z80:Math Routines

From Learn @ Cemetech
Revision as of 06:18, 5 February 2016 by KermMartian (talk | contribs) (Created page with "If 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 n...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigationJump to search

If 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.


Multiplication

DE_Times_A

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 and A are factors
   ;Outputs:
   ;     A is not changed
   ;     B is 0
   ;     C is not changed
   ;     DE is not changed
   ;     HL is the product
   ;Time:
   ;     342+6x
   ;
        ld b,8          ;7           7
        ld hl,0         ;10         10
          add hl,hl     ;11*8       88
          rlca          ;4*8        32
          jr nc,$+3     ;(12|18)*8  96+6x
            add hl,de   ;--         --
          djnz $-5      ;13*7+8     99
        ret             ;10         10


== DE_Times_A ==_Unrolled 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 and A are factors
   ;Outputs:
   ;     A is unchanged
   ;     BC is unchanged
   ;     DE is unchanged
   ;     HL is the product
   ;speed: min 199 cycles
   ;       max 261 cycles
   ;        212+6b cycles +15 if odd, -11 if non-negative
   ;=====================================Cycles====================
   ;1
        ld hl,0            ;210000           10      10
        rlca                   ;07             4
        jr nc,$+5 \ ld h,d \ ld e,l  ;3002626B   12+14p
   ;2
        add hl,hl              ;29            --
        rlca                   ;07             4
        jr nc,$+3 \ add hl,de  ;300119     12+6b
   ;3
        add hl,hl              ;29            11
        rlca                   ;07             4
        jr nc,$+3 \ add hl,de  ;300119     12+6b
   ;4
        add hl,hl              ;29            11
        rlca                   ;07             4
        jr nc,$+3 \ add hl,de  ;300119     12+6b
   ;5
        add hl,hl              ;29            11
        rlca                   ;07             4
        jr nc,$+3 \ add hl,de  ;300119     12+6b
   ;6
        add hl,hl              ;29            11
        rlca                   ;07             4
        jr nc,$+3 \ add hl,de  ;300119     12+6b
   ;7
        add hl,hl              ;29            11
        rlca                   ;07             4
        jr nc,$+3 \ add hl,de  ;300119     12+6b
   ;8
        add hl,hl              ;29            11
        rlca                   ;07             4
        ret nc                 ;D0         11-6b
        add hl,de              ;300119     12+6b
        ret                    ;C9            10


A_Times_DE

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.


   A_Times_DE:
   ;211 for times 1
   ;331 tops
   ;Outputs:
   ;     HL is the product
   ;     B is 0
   ;     A,C,DE are not changed
   ;     z flag set
   ;
        ld hl,0
        or a
        ld b,h    ;remove this if you don't need b=0 for output. Saves 4 cycles, 1 byte
        ret z
        ld b,9
          rlca
          dec b
          jr nc,$-2
   Loop1:
        add hl,de
   Loop2:
        dec b
        ret z
        add hl,hl
        rlca
        jp c,Loop1      ;21|20
        jp Loop2
   ;22 bytes


DE_Times_BC

   DE_Times_BC:
   ;Inputs:
   ;     DE and BC are factors
   ;Outputs:
   ;     A is 0
   ;     BC is not changed
   ;     DEHL is the product
   ;
          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

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


D_Times_C

This routine returns a 16-bit value with C as the overflow.


   ;Returns a 16-bit result
   ;
   ;===============================================================
   D_Times_C:
   ;===============================================================
   ;Inputs:
   ;     D and C are factors
   ;Outputs:
   ;     A is the product (lower 8 bits)
   ;     B is 0
   ;     C is the overflow (upper 8 bits)
   ;     DE, HL are not changed
   ;Size:  15 bytes
   ;Speed: 312+12z-y
   ;     See Speed Summary below
   ;===============================================================
        xor a         ;This is an optimised way to set A to zero. 4 cycles, 1 byte.
        ld b,8        ;Number of bits in E, so number of times we will cycle through
   Loop:
        add a,a       ;We double A, so we shift it left. Overflow goes into the c flag.
        rl c          ;Rotate overflow in and get the next bit of C in the c flag
        jr nc,$+6     ;If it is 0, we don't need to add anything to A
          add a,d     ;Since it was 1, we do A+1*D
          jr nc,$+3   ;Check if there was overflow
            inc c     ;If there was overflow, we need to increment E
        djnz Loop     ;Decrements B, if it isn't zero yet, jump back to Loop:
        ret
   
   
   
   ;Speed Summary
   ;     xor a         ;  4
   ;     ld b,8        ;  7
   ;Loop:              ;
   ;     add a,a       ; 32
   ;     rl c          ; 64
   ;     jr nc,$+6     ; 96+12z-y   z=number of bits in C, y is overflow, so at most one less than z
   ;       add a,d     ; --
   ;       jr nc,$+3   ; --
   ;         inc c     ; --
   ;     djnz Loop     ; 99
   ;     ret           ; 10
   ;
   ;312+12z-y
   ;     z is the number of bits in C
   ;     y is the number of overflows in the branch. This is at most z-1.
   ;Max: 415 cycles


DEHL_Mul_IXBC

32-bit multiplication

   ;***********
   ;**  RAM  **
   ;***********
   ;This uses Self Modifying Code to get a speed boost. This must be
   ;used from RAM.
   DEHL_Mul_IXBC:
   ;Inputs:
   ;    DEHL
   ;    IXBC
   ;Outputs:
   ;    AF is the return address
   ;    IXHLDEBC is the result
   ;    4 bytes at TempWord1 contain the upper 32-bits of the result
   ;    4 bytes at TempWord3 contain the value of the input stack values
   ;Comparison/Perspective:
   ;    At 6MHz, this can be executed at the slowest more than 726
   ;    times per second.
   ;    At 15MHz, this can be executed at the slowest more than
   ;    1815 times per second.
   ;===============================================================
        ld (TempWord1),hl
        ld (TempWord2),de
        ld (TempWord3),bc
        ld (TempWord4),ix
        ld a,32
        ld bc,0
        ld d,b \ ld e,b
   Mult32StackLoop:
        sla c \ rl b \ rl e \ rl d
        .db 21h                     ;ld hl,**
   TempWord1:
        .dw 0
        adc hl,hl
        .db 21h                     ;ld hl,**
   TempWord2:
        .dw 0
        adc hl,hl
        jr nc,OverFlowDone
          .db 21h
   TempWord3:
          .dw 0
          add hl,bc
          ld b,h \ ld c,l
          .db 21h
   TempWord4:
          .dw 0
          adc hl,de
          ex de,hl
          jr nc,OverFlowDone
            ld hl,TempWord1
            inc (hl) \ jr nz,OverFlowDone
            inc hl \ inc (hl) \ jr nz,OverFlowDone
            inc hl \ inc (hl) \ jr nz,OverFlowDone
            inc hl \ inc (hl)
   OverFlowDone:
        dec a
        jr nz,Mult32StackLoop
        ld ix,(TempWord2)
        ld hl,(TempWord1)
        ret


DEHL_Times_A

   ;===============================================================
   DEHL_Times_A:
   ;===============================================================
   ;Inputs:
   ;     DEHL is a 32 bit factor
   ;     A is an 8 bit factor
   ;Outputs:
   ;     interrupts disabled
   ;     BC is not changed
   ;     AHLDE is the 40-bit result
   ;     D'E' is the lower 16 bits of the input
   ;     H'L' is the lower 16 bits of the output
   ;     B' is 0
   ;     C' is not changed
   ;     A' is not changed
   ;===============================================================
        di
        push hl
        or a
        sbc hl,hl
        exx
        pop de
        sbc hl,hl
        ld b,8
   mul32Loop:
          add hl,hl
          rl e \ rl d
          add a,a
          jr nc,$+8
            add hl,de
            exx
            adc hl,de
            inc a
            exx
          djnz mul32Loop
          push hl
          exx
          pop de
          ret

H_Times_E

This is the fastest and smallest rolle 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:
        bit 7,h
        ret z
        xor a \ sub l \ ld l,a
        sbc a,a \ sub h \ ld h,a
        ret
   absDE:
        bit 7,d
        ret z
        xor a \ sub e \ ld e,a
        sbc a,a \ sub d \ ld d,a
        ret
   absBC:
        bit 7,b
        ret z
        xor a \ sub c \ ld c,a
        sbc a,a \ sub b \ ld b,a
        ret
   absA:
        or a
        ret p
        neg         ;or you can use      cpl \ inc 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

DE_Div_BC

This divides DE by BC, storing the result in DE, remainder in HL

   DE_Div_BC:          ;1281-2x, x is at most 16
        ld a,16        ;7
        ld hl,0        ;10
        jp $+5         ;10
   DivLoop:
          add hl,bc    ;--
          dec a        ;64
          ret z        ;86
   
          sla e        ;128
          rl d         ;128
          adc hl,hl    ;240
          sbc hl,bc    ;240
          jr nc,DivLoop ;23|21
          inc e        ;--
          jp DivLoop+1


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: [code]

    ld b,-1
    sub c
    inc b
    jr nc,$-2
    add a,c

[/code] 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



ConvFPAtHL

This converts a floating point number pointed to by HL 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 HL):


   ConvDecAtHL:
   ;Inputs:
   ;     HL points to the FP number to convert
   ;Outputs:
   ;      A is the 8-bit result
   ;      B is 0
   ;     DE is the 16-bit result
   ;     HL is incremented by 9
   ;     c flag reset
   ;     z flag reset
   ;Destroys:
   ;     C
        ld a,(hl)
        ld b,a
        inc hl
        ld a,(hl)
        ld de,8
        add hl,de
        rla \ jr c,$+6
          ld b,d
          ld e,d
          ld a,e
          ret
        push hl
        ccf
        sbc hl,de
        ld e,d
        rra \ and 0Fh
        push bc
        ld b,a \ inc b
          inc hl
          ld c,(hl)
          call ConvDecSubLoop
          dec b \ jr z,$+7
          call ConvDecSubLoop
          djnz $-11
          pop af
          pop hl
          or b \ ld a,e
          ret p
          xor a
          sub e
          ld e,a
          sbc a,a
          sub d
          ld d,a
          ld a,e
          ret
   ConvDecSubLoop:
          push hl
          ld h,d \ ld l,e
          add hl,hl
          add hl,hl
          add hl,de
          add hl,hl
          ex de,hl
          xor a
          sla c \ rla
          sla c \ rla
          sla c \ rla
          sla c \ rla
          add a,e
          pop hl
          ld e,a
          ret nc
          inc d
          ret


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_BC

This computes the Greatest Common Divisor of HL and BC:

   GCDHL_BC:
   ;Inputs:
   ;     HL,BC
   ;Outputs:
   ;     A is 0
   ;     BC,DE are both the GCD
   ;     HL is 0
        ld a,16
        ld de,0
        add hl,hl
        ex de,hl
        adc hl,hl
        or a
        sbc hl,bc
        jr c,$+3
        add hl,bc
        ex de,hl
        dec a
        jr nz,GCDHL_BC+5
        ld h,b
        ld l,c
        ld b,d
        ld c,e
        ld a,d \ or e
        jr nz,GCDHL_BC
        ret

Modulus

L_mod_3

Computes L mod 3 (essentially, the remainder of L after division by 3): L_mod_3:

   ;Outputs:
   ;     HL is preserved
   ;     A is the remainder
   ;     destroys DE,BC
   ;     z flag if divisible by 3, else nz
        ld bc,030Fh
   ;Now we need to add the upper and lower nibble in a
        ld a,l
        and c
        ld e,a
        ld a,l
        rlca
        rlca
        rlca
        rlca
        and c
   
        add a,e
        sub c
        jr nc,$+3
        add a,c
   ;add the lower half nibbles    ;at this point, we have l_mod_15
   
        ld d,a
        sra d
        sra d
        and b
        add a,d
        sub b
        ret nc
        add a,b
        ret
   ;at most 117 cycles, at least 108, 28 bytes

HL_mod_3

   HL_mod_3:
   ;Outputs:
   ;     Preserves HL
   ;     A is the remainder
   ;     destroys DE,BC
   ;     z flag if divisible by 3, else nz
        ld bc,030Fh
        ld a,h
        add a,l
        sbc a,0   ;conditional decrement
   ;Now we need to add the upper and lower nibble in a
        ld d,a
        and c
        ld e,a
        ld a,d
        rlca
        rlca
        rlca
        rlca
        and c
   
        add a,e
        sub c
        jr nc,$+3
        add a,c
   ;add the lower half nibbles
   
        ld d,a
        sra d
        sra d
        and b
        add a,d
        sub b
        ret nc
        add a,b
        ret
   ;at most 132 cycles, at least 123

DEHL_mod_3

Same as HLDE_mod_3

HLDE_mod_3

   DEHL_mod_3:
   HLDE_mod_3:
   ;Outputs:
   ;     A is the remainder
   ;     destroys DE,BC
   ;     z flag if divisible by 3, else nz
        ld bc,030Fh
        add hl,de
        jr nc,$+3
        dec hl
        ld a,h
        add a,l
        sbc a,0   ;conditional decrement
   ;Now we need to add the upper and lower nibble in a
        ld d,a
        and c
        ld e,a
        ld a,d
        rlca
        rlca
        rlca
        rlca
        and c
   
        add a,e
        sub c
        jr nc,$+3
        add a,c
   ;add the lower half nibbles
   
        ld d,a
        sra d
        sra d
        and b
        add a,d
        sub b
        ret nc
        add a,b
        ret
   ;at most 156 cycles, at least 146


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


PseudoRandByte_0

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.

   PseudoRandByte:
   ;f(n+1)=13f(n)+83
   ;97 cycles
        .db 3Eh     ;start of ld a,*
   randSeed:
        .db 0
        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


PseudoRandWord_0:

Similar to the PseudoRandByte_0, this generates a a sequence of pseudo-random values that has a cycle of 65536 (so it will hit every single number):

   PseudoRandWord:
   ;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
        .db 21h    ;start of ld hl,**
   randSeed:
        .dw 0
        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

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