Help! My calculator malfunctions!

JavaScript (ES7), 159 158 bytes

Edit: new version to comply with the updated rules regarding --
Saved 1 byte thanks to @Shaggy

Takes input in currying syntax (e)(r) where e is the array of equations and r is the array of expected results. Returns an array of characters.

e=>r=>(l=[...2**29+'4+-*/']).filter(x=>l.some(y=>eval("try{eval((S=(s=`[${e}]`).replace(/./g,c=>c==x?y:c==y?x:c)).split`--`.join`+`)+''==r&S!=s}catch(e){}")))

Test cases

let f =

e=>r=>(l=[...2**29+'4+-*/']).filter(x=>l.some(y=>eval("try{eval((S=(s=`[${e}]`).replace(/./g,c=>c==x?y:c==y?x:c)).split`--`.join`+`)+''==r&S!=s}catch(e){}")))

console.log(JSON.stringify(f
  ([ '123', '8423', '4+4', '4*7-10', '9/3', '42-9' ])
  ([ 3, 252, 8, 417, 3, -36 ])
));

console.log(JSON.stringify(f
  ([ '4/2', '3/0', '0/8+2', '95-5', '4+2' ])
  ([ 6, 3, 4, 90, 2 ])
));

console.log(JSON.stringify(f
  ([ '7+4', '5-15', '212-23' ])
  ([ 11, 46, -2121 ])
));

console.log(JSON.stringify(f
  ([ '4+8/2-9*1', '99/3-13', '1+2+3+4', '4-3-2-1' ])
  ([ -5, 20, 10, -6 ])
));

console.log(JSON.stringify(f
  ([ '18/18', '98-8', '55*88', '-5--15' ])
  ([ 1, 90, 4840, 10 ])
));

console.log(JSON.stringify(f
  ([ '9119', '5-3', '8*-9', '13116/3' ])
  ([ 18, 513, 152, -1 ])
));

Formatted and commented

e => r =>                                  // given e and r
  (l = [...2 ** 29 + '4+-*/'])             // generate l = [...'5368709124+-*/']
  .filter(x =>                             // for each character x of l
    l.some(y =>                            // for each character y of l
      eval("try {                          // we need to 'try', because we don't know
        eval(                              // whether the following expression is valid
          (S = (s = `[${e}]`).             // s = list of equations coerced to a string
            replace(/./g, c =>             // S =
              c == x ? y : c == y ? x : c  //   s with x and y exchanged
            )                              // end of replace()
          ).split`--`.join`+`              // replace '--' with '+'
        ) + '' == r                        // does the resulting list match r?
        & S != s                           // and was at least one character modified?
      } catch(e){}")                       // if we try, we oughta catch
    )                                      // end of some()
  )                                        // end of filter()

Perl 6, 132 113 bytes

Thanks to Jo King for -19 bytes.

->\e,$r {first {($!=e.trans($_=>.flip))ne e&&try "all {$!.&{S:g/\-/- /}} Z==$r".EVAL},[X~] (|^10,|<+ - * />)xx 2}

Try it online!

Input is a comma-separated string of equations and a comma-separated string of results (hope this is OK). Output is a string containing the two swapped buttons.

Correctly handles --. Might product false positives for ---, ++, **, or //, but I couldn't come up with a test case.


Oracle SQL & PL/SQL, 458 Bytes

Input can be in any reasonable format. [...] a list with equations and another list with the correct results.

Compile the PL/SQL function (210 bytes):

CREATE FUNCTION f(x CHAR,y CHAR)RETURN NUMBER IS o NUMBER;BEGIN EXECUTE IMMEDIATE 'BEGIN :1:='||REPLACE(x,'--','- -')||';END;'USING OUT o;RETURN CASE o WHEN y THEN 1 END;EXCEPTION WHEN OTHERS THEN RETURN 0;END;

Run the SQL (248 bytes):

WITH r(v)AS(SELECT SUBSTR('1234567890-+*/',LEVEL,1)FROM DUAL CONNECT BY LEVEL<15)SELECT r.v,s.v FROM T,r,r s WHERE r.v<>s.v GROUP BY r.v,s.v HAVING SUM(f(TRANSLATE(x,r.v||s.v,s.v||r.v),y))=(SELECT COUNT(1)FROM T)AND SUM(INSTR(x,r.v)+INSTR(x,s.v))>0

After having create a table T with the test data:

CREATE TABLE T(X,Y) AS
  SELECT '123',    3     FROM DUAL UNION ALL
  SELECT '8423',   252   FROM DUAL UNION ALL
  SELECT '4+4',    8     FROM DUAL UNION ALL
  SELECT '4*7-10', 417   FROM DUAL UNION ALL
  SELECT '9/3',    3     FROM DUAL UNION ALL
  SELECT '42-9',   -36   FROM DUAL

Output:

V V_1
- ---
2 *
* 2

Previous Version:

Assumed a string input like '123 = 3':

Same PL/SQL function and the SQL (322 bytes):

WITH r(v)AS(SELECT SUBSTR('1234567890-+*/',LEVEL,1)FROM DUAL CONNECT BY LEVEL<15),y(x,y)AS(SELECT REGEXP_SUBSTR(t,'[^=]+'),REGEXP_SUBSTR(t,'-?\d+$')FROM T)SELECT r.v,s.v FROM y,r,r s WHERE r.v<>s.v GROUP BY r.v,s.v HAVING SUM(f(TRANSLATE(x,r.v||s.v,s.v||r.v),y))=(SELECT COUNT(1)FROM T)AND SUM(INSTR(x,r.v)+INSTR(x,s.v))>0

After having create a table T with the test data:

CREATE TABLE T(T) AS
  SELECT '123    = 3'   FROM DUAL UNION ALL
  SELECT '8423   = 252' FROM DUAL UNION ALL
  SELECT '4+4    = 8'   FROM DUAL UNION ALL
  SELECT '4*7-10 = 417' FROM DUAL UNION ALL
  SELECT '9/3    = 3'   FROM DUAL UNION ALL
  SELECT '42-9   = -36' FROM DUAL;

Output:

V V_1
- ---
2 *
* 2

Update - Testing:

SQL Fiddle

Oracle 11g R2 Schema Setup:

CREATE FUNCTION F(x CHAR,y CHAR)RETURN NUMBER IS o NUMBER;BEGIN EXECUTE IMMEDIATE 'BEGIN :1:='||REPLACE(x,'--','- -')||';END;'USING OUT o;RETURN CASE o WHEN y THEN 1 END;EXCEPTION WHEN OTHERS THEN RETURN 0;END;
/

CREATE TABLE A(X,Y) AS
  SELECT '123',    3     FROM DUAL UNION ALL
  SELECT '8423',   252   FROM DUAL UNION ALL
  SELECT '4+4',    8     FROM DUAL UNION ALL
  SELECT '4*7-10', 417   FROM DUAL UNION ALL
  SELECT '9/3',    3     FROM DUAL UNION ALL
  SELECT '42-9',   -36   FROM DUAL
/

CREATE TABLE B(X,Y) AS
  SELECT '4/2',    6     FROM DUAL UNION ALL
  SELECT '3/0',    3     FROM DUAL UNION ALL
  SELECT '0/8+2',  4     FROM DUAL UNION ALL
  SELECT '95-5',   90    FROM DUAL UNION ALL
  SELECT '4+2',    2     FROM DUAL
/

CREATE TABLE C(X,Y) AS
  SELECT '7+4',    11    FROM DUAL UNION ALL
  SELECT '5-15',   46    FROM DUAL UNION ALL
  SELECT '212-23', -2121 FROM DUAL
/

CREATE TABLE D(X,Y) AS
  SELECT '4+8/2-9*1', -5 FROM DUAL UNION ALL
  SELECT '99/3-13',   20 FROM DUAL UNION ALL
  SELECT '1+2+3+4',   10 FROM DUAL UNION ALL
  SELECT '4-3-2-1',   -6 FROM DUAL
/

CREATE TABLE E(X,Y) AS
  SELECT '18/18',  1     FROM DUAL UNION ALL
  SELECT '98-8',   90    FROM DUAL UNION ALL
  SELECT '55*88',  4840  FROM DUAL UNION ALL
  SELECT '-5--15', 10    FROM DUAL
/

CREATE TABLE G(X,Y) AS
  SELECT '9119',    18   FROM DUAL UNION ALL
  SELECT '5-3',     513  FROM DUAL UNION ALL
  SELECT '8*-9',    152  FROM DUAL UNION ALL
  SELECT '13116/3', -1   FROM DUAL
/

Query 1:

WITH r(v)AS(SELECT SUBSTR('1234567890-+*/',LEVEL,1)FROM DUAL CONNECT BY LEVEL<15)SELECT r.v,s.v FROM A,r,r s WHERE r.v<>s.v GROUP BY r.v,s.v HAVING SUM(f(TRANSLATE(x,r.v||s.v,s.v||r.v),y))=(SELECT COUNT(1)FROM A)AND SUM(INSTR(x,r.v)+INSTR(x,s.v))>0

Results:

| V | V |
|---|---|
| 2 | * |
| * | 2 |

Query 2:

WITH r(v)AS(SELECT SUBSTR('1234567890-+*/',LEVEL,1)FROM DUAL CONNECT BY LEVEL<15)SELECT r.v,s.v FROM B,r,r s WHERE r.v<>s.v GROUP BY r.v,s.v HAVING SUM(f(TRANSLATE(x,r.v||s.v,s.v||r.v),y))=(SELECT COUNT(1)FROM B)AND SUM(INSTR(x,r.v)+INSTR(x,s.v))>0

Results:

| V | V |
|---|---|
| + | / |
| / | + |

Query 3:

WITH r(v)AS(SELECT SUBSTR('1234567890-+*/',LEVEL,1)FROM DUAL CONNECT BY LEVEL<15)SELECT r.v,s.v FROM C,r,r s WHERE r.v<>s.v GROUP BY r.v,s.v HAVING SUM(f(TRANSLATE(x,r.v||s.v,s.v||r.v),y))=(SELECT COUNT(1)FROM C)AND SUM(INSTR(x,r.v)+INSTR(x,s.v))>0

Results:

| V | V |
|---|---|
| 1 | - |
| - | 1 |

Query 4:

WITH r(v)AS(SELECT SUBSTR('1234567890-+*/',LEVEL,1)FROM DUAL CONNECT BY LEVEL<15)SELECT r.v,s.v FROM D,r,r s WHERE r.v<>s.v GROUP BY r.v,s.v HAVING SUM(f(TRANSLATE(x,r.v||s.v,s.v||r.v),y))=(SELECT COUNT(1)FROM D)AND SUM(INSTR(x,r.v)+INSTR(x,s.v))>0

Results:

| V | V |
|---|---|
| 2 | 4 |
| 4 | 2 |

Query 5:

WITH r(v)AS(SELECT SUBSTR('1234567890-+*/',LEVEL,1)FROM DUAL CONNECT BY LEVEL<15)SELECT r.v,s.v FROM E,r,r s WHERE r.v<>s.v GROUP BY r.v,s.v HAVING SUM(f(TRANSLATE(x,r.v||s.v,s.v||r.v),y))=(SELECT COUNT(1)FROM E)AND SUM(INSTR(x,r.v)+INSTR(x,s.v))>0

Results:

| V | V |
|---|---|
| 5 | 8 |
| 8 | 5 |

Query 6:

WITH r(v)AS(SELECT SUBSTR('1234567890-+*/',LEVEL,1)FROM DUAL CONNECT BY LEVEL<15)SELECT r.v,s.v FROM G,r,r s WHERE r.v<>s.v GROUP BY r.v,s.v HAVING SUM(f(TRANSLATE(x,r.v||s.v,s.v||r.v),y))=(SELECT COUNT(1)FROM G)AND SUM(INSTR(x,r.v)+INSTR(x,s.v))>0

Results:

| V | V |
|---|---|
| 1 | - |
| - | 1 |