## Java Performance: Efficiently Formatting Doubles

by Jack Shirazi12/15/2000

In my book Java
Performance Tuning, I considered efficient ways to convert numbers
to strings, and I provided conversion algorithms that are faster than
those released with the Java SDK (up to and including version 1.3). In
this article, I'll consider how to format `double`

s using a
variation of that conversion algorithm, and show that the conversion
of numbers with formatting can be faster than the original unformatted
conversion.

### The SDK conversion methodology

First, let's look at how the SDK converts `double`

s to
strings and see why that procedure is slow. Calling
`Double.toString(double)`

to convert a `double`

value to a string immediately creates an instance of
`java.lang.FloatingDecimal`

, a class that is private to the
`java.lang`

package. `FloatingDecimal`

is the
internal SDK class that provides all the logic necessary to handle
floating point numbers. If you look at the source code for this class,
you will see that converting floating point numbers to strings appears
to be horrendously complicated. The algorithm needs to deal with
several special cases: infinity, -infinity, not-a-number, -0.0, as
well as digits before and after the decimal point, exponential values,
and rounding accuracy. In addition, the SDK conversion algorithm works
out the minumum number of digits necessary to identify uniquely the
underlying bit-storage value.

This last factor is a little subtle: I'll illustrate it with an
example. The number 5.39E-322 is relatively close to the smallest
IEEE-754 `double`

value. The underlying `double`

representation cannot actually hold the full accuracy of a number that
small (note that this representation is not Java-specific: all
IEEE-754 floating point numbers in all languages have the same
limitations). Instead the `double`

value is held as
5.4E-322. The number 5.41E-322 is also stored to the same accuracy,
resulting in the same number, i.e. the code

```
double d1 = 5.39E-322D;
double d2 = 5.41E-322D;
System.out.println(d1);
System.out.println(d2);
System.out.println(d2==d1);
```

produces the output

```
5.4E-322
5.4E-322
true
```

In fact, the underlying bit-stored number is closer to
5.396039603960396E-322. However, the SDK algorithm correctly
identifies that all these numbers are stored as the same underlying
value, and that there is no other close `double`

value that
can be stored which requires more than one digit of accuracy to be
shown, i.e. 5.4E-322 uniquely identifies the underlying
`double`

value, while using the minimum number of
digits. (The next nearest `double`

value is 5.4455E-322
which is represented by 5.43E-322 using the SDK conversion algorithm,
and this `double`

would be assigned for all values between
5.42E-322 and 5.45E-322.)

### A faster conversion methodology

As you can see, converting `double`

s to strings is a
nuanced affair requiring lots of work. That explains why the SDK is so
slow to convert them. In my book, I presented an alternative algorithm
to convert `double`

s to strings. The algorithm works by
determining the magnitude of the `double`

(there is a rapid
algorithm to do this using some of the bits of the
`double`

), then scaling the `double`

value into
a `long`

and converting the `long`

value using
another efficient algorithm. Essentially, the algorithm looks similar
to

```
double example = 3.14159D;
//magnitude is 0 for 3.14159
long magnitude = magnitude(example);
// scaling_factor could be 1.0E18 here
long scaled = example * scaling_factor;
//now print the long, inserting a decimal point
//after the (magnitude+1)th digit.
...
```

The `long`

-to-string conversion algorithm I use is also
faster than the SDK version, as I use an algorithm which successively
strips off the highest digit, thus avoiding the use of an intermediate
`char`

array or `StringBuffer`

to hold a
partially built string. (The documented code for all algorithms is
included in the source code for this article.) The *n*th digit
of a particular `long`

can be obtained using a relatively
simple formula

```
//Input assumes l is positive. If negative, then pass -l.
//Note that if negating, should treat Long.MAX_VALUE specially,
//as that would overflow when negated.
public static long getNthDigit(long l, int n)
{
//Obviously, if you are successively accessing digits, you should
//inline this code, and avoid repeatedly calling tenthpower(long)
//since you only need to determine the tenthpower(long) for the
//first digit, then each subsequent digit has a tenthpower(long)
//which is the previous tenthpower(long) divided by 10.
return (l/(tenthPower(l)/l_tenthPower[n-1]))%10;
}
static long[] l_tenthPower = {1, 10L, 100L, 1000L, 10000L, 100000L,
1000000L, 10000000L, 100000000L, 1000000000L, 10000000000L,
100000000000L, 1000000000000L, 10000000000000L, 100000000000000L,
1000000000000000L, 10000000000000000L, 100000000000000000L,
1000000000000000000L,
};
private static long tenthPower(long l)
{
if (l < 10L) return 1;
else if (l < 100L) return 10L;
else if (l < 1000L) return 100L;
else if (l < 10000L) return 1000L;
else if (l < 100000L) return 10000L;
else if (l < 1000000L) return 100000L;
else if (l < 10000000L) return 1000000L;
else if (l < 100000000L) return 10000000L;
else if (l < 1000000000L) return 100000000L;
else if (l < 10000000000L) return 1000000000L;
else if (l < 100000000000L) return 10000000000L;
else if (l < 1000000000000L) return 100000000000L;
else if (l < 10000000000000L) return 1000000000000L;
else if (l < 100000000000000L) return 10000000000000L;
else if (l < 1000000000000000L) return 100000000000000L;
else if (l < 10000000000000000L) return 1000000000000000L;
else if (l < 100000000000000000L) return 10000000000000000L;
else if (l < 1000000000000000000L) return 100000000000000000L;
else return 1000000000000000000L;
}
```

By successively stripping off the highest digit, the numbers are printed or appended in order. This means that there is no need for any intermediate temporary structures when printing numbers using my method. This can be a great help when printing large amounts of data to streams. Avoiding the temporary objects that are generated during the normal SDK number-to-string conversions both speeds up the conversion and avoids garbage collection further down the line.

In my original `double`

-to-string conversion algorithm,
I didn't bother with printing the shortest unique representation of
the `double`

. In fact the algorithm always prints out all
the digits available, up to about 15 decimal places. During the
conversion, however, rounding errors mean the last few decimal places
are not necessarily correct. Therefore, if you need full accuracy to
the 15th decimal place, then you should not be using my conversion
algorithm to print `double`

values. I'd also argue that you
shouldn't be using `double`

s at all if that is your
requirement: instead math manipulation packages and
`BigDecimal`

are probably more appropriate for your
problem.

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