Generalised floating point multiplication: Difference between revisions

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10| +000 | +-000 | +0000 | ++000 | +--000 | +-0000 | +-+000 | +0-000 | +00000 | +0+000 | ++-000 | ++0000 |
10| +000 | +-000 | +0000 | ++000 | +--000 | +-0000 | +-+000 | +0-000 | +00000 | +0+000 | ++-000 | ++0000 |
</pre>
</pre>

=={{header|Julia}}==
<lang julia>using Formatting
import Base.BigInt, Base.BigFloat, Base.print, Base.+, Base.-, Base.*

abstract type BalancedBaseDigitArray end

mutable struct BalancedTernary <: BalancedBaseDigitArray
dig::Vector{Int8}
p::Int
BalancedTernary(arr::Vector, i) = new(Int8.(arr), i)
end

const MAX_PRECISION = 81

function BalancedTernary(s::String)
if (i = findfirst(x -> x == '.', s)) != nothing
p = length(s) - i
s = s[1:i-1] * s[i+1:end]
else
p = 0
end
b = BalancedTernary([c == '-' ? -1 : c == '0' ? 0 : 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))
b = BalancedTernary(arr, MAX_PRECISION)
canonicalize!(b)
return b
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] -= 1
end
end
end
if (i = findfirst(x -> x != 0, b.dig)) != nothing
if i > 1
b.dig = b.dig[i:end]
end
else
b.dig = [0]
end
if b.p > 0 && (i = findlast(x -> x != 0, b.dig)) != nothing
removable = min(b.p, length(b.dig) - i)
b.dig = b.dig[1:end-removable]
b.p -= removable
end
return b
end

# The following should work with any base number where dig, p are a similar array and Int
# and the proper constructors, canon, and conversion routines are defined # 6

Base.print(io::IO, b::BalancedBaseDigitArray) = print(io, String(b))

function +(b1::T, b2::T) where T <: BalancedBaseDigitArray # 4
if all(x -> x == 0, b1.dig)
return deepcopy(b2)
elseif all(x -> x == 0, b2.dig)
return deepcopy(b1)
end
ldigits1 = length(b1.dig) - b1.p
arr = b1.dig[1:ldigits1]
ldigits2 = length(b2.dig) - b2.p
arr2 = b2.dig[1:ldigits2]
if (i = ldigits1 - ldigits2) > 0
arr2 = [zeros(Int8, i); arr2]
elseif i < 0
arr = [zeros(Int8, -i); arr]
end
if (i = b1.p - b2.p) > 0
arr = [arr; b1.dig[ldigits1+1:end]]
arr2 = [arr2; b2.dig[ldigits2+1:end]; zeros(Int8, i)]
elseif i < 0
arr = [arr; b1.dig[ldigits1+1:end]; zeros(Int8, -i)]
arr2 = [arr2; b2.dig[ldigits2+1:end]]
end
arr .+= arr2
return canonicalize!(T(arr, max(b1.p, b2.p)))
end

-(b1::T) where T <: BalancedBaseDigitArray = T(b1.dig .* -1, b1.p) # 4
-(b1::T, b2::T) where T <: BalancedBaseDigitArray = +(b1, -b2) # 4

function *(b1::T, b2::T) where T <: BalancedBaseDigitArray # 4
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 code_reuse_task(T::Type) # test cases 1 and 2
a = T("+-0++0+.+-0++0+")
b = T(-436.436)
c = T("+-++-.+-++-")
println(" a = ", a, " = ", format(BigFloat(a)))
println(" b = ", b, " = ", format(BigFloat(b)))
println(" c = ", c, " = ", format(BigFloat(c)))
println("\na * (b - c) = ", String(a * (b - c)), "\n = ", format(BigFloat(a * (b - c))))

println("\n ---- Multiplication Table ----")
println(" x|+ (1) |+- (2) |+0 (3) |++ (4) |+-- (5)|+-0 (6)|+-+ (7)|+0- (8)|+e+-(9)|+0+(10)|++-(11)|++0(12)|")
for i in 1:27
print(lpad(i, 2), "|")
for j in 1:12
print(lpad(String(T(i * j)), 7), "|")
end
print("\n")
end
end

code_reuse_task(BalancedTernary)
</julia>{{out}}
<pre>
a = +-0++0+.+-0++0+ = 523.23914
b = -++-0--.--0+-00+++-0-+---0-+0++++0--++++0-+0+-0+0+-000-0----+0--0---+-000++--++-+-0--0-+ = -436.436
c = +-++-.+-++- = 65.26749

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+-+-+
= -262510.90268

---- Multiplication Table ----
x|+ (1) |+- (2) |+0 (3) |++ (4) |+-- (5)|+-0 (6)|+-+ (7)|+0- (8)|+e+-(9)|+0+(10)|++-(11)|++0(12)|
1| +| +-| +0| ++| +--| +-0| +-+| +0-| +00| +0+| ++-| ++0|
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}}==
=={{header|Phix}}==

Revision as of 19:04, 16 January 2020

Generalised floating point multiplication is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Use the Generalised floating point addition template to implement generalised floating point multiplication for a Balanced ternary test case.

Test case details: Balanced ternary is a way of representing numbers. Unlike the prevailing binary representation, a balanced ternary "real" is in base 3, and each digit can have the values 1, 0, or −1. For example, decimal 11 = 32 + 31 − 30, thus can be written as "++−", while 6 = 32 − 31 + 0 × 30, i.e., "+−0" and for an actual real number 6⅓ the exact representation is 32 − 31 + 0 × 30 + 1 × 3-1 i.e., "+−0.+"

For this task, implement balanced ternary representation of real numbers with the following:

Requirements

  1. Support arbitrary precision real numbers, both positive and negative;
  2. Provide ways to convert to and from text strings, using digits '+', '-' and '0' (unless you are already using strings to represent balanced ternary; but see requirement 5).
  3. Provide ways to convert to and from native integer and real type (unless, improbably, your platform's native integer type is balanced ternary). If your native integers can't support arbitrary length, overflows during conversion must be indicated.
  4. Provide ways to perform addition, negation and multiplication directly on balanced ternary integers; do not convert to native integers first.
  5. Make your implementation efficient, with a reasonable definition of "efficient" (and with a reasonable definition of "reasonable").
  6. The Template should successfully handle these multiplications in other bases. In particular Septemvigesimal and "Balanced base-27".

Optionally:

  • For faster long multiplication use Karatsuba algorithm.
  • Using the Karatsuba algorithm, spread the computation across multiple CPUs.

Test case 1 - With balanced ternaries a from string "+-0++0+.+-0++0+", b from native real -436.436, c "+-++-.+-++-":

  • write out a, b and c in decimal notation.
  • calculate a × (b − c), write out the result in both ternary and decimal notations.
  • In the above limit the precision to 81 ternary digits after the point.

Test case 2 - Generate a multiplication table of balanced ternaries where the rows of the table are for a 1st factor of 1 to 27, and the column of the table are for the second factor of 1 to 12.

Implement the code in a generalised form (such as a Template, Module or Mixin etc) that permits reusing of the code for different Bases.

If it is not possible to implement code in syntax of the specific language then:

  • note the reason.
  • perform the test case using a built-in or external library.

ALGOL 68

Works with: ALGOL 68 version Revision 1 - one minor extension to language used - PRAGMA READ, similar to C's #include directive.
Works with: ALGOL 68G version Any - tested with release algol68g-2.3.3.

File: Template.Big_float.Multiplication.a68<lang algol68>##########################################

  1. TASK CODE #
  2. Actual generic mulitplication operator #
  3. Alternatively use http://en.wikipedia.org/wiki/Karatsuba_algorithm #

OP * = (DIGITS a, b)DIGITS: ( 

 DIGITS minus one = -IDENTITY LOC DIGITS,
        zero = ZERO LOC DIGITS,
        one = IDENTITY LOC DIGITS;
 INT order = digit order OF arithmetic;
 IF SIGN a = 0 OR SIGN b = 0 THEN zero

CO # Note: The following require the inequality operators # 

 ELIF a = one THEN b
 ELIF b = one THEN a
 ELIF a = minus one THEN -b
 ELIF b = minus one THEN -a

END CO 

 ELSE
   DIGIT zero = ZERO LOC DIGIT;
   DIGIT one =  IDENTITY LOC DIGIT;
   [order + MSD a+MSD b: LSD a+LSD b]DIGIT a x b;

   FOR place FROM LSD a+LSD b BY order TO LSD a+MSD b DO
     a x b[place] := zero # pad the MSDs of the result with Zero #
   OD;
   FOR place a FROM LSD a BY order TO MSD a DO
     DIGIT digit a = a[place a];
     DIGIT carry := zero;
     FOR place b FROM LSD b BY order TO MSD b DO
       DIGIT digit b = b[place b];
       REF DIGIT digit ab = a x b[place a + place b];
       IF carry OF arithmetic THEN # used for big number arithmetic #
         MOID(carry := ( digit ab +:= carry ));
         DIGIT prod := digit a;
         MOID(carry +:= ( prod *:= digit b ));
         MOID(carry +:= ( digit ab +:= prod ))
       ELSE # carry = 0 so we can just ignore the carry #
         DIGIT prod := digit a;
         MOID(prod *:= digit b);
         MOID(digit ab +:= prod)
       FI
     OD;
     a x b[place a + MSD b + order] := carry
   OD;
   INITDIGITS a x b # normalise #
 FI

);

  2. Define the hybrid multiplication #
  3. operators for the generalised base #

OP * = (DIGIT a, DIGITS b)DIGITS: INITDIGITS a * b; OP * = (DIGITS a, DIGIT b)DIGITS: a * INITDIGITS b;

OP *:= = (REF DIGITS lhs, DIGIT arg)DIGITS: lhs := lhs * INITDIGITS arg; </lang>File: Template.Balanced_ternary_float.Base.a68<lang algol68>PR READ "Template.Big_float_BCD.Base.a68" PR # rc:Generalised floating point addition #

  2. First: define the attributes of the arithmetic we are using. #

arithmetic := ( 

 # balanced = # TRUE, 
 # carry = # TRUE, 
 # base = # 3, # width = # 1, # places = # 81, # order = # -1, 
 # repr = # USTRING("-","0","+")[@-1]

);

OP INITDIGIT = (CHAR c)DIGIT: ( 

 DIGIT out;
 digit OF out :=
   IF   c = "+" THEN +1
   ELIF c = "0" THEN  0
   ELIF c = "-" THEN -1
   ELSE raise value error("Unknown digit :"""+c+""""); SKIP
   FI;
 out

);

OP INITBIGREAL = (STRING s)BIGREAL: ( 

 BIGREAL out;
 BIGREAL base of arithmetic = INITBIGREAL base OF arithmetic; # Todo: Opt #
 INT point := UPB s; # put the point on the extreme right #
 FOR place FROM LWB s TO UPB s DO
   IF s[place]="." THEN
     point := place
   ELSE
     out := out SHR digit order OF arithmetic + INITDIGIT s[place]
   FI
 OD;
 out SHR (UPB s-point)

);</lang>File: test.Balanced_ternary_float.Multiplication.a68<lang algol68>#!/usr/local/bin/a68g --script #

  2. A program to test arbitrary length floating point multiplication #

PR READ "prelude/general.a68" PR # rc:Template:ALGOL 68/prelude #

PR READ "Template.Big_float.Multiplication.a68" PR

  1. include the basic axioms of the digits being used #

PR READ "Template.Balanced_ternary_float.Base.a68" PR

PR READ "Template.Big_float.Addition.a68" PR # rc:Generalised floating point addition # PR READ "Template.Big_float.Subtraction.a68" PR # rc:Generalised floating point addition #

test1:( # Basic arithmetic # 

 INT rw = long real width;
 BIGREAL a = INITBIGREAL "+-0++0+.+-0++0+", # 523.239... #
         b = INITBIGREAL - LONG 436.436,
         c = INITBIGREAL "+-++-.+-++-"; # 65.267... #
 printf(($g 9k g(rw,rw-5)39kgl$,
   "a =",INITLONGREAL a, REPR a,
   "b =",INITLONGREAL b, REPR b,
   "c =",INITLONGREAL c, REPR c,
   "a*(b-c)",INITLONGREAL(a*(b-c)), REPR(a*(b-c)),
 $l$))

);

test2:( # A floating point Ternary multiplication table # 

 FORMAT s = $"|"$; # field seperator #

 INT lwb = 1, tab = 8, upb = 12;

 printf($"# "f(s)" *   "f(s)$);
 FOR j FROM lwb TO upb DO
   FORMAT col = $n(tab)k f(s)$;
   printf(($g" #"g(0)f(col)$, REPR INITBIGREAL j,j))
 OD;
 printf($l$);
 FOR i FROM lwb TO 27 DO
   printf(($g(0) 3k f(s) g 9k f(s)$,i,REPR INITBIGREAL i));
   FOR j FROM lwb TO i MIN upb DO
     FORMAT col = $n(tab)k f(s)$;
     BIGREAL product = INITBIGREAL i * INITBIGREAL j;
     printf(($gf(col)$, REPR product))
   OD;
   IF upb > i THEN printf($n(upb-i)(n(tab-1)x f(s))$) FI;
   printf($l$)
 OD

)</lang>Output: 

a =     +523.23914037494284407864655  +-0++0+.+-0++0+
b =     -436.43600000000000000000000  -++-0--.--0+-00+++-0-+---0-+0++++0--0000+00-+-+--+0-0-00--++0-+00---+0+-+++0+-0----0++
c =      +65.26748971193415637860082  +-++-.+-++-
a*(b-c) -262510.90267998140903693919  ----000-0+0+.0+0-0-00---00--0-0+--+--00-0++-000++0-000-+0+-----+++-+-0+-+0+0++0+0-++-++0+---00++++

# | *   |+ #1   |+- #2  |+0 #3  |++ #4  |+-- #5 |+-0 #6 |+-+ #7 |+0- #8 |+e+- #9|+0+ #10|++- #11|++0 #12|
1 |+    |+      |       |       |       |       |       |       |       |       |       |       |       |
2 |+-   |+-     |++     |       |       |       |       |       |       |       |       |       |       |
3 |+0   |+0     |+-0    |+e+-   |       |       |       |       |       |       |       |       |       |
4 |++   |++     |+0-    |++0    |+--+   |       |       |       |       |       |       |       |       |
5 |+--  |+--    |+0+    |+--0   |+-+-   |+0-+   |       |       |       |       |       |       |       |
6 |+-0  |+-0    |++0    |+-e+-  |+0-0   |+0+0   |++e+-  |       |       |       |       |       |       |
7 |+-+  |+-+    |+---   |+-+0   |+00+   |++0-   |+---0  |+--++  |       |       |       |       |       |
8 |+0-  |+0-    |+--+   |+0-0   |++--   |++++   |+--+0  |+-0+-  |+-+0+  |       |       |       |       |
9 |+e+- |+e+-   |+-e+-  |+e+0   |++e+-  |+--e+- |+-e+0  |+-+e+- |+0-e+- |+e++   |       |       |       |
10|+0+  |+0+    |+-+-   |+0+0   |++++   |+-0--  |+-+-0  |+0--+  |+000-  |+0+e+- |++-0+  |       |       |
11|++-  |++-    |+-++   |++-0   |+--0-  |+-00+  |+-++0  |+00--  |+0+-+  |++-e+- |++0+-  |+++++  |       |
12|++0  |++0    |+0-0   |++e+-  |+--+0  |+-+-0  |+0-e+- |+00+0  |++--0  |++e+0  |++++0  |+--0-0 |+--+e+-|
13|+++  |+++    |+00-   |+++0   |+-0-+  |+-++-  |+00-0  |+0+0+  |++0--  |+++e+- |+---++ |+--+0- |+-0-+0 |
14|+--- |+---   |+00+   |+---0  |+-0+-  |+0--+  |+00+0  |++-0-  |++0++  |+---e+-|+--+-- |+-0-0+ |+-0+-0 |
15|+--0 |+--0   |+0+0   |+--e+- |+-+-0  |+0-+0  |+0+e+- |++0-0  |++++0  |+--e+0 |+-0--0 |+-00+0 |+-+-e+-|
16|+--+ |+--+   |++--   |+--+0  |+-+0+  |+000-  |++--0  |++0++  |+---+- |+--+e+-|+-00-+ |+-+--- |+-+0+0 |
17|+-0- |+-0-   |++-+   |+-0-0  |+0---  |+00++  |++-+0  |++++-  |+--00+ |+-0-e+-|+-0+0- |+-+0-+ |+0---0 |
18|+-e+-|+-e+-  |++e+-  |+-e+0  |+0-e+- |+0+e+- |++e+0  |+---e+-|+--+e+-|+-e++  |+-+-e+-|+-++e+-|+0-e+0 |
19|+-0+ |+-0+   |+++-   |+-0+0  |+0-++  |++---  |+++-0  |+--0-+ |+-0-0- |+-0+e+-|+-+00+ |+0--+- |+0-++0 |
20|+-+- |+-+-   |++++   |+-+-0  |+000-  |++-0+  |++++0  |+--+-- |+-00-+ |+-+-e+-|+-+++- |+0-0++ |+000-0 |
21|+-+0 |+-+0   |+---0  |+-+e+- |+00+0  |++0-0  |+---e+-|+--++0 |+-0+-0 |+-+e+0 |+0--+0 |+00--0 |+00+e+-|
22|+-++ |+-++   |+--0-  |+-++0  |+0+-+  |++0+-  |+--0-0 |+-0-0+ |+-+--- |+-++e+-|+0-0++ |+0000- |+0+-+0 |
23|+0-- |+0--   |+--0+  |+0--0  |+0++-  |+++-+  |+--0+0 |+-000- |+-+-++ |+0--e+-|+00--- |+00+0+ |+0++-0 |
24|+0-0 |+0-0   |+--+0  |+0-e+- |++--0  |++++0  |+--+e+-|+-0+-0 |+-+0+0 |+0-e+0 |+000-0 |+0+-+0 |++--e+-|
25|+0-+ |+0-+   |+-0--  |+0-+0  |++-0+  |+---0- |+-0--0 |+-0+++ |+-+++- |+0-+e+-|+00+-+ |+0++-- |++-0+0 |
26|+00- |+00-   |+-0-+  |+00-0  |++0--  |+---++ |+-0-+0 |+-+-+- |+0--0+ |+00-e+-|+0+-0- |++---+ |++0--0 |
27|+e+0 |+e+0   |+-e+0  |+e++   |++e+0  |+--e+0 |+-e++  |+-+e+0 |+0-e+0 |+e+--  |+0+e+0 |++-e+0 |++e++  |

Go

Translation of: Phix

In the interests of brevity many of the comments and all of the commented-out code has been omitted. <lang go>package main

import ( 

   "fmt"
   "log"
   "math"
   "strings"

)

const ( 

   maxdp           = 81
   binary          = "01"
   ternary         = "012"
   balancedTernary = "-0+"
   decimal         = "0123456789"
   hexadecimal     = "0123456789ABCDEF"
   septemVigesimal = "0123456789ABCDEFGHIJKLMNOPQ"
   balancedBase27  = "ZYXWVUTSRQPON0ABCDEFGHIJKLM"
   base37          = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"

)

/* helper functions */

func changeByte(s string, idx int, c byte) string { 

   bytes := []byte(s)
   bytes[idx] = c
   return string(bytes)

}

func removeByte(s string, idx int) string { 

   le := len(s)
   bytes := []byte(s)
   copy(bytes[idx:], bytes[idx+1:])
   return string(bytes[0 : le-1])

}

func insertByte(s string, idx int, c byte) string { 

   le := len(s)
   t := make([]byte, le+1)
   copy(t, s)
   copy(t[idx+1:], t[idx:])
   t[idx] = c
   return string(t)

}

func prependByte(s string, c byte) string { 

   le := len(s)
   bytes := make([]byte, le+1)
   copy(bytes[1:], s)
   bytes[0] = c
   return string(bytes)

}

func abs(i int) int { 

   if i < 0 {
       return -i
   }
   return i

}

// converts Phix indices to Go func gIndex(pIndex, le int) int { 

   if pIndex < 0 {
       return pIndex + le
   }
   return pIndex - 1

}

func getCarry(digit, base int) int { 

   if digit > base {
       return 1
   } else if digit < 1 {
       return -1
   }
   return 0

}

// convert string 'b' to a decimal floating point number func b2dec(b, alphabet string) float64 { 

   res := 0.0
   base := len(alphabet)
   zdx := strings.IndexByte(alphabet, '0') + 1
   signed := zdx == 1 && b[0] == '-'
   if signed {
       b = b[1:]
   }
   le := len(b)
   ndp := strings.IndexByte(b, '.') + 1
   if ndp != 0 {
       b = removeByte(b, ndp-1) // remove decimal point
       ndp = le - ndp
   }
   for i := 1; i <= len(b); i++ {
       idx := strings.IndexByte(alphabet, b[i-1]) + 1
       res = float64(base)*res + float64(idx) - float64(zdx)
   }
   if ndp != 0 {
       res /= math.Pow(float64(base), float64(ndp))
   }
   if signed {
       res = -res
   }
   return res

}

// string 'b' can be balanced or unbalanced func negate(b, alphabet string) string { 

   if alphabet[0] == '0' {
       if b != "0" {
           if b[0] == '-' {
               b = b[1:]
           } else {
               b = prependByte(b, '-')
           }
       }
   } else {
       for i := 1; i <= len(b); i++ {
           if b[i-1] != '.' {
               idx := strings.IndexByte(alphabet, b[i-1]) + 1
               gi := gIndex(-idx, len(alphabet))
               b = changeByte(b, i-1, alphabet[gi])
           }
       }
   }
   return b

}

func bTrim(b string) string { 

   // trim  trailing ".000"
   idx := strings.IndexByte(b, '.') + 1
   if idx != 0 {
       b = strings.TrimRight(strings.TrimRight(b, "0"), ".")
   }
   // trim leading zeros but not "0.nnn"
   for len(b) > 1 && b[0] == '0' && b[1] != '.' {
       b = b[1:]
   }
   return b

}

// for balanced number systems only func bCarry(digit, base, idx int, n, alphabet string) (int, string) { 

   carry := getCarry(digit, base)
   if carry != 0 {
       for i := idx; i >= 1; i-- {
           if n[i-1] != '.' {
               k := strings.IndexByte(alphabet, n[i-1]) + 1
               if k < base {
                   n = changeByte(n, i-1, alphabet[k])
                   break
               }
               n = changeByte(n, i-1, alphabet[0])
           }
       }
       digit -= base * carry
   }
   return digit, n

}

// convert a string from alphabet to alphabet2 func b2b(n, alphabet, alphabet2 string) string { 

   res, m := "0", ""
   if n != "0" {
       base := len(alphabet)
       base2 := len(alphabet2)
       zdx := strings.IndexByte(alphabet, '0') + 1
       zdx2 := strings.IndexByte(alphabet2, '0') + 1
       var carry, q, r, digit int
       idx := strings.IndexByte(alphabet, n[0]) + 1
       negative := (zdx == 1 && n[0] == '-') || (zdx != 1 && idx < zdx)
       if negative {
           n = negate(n, alphabet)
       }
       ndp := strings.IndexByte(n, '.') + 1
       if ndp != 0 {
           n, m = n[0:ndp-1], n[ndp:]
       }
       res = ""
       for len(n) > 0 {
           q = 0
           for i := 1; i <= len(n); i++ {
               digit = strings.IndexByte(alphabet, n[i-1]) + 1 - zdx
               q = q*base + digit
               r = abs(q) % base2
               digit = abs(q)/base2 + zdx
               if q < 0 {
                   digit--
               }
               if zdx != 1 {
                   digit, n = bCarry(digit, base, i-1, n, alphabet)
               }
               n = changeByte(n, i-1, alphabet[digit-1])
               q = r
           }
           r += zdx2
           if zdx2 != 1 {
               r += carry
               carry = getCarry(r, base2)
               r -= base2 * carry
           }
           res = prependByte(res, alphabet2[r-1])
           n = strings.TrimLeft(n, "0")
       }
       if carry != 0 {
           res = prependByte(res, alphabet2[carry+zdx2-1])
       }
       if len(m) > 0 {
           res += "."
           ndp = 0
           if zdx != 1 {
               lm := len(m)
               alphaNew := base37[0:len(alphabet)]
               m = b2b(m, alphabet, alphaNew)
               m = strings.Repeat("0", lm-len(m)) + m
               alphabet = alphaNew
               zdx = 1
           }
           for len(m) > 0 && ndp < maxdp {
               q = 0
               for i := len(m); i >= 1; i-- {
                   digit = strings.IndexByte(alphabet, m[i-1]) + 1 - zdx
                   q += digit * base2
                   r = abs(q)%base + zdx
                   q /= base
                   if q < 0 {
                       q--
                   }
                   m = changeByte(m, i-1, alphabet[r-1])
               }
               digit = q + zdx2
               if zdx2 != 1 {
                   digit, res = bCarry(digit, base2, len(res), res, alphabet2)
               }
               res += string(alphabet2[digit-1])
               m = strings.TrimRight(m, "0")
               ndp++
           }
       }
       res = bTrim(res)
       if negative {
           res = negate(res, alphabet2)
       }
   }
   return res

}

// convert 'd' to a string in the specified base func float2b(d float64, alphabet string) string { 

   base := len(alphabet)
   zdx := strings.Index(alphabet, "0") + 1
   carry := 0
   neg := d < 0
   if neg {
       d = -d
   }
   res := ""
   whole := int(d)
   d -= float64(whole)
   for {
       ch := whole%base + zdx
       if zdx != 1 {
           ch += carry
           carry = getCarry(ch, base)
           ch -= base * carry
       }
       res = prependByte(res, alphabet[ch-1])
       whole /= base
       if whole == 0 {
           break
       }
   }
   if carry != 0 {
       res = prependByte(res, alphabet[carry+zdx-1])
       carry = 0
   }
   if d != 0 {
       res += "."
       ndp := 0
       for d != 0 && ndp < maxdp {
           d *= float64(base)
           digit := int(d) + zdx
           d -= float64(digit)
           if zdx != 1 {
               digit, res = bCarry(digit, base, len(res), res, alphabet)
           }
           res += string(alphabet[digit-1])
           ndp++
       }
   }
   if neg {
       res = negate(res, alphabet)
   }
   return res

}

func bAdd(a, b, alphabet string) string { 

   base := len(alphabet)
   zdx := strings.IndexByte(alphabet, '0') + 1
   var carry, da, db, digit int
   if zdx == 1 {
       if a[0] == '-' {
           return bSub(b, negate(a, alphabet), alphabet)
       }
       if b[0] == '-' {
           return bSub(a, negate(b, alphabet), alphabet)
       }
   }
   adt := strings.IndexByte(a, '.') + 1
   bdt := strings.IndexByte(b, '.') + 1
   if adt != 0 || bdt != 0 {
       if adt != 0 {
           adt = len(a) - adt + 1
           gi := gIndex(-adt, len(a))
           a = removeByte(a, gi)
       }
       if bdt != 0 {
           bdt = len(b) - bdt + 1
           gi := gIndex(-bdt, len(b))
           b = removeByte(b, gi)
       }
       if bdt > adt {
           a += strings.Repeat("0", bdt-adt)
           adt = bdt
       } else if adt > bdt {
           b += strings.Repeat("0", adt-bdt)
       }
   }
   if len(a) < len(b) {
       a, b = b, a
   }
   for i := -1; i >= -len(a); i-- {
       if i < -len(a) {
           da = 0
       } else {
           da = strings.IndexByte(alphabet, a[len(a)+i]) + 1 - zdx
       }
       if i < -len(b) {
           db = 0
       } else {
           db = strings.IndexByte(alphabet, b[len(b)+i]) + 1 - zdx
       }
       digit = da + db + carry + zdx
       carry = getCarry(digit, base)
       a = changeByte(a, i+len(a), alphabet[digit-carry*base-1])
       if i < -len(b) && carry == 0 {
           break
       }
   }
   if carry != 0 {
       a = prependByte(a, alphabet[carry+zdx-1])
   }
   if adt != 0 {
       gi := gIndex(-adt+1, len(a))
       a = insertByte(a, gi, '.')
   }
   a = bTrim(a)
   return a

}

func aSmaller(a, b, alphabet string) bool { 

   if len(a) != len(b) {
       log.Fatal("strings should be equal in length")
   }
   for i := 1; i <= len(a); i++ {
       da := strings.IndexByte(alphabet, a[i-1]) + 1
       db := strings.IndexByte(alphabet, b[i-1]) + 1
       if da != db {
           return da < db
       }
   }
   return false

}

func bSub(a, b, alphabet string) string { 

   base := len(alphabet)
   zdx := strings.IndexByte(alphabet, '0') + 1
   var carry, da, db, digit int
   if zdx == 1 {
       if a[0] == '-' {
           return negate(bAdd(negate(a, alphabet), b, alphabet), alphabet)
       }
       if b[0] == '-' {
           return bAdd(a, negate(b, alphabet), alphabet)
       }
   }
   adt := strings.Index(a, ".") + 1
   bdt := strings.Index(b, ".") + 1
   if adt != 0 || bdt != 0 {
       if adt != 0 {
           adt = len(a) - adt + 1
           gi := gIndex(-adt, len(a))
           a = removeByte(a, gi)
       }
       if bdt != 0 {
           bdt = len(b) - bdt + 1
           gi := gIndex(-bdt, len(b))
           b = removeByte(b, gi)
       }
       if bdt > adt {
           a += strings.Repeat("0", bdt-adt)
           adt = bdt
       } else if adt > bdt {
           b += strings.Repeat("0", adt-bdt)
       }
   }
   bNegate := false
   if len(a) < len(b) || (len(a) == len(b) && aSmaller(a, b, alphabet)) {
       bNegate = true
       a, b = b, a
   }
   for i := -1; i >= -len(a); i-- {
       if i < -len(a) {
           da = 0
       } else {
           da = strings.IndexByte(alphabet, a[len(a)+i]) + 1 - zdx
       }
       if i < -len(b) {
           db = 0
       } else {
           db = strings.IndexByte(alphabet, b[len(b)+i]) + 1 - zdx
       }
       digit = da - db - carry + zdx
       carry = 0
       if digit <= 0 {
           carry = 1
       }
       a = changeByte(a, i+len(a), alphabet[digit+carry*base-1])
       if i < -len(b) && carry == 0 {
           break
       }
   }
   if carry != 0 {
       log.Fatal("carry should be zero")
   }
   if adt != 0 {
       gi := gIndex(-adt+1, len(a))
       a = insertByte(a, gi, '.')
   }
   a = bTrim(a)
   if bNegate {
       a = negate(a, alphabet)
   }
   return a

}

func bMul(a, b, alphabet string) string { 

   zdx := strings.IndexByte(alphabet, '0') + 1
   dpa := strings.IndexByte(a, '.') + 1
   dpb := strings.IndexByte(b, '.') + 1
   ndp := 0
   if dpa != 0 {
       ndp += len(a) - dpa
       a = removeByte(a, dpa-1)
   }
   if dpb != 0 {
       ndp += len(b) - dpb
       b = removeByte(b, dpb-1)
   }
   pos, res := a, "0"
   if zdx != 1 {
       // balanced number systems
       neg := negate(pos, alphabet)
       for i := len(b); i >= 1; i-- {
           m := strings.IndexByte(alphabet, b[i-1]) + 1 - zdx
           for m != 0 {
               temp, temp2 := pos, -1
               if m < 0 {
                   temp = neg
                   temp2 = 1
               }
               res = bAdd(res, temp, alphabet)
               m += temp2
           }
           pos += "0"
           neg += "0"
       }
   } else {
       // non-balanced number systems
       negative := false
       if a[0] == '-' {
           a = a[1:]
           negative = true
       }
       if b[0] == '-' {
           b = b[1:]
           negative = !negative
       }
       for i := len(b); i >= 1; i-- {
           m := strings.IndexByte(alphabet, b[i-1]) + 1 - zdx
           for m > 0 {
               res = bAdd(res, pos, alphabet)
               m--
           }
           pos += "0"
       }
       if negative {
           res = negate(res, alphabet)
       }
   }
   if ndp != 0 {
       gi := gIndex(-ndp, len(res))
       res = insertByte(res, gi, '.')
   }
   res = bTrim(res)
   return res

}

func multTable() { 

   fmt.Println("multiplication table")
   fmt.Println("====================")
   fmt.Printf("* |")
   for j := 1; j <= 12; j++ {
       fj := float64(j)
       fmt.Printf(" #%s %3s |", float2b(fj, hexadecimal), float2b(fj, balancedTernary))
   }
   for i := 1; i <= 27; i++ {
       fi := float64(i)
       a := float2b(fi, balancedTernary)
       fmt.Printf("\n%-2s|", float2b(fi, septemVigesimal))
       for j := 1; j <= 12; j++ {
           if j > i {
               fmt.Printf("        |")
           } else {
               fj := float64(j)
               b := float2b(fj, balancedTernary)
               m := bMul(a, b, balancedTernary)
               fmt.Printf(" %6s |", m)
           }
       }
   }
   fmt.Println()

}

func test(name, alphabet string) { 

   a := b2b("+-0++0+.+-0++0+", balancedTernary, alphabet)
   b := b2b("-436.436", decimal, alphabet)
   c := b2b("+-++-.+-++-", balancedTernary, alphabet)
   d := bSub(b, c, alphabet)
   r := bMul(a, d, alphabet)
   fmt.Printf("%s\n%s\n", name, strings.Repeat("=", len(name)))
   fmt.Printf("      a = %.16g  %s\n", b2dec(a, alphabet), a)
   fmt.Printf("      b = %.16g  %s\n", b2dec(b, alphabet), b)
   fmt.Printf("      c = %.16g  %s\n", b2dec(c, alphabet), c)
   fmt.Printf("a*(b-c) = %.16g  %s\n\n", b2dec(r, alphabet), r)

}

func main() { 

   test("balanced ternary", balancedTernary)
   test("balanced base 27", balancedBase27)
   test("decimal", decimal)
   test("binary", binary)
   test("ternary", ternary)
   test("hexadecimal", hexadecimal)
   test("septemvigesimal", septemVigesimal)
   multTable()

}</lang>

Output:
balanced ternary
================
      a = 523.2391403749428  +-0++0+.+-0++0+
      b = -436.4359999999999  -++-0--.--0+-00+++-0-+---0-+0++++0--0000+00-+-+--+0-0-00--++0-+00---+0+-+++0+-0----0++
      c = 65.26748971193416  +-++-.+-++-
a*(b-c) = -262510.9026799813  ----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 = 523.2391403749428  AUJ.FLI
      b = -436.4359999999999  NKQ.YFDFTYSMHVANGXPVXHIZJRJWZD0PBGFJAEBAKOZODLY0ITEHPQLSQSGLFZUINATKCIKUVMWEWJMQ0COTS
      c = 65.26748971193416  BK.GF
a*(b-c) = -262510.9026799812  ZVPJ.CWNYQPEENDVDPNJZXKFGCLHKLCX0YIBOMETHFWWBTVUFAH0SEZMTBJDCRRAQIQCAWMKXSTPYUXYPK0LODUO

decimal
=======
      a = 523.2391403749428  523.239140374942844078646547782350251486053955189757658893461362597165066300868770004
      b = -436.436  -436.436
      c = 65.26748971193413  65.267489711934156378600823045267489711934156378600823045267489711934156378600823045
a*(b-c) = -262510.9026799813  -262510.90267998140903693918986303277315826215892262734715612833785876513103053772667101895163734826631742752252837097627017862754285047634638652268078676654605120794218

binary
======
      a = 523.2391403749427  1000001011.001111010011100001001101101110011000100001011110100101001010100100000111001000111
      b = -436.436  -110110100.011011111001110110110010001011010000111001010110000001000001100010010011011101001
      c = 65.26748971193416  1000001.01000100011110100011010010101100110001100000111010111111101111001001001101111101
a*(b-c) = -262510.9026799814  -1000000000101101110.111001110001011000001001000001101110011111011100000100000100001000101011100011110010110001010100110111001011101001010000001110110100111110001101000000001111110101

ternary
=======
      a = 523.2391403749428  201101.0201101
      b = -436.4360000000002  -121011.102202211210021110012111201022222000202102010100101200200110122011122101110212
      c = 65.26748971193416  2102.02102
a*(b-c) = -262510.9026799813  -111100002121.2201010011100110022102110002120222120100001221111011202022012121122001201122110221112

hexadecimal
===========
      a = 523.2391403749427  20B.3D384DB9885E94A90723EF9CBCB174B443E45FFC41152FE0293416F15E3AC303A0F3799ED81589C62
      b = -436.436  -1B4.6F9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB2
      c = 65.26748971193416  41.447A34ACC60EBFBC937D5DC2E5A99CF8A021B641511E8D2B3183AFEF24DF5770B96A673E28086D905
a*(b-c) = -262510.9026799814  -4016E.E7160906E7DC10422DA508321819F4A637E5AEE668ED5163B12FCB17A732442F589975B7F24112B2E8F6E95EAD45803915EE26D20DF323D67CAEEC75D7BED68AA34E02F2B492257D66F028545FB398F60E

septemvigesimal
===============
      a = 523.2391403749428  JA.6C9
      b = -436.4359999999999  -G4.BKML7C5DJ8Q0KB39AIICH4HACN02OJKGPLOPG2D1MFBQI6LJ33F645JELD7I0Q6FNHG88E9M9GE3QO276
      c = 65.26748971193416  2B.76
a*(b-c) = -262510.9026799812  -D92G.OA1C42LM0N8N30HDAFKJNEIFEOB0BHP1DM6ILA9P797KPJ05MCE6OGMO54Q3I3NQ9DGB673C8BC2FQF1N82

multiplication table
====================
* | #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 |

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.

<lang Phix>-- demo\rosetta\Generic_multiplication.exw constant MAX_DP = 81

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. --

function b2dec(string b, alphabet) -- -- convert string b back into a normal (decimal) atom, -- eg b2dec("+0-",balancedternary) yields 8 -- 

   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

function negate(string b, alphabet) -- -- negate b (can be balanced or unbalanced) -- 

   if alphabet[1]='0' then
       -- traditional: add/remove a leading '-'
       -- eg "-123" <==> "123"
       if b!="0" then
           if b[1]='-' then
               b = b[2..$]
           else
               b = "-"&b
           end if
       end if
   else
       -- balanced: mirror [non-0] digits
       -- eg "-0+" (ie -8) <==> "+0-" (ie +8)
       for i=1 to length(b) do
           if b[i]!='.' then
               b[i] = alphabet[-find(b[i],alphabet)]
           end if
       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

function b_carry(integer digit, base, idx, string n, alphabet) -- (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

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 atm2b(atom d, string alphabet) -- -- convert d to a string in the specified base, -- eg atm2b(65535,hexadecimal) => "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(). -- 

   integer base = length(alphabet),
           zdx = find('0',alphabet),
           carry = 0
   bool neg = d<0
   if neg then d = -d end if
   string res = ""
   integer whole = floor(d)
   d -= whole
   while true do
       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

-- 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>b: a-b == -(b-a) [avoid running off end]

forward function b_sub(string a, b, alphabet)

function b_add(string a, b, alphabet) 

   integer base = length(alphabet),
           zdx = find('0',alphabet),
           carry = 0, da, db, digit
   if zdx=1 then
       if a[1]='-' then    -- (-a)+b == b-a
           return b_sub(b,negate(a,alphabet),alphabet)
       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

function a_smaller(string a, b, alphabet) -- return true if a is smaller than b -- if not balanced then both are +ve 

   if length(a)!=length(b) then ?9/0 end if -- sanity check
   for i=1 to length(a) do
       integer da = find(a[i],alphabet),
               db = find(b[i],alphabet),
               c = compare(a,b)
       if c!=0 then return c<0 end if
   end for
   return false -- (=, which is not <)

end function

function b_sub(string a, b, alphabet) 

   integer base = length(alphabet),
           zdx = find('0',alphabet),
           carry = 0, da, db, digit
   if zdx=1 then
       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

function b_mul(string a, b, alphabet) 

   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

-- [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.

procedure test(string name, alphabet) --string a = b2b("523.2391403749428",decimal,alphabet), -- [see note 1] string a = b2b("+-0++0+.+-0++0+",balancedternary,alphabet), 

      b = b2b("-436.436",decimal,alphabet),

-- b = b2b("-++-0--.--0+-00+++-",balancedternary,alphabet), -- c = b2b("65.26748971193416",decimal,alphabet), -- [see note 1] 

      c = b2b("+-++-.+-++-",balancedternary,alphabet),
      d = b_add(b,c,alphabet),
      r = b_mul(a,d,alphabet)
   printf(1,"%s\n%s\n",{name,repeat('=',length(name))})
   printf(1,"      a = %.16g  %s\n",{b2dec(a,alphabet),a})
   printf(1,"      b = %.16g  %s\n",{b2dec(b,alphabet),b})
   printf(1,"      c = %.16g  %s\n",{b2dec(c,alphabet),c})

-- printf(1," d = %.16g %s\n",{b2dec(d,alphabet),d}) 

   printf(1,"a*(b-c) = %.16g  %s\n\n",{b2dec(r,alphabet),r})

end procedure test("balanced ternary", balancedternary) test("balanced base 27", balanced_base27) test("decimal", decimal) test("binary", binary) test("ternary", ternary) test("hexadecimal", hexadecimal) test("septemvigesimal", septemvigesimal)</lang> 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] above.

Output:
balanced ternary
================
      a = 523.2391403749428  +-0++0+.+-0++0+
      b = -436.4359999999999  -++-0--.--0+-00+++-0-+---0-+0++++0--0000+00-+-+--+0-0-00--++0-+00---+0+-+++0+-0----0++
      c = 65.26748971193416  +-++-.+-++-
a*(b-c) = -262510.9026799813  ----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 = 523.2391403749428  AUJ.FLI
      b = -436.436  NKQ.YFDFTYSMHVANGXPVXHIZJRJWZD0PBGFJAEBAKOZODLY0ITEHPQLSQSGLFZUINATKCIKUVMWEWJMQ0COTS
      c = 65.26748971193416  BK.GF
a*(b-c) = -262510.9026799813  ZVPJ.CWNYQPEENDVDPNJZXKFGCLHKLCX0YIBOMETHFWWBTVUFAH0SEZMTBJDCRRAQIQCAWMKXSTPYUXYPK0LODUO

decimal
=======
      a = 523.239140374943  523.239140374942844078646547782350251486053955189757658893461362597165066300868770004
      b = -436.436  -436.436
      c = 65.26748971193415  65.267489711934156378600823045267489711934156378600823045267489711934156378600823045
a*(b-c) = -262510.9026799814  -262510.90267998140903693918986303277315826215892262734715612833785876513103053772667101895163734826631742752252837097627017862754285047634638652268078676654605120794218

binary
======
      a = 523.2391403749427  1000001011.001111010011100001001101101110011000100001011110100101001010100100000111001000111
      b = -436.436  -110110100.011011111001110110110010001011010000111001010110000001000001100010010011011101001
      c = 65.26748971193416  1000001.01000100011110100011010010101100110001100000111010111111101111001001001101111101
a*(b-c) = -262510.9026799814  -1000000000101101110.111001110001011000001001000001101110011111011100000100000100001000101011100011110010110001010100110111001011101001010000001110110100111110001101000000001111110101

ternary
=======
      a = 523.2391403749428  201101.0201101
      b = -436.4360000000001  -121011.102202211210021110012111201022222000202102010100101200200110122011122101110212
      c = 65.26748971193416  2102.02102
a*(b-c) = -262510.9026799813  -111100002121.2201010011100110022102110002120222120100001221111011202022012121122001201122110221112

hexadecimal
===========
      a = 523.2391403749427  20B.3D384DB9885E94A90723EF9CBCB174B443E45FFC41152FE0293416F15E3AC303A0F3799ED81589C62
      b = -436.436  -1B4.6F9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB2
      c = 65.26748971193416  41.447A34ACC60EBFBC937D5DC2E5A99CF8A021B641511E8D2B3183AFEF24DF5770B96A673E28086D905
a*(b-c) = -262510.9026799814  -4016E.E7160906E7DC10422DA508321819F4A637E5AEE668ED5163B12FCB17A732442F589975B7F24112B2E8F6E95EAD45803915EE26D20DF323D67CAEEC75D7BED68AA34E02F2B492257D66F028545FB398F60E

septemvigesimal
===============
      a = 523.2391403749428  JA.6C9
      b = -436.436  -G4.BKML7C5DJ8Q0KB39AIICH4HACN02OJKGPLOPG2D1MFBQI6LJ33F645JELD7I0Q6FNHG88E9M9GE3QO276
      c = 65.26748971193416  2B.76
a*(b-c) = -262510.9026799813  -D92G.OA1C42LM0N8N30HDAFKJNEIFEOB0BHP1DM6ILA9P797KPJ05MCE6OGMO54Q3I3NQ9DGB673C8BC2FQF1N82

multiplication table

Without e notation, with hexadecimal across, septemvigesimal down, and balanced ternary contents! <lang Phix>printf(1,"* |") for j=1 to 12 do 

   printf(1," #%s %3s |",{atm2b(j,hexadecimal),atm2b(j,balancedternary)})

end for for i=1 to 27 do 

   string a = atm2b(i,balancedternary)
   printf(1,"\n%-2s|",{atm2b(i,septemvigesimal)})
   for j=1 to 12 do
       if j>i then
           printf(1,"        |")
       else
           string b = atm2b(j,balancedternary)
           string m = b_mul(a,b,balancedternary)
           printf(1," %6s |",{m})
       end if
   end for

end for printf(1,"\n")</lang>

Output:
* | #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 |
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