\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - Number Types

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CGAL::Gmpfr Class Reference
GMP

`#include <CGAL/Gmpfr.h>`

## Definition

An object of the class `Gmpfr` is a fixed precision floating-point number, based on the Mpfr library.

This type is inexact, due to the fact that the mantissa of each number is represented by a fixed amount of bits (this amount is called precision). If an operation needs more bits than the precision of the result number, the results are rounded following different possible criteria (called rounding modes).

Currently, the `Gmpfr` interface supports four rounding modes: round to nearest, round toward zero, round down (or toward \( (-\infty)\)) and round up (or toward \( (+\infty)\)). When not specified explicitly, the operations use the default rounding mode, which is in practice a variable local to each execution thread. The default rounding mode can be set to any of the four rounding modes (initially, it is set to nearest). To specify rounding modes for operations, the type used is `std::float_round_style`.

This type is `ImplicitInteroperable` with `Gmpz`, `long`, `unsigned long`, `int`, `double` and `long double`.

Is Model Of:

  `FieldWithKthRoot`

  `RealEmbeddable`

Comparisons

Comparison operators `==`, `!=`, `>`, `<`, `>=` and `<=` are also overloaded. A `Gmpfr` can be compared with other `Gmpfr`, as well as with a `Gmpz`, `long`, `unsigned long`, `int`, `double` or `long double`. It is worth noting that the numbers are never converted nor rounded before comparison. In the case where one of the compared numbers is `NaN`, the `erange` flag is set.

Implementation

Since the Mpfr library can be compiled to be thread-safe, this interface is designed to keep the thread-safety.

`Gmpfr`s are reference counted. This behavior may be changed, by setting the flag `CGAL_GMPFR_NO_REFCOUNT`. A non-reference-counted class is slightly more efficient in case the implementation does not need to copy numbers (this is not usually the case). Nevertheless, setting this flag may be useful for debugging purposes.

See Also
  `RealEmbeddable`
  `FieldWithKthRoot`

## Related Functions
(Note that these are not member functions.)
std::istream & | operator>> (std::istream &in, Gmpfr &f)
| Reads a floating-point number from `in`. More...
std::ostream & | operator<< (std::ostream &out, const Gmpfr &f)
| If the ostream `out` is in pretty-print mode, writes a decimal approximation of `f` to `out`. More...

## Types
typedef unspecified_type | Precision_type
| Type representing the precision (number of bits used to represent the mantissa) of a number.

## Creation
Note that all constructors can be called with two optional parameters. One can specify as second parameter the rounding mode desired for the conversion from the source number and as a third parameter the precision with which this `Gmpfr` will be created. If only one optional parameter is specified, it can be either the rounding mode or the precision. If no optional parameters are specified, the precision of the created object is chosen in such a way that the conversion is exact (i.e., no rounding is performed). These optional parameters, along with other functions which will be explained below, allow users to control the rounding and precision. For example, being `z` a `Gmpz`, `Gmpfr g(z,53,std::round_toward_neg_infinity)` will construct a `Gmpfr` `g` having as value the biggest 53-bit floating-point number that is equal or smaller than to `z`.
| Gmpfr ()
| Creates an uninitialized `Gmpfr` `f`.
| Gmpfr (const Gmpfr &n)
| Copy constructor. More...
| Gmpfr (long si)
| Creates a `Gmpfr`, initialized with the value of `si`.
| Gmpfr (unsigned long ui)
| Creates a `Gmpfr`, initialized with the value of `ui`.
| Gmpfr (int i)
| Creates a `Gmpfr`, initialized with the value of `i`.
| Gmpfr (double d)
| Creates a `Gmpfr`, initialized with the value of `d`.
| Gmpfr (long double ld)
| Creates a `Gmpfr`, initialized with the value of `ld`.
| Gmpfr (const Gmpz &z)
| Creates a `Gmpfr`, initialized with the value of `z`.
| Gmpfr (const Gmpzf &zf)
| Creates a `Gmpfr`, initialized with the value of `zf`.
| Gmpfr (const std::pair< Gmpz, long > &ie)
| Creates a `Gmpfr`, initialized with the value of `ie.first` \( \times2^{\mathrm{ie.second}} \) .
static Precision_type | get_default_precision ()
| This returns the current precision used in `Gmpfr` creation by default.
static Precision_type | set_default_precision (Precision_type p)
| This function sets the default Mpfr precision to p, and returns the old one.

## Controlling the Precision
Each Gmpfr object has a precision associated to it. The precision is the amount of bits needed to represent the mantissa. Mpfr has a default precision value, which can be controlled by static functions of the Gmpfr class (in practice, this default value is a variable local to each execution thread). There are also functions to get and set the precision of each Gmpfr object.
Precision_type | get_precision () const
| Returns the precision of `f`.
Gmpfr | round (Precision_type p, std::float_round_style r) const
| Returns the value of `f`, rounded with precision `p` in the direction `r`.
static std::float_round_style | get_default_rndmode ()
| This function returns the current rounding mode used by Mpfr.
static std::float_round_style | set_default_rndmode (std::float_round_style r)
| This function sets the Mpfr rounding mode to `r` and returns the old one.

## Flags
Mpfr provides some flags to know whether performed operations were exact or not, or they incurred in overflow or underflow, if the exponent is out of range, or the result was `NaN` (not-a-number). One can clear the flags before a set of operations and inspect them afterward, in order to see if something unexpected happened during the operations. The static functions used to handle flags are:
static void | clear_flags ()
| Clears all the flags set by Mpfr(they are not cleared automatically).
static bool | underflow_flag ()
| Shows whether an operation incurred in underflow.
static bool | overflow_flag ()
| Shows whether an operation incurred in overflow.
static bool | nan_flag ()
| Shows whether the result of an operation was `NaN`.
static bool | inex_flag ()
| Shows whether an operation was inexact.
static bool | erange_flag ()
| Returns `true` iff a range error occurred. More...

## Arithmetic Operations
Arithmetic operators `+` , `-` , and `/` are overloaded, but special care must be taken when applying them. The precision of an operation between two `Gmpfr`s is defined as the maximum of the operands precision and the default precision. The second operand of the former operations can be a `Gmpfr`, `int`, `long`, `unsigned`, `unsigned long`, or `Gmpz`. The precision of an operation between a `Gmpfr` and a number of another type is defined as the maximum between the number's precision and the default precision. To specify the rounding mode and/or the precision to perform an operation, this class provides the four static functions `add`, `sub`, `mul` and `div`. Only one of them is shown here, since their interfaces are similar: When the precision is not specified in this family of functions, it is defined as in the overloaded operators. When the rounding mode is not specified, the default is used.
Gmpfr | abs (Precision_type p, std::float_round_style r=get_default_rndmode()) const
| Returns the absolute value of `f`, rounded with precision `p` in the direction `r`. More...
Gmpfr | sqrt (Precision_type p, std::float_round_style r=get_default_rndmode()) const
| Returns the square root of `f`, rounded with precision `p` in the direction `r`. More...
Gmpfr | kthroot (int k, Precision_type p, std::float_round_style r=get_default_rndmode()) const
| Returns the k-th root of `f`, rounded with precision `p` in the direction `r`. More...
Gmpfr | square (Precision_type p, std::float_round_style r=get_default_rndmode()) const
| Returns the square of `f`, rounded with precision `p` in the direction `r`. More...
double | to_double (std::float_round_style r=get_default_rndmode())
| Returns a double precision approximation of `f` using the rounding mode `r`.
std::pair< double, double > | to_interval ()
| Returns an interval of doubles which contains `f`. More...
std::pair< double, long > | to_double_exp (std::float_round_style r=get_default_rndmode())
| Returns the pair \( (d,e) \) such that \( 0.5 \le|d| < 1 \) and \( d \times2^e \) equals `f` rounded to double precision, using the rounding mode `r`. More...
std::pair< std::pair< double, double >, long > | to_interval_exp ()
| Returns \( ((m,M),e) \) such that \( m \times2^e \le f \le M \times2^e \). More...
std::pair< Gmpz, long > | to_integer_exp ()
| Returns a pair of integers \( (m,e) \), such that \( f = m \times2^e \). More...
static Gmpfr | add (const Gmpfr &a, const Gmpfr &b)
static Gmpfr | add (const Gmpfr &a, const Gmpfr &b, std::float_round_style r)
static Gmpfr | add (const Gmpfr &a, const Gmpfr &b, Precision_type p)
static Gmpfr | add (const Gmpfr &a, const Gmpfr &b, Precision_type p, std::float_round_style r)

## Query Functions
Sign | sign ()
| Returns the sign of `f`.
bool | is_zero ()
| Returns `true` iff `f` is zero.
bool | is_one ()
| Returns `true` iff `f` is one.
bool | is_nan ()
| Returns `true` iff `f` is NaN (not-a-number).
bool | is_inf ()
| Returns `true` iff `f` is plus or minus infinity.
bool | is_number ()
| Returns `true` iff `f` is a valid number.
bool | is_square ()
| Returns `true` iff `f` is the square of a number representable by an object of this type.
bool | is_square (const Gmpfr &y)
| Returns `true` iff `f` is the square of a number representable by an object of this type, computing and storing it in `y`.

## Constructor & Destructor Documentation

CGAL::Gmpfr::Gmpfr | ( | const Gmpfr & | n) |  |

Copy constructor.

The copied object inherits the precision of `n`, and thus it is not rounded.

## Member Function Documentation

Gmpfr CGAL::Gmpfr::abs | ( | Precision_type | p,
|  | std::float_round_style | r = `get_default_rndmode()`
| ) |  | const

Returns the absolute value of `f`, rounded with precision `p` in the direction `r`.

If `p` is not specified, the precision used is the maximum between `f`'s precision and the default.

static bool CGAL::Gmpfr::erange_flag | ( | ) |  |
| static

Returns `true` iff a range error occurred.

Such an exception occurs when some function which does not return a `Gmpfr` has an invalid result. For example, this flag will be set if one of the operands of a comparison is `NaN`.

Gmpfr CGAL::Gmpfr::kthroot | ( | int | k,
|  | Precision_type | p,
|  | std::float_round_style | r = `get_default_rndmode()`
| ) |  | const

Returns the k-th root of `f`, rounded with precision `p` in the direction `r`.

If `p` is not specified, the precision used is the maximum between `f`'s precision and the default.

Gmpfr CGAL::Gmpfr::sqrt | ( | Precision_type | p,
|  | std::float_round_style | r = `get_default_rndmode()`
| ) |  | const

Returns the square root of `f`, rounded with precision `p` in the direction `r`.

If `p` is not specified, the precision used is the maximum between `f`'s precision and the default.

Gmpfr CGAL::Gmpfr::square | ( | Precision_type | p,
|  | std::float_round_style | r = `get_default_rndmode()`
| ) |  | const

Returns the square of `f`, rounded with precision `p` in the direction `r`.

If `p` is not specified, the precision used is the maximum between `f`'s precision and the default.

std::pair<double,long> CGAL::Gmpfr::to_double_exp | ( | std::float_round_style | r = `get_default_rndmode()`) |  |

Returns the pair \( (d,e) \) such that \( 0.5 \le|d| < 1 \) and \( d \times2^e \) equals `f` rounded to double precision, using the rounding mode `r`.

If `f` is `NaN` or infinity, then the corresponding double is returned, leaving the exponent undefined and setting the appropriate error flag.

std::pair<Gmpz,long> CGAL::Gmpfr::to_integer_exp | ( | ) |  |

Returns a pair of integers \( (m,e) \), such that \( f = m \times2^e \).

Note that the returned value of \( m\) is not necessarily the smallest possible value of \( m\) (that is, it might be that \( 2|m\)).

std::pair<double,double> CGAL::Gmpfr::to_interval | ( | ) |  |

Returns an interval of doubles which contains `f`.

If a rounded endpoint does not fit in a double, the double is set to plus or minus infinity and the `overflow` or `underflow` flag.

std::pair<std::pair<double,double>,long> CGAL::Gmpfr::to_interval_exp | ( | ) |  |

Returns \( ((m,M),e) \) such that \( m \times2^e \le f \le M \times2^e \).

If `f` is `NaN` or infinity, then the corresponding doubles are returned, leaving the exponent undefined and setting the appropriate error flag.

## Friends And Related Function Documentation

std::ostream & operator<< | ( | std::ostream & | out,
|  | const Gmpfr & | f
| ) |  |
| related

If the ostream `out` is in pretty-print mode, writes a decimal approximation of `f` to `out`.

Otherwise, writes `f` to `out` in the form \( MeE\), where \( M\) is its mantissa and \( E\) is its exponent, both in base 10.

std::istream & operator>> | ( | std::istream & | in,
|  | Gmpfr & | f
| ) |  |
| related

Reads a floating-point number from `in`.

The number \( M \times2^E\) must be in the form \( MeE\), where the mantissa \( M\) and the exponent \( E\) are integers in base 10.