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Fast Math Library for Delphi
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erikvanbilsen Added Win64 support for Delphi 10.3 Rio.
Works now with Rio for Win32, Win64, iOS32, iOS64, macOS32 and Android32.
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README.md

FastMath - Fast Math Library for Delphi

FastMath is a Delphi math library that is optimized for fast performance (sometimes at the cost of not performing error checking or losing a little accuracy). It uses hand-optimized assembly code to achieve much better performance then the equivalent functions provided by the Delphi RTL.

This makes FastMath ideal for high-performance math-intensive applications such as multi-media applications and games. For even better performance, the library provides a variety of "approximate" functions (which all start with a Fast-prefix). These can be very fast, but you will lose some (sometimes surprisingly little) accuracy. For gaming and animation, this loss in accuracy is usually perfectly acceptable and outweighed by the increase in speed. Don't use them for scientific calculations though...

You may want to call DisableFloatingPointExceptions at application startup to suppress any floating-point exceptions. Instead, it will return extreme values (like Nan or Infinity) when an operation cannot be performed. If you use FastMath in multiple threads, you should call DisableFloatingPointExceptions in the Execute block of those threads.

Table of Contents

Superior Performance

Most operations can be performed on both singular values (scalars) as well as vectors (consisting of 2, 3 or 4 values). SIMD optimized assembly code is used to calculate multiple outputs at the same time. For example, adding two 4-value vectors together is almost as fast as adding two single values together, resulting in a 4-fold speed increase. Many functions are written in such a way that the performance is even better.

Here are some examples of speed up factors you can expect on different platforms:

RTL FastMath x86-32 x86-64 Arm32 Arm64
TVector3D + TVector3D TVector4 + TVector4 1.2x 1.6x 2.8x 2.5x
Single * TVector3D Single * TVector4 2.2x 2.1x 5.6x 3.7x
TVector3D.Length TVector4.Length 3.0x 5.6x 19.9x 17.1x
TVector3D.Normalize TVector4.Normalize 4.1x 5.1x 7.4x 11.7x
TVector3D * TMatrix3D TVector4 * TMatrix4 1.3x 4.0x 6.5x 4.2x
TMatrix3D * TMatrix3D TMatrix4 * TMatrix4 2.2x 7.2x 5.4x 8.0x
TMatrix3D.Inverse TMatrix4.Inverse 9.8x 9.2x 8.0x 9.8x
Sin(Single) (x4) FastSin(TVector4) 14.8x 7.7x 42.6x 40.1x
SinCos(Single) (x4) FastSinCos(TVector4) 19.1x 9.0x 67.9x 93.3x
Exp2(Single) (x4) FastExp2(TVector4) 22.4x 32.7x 275.0x 302.4x

As you can see, some very common (3D) operations like matrix multiplication and inversion can be almost 10 times faster than their corresponding RTL versions. In addition, FastMath includes a number of Fast* approximation functions that sacrifice a little accuracy for an enormous speed increase. For example, using FastSinCos to calculate 4 sine and cosine functions in parallel can be up to 90 times faster than calling the RTL SinCos function 4 times, while still providing excellent accuracy for angles up to +/4000 radians (or +/- 230,000 degrees).

On 32-bit and 64-bit desktop platforms (Windows and OS X), this performance is achieved by using the SSE2 instruction set. This means that the computer must support SSE2. However, since SSE2 was introduced back in 2001, the vast majority of computers in use today will support it. All 64-bit desktop computers have SSE2 support by default. However, you can always compile this library with the FM_NOSIMD define to disable SIMD optimization and use plain Pascal versions. This can also be useful to compare the speed of the Pascal versions with the SIMD optimized versions.

On 32-bit mobile platforms (iOS and Android), the NEON instruction set is used for SIMD optimization. This means that your device needs to support NEON. But since Delphi already requires this, this poses no further restrictions.

On 64-bit mobile platforms (iOS), the Arm64/AArch64 SIMD instruction set is used.

There is no hardware accelerated support for the iOS simulator (it will use Pascal versions for all calculations).

Architecture and Design Decisions

FastMath operations on single-precision floating-point values only. Double-precision floating-point arithmetic is (currently) unsupported.

Most functions operate on single values (of type Single) and 2-, 3- and 4-dimensional vectors (of types TVector2, TVector3 and TVector4 respectively). Vectors are not only used to represent points or directions in space, but can also be regarded as arrays of 2, 3 or 4 values that can be used to perform calculations in parallel. In addition to floating-point vectors, there are also vectors that operator on integer values (TIVector2, TIVector3 and TIVector4).

There is also support for 2x2, 3x3 and 4x4 matrices (called TMatrix2, TMatrix3 and TMatrix4). By default, matrices are stored in row-major order, like those in the RTL's System.Math.Vectors unit. However, you can change this layout with the FM_COLUMN_MAJOR define. This will store matrices in column-major order instead, which is useful for OpenGL applications (which work best with this layout). In addition, this define will also clip the depth of camera matrices to -1..1 instead of the default 0..1. Again, this is more in line with the default for OpenGL applications.

For representing rotations in 3D space, there is also a TQuaternion, which is similar to the RTL's TQuaternion3D type.

The operation of the library is somewhat inspired by shader languages (such as GLSL and HLSL). In those languages you can also treat single values and vectors similarly. For example, you can use the Sin function to calculate a single sine value, but you can also use it with a TVector4 type to calculate 4 sine values in one call. When combined with the approximate Fast* functions, this can result in an enormous performance boost, as shown earlier.

Overloaded Operators

All vector and matrix types support overloaded operators which allow you to negate, add, subtract, multiply and divide scalars, vectors and matrices.

There are also overloaded operators that compare vectors and matrices for equality. These operators check for exact matches (like Delphi's = operator). They don't allow for very small variations (like Delphi's SameValue functions).

The arithmetic operators +, -, * and / usually work component-wise when applied to vectors. For example if A and B are of type TVector4, then C := A * B will set C to (A.X * B.X, A.Y * B.Y, A.Z * B.Z, A.W * B.W). It will not perform a dot or cross product (you can use the Dot and Cross functions to compute those).

For matrices, the + and - operators also operate component-wise. However, when multiplying (or dividing) matrices with vectors or other matrices, then the usual linear algebraic multiplication (or division) is used. For example:

  • M := M1 * M2 performs a linear algebraic matrix multiplication
  • V := M1 * V1 performs a matrix * row vector linear algebraic multiplication
  • V := V1 * M1 performs a column vector * matrix linear algebraic multiplication

To multiply matrices component-wise, you can use the CompMult method.

Interoperability with the Delphi RTL

FastMath provides its own vector and matrix types for superior performance. Most of them are equivalent in functionality and data storage to the Delphi RTL types. You can typecast between them or implicitly convert from the FastMath type to the RTL type or vice versa (eg. MyVector2 := MyPointF). The following table shows the mapping:

Purpose FastMath Delphi RTL
2D point/vector TVector2 TPointF
3D point/vector TVector3 TPoint3D
4D point/vector TVector4 TVector3D
2x2 matrix TMatrix2 N/A
3x3 matrix TMatrix3 TMatrix
4x4 matrix TMatrix4 TMatrix3D
quaternion TQuaternion TQuaternion3D

Documentation

Documentation can be found in the HTML Help file FastMath.chm in the Doc directory.

Alternatively, you can read the documentation on-line.

Directory Organization

The FastMath repository hold the following directories:

  • Doc: documentation in HtmlHelp format. Also contains a spreadsheet (Benchmarks.xlsx) with results of performance tests on my devices (a Core i7 desktop and iPad3).
  • DocSource: contains batch files for generating the documentation. You need PasDocEx to generate the documentation yourself if you want to.
  • FastMath: contains the main Neslib.FastMath unit as well as various include files with processor specific optimizations, and static libraries with Arm optimized versions for iOS and Android.
    • Arm: Arm specific source code and scripts.
      • Arm32: contains the assembly source code for Arm Neon optimized functions.
      • Arm64: contains the assembly source code for Arm64 optimized functions.
      • fastmath-android: contains a batch file and helper files to build the static library for Android using the Android NDK.
      • fastmath-ios: contains a macOS shell script to build a universal static library for iOS.
  • Tests: contains a FireMonkey application that runs unit tests and performance tests.

License

FastMath is licensed under the Simplified BSD License. Some of its functions are based on other people's code licensed under the MIT, New BSD and ZLib licenses. Those licenses are as permissive as the Simplified BSD License used for the entire project.

See License.txt for details.

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