C, 24 23 bytes

Thanks to @aross for saving a byte!

f(n){return n>>31|!!n;}

Sigh.

Not guaranteed to work on all systems, works on TIO.

Retina, 9 bytes

[1-9].*
1

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Replaces a non-zero digit and everything after it with 1. This leaves a potential leading - intact and changes all numbers except 0 itself to absolute value 1.

COW, 225 213 201 bytes

oomMOOmoOmoOmoOmoOMoOMoOmOomOomOoMoOMMMmoOMMMMOOMOomOo
mOoMOomoOmoOmoomOomOoMMMmoOmoOmoOMMMMOOOOOmoOMOoMOomOo
mOomOoMoOMMMmoOMMMMOOMOomOomOoMoOmoOmoOmoomoOmoomOomOo
mOomoomoOMOOmoOmoOmoOMOoMMMOOOmooMMMOOM

Try it online!

The way that this code works is that it determines the sign by alternating adding and subtracting bigger numbers, and seeing which one was the last one that worked. Given any non-zero integer, first subtract 1, then add 2, then subtract 3, etc. and you'll eventually reach 0. Keep track of your state by alternating adding and subtracting 2 to a value that starts off at 0. For example:

-5  - 1  = -6  (current state: 0 + 2 = 2)
-6  + 2  = -4  (current state: 2 - 2 = 0)
-4  - 3  = -7  (current state: 0 + 2 = 2)
-7  + 4  = -3  (current state: 2 - 2 = 0)
-3  - 5  = -8  (current state: 0 + 2 = 2)
-8  + 6  = -2  (current state: 2 - 2 = 0)
-2  - 7  = -9  (current state: 0 + 2 = 2)
-9  + 8  = -1  (current state: 2 - 2 = 0)
-1  - 9  = -10 (current state: 0 + 2 = 2)
-10 + 10 =  0  (current state: 2 - 2 = 0)
value is now at 0.  state - 1 = 0 - 1 = -1
sign of original number is -1

When you're done, subtract 1 from your state and you get the sign, positive or negative. If the original number is 0, then don't bother doing any of this and just print 0.

Detailed Explanation:

oom                                        ;Read an integer into [0]
MOO                                        ;Loop while [0] is non-empty
    moOmoOmoOmoOMoOMoOmOomOomOo            ;    Decrement [4] twice
    MoOMMMmoOMMM                           ;    Increment [1], then copy [1] to [2]
    MOO                                    ;    Loop while [2] is non-empty
        MOomOomOoMOomoOmoO                 ;        Decrement [0] and [2]
    moo                                    ;    End loop now that [2] is empty
    mOomOoMMMmoOmoOmoOMMM                  ;    Navigate to [0], and copy to [3]
    MOO                                    ;    Perform the next steps only if [3] is non-zero
        OOOmoOMOoMOomOomOomOoMoOMMMmoOMMM  ;        Clear [3], increment [4] twice, increment [1], and copy it to [2]
        MOO                                ;        Loop while [2] is non-empty
            MOomOomOoMoOmoOmoO             ;            Decrement [2] and increment [0]
        moo                                ;        End loop now that [2] is empty
    moO                                    ;        Navigate back to [3]
    moo                                    ;    End the condition
    mOomOomOo                              ;    Navigate back to [0]
moo                                        ;End loop once [0] is empty.
moO                                        ;Navigate to [1]. If [1] is 0, then input was 0.  Otherwise, [4] contains (sign of [0] + 1)
MOO                                        ;Perform the next steps only if [1] is non-zero
    moOmoOmoOMOoMMMOOO                     ;    Navigate to [4], copy it to the register, and clear [4].
moo                                        ;End condition
MMMOOM                                     ;If the register contains something (which is true iff the condition ran), paste it and print it.  Otherwise, no-op and print 0.

I'm still experimenting with golfing it (you will be shocked to discover that golfing in COW is rather difficult), so this may come down a few more bytes in the future.

Mathematica, 4 bytes

Clip

How about not using the built-in Sign and still scoring 4 bytes? ;)

Clip with a single argument clips (or clamps) the input value between -1 and 1. Since the inputs will only be integers, this is the same as using Sign.

Cubix, 10 bytes

(W0^I?>O2@

Test it online!

This code is wrapped to the following cube net:

    ( W
    0 ^
I ? > O 2 @ . .
. . . . . . . .
    . .
    . .

The code is then run with the IP (instruction pointer) starting on the I, facing east. I inputs a signed integer from STDIN, pushing it onto the stack.

The next command is ?, which changes the direction of the IP depending on the sign of the top item. If the input is 0, it keeps moving in same direction, running through the following code:

• > - Point the IP to the east. (No-op since we're already going east.)
• O - Output the top item as an integer.
• 2 - Push 2 to the stack. This is practically a no-op, because...
• @ - Terminates the program.

If the input is negative, the IP turns left at the ?; because this is a cube, the IP moves onto the 0 in the second row, heading east. 0 pushes a literal 0, then this code is run:

• ^ - Point the IP north.
• W - "Sidestep" the IP one spot to the left.
• ( - Decrement the top item.

The TOS is now -1, and the IP wraps around the cube through a bunch of no-ops . until it hits the >. This runs the same output code mentioned above, outputting -1.

If the input is positive, the same thing happens as with negative inputs, with one exception: the IP turns right instead of left at the ?, and wraps around the cube to the 2, which pushes a literal 2. This is then decremented to 1 and sent to output.

JavaScript (ES6), 9 bytes

Math.sign

Straightforward.

The shortest non-builtin is 13 bytes:

n=>n>0|-(n<0)

Thanks to @Neil, this can be golfed by a byte, but at the cost of only working on 32-bit integers:

n=>n>0|n>>31

Or you could do

n=>n>0?1:!n-1

which seems more golfable, but I'm not sure how.

Labyrinth, 10 bytes

?:+:)%:(%!

Try it online!

Explanation

Labyrinth's control flow semantics actually give you a "free" way to determine a number's sign, because the chosen path at a 3-way fork depends on whether the sign is negative, zero or positive. However, I haven't been able to fit a program with junctions into less than 12 bytes so far (although it may be possible).

Instead, here's a closed-form solution, that doesn't require any branches:

Code    Comment             Example -5      Example 0       Example 5
?       Read input.         [-5]            [0]             [5]
:+      Double.             [-10]           [0]             [10]
:)      Copy, increment.    [-10 -9]        [0 1]           [10 11]
%       Modulo.             [-1]            [0]             [10]
:(      Copy, decrement.    [-1 -2]         [0 -1]          [10 9]
%       Modulo.             [-1]            [0]             [1]
!       Print.              []              []              []

The instruction pointer then hits a dead end, turns around and terminates when % now attempts a division by zero.

Doubling the input is necessary to make this work with inputs 1 and -1, otherwise one of the two modulo operations would already attempt a division by zero.

><>, 9 8 bytes

Thanks to Sp3000 for saving a byte.

'i$-%n
/

There's an unprintable 0x01 before the /.

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Explanation

This is a port of my character code-based Labyrinth answer.

'     Push the entire program (except ' itself) onto the stack, which ends 
      with [... 1 47].
i     Read the first character of the input.
$-    Subtract the 47.
%     Take the 1 modulo this value.
n     Output the result as an integer.
0x01  Unknown command, terminates the program.

Dyalog APL, 1 byte

Works for complex numbers too, returning 1∠θ:

×

TryAPL online!

Without that built-in:

>∘0-<∘0

>∘0 more-than-zero

- minus

<∘0 less-than-zero

TryAPL online!

Vim, 22 bytes

xVp:s/-/-1^M:s/[1-9]/1^M

Saved one byte thanks to @DJMcMayhem!

Here, ^M is a literal newline.

As @nmjcman101 pointed out in the comments, a single regex can be used (:s/\v(-)=[^0].*/\11^M, 20 bytes) instead, but since this is basically the same as a Retina answer would be, I'm sticking to my own method.

Explanation:

xVp                        Delete everything except the first character. If the number is negative, this leaves a -, a positive leaves any number between 1 and 9, and 0 leaves 0.
   :s/-/-1^M               Replace a - with a -1
            :s/[1-9]/1^M   Replace any number between 1 and 9 with 1.

Here's a gif of it running with a negative number (old version):

Here's it running with 0:

Running with positive:

Python 2, 17 bytes

lambda n:cmp(n,0)

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PHP, 16 bytes

Uses the new spaceship operator.

<?=$argv[1]<=>0;

Brain-Flak 74 42 40 Bytes

Saved 2 bytes thanks to 1000000000

{([({}<([()])>)]<>(())){({}())<>}}{}({})

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Explanation:

{([({}<([()])>)]<>(())){({}())<>}}{}({})
{                                }       # if 0 do nothing
   (          )                          # push:                           
    {}<     >                            # the input, after 
       (    )                            # pushing:
        [  ]                             # negative:
         ()                              # 1

 (                    )                  # Then push:
  [            ]                         # the negative of the input
                <>                       # on the other stack with:
                   ()                    # a 1 
                  (  )                   # pushed under it

                       {        }        # while 1: 
                        ({}())           # increment this stack and...
                              <>         # switch stacks

                                 {}      # pop the top (the counter or 0 from input)
                                   (  )  # push:
                                    {}   # the top (this is a no-op, or pushes a 0)

Perl, 9 bytes

Requires -E at no extra cost.

say<><=>0

Usage

perl -E 'say<><=>0' <<< -9999
-1
perl -E 'say<><=>0' <<< 9999
1
perl -E 'say<><=>0' <<< -0
0

I'm happy with the fish operator!

///, 52 bytes

/2/1//3/1//4/1//5/1//6/1//7/1//8/1//9/1//10/1//11/1/

Ungolfed, explanation:

/2/1/
/3/1/
/4/1/
/5/1/
/6/1/
/7/1/
/8/1/
/9/1/
/10/1/
/11/1/

It's basically a MapReduce implemenatation, i.e. there are two phases:

• Replace all occurrences of digits 2-9 by 1, e.g. 1230405 -> 1110101
• Reduce pairs of 11 or 10 to 1 repeatedly, e.g. 1110101-> 1

If there was a - in front initially, it will remain and the output will be -1. A single 0 is never replaced, thus resulting in itself.

Try it online, with test cases

MATL, 6 bytes

0>EGg-

Input may be a number or an array. The result is number or an array with the corresponding values.

Try it online! Or test several cases using array input.

Explanation

This avoids using the builtin sign function (ZS).

0>   % Take input implicitly. Push 1 if positive, 0 otherwise
E    % Multiply by 2
Gg   % Push input converted to logical: 1 if nonzero, 0 otherwise
-    % Subtract. Implicitly display

J, 1 byte

*

Try it online (with test cases)!

R, 25 bytes

'if'(x<-scan(),x/abs(x),0)

Takes the number to STDIN. Then checks if it's zero, if not, returns x/|x| which is either 1 of -1, and outputs 0 if x=0.

This is without using the builtin sign of course.

Octave, 26 24 bytes

f=@(x)real(asin(x))/pi*2

This is my first code-golf Octave answer, any golfing tips are appreciated!

Try it online!

The idea for taking the asin comes from the question where it says output the sign :)

Explanation

Note: dividing the number by pi and multiplying it by 2 is the equivalent of dividing the entire number by pi/2

Case 0:

asin(0) yields 0. Taking the real part of it and dividing it by pi/2 makes no difference to the output.

Case positive:

asin(1) yields pi/2. asin of any number bigger than 1 will give pi/2 + complex number. Taking the real part of it gives pi/2 and dividing it by pi/2 gives 1

Case negative:

asin(-1) yields -pi/2. asin of any number smaller than -1 will give -pi/2 + complex number. Taking the real part of it gives -pi/2 and dividing it by pi/2 gives -1

Stack Cats, 6 + 4 = 10 bytes

_[:I!:

+4 bytes for the ​ -nm flags. n is for numeric I/O, and since Stack Cats requires programs to be palindromic, m implicitly mirrors the source code to give the original source

_[:I!:!I:]_

Try it online! As with basically all good Stack Cats golfs, this was found by brute force, beat any manual attempts by a long shot, and can't easily be incorporated into a larger program.

Add a D flag if you'd like to see a step-by-step program trace, i.e. run with -nmD and check STDERR/debug.

Stack Cats uses a tape of stacks which are implicitly filled with zeroes at the bottom. At the start of the program, all input is pushed onto the input stack, with a -1 at the base to separate the input from the implicit zeroes. At the end of the program, the current stack is output, except a base -1 if present.

The relevant commands here are:

_           Perform subtraction [... y x] -> [... y y-x], where x is top of stack
[           Move left one stack, taking top of stack with you
]           Move right one stack, taking top of stack with you
:           Swap top two of stack
I           Perform [ if top is negative, ] if positive or don't move if zero. Then
                negate the top of stack.
!           Bitwise negate top of stack (n -> -n-1)

Note that all of these commands are invertible, with its inverse being the mirror of the command. This is the premise of Stack Cats — all nontrivial terminating programs are of odd length, since even length programs self-cancel.

We start with

               v
               n
              -1
...  0    0    0    0    0  ...

_ subtracts, making the top -1-n, and [ moves the result left one stack:

           v
       -1-n   -1
...  0    0    0    0    0  ...

: swaps top two and I does nothing, since the top of stack is now zero. ! then bitwise negates the top zero into a -1 and : swaps the top two back. ! then bitwise negates the top, turning -1-n back into n again:

          v
          n
         -1   -1
...  0    0    0    0    0  ...

Now we branch based on I, which is applied to our original n:

• If n is negative, we move left one stack and end with -n on an implicit zero. : swaps, putting a zero on top, and ] moves the zero on top of the -1 we just moved off. _ then subtracts, leaving the final stack like [-1 -1], and only one -1 is output since the base -1 is ignored.

• If n is zero, we don't move and : swaps, putting -1 on top. ] then moves this left -1 on top of the right -1, and _ subtracts, leaving the final stack like [-1 0], outputting the zero and ignoring the base -1.

• If n is positive, we move right one stack and end with -n on a -1. : swaps, putting the -1 on top, and ] moves this -1 right, on top of an implicit zero. _ then subtracts, giving 0 - (-1) = 1 and leaving the final stack like [1], which is output.

V 14 12 bytes

Thanks @DJMcMayhem for 2 bytes. Uses a reg-ex to do the substitution. Kind of fun, because it's not a built-in. I have a more fun function, but it's not working the way I expected.

ͨ-©½0]/±1

Verify Test Cases

This just translates to :%s/\v(-)=[^0].*/\11 which matches one or more - followed by anything but 0, followed by anything any number of times. It's replaced with the first match (so either a - or nothing) and a 1. The regex doesn't match 0, so that stays itself.

The More Fun Way (21 bytes)

é
Àé12|DkJòhé-òó^$/a

TryItOnline

This accepts the input as an argument rather than in the buffer.

é<CR> Insert a new line.

À run the argument as V code. a - will move the cursor to the previous line, and any number will become the count for the next command

é1 insert (count)1's

2| move to the second column

D delete everything from the second column onwards (leaving only one character)

kJ Join the two lines together.

òhé-ò translates to: "run hé- until breaking". If the 1 was on the second line, this breaks immediately after the h. If it was on the first line, it will insert a - before breaking.

ó^$/a This fixes the fact that -1,0,1 will leave a blank, and replaces a blank with the argument register.

C#, 16 15 bytes

Improved solution thanks to Neil

n=>n>0?1:n>>31;

Alternatively, the built-in method is 1 byte longer:

n=>Math.Sign(n);

Full program with test cases:

using System;

public class P
{
    public static void Main()
    {
        Func<int,int> f =
        n=>n>0?1:n>>31;

        // test cases:
        for (int i=-5; i<= 5; i++)
            Console.WriteLine(i + " -> " + f(i));
    }
}

Ruby, 10 bytes

->x{x<=>0}

Jelly, 1 byte

Ṡ

TryItOnline!

The monadic sign atom, Ṡ, does exactly what is specified for an integer input, either as a full program or as a monadic link (function taking one argument).

TI-Basic, 8 bytes

median({1,Ans,~1

Alternative solutions (feel free to suggest more):

max(~1,min(Ans,1               8  bytes
0:If Ans:Ans/abs(Ans           9  bytes
(Ans>0)-(Ans<0                 10 bytes

Actually, 1 byte

s

Try it online!

Another case of exactly what it says on the tin - s is the sign function.

Without the builtin (4 bytes):

;A\+

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;A\ divides the absolute value of the input by the input. This results -1 for negative inputs and 1 for positive inputs. Unfortunately, due to Actually's error handling (if something goes wrong, the command is ignored), 0 as input leaves two 0s on the stack. + rectifies this by adding them (which causes an error with anything else, so it's ignored).

Mathematica, 4 bytes

Sign

Exactly what it says on the tin

05AB1E, 3 bytes

0.S

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Labyrinth, 8 bytes

1,_47-%!

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I'm posting this as a separate answer because my other Labyrinth answer is based on arithmetic on the actual numerical input value, whereas this mostly ignores the number and works with the character code of the first character instead.

Explanation

So yeah, this reads the first character code which is either 45 (-, which should yield -1), 48 (0, which should yield 0) or 49 to 57 (1-9, which should yield 1). This mapping can be accomplished via the simple formula 1 % (x - 47). To see why this works, here is the breakdown of the code for 3 different examples:

Code    Comment             Example -5      Example 0       Example 5
1       Push 1.             [1]             [1]             [1]
,       Read character.     [1 45]          [1 48]          [1 53]
_47-    Subtract 47.        [1 -2]          [1 1]           [1 6]
%       Modulo.             [-1]            [0]             [1]
!       Print.              []              []              []

The instruction pointer then hits a dead end, turns around and terminates when % now attempts a division by zero.

Another simple computation that works:

x -= 46
x %= x-1

Mathematica, 28 bytes

Limit[2ArcTan@x#/Pi,x->∞]&

Mathematica, 30 bytes

2HeavisideTheta@#-1/._[_]->1/2&

Mathematica, 76 bytes

Round@Integrate[E^(2#+I t#)/(2+I t)/Pi,{t,-∞,∞},PrincipalValue->True]-1&

Just to be different :)