Generalised floating point multiplication: Difference between revisions
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<lang ecmascript>import "/str" for Str |
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import "/trait" for Stepped |
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Revision as of 17:29, 21 June 2021
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
- Support arbitrary precision real numbers, both positive and negative;
- 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).
- 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.
- Provide ways to perform addition, negation and multiplication directly on balanced ternary integers; do not convert to native integers first.
- Make your implementation efficient, with a reasonable definition of "efficient" (and with a reasonable definition of "reasonable").
- 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
File: Template.Big_float.Multiplication.a68<lang algol68>##########################################
- 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 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
);
- 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; </lang>File: Template.Balanced_ternary_float.Base.a68<lang algol68>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)
);</lang>File: test.Balanced_ternary_float.Multiplication.a68<lang algol68>#!/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
)</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
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 |
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))
return canonicalize!(BalancedTernary(arr, MAX_PRECISION))
end
function String(b::BalancedTernary) # 3
canonicalize!(b)
s = String([['-', '0', '+'][c + 2] for c in b.dig])
if b.p > 0
if b.p < length(s)
s = s[1:end-b.p] * "." * s[end-b.p+1:end]
elseif b.p == length(s)
s = "0." * s
else
s = "0." * "0"^(b.p - length(s)) * s
end
end
return s
end
function BigInt(b::BalancedTernary)
canonicalize!(b)
if b.p > 0
throw(InexactError("$(b.p) places after decimal point"))
end
return sum(t -> BigInt(3)^(t[1] - 1) * t[2], enumerate(reverse(b.dig))) # 3
end
BigFloat(b::BalancedTernary) = BigInt(BalancedTernary(b.dig, 0)) / big"3.0"^(b.p)
function canonicalize!(b::BalancedTernary)
for i in length(b.dig):-1:1
if b.dig[i] > 1
b.dig[i] -= 3
if i == 1
pushfirst!(b.dig, 1)
else
b.dig[i - 1] += 1
end
elseif b.dig[i] < -1
b.dig[i] += 3
if i == 1
pushfirst!(b.dig, -1)
else
b.dig[i - 1] -= 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]]
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")
end
end
code_reuse_task(BalancedTernary)
</lang>
- 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.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(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 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)
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
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!
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")
- 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 |
Wren
Translated via the Go entry. <lang ecmascript>import "/str" for Str import "/trait" for Stepped import "/fmt" for Fmt
var maxdp = 81 var binary = "01" var ternary = "012" var balancedTernary = "-0+" var decimal = "0123456789" var hexadecimal = "0123456789ABCDEF" var septemVigesimal = "0123456789ABCDEFGHIJKLMNOPQ" var balancedBase27 = "ZYXWVUTSRQPON0ABCDEFGHIJKLM" var base37 = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
// converts Phix indices to Wren var wIndex = Fn.new { |pIndex, le|
if (pIndex < 0) return pIndex + le return pIndex - 1
}
var getCarry = Fn.new { |digit, base|
if (digit > base) {
return 1
} else if (digit < 1) {
return -1
}
return 0
}
// convert string 'b' to a decimal floating point number var b2dec = Fn.new { |b, alphabet|
var res = 0
var base = alphabet.count
var zdx = alphabet.indexOf("0") + 1
var signed = zdx == 1 && b[0] == "-"
if (signed) b = b[1..-1]
var le = b.count
var ndp = b.indexOf(".") + 1
if (ndp != 0) {
b = Str.delete(b, ndp - 1) // remove decimal point
ndp = le - ndp
}
for (i in Stepped.ascend(1..b.count)) {
var idx = alphabet.indexOf(b[i-1]) + 1
res = base * res + idx - zdx
}
if (ndp != 0) res = res / base.pow(ndp)
if (signed) res = -res
return res
}
// string 'b' can be balanced or unbalanced var negate = Fn.new { |b, alphabet|
if (alphabet[0] == "0") {
if (b != "0") b = (b[0] == "-") ? b[1..-1] : Str.insert(b, 0, "-")
} else {
for (i in Stepped.ascend(1..b.count)) {
if (b[i-1] != ".") {
var idx = alphabet.indexOf(b[i-1]) + 1
var wi = wIndex.call(-idx, alphabet.count)
b = Str.change(b, i-1, alphabet[wi])
}
}
}
return b
}
var bTrim = Fn.new { |b|
// trim trailing ".000"
var idx = b.indexOf(".") + 1
if (idx != 0) b = b.trimEnd("0").trimEnd(".")
// trim leading zeros but not "0.nnn"
while (b.count > 1 && b[0] == "0" && b[1] != ".") b = b[1..-1]
return b
}
// for balanced number systems only var bCarry = Fn.new { |digit, base, idx, n, alphabet|
var carry = getCarry.call(digit, base)
if (carry != 0) {
for (i in Stepped.descend(idx..1)) {
if (n[i-1] != ".") {
var k = alphabet.indexOf(n[i-1]) + 1
if (k < base) {
n = Str.change(n, i-1, alphabet[k])
break
}
n = Str.change(n, i-1, alphabet[0])
}
}
digit = digit - base * carry
}
return [digit, n]
}
// convert a string from alphabet to alphabet2 var b2b // recursive function b2b = Fn.new { |n, alphabet, alphabet2|
var res = "0"
var m = ""
if (n != "0") {
var base = alphabet.count
var base2 = alphabet2.count
var zdx = alphabet.indexOf("0") + 1
var zdx2 = alphabet2.indexOf("0") + 1
var carry = 0
var q = 0
var r = 0
var digit = 0
var idx = alphabet.indexOf(n[0]) + 1
var negative = (zdx == 1 && n[0] == "-") || (zdx != 1 && idx < zdx)
if (negative) n = negate.call(n, alphabet)
var ndp = n.indexOf(".") + 1
if (ndp != 0) {
var t = n
n = n[0...ndp-1]
m = t[ndp..-1]
}
res = ""
while (n.count > 0) {
q = 0
for (i in Stepped.ascend(1..n.count)) {
digit = alphabet.indexOf(n[i-1]) + 1 - zdx
q = q*base + digit
r = q.abs % base2
digit = (q.abs/base2).floor + zdx
if (q < 0) digit = digit - 1
if (zdx != 1) {
var p = bCarry.call(digit, base, i-1, n, alphabet)
digit = p[0]
n = p[1]
}
n = Str.change(n, i-1, alphabet[digit-1])
q = r
}
r = r + zdx2
if (zdx2 != 1) {
r = r + carry
carry = getCarry.call(r, base2)
r = r - base2 * carry
}
res = Str.insert(res, 0, alphabet2[r-1])
n = n.trimStart("0")
}
if (carry != 0) res = Str.insert(res, 0, alphabet2[carry+zdx2-1])
if (m.count > 0) {
res = res + "."
ndp = 0
if (zdx != 1) {
var lm = m.count
var alphaNew = base37[0...alphabet.count]
m = b2b.call(m, alphabet, alphaNew)
m = ("0" * (lm-m.count)) + m
alphabet = alphaNew
zdx = 1
}
while (m.count > 0 && ndp < maxdp) {
q = 0
for (i in Stepped.descend(m.count..1)) {
digit = alphabet.indexOf(m[i-1]) + 1 - zdx
q = q + digit * base2
r = q.abs % base + zdx
q = (q / base).truncate
if (q < 0) q = q - 1
m = Str.change(m, i-1, alphabet[r-1])
}
digit = q + zdx2
if (zdx2 != 1) {
var p = bCarry.call(digit, base2, res.count, res, alphabet2)
digit = p[0]
res = p[1]
}
res = res + alphabet2[digit-1]
m = m.trimEnd("0")
ndp = ndp + 1
}
}
res = bTrim.call(res)
if (negative) res = negate.call(res, alphabet2)
}
return res
}
// convert 'd' to a string in the specified base var float2b = Fn.new { |d, alphabet|
var base = alphabet.count
var zdx = alphabet.indexOf("0") + 1
var carry = 0
var neg = d < 0
if (neg) d = -d
var res = ""
var whole = d.floor
d = d - whole
while (true) {
var ch = whole % base + zdx
if (zdx != 1) {
ch = ch + carry
carry = getCarry.call(ch, base)
ch = ch - base * carry
}
res = Str.insert(res, 0, alphabet[ch-1])
whole = (whole / base).truncate
if (whole == 0) break
}
if (carry != 0) {
res = Str.insert(res, 0, alphabet[carry+zdx-1])
carry = 0
}
if (d != 0) {
res = res + "."
var ndp = 0
while (d != 0 && ndp < maxdp) {
d = d * base
var digit = d.truncate + zdx
d = d - digit
if (zdx != 1) {
var p = bCarry.call(digit, base, res.count, res, alphabet)
digit = p[0]
res = p[1]
}
res = res + alphabet[digit-1]
ndp = ndp + 1
}
}
if (neg) res = negate.call(res, alphabet)
return res
}
var bSub // forwward declaration
var bAdd = Fn.new { |a, b, alphabet|
var base = alphabet.count
var zdx = alphabet.indexOf("0") + 1
var carry = 0
var da = 0
var db = 0
var digit = 0
if (zdx == 1) {
if (a[0] == "-") {
return bSub.call(b, negate.call(a, alphabet), alphabet)
}
if (b[0] == "-") {
return bSub.call(a, negate.call(b, alphabet), alphabet)
}
}
var adt = a.indexOf(".") + 1
var bdt = b.indexOf(".") + 1
if (adt != 0 || bdt != 0) {
if (adt != 0) {
adt = a.count - adt + 1
var wi = wIndex.call(-adt, a.count)
a = Str.delete(a, wi)
}
if (bdt != 0) {
bdt = b.count - bdt + 1
var wi = wIndex.call(-bdt, b.count)
b = Str.delete(b, wi)
}
if (bdt > adt) {
a = a + ("0" * (bdt-adt))
adt = bdt
} else if (adt > bdt) {
b = b + ("0" * (adt-bdt))
}
}
if (a.count < b.count) {
var t = a
a = b
b = t
}
for (i in Stepped.descend(-1..-a.count)) {
if (i < -a.count) {
da = 0
} else {
da = alphabet.indexOf(a[a.count + i]) + 1 - zdx
}
if (i < -b.count) {
db = 0
} else {
db = alphabet.indexOf(b[b.count + i]) + 1 - zdx
}
digit = da + db + carry + zdx
carry = getCarry.call(digit, base)
a = Str.change(a, i + a.count, alphabet[digit-carry*base-1])
if (i < -b.count && carry == 0) break
}
if (carry != 0) {
a = Str.insert(a, 0, alphabet[carry+zdx-1])
}
if (adt != 0) {
var wi = wIndex.call(-adt+1, a.count)
a = Str.insert(a, wi, ".")
}
a = bTrim.call(a)
return a
}
var aSmaller = Fn.new { |a, b, alphabet|
if (a.count != b.count) Fiber.abort("strings should be equal in length")
for (i in Stepped.ascend(1..a.count)) {
var da = alphabet.indexOf(a[i-1]) + 1
var db = alphabet.indexof(b[i-1]) + 1
if (da != db) return da < db
}
return false
}
// declared earlier bSub = Fn.new { |a, b, alphabet|
var base = alphabet.count
var zdx = alphabet.indexOf("0") + 1
var carry = 0
var da = 0
var db = 0
var digit = 0
if (zdx == 1) {
if (a[0] == "-") {
return negate.call(bAdd.call(negate.call(a, alphabet), b, alphabet), alphabet)
}
if (b[0] == "-") {
return bAdd.call(a, negate.call(b, alphabet), alphabet)
}
}
var adt = a.indexOf(".") + 1
var bdt = b.indexOf(".") + 1
if (adt != 0 || bdt != 0) {
if (adt != 0) {
adt = a.count - adt + 1
var wi = wIndex.call(-adt, a.count)
a = Str.delete(a, wi)
}
if (bdt != 0) {
bdt = b.count - bdt + 1
var wi = wIndex.call(-bdt, b.count)
b = Str.delete(b, wi)
}
if (bdt > adt) {
a = a + ("0" * (bdt-adt))
adt = bdt
} else if (adt > bdt) {
b = b + ("0" * (adt-bdt))
}
}
var bNegate = false
if (a.count < b.count || (a.count == b.count && aSmaller.call(a, b, alphabet))) {
bNegate = true
var t = a
a = b
b = t
}
for (i in Stepped.descend(-1..-a.count)) {
if (i < -a.count) {
da = 0
} else {
da = alphabet.indexOf(a[a.count+i]) + 1 - zdx
}
if (i < -b.count) {
db = 0
} else {
db = alphabet.indexOf(b[b.count+i]) + 1 - zdx
}
digit = da - db - carry + zdx
carry = 0
if (digit <= 0) carry = 1
a = Str.change(a, i+a.count, alphabet[digit+carry*base-1])
if (i < -b.count && carry == 0) break
}
if (carry != 0) Fiber.abort("carry should be zero")
if (adt != 0) {
var wi = wIndex.call(-adt+1, a.count)
a = Str.insert(a, wi, ".")
}
a = bTrim.call(a)
if (bNegate) a = negate.call(a, alphabet)
return a
}
var bMul = Fn.new { |a, b, alphabet|
var zdx = alphabet.indexOf("0") + 1
var dpa = a.indexOf(".") + 1
var dpb = b.indexOf(".") + 1
var ndp = 0
if (dpa != 0) {
ndp = ndp + a.count - dpa
a = Str.delete(a, dpa-1)
}
if (dpb != 0) {
ndp = ndp + b.count - dpb
b = Str.delete(b, dpb-1)
}
var pos = a
var res = "0"
if (zdx != 1) {
// balanced number systems
var neg = negate.call(pos, alphabet)
for (i in Stepped.descend(b.count..1)) {
var m = alphabet.indexOf(b[i-1]) + 1 - zdx
while (m != 0) {
var temp = pos
var temp2 = -1
if (m < 0) {
temp = neg
temp2 = 1
}
res = bAdd.call(res, temp, alphabet)
m = m + temp2
}
pos = pos + "0"
neg = neg + "0"
}
} else {
// non-balanced number systems
var negative = false
if (a[0] == "-") {
a = a[1..-1]
negative = true
}
if (b[0] == "-") {
b = b[1..-1]
negative = !negative
}
for (i in Stepped.descend(b.count..1)) {
var m = alphabet.indexOf(b[i-1]) + 1 - zdx
while (m > 0) {
res = bAdd.call(res, pos, alphabet)
m = m - 1
}
pos = pos + "0"
}
if (negative) res = negate.call(res, alphabet)
}
if (ndp != 0) {
var wi = wIndex.call(-ndp, res.count)
res = Str.insert(res, wi, ".")
}
res = bTrim.call(res)
return res
}
var multTable = Fn.new {
System.print("multiplication table")
System.print("====================")
System.write("* |")
for (j in 1..12) {
Fmt.write(" #$s $3s |", float2b.call(j, hexadecimal), float2b.call(j, balancedTernary))
}
for (i in 1..27) {
var a = float2b.call(i, balancedTernary)
Fmt.write("\n$-2s|", float2b.call(i, septemVigesimal))
for (j in 1..12) {
if (j > i) {
System.write(" |")
} else {
var b = float2b.call(j, balancedTernary)
var m = bMul.call(a, b, balancedTernary)
Fmt.write(" $6s |", m)
}
}
}
System.print()
}
var test = Fn.new {|name, alphabet|
var a = b2b.call("+-0++0+.+-0++0+", balancedTernary, alphabet)
var b = b2b.call("-436.436", decimal, alphabet)
var c = b2b.call("+-++-.+-++-", balancedTernary, alphabet)
var d = bSub.call(b, c, alphabet)
var r = bMul.call(a, d, alphabet)
Fmt.print("$s\n$s", name, "=" * name.count)
Fmt.print(" a = $.14f $s", b2dec.call(a, alphabet), a)
Fmt.print(" b = $.14f $s", b2dec.call(b, alphabet), b)
Fmt.print(" c = $.14f $s", b2dec.call(c, alphabet), c)
Fmt.print("a*(b-c) = $.14f $s\n", b2dec.call(r, alphabet), r)
}
test.call("balanced ternary", balancedTernary) test.call("balanced base 27", balancedBase27) test.call("decimal", decimal) test.call("binary", binary) test.call("ternary", ternary) test.call("hexadecimal", hexadecimal) test.call("septemvigesimal", septemVigesimal) multTable.call()</lang>
- Output:
balanced ternary
================
a = 523.23914037494285 +-0++0+.+-0++0+
b = -436.43599999999992 -++-0--.--0+-00+++-0-+---0-+0++++0--0000+00-+-+--+0-0-00--++0-+00---+0+-+++0+-0----0++
c = 65.26748971193416 +-++-.+-++-
a*(b-c) = -262510.90267998125637 ----000-0+0+.0+0-0-00---00--0-0+--+--00-0++-000++0-000-+0+-----+++-+-0+-+0+0++0+0-++-++0+---00++++
balanced base 27
================
a = 523.23914037494285 AUJ.FLI
b = -436.43600000000004 NKQ.YFDFTYSMHVANGXPVXHIZJRJWZD0PBGFJAEBAKOZODLY0ITEHPQLSQSGLFZUINATKCIKUVMWEWJMQ0COTS
c = 65.26748971193416 BK.GF
a*(b-c) = -262510.90267998125637 ZVPJ.CWNYQPEENDVDPNJZXKFGCLHKLCX0YIBOMETHFWWBTVUFAH0SEZMTBJDCRRAQIQCAWMKXSTPYUXYPK0LODUO
decimal
=======
a = 523.23914037494296 523.239140374942844078646547782350251486053955189757658893461362597165066300868770004
b = -436.43599999999998 -436.436
c = 65.26748971193415 65.267489711934156378600823045267489711934156378600823045267489711934156378600823045
a*(b-c) = -262510.90267998137278 -262510.90267998140903693918986303277315826215892262734715612833785876513103053772667101895163734826631742752252837097627017862754285047634638652268078676654605120794218
binary
======
a = 523.23914037494274 1000001011.001111010011100001001101101110011000100001011110100101001010100100000111001000111
b = -436.43600000000004 -110110100.011011111001110110110010001011010000111001010110000001000001100010010011011101001
c = 65.26748971193416 1000001.01000100011110100011010010101100110001100000111010111111101111001001001101111101
a*(b-c) = -262510.90267998143099 -1000000000101101110.111001110001011000001001000001101110011111011100000100000100001000101011100011110010110001010100110111001011101001010000001110110100111110001101000000001111110101
ternary
=======
a = 523.23914037494285 201101.0201101
b = -436.43600000000021 -121011.102202211210021110012111201022222000202102010100101200200110122011122101110212
c = 65.26748971193416 2102.02102
a*(b-c) = -262510.90267998125637 -111100002121.2201010011100110022102110002120222120100001221111011202022012121122001201122110221112
hexadecimal
===========
a = 523.23914037494274 20B.3D384DB9885E94A90723EF9CBCB174B443E45FFC41152FE0293416F15E3AC303A0F3799ED81589C62
b = -436.43599999999998 -1B4.6F9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB22D0E5604189374BC6A7EF9DB2
c = 65.26748971193416 41.447A34ACC60EBFBC937D5DC2E5A99CF8A021B641511E8D2B3183AFEF24DF5770B96A673E28086D905
a*(b-c) = -262510.90267998143099 -4016E.E7160906E7DC10422DA508321819F4A637E5AEE668ED5163B12FCB17A732442F589975B7F24112B2E8F6E95EAD45803915EE26D20DF323D67CAEEC75D7BED68AA34E02F2B492257D66F028545FB398F60E
septemvigesimal
===============
a = 523.23914037494285 JA.6C9
b = -436.43600000000004 -G4.BKML7C5DJ8Q0KB39AIICH4HACN02OJKGPLOPG2D1MFBQI6LJ33F645JELD7I0Q6FNHG88E9M9GE3QO276
c = 65.26748971193416 2B.76
a*(b-c) = -262510.90267998125637 -D92G.OA1C42LM0N8N30HDAFKJNEIFEOB0BHP1DM6ILA9P797KPJ05MCE6OGMO54Q3I3NQ9DGB673C8BC2FQF1N82
multiplication table
====================
* | #1 + | #2 +- | #3 +0 | #4 ++ | #5 +-- | #6 +-0 | #7 +-+ | #8 +0- | #9 +00 | #A +0+ | #B ++- | #C ++0 |
1 | + | | | | | | | | | | | |
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 |