lhprojects / qdpp

Port qd(Quad-double/256bits floating point) library to Windows/Visual Studio and add some new functions

What is this library?

This library qdpp supports double-double (104bits) and quad-double (209bits) floating points.

This library is a modification of QD library. https://www.davidhbailey.com/dhbsoftware/

Why is this library?

qdpp is based on the QD library.

So why QD?

QD supports double-double (104bits) and quad-double (206bits) floating points. The speed of double-double and quad-double are much faster than the normal multiple-precision library. Because

• Dynamic memory allocation is not needed. The memory for bits is allocated on stack.
• Easy to implement compiling time evaluation math functions (constexpr).

qdpp compared to QD

• Make the library header-only for c++.
• Add constexpr to math functions for dd_real and qd_real.
• Improvements for some math and IO functions.
• constexpr math functions have been implemented for double too.
• qdpp is not 100% compatible to the QD library.

Compute constants in compiling time

constinit dd_real dd_sqrt2 = sqrt(dd_real(2.));

constinit double double_sqrt2 = fb::sqrt(2.);

// 1.4142135623730951
constexpr double double_sqrt2_constexpr = fb::sqrt(2.);
// 2.0000000000000004
constinit double double_sqrt2_sqr = double_sqrt2_constexpr*double_sqrt2_constexpr;

// 1.4142135623730951 -9.6672933134529159E-17
constexpr dd_real dd_sqrt2_constexpr = sqrt(dd_real(2.));;
// 2.
constinit double dd_sqrt2_sqr_to_double = to_double(dd_sqrt2_constexpr*dd_sqrt2_constexpr);

constexpr double dd_sqr2_sqr_to_double_constexpr = to_double(dd_sqrt2_constexpr*dd_sqrt2_constexpr);

// can't use constinit
double double_stdsqrt2 = sqrt(2);

You can play with qdpp on godbolt https://godbolt.org/z/xx7eznKTK.

Note:

• Functions in namespace fb supports compiling time evaluation math functions for double
• qdpp supports compiling time evaluation for dd_real/qd_real.
• It's more accuracy to use dd_real as temporary variables.

Limitation

• Fast math must be turn off for compiling.
• Floating operation mode must be round-to-nearest.

Compiler&Platform supports

GCC & Clang & MSVC are OK to use.

C++20 is needed because of the use of std::bit_cast. For Clang we use __buildin_bit_cast as a workaround.

Both Windows and Linux (not tested) should be OK.

Build

• qdpp is header only project for c++.

  • All files of c++ parts have been clone into a single file, <qd_single.h>, you can include it for all functions. However, you can also include individual parts only.
    • #include <qd/double_math.h> for all stuff about double.
    • #include <qd/dd.h> for all stuff about dd_real.
      • For only dd_real type definition and basic operations, include #include <qd/dd_real.h>.
    • #include <qd/dd_real.h> for all stuff about qd_real.

• If you want to use qdpp in C , the simplest way to do that include the source filesc_*.cpp in your projects.

  Build tests (Cmake)

  mkdir build
  cd build
  cmake -S ../tests

Reference

(Sorry! Far away to complete.)

double

Any function in namespace fb is implemented like this

double foo() noexcept {
    if(std::is_constant_evaluated()) {
        my_foo();
    } else {
        return std::foo();
    }
}

It doesn't touch errno or throw exceptions in constant evaluation context. If any error occur, It just returns nan.

Precision

See the precision.txt for details.

double

• round, ceil, floor, abs, and fmod should be exact.
• sqrt should exactly round to nearest. (ups <= 1/2 ups)
• exp, log, log10, pow, sin, cos , tan, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh has an error up to (0.5+tiny number) ups. Almost all results round correctly to nearest.

qd_real and dd_real

For basic operations like +,-,*, /, the errors are about 0.5-2 ups for qd_real and 1-4 ups for dd_ral.

For most math functions (except fmod, sin, cos and tan), rounding errors are < 10 ups for qd_real and <20 ups for dd_real.

However, the accuracy for fmod, sin, cos, and tan can be very bad. The rounding error can be thousands or millions ups.

Some notes on changes to QD

(Not completed)

• Improve the accuracy for various math functions.

• Add expm1 and log1p.

• ddrand and qdrand now need parameter of generator, lrand48 is not portable, and std::rand() is too bad. Add real_rand function template for ddrand and qdrand

• Remove *d_real::*_real(char const *). Add string literals _dd and _qd.

• Test the accuracy.

• Test the performance for more operations. Use ` for timing. It's portable, precision, and performant.