/**************************************************************************\MODULE: RRSUMMARY:The class RR is used to represent arbitrary-precision floating pointnumbers.The functions in this module guarantee very strong accuracy conditionswhich make it easy to reason about the behavior of programs usingthese functions.The arithmetic operations always round their results to p bits, wherep is the current precision.  The current precision can be changedusing RR::SetPrecision(), and can be read using RR::precision().  The minimum precision that can be set is 53 bits (but see IMPLEMENTATION NOTES below).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 theresult was computed exactly, and then rounded to p bits.  If a numberlies 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 errorof at most 2^{-p}.The above rounding rules apply to all arithmetic operations in thismodule, 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 thenrounded to one of the two neighboring p-bit numbers (but not necessarilythe 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 withto a high precision p', and then be used as input to an arithmetic operationwhen 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 willbe computed correctly to 100 bits, even if, say, x and y agreein their high-order 50 bits.  If x and y had been rounded to100 bits before the subtraction, then the difference wouldonly 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 earlierin 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 easiestway 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 "denormalizednumbers".  Overflow, underflow, or taking a square root of a negativenumber all result in an error being raised.An RR is represented as a mantissa/exponent pair (x, e), where x is aZZ 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 isalways odd.CONVERSIONS AND PROMOTIONS:The complete set of conversion routines between RR and other types isdocumented in the file "conversions.txt". Conversion from any typeto 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(); // = 0RR(const RR& a); // copy constructorRR& 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.  This is a change in semantics// from versions 2.0 and earlier, in which the outputs were // rounded to current precision.RR& operator=(double a); // convert and assign~RR(); // destructorconst ZZ& mantissa() const;  // read the mantissalong exponent() const;  // read the exponentstatic void SetPrecision(long p);  // set current precision to max(p, 53) bits.// The default is 150static long precision();  // read current value of precisionstatic void SetOutputPrecision(long p);  // set the number of output decimal digits to max(p, 1).// The default is 10static 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 0long IsOne(const RR& a); // test if 1long 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);  // prefixvoid operator++(RR& x, int);  // postfixRR& operator--(RR& x);  // prefixvoid operator--(RR& x, int);  // postfix// procedural versions:void add(RR& z, const RR& a, const RR& b); // z = a+bvoid sub(RR& z, const RR& a, const RR& b); // z = a-bvoid 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*bvoid sqr(RR& z, const RR& a); // z = a * aRR 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/bvoid inv(RR& z, const RR& a); // z = 1 / aRR 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^xRR 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 smallvoid log1p(RR& res, const RR& x);RR log1p(const RR& x); // log(1 + x); more accurate than log(1 + x) when |x| is smallvoid pow(RR& res, const RR& x, const RR& y);  // x^yRR pow(const RR& x, const RR& y); void sin(RR& res, const RR& x);  // sin(x); restriction: |x| < 2^1000RR sin(const RR& x); void cos(RR& res, const RR& x);  // cos(x); restriction: |x| < 2^1000RR cos(const RR& x); void ComputePi(RR& pi); // approximate pi to current precisionRR ComputePi_RR();/**************************************************************************\                         Rounding to integer values        \**************************************************************************//*** RR output ***/void trunc(RR& z, const RR& a);  // z = a, truncated to 0RR trunc(const RR& a);void floor(RR& z, const RR& a);  // z = a, truncated to -infinityRR floor(const RR& a);void ceil(RR& z, const RR& a);   // z = a, truncated to +infinityRR ceil(const RR& a);void round(RR& z, const RR& a);   // z = a, truncated to nearest integerRR round(const RR& a);            // ties are rounded to an even integer/*** ZZ output ***/void TruncToZZ(ZZ& z, const RR& a);  // z = a, truncated to 0ZZ TruncToZZ(const RR& a);void FloorToZZ(ZZ& z, const RR& a);  // z = a, truncated to -infinityZZ FloorToZZ(const RR& a);           // same as RR to ZZ conversionvoid CeilToZZ(ZZ& z, const RR& a);   // z = a, truncated to +infinityZZ CeilToZZ(const ZZ& a);void RoundToZZ(ZZ& z, const RR& a);   // z = a, truncated to nearest integerZZ RoundToZZ(const RR& a);            // ties are rounded to an even integer/**************************************************************************\                                 Miscelaneous\**************************************************************************/void RoundToPrecision(RR& z, const RR& a, long p);RR RoundToPrecision(const RR& a, long p);// z = (a rounded to p bits of precsion);// must have p > 0.void MakeRR(RR& z, const ZZ& a,  long e);RR MakeRR(const ZZ& a,  long e);// z = a*2^e, rounded to current precisionvoid 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 negativeRR power(const RR& a, long e); void power2(RR& z, long e); // z = 2^e, e may be negativeRR power2_RR(long e); void clear(RR& z);  // z = 0void set(RR& z);  // z = 1void swap(RR& a, RR& b);  // swaps a and b (by swapping pointers)/**************************************************************************\                               Input/OutputInput 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 .5E10Note that the number of decimal digits of precision that are usedfor output can be set to any number p >= 1 by callingthe 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);/**************************************************************************\                        IMPLEMENTATION NOTESThe current working precision is stored in a global variablecalled RR::prec.When you call RR::SetPrecision(p), then RR::prec is setto max(53, p)."Casual users" should only set RR::prec through the SetPrecision routine.At some point in time, I decided to make the routine SetPrecision enforcethe invariant RR::prec >= 53, which seemed convenient for several reasons.I don't know if this is still a good idea, but it seems like aneven worse idea to change the semantics now.RR::prec can be set (with an assignment statement) to any value psuch that    0 < p < (1L << (NTL_BITS_PER_LONG-4))and any of the basic functions, including arithmetic (+,-,*,/), sqrt, input, conversion, rounding,will work correctly.Indeed, a number of routines in RR.c exploit this "feature".However, it is not necessarily safe to call any other routineswhen RR::prec is so small.A number of strange things can happening, and so it is bestto simply avoid this.\**************************************************************************/