Z80 Routines:Math:Advance Math

Introduction

These are routines designed for math of a slightly higher level. These don't necessarily contribute to everyday coding, but might be useful for an OS that handles such math (or programming language).

nCr Algorithm

This computes "n choose r" in such a way as to avoid overflow unless the final result would overflow 16 bits.

;Written by Zeda

; Requires
;    mul16          ;BC*DE ==> DEHL
;    DEHL_Div_BC    ;DEHL/BC ==> DEHL

ncr_HL_DE:
;"n choose r", defined as n!/(r!(n-r)!)
;Computes "HL choose DE"
;Inputs: HL,DE
;Outputs:
;   HL is the result
;       "HL choose DE"
;   carry flag reset means overflow
;Destroys:
;   A,BC,DE,IX
;Notes:
;   Overflow is returned as 0
;   Overflow happens if HL choose DE exceeds 65535
;   This algorithm is constructed in such a way that intermediate
;   operations won't erroneously trigger overflow.
;66 bytes
  ld bc,1
  or a
  sbc hl,de
  jr c,ncr_oob
  jr z,ncr_exit
  sbc hl,de
  add hl,de
  jr c,$+3
  ex de,hl
  ld a,h
  or l
  push hl
  pop ix
ncr_exit:
  ld h,b
  ld l,c
  scf
  ret z
ncr_loop:
  push bc \ push de
  push hl \ push bc
  ld b,h
  ld c,l
  call mul16          ;BC*DE ==> DEHL
  pop bc
  call DEHL_Div_BC    ;result in DEHL
  ld a,d
  or e
  pop bc
  pop de
  jr nz,ncr_overflow
  add hl,bc
  jr c,ncr_overflow
  pop bc
  inc bc
  ld a,b
  cp ixh
  jr c,ncr_loop
  ld a,ixl
  cp c
  jr nc,ncr_loop
  ret
ncr_overflow:
  pop bc
  xor a
  ld b,a
ncr_oob:
  ld h,b
  ld l,b
  ret

 

GCDHL_DE

This finds the greatest common divisor (GCD) of HL and DE using the binary GCD algorithm to improve performance.

gcdHL_DE:
;gcd(HL,DE)->HL
;Output:
;   B=0
;   HL is the GCD of the inputs
;Destroys:
;   A,DE
;     DE is guaranteed 0 unless the output is 0 (which only happens if one of the inputs is 0).
;Uses the binary GCD algorithm
;65 bytes

;B is our cofactor-of-2 counter
    ld b,0

;If HL=0, return 0
    ld a,h \ or l \ ret z

;If DE=0, return 0
    ex de,hl
    ld a,h \ or l \ jr nz,gcd_test_cofactor_of_2
    ret

gcd_cofactor_2_loop:
    inc b
    srl h \ rr l
    srl d \ rr e
gcd_test_cofactor_of_2:
    inc b
    ld a,e
    or l
    rra
    jr nc,gcd_cofactor_2_loop

gcd_remove_factors_of_2_op2:
    srl h \ rr l \ jr nc,gcd_remove_factors_of_2_op2
    adc hl,hl
    jr gcd_swap_ops

gcd_swap_ops_negate:
;At this point, HL needs to be negated and swapped with DE
    xor a \ sub l \ ld l,a \ sbc a,a \ sub h \ ld h,a
gcd_swap_ops:
    ex de,hl
gcd_remove_factors_of_2_op1:
    srl h \ rr l \ jr nc,gcd_remove_factors_of_2_op1
    adc hl,hl
    sbc hl,de
    jr c,gcd_swap_ops_negate
    jp nz,gcd_remove_factors_of_2_op1

;DE is the GCD, need to shift it left B-1 times.
    ex de,hl
    dec b
    ret z
    add hl,hl \ djnz $-1
    ret
 

LCM

This is as simple as multiplying the two numbers and dividing by the GCD.

Credits and Contributions

• Zeda (Xeda) Elnara for the nCr and GCD algorithm