squares

Improves runtime from 16.24 ms/px to 14.44 ms/px

This uses a routine found on Everything2:
https://everything2.com/title/Fast+6502+multiplication

which uses a lookup table of squares to do 8-bit imuls,
which are then composed into a 16-bit imul

.gitignore

@@ -1,3 +1,4 @@
 *.o
 *.xex
+tables.s
 .DS_Store

Makefile

@@ -2,13 +2,17 @@
 
 all : mandel.xex
 
-%.xex : %.o
-	ld65 -C atari-asm-xex.cfg -o $@ $<
+mandel.xex : mandel.o tables.o
+	ld65 -C ./atari-asm-xex.cfg -o $@ $+
 
 %.o : %.s
 	ca65 -o $@ $<
 
+tables.s : tables.js
+	node tables.js > tables.s
+
 clean :
+	rm -f tables.s
 	rm -f *.o
 	rm -f *.xex
 

mandel.s

@@ -25,14 +25,14 @@ z_buffer_active = $b0 ; boolean: 1 if we triggered the lake, 0 if not
 z_buffer_start  = $b1 ; u8: index into z_buffer
 z_buffer_end    = $b2 ; u8: index into z_buffer
 temp            = $b4 ; u16
-
-pixel_ptr       = $b6 ; u16
-pixel_color     = $b8 ; u8
-pixel_mask      = $b9 ; u8
-pixel_shift     = $ba ; u8
-pixel_offset    = $bb ; u8
-fill_level      = $bc ; u8
-palette_offset  = $bd ; u8
+temp2           = $b6 ; u16
+pixel_ptr       = $b8 ; u16
+pixel_color     = $ba ; u8
+pixel_mask      = $bb ; u8
+pixel_shift     = $bc ; u8
+pixel_offset    = $bd ; u8
+fill_level      = $be ; u8
+palette_offset  = $bf ; u8
 
 ; FP registers in zero page
 FR0    = $d4 ; float48
@@ -107,6 +107,10 @@ KEY_RIGHT = $87
     mantissa .byte 6
 .endstruct
 
+.import mul_lobyte256
+.import mul_hibyte256
+.import mul_hibyte512
+
 .data
 
 strings:
@@ -257,6 +261,12 @@ fill_masks:
     add 4, dest, arg2, dest
 .endmacro
 
+.macro add_carry dest
+    lda dest
+    adc #0
+    sta dest
+.endmacro
+
 ; 2 + 9 * byte cycles
 .macro sub bytes, dest, arg1, arg2
     sec ; 2 cyc
@@ -334,65 +344,15 @@ fill_masks:
     neg 4, arg
 .endmacro
 
-; inner loop for imul16
-; bitnum < 8: 25 or 41 cycles
-; bitnum >= 8: 30 or 46 cycles
-.macro bitmul16 arg1, arg2, result, bitnum
-    .local zero
-    .local one
-    .local next
-
-    ; does 16-bit adds
-    ; arg1 and arg2 are treated as unsigned
-    ; negative signed inputs must be flipped first
-
-    ; 7 cycles up to the branch
-
-    ; check if arg1 has 0 or 1 bit in this place
-    ; 5 cycles either way
-    .if bitnum < 8
-        lda arg1                 ; 3 cyc
-        and #(1 << (bitnum))       ; 2 cyc
-    .else
-        lda arg1 + 1             ; 3 cyc
-        and #(1 << ((bitnum) - 8)) ; 2 cyc
-    .endif
-    bne one ; 2 cyc
-
-zero: ; 18 cyc, 23 cyc
-    lsr result + 3 ; 5 cyc
-    jmp next       ; 3 cyc
-
-one: ; 32 cyc, 37 cyc
-    ; 16-bit add on the top bits
-    clc            ; 2 cyc
-    lda result + 2 ; 3 cyc
-    adc arg2       ; 3 cyc
-    sta result + 2 ; 3 cyc
-    lda result + 3 ; 3 cyc
-    adc arg2 + 1   ; 3 cyc
-    ror a          ; 2 cyc - get a jump on the shift
-    sta result + 3 ; 3 cyc
-next:
-    ror result + 2 ; 5 cyc
-    ror result + 1 ; 5 cyc
-    .if bitnum >= 8
-        ; we can save 5 cycles * 8 bits = 40 cycles total by skipping this byte
-        ; when it's all uninitialized data
-        ror result ; 5 cyc
-    .endif
-
-.endmacro
-
 ; 5 to 25 cycles
 .macro check_sign arg
     ; Check sign bit and flip argument to postive,
-    ; keeping a count of sign bits in the X register.
+    ; keeping a count of sign bits in the Y register.
     .local positive
     lda arg + 1   ; 3 cyc
     bpl positive  ; 2 cyc
     neg16 arg     ; 18 cyc
-    inx           ; 2 cyc
+    iny           ; 2 cyc
 positive:
 .endmacro
 
@@ -419,35 +379,93 @@ positive:
     copy16 dest, FR2 + 2  ; 12 cyc
 .endmacro
 
-; min 470 cycles
-; max 780 cycles
+; Adapted from https://everything2.com/title/Fast+6502+multiplication
+.macro imul8 dest, arg1, arg2
+    .local under256
+    .local next
+    .local small_product
+    .scope
+        mul_factor_a   = arg1
+        mul_factor_x   = arg2
+        mul_product_lo = dest
+        mul_product_hi = dest + 1
+
+        lda mul_factor_a      ; setup: 6 cycles
+        ;ldx mul_factor_x
+
+        clc                   ; (a + x)^2/2: 23 cycles
+        adc mul_factor_x
+        tax
+        bcc under256
+        lda mul_hibyte512,x
+        bcs next
+    under256:
+        lda mul_hibyte256,x
+        sec
+    next:
+        sta mul_product_hi
+        lda mul_lobyte256,x
+
+        ldx mul_factor_a      ; - a^2/2: 20 cycles
+        sbc mul_lobyte256,x
+        sta mul_product_lo
+        lda mul_product_hi
+        sbc mul_hibyte256,x
+        sta mul_product_hi
+
+        ldx mul_factor_x      ; + x & a & 1: 22 cycles
+        txa                   ; (this is a kludge to correct a
+        and mul_factor_a      ; roundoff error that makes odd * odd too low)
+        and #1
+
+        clc
+        adc mul_product_lo
+        bcc small_product
+        inc mul_product_hi
+    small_product:
+        sec                   ; - x^2/2: 25 cycles
+        sbc mul_lobyte256,x
+        sta mul_product_lo
+        lda mul_product_hi
+        sbc mul_hibyte256,x
+        sta mul_product_hi
+    .endscope
+.endmacro
+
 .proc imul16_func
     arg1 = FR0   ; 16-bit arg (clobbered)
     arg2 = FR1   ; 16-bit arg (clobbered)
     result = FR2 ; 32-bit result
+    inter = temp2
 
-    ldx #0          ; 2 cyc
-    ; counts the number of sign bits in X
+    ldy #0          ; 2 cyc
+    ; counts the number of sign bits in Y
     check_sign arg1 ; 5 to 25 cyc
     check_sign arg2 ; 5 to 25 cyc
 
-    ; zero out the 32-bit temp's top 16 bits
-    lda #0          ; 2 cyc
-    sta result + 2  ; 3 cyc
-    sta result + 3  ; 3 cyc
-    ; the bottom two bytes will get cleared by the shifts
+    ; h1l1 * h2l2
+    ; (h1*256 + l1) * (h2*256 + l2)
+    ; h1*256*(h2*256 + l2) + l1*(h2*256 + l2)
+    ; h1*h2*256*256 + h1*l2*256 + h2*l1*256 + l1*l2
 
-    ; unrolled loop for maximum speed, at the cost
-    ; of a larger routine
-    ; 440 to 696 cycles
-    .repeat 16, bitnum
-        ; bitnum < 8: 25 or 41 cycles
-        ; bitnum >= 8: 30 or 46 cycles
-        bitmul16 arg1, arg2, result, bitnum
-    .endrepeat
+    imul8 result, arg1, arg2
+    lda #0
+    sta result + 2
+    sta result + 3
+
+    imul8 inter, arg1 + 1, arg2
+    add16 result + 1, result + 1, inter
+    add_carry result + 3
+
+    imul8 inter, arg1, arg2 + 1
+    add16 result + 1, result + 1, inter
+    add_carry result + 3
+
+    imul8 inter, arg1 + 1, arg2 + 1
+    add16 result + 2, result + 2, inter
 
     ; In case of mixed input signs, return a negative result.
-    cpx #1              ; 2 cyc
+    cpy #1              ; 2 cyc
     bne positive_result ; 2 cyc
     neg32 result        ; 34 cyc
 positive_result:

readme.md

@@ -37,6 +37,7 @@ Add a running counter of ms/px using the vertical blank interrupts as a timer. T
 Check for cycles in (zx,zy) output when in the 'lake'; if values repeat, they cannot escape. This is a big time saver in fractint.
 
 I may be able to do a faster multiply using tables of squares for 8-bit component multiplication.
+(done)
 
 ## Deps and build instructions
 

tables.js

@@ -0,0 +1,38 @@
+function db(func) {
+    let lines = [];
+    for (let i = 0; i < 256; i += 16) {
+        let items = [];
+        for (let j = 0; j < 16; j++) {
+            let x = i + j;
+            items.push(func(x));
+        }
+        lines.push('    .byte ' + items.join(', '));
+    }
+    return lines.join('\n');
+}
+
+let squares = [];
+for (let i = 0; i < 512; i++) {
+    squares.push(Math.trunc((i * i + 1) / 2));
+}
+
+console.log(
+`.segment "TABLES"
+
+.export mul_lobyte256
+.export mul_hibyte256
+.export mul_hibyte512
+
+.align 256
+mul_lobyte256:
+${db((i) => squares[i] & 0xff)}
+
+.align 256
+mul_hibyte256:
+${db((i) => (squares[i] >> 8) & 0xff)}
+
+.align 256
+mul_hibyte512:
+${db((i) => (squares[i + 256] >> 8) & 0xff)}
+
+`);

testme.js

@@ -0,0 +1,41 @@
+// ax = (a + x)2/2 - a2/2 - x2/2 
+
+function half_square(x) {
+    return Math.round(x * x / 2) & 0xffff >>> 0;
+}
+
+function mul8(a, b) {
+    let result = half_square(a + b) & 0xffff;
+    result = (result - half_square(a)) & 0xffff;
+    result = (result - half_square(b)) & 0xffff;
+    result = (result + (b & a & 1)) & 0xffff;
+    return result >>> 0;
+}
+
+function mul16(a, b) {
+    let ah = (a & 0xff00) >>> 8;
+    let al = (a & 0x00ff) >>> 0;
+    let bh = (b & 0xff00) >>> 8;
+    let bl = (b & 0x00ff) >>> 0;
+    let result = (mul8(al, bl) & 0xffff) >>> 0;
+    result = ((result + (mul8(ah, bl) << 8)) & 0x00ffffff) >>> 0;
+    result = ((result + (mul8(al, bh) << 8)) & 0x01ffffff) >>> 0;
+    result = ((result + (mul8(ah, bh) << 16)) & 0xffffffff) >>> 0;
+    return result;
+}
+
+let max = 65536;
+//let max = 256;
+//let max = 128;
+//let max = 8;
+
+for (let a = 0; a < max; a++) {
+    for (let b = 0; b < max; b++) {
+        let expected = Math.imul(a, b) >>> 0;
+        //let actual = mul8(a, b);
+        let actual = mul16(a, b);
+        if (expected !== actual) {
+            console.log(`wrong! ${a} * ${b} expected ${expected} got ${actual}`);
+        }
+    }
+}