For example FLOAT type have default numeric precision 53
SELECT * FROM sys.types
and TDS protocol sends precision 53, but through OLEDB precision returns as 18.
What exactly precision 53 means, it seems impossible to put as many decimal digits into 8 bytes? How OLEDB gets 18 from that number? I found MaxLenFromPrecision in Microsoft sources but this array up to precision 37 for numeric types only. FreeTDS have array up to 77 tds_numeric_bytes_per_prec but isn't similar with Microsoft's data and item by index 53 not 18 as OLEDB returns. By that is 53 is some virtual number while 18 real precision as described at http://msdn.microsoft.com/en-us/library/ms190476.aspx ?
What exactly precision 53 means, it seems impossible to put as many decimal digits into 8 bytes?
Precision 53 is 53 bits. 8 bytes contain 64 bits.
A double precision floating point number has 1 sign bit, 11 exponent bits, and 53 mantissa bits.
From BOL: http://msdn.microsoft.com/en-us/library/ms173773.aspx
Where n is the number of bits that are used to store the mantissa of
the float number in scientific notation and, therefore, dictates the
precision and storage size. If n is specified, it must be a value
between 1 and 53. The default value of n is 53.
53 is the precision set at the database level if you don't explicitly set it when declaring your float data type.
Related
I see this in Wikipedia log 224 = 7.22.
I have no idea why we should calculate 2^24 and why we should take log10......I really really need your help.
why floating-points number's significant numbers is 7 or 6 (?)
Consider some thoughts employing the Pigeonhole principle:
binary32 float can encode about 232 different numbers exactly. The numbers one can write in text like 42.0, 1.0, 3.1415623... are infinite, even if we restrict ourselves to a range like -1038 ... +1038. Any time code has a textual value like 0.1f, it is encoded to a nearby float, which may not be the exact same text value. The question is: how many digits can we code and still maintain distinctive float?
For the various powers-of-2 range, 223 (8,388,608) values are normally linearly encoded.
Example: In the range [1.0 ... 2.0), 223 (8,388,608) values are linearly encoded.
In the range [233 or 8,589,934,592 ... 234 or 17,179,869,184), again, 223 (8,388,608) values are linearly encoded: 1024.0 apart from each other. In the sub range [9,000,000,000 and 10,000,000,000), there are about 976,562 different values.
Put this together ...
As text, the range [1.000_000 ... 2.000_000), using 1 lead digit and 6 trailing ones, there are 1,000,000 different values. Per #3, In the same range, with 8,388,608 different float exist, allowing each textual value to map to a different float. In this range we can use 7 digits.
As text, the range [9,000,000 × 103 and 10,000,000 × 103), using 1 lead digit and 6 trailing ones, there are 1,000,000 different values. Per #4, In the same range, there are less than 1,000,000 different float values. Thus some decimal textual values will convert to the same float. In this range we can use 6, not 7, digits for distinctive conversions.
The worse case for typical float is 6 significant digits. To find the limit for your float:
#include <float.h>
printf("FLT_DIG = %d\n", FLT_DIG); // this commonly prints 6
... no idea why we should calculate 2^24 and why we should take log10
224 is a generalization as with common float and its 24 bits of binary precision, that corresponds to fanciful decimal system with 7.22... digits. We take log10 to compare the binary float to decimal text.
224 == 107.22...
Yet we should not take 224. Let us look into how FLT_DIG is defined from C11dr §5.2.4.2.2 11:
number of decimal digits, q, such that any floating-point number with q decimal digits can be rounded into a floating-point number with p radix b digits and back again without change to the q decimal digits,
p log10 b ............. if b is a power of 10
⎣(p − 1) log10 _b_⎦.. otherwise
Notice "log10 224" is same as "24 log10 2".
As a float, the values are distributed linearly between powers of 2 as shown in #2,3,4.
As text, values are distributed linearly between powers of 10 like a 7 significant digit values of [1.000000 ... 9.999999]*10some_exponent.
The transition of these 2 groups happen at different values. 1,2,4,8,16,32... versus 1,10,100, ... In determining the worst case, we subtract 1 from the 24 bits to account for the mis-alignment.
⎣(p − 1) log10 _b_⎦ --> floor((24 − 1) log10(2)) --> floor(6.923...) --> 6.
Had our float used base 10, 100, or 1000, rather than very common 2, the transition of these 2 groups happen at same values and we would not subtract one.
An IEEE 754 single-precision float has a 24-bit mantissa. This means it has 24 binary bits' worth of precision.
But we might be interested in knowing how many decimal digits worth of precision it has.
One way of computing this is to consider how many 24-bit binary numbers there are. The answer, of course, is 224. So these binary numbers go from 0 to 16777215.
How many decimal digits is that? Well, log10 gives you the number of decimal digits. log10(224) is 7.2, or a little more than 7 decimal digits.
And look at that: 16777215 has 8 digits, but the leading digit is just 1, so in fact it's only a little more than 7 digits.
(Of course this doesn't mean we can represent only numbers from 0 to 16777215! It means we can represent numbers from 0 to 16777215 exactly. But we've also got the exponent to play with. We can represent numbers from 0 to 1677721.5 more or less exactly to one place past the decimal, numbers from 0 to 167772.15 more or less exactly to two decimal points, etc. And we can represent numbers from 0 to 167772150, or 0 to 1677721500, but progressively less exactly -- always with ~7 digits' worth of precision, meaning that we start losing precision in the low-order digits to the left of the decimal point.)
The other way of doing this is to note that log10(2) is about 0.3. This means that 1 bit corresponds to about 0.3 decimal digits. So 24 bits corresponds to 24 × 0.3 = 7.2.
(Actually, IEEE 754 single-precision floating point explicitly stores only 23 bits, not 24. But there's an implicit leading 1 bit in there, so we do get the effect of 24 bits.)
Let's start a little smaller. With 10 bits (or 10 base-2 digits), you can represent the numbers 0 upto 1023. So you can represent up to 4 digits for some values, but 3 digits for most others (the ones below 1000).
To find out how many base-10 (decimal) digits can be represented by a bunch of base-2 digits (bits), you can use the log10() of the maximum representable value, i.e. log10(2^10) = log10(2) * 10 = 3.01....
The above means you can represent all 3 digit — or smaller — values and a few 4 digits ones. Well, that is easily verified: 0-999 have at most 3 digits, and 1000-1023 have 4.
Now take 24 bits. In 24 bits you can store log10(2^24) = 24 * log(2) base-10 digits. But because the top bit is always the same, you can in fact only store log10(2^23) = log10(8388608) = 6.92. This means you can represent most 7 digits numbers, but not all. Some of the numbers you can represent faithfully can only have 6 digits.
The truth is a bit more complicated though, because exponents play role too, and some of the many possible larger values can be represented too, so 6.92 may not be the exact value. But it gets close, and can nicely serve as a rule of thumb, and that is why they say that single precision can represent 6 to 7 digits.
This question already has answers here:
What is the default Precision and Scale for a Number in Oracle?
(6 answers)
Closed 7 years ago.
i know basic difference between them ,i wanted to know
if i do not specify precision and scale and define the datatype as number
what are the default values for precision and scale assigned?
create table a(id number);
create table b(id number(3));
both above queries creates a table with column and number datatype but what is the difference from
1)performance point of view
2)How it is handled internally by database
3)Is there any advantage of specifying number as datatype over number(3)
Answer from Oracle perspective...
NUMBER Datatype
The NUMBER datatype stores zero as well as positive and negative fixed numbers with absolute values from 1.0 x 10-130 to (but not including) 1.0 x 10126. If you specify an arithmetic expression whose value has an absolute value greater than or equal to 1.0 x 10126, then Oracle returns an error. Each NUMBER value requires from 1 to 22 bytes.
Specify a fixed-point number using the following form:
NUMBER(p,s)
where:
p is the precision, or the total number of significant decimal digits, where the most significant digit is the left-most nonzero digit, and the least significant digit is the right-most known digit. Oracle guarantees the portability of numbers with precision of up to 20 base-100 digits, which is equivalent to 39 or 40 decimal digits depending on the position of the decimal point.
s is the scale, or the number of digits from the decimal point to the least significant digit. The scale can range from -84 to 127.
Positive scale is the number of significant digits to the right of the decimal point to and including the least significant digit.
Negative scale is the number of significant digits to the left of the decimal point, to but not including the least significant digit. For negative scale the least significant digit is on the left side of the decimal point, because the actual data is rounded to the specified number of places to the left of the decimal point. For example, a specification of (10,-2) means to round to hundreds.
Scale can be greater than precision, most commonly when e notation is used. When scale is greater than precision, the precision specifies the maximum number of significant digits to the right of the decimal point. For example, a column defined as NUMBER(4,5) requires a zero for the first digit after the decimal point and rounds all values past the fifth digit after the decimal point.
It is good practice to specify the scale and precision of a fixed-point number column for extra integrity checking on input. Specifying scale and precision does not force all values to a fixed length. If a value exceeds the precision, then Oracle returns an error. If a value exceeds the scale, then Oracle rounds it.
Specify an integer using the following form:
NUMBER(p)
This represents a fixed-point number with precision p and scale 0 and is equivalent to NUMBER(p,0).
Specify a floating-point number using the following form:
NUMBER
The absence of precision and scale designators specifies the maximum range and precision for an Oracle number.
And
NUMBER[(precision [, scale]])
Number having precision p and scale s. The precision p can range from 1 to 38. The scale s can range from -84 to 127
How many bits out of 64 is assigned to integer part and fractional part in double. Or is there any rule to specify it?
Note: I know I already replied with a comment. This is for my own benefit as much as the OPs; I always learn something new when I try to explain it.
Floating-point values (regardless of precision) are represented as follows:
sign * significand * βexp
where sign is 1 or -1, β is the base, exp is an integer exponent, and significand is a fraction. In this case, β is 2. For example, the real value 3.0 can be represented as 1.102 * 21, or 0.112 * 22, or even 0.0112 * 23.
Remember that a binary number is a sum of powers of 2, with powers decreasing from the left. For example, 1012 is equivalent to 1 * 22 + 0 * 21 + 1 * 20, which gives us the value 5. You can extend that past the radix point by using negative powers of 2, so 101.112 is equivalent to
1 * 22 + 0 * 21 + 1 * 20 + 1 * 2-1 + 1 * 2-2
which gives us the decimal value 5.75. A floating-point number is normalized such that there's a single non-zero digit prior to the radix point, so instead of writing 5.75 as 101.112, we'd write it as 1.01112 * 22
How is this encoded in a 32-bit or 64-bit binary format? The exact format depends on the platform; most modern platforms use the IEEE-754 specification (which also specifies the algorithms for floating-point arithmetic, as well as special values as infinity and Not A Number (NaN)), however some older platforms may use their own proprietary format (such as VAX G and H extended-precision floats). I think x86 also has a proprietary 80-bit format for intermediate calculations.
The general layout looks something like the following:
seeeeeeee...ffffffff....
where s represents the sign bit, e represents bits devoted to the exponent, and f represents bits devoted to the significand or fraction. The IEEE-754 32-bit single-precision layout is
seeeeeeeefffffffffffffffffffffff
This gives us an 8-bit exponent (which can represent the values -126 through 127) and a 22-bit significand (giving us roughly 6 to 7 significant decimal digits). A 0 in the sign bit represents a positive value, 1 represents negative. The exponent is encoded such that 000000012 represents -126, 011111112 represents 0, and 111111102 represents 127 (000000002 is reserved for representing 0 and "denormalized" numbers, while 111111112 is reserved for representing infinity and NaN). This format also assumes a hidden leading fraction bit that's always set to 1. Thus, our value 5.75, which we represent as 1.01112 * 22, would be encoded in a 32-bit single-precision float as
01000000101110000000000000000000
|| || |
|| |+----------+----------+
|| | |
|+--+---+ +------------ significand (1.0111, hidden leading bit)
| |
| +---------------------------- exponent (2)
+-------------------------------- sign (0, positive)
The IEEE-754 double-precision float uses 11 bits for the exponent (-1022 through 1023) and 52 bits for the significand. I'm not going to bother writing that out (this post is turning into a novel as it is).
Floating-point numbers have a greater range than integers because of the exponent; the exponent 127 only takes 8 bits to encode, but 2127 represents a 38-digit decimal number. The more bits in the exponent, the greater the range of values that can be represented. The precision (the number of significant digits) is determined by the number of bits in the significand. The more bits in the significand, the more significant digits you can represent.
Most real values cannot be represented exactly as a floating-point number; you cannot squeeze an infinite number of values into a finite number of bits. Thus, there are gaps between representable floating point values, and most values will be approximations. To illustrate the problem, let's look at an 8-bit "quarter-precision" format:
seeeefff
This gives us an exponent between -7 and 8 (we're not going to worry about special values like infinity and NaN) and a 3-bit significand with a hidden leading bit. The larger our exponent gets, the wider the gap between representable values gets. Here's a table showing the issue. The left column is the significand; each additional column shows the values we can represent for the given exponent:
sig -1 0 1 2 3 4 5
--- ---- ----- ----- ----- ----- ----- ----
000 0.5 1 2 4 8 16 32
001 0.5625 1.125 2.25 4.5 9 18 36
010 0.625 1.25 2.5 5 10 20 40
011 0.6875 1.375 2.75 5.5 11 22 44
100 0.75 1.5 3 6 12 24 48
101 0.8125 1.625 3.25 6.5 13 26 52
110 0.875 1.75 3.5 7 14 28 56
111 0.9375 1.875 3.75 7.5 15 30 60
Note that as we move towards larger values, the gap between representable values gets larger. We can represent 8 values between 0.5 and 1.0, with a gap of 0.0625 between each. We can represent 8 values between 1.0 and 2.0, with a gap of 0.125 between each. We can represent 8 values between 2.0 and 4.0, with a gap of 0.25 in between each. And so on. Note that we can represent all the positive integers up to 16, but we cannot represent the value 17 in this format; we simply don't have enough bits in the significand to do so. If we add the values 8 and 9 in this format, we'll get 16 as a result, which is a rounding error. If that result is used in any other computation, that rounding error will be compounded.
Note that some values cannot be represented exactly no matter how many bits you have in the significand. Just like 1/3 gives us the non-terminating decimal fraction 0.333333..., 1/10 gives us the non-terminating binary fraction 1.10011001100.... We would need an infinite number of bits in the significand to represent that value.
a double on a 64 bit machine, has one sign bit, 11 exponent bits and 52 fractional bits.
think (1 sign bit) * (52 fractional bits) ^ (11 exponent bits)
I was looking at documentation that a number type in oracle db can store range from 10 raise to -130 to 10 raise to 126.
Was wondering how many positive numbers a field NUMBER(18) can store?
Integer numbers with up to 18 digits (Integers between -10^18+1 and 10^18-1)
According to Oracle documentation, the NUMBER datatype stores fixed and floating-point numbers. Optionally, you can also specify a precision (total number of digits) and scale (number of digits to the right of the decimal point):
NUMBER (precision, scale)
If no scale is specified, the scale is zero.
In your case, NUMBER(18), you specified a precision of 18 digits and did not specify any scale so 0 is used (no numbers after the decimal point).
According to the entry for decimal and numeric data types in SQL Server 2008 Books Online, precision is:
p (precision)
The maximum total number of decimal digits that can be stored, both to the left and to the right of the decimal point. The precision must be a value from 1 through the maximum precision of 38. The default precision is 18.
However, the second select below fails with "Arithmetic overflow error converting int to data type numeric."
SELECT CAST(123456789 as decimal(9,0))
SELECT CAST(123456789 as decimal(9,1))
see here: http://msdn.microsoft.com/en-us/library/aa258832(SQL.80).aspx
decimal[(p[, s])]
p (precision) Specifies the maximum total number of decimal digits
that can be stored, both to the left
and to the right of the decimal point.
The precision must be a value from 1
through the maximum precision. The
maximum precision is 38. The default
precision is 18.
s (scale) Specifies the maximum number of decimal digits that can be
stored to the right of the decimal
point. Scale must be a value from 0
through p. Scale can be specified only
if precision is specified. The default
scale is 0; therefore, 0 <= s <= p.
Maximum storage sizes vary, based on
the precision.
when using: decimal(p,s), think of p as how many total digits (regardless of left or right of the decimal point) you want to store, and s as how many of those p digits should be to the right of the decimal point.
DECIMAL(10,5)= 12345.12345
DECIMAL(10,2)= 12345678.12
DECIMAL(10,10)= .1234567891
DECIMAL(11,10)= 1.1234567891
your sample code fails:
SELECT CAST(123456789 as decimal(9,1))
because:
9=precision (total number of digits to left and right of decimal)
1=scale (total number of digits to the right of the decimal)
(9-1)=8 (total digits to the left of the decimal)
and your value 123456789 requires 9 digits to the left of the decimal. you will need decimal(10,1) or just decimal(9,0)
Correct. Since you're doing decimal(9,1) that means you have 9 total digits, but the ,1 is reserving one of them for the right of the decimal place, so you can do at most 8 to the left and 1 to the right.
try
SELECT CAST(123456789 as decimal(10,1))