Dfp (Apache Commons Math 3.3 API)

org.apache.commons.math3.dfp

## Class Dfp

- java.lang.Object
- - org.apache.commons.math3.dfp.Dfp

- All Implemented Interfaces:
    FieldElement<Dfp>, RealFieldElement<Dfp>
  Direct Known Subclasses:
    DfpDec

      public class Dfp
      extends Object
      implements RealFieldElement<Dfp>

  Decimal floating point library for Java

  Another floating point class. This one is built using radix 10000 which is 104, so its almost decimal.

  The design goals here are:

  1. Decimal math, or close to it
  2. Settable precision (but no mix between numbers using different settings)
  3. Portability. Code should be kept as portable as possible.
  4. Performance
  5. Accuracy - Results should always be +/- 1 ULP for basic algebraic operation
  6. Comply with IEEE 854-1987 as much as possible. (See IEEE 854-1987 notes below)

  Trade offs:

  1. Memory foot print. I'm using more memory than necessary to represent numbers to get better performance.
  2. Digits are bigger, so rounding is a greater loss. So, if you really need 12 decimal digits, better use 4 base 10000 digits there can be one partially filled.

  Numbers are represented in the following form:

        n  =  sign × mant × (radix)exp;

  where sign is ±1, mantissa represents a fractional number between zero and one. mant[0] is the least significant digit. exp is in the range of -32767 to 32768

  IEEE 854-1987 Notes and differences

  IEEE 854 requires the radix to be either 2 or 10. The radix here is 10000, so that requirement is not met, but it is possible that a subclassed can be made to make it behave as a radix 10 number. It is my opinion that if it looks and behaves as a radix 10 number then it is one and that requirement would be met.

  The radix of 10000 was chosen because it should be faster to operate on 4 decimal digits at once instead of one at a time. Radix 10 behavior can be realized by adding an additional rounding step to ensure that the number of decimal digits represented is constant.

  The IEEE standard specifically leaves out internal data encoding, so it is reasonable to conclude that such a subclass of this radix 10000 system is merely an encoding of a radix 10 system.

  IEEE 854 also specifies the existence of "sub-normal" numbers. This class does not contain any such entities. The most significant radix 10000 digit is always non-zero. Instead, we support "gradual underflow" by raising the underflow flag for numbers less with exponent less than expMin, but don't flush to zero until the exponent reaches MIN_EXP-digits. Thus the smallest number we can represent would be: 1E(-(MIN_EXP-digits-1)*4), eg, for digits=5, MIN_EXP=-32767, that would be 1e-131092.

  IEEE 854 defines that the implied radix point lies just to the right of the most significant digit and to the left of the remaining digits. This implementation puts the implied radix point to the left of all digits including the most significant one. The most significant digit here is the one just to the right of the radix point. This is a fine detail and is really only a matter of definition. Any side effects of this can be rendered invisible by a subclass.

  Since:
    2.2
  Version:
    $Id: Dfp.java 1539704 2013-11-07 16:34:51Z tn $
  See Also:
    `DfpField`

- - ### Field Summary

    Fields
    Modifier and Type | Field and Description
    --- | ---
    `static int` | `ERR_SCALE` The amount under/overflows are scaled by before going to trap handler
    `protected int` | `exp` Exponent.
    `static byte` | `FINITE` Indicator value for normal finite numbers.
    `static byte` | `INFINITE` Indicator value for Infinity.
    `protected int[]` | `mant` Mantissa.
    `static int` | `MAX_EXP` The maximum exponent before overflow is signaled and results flushed to infinity
    `static int` | `MIN_EXP` The minimum exponent before underflow is signaled.
    `protected byte` | `nans` Indicator for non-finite / non-number values.
    `static byte` | `QNAN` Indicator value for quiet NaN.
    `static int` | `RADIX` The radix, or base of this system.
    `protected byte` | `sign` Sign bit: 1 for positive, -1 for negative.
    `static byte` | `SNAN` Indicator value for signaling NaN.

  - ### Constructor Summary

    Constructors
    Modifier | Constructor and Description
    --- | ---
    `` | `Dfp(Dfp d)` Copy constructor.
    `protected` | `Dfp(DfpField field)` Makes an instance with a value of zero.
    `protected` | `Dfp(DfpField field, byte x)` Create an instance from a byte value.
    `protected` | `Dfp(DfpField field, byte sign, byte nans)` Creates an instance with a non-finite value.
    `protected` | `Dfp(DfpField field, double x)` Create an instance from a double value.
    `protected` | `Dfp(DfpField field, int x)` Create an instance from an int value.
    `protected` | `Dfp(DfpField field, long x)` Create an instance from a long value.
    `protected` | `Dfp(DfpField field, String s)` Create an instance from a String representation.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `Dfp` | `abs()` Get the absolute value of instance.
    `Dfp` | `acos()` Arc cosine operation.
    `Dfp` | `acosh()` Inverse hyperbolic cosine operation.
    `Dfp` | `add(Dfp x)` Add x to this.
    `Dfp` | `add(double a)` '+' operator.
    `protected int` | `align(int e)` Make our exp equal to the supplied one, this may cause rounding.
    `Dfp` | `asin()` Arc sine operation.
    `Dfp` | `asinh()` Inverse hyperbolic sine operation.
    `Dfp` | `atan()` Arc tangent operation.
    `Dfp` | `atan2(Dfp x)` Two arguments arc tangent operation.
    `Dfp` | `atanh()` Inverse hyperbolic tangent operation.
    `Dfp` | `cbrt()` Cubic root.
    `Dfp` | `ceil()` Round to an integer using the round ceil mode.
    `int` | `classify()` Returns the type - one of FINITE, INFINITE, SNAN, QNAN.
    `protected int` | `complement(int extra)` Negate the mantissa of this by computing the complement.
    `Dfp` | `copySign(Dfp s)` Returns the instance with the sign of the argument.
    `static Dfp` | `copysign(Dfp x, Dfp y)` Creates an instance that is the same as x except that it has the sign of y.
    `Dfp` | `copySign(double s)` Returns the instance with the sign of the argument.
    `Dfp` | `cos()` Cosine operation.
    `Dfp` | `cosh()` Hyperbolic cosine operation.
    `protected String` | `dfp2sci()` Convert an instance to a string using scientific notation.
    `protected String` | `dfp2string()` Convert an instance to a string using normal notation.
    `Dfp` | `divide(Dfp divisor)` Divide this by divisor.
    `Dfp` | `divide(double a)` '÷' operator.
    `Dfp` | `divide(int divisor)` Divide by a single digit less than radix.
    `Dfp` | `dotrap(int type, String what, Dfp oper, Dfp result)` Raises a trap.
    `boolean` | `equals(Object other)` Check if instance is equal to x.
    `Dfp` | `exp()` Exponential.
    `Dfp` | `expm1()` Exponential minus 1.
    `Dfp` | `floor()` Round to an integer using the round floor mode.
    `DfpField` | `getField()` Get the `Field` (really a `DfpField`) to which the instance belongs.
    `Dfp` | `getOne()` Get the constant 1.
    `int` | `getRadixDigits()` Get the number of radix digits of the instance.
    `double` | `getReal()` Get the real value of the number.
    `Dfp` | `getTwo()` Get the constant 2.
    `Dfp` | `getZero()` Get the constant 0.
    `boolean` | `greaterThan(Dfp x)` Check if instance is greater than x.
    `int` | `hashCode()` Gets a hashCode for the instance.
    `Dfp` | `hypot(Dfp y)` Returns the hypotenuse of a triangle with sides `this` and `y` - sqrt(this2 +y2) avoiding intermediate overflow or underflow.
    `int` | `intLog10()` Get the exponent of the greatest power of 10 that is less than or equal to abs(this).
    `int` | `intValue()` Convert this to an integer.
    `boolean` | `isInfinite()` Check if instance is infinite.
    `boolean` | `isNaN()` Check if instance is not a number.
    `boolean` | `isZero()` Check if instance is equal to zero.
    `boolean` | `lessThan(Dfp x)` Check if instance is less than x.
    `Dfp` | `linearCombination(Dfp[] a, Dfp[] b)` Compute a linear combination.
    `Dfp` | `linearCombination(Dfp a1, Dfp b1, Dfp a2, Dfp b2)` Compute a linear combination.
    `Dfp` | `linearCombination(Dfp a1, Dfp b1, Dfp a2, Dfp b2, Dfp a3, Dfp b3)` Compute a linear combination.
    `Dfp` | `linearCombination(Dfp a1, Dfp b1, Dfp a2, Dfp b2, Dfp a3, Dfp b3, Dfp a4, Dfp b4)` Compute a linear combination.
    `Dfp` | `linearCombination(double[] a, Dfp[] b)` Compute a linear combination.
    `Dfp` | `linearCombination(double a1, Dfp b1, double a2, Dfp b2)` Compute a linear combination.
    `Dfp` | `linearCombination(double a1, Dfp b1, double a2, Dfp b2, double a3, Dfp b3)` Compute a linear combination.
    `Dfp` | `linearCombination(double a1, Dfp b1, double a2, Dfp b2, double a3, Dfp b3, double a4, Dfp b4)` Compute a linear combination.
    `Dfp` | `log()` Natural logarithm.
    `int` | `log10()` Deprecated. as of 3.2, replaced by `intLog10()`, in 4.0 the return type will be changed to Dfp
    `int` | `log10K()` Get the exponent of the greatest power of 10000 that is less than or equal to the absolute value of this.
    `Dfp` | `log1p()` Shifted natural logarithm.
    `Dfp` | `multiply(Dfp x)` Multiply this by x.
    `Dfp` | `multiply(double a)` '×' operator.
    `Dfp` | `multiply(int x)` Multiply this by a single digit x.
    `Dfp` | `negate()` Returns a number that is this number with the sign bit reversed.
    `boolean` | `negativeOrNull()` Check if instance is less than or equal to 0.
    `Dfp` | `newInstance()` Create an instance with a value of 0.
    `Dfp` | `newInstance(byte x)` Create an instance from a byte value.
    `Dfp` | `newInstance(byte sig, byte code)` Creates an instance with a non-finite value.
    `Dfp` | `newInstance(Dfp d)` Create an instance by copying an existing one.
    `Dfp` | `newInstance(double x)` Create an instance from a double value.
    `Dfp` | `newInstance(int x)` Create an instance from an int value.
    `Dfp` | `newInstance(long x)` Create an instance from a long value.
    `Dfp` | `newInstance(String s)` Create an instance from a String representation.
    `Dfp` | `nextAfter(Dfp x)` Returns the next number greater than this one in the direction of x.
    `boolean` | `positiveOrNull()` Check if instance is greater than or equal to 0.
    `Dfp` | `pow(Dfp e)` Power operation.
    `Dfp` | `pow(double p)` Power operation.
    `Dfp` | `pow(int n)` Integer power operation.
    `Dfp` | `power10(int e)` Return the specified power of 10.
    `Dfp` | `power10K(int e)` Get the specified power of 10000.
    `Dfp` | `reciprocal()` Returns the multiplicative inverse of `this` element.
    `Dfp` | `remainder(Dfp d)` Returns the IEEE remainder.
    `Dfp` | `remainder(double a)` IEEE remainder operator.
    `Dfp` | `rint()` Round to nearest integer using the round-half-even method.
    `Dfp` | `rootN(int n)` Nth root.
    `long` | `round()` Get the closest long to instance value.
    `protected int` | `round(int n)` Round this given the next digit n using the current rounding mode.
    `Dfp` | `scalb(int n)` Multiply the instance by a power of 2.
    `protected void` | `shiftLeft()` Shift the mantissa left, and adjust the exponent to compensate.
    `protected void` | `shiftRight()` Shift the mantissa right, and adjust the exponent to compensate.
    `Dfp` | `signum()` Compute the signum of the instance.
    `Dfp` | `sin()` Sine operation.
    `Dfp` | `sinh()` Hyperbolic sine operation.
    `Dfp` | `sqrt()` Compute the square root.
    `boolean` | `strictlyNegative()` Check if instance is strictly less than 0.
    `boolean` | `strictlyPositive()` Check if instance is strictly greater than 0.
    `Dfp` | `subtract(Dfp x)` Subtract x from this.
    `Dfp` | `subtract(double a)` '-' operator.
    `Dfp` | `tan()` Tangent operation.
    `Dfp` | `tanh()` Hyperbolic tangent operation.
    `double` | `toDouble()` Convert the instance into a double.
    `double[]` | `toSplitDouble()` Convert the instance into a split double.
    `String` | `toString()` Get a string representation of the instance.
    `protected Dfp` | `trap(int type, String what, Dfp oper, Dfp def, Dfp result)` Trap handler.
    `protected Dfp` | `trunc(DfpField.RoundingMode rmode)` Does the integer conversions with the specified rounding.
    `boolean` | `unequal(Dfp x)` Check if instance is not equal to x.

    - ### Methods inherited from class java.lang.Object

      `clone, finalize, getClass, notify, notifyAll, wait, wait, wait`

- - ### Field Detail

    - #### RADIX

          public static final int RADIX

      The radix, or base of this system. Set to 10000
      See Also:
        Constant Field Values

    - #### MIN_EXP

          public static final int MIN_EXP

      The minimum exponent before underflow is signaled. Flush to zero occurs at minExp-DIGITS
      See Also:
        Constant Field Values

    - #### MAX_EXP

          public static final int MAX_EXP

      The maximum exponent before overflow is signaled and results flushed to infinity
      See Also:
        Constant Field Values

    - #### ERR_SCALE

          public static final int ERR_SCALE

      The amount under/overflows are scaled by before going to trap handler
      See Also:
        Constant Field Values

    - #### FINITE

          public static final byte FINITE

      Indicator value for normal finite numbers.
      See Also:
        Constant Field Values

    - #### INFINITE

          public static final byte INFINITE

      Indicator value for Infinity.
      See Also:
        Constant Field Values

    - #### SNAN

          public static final byte SNAN

      Indicator value for signaling NaN.
      See Also:
        Constant Field Values

    - #### QNAN

          public static final byte QNAN

      Indicator value for quiet NaN.
      See Also:
        Constant Field Values

    - #### mant

          protected int[] mant

      Mantissa.

    - #### sign

          protected byte sign

      Sign bit: 1 for positive, -1 for negative.

    - #### exp

          protected int exp

      Exponent.

    - #### nans

          protected byte nans

      Indicator for non-finite / non-number values.

  - ### Constructor Detail

    - #### Dfp

          protected Dfp(DfpField field)

      Makes an instance with a value of zero.
      Parameters:
        `field` - field to which this instance belongs

    - #### Dfp

          protected Dfp(DfpField field,
             byte x)

      Create an instance from a byte value.
      Parameters:
        `field` - field to which this instance belongs
        `x` - value to convert to an instance

    - #### Dfp

          protected Dfp(DfpField field,
             int x)

      Create an instance from an int value.
      Parameters:
        `field` - field to which this instance belongs
        `x` - value to convert to an instance

    - #### Dfp

          protected Dfp(DfpField field,
             long x)

      Create an instance from a long value.
      Parameters:
        `field` - field to which this instance belongs
        `x` - value to convert to an instance

    - #### Dfp

          protected Dfp(DfpField field,
             double x)

      Create an instance from a double value.
      Parameters:
        `field` - field to which this instance belongs
        `x` - value to convert to an instance

    - #### Dfp

          public Dfp(Dfp d)

      Copy constructor.
      Parameters:
        `d` - instance to copy

    - #### Dfp

          protected Dfp(DfpField field,
             String s)

      Create an instance from a String representation.
      Parameters:
        `field` - field to which this instance belongs
        `s` - string representation of the instance

    - #### Dfp

          protected Dfp(DfpField field,
             byte sign,
             byte nans)

      Creates an instance with a non-finite value.
      Parameters:
        `field` - field to which this instance belongs
        `sign` - sign of the Dfp to create
        `nans` - code of the value, must be one of `INFINITE`, `SNAN`, `QNAN`

  - ### Method Detail

    - #### newInstance

          public Dfp newInstance()

      Create an instance with a value of 0. Use this internally in preference to constructors to facilitate subclasses
      Returns:
        a new instance with a value of 0

    - #### newInstance

          public Dfp newInstance(byte x)

      Create an instance from a byte value.
      Parameters:
        `x` - value to convert to an instance
      Returns:
        a new instance with value x

    - #### newInstance

          public Dfp newInstance(int x)

      Create an instance from an int value.
      Parameters:
        `x` - value to convert to an instance
      Returns:
        a new instance with value x

    - #### newInstance

          public Dfp newInstance(long x)

      Create an instance from a long value.
      Parameters:
        `x` - value to convert to an instance
      Returns:
        a new instance with value x

    - #### newInstance

          public Dfp newInstance(double x)

      Create an instance from a double value.
      Parameters:
        `x` - value to convert to an instance
      Returns:
        a new instance with value x

    - #### newInstance

          public Dfp newInstance(Dfp d)

      Create an instance by copying an existing one. Use this internally in preference to constructors to facilitate subclasses.
      Parameters:
        `d` - instance to copy
      Returns:
        a new instance with the same value as d

    - #### newInstance

          public Dfp newInstance(String s)

      Create an instance from a String representation. Use this internally in preference to constructors to facilitate subclasses.
      Parameters:
        `s` - string representation of the instance
      Returns:
        a new instance parsed from specified string

    - #### newInstance

          public Dfp newInstance(byte sig,
                        byte code)

      Creates an instance with a non-finite value.
      Parameters:
        `sig` - sign of the Dfp to create
        `code` - code of the value, must be one of `INFINITE`, `SNAN`, `QNAN`
      Returns:
        a new instance with a non-finite value

    - #### getField

          public DfpField getField()

      Get the `Field` (really a `DfpField`) to which the instance belongs.

      The field is linked to the number of digits and acts as a factory for `Dfp` instances.

      Specified by:
        `getField` in interface `FieldElement<Dfp>`
      Returns:
        `Field` (really a `DfpField`) to which the instance belongs

    - #### getRadixDigits

          public int getRadixDigits()

      Get the number of radix digits of the instance.
      Returns:
        number of radix digits

    - #### getZero

          public Dfp getZero()

      Get the constant 0.
      Returns:
        a Dfp with value zero

    - #### getOne

          public Dfp getOne()

      Get the constant 1.
      Returns:
        a Dfp with value one

    - #### getTwo

          public Dfp getTwo()

      Get the constant 2.
      Returns:
        a Dfp with value two

    - #### shiftLeft

          protected void shiftLeft()

      Shift the mantissa left, and adjust the exponent to compensate.

    - #### shiftRight

          protected void shiftRight()

      Shift the mantissa right, and adjust the exponent to compensate.

    - #### align

          protected int align(int e)

      Make our exp equal to the supplied one, this may cause rounding. Also causes de-normalized numbers. These numbers are generally dangerous because most routines assume normalized numbers. Align doesn't round, so it will return the last digit destroyed by shifting right.
      Parameters:
        `e` - desired exponent
      Returns:
        last digit destroyed by shifting right

    - #### lessThan

          public boolean lessThan(Dfp x)

      Check if instance is less than x.
      Parameters:
        `x` - number to check instance against
      Returns:
        true if instance is less than x and neither are NaN, false otherwise

    - #### greaterThan

          public boolean greaterThan(Dfp x)

      Check if instance is greater than x.
      Parameters:
        `x` - number to check instance against
      Returns:
        true if instance is greater than x and neither are NaN, false otherwise

    - #### negativeOrNull

          public boolean negativeOrNull()

      Check if instance is less than or equal to 0.
      Returns:
        true if instance is not NaN and less than or equal to 0, false otherwise

    - #### strictlyNegative

          public boolean strictlyNegative()

      Check if instance is strictly less than 0.
      Returns:
        true if instance is not NaN and less than or equal to 0, false otherwise

    - #### positiveOrNull

          public boolean positiveOrNull()

      Check if instance is greater than or equal to 0.
      Returns:
        true if instance is not NaN and greater than or equal to 0, false otherwise

    - #### strictlyPositive

          public boolean strictlyPositive()

      Check if instance is strictly greater than 0.
      Returns:
        true if instance is not NaN and greater than or equal to 0, false otherwise

    - #### abs

          public Dfp abs()

      Get the absolute value of instance.
      Specified by:
        `abs` in interface `RealFieldElement<Dfp>`
      Returns:
        absolute value of instance
      Since:
        3.2

    - #### isInfinite

          public boolean isInfinite()

      Check if instance is infinite.
      Returns:
        true if instance is infinite

    - #### isNaN

          public boolean isNaN()

      Check if instance is not a number.
      Returns:
        true if instance is not a number

    - #### isZero

          public boolean isZero()

      Check if instance is equal to zero.
      Returns:
        true if instance is equal to zero

    - #### equals

          public boolean equals(Object other)

      Check if instance is equal to x.
      Overrides:
        `equals` in class `Object`
      Parameters:
        `other` - object to check instance against
      Returns:
        true if instance is equal to x and neither are NaN, false otherwise

    - #### hashCode

          public int hashCode()

      Gets a hashCode for the instance.
      Overrides:
        `hashCode` in class `Object`
      Returns:
        a hash code value for this object

    - #### unequal

          public boolean unequal(Dfp x)

      Check if instance is not equal to x.
      Parameters:
        `x` - number to check instance against
      Returns:
        true if instance is not equal to x and neither are NaN, false otherwise

    - #### rint

          public Dfp rint()

      Round to nearest integer using the round-half-even method. That is round to nearest integer unless both are equidistant. In which case round to the even one.
      Specified by:
        `rint` in interface `RealFieldElement<Dfp>`
      Returns:
        rounded value
      Since:
        3.2

    - #### floor

          public Dfp floor()

      Round to an integer using the round floor mode. That is, round toward -Infinity
      Specified by:
        `floor` in interface `RealFieldElement<Dfp>`
      Returns:
        rounded value
      Since:
        3.2

    - #### ceil

          public Dfp ceil()

      Round to an integer using the round ceil mode. That is, round toward +Infinity
      Specified by:
        `ceil` in interface `RealFieldElement<Dfp>`
      Returns:
        rounded value
      Since:
        3.2

    - #### remainder

          public Dfp remainder(Dfp d)

      Returns the IEEE remainder.
      Specified by:
        `remainder` in interface `RealFieldElement<Dfp>`
      Parameters:
        `d` - divisor
      Returns:
        this less n × d, where n is the integer closest to this/d
      Since:
        3.2

    - #### trunc

          protected Dfp trunc(DfpField.RoundingMode rmode)

      Does the integer conversions with the specified rounding.
      Parameters:
        `rmode` - rounding mode to use
      Returns:
        truncated value

    - #### intValue

          public int intValue()

      Convert this to an integer. If greater than 2147483647, it returns 2147483647. If less than -2147483648 it returns -2147483648.
      Returns:
        converted number

    - #### log10K

          public int log10K()

      Get the exponent of the greatest power of 10000 that is less than or equal to the absolute value of this. I.E. if this is 106 then log10K would return 1.
      Returns:
        integer base 10000 logarithm

    - #### power10K

          public Dfp power10K(int e)

      Get the specified power of 10000.
      Parameters:
        `e` - desired power
      Returns:
        10000e

    - #### intLog10

          public int intLog10()

      Get the exponent of the greatest power of 10 that is less than or equal to abs(this).
      Returns:
        integer base 10 logarithm
      Since:
        3.2

    - #### power10

          public Dfp power10(int e)

      Return the specified power of 10.
      Parameters:
        `e` - desired power
      Returns:
        10e

    - #### complement

          protected int complement(int extra)

      Negate the mantissa of this by computing the complement. Leaves the sign bit unchanged, used internally by add. Denormalized numbers are handled properly here.
      Parameters:
        `extra` - ???
      Returns:
        ???

    - #### add

          public Dfp add(Dfp x)

      Add x to this.
      Specified by:
        `add` in interface `FieldElement<Dfp>`
      Parameters:
        `x` - number to add
      Returns:
        sum of this and x

    - #### negate

          public Dfp negate()

      Returns a number that is this number with the sign bit reversed.
      Specified by:
        `negate` in interface `FieldElement<Dfp>`
      Returns:
        the opposite of this

    - #### subtract

          public Dfp subtract(Dfp x)

      Subtract x from this.
      Specified by:
        `subtract` in interface `FieldElement<Dfp>`
      Parameters:
        `x` - number to subtract
      Returns:
        difference of this and a

    - #### round

          protected int round(int n)

      Round this given the next digit n using the current rounding mode.
      Parameters:
        `n` - ???
      Returns:
        the IEEE flag if an exception occurred

    - #### multiply

          public Dfp multiply(Dfp x)

      Multiply this by x.
      Specified by:
        `multiply` in interface `FieldElement<Dfp>`
      Parameters:
        `x` - multiplicand
      Returns:
        product of this and x

    - #### multiply

          public Dfp multiply(int x)

      Multiply this by a single digit x.
      Specified by:
        `multiply` in interface `FieldElement<Dfp>`
      Parameters:
        `x` - multiplicand
      Returns:
        product of this and x

    - #### divide

          public Dfp divide(Dfp divisor)

      Divide this by divisor.
      Specified by:
        `divide` in interface `FieldElement<Dfp>`
      Parameters:
        `divisor` - divisor
      Returns:
        quotient of this by divisor

    - #### divide

          public Dfp divide(int divisor)

      Divide by a single digit less than radix. Special case, so there are speed advantages. 0 <= divisor < radix
      Parameters:
        `divisor` - divisor
      Returns:
        quotient of this by divisor

    - #### reciprocal

          public Dfp reciprocal()

      Returns the multiplicative inverse of `this` element.
      Specified by:
        `reciprocal` in interface `FieldElement<Dfp>`
      Specified by:
        `reciprocal` in interface `RealFieldElement<Dfp>`
      Returns:
        the inverse of `this`.

    - #### sqrt

          public Dfp sqrt()

      Compute the square root.
      Specified by:
        `sqrt` in interface `RealFieldElement<Dfp>`
      Returns:
        square root of the instance
      Since:
        3.2

    - #### toString

          public String toString()

      Get a string representation of the instance.
      Overrides:
        `toString` in class `Object`
      Returns:
        string representation of the instance

    - #### dfp2sci

          protected String dfp2sci()

      Convert an instance to a string using scientific notation.
      Returns:
        string representation of the instance in scientific notation

    - #### dfp2string

          protected String dfp2string()

      Convert an instance to a string using normal notation.
      Returns:
        string representation of the instance in normal notation

    - #### dotrap

          public Dfp dotrap(int type,
                   String what,
                   Dfp oper,
                   Dfp result)

      Raises a trap. This does not set the corresponding flag however.
      Parameters:
        `type` - the trap type
        `what` - - name of routine trap occurred in
        `oper` - - input operator to function
        `result` - - the result computed prior to the trap
      Returns:
        The suggested return value from the trap handler

    - #### trap

          protected Dfp trap(int type,
                 String what,
                 Dfp oper,
                 Dfp def,
                 Dfp result)

      Trap handler. Subclasses may override this to provide trap functionality per IEEE 854-1987.
      Parameters:
        `type` - The exception type - e.g. FLAG_OVERFLOW
        `what` - The name of the routine we were in e.g. divide()
        `oper` - An operand to this function if any
        `def` - The default return value if trap not enabled
        `result` - The result that is specified to be delivered per IEEE 854, if any
      Returns:
        the value that should be return by the operation triggering the trap

    - #### classify

          public int classify()

      Returns the type - one of FINITE, INFINITE, SNAN, QNAN.
      Returns:
        type of the number

    - #### copysign

          public static Dfp copysign(Dfp x,
                     Dfp y)

      Creates an instance that is the same as x except that it has the sign of y. abs(x) = dfp.copysign(x, dfp.one)
      Parameters:
        `x` - number to get the value from
        `y` - number to get the sign from
      Returns:
        a number with the value of x and the sign of y

    - #### nextAfter

          public Dfp nextAfter(Dfp x)

      Returns the next number greater than this one in the direction of x. If this==x then simply returns this.
      Parameters:
        `x` - direction where to look at
      Returns:
        closest number next to instance in the direction of x

    - #### toDouble

          public double toDouble()

      Convert the instance into a double.
      Returns:
        a double approximating the instance
      See Also:
        `toSplitDouble()`

    - #### toSplitDouble

          public double[] toSplitDouble()

      Convert the instance into a split double.
      Returns:
        an array of two doubles which sum represent the instance
      See Also:
        `toDouble()`

    - #### getReal

          public double getReal()

      Get the real value of the number.
      Specified by:
        `getReal` in interface `RealFieldElement<Dfp>`
      Returns:
        real value
      Since:
        3.2

    - #### add

          public Dfp add(double a)

      '+' operator.
      Specified by:
        `add` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this+a
      Since:
        3.2

    - #### subtract

          public Dfp subtract(double a)

      '-' operator.
      Specified by:
        `subtract` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this-a
      Since:
        3.2

    - #### multiply

          public Dfp multiply(double a)

      '×' operator.
      Specified by:
        `multiply` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this×a
      Since:
        3.2

    - #### divide

          public Dfp divide(double a)

      '÷' operator.
      Specified by:
        `divide` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this÷s;a
      Since:
        3.2

    - #### remainder

          public Dfp remainder(double a)

      IEEE remainder operator.
      Specified by:
        `remainder` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a` - right hand side parameter of the operator
      Returns:
        this - n × a where n is the closest integer to this/a (the even integer is chosen for n if this/a is halfway between two integers)
      Since:
        3.2

    - #### round

          public long round()

      Get the closest long to instance value.
      Specified by:
        `round` in interface `RealFieldElement<Dfp>`
      Returns:
        closest long to `RealFieldElement.getReal()`
      Since:
        3.2

    - #### signum

          public Dfp signum()

      Compute the signum of the instance. The signum is -1 for negative numbers, +1 for positive numbers and 0 otherwise
      Specified by:
        `signum` in interface `RealFieldElement<Dfp>`
      Returns:
        -1.0, -0.0, +0.0, +1.0 or NaN depending on sign of a
      Since:
        3.2

    - #### copySign

          public Dfp copySign(Dfp s)

      Returns the instance with the sign of the argument. A NaN `sign` argument is treated as positive.
      Specified by:
        `copySign` in interface `RealFieldElement<Dfp>`
      Parameters:
        `s` - the sign for the returned value
      Returns:
        the instance with the same sign as the `sign` argument
      Since:
        3.2

    - #### copySign

          public Dfp copySign(double s)

      Returns the instance with the sign of the argument. A NaN `sign` argument is treated as positive.
      Specified by:
        `copySign` in interface `RealFieldElement<Dfp>`
      Parameters:
        `s` - the sign for the returned value
      Returns:
        the instance with the same sign as the `sign` argument
      Since:
        3.2

    - #### scalb

          public Dfp scalb(int n)

      Multiply the instance by a power of 2.
      Specified by:
        `scalb` in interface `RealFieldElement<Dfp>`
      Parameters:
        `n` - power of 2
      Returns:
        this × 2n
      Since:
        3.2

    - #### hypot

          public Dfp hypot(Dfp y)

      Returns the hypotenuse of a triangle with sides `this` and `y` - sqrt(this2 +y2)
      avoiding intermediate overflow or underflow.
      - If either argument is infinite, then the result is positive infinity.
      - else, if either argument is NaN then the result is NaN.
      Specified by:
        `hypot` in interface `RealFieldElement<Dfp>`
      Parameters:
        `y` - a value
      Returns:
        sqrt(this2 +y2)
      Since:
        3.2

    - #### cbrt

          public Dfp cbrt()

      Cubic root.
      Specified by:
        `cbrt` in interface `RealFieldElement<Dfp>`
      Returns:
        cubic root of the instance
      Since:
        3.2

    - #### rootN

          public Dfp rootN(int n)

      Nth root.
      Specified by:
        `rootN` in interface `RealFieldElement<Dfp>`
      Parameters:
        `n` - order of the root
      Returns:
        nth root of the instance
      Since:
        3.2

    - #### pow

          public Dfp pow(double p)

      Power operation.
      Specified by:
        `pow` in interface `RealFieldElement<Dfp>`
      Parameters:
        `p` - power to apply
      Returns:
        thisp
      Since:
        3.2

    - #### pow

          public Dfp pow(int n)

      Integer power operation.
      Specified by:
        `pow` in interface `RealFieldElement<Dfp>`
      Parameters:
        `n` - power to apply
      Returns:
        thisn
      Since:
        3.2

    - #### pow

          public Dfp pow(Dfp e)

      Power operation.
      Specified by:
        `pow` in interface `RealFieldElement<Dfp>`
      Parameters:
        `e` - exponent
      Returns:
        thise
      Since:
        3.2

    - #### exp

          public Dfp exp()

      Exponential.
      Specified by:
        `exp` in interface `RealFieldElement<Dfp>`
      Returns:
        exponential of the instance
      Since:
        3.2

    - #### expm1

          public Dfp expm1()

      Exponential minus 1.
      Specified by:
        `expm1` in interface `RealFieldElement<Dfp>`
      Returns:
        exponential minus one of the instance
      Since:
        3.2

    - #### log

          public Dfp log()

      Natural logarithm.
      Specified by:
        `log` in interface `RealFieldElement<Dfp>`
      Returns:
        logarithm of the instance
      Since:
        3.2

    - #### log1p

          public Dfp log1p()

      Shifted natural logarithm.
      Specified by:
        `log1p` in interface `RealFieldElement<Dfp>`
      Returns:
        logarithm of one plus the instance
      Since:
        3.2

    - #### log10

          @Deprecated
          public int log10()

      Deprecated. as of 3.2, replaced by `intLog10()`, in 4.0 the return type will be changed to Dfp
      Get the exponent of the greatest power of 10 that is less than or equal to abs(this).
      Returns:
        integer base 10 logarithm

    - #### cos

          public Dfp cos()

      Cosine operation.
      Specified by:
        `cos` in interface `RealFieldElement<Dfp>`
      Returns:
        cos(this)
      Since:
        3.2

    - #### sin

          public Dfp sin()

      Sine operation.
      Specified by:
        `sin` in interface `RealFieldElement<Dfp>`
      Returns:
        sin(this)
      Since:
        3.2

    - #### tan

          public Dfp tan()

      Tangent operation.
      Specified by:
        `tan` in interface `RealFieldElement<Dfp>`
      Returns:
        tan(this)
      Since:
        3.2

    - #### acos

          public Dfp acos()

      Arc cosine operation.
      Specified by:
        `acos` in interface `RealFieldElement<Dfp>`
      Returns:
        acos(this)
      Since:
        3.2

    - #### asin

          public Dfp asin()

      Arc sine operation.
      Specified by:
        `asin` in interface `RealFieldElement<Dfp>`
      Returns:
        asin(this)
      Since:
        3.2

    - #### atan

          public Dfp atan()

      Arc tangent operation.
      Specified by:
        `atan` in interface `RealFieldElement<Dfp>`
      Returns:
        atan(this)
      Since:
        3.2

    - #### atan2

          public Dfp atan2(Dfp x)
                    throws DimensionMismatchException

      Two arguments arc tangent operation.
      Specified by:
        `atan2` in interface `RealFieldElement<Dfp>`
      Parameters:
        `x` - second argument of the arc tangent
      Returns:
        atan2(this, x)
      Throws:
        `DimensionMismatchException` - if number of free parameters or orders are inconsistent
      Since:
        3.2

    - #### cosh

          public Dfp cosh()

      Hyperbolic cosine operation.
      Specified by:
        `cosh` in interface `RealFieldElement<Dfp>`
      Returns:
        cosh(this)
      Since:
        3.2

    - #### sinh

          public Dfp sinh()

      Hyperbolic sine operation.
      Specified by:
        `sinh` in interface `RealFieldElement<Dfp>`
      Returns:
        sinh(this)
      Since:
        3.2

    - #### tanh

          public Dfp tanh()

      Hyperbolic tangent operation.
      Specified by:
        `tanh` in interface `RealFieldElement<Dfp>`
      Returns:
        tanh(this)
      Since:
        3.2

    - #### acosh

          public Dfp acosh()

      Inverse hyperbolic cosine operation.
      Specified by:
        `acosh` in interface `RealFieldElement<Dfp>`
      Returns:
        acosh(this)
      Since:
        3.2

    - #### asinh

          public Dfp asinh()

      Inverse hyperbolic sine operation.
      Specified by:
        `asinh` in interface `RealFieldElement<Dfp>`
      Returns:
        asin(this)
      Since:
        3.2

    - #### atanh

          public Dfp atanh()

      Inverse hyperbolic tangent operation.
      Specified by:
        `atanh` in interface `RealFieldElement<Dfp>`
      Returns:
        atanh(this)
      Since:
        3.2

    - #### linearCombination

          public Dfp linearCombination(Dfp[] a,
                              Dfp[] b)
                                throws DimensionMismatchException

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a` - Factors.
        `b` - Factors.
      Returns:
        `Σi ai bi`.
      Throws:
        `DimensionMismatchException` - if arrays dimensions don't match
      Since:
        3.2

    - #### linearCombination

          public Dfp linearCombination(double[] a,
                              Dfp[] b)
                                throws DimensionMismatchException

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a` - Factors.
        `b` - Factors.
      Returns:
        `Σi ai bi`.
      Throws:
        `DimensionMismatchException` - if arrays dimensions don't match
      Since:
        3.2

    - #### linearCombination

          public Dfp linearCombination(Dfp a1,
                              Dfp b1,
                              Dfp a2,
                              Dfp b2)

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
      Returns:
        a1×b1 + a2×b2
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(Object, Object, Object, Object, Object, Object)`, `RealFieldElement.linearCombination(Object, Object, Object, Object, Object, Object, Object, Object)`

    - #### linearCombination

          public Dfp linearCombination(double a1,
                              Dfp b1,
                              double a2,
                              Dfp b2)

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
      Returns:
        a1×b1 + a2×b2
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(double, Object, double, Object, double, Object)`, `RealFieldElement.linearCombination(double, Object, double, Object, double, Object, double, Object)`

    - #### linearCombination

          public Dfp linearCombination(Dfp a1,
                              Dfp b1,
                              Dfp a2,
                              Dfp b2,
                              Dfp a3,
                              Dfp b3)

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
        `a3` - first factor of the third term
        `b3` - second factor of the third term
      Returns:
        a1×b1 + a2×b2 + a3×b3
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(Object, Object, Object, Object)`, `RealFieldElement.linearCombination(Object, Object, Object, Object, Object, Object, Object, Object)`

    - #### linearCombination

          public Dfp linearCombination(double a1,
                              Dfp b1,
                              double a2,
                              Dfp b2,
                              double a3,
                              Dfp b3)

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
        `a3` - first factor of the third term
        `b3` - second factor of the third term
      Returns:
        a1×b1 + a2×b2 + a3×b3
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(double, Object, double, Object)`, `RealFieldElement.linearCombination(double, Object, double, Object, double, Object, double, Object)`

    - #### linearCombination

          public Dfp linearCombination(Dfp a1,
                              Dfp b1,
                              Dfp a2,
                              Dfp b2,
                              Dfp a3,
                              Dfp b3,
                              Dfp a4,
                              Dfp b4)

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
        `a3` - first factor of the third term
        `b3` - second factor of the third term
        `a4` - first factor of the third term
        `b4` - second factor of the third term
      Returns:
        a1×b1 + a2×b2 + a3×b3 + a4×b4
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(Object, Object, Object, Object)`, `RealFieldElement.linearCombination(Object, Object, Object, Object, Object, Object)`

    - #### linearCombination

          public Dfp linearCombination(double a1,
                              Dfp b1,
                              double a2,
                              Dfp b2,
                              double a3,
                              Dfp b3,
                              double a4,
                              Dfp b4)

      Compute a linear combination.
      Specified by:
        `linearCombination` in interface `RealFieldElement<Dfp>`
      Parameters:
        `a1` - first factor of the first term
        `b1` - second factor of the first term
        `a2` - first factor of the second term
        `b2` - second factor of the second term
        `a3` - first factor of the third term
        `b3` - second factor of the third term
        `a4` - first factor of the third term
        `b4` - second factor of the third term
      Returns:
        a1×b1 + a2×b2 + a3×b3 + a4×b4
      Since:
        3.2
      See Also:
        `RealFieldElement.linearCombination(double, Object, double, Object)`, `RealFieldElement.linearCombination(double, Object, double, Object, double, Object)`

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