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base:exponentiation

# Exponentiation

This routine computes the exponentiation of a 16 bit value. It handles only integer values. The largest result is 2^32-1 (32 bits); that makes 31 the largest possible exponent. Results larger than 2^32-1 will overflow.

The algorithm is recursive and at each iteration breaks the exponentiation in a simpler product: if the exponent is even, it will compute the exponentiation with half the exponent and square it, while if it's odd it will compute the product of the value by the value raised at the exponent minus one. The number of multiplications to be computed varies with the exponent, and the maximum is eight for the exponent 31 (31, 30, 15, 14, 7, 6, 3, 2).

The multiplication algorithm provided is tailored for this routine: it accepts 32bit values and will produce a 32bit result.

```Num1=25
Num2=5

example lda #>Num1
sta B+1
lda #<Num1
sta B
lda #Num2
jmp Exponent

; ************************************
;
;       Exponent
;
;       input:  B value to be raised
;               .A exponent
;
; algo:  if .A=0 res=1
;        if .A=1 res=B
;             _
;            | B if E=1
;  Exp(B,E)= | B*Exp(B,E-1) if E is odd
;            |_Exp(B,E/2)*Exp(B,E/2) if E is even
;
; ************************************

P = \$fb
M = \$62
N = \$6a
;E = \$2
B = \$3

Exponent
tax
beq res1        ; is E==0 ?
lda B
lsr
ora B+1
beq resB        ; if B==0 or B==1 then result=B
txa
cmp #1
bne ExpSub

resB    lda #0          ; E==1 | B==1 | B==0, result=B
sta P+2
sta P+3
lda B
sta P
lda B+1
sta P+1
rts

res1    sta P+1         ; E=0, result=1
sta P+2
sta P+3
lda #1
sta P
rts

ExpSub  lsr             ; E = int(E/2)
beq resB        ; E is 1
bcs ExpOdd      ; E is Odd

ExpEven jsr ExpSub      ; E is Even
ldx #\$3
_ldP    lda p,x         ; multiply P by itself
sta m,x         ; P is the result of a previous mult
sta n,x         ; copy P in M and N
dex
bpl _ldP
jmp Mult32

ExpOdd  asl             ; E = 2*int(E/2) (=E-1)
jsr ExpSub
ldx #\$4
_ldD    lda <p-1,x      ; multiply P by B
sta <m-1,x      ; P is the result of a previous mult
dex             ; copy P in M
bne _ldD
lda B           ; copy B in N
sta N
lda B+1
sta N+1
;lda #0
stx N+2
stx N+3
jmp Mult32

Mult32          ; 32=32*32
lda #0
sta P
sta P+1
sta P+2
sta P+3
ldy #\$20
_loop   asl p
rol p+1
rol p+2
rol p+3
asl N
rol N+1
rol N+2
rol N+3
bcc _skip
clc
ldx #\$fc