Normally, 8-bit processors perform multiplication with the long multiplication method implemented in software. This is slow, to the tune of 200 or so cycles per multiply. Often, the best optimization is a new algorithm. Note that

(`a` + `x`)^{2} = `a`^{2} + `x`^{2} + 2`ax`

Solve for

`ax`, and a multiplication becomes

`ax` = (`a` + `x`)^{2}/2 - `a`^{2}/2 + `x`^{2}/2

that is, a couple adds and squares. Squares are easy to optimize into

lookup tables, and this is where it gets fast. Here's the code for the

6502 processor (and variants such as

6510,

65c02, and

2a03):

; You can change these for whatever system you're working on.
mul_factor_a = $f0
mul_factor_x = $f1
mul_product_lo = $f2
mul_product_hi = $f3
;
; mul
; Wicked fast LUT-based multiplication
; input: a = factor a; x = factor b
; output: mul_factor_a = factor a; mul_factor_x = x = factor x;
; mul_product_lo = low byte of product a*b;
; mul_product_hi = a = high byte of product
; preserved: x, y
; max cycles: under 90
;
mul
sta mul_factor_a ; setup: 6 cycles
stx mul_factor_x
clc ; (a + x)^2/2: 23 cycles
adc mul_factor_x
tax
bcc +
lda mul_hibyte512,x
bcs ++
+
lda mul_hibyte256,x
sec
++
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 +
inc mul_product_hi
+
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
rts
; here are the big fat lookup tables
mul_lobyte256
.db 0, 1, 2, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98,113
.db 128,145,162,181,200,221,242, 9, 32, 57, 82,109,136,165,194,225
.db 0, 33, 66,101,136,173,210,249, 32, 73,114,157,200,245, 34, 81
.db 128,177,226, 21, 72,125,178,233, 32, 89,146,205, 8, 69,130,193
.db 0, 65,130,197, 8, 77,146,217, 32,105,178,253, 72,149,226, 49
.db 128,209, 34,117,200, 29,114,201, 32,121,210, 45,136,229, 66,161
.db 0, 97,194, 37,136,237, 82,185, 32,137,242, 93,200, 53,162, 17
.db 128,241, 98,213, 72,189, 50,169, 32,153, 18,141, 8,133, 2,129
.db 0,129, 2,133, 8,141, 18,153, 32,169, 50,189, 72,213, 98,241
.db 128, 17,162, 53,200, 93,242,137, 32,185, 82,237,136, 37,194, 97
.db 0,161, 66,229,136, 45,210,121, 32,201,114, 29,200,117, 34,209
.db 128, 49,226,149, 72,253,178,105, 32,217,146, 77, 8,197,130, 65
.db 0,193,130, 69, 8,205,146, 89, 32,233,178,125, 72, 21,226,177
.db 128, 81, 34,245,200,157,114, 73, 32,249,210,173,136,101, 66, 33
.db 0,225,194,165,136,109, 82, 57, 32, 9,242,221,200,181,162,145
.db 128,113, 98, 85, 72, 61, 50, 41, 32, 25, 18, 13, 8, 5, 2, 1
mul_hibyte256
.db 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
.db 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1
.db 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4
.db 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7
.db 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12
.db 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17
.db 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24
.db 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31
.db 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39
.db 40, 41, 41, 42, 42, 43, 43, 44, 45, 45, 46, 46, 47, 48, 48, 49
.db 50, 50, 51, 51, 52, 53, 53, 54, 55, 55, 56, 57, 57, 58, 59, 59
.db 60, 61, 61, 62, 63, 63, 64, 65, 66, 66, 67, 68, 69, 69, 70, 71
.db 72, 72, 73, 74, 75, 75, 76, 77, 78, 78, 79, 80, 81, 82, 82, 83
.db 84, 85, 86, 86, 87, 88, 89, 90, 91, 91, 92, 93, 94, 95, 96, 97
.db 98, 98, 99,100,101,102,103,104,105,106,106,107,108,109,110,111
.db 112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127
mul_hibyte512
.db 128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143
.db 144,145,146,147,148,149,150,152,153,154,155,156,157,158,159,160
.db 162,163,164,165,166,167,168,169,171,172,173,174,175,176,178,179
.db 180,181,182,184,185,186,187,188,190,191,192,193,195,196,197,198
.db 200,201,202,203,205,206,207,208,210,211,212,213,215,216,217,219
.db 220,221,223,224,225,227,228,229,231,232,233,235,236,237,239,240
.db 242,243,244,246,247,248,250,251,253,254,255, 1, 2, 4, 5, 7
.db 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30
.db 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54
.db 56, 58, 59, 61, 62, 64, 65, 67, 69, 70, 72, 73, 75, 77, 78, 80
.db 82, 83, 85, 86, 88, 90, 91, 93, 95, 96, 98,100,101,103,105,106
.db 108,110,111,113,115,116,118,120,122,123,125,127,129,130,132,134
.db 136,137,139,141,143,144,146,148,150,151,153,155,157,159,160,162
.db 164,166,168,169,171,173,175,177,179,180,182,184,186,188,190,192
.db 194,195,197,199,201,203,205,207,209,211,212,214,216,218,220,222
.db 224,226,228,230,232,234,236,238,240,242,244,246,248,250,252,254

You can find test code for this subroutine, as well as C code to generate the lookup tables, at **http://www.cs.rose-hulman.edu/~yerricde/**.

Squeezing a multiplication into 90 cycles is a Good Thing. Unfortunately, this algorithm is not good for microcontrollers with limited ROM space, but for a few more cycles, the tables' size could be cut in half (the first is symmetrical, and the third is the second plus 128 plus 2`i`). But for microcontrollers that need fast multiplication that cannot be strength-reduced to addition, it'd probably be best to put a multiplier in the chipset. This is the approach taken by the customized 65c816 processor in the Super NES game console.

I also don't see a generalization to division.