# Math Utilities

Java supports integer and floating-point arithmetic directly in the language. Higher-level math operations are supported through the `java.lang.Math` class. As you may have seen by now, wrapper classes for primitive data types allow you to treat them as objects. Wrapper classes also hold some methods for basic conversions.

First, a few words about built-in arithmetic in Java. Java handles errors in integer arithmetic by throwing an `ArithmeticException`:

```    `int` `zero` `=` `0``;`

`try` `{`
`int` `i` `=` `72` `/` `zero``;`
`}` `catch` `(` `ArithmeticException` `e` `)` `{`
`// division by zero`
`}````

To generate the error in this example, we created the intermediate variable `zero`. The compiler is somewhat crafty and would have caught us if we had blatantly tried to perform division by a literal zero.

Floating-point arithmetic expressions, on the other hand, don’t throw exceptions. Instead, they take on the special out-of-range values shown in Table 11-1.

Table 11-1. Special floating-point values

Value

Mathematical representation

`POSITIVE_INFINITY`

1.0/0.0

`NEGATIVE_INFINITY`

-1.0/0.0

`NaN`

0.0/0.0

The following example generates an infinite result:

```    `double` `zero` `=` `0.0``;`
`double` `d` `=` `1.0``/``zero``;`

`if` `(` `d` `==` `Double``.``POSITIVE_INFINITY` `)`
`System``.``out``.``println``(` `"Division by zero"` `);````

The special value `NaN` (not a number) indicates the result of dividing zero by zero. This value has the special mathematical distinction of not being equal to itself (`NaN != NaN` evaluates to `true`). Use `Float.isNaN()` or `Double.isNaN()` to test for `NaN`.

## The java.lang.Math Class

The `java.lang.Math` class is Java’s math library. It holds a suite of static methods covering all of the usual mathematical operations like `sin()`, `cos()`, and `sqrt()`. The `Math` class isn’t very object-oriented (you can’t create an instance of `Math`). Instead, it’s really just a convenient holder for static methods that are more like global functions. As we saw in Chapter 6, it’s possible to use the static import functionality to import the names of static methods and constants like this directly into the scope of our class and use them by their simple, unqualified names.

Table 11-2 summarizes the methods in `java.lang.Math`.

Table 11-2. Methods in java.lang.Math

Method

Argument type(s)

Functionality

`Math.abs(a)`

`int`, `long`, `float`, `double`

Absolute value

`Math.acos(a)`

`double`

Arc cosine

`Math.asin(a)`

`double`

Arc sine

`Math.atan(a)`

`double`

Arc tangent

`Math.atan2(a,b)`

`double`

Angle part of rectangular-to-polar coordinate transform

`Math.ceil(a)`

`double`

Smallest whole number greater than or equal to `a`

`Math.cbrt(a)`

`double`

Cube root of `a`

`Math.cos(a)`

`double`

Cosine

`Math.cosh(a)`

`double`

Hyperbolic cosine

`Math.exp(a)`

`double`

`Math.E` to the power `a`

`Math.floor(a)`

`double`

Largest whole number less than or equal to `a`

`Math.hypot(a,b)`

`double`

Precision calculation of the `sqrt()` of `a`2 + `b`2

`Math.log(a)`

`double`

Natural logarithm of `a`

`Math.log10(a)`

`double`

Log base 10 of `a`

```Math.max(a, b)```

`int`, `long`, `float`, `double`

The value `a` or `b` closer to `Long.MAX_VALUE`

`Math.min(a, b)`

`int`, `long`, `float`, `double`

The value `a` or `b` closer to `Long.MIN_VALUE`

`Math.pow(a, b)`

`double`

`a` to the power `b`

`Math.random()`

`None`

Random-number generator

`Math.rint(a)`

`double`

Converts double value to integral value in double format

`Math.round(a)`

`float`, `double`

Rounds to whole number

`Math.signum(a)`

`double`, `float`

Get the sign of the number at 1.0, –1.0, or 0

`Math.sin(a)`

`double`

Sine

`Math.sinh(a)`

`double`

Hyperbolic sine

`Math.sqrt(a)`

`double`

Square root

`Math.tan(a)`

`double`

Tangent

`Math.tanh(a)`

`double`

Hyperbolic tangent

`Math.toDegrees(a)`

`double`

`Math.toRadians(a)`

`double`

`log()`, `pow()`, and `sqrt()` can throw a runtime `ArithmeticException`. `abs()`, `max()`, and `min()` are overloaded for all the scalar values, `int`, `long`, `float`, or `double`, and return the corresponding type. Versions of `Math.round()` accept either `float` or `double` and return `int` or `long`, respectively. The rest of the methods operate on and return `double` values:

```    `double` `irrational` `=` `Math``.``sqrt``(` `2.0` `);` `// 1.414...`
`int` `bigger` `=` `Math``.``max``(` `3``,` `4` `);`  `// 4`
`long` `one` `=` `Math``.``round``(` `1.125798` `);` `// 1````

For convenience, `Math` also contains the static final double values `E` and `PI`:

`    `double` `circumference` `=` `diameter`  `*` `Math``.``PI``;``

## Big/Precise Numbers

If the `long` and `double` types are not large or precise enough for you, the `java.math` package provides two classes, `BigInteger` and `BigDecimal`, that support arbitrary-precision numbers. These full-featured classes have a bevy of methods for performing arbitrary-precision math and precisely controlling rounding of remainders. In the following example, we use `BigDecimal` to add two very large numbers and then create a fraction with a 100-digit result:

```    `long` `l1` `=` `9223372036854775807L``;` `// Long.MAX_VALUE`
`long` `l2` `=` `9223372036854775807L``;`
`System``.``out``.``println``(` `l1` `+` `l2` `);` `// -2 ! Not good.`
` `
`try` `{`
`BigDecimal` `bd1` `=` `new` `BigDecimal``(` `"9223372036854775807"` `);`
`BigDecimal` `bd2` `=` `new` `BigDecimal``(` `9223372036854775807L` `);`
`System``.``out``.``println``(` `bd1``.``add``(` `bd2` `)` `);` `// 18446744073709551614`
` `
`BigDecimal` `numerator` `=` `new` `BigDecimal``(``1``);`
`BigDecimal` `denominator` `=` `new` `BigDecimal``(``3``);`
`BigDecimal` `fraction` `=`
`numerator``.``divide``(` `denominator``,` `100``,` `BigDecimal``.``ROUND_UP` `);`
`// 100 digit fraction = 0.333333 ... 3334`
`}`
`catch` `(``NumberFormatException` `nfe``)` `{` `}`
`catch` `(``ArithmeticException` `ae``)` `{` `}````

If you implement cryptographic or scientific algorithms for fun, `BigInteger` is crucial. Other than that, you’re not likely to need these classes.

## Floating-Point Components

As we mentioned in Chapter 4, Java uses the IEEE 754 standard to represent floating-point numbers (float and double types) internally. Those of you familiar with how floating-point math works will already know that “decimal” numbers are represented in binary in this standard by separating the number into three components: a sign (positive or negative), an exponent representing the magnitude in powers of 2 of the number, and a mantissa using up most of the bits to represent the precise value irrespective of its magnitude. While for most applications the precision of float and double-type floating-point numbers is sufficient enough that we don’t need to worry about running into limitations, there are times when specialized apps may wish to work with the floating-point values more directly.

By definition, floating-point numbers trade off precision and scale. Even the smallest Java floating-point type, `float`, can represent (literally) astronomical numbers ranging from negative 10–45 to positive 1038. This is accomplished, put in decimal terms, by having the mantissa part of the floating-point value represent a fixed number of “digits” and the exponent tell us where to put the decimal point. As the numbers get larger in magnitude, the “precision” therefore gets shifted to the “left” as more digits appear to the left of the decimal point. What this means is that floating-point numbers can very precisly (with a large number of digits) represent small values like pi, but for bigger numbers (in the billions and trillions) those digits will be taken up with the more signifcant digits. Therefore, the gap between any two consecutive numbers that can be represented by a floating-point value grows larger as the numbers get bigger.

For some applications, knowing the limitations may be important. The `java.lang.Math` class therefore provides a few methods for interrogating floats and doubles about their precision. The `Math.ulp()` method retrieves the “unit of least precision” for a given floating-point number, which is the smallest value that bits in the mantissa represent at their current exponent. Another way to say this is that the `ulp()` is the approximate distance from the floating-point number to the next closest higher or lower floating-point number that can be represented. Adding positive values smaller than half the ULP to a float will not yield a new number. Adding values between half and the full ULP will result in the value plus the ULP. The `Math.nextUp()` method is a convenience that will take a float and tell you the next number that can be represented by adding the ULP.

```        `float` `trillionish` `=` `(``float``)``1``e12``;` `// trillionish ~= 999,999,995,904`
`float` `ulp` `=` `Math``.``ulp``(` `f` `);` `// ulp = 65536`
`float` `next` `=` `Math``.``nextUp``(` `f` `);` `// next ~= 1000000061440`
`trillionish` `+=` `32767``;` `// trillionish still ~= 999,999,995,904. No change!````

Additionally, the `java.lang.Math` class contains the method `getExponent()`, which retrieves the exponent part of a floating-point number (and from there one could determine the mantissa by division). It is also possible to get the raw bits of a float or double using their corresponding wrapper class methods `floatToIntBits()` and `doubleTo``RawLongBits()` and pick out the (IEEE standard) bits yourself.

## Random Numbers

You can use the `java.util.Random` class to generate random values. It’s a pseudorandom-number generator that is initialized with a 48-bit seed.[31] Because it’s a pseudorandom algorithm, you’ll get the same series of values every time you use the same seed value. The default constructor uses the current time to produce a seed, but you can specify your own value in the constructor:

```    `long` `seed` `=` `mySeed``;`
`Random` `rnums` `=` `new` `Random``(` `seed` `);````

After you have a generator, you can ask for one or more random values of various types using the methods listed in Table 11-3.

Table 11-3. Random-number methods

Method

Range

`nextBoolean()`

`true` or `false`

`nextInt()`

–2147483648 to 2147483647

`nextInt(int` `n` `)`

0 to (n – 1) inclusive

`nextLong()`

–9223372036854775808 to 9223372036854775807

`nextFloat()`

0.0 inclusive to 1.0 exclusive

`nextDouble()`

0.0 inclusive to 1.0 exclusive

`nextGaussian()`

Gaussian distributed double with mean 0.0 and standard deviation of 1.0

By default, the values are uniformly distributed. You can use the `nextGaussian()` method to create a Gaussian (bell curve) distribution of `double` values, with a mean of 0.0 and a standard deviation of 1.0. (Lots of natural phenomena follow a Gaussian distribution rather than a strictly uniform random one.)

The `static` method `Math.random()` retrieves a random `double` value. This method initializes a `private` random-number generator in the `Math` class the first time it’s used, using the default `Random` constructor. Thus, every call to `Math.random()` corresponds to a call to `nextDouble()` on that random-number generator.

[31] The generator uses a linear congruential formula. See The Art of Computer Programming, Volume 2: Semi-numerical Algorithms by Donald Knuth (Addison-Wesley).

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