# Why does “SELECT POWER(10.0, 38.0);” throw an arithmetic overflow error?

As part of the check I compute the size of the data type's range for every `IDENTITY` column; I use that to calculate what percentage of that range has been exhausted. For `DECIMAL` and `NUMERIC` the size of that range is `2 * 10^p - 2` where `p` is the precision.

I created a bunch of test tables with `DECIMAL` and `NUMERIC` `IDENTITY` columns and attempted to calculate their ranges as follows:

``````SELECT POWER(10.0, precision)
FROM sys.columns
WHERE
is_identity = 1
AND type_is_decimal_or_numeric
;
``````

This threw the following error:

``````Msg 8115, Level 16, State 6, Line 1
Arithmetic overflow error converting float to data type numeric.
``````

I narrowed it down to the `IDENTITY` columns of type `DECIMAL(38, 0)` (i.e. with the maximum precision), so I then tried the `POWER()` calculation directly on that value.

All of the following queries

``````SELECT POWER(10.0, 38.0);
SELECT CONVERT(FLOAT, (POWER(10.0, 38.0)));
SELECT CAST(POWER(10.0, 38.0) AS FLOAT);
``````

also resulted in the same error.

• Why does SQL Server try to convert the output of `POWER()`, which is of type `FLOAT`, to `NUMERIC` (especially when `FLOAT` has a higher precedence)?
• How can I dynamically calculate the range of a `DECIMAL` or `NUMERIC` column for all possible precisions (including `p = 38`, of course)?

From the `POWER` documentation:

### Syntax

`POWER ( float_expression , y )`

### Arguments

float_expression
Is an expression of type float or of a type that can be implicitly converted to float.

y
Is the power to which to raise float_expression. y can be an expression of the exact numeric or approximate numeric data type category, except for the bit data type.

### Return Types

Returns the same type as submitted in float_expression. For example, if a decimal(2,0) is submitted as float_expression, the result returned is decimal(2,0).

The first input is implicitly cast to `float` if necessary.

The internal calculation is performed using `float` arithmetic by the standard C Runtime Library (CRT) function `pow`.

The `float` output from `pow` is then cast back to the type of the left hand operand (implied to be `numeric(3,1)` when you use the literal value 10.0).

Using an explicit `float` works fine in your case:

``````SELECT POWER(1e1, 38);
SELECT POWER(CAST(10 as float), 38.0);
``````

An exact result for 1038 cannot be stored in a SQL Server `decimal/numeric` because it would require 39 digits of precision (1 followed by 38 zeros). The maximum precision is 38.

Instead of meddling with Martin's answer any further, I'll add the rest of my findings regarding `POWER()` here.

Hold on to your knickers.

# Preamble

First, I present to you exhibit A, the MSDN documentation for `POWER()`:

# Syntax

`POWER ( float_expression , y )`

# Arguments

`float_expression` Is an expression of type float or of a type that can be implicitly converted to float.

# Return Types

Same as `float_expression`.

You may conclude from reading that last line that `POWER()`'s return type is `FLOAT`, but read again. `float_expression` is "of type float or of a type that can be implicitly converted to float". So, despite its name, `float_expression` may actually be a `FLOAT`, a `DECIMAL`, or an `INT`. Since the output of `POWER()` is the same as that of `float_expression`, it too may also be one of those types.

So we have a scalar function with return types that depend on the input. Could it be?

# Observations

I present to you exhibit B, a test demonstrating that `POWER()` casts its output to different data types depending on its input. Please read this script with a faux Jamaican accent for maximum derived benefit.

``````SELECT
POWER(10, 3)             AS int_man
, POWER(1000000000000, 3)  AS numeric0_man     -- one trillion
, POWER(10.0, 3)           AS numeric1_man
, POWER(10.12305, 3)       AS numeric5_man
, POWER(1e1, 3)            AS float_man
INTO power_test_man;

EXECUTE sp_help power_test_man;

DROP TABLE power_test_man;
``````

The relevant results are:

``````Column_name    Type      Length    Prec     Scale
-------------------------------------------------
int_man        int       4         10       0
numeric0_man   numeric   17        38       0
numeric1_man   numeric   17        38       1
numeric5_man   numeric   17        38       5
float_man      float     8         53       NULL
``````

What appears to be happening is that `POWER()` casts `float_expression` into the smallest type that fits it, not including `BIGINT`.

Therefore, `SELECT POWER(10.0, 38);` fails with an overflow error because `10.0` gets cast to `NUMERIC(38, 1)` which isn't big enough to hold the result of 1038. That's because 1038 expands to take 39 digits before the decimal, whereas `NUMERIC(38, 1)` can store 37 digits before the decimal plus one after it. Therefore, the maximum value `NUMERIC(38, 1)` can hold is 1037 - 0.1.

Armed with this understanding I can concoct another overflow failure as follows.

``````SELECT POWER(1000000000, 3);    -- one billion
``````

One billion (as opposed to the one trillion from the first example, which is cast to `NUMERIC(38, 0)`) is just small enough to fit in an `INT`. One billion raised to the third power, however, is too big for `INT`, hence the overflow error.

Several other functions exhibit similar behavior, where their output type is dependent on their input:

• Mathematical functions: `POWER()`, `CEILING()`, `FLOOR()`, `RADIANS()`, `DEGREES()`, and `ABS()`
• System functions and expressions: `NULLIF()`, `ISNULL()`, `COALESCE()`, `IIF()`, `CHOOSE()`, and `CASE` expressions
• Arithmetic operators: Both `SELECT 2 * @MAX_INT;` and `SELECT @MAX_SMALLINT + @MAX_SMALLINT;`, for example, result in arithmetic overflows when the variables are of the named data type.

# Conclusion

In this particular case, the solution is to use `SELECT POWER(1e1, precision)...`. This will work for all possible precisions since `1e1` gets cast to `FLOAT`, which can hold ridiculously large numbers.

Since these functions are so commonplace, it's important to understand that your results may be rounded or may cause overflow errors due to their behavior. If you expect or rely on a specific data type for your output, explicitly cast the relevant input as necessary.

So kids, now that you know this, you may go forth and prosper.

Yeah man.