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:
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 5.2.6, “SQL Modes”.
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:
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 any digits left
over 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:
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 5.2.6, “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 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 Section 21.4, “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:
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.
This section discusses precision math rounding for the
ROUND()
function and for inserts into columns
with exact-value types (DECIMAL
and integer).
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
or integer 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 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)
.)