Neslib.MultiPrecision

High-Precision Floating-Point Types for Delphi

Neslib.MultiPrecision adds two additional floating-point types for use with Delphi. These offer a precision that is up to 4 times larger than that of the Double type.

Neslib.MultiPrecision is build on top of the QD 2.3.22 and works on:

• Windows (32-bit and 64-bit)
• MacOS (64-bit)
• iOS (64-bit, no simulator)
• Android (32-bit and 64-bit)

The algorithms used for these types were developed by David H. Bailey, Yozo Hida and Xiaoye S. Li. Take a look the the QD.pdf file in the C subdirectory if you are interested in the details.

Installation

To install:

> git clone https://github.com/neslib/Neslib.MultiPrecision

This library has no dependencies on other libraries. There are also no run-time dependencies. The underlying C/C++ QD library is linked into the executable using object files or static libraries.

Two new Floating-Point Types

This library defines two additional floating-point types:

• DoubleDouble: this is a 128-bit type that has twice the precision of the Double type.
• QuadDouble: this is a 256-bit type that has four times the precision of the Double type.

Both types have the same range as the Double type (about ± 10³⁰⁸) but much higher precision. Compared to Single and Double:

Type	Exponent Bits	Mantissa Bits	Precision (decimal digits)
Single	9	24	7
Double	12	53	16
DoubleDouble	12	106	32
QuadDouble	12	212	64

The names DoubleDouble and QuadDouble come from the underlying QD library. I considered naming these Float128 and Float256 instead, but there are already official IEEE specifications for such (hardware) types. And since these are not compatible with DoubleDouble and QuadDouble, I didn't want to add confusion.

The underlying QD library does not use emulation for calculations. Instead, it uses the existing floating-point capabilities of the CPU to use either 2 or 4 Double values to increase the precision. As a result, these types are much faster than other arbitrary/high precision math libraries using the same precision. Also, as opposed to many other libraries, these types don't require dynamic memory allocations, which further helps performance and also reduces memory fragmentation.

Usage

The DoubleDouble and QuadDouble types can be used much the same way as the Double type. They support the usual operators (+, -, *, /, =, <>, <, <=, > and >=) as well as most methods available for the record helpers for the Double type (IsNan, IsInfinity, IsNegativeInfinity, IsPositiveInfinity, ToString, Parse and TryParse). There are methods and operators to convert to and from Double, DoubleDouble, QuadDouble and String.

Since the Delphi language doesn't allow you to enter high-precision floating-point literals in source code, the value of a DoubleDouble or QuadDouble variable must be initialized in some other way. There are various options (assuming DD is of type DoubleDouble here):

• Use one of the Init overloads. For example, DD.Init(1.23); to initialize a DoubleDouble from a Double.
• Use an explicit cast, as in DD := DoubleDouble(1.23);.
• Use one of the predefined constants, as in DD := DoubleDouble.Pi;.
• Or perhaps the easiest way is to just assign a string value, as in DD := '3.1415926535897932384626433832795';.

I didn't add implicit operators, such as DD := 1.23;, since those can lead to unintentional conversions that impact performance.

It's important that you call MultiPrecisionInit before performing any DoubleDouble/QuadDouble calculations. This prepares the FPU/CPU for high-precision math. You can use MultiPrecisionReset to restore the FPU/CPU to its previous state.

Customization

The default configuration of the library is suitable for most applications. This configuration sacrifices a bit of accuracy for increased speed. If accuracy is more important than speed for your purposes, then you can compile the library with the MP_ACCURATE define. This will make many calculations a bit slower but more accurate.

Mathematical Functions

The underlying QD library (and thus this library) supports a variety of common mathematical functions. In addition, the Neslib.MultiPrecision library adds numerous equivalents of functions found in the System.SysUtils and System.Math units.

Name	Description
StrToDoubleDouble, StrToQuadDouble, StrToDoubleDoubleDef, StrToQuadDoubleDef, TryStrToFloat	Convert from string
FloatToStr, FloatToStrF	Convert to string
Inverse	Calculate reciprocal
Rem	Calculate remainder, rounding to nearest
DivRem	Calculate result of division, including remainder
FMod	Calculate remainder, rounded towards zero
Sqrt, Sqr	Square root and Square
Trunc, Floor, Ceil, Round	Various rounding methods
Abs	Absolute value
Min, Max	Return minimum or maximum value
InRange, EnsureRange	Compare against range
SameValue	Approximate equality check
Power, IntPower, NRoot, Ldexp, Exp	Exponential functions
Ln, LnXP1, Log2, Log10, LogN	Logarithmic functions
Sin, Cos, SinCos, Tan	Trigonometric functions
ArcSin, ArcCos, ArcTan, ArcTan2	Inverse trigonometric functions
Sinh, Cosh, SinCosh, Tanh	Hyperbolic functions
ArcSinh, ArcCosh, ArcTanh	Inverse hyperbolic functions
Cotan, Cot, Secant, Sec, Cosecant, Csc	Reciprocal trigonometric functions
CotH, SecH, CscH	Reciprocal hyperbolic functions
ArcCot, ArcSec, ArcCsc	Reciprocal inverse trigonometric functions
ArcCotH, ArcSecH, ArcCscH	Reciprocal inverse hyperbolic functions
RadToDeg, RadToGrad, RadToCycle	Convert from radians
DegToRad, DegToGrad, DegToCycle	Convert from degrees
GradToRad, GradToDeg, GradToCycle	Convert from grads
CycleToRad, CycleToDeg, CycleToGrad	Convert from cycles

Samples

A fun way to demonstrate high-precision math is by calculating the Mandelbrot fractal. As you zoom into the fractal, you need more and more precision. The Samples subdirectory contains a FireMonkey application that generates the Mandelbrot fractal at 4 levels of precision (Single, Double, DoubleDouble and QuadDouble).

The following image shows a tiny section of the fractal using a magnification of one quadrillion (10¹⁵, aka one billiard in Europe) and Double precision:

You can clearly see that the Double type doesn't provide enough precision at this magnification level. The DoubleDouble type offers more than enough precision though:

It's not until you reach a magnification level of 10³¹, that you need to switch to QuadDouble.

More Information

There is more to Neslib.MultiPrecision than described above. For more details you can look at the well-documented Neslib.MultiPrecision.pas source file. Additional usage samples can be found in the UnitTests subdirectory and the Mandelbrot sample application.

If you are interested in the technical details and algorithms used for these types, you can take a look at the QD.pdf file in the C subdirectory.

Building the C/C++ Library

As said, this library is build on top of the QD library. This is a C/C++ library that is linked into your Delphi executable using object files or static libraries. If you ever need or want to build these object files and static libraries yourself, then take a look at the readme.txt file in the C subdirectory for instructions.

License

Neslib.MultiPrecision is licensed under the Simplified BSD License. See License.txt for details.

The underlying QD library is licensed under the BSD-LBNL License. See the document BSD-LBNL-License.doc in the C subdirectory for details.