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Generalised floating point multiplication (view source)

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{{works with|ALGOL 68G|Any - tested with release [http://sourceforge.net/projects/algol68/files/algol68g/algol68g-2.3.3 algol68g-2.3.3].}}
{{wont work with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d] - due to extensive use of '''format'''[ted] ''transput''.}}
'''File: Template.Big_float.Multiplication.a68'''<langsyntaxhighlight lang="algol68">##########################################
# TASK CODE #
# Actual generic mulitplication operator #
Line 116:
 
OP *:= = (REF DIGITS lhs, DIGIT arg)DIGITS: lhs := lhs * INITDIGITS arg;
</langsyntaxhighlight>'''File: Template.Balanced_ternary_float.Base.a68'''<langsyntaxhighlight lang="algol68">PR READ "Template.Big_float_BCD.Base.a68" PR # [[rc:Generalised floating point addition]] #
 
################################################################
Line 151:
OD;
out SHR (UPB s-point)
);</langsyntaxhighlight>'''File: test.Balanced_ternary_float.Multiplication.a68'''<langsyntaxhighlight lang="algol68">#!/usr/local/bin/a68g --script #
####################################################################
# A program to test arbitrary length floating point multiplication #
Line 200:
printf($l$)
OD
)</langsyntaxhighlight>'''Output:'''
<pre>
a = +523.23914037494284407864655 +-0++0+.+-0++0+
Line 241:
{{trans|Phix}}
In the interests of brevity many of the comments and all of the commented-out code has been omitted.
<langsyntaxhighlight lang="go">package main
 
import (
Line 796:
test("septemvigesimal", septemVigesimal)
multTable()
}</langsyntaxhighlight>
 
{{out}}
Line 881:
</pre>
 
=={{header|Phix}}==
Note regarding requirement #5: While this meets my definition of "reasonably efficient",
it should not shock anyone that this kind of "string maths" which works digit-by-digit
and uses repeated addition (eg *999 performs 27 additions) could easily be 10,000 times
slower than raw hardware or a carefully optimised library such as gmp. However this does
offer perfect accuracy in any given base, whereas gmp, for all it's brilliance, can hold
0.1 accurate to several million decimal places, but just never quite exact.
 
=={{header|Julia}}==
<lang Phix>-- demo\rosetta\Generic_multiplication.exw
<syntaxhighlight lang="julia">using Formatting
constant MAX_DP = 81
import Base.BigInt, Base.BigFloat, Base.print, Base.+, Base.-, Base.*
 
abstract type BalancedBaseDigitArray end
constant binary = "01",
ternary = "012",
balancedternary = "-0+",
decimal = "0123456789",
hexadecimal = "0123456789ABCDEF",
septemvigesimal = "0123456789ABCDEFGHIJKLMNOPQ",
-- heptavintimal = "0123456789ABCDEFGHKMNPRTVXZ", -- ??
-- wonky_donkey_26 = "0ABCDEFGHIJKLMNOPQRSTUVWXY",
-- wonky_donkey_27 = "0ABCDEFGHIJKLMNOPQRSTUVWXYZ",
balanced_base27 = "ZYXWVUTSRQPON0ABCDEFGHIJKLM",
base37 = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
--
--Note: I have seen some schemes where balanced-base-27 uses
--==== the same character set as septemvigesimal, with 'D'
-- representing 0, and wonky_donkey_27 with 'M'==0(!).
-- These routines do not support that directly, except
-- (perhaps) via a simple mapping on all inputs/outputs.
-- It may be possible to add a defaulted parameter such
-- as zero='0' - left as an exercise for the reader.
-- Admittedly that balanced_base27 is entirely my own
-- invention, just for this specific task.
--
 
mutable struct BalancedTernary <: BalancedBaseDigitArray
function b2dec(string b, alphabet)
dig::Vector{Int8}
--
p::Int
-- convert string b back into a normal (decimal) atom,
BalancedTernary(arr::Vector, i) = new(Int8.(arr), i)
-- eg b2dec("+0-",balancedternary) yields 8
end
--
atom res = 0
integer base = length(alphabet),
zdx = find('0',alphabet)
bool signed = (zdx=1 and b[1]='-')
if signed then b = b[2..$] end if
integer len = length(b),
ndp = find('.',b)
if ndp!=0 then
b[ndp..ndp] = "" -- remove '.'
ndp = len-ndp
end if
for i=1 to length(b) do
res = base*res+find(b[i],alphabet)-zdx
end for
if ndp!=0 then res /= power(base,ndp) end if
if signed then res = -res end if
return res
end function
 
const MAX_PRECISION = 81
function negate(string b, alphabet)
 
--
function BalancedTernary(s::String)
-- negate b (can be balanced or unbalanced)
if (i = findfirst(x -> x == '.', s)) != nothing
--
p = length(s) - i
if alphabet[1]='0' then
--s traditional= s[1:i-1] add/remove* a leading '-'s[i+1:end]
else
-- eg "-123" <==> "123"
ifp b!="0" then0
end
if b[1]='-' then
b = BalancedTernary([c == '-' ? -1 : c == '0' ? b0 =: b[2..$1 for c in s], p) # 2
end
 
function BalancedTernary(n::Integer) # 1, 3
if n < 0
return -BalancedTernary(-n)
elseif n == 0
return BalancedTernary([0], 0)
else
return canonicalize!(BalancedTernary(reverse(digits(n, base=3)), 0))
end
end
BalancedTernary() = BalancedTernary(0)
 
function BalancedTernary(x::Real) # 1, 3
if x < 0
return -BalancedTernary(-x)
end
arr = reverse(digits(BigInt(round(x * big"3.0"^MAX_PRECISION)), base=3))
return canonicalize!(BalancedTernary(arr, MAX_PRECISION))
end
 
function String(b::BalancedTernary) # 3
canonicalize!(b)
s = String([['-', '0', '+'][c + 2] for c in b.dig])
if b.p > 0
if b.p < length(s)
s = s[1:end-b.p] * "." * s[end-b.p+1:end]
elseif b.p == length(s)
s = "0." * s
else
s = "0." * "0"^(b.p - length(s)) * s
end
end
return s
end
 
function BigInt(b::BalancedTernary)
canonicalize!(b)
if b.p > 0
throw(InexactError("$(b.p) places after decimal point"))
end
return sum(t -> BigInt(3)^(t[1] - 1) * t[2], enumerate(reverse(b.dig))) # 3
end
 
BigFloat(b::BalancedTernary) = BigInt(BalancedTernary(b.dig, 0)) / big"3.0"^(b.p)
 
function canonicalize!(b::BalancedTernary)
for i in length(b.dig):-1:1
if b.dig[i] > 1
b.dig[i] -= 3
if i == 1
pushfirst!(b.dig, 1)
else
b.dig[i - 1] += 1
end
elseif b.dig[i] < -1
b.dig[i] += 3
if i == 1
pushfirst!(b.dig, -1)
else
b.dig[i =- 1] "-"&b= 1
end if
end if
end
if (i = findfirst(x -> x != 0, b.dig)) != nothing
if i > 1
b.dig = b.dig[i:end]
end
else
--b.dig balanced: mirror= [non-0] digits
end
-- eg "-0+" (ie -8) <==> "+0-" (ie +8)
if b.p > 0 for&& (i =1 to lengthfindlast(x -> x != 0, b.dig)) != donothing
removable = min(b.p, if length(b[i]!='.'dig) then- i)
b[i].dig = alphabetb.dig[1:end-find(b[i],alphabet)removable]
b.p -= end ifremovable
end for
end if
return b
end function
function b_trim(string b)
-- (common code)
-- trim trailing ".000"
if find('.',b) then
b = trim_tail(trim_tail(b,'0'),'.')
end if
-- trim leading zeroes, but not "0.nnn" -> ".nnn"
-- [hence we cannot use the standard trim_head()]
while length(b)>1 and b[1]='0' and b[2]!='.' do
b = b[2..$]
end while
return b
end function
 
# The following should work with any base number where dig, p are a similar array and Int
function b_carry(integer digit, base, idx, string n, alphabet)
# and the proper constructors, canon, and conversion routines are defined # 6
-- (common code, for balanced number systems only)
integer carry = iff(digit>base?+1:iff(digit<1?-1:0))
if carry then
for i=idx to 0 by -1 do
if n[i]!='.' then
integer k = find(n[i],alphabet)
if k<base then
n[i] = alphabet[k+1]
exit
end if
n[i]=alphabet[1]
end if
end for
digit -= base*carry
end if
return {digit,n}
end function
 
Base.print(io::IO, b::BalancedBaseDigitArray) = print(io, String(b))
function b2b(string n, alphabet, alphabet2)
--
-- convert a string from alphabet to alphabet2,
-- eg b2b("8",decimal,balancedternary) yields "+0-",
-- & b2b("+0-",balancedternary,decimal) yields "8",
--
string res = "0", m = ""
if n!="0" then
integer base = length(alphabet),
base2 = length(alphabet2),
zdx = find('0',alphabet),
zdx2 = find('0',alphabet2),
carry = 0, q, r, digit
bool negative = ((zdx=1 and n[1]='-') or
(zdx!=1 and find(n[1],alphabet)<zdx))
if negative then n = negate(n,alphabet) end if
integer ndp = find('.',n)
if ndp!=0 then
{n,m} = {n[1..ndp-1],n[ndp+1..$]}
end if
res = ""
while length(n) do
q = 0
for i=1 to length(n) do
--
-- this is a digit-by-digit divide (/mod) loop
-- eg for hex->decimal we would want:
-- this loop/modrem("FFFF",10) --> "1999" rem 5,
-- this loop/modrem("1999",10) --> "28F" rem 3,
-- this loop/modrem("28F",10) --> "41" rem 5,
-- this loop/modrem("41",10) --> "6" rem 5,
-- this loop/modrem("6",10) --> "0" rem 6,
-- ==> res:="65535" (in 5 full iterations over n).
--
digit = find(n[i],alphabet)-zdx
q = q*base+digit
r = mod(q,base2)
digit = floor(q/base2)+zdx
if zdx!=1 then
{digit,n} = b_carry(digit,base,i-1,n,alphabet)
end if
n[i] = alphabet[digit]
q = r
end for
r += zdx2
if zdx2!=1 then
r += carry
carry = iff(r>base2?+1:iff(r<1?-1:0))
r -= base2*carry
end if
res = alphabet2[r]&res
n = trim_head(n,'0')
end while
if carry then
res = alphabet2[carry+zdx2]&res
end if
if length(m) then
res &= '.'
ndp = 0
if zdx!=1 then
-- convert fraction to unbalanced, to simplify the (other-base) multiply.
integer lm = length(m)
string alphanew = base37[1..length(alphabet)]
m = b2b(m,alphabet,alphanew) -- (nb: no fractional part!)
m = repeat('0',lm-length(m))&m -- zero-pad if required
alphabet = alphanew
zdx = 1
end if
while length(m) and ndp<MAX_DP do
q = 0
for i=length(m) to 1 by -1 do
--
-- this is a digit-by-digit multiply loop
-- eg for [.]"1415" decimal->decimal we
-- would repeatedly multiply by 10, giving
-- 1 and "4150", then 4 and "1500", then
-- 1 and "5000", then 5 and "0000". We
-- strip zeroes between each output digit
-- & obviously normally alphabet in!=out.
--
digit = find(m[i],alphabet)-zdx
q += digit*base2
r = mod(q,base)+zdx
q = floor(q/base)
m[i] = alphabet[r]
end for
digit = q + zdx2
if zdx2!=1 then
{digit,res} = b_carry(digit,base2,length(res),res,alphabet2)
end if
res &= alphabet2[digit]
m = trim_tail(m,'0')
ndp += 1
end while
end if
res = b_trim(res)
if negative then res = negate(res,alphabet2) end if
end if
return res
end function
 
function +(b1::T, b2::T) where T <: BalancedBaseDigitArray # 4
function atm2b(atom d, string alphabet)
if all(x -> x == 0, b1.dig)
--
return deepcopy(b2)
-- convert d to a string in the specified base,
elseif all(x -> x == 0, b2.dig)
-- eg atm2b(65535,hexadecimal) => "FFFF"
return deepcopy(b1)
--
end
-- As a standard feature of phix, you can actually specify
ldigits1 = length(b1.dig) - b1.p
-- d in any number base between 2 and 36, eg 0(13)168 is
arr = b1.dig[1:ldigits1]
-- equivalent to 255 (see test\t37misc.exw for more), but
ldigits2 = length(b2.dig) - b2.p
-- not (yet) in balanced number bases, or with fractions,
arr2 = b2.dig[1:ldigits2]
-- except (of course) for normal decimal fractions.
if (i = ldigits1 - ldigits2) > 0
--
arr2 = [zeros(Int8, i); arr2]
-- Note that eg b2b("-436.436",decimal,balancedternary) is
elseif i < 0
-- more acccurate that atm2b(-436.436,balancedternary) due
arr = [zeros(Int8, -i); arr]
-- to standard IEEE 754 floating point limitations.
end
-- For integers, discrepancies only creep in for values
if (i = b1.p - b2.p) > 0
-- outside the range +/-9,007,199,254,740,992 (on 32-bit).
arr = [arr; b1.dig[ldigits1+1:end]]
-- However, this is much simpler and faster than b2b().
arr2 = [arr2; b2.dig[ldigits2+1:end]; zeros(Int8, i)]
--
elseif i < 0
integer base = length(alphabet),
arr = [arr; b1.dig[ldigits1+1:end]; zdx = findzeros('0'Int8,alphabet -i),]
arr2 = [arr2; carry = 0b2.dig[ldigits2+1:end]]
bool neg = d<0else
if neg then d arr = -d[arr; b1.dig[ldigits1+1:end if]]
arr2 = [arr2; b2.dig[ldigits2+1:end]]
string res = ""
end
integer whole = floor(d)
darr -.+= wholearr2
return canonicalize!(T(arr, max(b1.p, b2.p)))
while true do
end
integer ch = mod(whole,base) + zdx
if zdx!=1 then
ch += carry
carry = iff(ch>base?+1:iff(ch<1?-1:0))
ch -= base*carry
end if
res = alphabet[ch]&res
whole = floor(whole/base)
if whole=0 then exit end if
end while
if carry then
res = alphabet[carry+zdx]&res
carry = 0
end if
if d!=0 then
res &= '.'
integer ndp = 0
while d!=0 and ndp<MAX_DP do
d *= base
integer digit = floor(d) + zdx
d -= digit
if zdx!=1 then
{digit,res} = b_carry(digit,base,length(res),res,alphabet)
end if
res &= alphabet[digit]
ndp += 1
end while
end if
if neg then res = negate(res,alphabet) end if
return res
end function
 
-(b1::T) where T <: BalancedBaseDigitArray = T(b1.dig .* -1, b1.p) # 4
-- negative numbers in addition and subtraction
-(b1::T, b2::T) where T <: BalancedBaseDigitArray = +(b1, -b2) # 4
-- (esp. non-balanced) are treated as follows:
-- for -ve a: (-a)+b == b-a; (-a)-b == -(a+b)
-- for -ve b: a+(-b) == a-b; a-(-b) == a+b
-- for a>b: a-b == -(b-a) [avoid running off end]
 
function *(b1::T, b2::T) where T <: BalancedBaseDigitArray # 4
forward function b_sub(string a, b, alphabet)
len = length(b2.dig)
bsum = T()
for i in len:-1:1
bsum += T([b1.dig .* b2.dig[i]; zeros(Int8, len - i)], 0)
end
bsum.p = b1.p + b2.p
return canonicalize!(bsum)
end
 
function b_addcode_reuse_task(string a, b, alphabetT::Type)
a = T("+-0++0+.+-0++0+")
integer base = length(alphabet),
zdxb = findT('0',alphabet-436.436),
c = T("+-++-.+-++-")
carry = 0, da, db, digit
println(" a = ", a, " = ", format(BigFloat(a)))
if zdx=1 then
println(" b if a[1]='-' then", b, " = --", format(-aBigFloat(b)))+b == b-a
println(" c = ", c, " = ", return b_subformat(b,negateBigFloat(a,alphabetc)),alphabet)
println("\na * (b - c) = ", String(a * (b - c)), "\n = ", format(BigFloat(a * (b - c))))
end if
if b[1]='-' then -- a+(-b) == a-b
return b_sub(a,negate(b,alphabet),alphabet)
end if
end if
integer adt = find('.',a),
bdt = find('.',b)
if adt or bdt then
-- remove the '.'s and zero-pad the shorter as needed
-- (thereafter treat them as two whole integers)
-- eg "1.23"+"4.5" -> "123"+"450" (leaving adt==2)
if adt then adt = length(a)-adt+1; a[-adt..-adt] = "" end if
if bdt then bdt = length(b)-bdt+1; b[-bdt..-bdt] = "" end if
if bdt>adt then
a &= repeat('0',bdt-adt)
adt = bdt
elsif adt>bdt then
b &= repeat('0',adt-bdt)
end if
end if
if length(a)<length(b) then
{a,b} = {b,a} -- ensure b is the shorter
end if
for i=-1 to -length(a) by -1 do
da = iff(i<-length(a)?0:find(a[i],alphabet)-zdx)
db = iff(i<-length(b)?0:find(b[i],alphabet)-zdx)
digit = da + db + carry + zdx
carry = iff(digit>base?+1:iff(digit<1?-1:0))
a[i] = alphabet[digit-carry*base]
if i<-length(b) and carry=0 then exit end if
end for
if carry then
a = alphabet[carry+zdx]&a
end if
if adt then
a[-adt+1..-adt] = "."
end if
a = b_trim(a)
return a
end function
 
println("\n Multiplication 27 X 12")
function a_smaller(string a, b, alphabet)
println(" x|+ (1) |+- (2) |+0 (3) |++ (4) |+-- (5)|+-0 (6)|+-+ (7)|+0- (8)|+e+-(9)|+0+(10)|++-(11)|++0(12)|")
-- return true if a is smaller than b
for i in 1:27
-- if not balanced then both are +ve
print(lpad(i, 2), "|")
if length(a)!=length(b) then ?9/0 end if -- sanity check
for i=1 to length(a) dofor j in 1:12
integer da = find print(lpad(String(T(a[i] * j)),alphabet 7), "|")
end
db = find(b[i],alphabet),
c = compareprint(a,b"\n")
end
if c!=0 then return c<0 end if
end for
return false -- (=, which is not <)
end function
 
code_reuse_task(BalancedTernary)
function b_sub(string a, b, alphabet)
</syntaxhighlight>{{out}}
integer base = length(alphabet),
<pre>
zdx = find('0',alphabet),
a = +-0++0+.+-0++0+ = 523.23914
carry = 0, da, db, digit
b = -++-0--.--0+-00+++-0-+---0-+0++++0--++++0-+0+-0+0+-000-0----+0--0---+-000++--++-+-0--0-+ = -436.436
if zdx=1 then
c = +-++-.+-++- = 65.26749
if a[1]='-' then -- (-a)-b == -(a+b)
return negate(b_add(negate(a,alphabet),b,alphabet),alphabet)
end if
if b[1]='-' then -- a-(-b) == a+b
return b_add(a,negate(b,alphabet),alphabet)
end if
end if
integer adt = find('.',a),
bdt = find('.',b)
if adt or bdt then
-- remove the '.'s and zero-pad the shorter as needed
-- (thereafter treat them as two whole integers)
-- eg "1.23"+"4.5" -> "123"+"450" (leaving adt==2)
if adt then adt = length(a)-adt+1; a[-adt..-adt] = "" end if
if bdt then bdt = length(b)-bdt+1; b[-bdt..-bdt] = "" end if
if bdt>adt then
a &= repeat('0',bdt-adt)
adt = bdt
elsif adt>bdt then
b &= repeat('0',adt-bdt)
end if
end if
bool bNegate = false
if length(a)<length(b)
or (length(a)=length(b) and a_smaller(a,b,alphabet)) then
bNegate = true
{a,b} = {b,a} -- ensure b is the shorter/smaller
end if
for i=-1 to -length(a) by -1 do
da = iff(i<-length(a)?0:find(a[i],alphabet)-zdx)
db = iff(i<-length(b)?0:find(b[i],alphabet)-zdx)
digit = da - (db + carry) + zdx
carry = digit<=0
a[i] = alphabet[digit+carry*base]
if i<-length(b) and carry=0 then exit end if
end for
if carry then
?9/0 -- should have set bNegate above...
end if
if adt then
a[-adt+1..-adt] = "."
end if
a = b_trim(a)
if bNegate then
a = negate(a,alphabet)
end if
return a
end function
 
a * (b - c) = ----000-0+0+.0+0-0-00---00--0-0+--+0-0+0++-+-0--0--+0-++-0-+00-++0-0-0+++--0-+0--+-++-+-+-++-+0+-+-+
function b_mul(string a, b, alphabet)
= -262510.90268
integer base = length(alphabet),
zdx = find('0',alphabet),
dpa = find('.',a),
dpb = find('.',b),
ndp = 0
if dpa then ndp += length(a)-dpa; a[dpa..dpa] = "" end if
if dpb then ndp += length(b)-dpb; b[dpb..dpb] = "" end if
string pos = a, res = "0"
if zdx!=1 then
-- balanced number systems
string neg = negate(pos,alphabet)
for i=length(b) to 1 by -1 do
integer m = find(b[i],alphabet)-zdx
while m do
res = b_add(res,iff(m<0?neg:pos),alphabet)
m += iff(m<0?+1:-1)
end while
pos &= '0'
neg &= '0'
end for
else
-- non-balanced (normal) number systems
bool negative = false
if a[1]='-' then a = a[2..$]; negative = true end if
if b[1]='-' then b = b[2..$]; negative = not negative end if
for i=length(b) to 1 by -1 do
integer m = find(b[i],alphabet)-zdx
while m>0 do
res = b_add(res,pos,alphabet)
m -= 1
end while
pos &= '0'
end for
if negative then res = negate(res,alphabet) end if
end if
if ndp then
res[-ndp..-ndp-1] = "."
end if
res = b_trim(res)
return res
end function
 
Multiplication 27 X 12
-- [note 1] not surprisingly, the decimal output is somewhat cleaner/shorter when
x|+ (1) |+- (2) |+0 (3) |++ (4) |+-- (5)|+-0 (6)|+-+ (7)|+0- (8)|+e+-(9)|+0+(10)|++-(11)|++0(12)|
-- the decimal string inputs for a and c are used, whereas tests 1/2/5/7
1| +| +-| +0| ++| +--| +-0| +-+| +0-| +00| +0+| ++-| ++0|
-- (the 3-based ones) look much better with all ternary string inputs.
2| +-| ++| +-0| +0-| +0+| ++0| +---| +--+| +-00| +-+-| +-++| +0-0|
3| +0| +-0| +00| ++0| +--0| +-00| +-+0| +0-0| +000| +0+0| ++-0| ++00|
4| ++| +0-| ++0| +--+| +-+-| +0-0| +00+| ++--| ++00| ++++| +--0-| +--+0|
5| +--| +0+| +--0| +-+-| +0-+| +0+0| ++0-| ++++| +--00| +-0--| +-00+| +-+-0|
6| +-0| ++0| +-00| +0-0| +0+0| ++00| +---0| +--+0| +-000| +-+-0| +-++0| +0-00|
7| +-+| +---| +-+0| +00+| ++0-| +---0| +--++| +-0+-| +-+00| +0--+| +00--| +00+0|
8| +0-| +--+| +0-0| ++--| ++++| +--+0| +-0+-| +-+0+| +0-00| +000-| +0+-+| ++--0|
9| +00| +-00| +000| ++00| +--00| +-000| +-+00| +0-00| +0000| +0+00| ++-00| ++000|
10| +0+| +-+-| +0+0| ++++| +-0--| +-+-0| +0--+| +000-| +0+00| ++-0+| ++0+-| ++++0|
11| ++-| +-++| ++-0| +--0-| +-00+| +-++0| +00--| +0+-+| ++-00| ++0+-| +++++| +--0-0|
12| ++0| +0-0| ++00| +--+0| +-+-0| +0-00| +00+0| ++--0| ++000| ++++0| +--0-0| +--+00|
13| +++| +00-| +++0| +-0-+| +-++-| +00-0| +0+0+| ++0--| +++00| +---++| +--+0-| +-0-+0|
14| +---| +00+| +---0| +-0+-| +0--+| +00+0| ++-0-| ++0++| +---00| +--+--| +-0-0+| +-0+-0|
15| +--0| +0+0| +--00| +-+-0| +0-+0| +0+00| ++0-0| ++++0| +--000| +-0--0| +-00+0| +-+-00|
16| +--+| ++--| +--+0| +-+0+| +000-| ++--0| ++0++| +---+-| +--+00| +-00-+| +-+---| +-+0+0|
17| +-0-| ++-+| +-0-0| +0---| +00++| ++-+0| ++++-| +--00+| +-0-00| +-0+0-| +-+0-+| +0---0|
18| +-00| ++00| +-000| +0-00| +0+00| ++000| +---00| +--+00| +-0000| +-+-00| +-++00| +0-000|
19| +-0+| +++-| +-0+0| +0-++| ++---| +++-0| +--0-+| +-0-0-| +-0+00| +-+00+| +0--+-| +0-++0|
20| +-+-| ++++| +-+-0| +000-| ++-0+| ++++0| +--+--| +-00-+| +-+-00| +-+++-| +0-0++| +000-0|
21| +-+0| +---0| +-+00| +00+0| ++0-0| +---00| +--++0| +-0+-0| +-+000| +0--+0| +00--0| +00+00|
22| +-++| +--0-| +-++0| +0+-+| ++0+-| +--0-0| +-0-0+| +-+---| +-++00| +0-0++| +0000-| +0+-+0|
23| +0--| +--0+| +0--0| +0++-| +++-+| +--0+0| +-000-| +-+-++| +0--00| +00---| +00+0+| +0++-0|
24| +0-0| +--+0| +0-00| ++--0| ++++0| +--+00| +-0+-0| +-+0+0| +0-000| +000-0| +0+-+0| ++--00|
25| +0-+| +-0--| +0-+0| ++-0+| +---0-| +-0--0| +-0+++| +-+++-| +0-+00| +00+-+| +0++--| ++-0+0|
26| +00-| +-0-+| +00-0| ++0--| +---++| +-0-+0| +-+-+-| +0--0+| +00-00| +0+-0-| ++---+| ++0--0|
27| +000| +-000| +0000| ++000| +--000| +-0000| +-+000| +0-000| +00000| +0+000| ++-000| ++0000|
</pre>
 
=={{header|Phix}}==
Note regarding requirement #5: While this meets my definition of "reasonably efficient",
it should not shock anyone that this kind of "string maths" which works digit-by-digit
and uses repeated addition (eg *999 performs 27 additions) could easily be 10,000 times
slower than raw hardware or a carefully optimised library such as gmp. However this does
offer perfect accuracy in any given base, whereas gmp, for all it's brilliance, can hold
0.1 accurate to several million decimal places, but just never quite exact.
 
<!--<syntaxhighlight lang="phix">(phixonline)-->
procedure test(string name, alphabet)
<span style="color: #000080;font-style:italic;">-- demo\rosetta\Generic_multiplication.exw</span>
--string a = b2b("523.2391403749428",decimal,alphabet), -- [see note 1]
<span style="color: #008080;">constant</span> <span style="color: #000000;">MAX_DP</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">81</span>
string a = b2b("+-0++0+.+-0++0+",balancedternary,alphabet),
b = b2b("-436.436",decimal,alphabet),
<span style="color: #008080;">constant</span> <span style="color: #000000;">binary</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"01"</span><span style="color: #0000FF;">,</span>
-- b = b2b("-++-0--.--0+-00+++-",balancedternary,alphabet),
<span style="color: #000000;">ternary</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"012"</span><span style="color: #0000FF;">,</span>
-- c = b2b("65.26748971193416",decimal,alphabet), -- [see note 1]
<span style="color: #000000;">balancedternary</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"-0+"</span><span style="color: #0000FF;">,</span>
c = b2b("+-++-.+-++-",balancedternary,alphabet),
<span style="color: #000000;">decimal</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0123456789"</span><span style="color: #0000FF;">,</span>
d = b_add(b,c,alphabet),
<span style="color: #000000;">hexadecimal</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0123456789ABCDEF"</span><span style="color: #0000FF;">,</span>
r = b_mul(a,d,alphabet)
<span style="color: #000000;">septemvigesimal</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0123456789ABCDEFGHIJKLMNOPQ"</span><span style="color: #0000FF;">,</span>
printf(1,"%s\n%s\n",{name,repeat('=',length(name))})
<span style="color: #000080;font-style:italic;">-- heptavintimal = "0123456789ABCDEFGHKMNPRTVXZ", -- ??
printf(1," a = %.16g %s\n",{b2dec(a,alphabet),a})
-- wonky_donkey_26 = "0ABCDEFGHIJKLMNOPQRSTUVWXY",
printf(1," b = %.16g %s\n",{b2dec(b,alphabet),b})
-- wonky_donkey_27 = "0ABCDEFGHIJKLMNOPQRSTUVWXYZ",</span>
printf(1," c = %.16g %s\n",{b2dec(c,alphabet),c})
<span style="color: #000000;">balanced_base27</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"ZYXWVUTSRQPON0ABCDEFGHIJKLM"</span><span style="color: #0000FF;">,</span>
-- printf(1," d = %.16g %s\n",{b2dec(d,alphabet),d})
<span style="color: #000000;">base37</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"</span>
printf(1,"a*(b-c) = %.16g %s\n\n",{b2dec(r,alphabet),r})
<span style="color: #000080;font-style:italic;">--
end procedure
--Note: I have seen some schemes where balanced-base-27 uses
test("balanced ternary", balancedternary)
--==== the same character set as septemvigesimal, with 'D'
test("balanced base 27", balanced_base27)
-- representing 0, and wonky_donkey_27 with 'M'==0(!).
test("decimal", decimal)
-- These routines do not support that directly, except
test("binary", binary)
-- (perhaps) via a simple mapping on all inputs/outputs.
test("ternary", ternary)
-- It may be possible to add a defaulted parameter such
test("hexadecimal", hexadecimal)
-- as zero='0' - left as an exercise for the reader.
test("septemvigesimal", septemvigesimal)</lang>
-- Admittedly that balanced_base27 is entirely my own
The printed decimal output is inherently limited to IEEE 754 precision, hence I
-- invention, just for this specific task.
deliberately limited output (%.16g) because it is silly to try and go any higher,
--</span>
whereas the output from b_mul() is actually perfectly accurate, see [note 1] above.
<span style="color: #008080;">function</span> <span style="color: #000000;">b2dec</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">--
-- convert string b back into a normal (decimal) atom,
-- eg b2dec("+0-",balancedternary) yields 8
--</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">signed</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">and</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">signed</span> <span style="color: #008080;">then</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$]</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">len</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">ndp</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">ndp</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ndp</span><span style="color: #0000FF;">..</span><span style="color: #000000;">ndp</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #000080;font-style:italic;">-- remove '.'</span>
<span style="color: #000000;">ndp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">len</span><span style="color: #0000FF;">-</span><span style="color: #000000;">ndp</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">base</span><span style="color: #0000FF;">*</span><span style="color: #000000;">res</span><span style="color: #0000FF;">+</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">ndp</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">/=</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">base</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ndp</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">signed</span> <span style="color: #008080;">then</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">res</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">--
-- negate b (can be balanced or unbalanced)
--</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'0'</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- traditional: add/remove a leading '-'
-- eg "-123" &lt;==&gt; "123"</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">!=</span><span style="color: #008000;">"0"</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$]</span>
<span style="color: #008080;">else</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"-"</span><span style="color: #0000FF;">&</span><span style="color: #000000;">b</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">else</span>
<span style="color: #000080;font-style:italic;">-- balanced: mirror [non-0] digits
-- eg "-0+" (ie -8) &lt;==&gt; "+0-" (ie +8)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]!=</span><span style="color: #008000;">'.'</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[-</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">b</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">b_trim</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- (common code)
-- trim trailing ".000"</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">trim_tail</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">trim_tail</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">),</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000080;font-style:italic;">-- trim leading zeroes, but not "0.nnn" -&gt; ".nnn"
-- [hence we cannot use the standard trim_head()]</span>
<span style="color: #008080;">while</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)></span><span style="color: #000000;">1</span> <span style="color: #008080;">and</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'0'</span> <span style="color: #008080;">and</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]!=</span><span style="color: #008000;">'.'</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">b</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">b_carry</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">digit</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">base</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">idx</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">string</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- (common code, for balanced number systems only)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">></span><span style="color: #000000;">base</span><span style="color: #0000FF;">?+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">digit</span><span style="color: #0000FF;"><</span><span style="color: #000000;">1</span><span style="color: #0000FF;">?-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #000000;">0</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">carry</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">idx</span> <span style="color: #008080;">to</span> <span style="color: #000000;">0</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]!=</span><span style="color: #008000;">'.'</span> <span style="color: #008080;">then</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">k</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">k</span><span style="color: #0000FF;"><</span><span style="color: #000000;">base</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">n</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">exit</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">n</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">digit</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">base</span><span style="color: #0000FF;">*</span><span style="color: #000000;">carry</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">b2b</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet2</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">--
-- convert a string from alphabet to alphabet2,
-- eg b2b("8",decimal,balancedternary) yields "+0-",
-- & b2b("+0-",balancedternary,decimal) yields "8",
--</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">!=</span><span style="color: #008000;">"0"</span> <span style="color: #008080;">then</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">base2</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet2</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zdx2</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet2</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">q</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">digit</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">negative</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">((</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">and</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">or</span>
<span style="color: #0000FF;">(</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)<</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">negative</span> <span style="color: #008080;">then</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">ndp</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">ndp</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">ndp</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span><span style="color: #000000;">n</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ndp</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..$]}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span>
<span style="color: #008080;">while</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #000080;font-style:italic;">--
-- this is a digit-by-digit divide (/mod) loop
-- eg for hex-&gt;decimal we would want:
-- this loop/modrem("FFFF",10) --&gt; "1999" rem 5,
-- this loop/modrem("1999",10) --&gt; "28F" rem 3,
-- this loop/modrem("28F",10) --&gt; "41" rem 5,
-- this loop/modrem("41",10) --&gt; "6" rem 5,
-- this loop/modrem("6",10) --&gt; "0" rem 6,
-- ==&gt; res:="65535" (in 5 full iterations over n).
--</span>
<span style="color: #000000;">digit</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span>
<span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">q</span><span style="color: #0000FF;">*</span><span style="color: #000000;">base</span><span style="color: #0000FF;">+</span><span style="color: #000000;">digit</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base2</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">digit</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">/</span><span style="color: #000000;">base2</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">zdx</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_carry</span><span style="color: #0000FF;">(</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">n</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">r</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">zdx2</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx2</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">carry</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">></span><span style="color: #000000;">base2</span><span style="color: #0000FF;">?+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;"><</span><span style="color: #000000;">1</span><span style="color: #0000FF;">?-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #000000;">0</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">base2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">carry</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet2</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">]&</span><span style="color: #000000;">res</span>
<span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">trim_head</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">carry</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet2</span><span style="color: #0000FF;">[</span><span style="color: #000000;">carry</span><span style="color: #0000FF;">+</span><span style="color: #000000;">zdx2</span><span style="color: #0000FF;">]&</span><span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">&=</span> <span style="color: #008000;">'.'</span>
<span style="color: #000000;">ndp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- convert fraction to unbalanced, to simplify the (other-base) multiply.</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">lm</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">alphanew</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">base37</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)]</span>
<span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b2b</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphanew</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (nb: no fractional part!)</span>
<span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">lm</span><span style="color: #0000FF;">-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">))&</span><span style="color: #000000;">m</span> <span style="color: #000080;font-style:italic;">-- zero-pad if required</span>
<span style="color: #000000;">alphabet</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphanew</span>
<span style="color: #000000;">zdx</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">while</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #000000;">ndp</span><span style="color: #0000FF;"><</span><span style="color: #000000;">MAX_DP</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #000080;font-style:italic;">--
-- this is a digit-by-digit multiply loop
-- eg for [.]"1415" decimal-&gt;decimal we
-- would repeatedly multiply by 10, giving
-- 1 and "4150", then 4 and "1500", then
-- 1 and "5000", then 5 and "0000". We
-- strip zeroes between each output digit
-- & obviously normally alphabet in!=out.
--</span>
<span style="color: #000000;">digit</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span>
<span style="color: #000000;">q</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">digit</span><span style="color: #0000FF;">*</span><span style="color: #000000;">base2</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">zdx</span>
<span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">/</span><span style="color: #000000;">base</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">m</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">digit</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">q</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">zdx2</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx2</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_carry</span><span style="color: #0000FF;">(</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base2</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">),</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet2</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">alphabet2</span><span style="color: #0000FF;">[</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">trim_tail</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">,</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">ndp</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">negative</span> <span style="color: #008080;">then</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet2</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">atm2b</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">string</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">--
-- convert d to a string in the specified base,
-- eg atm2b(65535,hexadecimal) =&gt; "FFFF"
--
-- As a standard feature of phix, you can actually specify
-- d in any number base between 2 and 36, eg 0(13)168 is
-- equivalent to 255 (see test\t37misc.exw for more), but
-- not (yet) in balanced number bases, or with fractions,
-- except (of course) for normal decimal fractions.
--
-- Note that eg b2b("-436.436",decimal,balancedternary) is
-- more acccurate that atm2b(-436.436,balancedternary) due
-- to standard IEEE 754 floating point limitations.
-- For integers, discrepancies only creep in for values
-- outside the range +/-9,007,199,254,740,992 (on 32-bit).
-- However, this is much simpler and faster than b2b().
--</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">neg</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">d</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">neg</span> <span style="color: #008080;">then</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">d</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">whole</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">d</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">whole</span>
<span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">ch</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">whole</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">zdx</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">ch</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">carry</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ch</span><span style="color: #0000FF;">></span><span style="color: #000000;">base</span><span style="color: #0000FF;">?+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ch</span><span style="color: #0000FF;"><</span><span style="color: #000000;">1</span><span style="color: #0000FF;">?-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #000000;">0</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">ch</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">base</span><span style="color: #0000FF;">*</span><span style="color: #000000;">carry</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ch</span><span style="color: #0000FF;">]&</span><span style="color: #000000;">res</span>
<span style="color: #000000;">whole</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">whole</span><span style="color: #0000FF;">/</span><span style="color: #000000;">base</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">whole</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">carry</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">carry</span><span style="color: #0000FF;">+</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">]&</span><span style="color: #000000;">res</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">&=</span> <span style="color: #008000;">'.'</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">ndp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">and</span> <span style="color: #000000;">ndp</span><span style="color: #0000FF;"><</span><span style="color: #000000;">MAX_DP</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">d</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">base</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">digit</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">zdx</span>
<span style="color: #000000;">d</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">digit</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_carry</span><span style="color: #0000FF;">(</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">),</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">ndp</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">neg</span> <span style="color: #008080;">then</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #000080;font-style:italic;">-- negative numbers in addition and subtraction
-- (esp. non-balanced) are treated as follows:
-- for -ve a: (-a)+b == b-a; (-a)-b == -(a+b)
-- for -ve b: a+(-b) == a-b; a-(-b) == a+b
-- for a&gt;b: a-b == -(b-a) [avoid running off end]</span>
<span style="color: #008080;">forward</span> <span style="color: #008080;">function</span> <span style="color: #000000;">b_sub</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">da</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">db</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">digit</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- (-a)+b == b-a</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">b_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- a+(-b) == a-b</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">b_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">bdt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">or</span> <span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- remove the '.'s and zero-pad the shorter as needed
-- (thereafter treat them as two whole integers)
-- eg "1.23"+"4.5" -&gt; "123"+"450" (leaving adt==2)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">then</span> <span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span> <span style="color: #000000;">bdt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">bdt</span><span style="color: #0000FF;">></span><span style="color: #000000;">adt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bdt</span>
<span style="color: #008080;">elsif</span> <span style="color: #000000;">adt</span><span style="color: #0000FF;">></span><span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)<</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">}</span> <span style="color: #000080;font-style:italic;">-- ensure b is the shorter</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=-</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #0000FF;">-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">da</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)?</span><span style="color: #000000;">0</span><span style="color: #0000FF;">:</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">db</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)?</span><span style="color: #000000;">0</span><span style="color: #0000FF;">:</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">digit</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">da</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">db</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">carry</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">zdx</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">></span><span style="color: #000000;">base</span><span style="color: #0000FF;">?+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">digit</span><span style="color: #0000FF;"><</span><span style="color: #000000;">1</span><span style="color: #0000FF;">?-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #000000;">0</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">-</span><span style="color: #000000;">carry</span><span style="color: #0000FF;">*</span><span style="color: #000000;">base</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #000000;">carry</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">carry</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">carry</span><span style="color: #0000FF;">+</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">]&</span><span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"."</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">a_smaller</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- return true if a is smaller than b
-- if not balanced then both are +ve</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)!=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- sanity check</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">da</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">db</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">compare</span><span style="color: #0000FF;">(</span><span style="color: #000000;">da</span><span style="color: #0000FF;">,</span><span style="color: #000000;">db</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">c</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #004600;">false</span> <span style="color: #000080;font-style:italic;">-- (=, which is not &lt;)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">b_sub</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">da</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">db</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">digit</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- (-a)-b == -(a+b)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- a-(-b) == a+b</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">bdt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">or</span> <span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- remove the '.'s and zero-pad the shorter as needed
-- (thereafter treat them as two whole integers)
-- eg "1.23"+"4.5" -&gt; "123"+"450" (leaving adt==2)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">then</span> <span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span> <span style="color: #000000;">bdt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">bdt</span><span style="color: #0000FF;">></span><span style="color: #000000;">adt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bdt</span>
<span style="color: #008080;">elsif</span> <span style="color: #000000;">adt</span><span style="color: #0000FF;">></span><span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">bNegate</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">false</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)<</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">or</span> <span style="color: #0000FF;">(</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #000000;">a_smaller</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">))</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">bNegate</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">}</span> <span style="color: #000080;font-style:italic;">-- ensure b is the shorter/smaller</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=-</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #0000FF;">-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">da</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)?</span><span style="color: #000000;">0</span><span style="color: #0000FF;">:</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">db</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)?</span><span style="color: #000000;">0</span><span style="color: #0000FF;">:</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">digit</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">da</span> <span style="color: #0000FF;">-</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">db</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">carry</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">zdx</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">digit</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">0</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">+</span><span style="color: #000000;">carry</span><span style="color: #0000FF;">*</span><span style="color: #000000;">base</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #000000;">carry</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">carry</span> <span style="color: #008080;">then</span>
<span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #000080;font-style:italic;">-- should have set bNegate above...</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"."</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">bNegate</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">b_mul</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">dpa</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">dpb</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">ndp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">dpa</span> <span style="color: #008080;">then</span> <span style="color: #000000;">ndp</span> <span style="color: #0000FF;">+=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">dpa</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">dpa</span><span style="color: #0000FF;">..</span><span style="color: #000000;">dpa</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">dpb</span> <span style="color: #008080;">then</span> <span style="color: #000000;">ndp</span> <span style="color: #0000FF;">+=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">dpb</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">dpb</span><span style="color: #0000FF;">..</span><span style="color: #000000;">dpb</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">pos</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0"</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- balanced number systems</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">neg</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">m</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span><span style="color: #0000FF;">?</span><span style="color: #000000;">neg</span><span style="color: #0000FF;">:</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">),</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">m</span> <span style="color: #0000FF;">+=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span><span style="color: #0000FF;">?+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #000000;">pos</span> <span style="color: #0000FF;">&=</span> <span style="color: #008000;">'0'</span>
<span style="color: #000000;">neg</span> <span style="color: #0000FF;">&=</span> <span style="color: #008000;">'0'</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">else</span>
<span style="color: #000080;font-style:italic;">-- non-balanced (normal) number systems</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">negative</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">false</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$];</span> <span style="color: #000000;">negative</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$];</span> <span style="color: #000000;">negative</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">not</span> <span style="color: #000000;">negative</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">></span><span style="color: #000000;">0</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">m</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #000000;">pos</span> <span style="color: #0000FF;">&=</span> <span style="color: #008000;">'0'</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">negative</span> <span style="color: #008080;">then</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">ndp</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">res</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">ndp</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">ndp</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"."</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #000080;font-style:italic;">-- [note 1] not surprisingly, the decimal output is somewhat cleaner/shorter when
-- the decimal string inputs for a and c are used, whereas tests 1/2/5/7
-- (the 3-based ones) look much better with all ternary string inputs.</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">name</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">--string a = b2b("523.2391403749428",decimal,alphabet), -- [see note 1]</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b2b</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"+-0++0+.+-0++0+"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">balancedternary</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b2b</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"-436.436"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">decimal</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000080;font-style:italic;">-- b = b2b("-++-0--.--0+-00+++-",balancedternary,alphabet),
-- c = b2b("65.26748971193416",decimal,alphabet), -- [see note 1]</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b2b</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"+-++-.+-++-"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">balancedternary</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%s\n%s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">name</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'='</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">name</span><span style="color: #0000FF;">))})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" a = %.16g %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">b2dec</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">a</span><span style="color: #0000FF;">})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" b = %.16g %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">b2dec</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">b</span><span style="color: #0000FF;">})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" c = %.16g %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">b2dec</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">c</span><span style="color: #0000FF;">})</span>
<span style="color: #000080;font-style:italic;">-- printf(1," d = %.16g %s\n",{b2dec(d,alphabet),d})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"a*(b-c) = %.16g %s\n\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">b2dec</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">r</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"balanced ternary"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">balancedternary</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"balanced base 27"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">balanced_base27</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"decimal"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">decimal</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"binary"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">binary</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"ternary"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">ternary</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"hexadecimal"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">hexadecimal</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"septemvigesimal"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">septemvigesimal</span><span style="color: #0000FF;">)</span>
<!--</syntaxhighlight>-->
The printed decimal output is inherently limited to IEEE 754 precision, hence I deliberately limited output (%.16g)
because it is silly to try and go any higher, whereas the output from b_mul() is actually perfectly accurate,
see [note 1] regarding decimal/ternary input, just above.
{{out}}
<pre>
Line 1,408 ⟶ 1,620:
=== multiplication table ===
Without e notation, with hexadecimal across, septemvigesimal down, and balanced ternary contents!
<!--<syntaxhighlight lang="phix">(phixonline)-->
<lang Phix>printf(1,"* |")
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"* |"</span><span style="color: #0000FF;">)</span>
for j=1 to 12 do
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">12</span> <span style="color: #008080;">do</span>
printf(1," #%s %3s |",{atm2b(j,hexadecimal),atm2b(j,balancedternary)})
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" #%s %3s |"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">atm2b</span><span style="color: #0000FF;">(</span><span style="color: #000000;">j</span><span style="color: #0000FF;">,</span><span style="color: #000000;">hexadecimal</span><span style="color: #0000FF;">),</span><span style="color: #000000;">atm2b</span><span style="color: #0000FF;">(</span><span style="color: #000000;">j</span><span style="color: #0000FF;">,</span><span style="color: #000000;">balancedternary</span><span style="color: #0000FF;">)})</span>
end for
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
for i=1 to 27 do
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">27</span> <span style="color: #008080;">do</span>
string a = atm2b(i,balancedternary)
<span style="color: #004080;">string</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">atm2b</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">balancedternary</span><span style="color: #0000FF;">)</span>
printf(1,"\n%-2s|",{atm2b(i,septemvigesimal)})
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n%-2s|"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">atm2b</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">septemvigesimal</span><span style="color: #0000FF;">)})</span>
for j=1 to 12 do
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">12</span> <span style="color: #008080;">do</span>
if j>i then
<span style="color: #008080;">if</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">></span><span style="color: #000000;">i</span> <span style="color: #008080;">then</span>
printf(1," |")
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" |"</span><span style="color: #0000FF;">)</span>
else
<span string b style="color: atm2b(j,balancedternary)#008080;">else</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">atm2b</span><span style="color: #0000FF;">(</span><span style="color: #000000;">j</span><span style="color: #0000FF;">,</span><span style="color: #000000;">balancedternary</span><span style="color: #0000FF;">)</span>
string m = b_mul(a,b,balancedternary)
<span style="color: #004080;">string</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">balancedternary</span><span style="color: #0000FF;">)</span>
printf(1," %6s |",{m})
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" %6s |"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">m</span><span style="color: #0000FF;">})</span>
end if
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
end for
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
end for
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
printf(1,"\n")</lang>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">)</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
* | #1 + | #2 +- | #3 +0 | #4 ++ | #5 +-- | #6 +-0 | #7 +-+ | #8 +0- | #9 +00 | #A +0+ | #B ++- | #C ++0 |
1 | + | | | | | | | | | | | |
2 | +- | ++ | | | | | | | | | | |
3 | +0 | +-0 | +00 | | | | | | | | | |
4 | ++ | +0- | ++0 | +--+ | | | | | | | | |
5 | +-- | +0+ | +--0 | +-+- | +0-+ | | | | | | | |
6 | +-0 | ++0 | +-00 | +0-0 | +0+0 | ++00 | | | | | | |
7 | +-+ | +--- | +-+0 | +00+ | ++0- | +---0 | +--++ | | | | | |
8 | +0- | +--+ | +0-0 | ++-- | ++++ | +--+0 | +-0+- | +-+0+ | | | | |
9 | +00 | +-00 | +000 | ++00 | +--00 | +-000 | +-+00 | +0-00 | +0000 | | | |
A | +0+ | +-+- | +0+0 | ++++ | +-0-- | +-+-0 | +0--+ | +000- | +0+00 | ++-0+ | | |
B | ++- | +-++ | ++-0 | +--0- | +-00+ | +-++0 | +00-- | +0+-+ | ++-00 | ++0+- | +++++ | |
C | ++0 | +0-0 | ++00 | +--+0 | +-+-0 | +0-00 | +00+0 | ++--0 | ++000 | ++++0 | +--0-0 | +--+00 |
D | +++ | +00- | +++0 | +-0-+ | +-++- | +00-0 | +0+0+ | ++0-- | +++00 | +---++ | +--+0- | +-0-+0 |
E | +--- | +00+ | +---0 | +-0+- | +0--+ | +00+0 | ++-0- | ++0++ | +---00 | +--+-- | +-0-0+ | +-0+-0 |
F | +--0 | +0+0 | +--00 | +-+-0 | +0-+0 | +0+00 | ++0-0 | ++++0 | +--000 | +-0--0 | +-00+0 | +-+-00 |
G | +--+ | ++-- | +--+0 | +-+0+ | +000- | ++--0 | ++0++ | +---+- | +--+00 | +-00-+ | +-+--- | +-+0+0 |
H | +-0- | ++-+ | +-0-0 | +0--- | +00++ | ++-+0 | ++++- | +--00+ | +-0-00 | +-0+0- | +-+0-+ | +0---0 |
I | +-00 | ++00 | +-000 | +0-00 | +0+00 | ++000 | +---00 | +--+00 | +-0000 | +-+-00 | +-++00 | +0-000 |
J | +-0+ | +++- | +-0+0 | +0-++ | ++--- | +++-0 | +--0-+ | +-0-0- | +-0+00 | +-+00+ | +0--+- | +0-++0 |
K | +-+- | ++++ | +-+-0 | +000- | ++-0+ | ++++0 | +--+-- | +-00-+ | +-+-00 | +-+++- | +0-0++ | +000-0 |
L | +-+0 | +---0 | +-+00 | +00+0 | ++0-0 | +---00 | +--++0 | +-0+-0 | +-+000 | +0--+0 | +00--0 | +00+00 |
M | +-++ | +--0- | +-++0 | +0+-+ | ++0+- | +--0-0 | +-0-0+ | +-+--- | +-++00 | +0-0++ | +0000- | +0+-+0 |
N | +0-- | +--0+ | +0--0 | +0++- | +++-+ | +--0+0 | +-000- | +-+-++ | +0--00 | +00--- | +00+0+ | +0++-0 |
O | +0-0 | +--+0 | +0-00 | ++--0 | ++++0 | +--+00 | +-0+-0 | +-+0+0 | +0-000 | +000-0 | +0+-+0 | ++--00 |
P | +0-+ | +-0-- | +0-+0 | ++-0+ | +---0- | +-0--0 | +-0+++ | +-+++- | +0-+00 | +00+-+ | +0++-- | ++-0+0 |
Q | +00- | +-0-+ | +00-0 | ++0-- | +---++ | +-0-+0 | +-+-+- | +0--0+ | +00-00 | +0+-0- | ++---+ | ++0--0 |
10| +000 | +-000 | +0000 | ++000 | +--000 | +-0000 | +-+000 | +0-000 | +00000 | +0+000 | ++-000 | ++0000 |
</pre>
 
=={{header|Wren}}==
{{trans|Phix}}
{{libheader|Wren-str}}
{{libheader|Wren-iterate}}
{{libheader|Wren-fmt}}
Translated via the Go entry.
<syntaxhighlight lang="wren">import "./str" for Str
import "./iterate" for Stepped
import "./fmt" for Fmt
 
var maxdp = 81
var binary = "01"
var ternary = "012"
var balancedTernary = "-0+"
var decimal = "0123456789"
var hexadecimal = "0123456789ABCDEF"
var septemVigesimal = "0123456789ABCDEFGHIJKLMNOPQ"
var balancedBase27 = "ZYXWVUTSRQPON0ABCDEFGHIJKLM"
var base37 = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
 
// converts Phix indices to Wren
var wIndex = Fn.new { |pIndex, le|
if (pIndex < 0) return pIndex + le
return pIndex - 1
}
 
var getCarry = Fn.new { |digit, base|
if (digit > base) {
return 1
} else if (digit < 1) {
return -1
}
return 0
}
 
// convert string 'b' to a decimal floating point number
var b2dec = Fn.new { |b, alphabet|
var res = 0
var base = alphabet.count
var zdx = alphabet.indexOf("0") + 1
var signed = zdx == 1 && b[0] == "-"
if (signed) b = b[1..-1]
var le = b.count
var ndp = b.indexOf(".") + 1
if (ndp != 0) {
b = Str.delete(b, ndp - 1) // remove decimal point
ndp = le - ndp
}
for (i in Stepped.ascend(1..b.count)) {
var idx = alphabet.indexOf(b[i-1]) + 1
res = base * res + idx - zdx
}
if (ndp != 0) res = res / base.pow(ndp)
if (signed) res = -res
return res
}
 
// string 'b' can be balanced or unbalanced
var negate = Fn.new { |b, alphabet|
if (alphabet[0] == "0") {
if (b != "0") b = (b[0] == "-") ? b[1..-1] : Str.insert(b, 0, "-")
} else {
for (i in Stepped.ascend(1..b.count)) {
if (b[i-1] != ".") {
var idx = alphabet.indexOf(b[i-1]) + 1
var wi = wIndex.call(-idx, alphabet.count)
b = Str.change(b, i-1, alphabet[wi])
}
}
}
return b
}
 
var bTrim = Fn.new { |b|
// trim trailing ".000"
var idx = b.indexOf(".") + 1
if (idx != 0) b = b.trimEnd("0").trimEnd(".")
// trim leading zeros but not "0.nnn"
while (b.count > 1 && b[0] == "0" && b[1] != ".") b = b[1..-1]
return b
}
 
// for balanced number systems only
var bCarry = Fn.new { |digit, base, idx, n, alphabet|
var carry = getCarry.call(digit, base)
if (carry != 0) {
for (i in Stepped.descend(idx..1)) {
if (n[i-1] != ".") {
var k = alphabet.indexOf(n[i-1]) + 1
if (k < base) {
n = Str.change(n, i-1, alphabet[k])
break
}
n = Str.change(n, i-1, alphabet[0])
}
}
digit = digit - base * carry
}
return [digit, n]
}
 
// convert a string from alphabet to alphabet2
var b2b // recursive function
b2b = Fn.new { |n, alphabet, alphabet2|
var res = "0"
var m = ""
if (n != "0") {
var base = alphabet.count
var base2 = alphabet2.count
var zdx = alphabet.indexOf("0") + 1
var zdx2 = alphabet2.indexOf("0") + 1
var carry = 0
var q = 0
var r = 0
var digit = 0
var idx = alphabet.indexOf(n[0]) + 1
var negative = (zdx == 1 && n[0] == "-") || (zdx != 1 && idx < zdx)
if (negative) n = negate.call(n, alphabet)
var ndp = n.indexOf(".") + 1
if (ndp != 0) {
var t = n
n = n[0...ndp-1]
m = t[ndp..-1]
}
res = ""
while (n.count > 0) {
q = 0
for (i in Stepped.ascend(1..n.count)) {
digit = alphabet.indexOf(n[i-1]) + 1 - zdx
q = q*base + digit
r = q.abs % base2
digit = (q.abs/base2).floor + zdx
if (q < 0) digit = digit - 1
if (zdx != 1) {
var p = bCarry.call(digit, base, i-1, n, alphabet)
digit = p[0]
n = p[1]
}
n = Str.change(n, i-1, alphabet[digit-1])
q = r
}
r = r + zdx2
if (zdx2 != 1) {
r = r + carry
carry = getCarry.call(r, base2)
r = r - base2 * carry
}
res = Str.insert(res, 0, alphabet2[r-1])
n = n.trimStart("0")
}
if (carry != 0) res = Str.insert(res, 0, alphabet2[carry+zdx2-1])
if (m.count > 0) {
res = res + "."
ndp = 0
if (zdx != 1) {
var lm = m.count
var alphaNew = base37[0...alphabet.count]
m = b2b.call(m, alphabet, alphaNew)
m = ("0" * (lm-m.count)) + m
alphabet = alphaNew
zdx = 1
}
while (m.count > 0 && ndp < maxdp) {
q = 0
for (i in Stepped.descend(m.count..1)) {
digit = alphabet.indexOf(m[i-1]) + 1 - zdx
q = q + digit * base2
r = q.abs % base + zdx
q = (q / base).truncate
if (q < 0) q = q - 1
m = Str.change(m, i-1, alphabet[r-1])
}
digit = q + zdx2
if (zdx2 != 1) {
var p = bCarry.call(digit, base2, res.count, res, alphabet2)
digit = p[0]
res = p[1]
}
res = res + alphabet2[digit-1]
m = m.trimEnd("0")
ndp = ndp + 1
}
}
res = bTrim.call(res)
if (negative) res = negate.call(res, alphabet2)
}
return res
}
 
// convert 'd' to a string in the specified base
var float2b = Fn.new { |d, alphabet|
var base = alphabet.count
var zdx = alphabet.indexOf("0") + 1
var carry = 0
var neg = d < 0
if (neg) d = -d
var res = ""
var whole = d.floor
d = d - whole
while (true) {
var ch = whole % base + zdx
if (zdx != 1) {
ch = ch + carry
carry = getCarry.call(ch, base)
ch = ch - base * carry
}
res = Str.insert(res, 0, alphabet[ch-1])
whole = (whole / base).truncate
if (whole == 0) break
}
if (carry != 0) {
res = Str.insert(res, 0, alphabet[carry+zdx-1])
carry = 0
}
if (d != 0) {
res = res + "."
var ndp = 0
while (d != 0 && ndp < maxdp) {
d = d * base
var digit = d.truncate + zdx
d = d - digit
if (zdx != 1) {
var p = bCarry.call(digit, base, res.count, res, alphabet)
digit = p[0]
res = p[1]
}
res = res + alphabet[digit-1]
ndp = ndp + 1
}
}
if (neg) res = negate.call(res, alphabet)
return res
}
 
var bSub // forwward declaration
 
var bAdd = Fn.new { |a, b, alphabet|
var base = alphabet.count
var zdx = alphabet.indexOf("0") + 1
var carry = 0
var da = 0
var db = 0
var digit = 0
if (zdx == 1) {
if (a[0] == "-") {
return bSub.call(b, negate.call(a, alphabet), alphabet)
}
if (b[0] == "-") {
return bSub.call(a, negate.call(b, alphabet), alphabet)
}
}
var adt = a.indexOf(".") + 1
var bdt = b.indexOf(".") + 1
if (adt != 0 || bdt != 0) {
if (adt != 0) {
adt = a.count - adt + 1
var wi = wIndex.call(-adt, a.count)
a = Str.delete(a, wi)
}
if (bdt != 0) {
bdt = b.count - bdt + 1
var wi = wIndex.call(-bdt, b.count)
b = Str.delete(b, wi)
}
if (bdt > adt) {
a = a + ("0" * (bdt-adt))
adt = bdt
} else if (adt > bdt) {
b = b + ("0" * (adt-bdt))
}
}
if (a.count < b.count) {
var t = a
a = b
b = t
}
for (i in Stepped.descend(-1..-a.count)) {
if (i < -a.count) {
da = 0
} else {
da = alphabet.indexOf(a[a.count + i]) + 1 - zdx
}
if (i < -b.count) {
db = 0
} else {
db = alphabet.indexOf(b[b.count + i]) + 1 - zdx
}
digit = da + db + carry + zdx
carry = getCarry.call(digit, base)
a = Str.change(a, i + a.count, alphabet[digit-carry*base-1])
if (i < -b.count && carry == 0) break
}
if (carry != 0) {
a = Str.insert(a, 0, alphabet[carry+zdx-1])
}
if (adt != 0) {
var wi = wIndex.call(-adt+1, a.count)
a = Str.insert(a, wi, ".")
}
a = bTrim.call(a)
return a
}
 
var aSmaller = Fn.new { |a, b, alphabet|
if (a.count != b.count) Fiber.abort("strings should be equal in length")
for (i in Stepped.ascend(1..a.count)) {
var da = alphabet.indexOf(a[i-1]) + 1
var db = alphabet.indexof(b[i-1]) + 1
if (da != db) return da < db
}
return false
}
 
// declared earlier
bSub = Fn.new { |a, b, alphabet|
var base = alphabet.count
var zdx = alphabet.indexOf("0") + 1
var carry = 0
var da = 0
var db = 0
var digit = 0
if (zdx == 1) {
if (a[0] == "-") {
return negate.call(bAdd.call(negate.call(a, alphabet), b, alphabet), alphabet)
}
if (b[0] == "-") {
return bAdd.call(a, negate.call(b, alphabet), alphabet)
}
}
var adt = a.indexOf(".") + 1
var bdt = b.indexOf(".") + 1
if (adt != 0 || bdt != 0) {
if (adt != 0) {
adt = a.count - adt + 1
var wi = wIndex.call(-adt, a.count)
a = Str.delete(a, wi)
}
if (bdt != 0) {
bdt = b.count - bdt + 1
var wi = wIndex.call(-bdt, b.count)
b = Str.delete(b, wi)
}
if (bdt > adt) {
a = a + ("0" * (bdt-adt))
adt = bdt
} else if (adt > bdt) {
b = b + ("0" * (adt-bdt))
}
}
var bNegate = false
if (a.count < b.count || (a.count == b.count && aSmaller.call(a, b, alphabet))) {
bNegate = true
var t = a
a = b
b = t
}
for (i in Stepped.descend(-1..-a.count)) {
if (i < -a.count) {
da = 0
} else {
da = alphabet.indexOf(a[a.count+i]) + 1 - zdx
}
if (i < -b.count) {
db = 0
} else {
db = alphabet.indexOf(b[b.count+i]) + 1 - zdx
}
digit = da - db - carry + zdx
carry = 0
if (digit <= 0) carry = 1
a = Str.change(a, i+a.count, alphabet[digit+carry*base-1])
if (i < -b.count && carry == 0) break
}
if (carry != 0) Fiber.abort("carry should be zero")
if (adt != 0) {
var wi = wIndex.call(-adt+1, a.count)
a = Str.insert(a, wi, ".")
}
a = bTrim.call(a)
if (bNegate) a = negate.call(a, alphabet)
return a
}
 
var bMul = Fn.new { |a, b, alphabet|
var zdx = alphabet.indexOf("0") + 1
var dpa = a.indexOf(".") + 1
var dpb = b.indexOf(".") + 1
var ndp = 0
if (dpa != 0) {
ndp = ndp + a.count - dpa
a = Str.delete(a, dpa-1)
}
if (dpb != 0) {
ndp = ndp + b.count - dpb
b = Str.delete(b, dpb-1)
}
var pos = a
var res = "0"
if (zdx != 1) {
// balanced number systems
var neg = negate.call(pos, alphabet)
for (i in Stepped.descend(b.count..1)) {
var m = alphabet.indexOf(b[i-1]) + 1 - zdx
while (m != 0) {
var temp = pos
var temp2 = -1
if (m < 0) {
temp = neg
temp2 = 1
}
res = bAdd.call(res, temp, alphabet)
m = m + temp2
}
pos = pos + "0"
neg = neg + "0"
}
} else {
// non-balanced number systems
var negative = false
if (a[0] == "-") {
a = a[1..-1]
negative = true
}
if (b[0] == "-") {
b = b[1..-1]
negative = !negative
}
for (i in Stepped.descend(b.count..1)) {
var m = alphabet.indexOf(b[i-1]) + 1 - zdx
while (m > 0) {
res = bAdd.call(res, pos, alphabet)
m = m - 1
}
pos = pos + "0"
}
if (negative) res = negate.call(res, alphabet)
}
if (ndp != 0) {
var wi = wIndex.call(-ndp, res.count)
res = Str.insert(res, wi, ".")
}
res = bTrim.call(res)
return res
}
 
var multTable = Fn.new {
System.print("multiplication table")
System.print("====================")
System.write("* |")
for (j in 1..12) {
Fmt.write(" #$s $3s |", float2b.call(j, hexadecimal), float2b.call(j, balancedTernary))
}
for (i in 1..27) {
var a = float2b.call(i, balancedTernary)
Fmt.write("\n$-2s|", float2b.call(i, septemVigesimal))
for (j in 1..12) {
if (j > i) {
System.write(" |")
} else {
var b = float2b.call(j, balancedTernary)
var m = bMul.call(a, b, balancedTernary)
Fmt.write(" $6s |", m)
}
}
}
System.print()
}
 
var test = Fn.new {|name, alphabet|
var a = b2b.call("+-0++0+.+-0++0+", balancedTernary, alphabet)
var b = b2b.call("-436.436", decimal, alphabet)
var c = b2b.call("+-++-.+-++-", balancedTernary, alphabet)
var d = bSub.call(b, c, alphabet)
var r = bMul.call(a, d, alphabet)
Fmt.print("$s\n$s", name, "=" * name.count)
Fmt.print(" a = $.14f $s", b2dec.call(a, alphabet), a)
Fmt.print(" b = $.14f $s", b2dec.call(b, alphabet), b)
Fmt.print(" c = $.14f $s", b2dec.call(c, alphabet), c)
Fmt.print("a*(b-c) = $.14f $s\n", b2dec.call(r, alphabet), r)
}
 
test.call("balanced ternary", balancedTernary)
test.call("balanced base 27", balancedBase27)
test.call("decimal", decimal)
test.call("binary", binary)
test.call("ternary", ternary)
test.call("hexadecimal", hexadecimal)
test.call("septemvigesimal", septemVigesimal)
multTable.call()</syntaxhighlight>
 
{{out}}
<pre>
balanced ternary
================
a = 523.23914037494285 +-0++0+.+-0++0+
b = -436.43599999999992 -++-0--.--0+-00+++-0-+---0-+0++++0--0000+00-+-+--+0-0-00--++0-+00---+0+-+++0+-0----0++
c = 65.26748971193416 +-++-.+-++-
a*(b-c) = -262510.90267998125637 ----000-0+0+.0+0-0-00---00--0-0+--+--00-0++-000++0-000-+0+-----+++-+-0+-+0+0++0+0-++-++0+---00++++
 
balanced base 27
================
a = 523.23914037494285 AUJ.FLI
b = -436.43600000000004 NKQ.YFDFTYSMHVANGXPVXHIZJRJWZD0PBGFJAEBAKOZODLY0ITEHPQLSQSGLFZUINATKCIKUVMWEWJMQ0COTS
c = 65.26748971193416 BK.GF
a*(b-c) = -262510.90267998125637 ZVPJ.CWNYQPEENDVDPNJZXKFGCLHKLCX0YIBOMETHFWWBTVUFAH0SEZMTBJDCRRAQIQCAWMKXSTPYUXYPK0LODUO
 
decimal
=======
a = 523.23914037494296 523.239140374942844078646547782350251486053955189757658893461362597165066300868770004
b = -436.43599999999998 -436.436
c = 65.26748971193415 65.267489711934156378600823045267489711934156378600823045267489711934156378600823045
a*(b-c) = -262510.90267998137278 -262510.90267998140903693918986303277315826215892262734715612833785876513103053772667101895163734826631742752252837097627017862754285047634638652268078676654605120794218
 
binary
======
a = 523.23914037494274 1000001011.001111010011100001001101101110011000100001011110100101001010100100000111001000111
b = -436.43600000000004 -110110100.011011111001110110110010001011010000111001010110000001000001100010010011011101001
c = 65.26748971193416 1000001.01000100011110100011010010101100110001100000111010111111101111001001001101111101
a*(b-c) = -262510.90267998143099 -1000000000101101110.111001110001011000001001000001101110011111011100000100000100001000101011100011110010110001010100110111001011101001010000001110110100111110001101000000001111110101
 
ternary
=======
a = 523.23914037494285 201101.0201101
b = -436.43600000000021 -121011.102202211210021110012111201022222000202102010100101200200110122011122101110212
c = 65.26748971193416 2102.02102
a*(b-c) = -262510.90267998125637 -111100002121.2201010011100110022102110002120222120100001221111011202022012121122001201122110221112
 
hexadecimal
===========
a = 523.23914037494274 20B.3D384DB9885E94A90723EF9CBCB174B443E45FFC41152FE0293416F15E3AC303A0F3799ED81589C62
b = -436.43599999999998 -1B4.6F9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB2
c = 65.26748971193416 41.447A34ACC60EBFBC937D5DC2E5A99CF8A021B641511E8D2B3183AFEF24DF5770B96A673E28086D905
a*(b-c) = -262510.90267998143099 -4016E.E7160906E7DC10422DA508321819F4A637E5AEE668ED5163B12FCB17A732442F589975B7F24112B2E8F6E95EAD45803915EE26D20DF323D67CAEEC75D7BED68AA34E02F2B492257D66F028545FB398F60E
 
septemvigesimal
===============
a = 523.23914037494285 JA.6C9
b = -436.43600000000004 -G4.BKML7C5DJ8Q0KB39AIICH4HACN02OJKGPLOPG2D1MFBQI6LJ33F645JELD7I0Q6FNHG88E9M9GE3QO276
c = 65.26748971193416 2B.76
a*(b-c) = -262510.90267998125637 -D92G.OA1C42LM0N8N30HDAFKJNEIFEOB0BHP1DM6ILA9P797KPJ05MCE6OGMO54Q3I3NQ9DGB673C8BC2FQF1N82
 
multiplication table
====================
* | #1 + | #2 +- | #3 +0 | #4 ++ | #5 +-- | #6 +-0 | #7 +-+ | #8 +0- | #9 +00 | #A +0+ | #B ++- | #C ++0 |
1 | + | | | | | | | | | | | |
PureFox
9,476

edits

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