rr.txt

NTL is a high-performance, portable C++ library providing data structures and algorithms for manipul

/**************************************************************************\

MODULE: RR

SUMMARY:

The class RR is used to represent arbitrary-precision floating point
numbers.

The functions in this module guarantee very strong accuracy conditions
which make it easy to reason about the behavior of programs using
these functions.

The arithmetic operations always round their results to p bits, where
p is the current precision.  The current precision can be changed
using RR::SetPrecision(), and can be read using RR::precision().  

The minimum precision that can be set is 53 bits.
The maximum precision is limited only by the word size of the machine.

All arithmetic operations are implemented so that the effect is as if the
result was computed exactly, and then rounded to p bits.  If a number
lies exactly half-way between two p-bit numbers, the "round to even"
rule is used.  So in particular, the computed result will have a relative error
of at most 2^{-p}.


The above rounding rules apply to all arithmetic operations in this
module, except for the following routines:

* The transcendental functions: 
     log, exp, log10, expm1, log1p, pow, sin, cos, ComputePi

* The power function

* The input and ascii to RR conversion functions when using "e"-notation 

For these functions, a very strong accuracy condition is still 
guaranteed: the computed result has a relative error of less than 2^{-p + 1}
(and actually much closer to 2^{-p}).
That is, it is as if the resulted were computed exactly, and then
rounded to one of the two neighboring p-bit numbers (but not necessarily
the closest).

The behavior of all functions in this module is completely platform 
independent: you should get *exactly* the same results on any platform
(the only exception to this rule is the random number generator).

Note that because precision is variable, a number may be computed with
to a high precision p', and then be used as input to an arithmetic operation
when the current precision is p < p'.  
The above accuracy guarantees still apply; in particular, 
no rounding is done until *after* the operation is performed.  

EXAMPLE: If x and y are computed to 200 bits of precision,
and then the precision is set to 100 bits, then x-y will
be computed correctly to 100 bits, even if, say, x and y agree
in their high-order 50 bits.  If x and y had been rounded to
100 bits before the subtraction, then the difference would
only be accurate to 50 bits of precision.

Note that the assignment operator and the copy constructor 
produce *exact* copies of their inputs---they are *never* rounded. 
This is a change in semantics from versions 2.0 and earlier
in which assignment and copy rounded their outputs.
This was deemed a design error and has been changed.

If you want to force rounding to current precision, the easiest
way to do this is with the RR to RR conversion routines:
   conv(x, a);
or
   x = to_RR(a); 
This will round a to current precision and store the result in x.
Note that writing
   x = a + 0;
or
   x = a*1;
also has the same effect.

Unlike IEEE standard floating point, there are no "special values",
like "infinity" or "not a number", nor are there any "denormalized
numbers".  Overflow, underflow, or taking a square root of a negative
number all result in an error being raised.

An RR is represented as a mantissa/exponent pair (x, e), where x is a
ZZ and e is a long.  The real number represented by (x, e) is x * 2^e.
Zero is always represented as (0, 0).  For all other numbers, x is
always odd.


CONVERSIONS AND PROMOTIONS:
The complete set of conversion routines between RR and other types is
documented in the file "conversions.txt". Conversion from any type
to RR always rounds the result to the current precision.

The basic operations also support the notion of "promotions", 
so that they promote a double to an RR.  For example, one can write 
   x = y + 1.5;
where x and y are RR's. One should be aware that these promotions are 
always implemented using the double to RR conversion routine.


SIZE INVARIANT: max(NumBits(x), |e|) < 2^(NTL_BITS_PER_LONG-4)

\**************************************************************************/




#include <NTL/ZZ.h>
#include <NTL/xdouble.h>
#include <NTL/quad_float.h>

class RR {

public:

RR(); // = 0

RR(const RR& a); // copy constructor
RR& operator=(const RR& a); // assignment operator

// NOTE: the copy constructor and assignment operator
// produce exact copies of their inputs, and do not round
// to current precision.  

RR& operator=(double a); // convert and assign

~RR(); // destructor

const ZZ& mantissa() const;  // read the mantissa
long exponent() const;  // read the exponent

static void SetPrecision(long p);  
// set current precision to max(p, 53) bits.
// The default is 150

static long precision();  // read current value of precision

static void SetOutputPrecision(long p);  
// set the number of output decimal digits to max(p, 1).
// The default is 10

static long OutputPrecision();
// read the current number of output decimal digits


};



/**************************************************************************\

                                  Comparison

\**************************************************************************/



// standard comparison operators:

long operator==(const RR& a, const RR& b);
long operator!=(const RR& a, const RR& b);
long operator<=(const RR& a, const RR& b);
long operator>=(const RR& a, const RR& b);
long operator <(const RR& a, const RR& b);
long operator >(const RR& a, const RR& b);


long IsZero(const RR& a); // test if 0
long IsOne(const RR& a); // test if 1

long sign(const RR& a);  // returns sign of a (+1, -1, 0)
long compare(const RR& a, const RR& b); // returns sign(a-b);

// PROMOTIONS: operators ==, ..., > and function compare
// promote double to RR on (a, b).



/**************************************************************************\

                                  Addition

\**************************************************************************/

// operator notation:

RR operator+(const RR& a, const RR& b);
RR operator-(const RR& a, const RR& b);
RR operator-(const RR& a); // unary -

RR& operator+=(RR& x, const RR& a);
RR& operator+=(RR& x, double a);

RR& operator-=(RR& x, const RR& a);
RR& operator-=(RR& x, double a);

RR& operator++(RR& x);  // prefix
void operator++(RR& x, int);  // postfix

RR& operator--(RR& x);  // prefix
void operator--(RR& x, int);  // postfix


// procedural versions:

void add(RR& z, const RR& a, const RR& b); // z = a+b
void sub(RR& z, const RR& a, const RR& b); // z = a-b
void negate(RR& z, const RR& a); // z = -a

// PROMOTIONS: operators +, -, and procedures add, sub promote double
// to RR on (a, b).

void abs(RR& z, const RR& a); // z = |a|
RR fabs(const RR& a);  
RR abs(const RR& a); 


/**************************************************************************\

                                  Multiplication

\**************************************************************************/


// operator notation:

RR operator*(const RR& a, const RR& b);

RR& operator*=(RR& x, const RR& a);
RR& operator*=(RR& x, double a);

// procedural versions:


void mul(RR& z, const RR& a, const RR& b); // z = a*b

void sqr(RR& z, const RR& a); // z = a * a
RR sqr(const RR& a); 

// PROMOTIONS: operator * and procedure mul promote double to RR on (a, b).


/**************************************************************************\

                               Division

\**************************************************************************/


// operator notation:

RR operator/(const RR& a, const RR& b);

RR& operator/=(RR& x, const RR& a);
RR& operator/=(RR& x, double a);


// procedural versions:


void div(RR& z, const RR& a, const RR& b); z = a/b

void inv(RR& z, const RR& a); // z = 1 / a
RR inv(const RR& a);