Real base conversion

Question

We've had a few challenges for base conversion, but all of them seem to apply to integer values. Let's do it with real numbers!

The challenge

Inputs:

• A real positive number x, expressed in base 10. This can be taken as a double-precision float or as a string. To avoid precision issues, the number can be assumed to be greater than 10⁻⁶ and less than 10¹⁵.
• A target base b. This will be an integer from 2 to 36.
• A number of fractional digits n. This will be an integer from 1 to 20.

Output: the representation of x in base b with n fractional digits.

When computing the output expression, the digits beyond the n-th should be truncated (not rounded). For example, x = 3.141592653589793 in base b = 3 is 10.0102110122..., so for n = 3 the output would be 10.010 (truncation), not 10.011 (rounding).

For x and b that produce a finite number of digits in the fractional part, the equivalent infinite representation (truncated to n digits) is also allowed. For example, 4.5 in decimal can also be represented as 4.49999....

Don't worry about floating point errors.

Input and output format

x will be given without leading zeros. If x happens to be an integer you can assume that it will be given with a zero decimal part (3.0), or without decimal part (3).

The output is flexible. For example, it can be:

• A string representing the number with a suitable separator (decimal point) between integer and fractional parts. Digits 11, 12 etc (for b beyond 10) can be represented as letters A, B as usual, or as any other distinct characters (please specify).
• A string for the integer part and another string for the fractional part.
• Two arrays/lists, one for each part, containing numbers from 0 to 35 as digits.

The only restrictions are that the integer and fractional parts can be told apart (suitable separator) and use the same format (for example, no [5, 11] for the list representing the integer part and ['5', 'B'] for the list representing the fractional part).

Additional rules

• Programs or functions are allowed, in any programming language. Standard loopholes are forbidden.
• Shortest code in bytes wins.

Test cases

Output is shown as a string with digits 0, ..., 9, A, ... , Z, using . as decimal separator.

x, b, n                    ->  output(s)

4.5, 10, 5                 ->  4.50000 or 4.49999
42, 13, 1                  ->  33.0 or 32.C
3.141592653589793, 3, 8    ->  10.01021101
3.141592653589793, 5, 10   ->  3.0323221430
1.234, 16, 12              ->  1.3BE76C8B4395
10.5, 2, 8                 ->  1010.10000000 or 1010.01111111
10.5, 3, 8                 ->  101.11111111
6.5817645, 20, 10          ->  6.BCE2680000 or 6.BCE267JJJJ
0.367879441171442, 25, 10  ->  0.94N2MGH7G8
12944892982609, 29, 9      ->  PPCGROCKS.000000000

Answer 1

Jelly, 16 bytes

*×⁵b⁸ḞðṖḣ⁹,ṫø⁹N‘

Try it online!

Note that singletons are printed as the element in the output.

Answer 2

JavaScript (ES8), 81 74 71 bytes

f=
(x,b,n,g=x=>x.toString(b))=>g(x-x%1)+'.'+g(x%1).substr(2,n).padEnd(n,0)

<div oninput=o.textContent=f(+x.value,b.value,n.value)><input id=x><input type=number min=2 max=36 value=10 id=b><input type=number min=1 max=20 value=10 id=n><pre id=o>

Works for x between 1e-6 and 1e21, b from 2 to 36 (exactly as required) and n from 1 to anything from 10 to 48 depending on the base before floating-point errors creep in. Edit: Saved 7 bytes with help from @Birjolaxew. Saved a further 3 bytes with help from @tsh. Previous 74-byte version also worked with negative numbers:

f=
(x,b,n,[i,d]=`${x.toString(b)}.`.split`.`)=>i+`.`+d.slice(0,n).padEnd(n,0)

<div oninput=o.textContent=f(+x.value,b.value,n.value)><input id=x><input type=number min=2 max=36 value=10 id=b><input type=number min=1 max=20 value=10 id=n><pre id=o>

Answer 3

Python 2, 153 149 144 137 135 109 bytes

def f(x,b,m):
 i=int(x);s=[];t=[]
 while i:s=[i%b]+s;i/=b
 while m:m-=1;x=x%1*b;t+=[int(x)]
 return s or[0],t

Hadn't noticed I can just return the digits as numbers, so that makes it a lot simpler. Returns two lists of digits, first for the integer part, second for the fractional.

Try it online!

Answer 4

Perl 6, 25 bytes

->\x,\b,\n{+x .base(b,n)}

Try it

Expanded:

-> \x, \b, \n {
  +x            # make sure it is a Numeric
  .base( b, n ) # do the base conversion
}

Note that the space is so that it is parsed as (+x).base(b,n)
not +( x.base(b,n) ).

Answer 5

Mathematica, 158 bytes

since this challenge allready got a very nice answer in mathematica by @KellyLowder, I tried to produce (with a different approach) the exact results as shown in the test cases

ToUpperCase[""<>Insert[StringReplace[ToString@BaseForm[#,p]&/@PadRight[#&@@(d=RealDigits[#,p=#2]),w=(#3+d[[2]])][[;;w]],"\n "<>ToString@p->""],".",d[[2]]+1]]&

input

[12944892982609, 29, 9]

output

PPCGROCKS.000000000

Answer 6

C (gcc), 157 152 bytes

Needs 64 bits long int for this to work with larger test cases.

-5 bytes thanks to Peter Cordes

#define P r=99;i=l=x;do{z[--r]=48+7*(l%b>9)+l%b;}while(l/=b);printf(z+r)
long i,r,l;char z[99];f(x,b,n)double x;{P;putchar(46);while(n--){x=(x-i)*b;P;}}

Try it online!

edit: a few bytes can be shaved if it's allowed to output two strings separated by a newline separator:

149 bytes:

#define P r=99;i=l=x;do{z[--r]=48+7*(l%b>9)+l%b;}while(l/=b);printf(z+r)
long i,r,l;char z[99];f(x,b,n)double x;{P;puts("");while(n--){x=(x-i)*b;P;}}

edit: this submission is not the longest one, yay!

Answer 7

Ruby, 45 bytes

->x,b,n{(x*b**n).round.to_s(b).insert(~n,?.)}

Why?

Since b^n in base b is 10^n, we multiply x by that number, and then add the decimal point where it belongs.

Try it online!

Answer 8

SageMath, 68 bytes

def f(n,b,k):y=n.str(b).split('.')+[''];return y[0],(y[1]+'0'*k)[:k]

Try it online!

Answer 9

Mathematica 47 Bytes

TakeDrop@@r[#,#2,#3+Last@(r=RealDigits)[#,#2]]&

Calling RealDigits twice to first figure out the number of digits to the left of the decimal.

Answer 10

Haskell, 188 bytes

f=fromIntegral
g 0 _=[]
g n p=g(div n p)p++[mod n p]
z=(!!)(['0'..'9']++['A'..'Z']++['.'])
h x p l|(i,d)<-properFraction x=z<$>(g i p++[36]++(last$g(floor$d*(f p**f l))p:[0<$[1..l]|d==0]))

Try it online!

g converts a number to a list representing that number in a given base

z maps integers to letters (36 = .)

h applies the previous functions to the integer and fractional part of a number.

Answer 11

Axiom, 127 bytes

g(a)==floor(a)::INT;f(a:Float,b:PI,n:NNI):Any==(b<2 or n>28=>%i;x:=g(a);radix(x,b)+radix(g((a-x)*b^n),b)::RadixExpansion b/b^n)

results

(4) -> f(%e,2,10)
   (4)  10.1011011111
                                                   Type: RadixExpansion 2
(5) -> f(%e,3,10)
   (5)  2.2011011212
                                                   Type: RadixExpansion 3
(6) -> f(%e,35,10)
   (6)  2.P4VBNEB51S
                                                  Type: RadixExpansion 35
(7) -> f(1.4,35,10)
   (7)  1.DYYYYYYYYY
                                                  Type: RadixExpansion 35
(8) -> f(%pi,3,8)
   (8)  10.01021101
                                                   Type: RadixExpansion 3
(9) -> f(%pi,5,10)
   (9)  3.032322143
                                                   Type: RadixExpansion 5
(10) -> f(1.234,16,12)
   (10)  1.3BE76C8B4395
                                                  Type: RadixExpansion 16

It has a little problem for final zero example

 f(4.5,10,5)

Would return '4.5' and not '4.50000'

Answer 12

Axiom, 566 bytes

c:=alphanumeric()::List Character
f(a:INT,b:PI):List Character==(r:=[];repeat(y:=a rem b;r:=cons(c.(y+1),r);a:=a quo b;a=0=>break);r)
g(x)==floor(x)::INT
F(x)==>for i in 1..#x repeat z:=concat(z,x.i)
w(a:Float,b:PI,n:NNI):String==
  z:="";b<2 or b>36 or a<0=>z
  ip:=g(a);    fp:=g((a-ip)*b^n)
  ipb:=f(ip,b);fpb:=f(fp,b);cnt:=n-#fpb
  for i in 1..cnt repeat fpb:=cons(c.1,fpb)
  F(ipb);z:=concat(z,".");F(fpb)
  z

h(a,b,n)==>(n>=0 and b>0=>(nd123:=10+g(n*log_2(b)/log_2(10));mxv123456:=digits(nd123::PI);res78484:=w(a,b,n);digits(mxv123456);res78484);"")

it was particular difficult this question; afther some time in write something, the right results it seems generate using one macro for preserve digits()... it is not golfed too much...results:

(7) -> h(4.5,10,5)
   (7)  "4.50000"
                                                             Type: String
(8) -> h(42,13,1)
   (8)  "33.0"
                                                             Type: String
(9) -> h(%pi,3,8)
   (9)  "10.01021101"
                                                             Type: String
(10) -> h(%pi,5,10)
   (10)  "3.0323221430"
                                                             Type: String
(11) -> h(1.234,16,12)
   (11)  "1.3BE76C8B4395"
                                                             Type: String
(12) -> h(0.367879441171442,25,10)
   (12)  "0.94N2MGH7G8"
                                                             Type: String
(13) -> h(12944892982609,29,9)
   (13)  "PPCGROCKS.000000000"
                                                             Type: String
(14) -> h(6.5817645,20,10)
   (14)  "6.BCE267JJJJ"
                                                             Type: String

the real target is one function that convert to base 2..36 each Float [that has k:=digits()] or each calculated number as %pi or %e or the division of two float/int as in 1./3. ['oo' digits]

(15) -> h(%pi,13,800)
   (15)
  "3.1AC1049052A2C77369C0BB89CC9883278298358B370160306133CA5ACBA57614B65B410020
  C22B4C71457A955A5155B04A6CB6CC2C494843A8BBBBA9A039B77B34CB0C036CAC761129B3168
  B8BAB860134C419787C911812985646C7AAA3025BAA118B3AB8265CB347852065667291482145
  6C533447BC53A5262177C9985455C395626091A2CC3126B395C91B65B654A1804226197528410
  29A8A4A55CC7937B347B77B5A914127B11C6A57A84510775A9A467819A468B6B74339CC1290B2
  24921C6A771BC2AB6AB41735119C2231545A86399483119AAA5AC34B46B7B5C9089946A364860
  9B26CB0BAC0ABCBA182C12881933AA93C3942C71AA664753989A3C82166BA2109796C4A134607
  59725A72C9117AC980556A147557C319438287226C94725B125753B009387A48AA45CB1960A04
  A064052C00A6069371949872B14590895C555CB01A39B7589824B8621618A8B1971841201A2AB
  B04B80C7534CC1CB079581491995B46C679555316288C82665645A1A600C1A669B865651B6B842470C018B03C1115B3C4306C015C0B45C"
                                                             Type: String