/**************************************************************************\ MODULE: quad_float SUMMARY: The class quad_float is used to represent quadruple precision numbers. Thus, with standard IEEE floating point, you should get the equivalent of about 106 bits of precision (but actually just a bit less). The interface allows you to treat quad_floats more or less as if they were "ordinary" floating point types. See below for more implementation details. \**************************************************************************/ #include <NTL/ZZ.h> class quad_float { public: quad_float(); // = 0 quad_float(const quad_float& a); // copy constructor explicit quad_float(double a); // promotion constructor quad_float& operator=(const quad_float& a); // assignment operator quad_float& operator=(double a); ~quad_float(); static void SetOutputPrecision(long p); // This sets the number of decimal digits to be output. Default is // 10. static long OutputPrecision(); // returns current output precision. }; /**************************************************************************\ Arithmetic Operations \**************************************************************************/ quad_float operator +(const quad_float& x, const quad_float& y); quad_float operator -(const quad_float& x, const quad_float& y); quad_float operator *(const quad_float& x, const quad_float& y); quad_float operator /(const quad_float& x, const quad_float& y); // PROMOTIONS: operators +, -, *, / promote double to quad_float // on (x, y). quad_float operator -(const quad_float& x); quad_float& operator += (quad_float& x, const quad_float& y); quad_float& operator += (quad_float& x, double y); quad_float& operator -= (quad_float& x, const quad_float& y); quad_float& operator -= (quad_float& x, double y); quad_float& operator *= (quad_float& x, const quad_float& y); quad_float& operator *= (quad_float& x, double y); quad_float& operator /= (quad_float& x, const quad_float& y); quad_float& operator /= (quad_float& x, double y); quad_float& operator++(quad_float& a); // prefix void operator++(quad_float& a, int); // postfix quad_float& operator--(quad_float& a); // prefix void operator--(quad_float& a, int); // postfix /**************************************************************************\ Comparison \**************************************************************************/ long operator> (const quad_float& x, const quad_float& y); long operator>=(const quad_float& x, const quad_float& y); long operator< (const quad_float& x, const quad_float& y); long operator<=(const quad_float& x, const quad_float& y); long operator==(const quad_float& x, const quad_float& y); long operator!=(const quad_float& x, const quad_float& y); long sign(const quad_float& x); // sign of x, -1, 0, +1 long compare(const quad_float& x, const quad_float& y); // sign of x - y // PROMOTIONS: operators >, ..., != and function compare // promote double to quad_float on (x, y). /**************************************************************************\ 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 quad_float::SetOutputPrecision(p). The default value of p is 10. The current value of p is returned by a call to quad_float::OutputPrecision(). \**************************************************************************/ istream& operator >> (istream& s, quad_float& x); ostream& operator << (ostream& s, const quad_float& x); /**************************************************************************\ Miscellaneous \**************************************************************************/ quad_float sqrt(const quad_float& x); quad_float floor(const quad_float& x); quad_float ceil(const quad_float& x); quad_float trunc(const quad_float& x); quad_float fabs(const quad_float& x); quad_float exp(const quad_float& x); quad_float log(const quad_float& x); void power(quad_float& x, const quad_float& a, long e); // x = a^e quad_float power(const quad_float& a, long e); void power2(quad_float& x, long e); // x = 2^e quad_float power2_quad_float(long e); quad_float ldexp(const quad_float& x, long e); // return x*2^e long IsFinite(quad_float *x); // checks if x is "finite" // pointer is used for compatability with // IsFinite(double*) void random(quad_float& x); quad_float random_quad_float(); // generate a random quad_float x with 0 <= x <= 1 /***********************************************************************\ IMPLEMENTATION DETAILS A quad_float x is represented as a pair of doubles, x.hi and x.lo, such that the number represented by x is x.hi + x.lo, where |x.lo| <= 0.5*ulp(x.hi), (*) and ulp(y) means "unit in the last place of y". For the software to work correctly, IEEE Standard Arithmetic is sufficient. That includes just about every modern computer; the only exception I'm aware of is Intel x86 platforms running Linux (but you can still use this platform--see below). Also sufficient is any platform that implements arithmetic with correct rounding, i.e., given double floating point numbers a and b, a op b is computed exactly and then rounded to the nearest double. The tie-breaking rule is not important. This is a rather weird representation; although it gives one essentially twice the precision of an ordinary double, it is not really the equivalent of quadratic precision (despite the name). For example, the number 1 + 2^{-200} can be represented exactly as a quad_float. Also, there is no real notion of "machine precision". Note that overflow/underflow for quad_floats does not follow any particularly useful rules, even if the underlying floating point arithmetic is IEEE compliant. Generally, when an overflow/underflow occurs, the resulting value is unpredicatble, although typically when overflow occurs in computing a value x, the result is non-finite (i.e., IsFinite(&x) == 0). Note, however, that some care is taken to ensure that the ZZ to quad_float conversion routine produces a non-finite value upon overflow. THE INTEL x86/x87 PROBLEM [The following discussion was written before the advent of SSE2 instructions, back when all floating point on x86 was done using the x87 FPU instruction set and registers. By now, it is mostly of historical interest, as modern x86 CPUs (since SSE2) use a new set of instructions and registers that avoid all of these problems, and by default, it seems that all modern C++ compilers (by default) avoid the x87 altogether. However, there are new problems (see THE FMA PROBLEM, below).] Although just about every modern processor implements the IEEE floating point standard, there still can be problems on processors that support IEEE extended double precision. The only processor I know of that supports this is the x86. While extended double precision may sound like a nice thing, it is not. Normal double precision has 53 bits of precision. Extended has 64. On x86s, the FP registers have 53 or 64 bits of precision---this can be set at run-time by modifying the cpu "control word" (something that can be done only in assembly code). However, doubles stored in memory always have only 53 bits. Compilers may move values between memory and registers whenever they want, which can effectively change the value of a floating point number even though at the C/C++ level, nothing has happened that should have changed the value. Is that sick, or what? Actually, the new C99 standard seems to outlaw such "spontaneous" value changes; however, this behavior is not necessarily universally implemented. This is a real headache, and if one is not just a bit careful, the quad_float code will break. This breaking is not at all subtle, and the program QuadTest will catch the problem if it exists. You should not need to worry about any of this, because NTL automatically detects and works around these problems as best it can, as described below. It shouldn't make a mistake, but if it does, you will catch it in the QuadTest program. If things don't work quite right, you might try setting NTL_FIX_X86 or NTL_NO_FIX_X86 flags in ntl_config.h, but this should not be necessary. Here are the details about how NTL fixes the problem. The first and best way is to have the default setting of the control word be 53 bits. However, you are at the mercy of your platform (compiler, OS, run-time libraries). Windows does this, and so the problem simply does not arise here, and NTL neither detects nor fixes the problem. Linux, however, does not do this, which really sucks. Can we talk these Linux people into changing this? The second way to fix the problem is by having NTL fiddle with control word itself. If you compile NTL using a GNU compiler on an x86, this should happen automatically. On the one hand, this is not a general, portable solution, since it will only work if you use a GNU compiler, or at least one that supports GNU 'asm' syntax. On the other hand, almost everybody who compiles C++ on x86/Linux platforms uses GNU compilers (although there are some commercial compilers out there that I don't know too much about). The third way to fix the problem is to 'force' all intermediate floating point results into memory. This is not an 'ideal' fix, since it is not fully equivalent to 53-bit precision (because of double rounding), but it works (although to be honest, I've never seen a full proof of correctness in this case). NTL's quad_float code does this by storing intermediate results in local variables declared to be 'volatile'. This is the solution to the problem that NTL uses if it detects the problem and can't fix it using the GNU 'asm' hack mentioned above. This solution should work on any platform that faithfully implements 'volatile' according to the ANSI C standard. THE FMA PROBLEM Some CPUs come equipped with a fused-multiply-add (FMA) instruction, which computes x + a*b with just a single rounding. While this generally is faster and more precise than performing this using two instructions and two roundings, FMA instructions can break the logic of quad_float. To mitigate this problem, NTL tries to detect whether the compiler emits FMA instructions when it builds the make_desc.h file. The macro NTL_FMA_DETECTED gets set to 1 if it detects this to be the case. Based on the setting of this macro, code in quad_float.cpp will strategically replace certain multiplications a*b with a*b + z, where z is an external variable that will always be zero (but hopefully the compiler will not figure this out). It is a bit of a hack, but it generally works. THE FLOATING POINT REASSOCIATION PROBLEM The C++ standard says that compilers must issue instructions that respect the grouping of floating point operations. So the compiler is not allowed to compile (a+b)+c as a+(b+c). Most compilers (at least by default) repect this rule. One exception is the Intel icc compiler. Because of this, the quad_float.cpp file includes a pragma that will force the Intel compiler to respect the standard. Note that gcc respects this rule by default, unless you pass -ffast-math as a compilation flag. As long as quad_float.cpp is not compiled with this flag, you should be OK. The quad_float.cpp file contains a preprocessor check to detect if it is compiled under gcc with -ffast-math, and if so, reports an error. This is the only file in NTL which requires this rule. Also, it should be OK to compile client code with -ffast-math, if that is desired. BACKGROUND INFO The code NTL uses algorithms designed by Knuth, Kahan, Dekker, and Linnainmaa. The original transcription to C++ was done by Douglas Priest. Enhancements and bug fixes were done by Keith Briggs --- see http://keithbriggs.info/doubledouble.html. The NTL version is a stripped down version of Briggs' code, with some bug fixes and portability improvements. Here is a brief annotated bibliography (compiled by Priest) of papers dealing with DP and similar techniques, arranged chronologically. Kahan, W., Further Remarks on Reducing Truncation Errors, {\it Comm.\ ACM\/} {\bf 8} (1965), 40. M{\o}ller, O., Quasi Double Precision in Floating-Point Addition, {\it BIT\/} {\bf 5} (1965), 37--50. The two papers that first presented the idea of recovering the roundoff of a sum. Dekker, T., A Floating-Point Technique for Extending the Available Precision, {\it Numer.\ Math.} {\bf 18} (1971), 224--242. The classic reference for DP algorithms for sum, product, quotient, and square root. Pichat, M., Correction d'une Somme en Arithmetique \`a Virgule Flottante, {\it Numer.\ Math.} {\bf 19} (1972), 400--406. An iterative algorithm for computing a protracted sum to working precision by repeatedly applying the sum-and-roundoff method. Linnainmaa, S., Analysis of Some Known Methods of Improving the Accuracy of Floating-Point Sums, {\it BIT\/} {\bf 14} (1974), 167--202. Comparison of Kahan and M{\o}ller algorithms with variations given by Knuth. Bohlender, G., Floating-Point Computation of Functions with Maximum Accuracy, {\it IEEE Trans.\ Comput.} {\bf C-26} (1977), 621--632. Extended the analysis of Pichat's algorithm to compute a multi-word representation of the exact sum of n working precision numbers. This is the algorithm Kahan has called "distillation". Linnainmaa, S., Software for Doubled-Precision Floating-Point Computations, {\it ACM Trans.\ Math.\ Soft.} {\bf 7} (1981), 272--283. Generalized the hypotheses of Dekker and showed how to take advantage of extended precision where available. Leuprecht, H., and W.~Oberaigner, Parallel Algorithms for the Rounding-Exact Summation of Floating-Point Numbers, {\it Computing} {\bf 28} (1982), 89--104. Variations of distillation appropriate for parallel and vector architectures. Kahan, W., Paradoxes in Concepts of Accuracy, lecture notes from Joint Seminar on Issues and Directions in Scientific Computation, Berkeley, 1989. Gives the more accurate DP sum I've shown above, discusses some examples. Priest, D., Algorithms for Arbitrary Precision Floating Point Arithmetic, in P.~Kornerup and D.~Matula, Eds., {\it Proc.\ 10th Symposium on Com- puter Arithmetic}, IEEE Computer Society Press, Los Alamitos, Calif., 1991. Extends from DP to arbitrary precision; gives portable algorithms and general proofs. Sorensen, D., and P.~Tang, On the Orthogonality of Eigenvectors Computed by Divide-and-Conquer Techniques, {\it SIAM J.\ Num.\ Anal.} {\bf 28} (1991), 1752--1775. Uses some DP arithmetic to retain orthogonality of eigenvectors computed by a parallel divide-and-conquer scheme. Priest, D., On Properties of Floating Point Arithmetics: Numerical Stability and the Cost of Accurate Computations, Ph.D. dissertation, University of California at Berkeley, 1992. More examples, organizes proofs in terms of common properties of fp addition/subtraction, gives other summation algorithms. Another relevant paper: X. S. Li, et al. Design, implementation, and testing of extended and mixed precision BLAS. ACM Trans. Math. Soft., 28:152-205, 2002. \***********************************************************************/