16-Bit Xorshift Pseudorandom Numbers in Z80 Assembly

Xorshift is a simple, fast pseudorandom number generator developed by George Marsaglia. The generator combines three xorshift operations where a number is exclusive-ored with a shifted copy of itself:

/* 16-bit xorshift PRNG */

unsigned xs = 1;

unsigned xorshift( )
{
    xs ^= xs << 7;
    xs ^= xs >> 9;
    xs ^= xs << 8;
    return xs;
}

There are 60 shift triplets with the maximum period 2¹⁶-1. Four triplets pass a series of lightweight randomness tests including randomly plotting various n × n matrices using the high bits, low bits, reversed bits, etc. These are: 6, 7, 13; 7, 9, 8; 7, 9, 13; 9, 7, 13.

7, 9, 8 is the most efficient when implemented in Z80, generating a number in 86 cycles. For comparison the example in C takes approx ~1200 cycles when compiled with HiSoft C v1.3.

; 16-bit xorshift pseudorandom number generator
; 20 bytes, 86 cycles (excluding ret)

; returns   hl = pseudorandom number
; corrupts   a

xrnd:
  ld hl,1       ; seed must not be 0

  ld a,h
  rra
  ld a,l
  rra
  xor h
  ld h,a
  ld a,l
  rra
  ld a,h
  rra
  xor l
  ld l,a
  xor h
  ld h,a

  ld (xrnd+1),hl

  ret

A Fast Z80 Integer Square Root

A project I'm working on needs a fast square root but I couldn't find anything suitable online. After implementing several versions of the bit-by-bit algorithm I discovered the following code is particularly efficient when unrolled:

/* Return the square root of numb */

int isqrt( numb )
{
    int root = 0, bit = 04000h;
    while ( bit != 0 )
    {
        if ( numb >= root + bit )
        {
            numb = numb - root - bit;
            root = root / 2 + bit;
        }
        else
            root = root / 2;
        bit = bit / 4;
    }
    return root;
}

First Make It Work

The looping version is small but clunky. It spends almost 400 t-states shifting bits. We'll be able to eliminate most of the shifting by hard-coding the bit positions when the loop is unrolled:

; 16-bit integer square root
; 34 bytes, 1005-1101 cycles (average 1053)

; call with de = number to square root
; returns   hl = square root
; corrupts  bc, de

  ld bc,08000h
  ld h,c
  ld l,c
sqrloop:
  srl b
  rr c
  add hl,bc
  ex de,hl
  sbc hl,de
  jr c,sqrbit
  ex de,hl
  add hl,bc
  jr sqrfi
sqrbit:
  add hl,de
  ex de,hl
  or a
  sbc hl,bc
sqrfi:
  srl h
  rr l
  srl b
  rr c
  jr nc,sqrloop

Then Make It Work Faster

First the loop is unrolled. The first 4 iterations are optimized to work on 8-bit values and bit positions are hard-coded. The first and last iteration are optimized as a special case, we work with the bitwise complement of the root instead of the root and small jumps are replace with overlapping code. The final code finds the root in an average of 362 t-states:

; fast 16-bit integer square root
; 92 bytes, 344-379 cycles (average 362)
; v2 - 3 t-state optimization spotted by Russ McNulty

; call with hl = number to square root
; returns    a = square root
; corrupts  hl, de

  ld a,h
  ld de,0B0C0h
  add a,e
  jr c,sq7
  ld a,h
  ld d,0F0h
sq7:

; ----------

  add a,d
  jr nc,sq6
  res 5,d
  db 254
sq6:
  sub d
  sra d

; ----------

  set 2,d
  add a,d
  jr nc,sq5
  res 3,d
  db 254
sq5:
  sub d
  sra d

; ----------

  inc d
  add a,d
  jr nc,sq4
  res 1,d
  db 254
sq4:
  sub d
  sra d
  ld h,a

; ----------

  add hl,de
  jr nc,sq3
  ld e,040h
  db 210
sq3:
  sbc hl,de
  sra d
  ld a,e
  rra

; ----------

  or 010h
  ld e,a
  add hl,de
  jr nc,sq2
  and 0DFh
  db 218
sq2:
  sbc hl,de
  sra d
  rra

; ----------

  or 04h
  ld e,a
  add hl,de
  jr nc,sq1
  and 0F7h
  db 218
sq1:
  sbc hl,de
  sra d
  rra

; ----------

  inc a
  ld e,a
  add hl,de
  jr nc,sq0
  and 0FDh
sq0:
  sra d
  rra
  cpl

ZX Spectrum BASIC Challenges

Recently I've entered a few of the programming challenges in the BASIC on the ZX Spectrum group on Facebook. They vary in difficultly but it's possible to write a program for most in under 30 minutes. If you're looking for a quick challenge or the opportunity to improve you BASIC, why not take a look.

Here are some examples of the challenges:

Japanese Pattern

Uwe Geiken asked us to recreate an intricate Japanese pattern. By mirroring, rotating and repeating a 4 line candy cane shape I squeezed this into 156 bytes.

Earth/Venus Orbits

David Saphier challenged us to write the fastest code to display the Earth and Venus orbit pattern in BASIC. Being a bit of a rebel I aimed to write the shortest code and managed to completely botch it before coming up with this working version:

Greenlandic Flag

Matthew Logue issued a challenge to accurately display the flag of Greenland. The flag is simple enough to be reduced to a formula x²+y² < 54² ⊻ y > 0:

Triangles

Uke Geiken showed a pattern of triangles and asked for the shortest code. The shortest implementation I found uses UDGs:

Grid

Matthew Logue asked us recreate a grid-like pattern with the shortest code. Surprisingly I actually managed to discover the smallest program:

Weaving

‎Uwe Geiken challenged us to recreate a pattern of weaving attributes. I found this one pretty tricky to reduce in size, but finally got it down to 109 bytes:

Flag

Matthew Logue asked for the shortest code to recreate a 31×21 attribute flag-like pattern. Uwe Geiken solved this is 67 bytes, easily beating my 74:

Rudimentary Gear

Matthew Logue challenged us again, this time to draw a rudimentary gear with 10 teeth and a circular radius. Here's what I came up with:

ZX Spectrum Scanline Flood Fill

A flood fill is a graphical algorithm to colour an area of screen bounded by pixels of another colour. The scanline technique is a fast, stack-efficient flood fill which can be implemented in 99 bytes of Z80, as demonstrated below:

; scanline fill by John Metcalf
; call with d=x-coord, e=y-coord

; set end marker

fill:
  ld l,255
  push hl

; calculate bit position of pixel

nextrun:
  ld a,d
  and 7
  inc a
  ld b,a
  ld a,1
bitpos:
  rrca
  djnz bitpos
  ld c,b
  ld b,a

; move left until hitting a set pixel or the screen edge

seekleft:
  ld a,d
  or a
  jr z,goright
  dec d
  rlc b
  call scrpos
  jr nz,seekleft

; move right until hitting a set pixel or the screen edge,
; setting pixels as we go. Check rows above and below and
; save their coordinates to fill later if necessary

seekright:  
  rrc b
  inc d
  jr z,rightedge
goright:
  call scrpos
  jr z,rightedge
  ld (hl),a
  inc e
  call checkadj
  dec e
  dec e
  call checkadj
  inc e
  jr seekright

; check to see if there's another row waiting to be filled

rightedge:
  pop de
  ld a,e
  inc a
  jr nz,nextrun
  ret  

; calculate the pixel address and whether or not it's set

scrpos:
  ld a,e
  and 248
  rra
  scf
  rra
  rra
  ld l,a
  xor e
  and 248
  xor e
  ld h,a
  ld a,l
  xor d
  and 7
  xor d
  rrca
  rrca
  rrca
  ld l,a
  ld a,b
  or (hl)
  cp (hl)
  ret

; check and save the coordinates of an adjacent row

checkadj:
  sla c
  ld a,e
  cp 192
  ret nc
  call scrpos+1
  ret z
  inc c
  bit 2,c
  ret nz
  pop hl
  push de
  jp (hl)

Divide and Conquer Line Algorithm for the ZX Spectrum

While attempting to write a game in 256 bytes I needed a routine to draw lines, but Bresenham's line algorithm weighs in at approx ~120 bytes. The only suitable alternative I'm aware of is recursive divide and conquer: divide a line into two smaller lines and call the draw routine with each in turn:

/* Draw a line from (ax,ay) to (bx,by) */

int draw ( ax, ay, bx, by )
{
    int midx, midy;
    midx = ( ax+bx ) / 2;
    midy = ( ay+by ) / 2;
    if ( midx != ax && midy != ay )
    {
        draw( midx, midy, ax, ay );
        draw( bx, by, midx, midy );
        plot( midx, midy );
    }
}

This is significantly smaller thank Bresenham's, 32 byte of Z80. However, there are a couple of compromises: it's slower and the lines aren't perfect because the rounding errors accumulate.

; draw lines using recursive divide and conquer
; from de = end1 (d = x-axis, e = y-axis)
; to   hl = end2 (h = x-axis, l = y-axis)

DRAW:
  call PLOT

  push hl

; calculate hl = centre pixel

  ld a,l
  add a,e
  rra
  ld l,a
  ld a,h
  add a,d
  rra
  ld h,a

; if de (end1) = hl (centre) then we're done

  or a
  sbc hl,de
  jr z,EXIT
  add hl,de

  ex de,hl
  call DRAW    ; de = centre, hl = end1
  ex (sp),hl
  ex de,hl
  call DRAW    ; de = end2, hl = centre

  ex de,hl
  pop de
  ret

EXIT:
  pop hl
  ret

; ---------------------------

; plot d = x-axis, e = y-axis

PLOT:
  push hl
  ld a,d
  and 7
  ld b,a
  inc b
  ld a,e
  rra
  scf
  rra
  or a
  rra
  ld l,a
  xor e
  and 248
  xor e
  ld h,a
  ld a,l
  xor d
  and 7
  xor d
  rrca
  rrca
  rrca
  ld l,a
  ld a,1
PLOTBIT:
  rrca
  djnz PLOTBIT
  or (hl)
  ld (hl),a
  pop hl
  ret

Alternatively the de(end1) = hl(centre) test can be replaced with a recursion depth count to create an even slower 28 byte routine:

; draw lines using recursive divide and conquer
; from de = end1 (d = x-axis, e = y-axis)
; to   hl = end2 (h = x-axis, l = y-axis)

DRAW:
  ld c,8

DRAW2:
  dec c
  jr z,EXIT

  push de

; calculate de = centre pixel

  ld a,l
  add a,e
  rra
  ld e,a
  ld a,h
  add a,d
  rra
  ld d,a

  call DRAW2   ; de = centre, hl = end1
  ex (sp),hl
  call DRAW2   ; de = centre, hl = end2

  call PLOT
  ex de,hl
  pop hl
EXIT:
  inc c
  ret

Langton's Ant for the ZX Spectrum

Langton's Ant is an automata which creates a complex pattern by following a couple of simple rules:

• If the ant is on an empty pixel, turn 90° right, set the pixel then move forward
• If the ant is on a set pixel, turn 90° left, reset the pixel then move forward

The ant's path appears chaotic at first before falling into a repetitive “highway” pattern, moving 2 pixels diagonally every 104 cycles.

Here's the code to display Langton's Ant on the ZX Spectrum in 61 bytes. It runs in just over a second so you might want to add a halt to slow things down:

  org 65472

  ld de,128*256+96 

ANT:
; halt
  ld a,c      ; check direction
  and 3
  rrca
  add a,a
  dec a
  jr nc,XMOVE

  add a,e     ; adjust y position +/-1
  ld e,a
  cp 192
  ret nc  
  xor a

XMOVE:
  add a,d     ; adjust x position +/-1
  ld d,a

; ----------
  and 7       ; calculate screen address
  ld b,a
  inc b
  ld a,e
  rra
  scf
  rra
  or a
  rra
  ld l,a
  xor e
  and 248
  xor e
  ld h,a
  ld a,d
  xor l
  and 7
  xor d
  rrca
  rrca
  rrca
  ld l,a
  ld a,1
PLOTBIT:
  rrca
  djnz PLOTBIT
; ----------

  ld b,a      ; test pixel
  and (hl)

  jr nz,LEFT  ; turn left/right
  inc c
  inc c
LEFT:
  dec c

  ld a,b      ; flip pixel
  xor (hl)
  ld (hl),a
  jr ANT

The Matrix Digital Rain for the ZX Spectrum

A few days ago I coded The Matrix digital rain effect, a fictional representation of the code for the virtual reality of The Matrix. The technique is simple: fill the screen with random characters and scroll down columns of attributes, occasionally switching between black and green.

Here's the final code - 147 bytes of Z80 using the default Sinclair font:

        org 08000h

; black border / black attributes

        xor a
        out (0FEh),a
        ld hl,05AFFh
attr:   ld (hl),a
        dec hl
        bit 2,h
        jr z,attr

; fill screen with random characters

        ld e,a
fillscr:ld d,040h
fill:   call rndchar
        ld a,d
        cp 058h
        jr nz,fill
        inc e
        jr nz,fillscr

; digital rain loop

frame:  ld b,06h
        halt
column: push bc

; randomize one character

        call random
        and 018h
        jr z,docol
        add a,038h
        ld d,a
        call random
        ld e,a
        call rndchar

; select a random column

docol:  call random
        and 01Fh
        ld l,a
        ld h,058h

; ~1% chance black -> white

        ld a,(hl)
        or a
        ld bc,0247h
        jr z,check

; white -> bright green

white:  cp c
        ld c,044h
        jr z,movecol

; bright green -> green

        cp c
        ld c,04h
        jr z,movecol

; ~6% chance green -> black

        ld bc,0F00h
check:  call random
        cp b
        jr c,movecol
        ld c,(hl)

; move column down

movecol:ld de,020h
        ld b,018h
down:   ld a,(hl)
        ld (hl),c
        ld c,a
        add hl,de
        djnz down
        pop bc
        djnz column

; test for keypress

        ld bc,07FFEh
        in a,(c)
        rrca
        jr c,frame
        ret

; display a random glyph

rndchar:call random
crange: sub 05Fh
        jr nc,crange
        add a,a
        ld l,a
        ld h,0
        add hl,hl
        add hl,hl
        ld bc,(05C36h)
        add hl,bc
        ld b,8
char:   ld a,(hl)
        ld (de),a
        inc d
        inc hl
        djnz char
        ret

; get a byte from the ROM

random: push hl
        ld hl,(seed)
        inc hl
        ld a,h
        and 01Fh
        ld h,a
        ld (seed),hl
        ld a,(hl)
        pop hl
        ret

seed: