The vdt mathematical library

vectorised math

• A collection of fast and inline implementations of mathematical functions.
• The functions can be used in autovectorised loops.
• Double and single precision implementations are available.
• No overhead present, no intrinsics used.
• A scalar (T(T)) and array signature (void(const unsigned int,T*,T*)) are provided.

Born and developed at CERN, it is used, among the others, by LHC experiments and the Geant4 simulation toolkit.

Much of the VDT code is inspired by the well known Cephes mathematical library.

How to get, compile and install it

The vdt functions are inline and contained in header files: they are ready to be used without compilation of an external library. In any case, there is the possibility to compile a shared library containing the array signatures void(const unsigned int,T*,T*). The makesystem chosen for vdt is CMake.

export INSTALLDIR=/path/to/mydir
git clone https://github.com/dpiparo/vdt.git
cd vdt
cmake -DCMAKE_INSTALL_PREFIX=$INSTALLDIR .
make
make install

How to use it

Good examples of vdt functions usage are located in the progs and progs/units directories.

The vdt functions

All vdt functions live in the vdt namespace. Their names are structured as follows:

vdt::fast_<function_name>[f][v]

Where:

• The function name is one of the list in the table below.
• The f letter stands for the single precision function (float).
• The v letter identifies the array function. You may wonder, why prepending fast_? This is done to allow the user to decide where a fast and approximate implementation of the function is to be used. It is not always obvious where the accuracy can be reduced: all the flexibility is needed.

These are the available functions:

Function	Scalar double precision	Scalar single precision	Array double precision	Array single precision
exponential	fast_exp	fast_expf	fast_expv	fast_expfv
sine	fast_sin	fast_sinf	fast_sinv	fast_sinfv
cosine	fast_cos	fast_cosf	fast_cosv	fast_cosfv
tangent	fast_tan	fast_tanf	fast_tanv	fast_tanfv
hyperbolic tangent	fast_tanh	fast_tanhf	fast_tanhv	fast_tanhfv
logarithm	fast_log	fast_logf	fast_logv	fast_logfv
arcsine	fast_asin	fast_asinf	fast_asinv	fast_asinfv
arccosine	fast_acos	fast_acosf	fast_acosv	fast_acosfv
arctangent	fast_atan	fast_atanf	fast_atanv	fast_atanfv
inverse square root (1/sqrt)	fast_isqrt	fast_isqrtf	fast_isqrtv	fast_isqrtfv

Other Cmake options

Other options for Cmake are available to steer the creation of the makefile:

• Enable AVX extensions -D AVX=1
• Enable NEON extensions on ARM -D NEON=1
• Benchmarking tools and unit tests (requires C++11 support by the compiler) -D DIAG=1
• Build static library -D BUILD_SHARED_LIBS=0
• Prepare the library to be pre-loaded in order to replace the calls to the default math lib at runtime -D PRELOAD=1

Supported Compilers

The vdt functions can be used with every compiler (icc and gcc were tested). To compile the benchmarking tools gcc4.7 (icc12) is at least needed because of the support of c++11. To vectorise the functions gcc4.7 (icc12) is at least needed.

Benchmarks

This section is for experts who want to study the details of the functions provided and/or to compare them to other implementations. vdt comes with a complete benchmark suite both for accuracy and speed measurements. To measure the speed of the functions, you should use the ''vdtPerfBenchmark''. To dump on disk the ascii files summarising the accuracy of the functions, you should use ''vdtArithmBenchmark'', while the tool to compare them is ''vdtArithmComparison''. In order to produce the plots of the different bits as a function of input, the script to be used is ''diffhisto.py'' (which depends on ROOT to produce plots).

Example of Performance

Double precision, Intel® Core™ i7-3930K CPU @ 3.20GHz running Scientific Linux 6. Operative interval of the input: [-5000,5000] ((0,5000] for isqrt and [-1,1] for Asin and Acos). Time in ns per call.

Function	libm	VDT	VDT SSE	VDT AVX
Exp	16.7	6.1	3.8	2.9
Log	34.9	12.5	5.7	4.2
Sin	33.7	16.2	6	5.7
Cos	34.4	13.4	5.4	5.1
Tan	46.6	12.5	6.3	5.6
Asin	23	10.3	8.6	8.1
Acos	23.7	11	8.2	8.1
Atan	19.7	11	8.3	8.3
Isqrt	9.3	6.7	3	2.1

Accuracy

Accuracy measured in terms of least significant bit. Average difference with respect to libm.

Function	AVG vdt
Acos	0.39
Asin	0.32
Atan	0.33
Cos	0.25
Exp	0.14
Isqrt	0.45
Log	0.42
Sin	0.25
Tan	0.35

Related Documents:

• A full characterisation of the accuracies please refer to this presentation.

Reference

If you want to cite vdt, please use your reference: D. Piparo, V.Innocente and T.Hauth 2014 J. Phys.: Conf. Ser. 513 052027 "Speeding up HEP experiment software with a library of fast and auto-vectorisable mathematical functions"

Mailing List

The VDT projects has a mailing list: VDTlibrary-talk at cern ch, linked to an e-group with the same name. The Infrastructure used is the one provided by CERN IT. If you don't have a CERN account, you can obtain an external one here. Alternatively, feel free to contact Danilo Piparo (danilo_dot_piparo_at_cern_dot_ch).

Licence

The VDT mathematical library is licenced under the LGPL3 licence