Inverse Ax+B/Cx+D sized function

Question

Let's assume that

$$ f(x) = \frac{Ax+B}{Cx+D} $$

Where, \$x\$ is a variable and \$A\$,\$B\$,\$C\$,\$D\$ are constants.

Now we have to find out the inverse function of \$f(x)\$, mathematically \$f^{-1}(x)\$, To do this first we assume,

$$ y = f(x) \\\rightarrow y=\frac{Ax+B}{Cx+D} \\\rightarrow Cxy+Dy=Ax+B \\\rightarrow Cxy-Ax=-Dy+B \\\rightarrow x(Cy-A)=-Dy+B \\\rightarrow x=\frac{-Dy+B}{Cy-A} $$

Then, we know that

$$ y=f(x) \\\rightarrow f^{-1}(y)=x \\\rightarrow f^{-1}(y)=\frac{-Dy+B}{Cy-A} ..... (i) $$

And from \$(i)\$ equation, we can write \$x\$ instead of \$y\$

$$ \\\rightarrow f^{-1}(x)=\frac{-Dx+B}{Cx-A} $$

So, \$\frac{-Dx+B}{Cx-A}\$ is the inverse function of \$f(x)\$

This is a very long official mathematical solution, but we have a "cool" shortcut to do this:

1. Swap the position of the first and last constant diagonally, in this example \$A\$ and \$D\$ will be swapped, so it becomes:

$$ \frac{Dx+B}{Cx+A} $$

2. Reverse the sign of the replaced constants, in this example \$A\$ is positive (\$+A\$) so it will be negative \$-A\$, \$D\$ is positive (\$+D\$) so it will be negative \$-D\$

$$ \frac{-Dx+B}{Cx-A} $$

And VOILA!! We got the inverse function \$\frac{Ax+B}{Cx+D}\$ in just two steps!!

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Challenge

(Input of \$\frac{Ax+B}{Cx+D}\$ is given like Ax+B/Cx+D)

Now, let's go back to the challenge.

Input of a string representation of a function of \$\frac{Ax+B}{Cx+D}\$ size, and output its inverse function in string representation.

I have just shown two ways to that (Second one will be easier for programs), there may be other ways to do this, good luck!

Test cases

(Input of \$\frac{Ax+B}{Cx+D}\$ is given like Ax+B/Cx+D)

4x+6/8x+7 -> -7x+6/8x-4
2x+3/2x-1 -> x+3/2x-2
-4x+6/8x+7 -> -7x+6/8x+4
2x+3/2x+1 -> x+3/2x+2

Or you can give it using list of A,B,C,D

4,6,8,7 -> -7x+6/8x-4

Or you can output -7,6,8,-4

Rules

• Input is always in \$\frac{Ax+B}{Cx+D}\$ size, and is guaranteed to be valid.
• Standard loopholes are forbidden.
• Trailing/Leading whitespace in output is allowed.
• If possible, please link to an online interpreter (e.g. TIO) to run your program on.
• Please explain your answer. This is not necessary, but it makes it easier for others to understand.
• Languages newer than the question are allowed. This means you could create your own language where the empty program calculates this number, but don't expect any upvotes.
• This is code-golf, so shortest code in bytes wins!

(Some terminology might be incorrect, feel free ask me if you have problems)

Answer 1

Perl 5, 27 bytes

sub{@_[0,3]=map-$_,@_[3,0]}

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Answer 2

Retina 0.8.2, 48 bytes

^-
+
^\d
-$&
(.\d+)(.*)([-+].+)
$3$2$1
^-

^\+
-

Try it online! Link includes test cases. Assumes that 1x is always written as such. Explanation:

^-
+
^\d
-$&

Negate and enforce the sign on A.

(.\d+)(.*)([-+].+)
$3$2$1

Swap A and D.

^-

^\+
-

Negate D, removing the sign if it was negative.

Answer 3

Vyxal, 9 bytes

÷↭⫙'N^N^W

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Explained

÷↭⫙'N^N^W
÷          # Push every item of the input onto the stack
 ↭        # And rotate the top three items: [a, b, c, d] → [a, d, b, c]
   ⫙'      # Rotate the entire stack left
     N^N^W # Negate the top and bottom of the stack and wrap into a list

Answer 4

brainfuck, 31 29 bytes

,[-<->],>,>,[->-<]>.<<<.>.<<.

Commented:

,            #input A into cell 0
[-<->]       #decrement cells 0 and -1 until cell 0 contains 0 and cell -1 contains -A
,>,>,        #input B,C,D into cells 0,1,2
[->-<]       #decrement cells 2 and 3 until cell 2 contains 0 and cell 3 contains -D
>.<<<.>.<<.  #output -D,B,C and -A

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Very rarely, a challenge comes along where Brainfuck is somewhat competitive.

Takes 4 bytes as input, gives 4 bytes as output.

The normal form of interface for Brainfuck is an ASCII terminal, so the input ABCD in the TIO link is intrepreted as [65,66,67,68]. Outputs are per the table below. Note that signed negative numbers are treated as their unsigned 8 bit counterparts, and give extended ASCII output in the range 128-255, in accordance with https://en.wikipedia.org/wiki/Extended_ASCII#/media/File:Table_ascii_extended.png

Input             Signed Output    Unsigned equivalent ASCII output  
A=65              -65              191                  ¿
B=66               66               66                  B
C=67               67               67                  C
D=68              -68              188                  ¼             

After swapping A and D, we arrive at the required output [-68,66,67,-65], which in ASCII looks like ¼BC¿

Answer 5

Keg, 7 6 bytes ^(SBCS)

Input as a list. Outputs by values on the stack.

Footer outputs entire stack joined by newlines.

÷±'±$"

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Explanation

        # Input as a list    (e.g. [4, 6, 8, 7])
÷       # Dump it onto stack (Stack = [ 4, 6, 8, 7])
 ±      # Negate TOS         (Stack = [ 4, 6, 8,-7])
  '     # Shift left         (Stack = [ 6, 8,-7, 4])
   ±    # Negate TOS         (Stack = [ 6, 8,-7,-4])
    $   # Exchange TOS       (Stack = [ 6, 8,-4,-7])
     "  # Shift right        (Stack = [-7, 6, 8,-4])

Footer explanation

^        # Reverse the stack (So that the first item of TOS is outputted first)
 ∑       # Apply to entire stack:
  .      # Output TOS as integer
   19+,  # Print a trailing newline

Credits

• Saved 1 byte thanks to @Lyxal

Answer 6

PowerShell, 43 bytes

Takes input as and outputs as an array.

$n=$args;-1,1,2,-4|%{$n[$_]*(2*($_-gt0)-1)}

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Less Fun PowerShell, 31 bytes

param($a,$b,$c,$d)-$d,$b,$c,-$a

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Most Unfun PowerShell, 30 bytes

-$args[3],$args[1,2],-$args[0]

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Least Useful PowerShell, 26 bytes

"-{3},{1},{2},-{0}"-f$args

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Answer 7

05AB1E, 7 bytes

À`(s()Á

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         # implicit input                [A, B, C, D]
À        # rotate the input left         [B, C, D, A]
 `       # dump all values on the stack  B, C, D, A
  (      # negate top of stack           B, C, D, -A
   s     # swap top two values           B, C, -A, D
    (    # negate top of stack           B, C, -A, -D
     )   # collect all values in a list  [B, C, -A, -D]
      Á  # rotate the list right         [-D, B, C, -A]

Answer 8

APL(Dyalog Unicode), ⁹ 7 bytes ^SBCS

-∘⌽@1 4

Try it on APLgolf!

A function submission which accepts an array.

-2 bytes from rak1507.

Answer 9

Ruby, 26 bytes

->n{a,b,c,d=n
[-d,b,c,-a]}

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Shorter than my brainfuck answer by a few bytes, but not as interesting.

Answer 10

jq, 22 bytes

[-.[3],.[1:3][],-.[0]]

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Answer 11

Java, 31 bytes

(a,b,c,d)->new int[]{-d,b,c,-a}

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Answer 12

JavaScript (Node.js), 22 bytes

(a,b,c,d)=>[-d,b,c,-a]

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Answer 13

Dash, 23 bytes

$((-$4)) $2 $3 $((-$1))

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Usage

Requires positional parameters for each of A, B, C, and D. Before evaluating the command. Also the code evaluates to just four strings, so run echo command with those code (may not portable when D is originally positive, as hyphen-parameter may be treated as an option).

How it works

• dollar-double-parentheses is arithmetic expansion.

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Polyglot for Bash, OSH, bosh, Dash, ksh, yash, and Zsh, 25 bytes

$((- $4)) $2 $3 $((- $1))

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Differences

• A space between hyphen and dollar. This is a workaround for negative values; if one of them was expanded to $((--1)), for example, Dash, OSH and Bash parse as minus of minus one, but other four don't; they parse to decrement a variable named one, although one is a constant.

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Polyglots for Bash and OSH, 19 bytes

Thanks, 2x-1!

$[-$4] $2 $3 $[-$1]

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What the heck

$[ ... ] is historically equivalent to $(( ... )), which is not adopted for POSIX.

OBTW adding a space betwee hyphen and dollar (which is +2 bytes), the code would be polyglot for Zsh, too.

Answer 14

convey, 16 + 6 = 22 or 53 bytes

Version 1: 16 (but takes extra input)

{?>>>>*}
 >>>>>^

After the four numbers, also input -1 1 1 -1 (+6 bytes). I can't figure out how to do it shortly any other way. Try it online!

Version 2, 53 bytes

{?>*}v<0
 v ^<<^-1
 ?>>*}
v<  ^v<1
v 0>^v
v1-^<<
>>*}

It's complicated. Multiplies the first and last numbers by -1. I'm a noob at convey and wait was crashing the program. Try it online!

Answer 15

dc, 19 bytes

0?0r-sbscsd-lcldlbf

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Explanation

                     # Say the input is 4 6 8 7.
                     # Effect:
0                    # [0]
 ?                   # [0, 4, 6, 8, 7]
  0r-                # [0, 4, 6, 8,-7]
     sbscsd          # [0, 4] (B=-7, C=8, D=4)
           -         # [-4]   (B=-7, C=8, D=4)
            lcldlb   # [-4, 8, 4, -7]
                  f  # Output in reverse (i.e. -7 6 8 -4)

Answer 16

MMIX, 44 bytes (11 instrs)

Inverts mobius *a into mobius *b. This is with typedef double mobius[4];.

Note that NaNs and zeroes have the sign negated; to not do so would cost an extra instruction (zeroing out $2 at start and replacing INCL $255,#8000 with FSUB $255,$2,$255).

(jxd)

00000000: 8fff0018 e4ff8000 afff0100 8fff0010  Ɓ”¡ðỵ”⁰¡Ḥ”¢¡Ɓ”¡Ñ
00000010: afff0108 8fff0008 afff0110 8fff0000  Ḥ”¢®Ɓ”¡®Ḥ”¢ÑƁ”¡¡
00000020: e4ff8000 afff0118 f8000000           ỵ”⁰¡Ḥ”¢ðẏ¡¡¡

Disassembled:

invmob  LDOU $255,$0,24
        INCH $255,#8000
        STOU $255,$1
        LDOU $255,$0,16
        STOU $255,$1,8
        LDOU $255,$0,8
        STOU $255,$1,16
        LDOU $255,$0
        INCH $255,#8000
        STOU $255,$1,24
        POP  0,0

Answer 17

Python 3, 26 bytes

lambda a,b,c,d:[-d,b,c,-a]

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4 input. Output is list.