop af
ld hl,0
ld b,h
add a,a \ jr nc,$+5 \ ld h,d \ ld l,e
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ jr nc,$+4 \ add hl,de \ adc a,b
add hl,hl \ rla \ ret nc \ add hl,de \ adc a,b
ret
source: Zeda's Pastebin/rand
Fixed Point Math
Fixed Point numbers are similar to Floating Point numbers in that they give the user a way to work with non-integers. For some terminology, an 8.8 Fixed Point number is 16 bits where the upper 8 bits is the integer part, the lower 8 bits is the fractional part. Both Floating Point and Fixed Point are abbreviated 'FP', but one can tell if Fixed Point is being referred to by context. The way one would interpret an 8.8 FP number would be to take the upper 8 bits as the integer part and divide the lower 8-bits by 256 (2[sup]8[/sup]) so if HL is an 8.8 FP number that is $1337, then its value is 19+55/256 = 19.21484375. In most cases, integers are enough for working in Z80 Assembly, but if that doesn't work, you will rarely need more than 16.16 FP precision (which is 32 bits in all). FP algorithms are generally pretty similar to their integer counterparts, so it isn't too difficult to convert.
FPLog88
This is an 8.8 fixed point natural log routine. This is extremely accurate. In the very worst case, it is off by 2/256, but on average, it is off by less than 1/256 (the smallest unit for an 8.8 FP number).
FPLog88:
;Input:
; HL is the 8.8 Fixed Point input. H is the integer part, L is the fractional part.
;Output:
; HL is the natural log of the input, in 8.8 Fixed Point format.
ld a,h
or l
dec hl
ret z
inc hl
push hl
ld b,15
add hl,hl
jr c,$+4
djnz $-3
ld a,b
sub 8
jr nc,$+4
neg
ld b,a
pop hl
push af
jr nz,lnx
jr nc,$+7
add hl,hl
djnz $-1
jr lnx
sra h
rr l
djnz $-4
lnx:
dec h ;subtract 1 so that we are doing ln((x-1)+1) = ln(x)
push hl ;save for later
add hl,hl ;we are doing the 4x/(4+4x) part
add hl,hl
ld d,h
ld e,l
inc h
inc h
inc h
inc h
call FPDE_Div_HL ;preserves DE, returns AHL as the 16.8 result
pop de ;DE is now x instead of 4x
inc h ;now we are doing x/(3+Ans)
inc h
inc h
call FPDE_Div_HL
inc h ;now we are doing x/(2+Ans)
inc h
call FPDE_Div_HL
inc h ;now we are doing x/(1+Ans)
call FPDE_Div_HL ;now it is computed to pretty decent accuracy
pop af ;the power of 2 that we divided the initial input by
ret z ;if it was 0, we don't need to add/subtract anything else
ld b,a
jr c,SubtLn2
push hl
xor a
ld de,$B172 ;this is approximately ln(2) in 0.16 FP format
ld h,a
ld l,a
add hl,de
jr nc,$+3
inc a
djnz $-4
pop de
rl l ;returns c flag if we need to round up
ld l,h
ld h,a
jr nc,$+3
inc hl
add hl,de
ret
SubtLn2:
ld de,$00B1
or a
sbc hl,de
djnz $-3
ret
FPDE_Div_HL:
;Inputs:
; DE,HL are 8.8 Fixed Point numbers
;Outputs:
; DE is preserved
; AHL is the 16.8 Fixed Point result (rounded to the least significant bit)
di
push de
ld b,h
ld c,l
ld a,16
ld hl,0
Loop1:
sla e
rl d
adc hl,hl
jr nc,$+8
or a
sbc hl,bc
jp incE
sbc hl,bc
jr c,$+5
incE:
inc e
jr $+3
add hl,bc
dec a
jr nz,Loop1
ex af,af'
ld a,8
Loop2:
ex af,af'
sla e
rl d
rl a
ex af,af'
add hl,hl
jr nc,$+8
or a
sbc hl,bc
jp incE_2
sbc hl,bc
jr c,$+5
incE_2:
inc e
jr $+3
add hl,bc
dec a
jr nz,Loop2
;round
ex af,af'
add hl,hl
jr c,$+6
sbc hl,de
jr c,$+9
inc e
jr nz,$+6
inc d
jr nz,$+3
inc a
ex de,hl
pop de
ret
FPDE_Div_BC88
This performs Fixed Point division for DE/BC where DE and BC are 8.8 FP numbers. This returns a little extra precision for the integer part (16-bit integer part, 8-bit fractional part).
FPDE_Div_BC88:
;Inputs:
; DE,BC are 8.8 Fixed Point numbers
;Outputs:
; ADE is the 16.8 Fixed Point result (rounded to the least significant bit)
di
ld a,16
ld hl,0
Loop1:
sla e
rl d
adc hl,hl
jr nc,$+8
or a
sbc hl,bc
jp incE
sbc hl,bc
jr c,$+5
incE:
inc e
jr $+3
add hl,bc
dec a
jr nz,Loop1
ex af,af'
ld a,8
Loop2:
ex af,af'
sla e
rl d
rla
ex af,af'
add hl,hl
jr nc,$+8
or a
sbc hl,bc
jp incE_2
sbc hl,bc
jr c,$+5
incE_2:
inc e
jr $+3
add hl,bc
dec a
jr nz,Loop2
;round
ex af,af'
add hl,hl
jr c,$+5
sbc hl,de
ret c
inc e
ret nz
inc d
ret nz
inc a
ret
Log_2_88
These computes log base 2 of the fixed point 8.8 number. This is much faster and smaller than the natural log routine above.
(size optimised)
Log_2_88_size:
;Inputs:
; HL is an unsigned 8.8 fixed point number.
;Outputs:
; HL is the signed 8.8 fixed point value of log base 2 of the input.
;Example:
; pass HL = 3.0, returns 1.58203125 (actual is ~1.584962501...)
;averages about 39 t-states slower than original
;62 bytes
ex de,hl
ld hl,0
ld a,d
ld c,8
or a
jr z,DE_lessthan_1
srl d
jr z,logloop-1
inc l
rr e
jr $-7
DE_lessthan_1:
ld a,e
dec hl
or a
ret z
inc l
dec l
add a,a
jr nc,$-2
ld e,a
inc d
logloop:
add hl,hl
push hl
ld h,d
ld l,e
ld a,e
ld b,8
add hl,hl
rla
jr nc,$+5
add hl,de
adc a,0
djnz $-7
ld e,h
ld d,a
pop hl
rr a ;this is right >_>
jr z,$+7
srl d
rr e
inc l
dec c
jr nz,logloop
ret
(speed optimised)
Log_2_88_speed:
;Inputs:
; HL is an unsigned 8.8 fixed point number.
;Outputs:
; HL is the signed 8.8 fixed point value of log base 2 of the input.
;Example:
; pass HL = 3.0, returns 1.58203125 (actual is ~1.584962501...)
;saves at least 688 t-states over regular (about 17% speed boost)
;98 bytes
ex de,hl
ld hl,0
ld a,d
ld c,8
or a
jr z,DE_lessthan_1
srl d
jr z,logloop-1
inc l
rr e
jp $-7
DE_lessthan_1:
ld a,e
dec hl
or a
ret z
inc l
dec l
add a,a
jr nc,$-2
ld e,a
inc d
logloop:
add hl,hl
push hl
ld h,d
ld l,e
ld a,e
ld b,7
add hl,hl
rla
jr nc,$+3
add hl,de
add hl,hl
rla
jr nc,$+3
add hl,de
add hl,hl
rla
jr nc,$+3
add hl,de
add hl,hl
rla
jr nc,$+3
add hl,de
add hl,hl
rla
jr nc,$+3
add hl,de
add hl,hl
rla
jr nc,$+3
add hl,de
add hl,hl
rla
jr nc,$+5
add hl,de
adc a,0
add hl,hl
rla
jr nc,$+5
add hl,de
adc a,0
ld e,h
ld d,a
pop hl
rr a
jr z,$+7
srl d
rr e
inc l
dec c
jr nz,logloop
ret
(balanced)
(this only saves about 40 cycles over the size optimised one)
Log_2_88:
;Inputs:
; HL is an unsigned 8.8 fixed point number.
;Outputs:
; HL is the signed 8.8 fixed point value of log base 2 of the input.
;Example:
; pass HL = 3.0, returns 1.58203125 (actual is ~1.584962501...)
;70 bytes
ex de,hl
ld hl,0
ld a,d
ld c,8
or a
jr z,DE_lessthan_1
srl d
jr z,logloop-1
inc l
rr e
jp $-7
DE_lessthan_1:
ld a,e
dec hl
or a
ret z
inc l
dec l
add a,a
jr nc,$-2
ld e,a
inc d
logloop:
add hl,hl
push hl
ld h,d
ld l,e
ld a,e
ld b,7
add hl,hl
rla
jr nc,$+3
add hl,de
djnz $-5
adc a,0
add hl,hl
rla
jr nc,$+5
add hl,de
adc a,0
ld e,h
ld d,a
pop hl
rr a
jr z,$+7
srl d
rr e
inc l
dec c
jr nz,logloop
ret
