## 21.5. Precision Math Examples

This section provides some examples that show how precision math improves query results in MySQL 5 compared to older versions.

Example 1. Numbers are used with their exact value as given when possible.

Before MySQL 5.0.3, numbers that are treated as floating-point values produce inexact results:

```mysql> `SELECT .1 + .2 = .3;`
+--------------+
| .1 + .2 = .3 |
+--------------+
|            0 |
+--------------+
```

As of MySQL 5.0.3, numbers are used as given when possible:

```mysql> `SELECT .1 + .2 = .3;`
+--------------+
| .1 + .2 = .3 |
+--------------+
|            1 |
+--------------+
```

For floating-point values, results are inexact:

```mysql> `SELECT .1E0 + .2E0 = .3E0;`
+--------------------+
| .1E0 + .2E0 = .3E0 |
+--------------------+
|                  0 |
+--------------------+
```

Another way to see the difference in exact and approximate value handling is to add a small number to a sum many times. Consider the following stored procedure, which adds `.0001` to a variable 1,000 times.

```CREATE PROCEDURE p ()
BEGIN
DECLARE i INT DEFAULT 0;
DECLARE d DECIMAL(10,4) DEFAULT 0;
DECLARE f FLOAT DEFAULT 0;
WHILE i < 10000 DO
SET d = d + .0001;
SET f = f + .0001E0;
SET i = i + 1;
END WHILE;
SELECT d, f;
END;
```

The sum for both `d` and `f` logically should be 1, but that is true only for the decimal calculation. The floating-point calculation introduces small errors:

```+--------+------------------+
| d      | f                |
+--------+------------------+
| 1.0000 | 0.99999999999991 |
+--------+------------------+
```

Example 2. Multiplication is performed with the scale required by standard SQL. That is, for two numbers `X1` and `X2` that have scale `S1` and `S2`, the scale of the result is ```S1 + S2```:

Before MySQL 5.0.3, this is what happens:

```mysql> `SELECT .01 * .01;`
+-----------+
| .01 * .01 |
+-----------+
|      0.00 |
+-----------+
```

The displayed value is incorrect. The value was calculated correctly in this case, but not displayed to the required scale. To see that the calculated value actually was .0001, try this:

```mysql> `SELECT .01 * .01 + .0000;`
+-------------------+
| .01 * .01 + .0000 |
+-------------------+
|            0.0001 |
+-------------------+
```

As of MySQL 5.0.3, the displayed scale is correct:

```mysql> `SELECT .01 * .01;`
+-----------+
| .01 * .01 |
+-----------+
| 0.0001    |
+-----------+
```

Example 3. Rounding behavior is well-defined.

Before MySQL 5.0.3, rounding behavior (for example, with the `ROUND()` function) is dependent on the implementation of the underlying C library. This results in inconsistencies from platform to platform. For example, you might get a different value on Windows than on Linux, or a different value on x86 machines than on PowerPC machines.

As of MySQL 5.0.3, rounding happens like this:

Rounding for exact-value columns (`DECIMAL` and integer) and exact-valued numbers uses the “round half up” rule. Values with a fractional part of .5 or greater are rounded away from zero to the nearest integer, as shown here:

```mysql> `SELECT ROUND(2.5), ROUND(-2.5);`
+------------+-------------+
| ROUND(2.5) | ROUND(-2.5) |
+------------+-------------+
| 3          | -3          |
+------------+-------------+
```

However, rounding for floating-point values uses the C library, which on many systems uses the “round to nearest even” rule. Values with any fractional part on such systems are rounded to the nearest even integer:

```mysql> `SELECT ROUND(2.5E0), ROUND(-2.5E0);`
+--------------+---------------+
| ROUND(2.5E0) | ROUND(-2.5E0) |
+--------------+---------------+
|            2 |            -2 |
+--------------+---------------+
```

Example 4. In strict mode, inserting a value that is too large results in overflow and causes an error, rather than truncation to a legal value.

Before MySQL 5.0.2 (or in 5.0.2 and later, without strict mode), truncation to a legal value occurs:

```mysql> `CREATE TABLE t (i TINYINT);`
Query OK, 0 rows affected (0.01 sec)

mysql> `INSERT INTO t SET i = 128;`
Query OK, 1 row affected, 1 warning (0.00 sec)

mysql> `SELECT i FROM t;`
+------+
| i    |
+------+
|  127 |
+------+
1 row in set (0.00 sec)
```

As of MySQL 5.0.2, overflow occurs if strict mode is in effect:

```mysql> `SET sql_mode='STRICT_ALL_TABLES';`
Query OK, 0 rows affected (0.00 sec)

mysql> `CREATE TABLE t (i TINYINT);`
Query OK, 0 rows affected (0.00 sec)

mysql> `INSERT INTO t SET i = 128;`
ERROR 1264 (22003): Out of range value adjusted for column 'i' at row 1

mysql> `SELECT i FROM t;`
Empty set (0.00 sec)
```

Example 5: In strict mode and with `ERROR_FOR_DIVISION_BY_ZERO` set, division by zero causes an error, and not a result of `NULL`.

Before MySQL 5.0.2 (or when not using strict mode in 5.0.2 or a later version), division by zero has a result of `NULL`:

```mysql> `CREATE TABLE t (i TINYINT);`
Query OK, 0 rows affected (0.01 sec)

mysql> `INSERT INTO t SET i = 1 / 0;`
Query OK, 1 row affected (0.00 sec)

mysql> `SELECT i FROM t;`
+------+
| i    |
+------+
| NULL |
+------+
1 row in set (0.00 sec)
```

As of MySQL 5.0.2, division by zero is an error if the proper SQL modes are in effect:

```mysql> `SET sql_mode='STRICT_ALL_TABLES,ERROR_FOR_DIVISION_BY_ZERO';`
Query OK, 0 rows affected (0.00 sec)

mysql> `CREATE TABLE t (i TINYINT);`
Query OK, 0 rows affected (0.00 sec)

mysql> `INSERT INTO t SET i = 1 / 0;`
ERROR 1365 (22012): Division by 0

mysql> `SELECT i FROM t;`
Empty set (0.01 sec)
```

Example 6. Prior to MySQL 5.0.3 (before precision math was introduced), exact-value and approximate-value literals both are converted to double-precision floating-point values:

```mysql> `SELECT VERSION();`
+------------+
| VERSION()  |
+------------+
| 4.1.18-log |
+------------+
1 row in set (0.01 sec)

mysql> `CREATE TABLE t SELECT 2.5 AS a, 25E-1 AS b;`
Query OK, 1 row affected (0.07 sec)
Records: 1  Duplicates: 0  Warnings: 0

mysql> `DESCRIBE t;`
+-------+-------------+------+-----+---------+-------+
| Field | Type        | Null | Key | Default | Extra |
+-------+-------------+------+-----+---------+-------+
| a     | double(3,1) |      |     | 0.0     |       |
| b     | double      |      |     | 0       |       |
+-------+-------------+------+-----+---------+-------+
2 rows in set (0.04 sec)
```

As of MySQL 5.0.3, the approximate-value literal still is converted to floating-point, but the exact-value literal is handled as `DECIMAL`:

```mysql> `SELECT VERSION();`
+------------+
| VERSION()  |
+------------+
| 5.0.19-log |
+------------+
1 row in set (0.17 sec)

mysql> `CREATE TABLE t SELECT 2.5 AS a, 25E-1 AS b;`
Query OK, 1 row affected (0.19 sec)
Records: 1  Duplicates: 0  Warnings: 0

mysql> `DESCRIBE t;`
+-------+-----------------------+------+-----+---------+-------+
| Field | Type                  | Null | Key | Default | Extra |
+-------+-----------------------+------+-----+---------+-------+
| a     | decimal(2,1) unsigned | NO   |     | 0.0     |       |
| b     | double                | NO   |     | 0       |       |
+-------+-----------------------+------+-----+---------+-------+
2 rows in set (0.02 sec)
```

Example 7. If the argument to an aggregate function is an exact numeric type, the result is also an exact numeric type, with a scale at least that of the argument.

Consider these statements:

```mysql> `CREATE TABLE t (i INT, d DECIMAL, f FLOAT);`
mysql> `INSERT INTO t VALUES(1,1,1);`
mysql> `CREATE TABLE y SELECT AVG(i), AVG(d), AVG(f) FROM t;`
```

Result before MySQL 5.0.3 (prior to the introduction of precision math in MySQL):

```mysql> `DESCRIBE y;`
+--------+--------------+------+-----+---------+-------+
| Field  | Type         | Null | Key | Default | Extra |
+--------+--------------+------+-----+---------+-------+
| AVG(i) | double(17,4) | YES  |     | NULL    |       |
| AVG(d) | double(17,4) | YES  |     | NULL    |       |
| AVG(f) | double       | YES  |     | NULL    |       |
+--------+--------------+------+-----+---------+-------+
```

The result is a double no matter the argument type.

Result as of MySQL 5.0.3:

```mysql> `DESCRIBE y;`
+--------+---------------+------+-----+---------+-------+
| Field  | Type          | Null | Key | Default | Extra |
+--------+---------------+------+-----+---------+-------+
| AVG(i) | decimal(14,4) | YES  |     | NULL    |       |
| AVG(d) | decimal(14,4) | YES  |     | NULL    |       |
| AVG(f) | double        | YES  |     | NULL    |       |
+--------+---------------+------+-----+---------+-------+
```

The result is a double only for the floating-point argument. For exact type arguments, the result is also an exact type. (From MySQL 5.0.3 to 5.0.6, the first two columns are `DECIMAL(64,0)`.)