Chapter 21. Precision Math
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## Chapter 21. Precision Math

MySQL 5.0 introduces precision math: numeric value handling that results in more accurate results and more control over invalid values than in earlier versions of MySQL. Precision math is based on two implementation changes:

- The introduction of SQL modes in MySQL 5.0 that control how strict the server is about accepting or rejecting invalid data.

- The introduction in MySQL 5.0.3 of a library for fixed-point arithmetic.

These changes have several implications for numeric operations:

- More precise calculations: For exact-value numbers, calculations do not introduce floating-point errors. Instead, exact precision is used. For example, a number such as `.0001` is treated as an exact value rather than as an approximation, and summing it 10,000 times produces a result of exactly `1`, not a value that merely “close” to 1.

- Well-defined rounding behavior: For exact-value numbers, the result of `ROUND()` depends on its argument, not on environmental factors such as how the underlying C library works.

- Improved platform independence: Operations on exact numeric values are the same across different platforms such as Windows and Unix.

- Control over handling of invalid values: Overflow and division by zero are detectable and can be treated as errors. For example, you can treat a value that is too large for a column as an error rather than having the value truncated to lie within the range of the column's data type. Similarly, you can treat division by zero as an error rather than as an operation that produces a result of `NULL`. The choice of which approach to take is determined by the setting of the `sql_mode` system variable.

An important result of these changes is that MySQL provides improved compliance with standard SQL.

The following discussion covers several aspects of how precision math works (including possible incompatibilities with older applications). At the end, some examples are given that demonstrate how MySQL 5.0 handles numeric operations precisely. For information about using the `sql_mode` system variable to control the SQL mode, see Section 5.2.5, “The Server SQL Mode”.

## 21.1. Types of Numeric Values

The scope of precision math for exact-value operations includes the exact-value data types (`DECIMAL` and integer types) and exact-value numeric literals. Approximate-value data types and numeric literals still are handled as floating-point numbers.

Exact-value numeric literals have an integer part or fractional part, or both. They may be signed. Examples: `1`, `.2`, `3.4`, `-5`, `-6.78`, `+9.10`.

Approximate-value numeric literals are represented in scientific notation with a mantissa and exponent. Either or both parts may be signed. Examples: `1.2E3`, `1.2E-3`, `-1.2E3`, `-1.2E-3`.

Two numbers that look similar need not be both exact-value or both approximate-value. For example, `2.34` is an exact-value (fixed-point) number, whereas `2.34E0` is an approximate-value (floating-point) number.

The `DECIMAL` data type is a fixed-point type and calculations are exact. In MySQL, the `DECIMAL` type has several synonyms: `NUMERIC`, `DEC`, `FIXED`. The integer types also are exact-value types.

The `FLOAT` and `DOUBLE` data types are floating-point types and calculations are approximate. In MySQL, types that are synonymous with `FLOAT` or `DOUBLE` are `DOUBLE PRECISION` and `REAL`.

## 21.2. `DECIMAL` Data Type Changes

This section discusses the characteristics of the `DECIMAL` data type (and its synonyms) as of MySQL 5.0.3, with particular regard to the following topics:

- Maximum number of digits

- Storage format

- Storage requirements

- The non-standard MySQL extension to the upper range of `DECIMAL` columns

Some of these changes result in possible incompatibilities for applications that are written for older versions of MySQL. These incompatibilities are noted throughout this section.

The declaration syntax for a `DECIMAL` column remains `DECIMAL(`M`,`D`)`, although the range of values for the arguments has changed somewhat:

- `M` is the maximum number of digits (the precision). It has a range of 1 to 65. This introduces a possible incompatibility for older applications, because previous versions of MySQL allow a range of 1 to 254.

  The precision of 65 digits actually applies as of MySQL 5.0.6. From 5.0.3 to 5.0.5, the precision is 64 digits.

- `D` is the number of digits to the right of the decimal point (the scale). It has a range of 0 to 30 and must be no larger than `M`.

The maximum value of 65 for `M` means that calculations on `DECIMAL` values are accurate up to 65 digits. This limit of 65 digits of precision also applies to exact-value numeric literals, so the maximum range of such literals is different from before. (Prior to MySQL 5.0.3, decimal values could have up to 254 digits. However, calculations were done using floating-point and thus were approximate, not exact.) This change in the range of literal values is another possible source of incompatibility for older applications.

Values for `DECIMAL` columns no longer are represented as strings that require one byte per digit or sign character. Instead, a binary format is used that packs nine decimal digits into four bytes. This change to `DECIMAL` storage format changes the storage requirements as well. The storage requirements for the integer and fractional parts of each value are determined separately. Each multiple of nine digits requires four bytes, and the "leftover" digits require some fraction of four bytes. For example, a `DECIMAL(18,9)` column has nine digits on either side of the decimal point, so the integer part and the fractional part each require four bytes. A `DECIMAL(20,10)` column has ten digits on either side of the decimal point. Each part requires four bytes for nine of the digits, and one byte for the remaining digit.

The storage required for leftover digits is given by the following table:

Leftover Digits | Number of Bytes
0 | 0
1 | 1
2 | 1
3 | 2
4 | 2
5 | 3
6 | 3
7 | 4
8 | 4
9 | 4

As a result of the change from string to numeric format for `DECIMAL` storage, `DECIMAL` columns no longer store a leading `+` character or leading `0` digits. Before MySQL 5.0.3, if you inserted `+0003.1` into a `DECIMAL(5,1)` column, it was stored as `+0003.1`. As of MySQL 5.0.3, it is stored as `3.1`. Applications that rely on the older behavior must be modified to account for this change.

The change of storage format also means that `DECIMAL` columns no longer support the non-standard extension that allowed values larger than the range implied by the column definition. Formerly, one byte was allocated for storing the sign character. For positive values that needed no sign byte, MySQL allowed an extra digit to be stored instead. For example, a `DECIMAL(3,0)` column must support a range of at least `–999` to `999`, but MySQL would allow storing values from `1000` to `9999` as well, by using the sign byte to store an extra digit. This extension to the upper range of `DECIMAL` columns no longer is allowed. In MySQL 5.0.3 and up, a `DECIMAL(`M`,`D`)` column allows at most `M`−`D` digits to the left of the decimal point. This can result in an incompatibility if an application has a reliance on MySQL allowing "too-large" values.

The SQL standard requires that the precision of `NUMERIC(`M`,`D`)` be exactly `M` digits. For `DECIMAL(`M`,`D`)`, the standard requires a precision of at least `M` digits but allows more. In MySQL, `DECIMAL(`M`,`D`)` and `NUMERIC(`M`,`D`)` are the same, and both have a precision of exactly `M` digits.

Summary of incompatibilities:

The following list summarizes the incompatibilities that result from changes to `DECIMAL` column and value handling. You can use it as guide when porting older applications for use with MySQL 5.0.3 and up.

- For `DECIMAL(`M`,`D`)`, the maximum `M` is 65, not 254.

- Calculations involving exact-value decimal numbers are accurate to 65 digits. This is fewer than the maximum number of digits allowed before MySQL 5.0.3 (254 digits), but the exact-value precision is greater. Calculations formerly were done with double-precision floating-point, which has a precision of 52 bits (about 15 decimal digits).

- The non-standard MySQL extension to the upper range of `DECIMAL` columns no longer is supported.

- Leading ‘`+`’ and ‘`0`’ characters are not stored.

The behavior used by the server for `DECIMAL` columns in a table depends on the version of MySQL used to create the table. If your server is from MySQL 5.0.3 or higher, but you have `DECIMAL` columns in tables that were created before 5.0.3, the old behavior still applies to those columns. To convert the tables to the newer `DECIMAL` format, dump them with mysqldump and reload them.

## 21.3. Expression Handling

With precision math, exact-value numbers are used as given whenever possible. For example, numbers in comparisons are used exactly as given without a change in value. In strict SQL mode, for `INSERT` into a column with an exact data type (`DECIMAL` or integer), a number is inserted with its exact value if it is within the column range. When retrieved, the value should be the same as what was inserted. (Without strict mode, truncation for `INSERT` is allowable.)

Handling of a numeric expression depends on what kind of values the expression contains:

- If any approximate values are present, the expression is approximate and is evaluated using floating-point arithmetic.

- If no approximate values are present, the expression contains only exact values. If any exact value contains a fractional part (a value following the decimal point), the expression is evaluated using `DECIMAL` exact arithmetic and has a precision of 65 digits. (The term “exact” is subject to the limits of what can be represented in binary. For example, `1.0/3.0` can be approximated in decimal notation as `.333...`, but not written as an exact number, so `(1.0/3.0)*3.0` does not evaluate to exactly `1.0`.)

- Otherwise, the expression contains only integer values. The expression is exact and is evaluated using integer arithmetic and has a precision the same as `BIGINT` (64 bits).

If a numeric expression contains any strings, they are converted to double-precision floating-point values and the expression is approximate.

Inserts into numeric columns are affected by the SQL mode, which is controlled by the `sql_mode` system variable. (See Section 5.2.5, “The Server SQL Mode”.) The following discussion mentions strict mode (selected by the `STRICT_ALL_TABLES` or `STRICT_TRANS_TABLES` mode values) and `ERROR_FOR_DIVISION_BY_ZERO`. To turn on all restrictions, you can simply use `TRADITIONAL` mode, which includes both strict mode values and `ERROR_FOR_DIVISION_BY_ZERO`:

    mysql> SET sql_mode='TRADITIONAL';

If a number is inserted into an exact type column (`DECIMAL` or integer), it is inserted with its exact value if it is within the column range.

If the value has too many digits in the fractional part, rounding occurs and a warning is generated. Rounding is done as described in Rounding Behavior.

If the value has too many digits in the integer part, it is too large and is handled as follows:

- If strict mode is not enabled, the value is truncated to the nearest legal value and a warning is generated.

- If strict mode is enabled, an overflow error occurs.

Underflow is not detected, so underflow handing is undefined.

By default, division by zero produces a result of `NULL` and no warning. With the `ERROR_FOR_DIVISION_BY_ZERO` SQL mode enabled, MySQL handles division by zero differently:

- If strict mode is not enabled, a warning occurs.

- If strict mode is enabled, inserts and updates involving division by zero are prohibited, and an error occurs.

In other words, inserts and updates involving expressions that perform division by zero can be treated as errors, but this requires `ERROR_FOR_DIVISION_BY_ZERO` in addition to strict mode.

Suppose that we have this statement:

    INSERT INTO t SET i = 1/0;

This is what happens for combinations of strict and `ERROR_FOR_DIVISION_BY_ZERO` modes:

`sql_mode` Value | Result
`''` (Default) | No warning, no error; `i` is set to `NULL`.
strict | No warning, no error; `i` is set to `NULL`.
`ERROR_FOR_DIVISION_BY_ZERO` | Warning, no error; `i` is set to `NULL`.
strict,`ERROR_FOR_DIVISION_BY_ZERO` | Error condition; no row is inserted.

For inserts of strings into numeric columns, conversion from string to number is handled as follows if the string has non-numeric contents:

- A string that does not begin with a number cannot be used as a number and produces an error in strict mode, or a warning otherwise. This includes the empty string.

- A string that begins with a number can be converted, but the trailing non-numeric portion is truncated. If the truncated portion contains anything other than spaces, this produces an error in strict mode, or a warning otherwise.

## 21.4. Rounding Behavior

This section discusses precision math rounding for the `ROUND()` function and for inserts into `DECIMAL` columns.

The `ROUND()` function rounds differently depending on whether its argument is exact or approximate:

- For exact-value numbers, `ROUND()` uses the “round half up” rule: A value with a fractional part of .5 or greater is rounded up to the next integer if positive or down to the next integer if negative. (In other words, it is rounded away from zero.) A value with a fractional part less than .5 is rounded down to the next integer if positive or up to the next integer if negative.

- For approximate-value numbers, the result depends on the C library. On many systems, this means that `ROUND()` uses the “round to nearest even” rule: A value with any fractional part is rounded to the nearest even integer.

The following example shows how rounding differs for exact and approximate values:

    mysql> SELECT ROUND(2.5), ROUND(25E-1);
    +------------+--------------+
    | ROUND(2.5) | ROUND(25E-1) |
    +------------+--------------+
    | 3          |            2 |
    +------------+--------------+

For inserts into a `DECIMAL` column, the target is an exact data type, so rounding uses "round half up," regardless of whether the value to be inserted is exact or approximate:

    mysql> CREATE TABLE t (d DECIMAL(10,0));
    Query OK, 0 rows affected (0.00 sec)

    mysql> INSERT INTO t VALUES(2.5),(2.5E0);
    Query OK, 2 rows affected, 2 warnings (0.00 sec)
    Records: 2  Duplicates: 0  Warnings: 2

    mysql> SELECT d FROM t;
    +------+
    | d    |
    +------+
    | 3    |
    | 3    |
    +------+

## 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 `DECIMAL` columns 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)`.)

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