# Generalised floating point multiplication

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 × (bc), 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
`###########################################               TASK CODE                ## Actual generic mulitplication operator ############################################ 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 zeroCO # 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 -aEND 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); #######################################  Define the hybrid multiplication  ## 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; `
File: Template.Balanced_ternary_float.Base.a68
`PR READ "Template.Big_float_BCD.Base.a68" PR # [[rc:Generalised floating point addition]] # ################################################################# 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));`
File: test.Balanced_ternary_float.Multiplication.a68
`#!/usr/local/bin/a68g --script ###################################################################### 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 # 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)`
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.

`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 Gofunc 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 numberfunc 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 unbalancedfunc 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 onlyfunc 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 alphabet2func 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 basefunc 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()}`
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

===========
a = 523.2391403749427  20B.3D384DB9885E94A90723EF9CBCB174B443E45FFC41152FE0293416F15E3AC303A0F3799ED81589C62
b = -436.436  -1B4.6F9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB2
c = 65.26748971193416  41.447A34ACC60EBFBC937D5DC2E5A99CF8A021B641511E8D2B3183AFEF24DF5770B96A673E28086D905

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 |
```

## Julia

`using Formattingimport 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)  # 2end 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))    endendBalancedTernary() = 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 send 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)))          # 3end 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 bend # 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]]    else        arr = [arr; b1.dig[ldigits1+1:end]]        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)    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 27 X 12")    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")    endend code_reuse_task(BalancedTernary) `
Output:
``` 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 27 X 12
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|
```

## 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.

`-- demo\rosetta\Generic_multiplication.exwconstant 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 resend 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 bend 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 bend 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 resend 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 resend 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 aend 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(da,db)        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 aend 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 resend 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 proceduretest("balanced ternary", balancedternary)test("balanced base 27", balanced_base27)test("decimal", decimal)test("binary", binary)test("ternary", ternary)test("hexadecimal", hexadecimal)test("septemvigesimal", septemvigesimal)`

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.

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

===========
a = 523.2391403749427  20B.3D384DB9885E94A90723EF9CBCB174B443E45FFC41152FE0293416F15E3AC303A0F3799ED81589C62
b = -436.436  -1B4.6F9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB2
c = 65.26748971193416  41.447A34ACC60EBFBC937D5DC2E5A99CF8A021B641511E8D2B3183AFEF24DF5770B96A673E28086D905

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!

`printf(1,"* |")for j=1 to 12 do    printf(1," #%s %3s |",{atm2b(j,hexadecimal),atm2b(j,balancedternary)})end forfor 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 forend forprintf(1,"\n")`
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 |
```