~/ntl-11.4.2/doc/RR.cpp.html

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

    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.

    A convenience class RRPush is provided to automatically save and
    restore the current precision.

    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

    explicit RR(double a);  // promotion 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

    RR(RR&& a);
    // move constructor (C++11 only)
    // declared noexcept unless NTL_EXCEPTIONS flag is set

    RR& operator=(RR&& a);
    // move assignment (C++11 only)
    // declared noexcept unless NTL_EXCEPTIONS flag is set

    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);

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

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

                           Transcendental functions

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

    void exp(RR& res, const RR& x);  // e^x
    RR exp(const RR& x);

    void log(RR& res, const RR& x); // log(x) (natural log)
    RR log(const RR& x);

    void log10(RR& res, const RR& x); // log(x)/log(10)
    RR log10(const RR& x);

    void expm1(RR& res, const RR&  x);
    RR expm1(const RR& x);
    // e^(x)-1; more accurate than exp(x)-1 when |x| is small

    void log1p(RR& res, const RR& x);
    RR log1p(const RR& x);
    // log(1 + x); more accurate than log(1 + x) when |x| is small

    void pow(RR& res, const RR& x, const RR& y);  // x^y
    RR pow(const RR& x, const RR& y);

    void sin(RR& res, const RR& x);  // sin(x); restriction: |x| < 2^1000
    RR sin(const RR& x);

    void cos(RR& res, const RR& x);  // cos(x); restriction: |x| < 2^1000
    RR cos(const RR& x);

    void ComputePi(RR& pi); // approximate pi to current precision
    RR ComputePi_RR();

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

                             Rounding to integer values

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

    /*** RR output ***/

    void trunc(RR& z, const RR& a);  // z = a, truncated to 0
    RR trunc(const RR& a);

    void floor(RR& z, const RR& a);  // z = a, truncated to -infinity
    RR floor(const RR& a);

    void ceil(RR& z, const RR& a);   // z = a, truncated to +infinity
    RR ceil(const RR& a);

    void round(RR& z, const RR& a);   // z = a, truncated to nearest integer
    RR round(const RR& a);            // ties are rounded to an even integer

    /*** ZZ output ***/

    void TruncToZZ(ZZ& z, const RR& a);  // z = a, truncated to 0
    ZZ TruncToZZ(const RR& a);

    void FloorToZZ(ZZ& z, const RR& a);  // z = a, truncated to -infinity
    ZZ FloorToZZ(const RR& a);           // same as RR to ZZ conversion

    void CeilToZZ(ZZ& z, const RR& a);   // z = a, truncated to +infinity
    ZZ CeilToZZ(const ZZ& a);

    void RoundToZZ(ZZ& z, const RR& a);   // z = a, truncated to nearest integer
    ZZ RoundToZZ(const RR& a);            // ties are rounded to an even integer

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

                     Saving and restoring the current precision

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

    class RRPush {
    public:
       RRPush();  // saves the cuurent precision
       ~RRPush(); // restores the saved precision

    private:
       RRPush(const RRPush&); // disable
       void operator=(const RRPush&); // disable
    };

    // Example:
    //
    // {
    //    RRPush push;  // don't forget to declare a variable!!
    //    RR::SetPrecsion(new_p);
    //    ...
    // } // old precsion restored when scope is exited

    class RROutputPush {
    public:
       RROutputPush();   // saves the cuurent output precision
       ~RROutputPush();  // restores the saved output precision

    private:
       RROutputPush(const RROutputPush&); // disable
       void operator=(const RROutputPush&); // disable
    };

    // Example:
    //
    // {
    //    RROutputPush push;  // don't forget to declare a variable!!
    //    RR::SetOutputPrecsion(new_op);
    //    ...
    // } // old output precsion restored when scope is exited

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

                                     Miscelaneous

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

    void MakeRR(RR& z, const ZZ& a,  long e);
    RR MakeRR(const ZZ& a,  long e);
    // z = a*2^e, rounded to current precision

    void random(RR& z);
    RR random_RR();
    // z = pseudo-random number in the range [0,1).
    // Note that the behaviour of this function is somewhat platform
    // dependent, because the underlying pseudo-ramdom generator is.

    void SqrRoot(RR& z, const RR& a); // z = sqrt(a);
    RR SqrRoot(const RR& a);
    RR sqrt(const RR& a);

    void power(RR& z, const RR& a, long e); // z = a^e, e may be negative
    RR power(const RR& a, long e);

    void power2(RR& z, long e); // z = 2^e, e may be negative
    RR power2_RR(long e);

    void clear(RR& z);  // z = 0
    void set(RR& z);  // z = 1

    void RR::swap(RR& a);
    void swap(RR& a, RR& b);
    // swap (pointer swap)

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

                                   Input/Output
    Input Syntax:

    <number>: [ "-" ] <unsigned-number>
    <unsigned-number>: <dotted-number> [ <e-part> ] | <e-part>
    <dotted-number>: <digits> | <digits> "." <digits> | "." <digits> | <digits> "."
    <digits>: <digit> <digits> | <digit>
    <digit>: "0" | ... | "9"
    <e-part>: ( "E" | "e" ) [ "+" | "-" ] <digits>

    Examples of valid input:

    17 1.5 0.5 .5 5.  -.5  e10 e-10 e+10 1.5e10 .5e10 .5E10

    Note that the number of decimal digits of precision that are used
    for output can be set to any number p >= 1 by calling
    the routine RR::SetOutputPrecision(p).  The default value of p is 10.
    The current value of p is returned by a call to RR::OutputPrecision().

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

    ostream& operator<<(ostream& s, const RR& a);
    istream& operator>>(istream& s, RR& x);

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

                Specialized routines with explicit precision parameter

    These routines take an explicit precision parameter p.  The value of p may be
    any positive integer.  All results are computed to *precisely* p bits of
    precision, regardless of the current precision (as set by RR::SetPrecision).

    These routines are provided both for convenience and for situations where the
    computation must be done with a precision that may be less than 53.

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

    void AddPrec(RR& z, const RR& a, const RR& b, long p); // z = a + b
    RR AddPrec(const RR& a, const RR& b, long p);

    void SubPrec(RR& z, const RR& a, const RR& b, long p); // z = a - b
    RR SubPrec(const RR& a, const RR& b, long p);

    void NegatePrec(RR& z, const RR& a, long p); // z = -a
    RR NegatePrec(const RR& a, long p);

    void AbsPrec(RR& z, const RR& a, long p); // z = |a|
    RR AbsPrec(const RR& a, long p);

    void MulPrec(RR& z, const RR& a, const RR& b, long p); // z = a*b
    RR MulPrec(const RR& a, const RR& b, long p);

    void SqrPrec(RR& z, const RR& a, long p); // z = a*a
    RR SqrPrec(const RR& a, long p);

    void DivPrec(RR& z, const RR& a, const RR& b, long p);  // z = a/b
    RR DivPrec(const RR& a, const RR& b, long p);

    void InvPrec(RR& z, const RR& a, long p);  // z = 1/a
    RR DivPrec(const RR& a, long p);

    void SqrRootPrec(RR& z, const RR& a, long p); // z = sqrt(a)
    RR SqrRootPrec(const RR& a, long p);

    void TruncPrec(RR& z, const RR& a, long p); // z = a, truncated to 0
    RR TruncPrec(const RR& a, long p);

    void FloorPrec(RR& z, const RR& a, long p); // z = a, truncated to -infinity
    RR FloorPrec(const RR& a, long p);

    void CeilPrec(RR& z, const RR& a, long p);  // z = a, truncated to +infinity
    RR CeilPrec(const RR& a, long p);

    void RoundPrec(RR& z, const RR& a, long p); // z = a,
                                                // truncated to nearest integer,
                                                // ties are roundec to an even
                                                // integer
    RR RoundPrec(const RR& a, long p);

    void ConvPrec(RR& z, const RR& a, long p); // z = a
    RR ConvPrec(const RR& a, long p);

    void ConvPrec(RR& z, const ZZ& a, long p); // z = a
    RR ConvPrec(const ZZ& a, long p);

    void ConvPrec(RR& z, long a, long p); // z = a
    RR ConvPrec(long a, long p);

    void ConvPrec(RR& z, int a, long p); // z = a
    RR ConvPrec(int a, long p);

    void ConvPrec(RR& z, unsigned long a, long p); // z = a
    RR ConvPrec(unsigned long a, long p);

    void ConvPrec(RR& z, unsigned int a, long p); // z = a
    RR ConvPrec(unsigned int a, long p);

    void ConvPrec(RR& z, double a, long p); // z = a
    RR ConvPrec(double a, long p);

    void ConvPrec(RR& z, const xdouble& a, long p); // z = a
    RR ConvPrec(const xdouble& a, long p);

    void ConvPrec(RR& z, const quad_float& a, long p); // z = a
    RR ConvPrec(const quad_float& a, long p);

    void ConvPrec(RR& z, const char *s, long p); // read z from s
    RR ConvPrec(const char *s, long p);

    istream& InputPrec(RR& z, istream& s, long p);  // read z from s
    RR InputPrec(istream& s, long p);
    // The functional variant raises an error if input
    // is missing or ill-formed, while procedural form
    // does not.

    void MakeRRPrec(RR& z, const ZZ& a, long e, long p); // z = a*2^e
    RR MakeRRPrec(const ZZ& a, long e, long p);

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

    COMPATABILITY NOTES:

     (1)  Prior to version 5.3, the documentation indicated that under certain
          circumstances, the value of the current precision could be directly set
          by setting the variable RR::prec.  Such usage is now considered
          obsolete.  To perform computations using a precision of less than 53
          bits, users should use the specialized routines AddPrec, SubPrec, etc.,
          documented above.

     (2)  The routine RoundToPrecision is obsolete, although for backward
          compatability, it is still declared (in both procedural and function
          forms), and is equivalent to ConvPrec.

     (3)  In versions 2.0 and earlier, the assignment operator and copy
          constructor for the class RR rounded their outputs to the current
          precision.  This is no longer the case:  their outputs are now exact
          copies of their inputs, regardless of the current precision.

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