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<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i>MODULE: RR</i></font><br>
<br>
<font color="#0000ed"><i>SUMMARY:</i></font><br>
<br>
<font color="#0000ed"><i>The class RR is used to represent arbitrary-precision floating point</i></font><br>
<font color="#0000ed"><i>numbers.</i></font><br>
<br>
<font color="#0000ed"><i>The functions in this module guarantee very strong accuracy conditions</i></font><br>
<font color="#0000ed"><i>which make it easy to reason about the behavior of programs using</i></font><br>
<font color="#0000ed"><i>these functions.</i></font><br>
<br>
<font color="#0000ed"><i>The arithmetic operations always round their results to p bits, where</i></font><br>
<font color="#0000ed"><i>p is the current precision. The current precision can be changed</i></font><br>
<font color="#0000ed"><i>using RR::SetPrecision(), and can be read using RR::precision(). </i></font><br>
<br>
<font color="#0000ed"><i>The minimum precision that can be set is 53 bits.</i></font><br>
<font color="#0000ed"><i>The maximum precision is limited only by the word size of the machine.</i></font><br>
<br>
<font color="#0000ed"><i>A convenience class RRPush is provided to automatically save and</i></font><br>
<font color="#0000ed"><i>restore the current precision.</i></font><br>
<br>
<font color="#0000ed"><i>All arithmetic operations are implemented so that the effect is as if the</i></font><br>
<font color="#0000ed"><i>result was computed exactly, and then rounded to p bits. If a number</i></font><br>
<font color="#0000ed"><i>lies exactly half-way between two p-bit numbers, the "round to even"</i></font><br>
<font color="#0000ed"><i>rule is used. So in particular, the computed result will have a relative error</i></font><br>
<font color="#0000ed"><i>of at most 2^{-p}.</i></font><br>
<br>
<br>
<font color="#0000ed"><i>The above rounding rules apply to all arithmetic operations in this</i></font><br>
<font color="#0000ed"><i>module, except for the following routines:</i></font><br>
<br>
<font color="#0000ed"><i>* The transcendental functions: </i></font><br>
<font color="#0000ed"><i> log, exp, log10, expm1, log1p, pow, sin, cos, ComputePi</i></font><br>
<br>
<font color="#0000ed"><i>* The power function</i></font><br>
<br>
<font color="#0000ed"><i>* The input and ascii to RR conversion functions when using "e"-notation </i></font><br>
<br>
<font color="#0000ed"><i>For these functions, a very strong accuracy condition is still </i></font><br>
<font color="#0000ed"><i>guaranteed: the computed result has a relative error of less than 2^{-p + 1}</i></font><br>
<font color="#0000ed"><i>(and actually much closer to 2^{-p}).</i></font><br>
<font color="#0000ed"><i>That is, it is as if the resulted were computed exactly, and then</i></font><br>
<font color="#0000ed"><i>rounded to one of the two neighboring p-bit numbers (but not necessarily</i></font><br>
<font color="#0000ed"><i>the closest).</i></font><br>
<br>
<font color="#0000ed"><i>The behavior of all functions in this module is completely platform </i></font><br>
<font color="#0000ed"><i>independent: you should get *exactly* the same results on any platform</i></font><br>
<font color="#0000ed"><i>(the only exception to this rule is the random number generator).</i></font><br>
<br>
<font color="#0000ed"><i>Note that because precision is variable, a number may be computed with</i></font><br>
<font color="#0000ed"><i>to a high precision p', and then be used as input to an arithmetic operation</i></font><br>
<font color="#0000ed"><i>when the current precision is p < p'. </i></font><br>
<font color="#0000ed"><i>The above accuracy guarantees still apply; in particular, </i></font><br>
<font color="#0000ed"><i>no rounding is done until *after* the operation is performed. </i></font><br>
<br>
<font color="#0000ed"><i>EXAMPLE: If x and y are computed to 200 bits of precision,</i></font><br>
<font color="#0000ed"><i>and then the precision is set to 100 bits, then x-y will</i></font><br>
<font color="#0000ed"><i>be computed correctly to 100 bits, even if, say, x and y agree</i></font><br>
<font color="#0000ed"><i>in their high-order 50 bits. If x and y had been rounded to</i></font><br>
<font color="#0000ed"><i>100 bits before the subtraction, then the difference would</i></font><br>
<font color="#0000ed"><i>only be accurate to 50 bits of precision.</i></font><br>
<br>
<font color="#0000ed"><i>Note that the assignment operator and the copy constructor </i></font><br>
<font color="#0000ed"><i>produce *exact* copies of their inputs---they are *never* rounded. </i></font><br>
<font color="#0000ed"><i>This is a change in semantics from versions 2.0 and earlier</i></font><br>
<font color="#0000ed"><i>in which assignment and copy rounded their outputs.</i></font><br>
<font color="#0000ed"><i>This was deemed a design error and has been changed.</i></font><br>
<br>
<font color="#0000ed"><i>If you want to force rounding to current precision, the easiest</i></font><br>
<font color="#0000ed"><i>way to do this is with the RR to RR conversion routines:</i></font><br>
<font color="#0000ed"><i> conv(x, a);</i></font><br>
<font color="#0000ed"><i>or</i></font><br>
<font color="#0000ed"><i> x = to_RR(a); </i></font><br>
<font color="#0000ed"><i>This will round a to current precision and store the result in x.</i></font><br>
<font color="#0000ed"><i>Note that writing</i></font><br>
<font color="#0000ed"><i> x = a + 0;</i></font><br>
<font color="#0000ed"><i>or</i></font><br>
<font color="#0000ed"><i> x = a*1;</i></font><br>
<font color="#0000ed"><i>also has the same effect.</i></font><br>
<br>
<font color="#0000ed"><i>Unlike IEEE standard floating point, there are no "special values",</i></font><br>
<font color="#0000ed"><i>like "infinity" or "not a number", nor are there any "denormalized</i></font><br>
<font color="#0000ed"><i>numbers". Overflow, underflow, or taking a square root of a negative</i></font><br>
<font color="#0000ed"><i>number all result in an error being raised.</i></font><br>
<br>
<font color="#0000ed"><i>An RR is represented as a mantissa/exponent pair (x, e), where x is a</i></font><br>
<font color="#0000ed"><i>ZZ and e is a long. The real number represented by (x, e) is x * 2^e.</i></font><br>
<font color="#0000ed"><i>Zero is always represented as (0, 0). For all other numbers, x is</i></font><br>
<font color="#0000ed"><i>always odd.</i></font><br>
<br>
<br>
<font color="#0000ed"><i>CONVERSIONS AND PROMOTIONS:</i></font><br>
<font color="#0000ed"><i>The complete set of conversion routines between RR and other types is</i></font><br>
<font color="#0000ed"><i>documented in the file "conversions.txt". Conversion from any type</i></font><br>
<font color="#0000ed"><i>to RR always rounds the result to the current precision.</i></font><br>
<br>
<font color="#0000ed"><i>The basic operations also support the notion of "promotions", </i></font><br>
<font color="#0000ed"><i>so that they promote a double to an RR. For example, one can write </i></font><br>
<font color="#0000ed"><i> x = y + 1.5;</i></font><br>
<font color="#0000ed"><i>where x and y are RR's. One should be aware that these promotions are </i></font><br>
<font color="#0000ed"><i>always implemented using the double to RR conversion routine.</i></font><br>
<br>
<br>
<font color="#0000ed"><i>SIZE INVARIANT: max(NumBits(x), |e|) < 2^(NTL_BITS_PER_LONG-4)</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<br>
<br>
<font color="#1773cc">#include </font><font color="#4a6f8b"><NTL/ZZ.h></font><br>
<font color="#1773cc">#include </font><font color="#4a6f8b"><NTL/xdouble.h></font><br>
<font color="#1773cc">#include </font><font color="#4a6f8b"><NTL/quad_float.h></font><br>
<br>
<font color="#008b00"><b>class</b></font> RR {<br>
<br>
<font color="#b02f60"><b>public</b></font>:<br>
<br>
RR(); <font color="#0000ed"><i>// = 0</i></font><br>
<br>
RR(<font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// copy constructor</i></font><br>
<br>
<br>
<font color="#008b00"><b>explicit</b></font> RR(<font color="#008b00"><b>double</b></font> a); <font color="#0000ed"><i>// promotion constructor</i></font><br>
<br>
RR& <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// assignment operator</i></font><br>
<br>
<font color="#0000ed"><i>// NOTE: the copy constructor and assignment operator</i></font><br>
<font color="#0000ed"><i>// produce exact copies of their inputs, and do not round</i></font><br>
<font color="#0000ed"><i>// to current precision. </i></font><br>
<br>
RR& <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>double</b></font> a); <font color="#0000ed"><i>// convert and assign</i></font><br>
<br>
~RR(); <font color="#0000ed"><i>// destructor</i></font><br>
<br>
<font color="#008b00"><b>const</b></font> ZZ& mantissa() <font color="#008b00"><b>const</b></font>; <font color="#0000ed"><i>// read the mantissa</i></font><br>
<font color="#008b00"><b>long</b></font> exponent() <font color="#008b00"><b>const</b></font>; <font color="#0000ed"><i>// read the exponent</i></font><br>
<br>
<font color="#008b00"><b>static</b></font> <font color="#008b00"><b>void</b></font> SetPrecision(<font color="#008b00"><b>long</b></font> p); <br>
<font color="#0000ed"><i>// set current precision to max(p, 53) bits.</i></font><br>
<font color="#0000ed"><i>// The default is 150</i></font><br>
<br>
<font color="#008b00"><b>static</b></font> <font color="#008b00"><b>long</b></font> precision(); <font color="#0000ed"><i>// read current value of precision</i></font><br>
<br>
<font color="#008b00"><b>static</b></font> <font color="#008b00"><b>void</b></font> SetOutputPrecision(<font color="#008b00"><b>long</b></font> p); <br>
<font color="#0000ed"><i>// set the number of output decimal digits to max(p, 1).</i></font><br>
<font color="#0000ed"><i>// The default is 10</i></font><br>
<br>
<font color="#008b00"><b>static</b></font> <font color="#008b00"><b>long</b></font> OutputPrecision();<br>
<font color="#0000ed"><i>// read the current number of output decimal digits</i></font><br>
<br>
<br>
};<br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Comparison</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>// standard comparison operators:</i></font><br>
<br>
<font color="#008b00"><b>long</b></font> <font color="#b02f60"><b>operator</b></font>==(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b);<br>
<font color="#008b00"><b>long</b></font> <font color="#b02f60"><b>operator</b></font>!=(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b);<br>
<font color="#008b00"><b>long</b></font> <font color="#b02f60"><b>operator</b></font><=(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b);<br>
<font color="#008b00"><b>long</b></font> <font color="#b02f60"><b>operator</b></font>>=(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b);<br>
<font color="#008b00"><b>long</b></font> <font color="#b02f60"><b>operator</b></font> <(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b);<br>
<font color="#008b00"><b>long</b></font> <font color="#b02f60"><b>operator</b></font> >(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b);<br>
<br>
<br>
<font color="#008b00"><b>long</b></font> IsZero(<font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// test if 0</i></font><br>
<font color="#008b00"><b>long</b></font> IsOne(<font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// test if 1</i></font><br>
<br>
<font color="#008b00"><b>long</b></font> sign(<font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// returns sign of a (+1, -1, 0)</i></font><br>
<font color="#008b00"><b>long</b></font> compare(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b); <font color="#0000ed"><i>// returns sign(a-b);</i></font><br>
<br>
<font color="#0000ed"><i>// PROMOTIONS: operators ==, ..., > and function compare</i></font><br>
<font color="#0000ed"><i>// promote double to RR on (a, b).</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Addition</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<font color="#0000ed"><i>// operator notation:</i></font><br>
<br>
RR <font color="#b02f60"><b>operator</b></font>+(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b);<br>
RR <font color="#b02f60"><b>operator</b></font>-(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b);<br>
RR <font color="#b02f60"><b>operator</b></font>-(<font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// unary -</i></font><br>
<br>
RR& <font color="#b02f60"><b>operator</b></font>+=(RR& x, <font color="#008b00"><b>const</b></font> RR& a);<br>
RR& <font color="#b02f60"><b>operator</b></font>+=(RR& x, <font color="#008b00"><b>double</b></font> a);<br>
<br>
RR& <font color="#b02f60"><b>operator</b></font>-=(RR& x, <font color="#008b00"><b>const</b></font> RR& a);<br>
RR& <font color="#b02f60"><b>operator</b></font>-=(RR& x, <font color="#008b00"><b>double</b></font> a);<br>
<br>
RR& <font color="#b02f60"><b>operator</b></font>++(RR& x); <font color="#0000ed"><i>// prefix</i></font><br>
<font color="#008b00"><b>void</b></font> <font color="#b02f60"><b>operator</b></font>++(RR& x, <font color="#008b00"><b>int</b></font>); <font color="#0000ed"><i>// postfix</i></font><br>
<br>
RR& <font color="#b02f60"><b>operator</b></font>--(RR& x); <font color="#0000ed"><i>// prefix</i></font><br>
<font color="#008b00"><b>void</b></font> <font color="#b02f60"><b>operator</b></font>--(RR& x, <font color="#008b00"><b>int</b></font>); <font color="#0000ed"><i>// postfix</i></font><br>
<br>
<br>
<font color="#0000ed"><i>// procedural versions:</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> add(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b); <font color="#0000ed"><i>// z = a+b</i></font><br>
<font color="#008b00"><b>void</b></font> sub(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b); <font color="#0000ed"><i>// z = a-b</i></font><br>
<font color="#008b00"><b>void</b></font> negate(RR& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = -a</i></font><br>
<br>
<font color="#0000ed"><i>// PROMOTIONS: operators +, -, and procedures add, sub promote double</i></font><br>
<font color="#0000ed"><i>// to RR on (a, b).</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> abs(RR& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = |a|</i></font><br>
RR fabs(<font color="#008b00"><b>const</b></font> RR& a); <br>
RR abs(<font color="#008b00"><b>const</b></font> RR& a); <br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Multiplication</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#0000ed"><i>// operator notation:</i></font><br>
<br>
RR <font color="#b02f60"><b>operator</b></font>*(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b);<br>
<br>
RR& <font color="#b02f60"><b>operator</b></font>*=(RR& x, <font color="#008b00"><b>const</b></font> RR& a);<br>
RR& <font color="#b02f60"><b>operator</b></font>*=(RR& x, <font color="#008b00"><b>double</b></font> a);<br>
<br>
<font color="#0000ed"><i>// procedural versions:</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> mul(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b); <font color="#0000ed"><i>// z = a*b</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> sqr(RR& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = a * a</i></font><br>
RR sqr(<font color="#008b00"><b>const</b></font> RR& a); <br>
<br>
<font color="#0000ed"><i>// PROMOTIONS: operator * and procedure mul promote double to RR on (a, b).</i></font><br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Division</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#0000ed"><i>// operator notation:</i></font><br>
<br>
RR <font color="#b02f60"><b>operator</b></font>/(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b);<br>
<br>
RR& <font color="#b02f60"><b>operator</b></font>/=(RR& x, <font color="#008b00"><b>const</b></font> RR& a);<br>
RR& <font color="#b02f60"><b>operator</b></font>/=(RR& x, <font color="#008b00"><b>double</b></font> a);<br>
<br>
<br>
<font color="#0000ed"><i>// procedural versions:</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> div(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b); z = a/b<br>
<br>
<font color="#008b00"><b>void</b></font> inv(RR& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = 1 / a</i></font><br>
RR inv(<font color="#008b00"><b>const</b></font> RR& a); <br>
<br>
<font color="#0000ed"><i>// PROMOTIONS: operator / and procedure div promote double to RR on (a, b).</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Transcendental functions </i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> exp(RR& res, <font color="#008b00"><b>const</b></font> RR& x); <font color="#0000ed"><i>// e^x</i></font><br>
RR exp(<font color="#008b00"><b>const</b></font> RR& x); <br>
<br>
<font color="#008b00"><b>void</b></font> log(RR& res, <font color="#008b00"><b>const</b></font> RR& x); <font color="#0000ed"><i>// log(x) (natural log)</i></font><br>
RR log(<font color="#008b00"><b>const</b></font> RR& x); <br>
<br>
<font color="#008b00"><b>void</b></font> log10(RR& res, <font color="#008b00"><b>const</b></font> RR& x); <font color="#0000ed"><i>// log(x)/log(10)</i></font><br>
RR log10(<font color="#008b00"><b>const</b></font> RR& x); <br>
<br>
<font color="#008b00"><b>void</b></font> expm1(RR& res, <font color="#008b00"><b>const</b></font> RR& x);<br>
RR expm1(<font color="#008b00"><b>const</b></font> RR& x); <br>
<font color="#0000ed"><i>// e^(x)-1; more accurate than exp(x)-1 when |x| is small</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> log1p(RR& res, <font color="#008b00"><b>const</b></font> RR& x);<br>
RR log1p(<font color="#008b00"><b>const</b></font> RR& x); <br>
<font color="#0000ed"><i>// log(1 + x); more accurate than log(1 + x) when |x| is small</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> pow(RR& res, <font color="#008b00"><b>const</b></font> RR& x, <font color="#008b00"><b>const</b></font> RR& y); <font color="#0000ed"><i>// x^y</i></font><br>
RR pow(<font color="#008b00"><b>const</b></font> RR& x, <font color="#008b00"><b>const</b></font> RR& y); <br>
<br>
<font color="#008b00"><b>void</b></font> sin(RR& res, <font color="#008b00"><b>const</b></font> RR& x); <font color="#0000ed"><i>// sin(x); restriction: |x| < 2^1000</i></font><br>
RR sin(<font color="#008b00"><b>const</b></font> RR& x); <br>
<br>
<font color="#008b00"><b>void</b></font> cos(RR& res, <font color="#008b00"><b>const</b></font> RR& x); <font color="#0000ed"><i>// cos(x); restriction: |x| < 2^1000</i></font><br>
RR cos(<font color="#008b00"><b>const</b></font> RR& x); <br>
<br>
<font color="#008b00"><b>void</b></font> ComputePi(RR& pi); <font color="#0000ed"><i>// approximate pi to current precision</i></font><br>
RR ComputePi_RR();<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Rounding to integer values </i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>** RR output **</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> trunc(RR& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = a, truncated to 0</i></font><br>
RR trunc(<font color="#008b00"><b>const</b></font> RR& a);<br>
<br>
<font color="#008b00"><b>void</b></font> floor(RR& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = a, truncated to -infinity</i></font><br>
RR floor(<font color="#008b00"><b>const</b></font> RR& a);<br>
<br>
<font color="#008b00"><b>void</b></font> ceil(RR& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = a, truncated to +infinity</i></font><br>
RR ceil(<font color="#008b00"><b>const</b></font> RR& a);<br>
<br>
<font color="#008b00"><b>void</b></font> round(RR& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = a, truncated to nearest integer</i></font><br>
RR round(<font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// ties are rounded to an even integer</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>** ZZ output **</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> TruncToZZ(ZZ& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = a, truncated to 0</i></font><br>
ZZ TruncToZZ(<font color="#008b00"><b>const</b></font> RR& a);<br>
<br>
<font color="#008b00"><b>void</b></font> FloorToZZ(ZZ& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = a, truncated to -infinity</i></font><br>
ZZ FloorToZZ(<font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// same as RR to ZZ conversion</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> CeilToZZ(ZZ& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = a, truncated to +infinity</i></font><br>
ZZ CeilToZZ(<font color="#008b00"><b>const</b></font> ZZ& a);<br>
<br>
<font color="#008b00"><b>void</b></font> RoundToZZ(ZZ& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = a, truncated to nearest integer</i></font><br>
ZZ RoundToZZ(<font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// ties are rounded to an even integer</i></font><br>
<br>
<br>
<br>
<a name="push"></a>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Saving and restoring the current precision</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#008b00"><b>class</b></font> RRPush {<br>
<font color="#b02f60"><b>public</b></font>:<br>
RRPush(); <font color="#0000ed"><i>// saves the cuurent precision</i></font><br>
~RRPush(); <font color="#0000ed"><i>// restores the saved precision</i></font><br>
<br>
<font color="#b02f60"><b>private</b></font>: <br>
RRPush(<font color="#008b00"><b>const</b></font> RRPush&); <font color="#0000ed"><i>// disable</i></font><br>
<font color="#008b00"><b>void</b></font> <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>const</b></font> RRPush&); <font color="#0000ed"><i>// disable</i></font><br>
};<br>
<br>
<br>
<font color="#0000ed"><i>// Example: </i></font><br>
<font color="#0000ed"><i>//</i></font><br>
<font color="#0000ed"><i>// {</i></font><br>
<font color="#0000ed"><i>// RRPush push; // don't forget to declare a variable!!</i></font><br>
<font color="#0000ed"><i>// RR::SetPrecsion(new_p);</i></font><br>
<font color="#0000ed"><i>// ...</i></font><br>
<font color="#0000ed"><i>// } // old precsion restored when scope is exited</i></font><br>
<br>
<br>
<font color="#008b00"><b>class</b></font> RROutputPush {<br>
<font color="#b02f60"><b>public</b></font>:<br>
RROutputPush(); <font color="#0000ed"><i>// saves the cuurent output precision</i></font><br>
~RROutputPush(); <font color="#0000ed"><i>// restores the saved output precision</i></font><br>
<br>
<font color="#b02f60"><b>private</b></font>: <br>
RROutputPush(<font color="#008b00"><b>const</b></font> RROutputPush&); <font color="#0000ed"><i>// disable</i></font><br>
<font color="#008b00"><b>void</b></font> <font color="#b02f60"><b>operator</b></font>=(<font color="#008b00"><b>const</b></font> RROutputPush&); <font color="#0000ed"><i>// disable</i></font><br>
};<br>
<br>
<br>
<font color="#0000ed"><i>// Example: </i></font><br>
<font color="#0000ed"><i>//</i></font><br>
<font color="#0000ed"><i>// {</i></font><br>
<font color="#0000ed"><i>// RROutputPush push; // don't forget to declare a variable!!</i></font><br>
<font color="#0000ed"><i>// RR::SetOutputPrecsion(new_op);</i></font><br>
<font color="#0000ed"><i>// ...</i></font><br>
<font color="#0000ed"><i>// } // old output precsion restored when scope is exited</i></font><br>
<br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Miscelaneous</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> MakeRR(RR& z, <font color="#008b00"><b>const</b></font> ZZ& a, <font color="#008b00"><b>long</b></font> e);<br>
RR MakeRR(<font color="#008b00"><b>const</b></font> ZZ& a, <font color="#008b00"><b>long</b></font> e);<br>
<font color="#0000ed"><i>// z = a*2^e, rounded to current precision</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> random(RR& z);<br>
RR random_RR(); <br>
<font color="#0000ed"><i>// z = pseudo-random number in the range [0,1).</i></font><br>
<font color="#0000ed"><i>// Note that the behaviour of this function is somewhat platform</i></font><br>
<font color="#0000ed"><i>// dependent, because the underlying pseudo-ramdom generator is.</i></font><br>
<br>
<br>
<font color="#008b00"><b>void</b></font> SqrRoot(RR& z, <font color="#008b00"><b>const</b></font> RR& a); <font color="#0000ed"><i>// z = sqrt(a);</i></font><br>
RR SqrRoot(<font color="#008b00"><b>const</b></font> RR& a);<br>
RR sqrt(<font color="#008b00"><b>const</b></font> RR& a);<br>
<br>
<font color="#008b00"><b>void</b></font> power(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> e); <font color="#0000ed"><i>// z = a^e, e may be negative</i></font><br>
RR power(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> e); <br>
<br>
<font color="#008b00"><b>void</b></font> power2(RR& z, <font color="#008b00"><b>long</b></font> e); <font color="#0000ed"><i>// z = 2^e, e may be negative</i></font><br>
RR power2_RR(<font color="#008b00"><b>long</b></font> e); <br>
<br>
<br>
<font color="#008b00"><b>void</b></font> clear(RR& z); <font color="#0000ed"><i>// z = 0</i></font><br>
<font color="#008b00"><b>void</b></font> set(RR& z); <font color="#0000ed"><i>// z = 1</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> RR::swap(RR& a);<br>
<font color="#008b00"><b>void</b></font> swap(RR& a, RR& b); <br>
<font color="#0000ed"><i>// swap (pointer swap)</i></font><br>
<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i> Input/Output</i></font><br>
<font color="#0000ed"><i>Input Syntax:</i></font><br>
<br>
<font color="#0000ed"><i><number>: [ "-" ] <unsigned-number></i></font><br>
<font color="#0000ed"><i><unsigned-number>: <dotted-number> [ <e-part> ] | <e-part></i></font><br>
<font color="#0000ed"><i><dotted-number>: <digits> | <digits> "." <digits> | "." <digits> | <digits> "."</i></font><br>
<font color="#0000ed"><i><digits>: <digit> <digits> | <digit></i></font><br>
<font color="#0000ed"><i><digit>: "0" | ... | "9"</i></font><br>
<font color="#0000ed"><i><e-part>: ( "E" | "e" ) [ "+" | "-" ] <digits></i></font><br>
<br>
<font color="#0000ed"><i>Examples of valid input:</i></font><br>
<br>
<font color="#0000ed"><i>17 1.5 0.5 .5 5. -.5 e10 e-10 e+10 1.5e10 .5e10 .5E10</i></font><br>
<br>
<font color="#0000ed"><i>Note that the number of decimal digits of precision that are used</i></font><br>
<font color="#0000ed"><i>for output can be set to any number p >= 1 by calling</i></font><br>
<font color="#0000ed"><i>the routine RR::SetOutputPrecision(p). The default value of p is 10.</i></font><br>
<font color="#0000ed"><i>The current value of p is returned by a call to RR::OutputPrecision().</i></font><br>
<br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<br>
ostream& <font color="#b02f60"><b>operator</b></font><<(ostream& s, <font color="#008b00"><b>const</b></font> RR& a);<br>
istream& <font color="#b02f60"><b>operator</b></font>>>(istream& s, RR& x);<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<br>
<font color="#0000ed"><i> Specialized routines with explicit precision parameter</i></font><br>
<br>
<font color="#0000ed"><i>These routines take an explicit precision parameter p. The value of p may be</i></font><br>
<font color="#0000ed"><i>any positive integer. All results are computed to *precisely* p bits of</i></font><br>
<font color="#0000ed"><i>precision, regardless of the current precision (as set by RR::SetPrecision).</i></font><br>
<br>
<font color="#0000ed"><i>These routines are provided both for convenience and for situations where the</i></font><br>
<font color="#0000ed"><i>computation must be done with a precision that may be less than 53.</i></font><br>
<br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
<br>
<br>
<font color="#008b00"><b>void</b></font> AddPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a + b</i></font><br>
RR AddPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> SubPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a - b</i></font><br>
RR SubPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> NegatePrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = -a</i></font><br>
RR NegatePrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> AbsPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = |a|</i></font><br>
RR AbsPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> MulPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a*b</i></font><br>
RR MulPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> SqrPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a*a</i></font><br>
RR SqrPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> DivPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a/b</i></font><br>
RR DivPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>const</b></font> RR& b, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> InvPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = 1/a</i></font><br>
RR DivPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> SqrRootPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = sqrt(a)</i></font><br>
RR SqrRootPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> TruncPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a, truncated to 0</i></font><br>
RR TruncPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> FloorPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a, truncated to -infinity</i></font><br>
RR FloorPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> CeilPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a, truncated to +infinity</i></font><br>
RR CeilPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> RoundPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a, </i></font><br>
<font color="#0000ed"><i>// truncated to nearest integer,</i></font><br>
<font color="#0000ed"><i>// ties are roundec to an even </i></font><br>
<font color="#0000ed"><i>// integer</i></font><br>
RR RoundPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> ConvPrec(RR& z, <font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a</i></font><br>
RR ConvPrec(<font color="#008b00"><b>const</b></font> RR& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> ConvPrec(RR& z, <font color="#008b00"><b>const</b></font> ZZ& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a</i></font><br>
RR ConvPrec(<font color="#008b00"><b>const</b></font> ZZ& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> ConvPrec(RR& z, <font color="#008b00"><b>long</b></font> a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a</i></font><br>
RR ConvPrec(<font color="#008b00"><b>long</b></font> a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> ConvPrec(RR& z, <font color="#008b00"><b>int</b></font> a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a</i></font><br>
RR ConvPrec(<font color="#008b00"><b>int</b></font> a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> ConvPrec(RR& z, <font color="#008b00"><b>unsigned</b></font> <font color="#008b00"><b>long</b></font> a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a</i></font><br>
RR ConvPrec(<font color="#008b00"><b>unsigned</b></font> <font color="#008b00"><b>long</b></font> a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> ConvPrec(RR& z, <font color="#008b00"><b>unsigned</b></font> <font color="#008b00"><b>int</b></font> a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a </i></font><br>
RR ConvPrec(<font color="#008b00"><b>unsigned</b></font> <font color="#008b00"><b>int</b></font> a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> ConvPrec(RR& z, <font color="#008b00"><b>double</b></font> a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a</i></font><br>
RR ConvPrec(<font color="#008b00"><b>double</b></font> a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> ConvPrec(RR& z, <font color="#008b00"><b>const</b></font> xdouble& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a</i></font><br>
RR ConvPrec(<font color="#008b00"><b>const</b></font> xdouble& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> ConvPrec(RR& z, <font color="#008b00"><b>const</b></font> quad_float& a, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a</i></font><br>
RR ConvPrec(<font color="#008b00"><b>const</b></font> quad_float& a, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<font color="#008b00"><b>void</b></font> ConvPrec(RR& z, <font color="#008b00"><b>const</b></font> <font color="#008b00"><b>char</b></font> *s, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// read z from s</i></font><br>
RR ConvPrec(<font color="#008b00"><b>const</b></font> <font color="#008b00"><b>char</b></font> *s, <font color="#008b00"><b>long</b></font> p);<br>
<br>
istream& InputPrec(RR& z, istream& s, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// read z from s</i></font><br>
RR InputPrec(istream& s, <font color="#008b00"><b>long</b></font> p);<br>
<font color="#0000ed"><i>// The functional variant raises an error if input</i></font><br>
<font color="#0000ed"><i>// is missing or ill-formed, while procedural form</i></font><br>
<font color="#0000ed"><i>// does not.</i></font><br>
<br>
<font color="#008b00"><b>void</b></font> MakeRRPrec(RR& z, <font color="#008b00"><b>const</b></font> ZZ& a, <font color="#008b00"><b>long</b></font> e, <font color="#008b00"><b>long</b></font> p); <font color="#0000ed"><i>// z = a*2^e</i></font><br>
RR MakeRRPrec(<font color="#008b00"><b>const</b></font> ZZ& a, <font color="#008b00"><b>long</b></font> e, <font color="#008b00"><b>long</b></font> p);<br>
<br>
<br>
<font color="#0000ed"><i>/*</i></font><font color="#0000ed"><i>*************************************************************************\</i></font><br>
<br>
<font color="#0000ed"><i>COMPATABILITY NOTES: </i></font><br>
<br>
<font color="#0000ed"><i> (1) Prior to version 5.3, the documentation indicated that under certain</i></font><br>
<font color="#0000ed"><i> circumstances, the value of the current precision could be directly set</i></font><br>
<font color="#0000ed"><i> by setting the variable RR::prec. Such usage is now considered</i></font><br>
<font color="#0000ed"><i> obsolete. To perform computations using a precision of less than 53</i></font><br>
<font color="#0000ed"><i> bits, users should use the specialized routines AddPrec, SubPrec, etc.,</i></font><br>
<font color="#0000ed"><i> documented above.</i></font><br>
<br>
<font color="#0000ed"><i> (2) The routine RoundToPrecision is obsolete, although for backward</i></font><br>
<font color="#0000ed"><i> compatability, it is still declared (in both procedural and function</i></font><br>
<font color="#0000ed"><i> forms), and is equivalent to ConvPrec.</i></font><br>
<br>
<font color="#0000ed"><i> (3) In versions 2.0 and earlier, the assignment operator and copy</i></font><br>
<font color="#0000ed"><i> constructor for the class RR rounded their outputs to the current</i></font><br>
<font color="#0000ed"><i> precision. This is no longer the case: their outputs are now exact</i></font><br>
<font color="#0000ed"><i> copies of their inputs, regardless of the current precision.</i></font><br>
<br>
<font color="#0000ed"><i>\*************************************************************************</i></font><font color="#0000ed"><i>*/</i></font><br>
<br>
<br>
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