Chapter 21. Precision Math

Table of Contents

21.1. Types of Numeric Values
21.2. DECIMAL Data Type Changes
21.3. Expression Handling
21.4. Rounding Behavior
21.5. Precision Math Examples

MySQL 5.0 introduces precision math, that is, 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:

These changes have several implications for numeric operations:

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 handles numeric operations more precisely than before.

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.

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

In MySQL 5.0.3, several changes were made to several aspects of the DECIMAL data type (and its synonyms):

  • 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:

Digits Left OverNumber of Bytes
00
11
21
32
42
53
63
74
84
94

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 MD 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.

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 expresssion 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 see Section 1.8.2, “Selecting SQL Modes”.) 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 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 should be 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 ValueResult
'' (Default)No warning, no error; i is set to NULL.
strictNo warning, no error; i is set to NULL.
ERROR_FOR_DIVISION_BY_ZEROWarning, no error; i is set to NULL.
strict,ERROR_FOR_DIVISION_BY_ZEROError condition; no row 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. 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 |
+--------------+

However, 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 (0.00 sec)

mysql> INSERT INTO t SET i = 128;
Query OK, 1 row affected, 1 warning (0.01 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='TRADITIONAL';
Query OK, 0 rows affected (0.00 sec)

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

mysql> SET sql_mode='STRICT_ALL_TABLES';
Query OK, 0 rows affected (0.10 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 (0.06 sec)

mysql> SELECT i FROM t;
+------+
| i    |
+------+
| NULL |
+------+
1 row in set (0.01 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='TRADITIONAL';
Query OK, 0 rows affected (0.00 sec)

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

mysql> SET sql_mode='STRICT_ALL_TABLES,ERROR_FOR_DIVISION_BY_ZERO';
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. In MySQL 4 (before precision math was introduced), both exact-value and approximate-value literals were converted to double-precision floating-point values:

mysql> SELECT VERSION();
+-----------------+
| VERSION()       |
+-----------------+
| 4.0.25-standard |
+-----------------+
1 row in set (0.00 sec)


mysql> CREATE TABLE t SELECT 2.5 AS a, 25E-1 AS b;

mysql> DESCRIBE t;
+-------+-------------+------+-----+---------+-------+
| Field | Type        | Null | Key | Default | Extra |
+-------+-------------+------+-----+---------+-------+
| a     | double(3,1) |      |     | 0.0     |       |
| b     | double      |      |     | 0       |       |
+-------+-------------+------+-----+---------+-------+

In MySQL 5.0, the approximate-value literal still is converted to floating-point, but the exact-value literal is handled as DECIMAL:

mysql> SELECT VERSION();
+-----------------+
| VERSION()       |
+-----------------+
| 5.0.15-standard |
+-----------------+
1 row in set (0.00 sec)

mysql> CREATE TABLE t SELECT 2.5 AS a, 25E-1 AS b;

mysql> DESCRIBE t;
+-------+--------------+------+-----+---------+-------+
| Field | Type         | Null | Key | Default | Extra |
+-------+--------------+------+-----+---------+-------+
| a     | decimal(2,1) | NO   |     | 0.0     |       |
| b     | double       | NO   |     | 0       |       |
+-------+--------------+------+-----+---------+-------+

Example 7. If the argument to an aggregate function is an exact numeric type, the result is also, 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 in MySQL 4.0 or 4.1 (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).)