Precision (yet again), Part I
I wrote about precision here and here, but they were pretty technical.
“Great,” coworkers inside StataCorp said to me, “but couldn’t you explain these issues in a way that doesn’t get lost in the details of how computers store binary and maybe, just maybe, write about floats and doubles from a user’s perspective instead of programmer’s perspective?”
“Mmmm,” I said clearly.
Later, when I tried, I liked the result. It contains new material, too. What follows is what I now wish I had written first. I’d would have still written the other two postings, but as technical appendices.
In Part 2 (forthcoming), I provide the mathematical derivations underlying what follows. There are a few interesting issues underlying what follows.
Please excuse the manualish style of what follows, but I suspect that what follows will eventually work its way into Stata’s help files or manuals, so I wrote it that way.
Syntax
Problem:
. generate x = 1.1 . list (Stata displays output showing x is 1.1 in all observations) . count if x==1.1 0
Solution 1:
. count if x==float(1.1) 100
Solution 2:
. generate double x = 1.1 . count if x==1.1 100
Solution 3:
. set type double . generate x = 1.1 . count if x==1.1 100
Description
Stata works in binary. Stata stores data in float precision by default. Stata preforms all calculations in double precision. Sometimes the combination results in surprises until you think more carefully about what happened.
Remarks
Remarks are presented under the headings
Summary
Why count==1.1 produces 0
How count==float(1.1) solves the problem
How storing data as double appears to solve the problem (and does)
Float is plenty accurate to store most data
Why don’t I have the problems using Excel?
Summary
Justifications for all statements made appear in the sections below. In summary,

It sometimes appears that Stata is inaccurate. That is not true and, in fact, the appearance of inaccuracy happens in part because Stata is so accurate.

You can cover up this appearance of inaccuracy by storing all your data in double precision. This will double or more the size of your dataset, and so I do not recommend the doubleprecision solution unless your dataset is small relative to the amount of memory on your computer. In that case, there is nothing wrong with storing all your data in double precision.
The easiest way to implement the doubleprecision solution is by typing set type double. After that, Stata will default to to creating all new variables as doubles, at least for the remainder of the session. If all your datasets are small relative to the amount of memory on your computer, you can set type double, permanently.

The doubleprecision solution is needlessly wasteful of memory. It is difficult to imagine data that are accurate to more than float precision. Regardless of how your data are stored, Stata does all calculations in double precision, and sometimes in quad precision.

The issue of 1.1 not being equal to 1.1 arises only with “nice” decimal numbers. You just have to remember to use Stata’s float() function when dealing with such numbers.
Why count x==1.1 produces 0
Let’s trace through what happens when you type the commands
. generate x = 1.1 . count if x==1.1 0
Here is how it works:

Some numbers have no exact finitedigit binary representation just as some numbers have no exact finitedigit decimal representation. Onethird, 0.3333… (base 10), is an example of a number with no exact finitedigit decimal representation. In base 12, onethird does have an exact finitedigit representation, namely 0.4 (base 12). In base 2 (binary), base 10 numbers such as 0.1, 0.2, 0.3, 0.4, 0.6, … have no exact finitedigit representation.

Computers store numbers with a finite number of binary digits. In float precision, numbers have 24 binary digits. In double precision, they have 53 binary digits.
The decimal number 1.1 in binary is 1.000110011001… (base 2). The 1001 on the end repeats forever. Thus, 1.1 (base 10) is stored by a computer as
1.00011001100110011001101
in float, or as
1.0001100110011001100110011001100110011001100110011010
in double. There are 24 and 53 digits in the numbers above.

Typing generate x = 1.1 results in 1.1 being interpreted as the longer binary number Stata performs all calculations in double precision. New variable x is created as a float by default. When the more precise number is stored in x, it is rounded to the shorter number.

Thus when you count if x==1.1 the result is 0 because 1.1 is again interpreted as the longer binary number and the longer number is compared to shorter number stored in x, and they are not equal.
How count x==float(1.1) solves the problem
One way to fix the problem is to change count if x==1.1 to read count if x==float(1.1):
. generate x = 1.1 . count if x==float(1.1) 100
Function float() rounds results to float precision. When you type float(1.1), the 1.1 is converted to binary, double precision, namely,
1.0001100110011001100110011001100110011001100110011010 (base 2)
and float() then rounds that long binary number to
1.00011001100110011001101 (base 2)
or more correctly, to
1.0001100110011001100110000000000000000000000000000000 (base 2)
because the number is still stored in double precision. Regardless, this new value is equal to the value stored in x, and so count reports that 100 observations contain float(1.1).
As an aside, when you typed generate x = 1.1, Stata acted as if you typed generate x = float(1.1). Whenever you type generate x = … and x is a float, Stata acts if if you typed generate x = float(…).
How storing data as double appears to solve the problem (and does)
When you type
. generate double x = 1.1 . count if x==1.1 100
it should be pretty obvious how the problem was solved. Stata stores
1.0001100110011001100110011001100110011001100110011010 (base 2)
in x, and then compares the stored result to
1.0001100110011001100110011001100110011001100110011010 (base 2)
and of course they are equal.
In the Summary above, I referred to this as a cover up. It is a cover up because 1.1 (base 10) is not what is stored in x. What is stored in x is the binary number just shown, and to be equal to 1.1 (base 10), the binary number needs to suffixed with 1001, and then another 1001, and then another, and so on without end.
Stata tells you that x is equal to 1.1 because Stata converted the 1.1 in count to the same inexact binary representation as Stata previously stored in x, and those two values are equal, but neither is equal to 1.1 (base 10). This leads to an important property of digital computers:
If storage and calculation are done to the same precision, it will appear to the user as if all numbers that the user types are stored without error.
That is, it appears to you as if there is no inaccuracy in storing 1.1 in x when x is a double because Stata performs calculations in double. And it is equally true that it would appear to you as if there were no accuracy issues storing 1.1 when x is stored in float precision if Stata, observing that x is float, performed calculations involving x in float. The fact is that there are accuracy issues in both cases.
“Wait,” you are probably thinking. “I understand your argument, but I’ve always heard that float is inaccurate and double is accurate. I understand from your argument that it is only a matter of degree but, in this case, those two degrees are on opposite sides of an important line.”
“No,” I respond.
What you have heard is right with respect to calculation. What you have heard might apply to data storage too, but that is unlikely. It turns out that float provides plenty of precision to store most real measurements.
Float is plenty accurate to store most data
The misconception that float precision is inaccurate comes from the true statement that float precision is not accurate enough when it comes to making calculations with stored values. Whether float precision is accurate enough for storing values depends solely on the accuracy with which the values are measured.
Float precision provides 24 base2 (binary) digits, and thus values stored in float precision have a maximum relative error error of plusorminus 2^(24) = 5.96e08, or less than +/1 part in 15 million.

The U.S. deficit in 2011 is projected to be $1.5 trillion. Stored as a float, the number has a (maximum) error of 2^(24) * 1.5e+12 = $89,407. That is, if the true number is 1.5 trillion, the number recorded in float precision is guaranteed to be somewhere in the range [(1.5e+12)89,407, (1.5e+14)+89,407]. The projected U.S. deficit is not known to an accuracy of +/$89,407.

People in the US work about 40 hours per week, or roughly 0.238 of the hours in the week. 2^(24) * 0.238 = 1.419e09 of a week, or 0.1 milliseconds. Time worked in a week is not known to an accuracy of +/0.1 milliseconds.

A cancer survivor might live 350 days. 2^(24) * 350 = .00002086, or 1.8 seconds. Time of death is rarely recorded to an accuracy of +/1.8 seconds. Time of diagnosis is never recorded to such accuracy, nor could it be.

The moon is said to be 384,401 kilometers from the Earth. 2^(24) * 348,401 = 0.023 kilometers, or 23 meters. At its closest and farthest, the moon is 356,400 and 406,700 kilometers from Earth.

Most fundamental constants of the universe are known to a few parts in a million, which is to say, less than 1 part in 15 million, the accuracy float precision can provide. An exception is the speed of light, measured to be 299,793.458 kilometers per second. Record that as a float and you will be off by 0.01 km/s.
In all the examples except the last, quoted are worstcase scenarios. The actual errors depend on the exact number and is a more tedious calculation (not shown):

For the U.S. deficit, the exact error for 1.5 trillion is $26,624, which is within the plus or minus $89,407 quoted.

For fraction of the week, at 0.238 the error is 0.04 milliseconds, which is within the +/0.1 milliseconds quoted.

For cancer survival time, at 350 days the actual error is 0, which is within the +/1.8 seconds quoted.

For the distance between the Earth and moon, the actual error is 0, which is within within the +/23 meters quoted.
The actual errors may be interesting, but the maximum errors are more useful. Remember the multiplier 2^(24). All you have to do is multiply a measurement by 2^(24) and compare the result with the inherent error in the measurement. If 2^(24) multiplied by the measurement is less than the inherent error, you can use float precision to store your data. Otherwise, you need to use double.
By the way, the formula
maximum_error = 2^(24) * x
is an approximation. The true formula is
maximum_error = 2^(24) * 2^(floor(log2(x)))
It can be readily proven that x ≥ 2^(floor(log2(x))) and thus the approximation formula overstates the maximum error. The approximation formula can overstate the maximum error by as much as a factor of 2. Float precision is adequate for most data. There is one kind of data, however, where float precision may not be adequate, and that is financial data such as sales data, general ledgers, and the like. People working with dollarandcent data, or EuroandEurocent data, or Pound Stirlingandpenny data, or any other currency data, usually find it best to use doubles. To avoid rounding issues, it is preferable to store the data as pennies. Float precision binary cannot store 0.01, 0.02, and the like, exactly. Integer values, however, can be stored exactly, at least up to certain 16,777,215.
Floats can store up to 16,777,215 exactly. If stored your data in pennies, that would correspond to $167,772.15.
Doubles can store up to 9,007,199,254,740,991 exactly. If you stored your data in pennies, the would correspond to $90,071,992,547,409.91, or just over $90 trillion.
Why don’t I have these problems using Excel?
You do not have these problems when you use Excel because Excel stores numeric values in double precision. As I explained in How float(1.1) solves the problem above,
If storage and calculation are done to the same precision, it will appear to the user as if all numbers that the user types are stored without error.
You can adopt the Excel solution in Stata by typing
. set type double, permanently
You will double (or more) the amount of memory Stata uses to store your data, but if that is not of concern to you, there are no other disadvantages to adopting this solution. If you adopt this solution and later wish to change your mind, type
. set type float, permanently
That’s all for today
If you enjoyed the above, you may want to see Part II (forthcoming). As I said, There are a few technical issues underlying what is written above that may interest those interested in computer science as it applies to statistical computing.
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