Accuracy Versus Performance Tradeoffs in Floating-Point Computations

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oneAPI GPU Optimization Guide

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ID 771772

Date 7/18/2024

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Accuracy Versus Performance Tradeoffs in Floating-Point Computations

Programmers of floating-point applications typically aim for the following two objectives:

Accuracy: Produce results that are “close” to the result of the exact calculation.
Performance: Produce an application that runs as fast as possible.

The two objectives usually conflict. However, good programming practices and judicious use of compiler options allow you to control the tradeoffs.

For more information, see the article: Consistency of Floating-Point Results using the Intel Compiler or Why doesn’t my application always give the same answer?, by Dr. Martyn J. Corden and David Kreitzer (2018)

In this section, we present some mechanisms (compiler options and source-level changes) that allow the programmer to control the semantics (and hence the accuracy and performance) of floating-point computations on the host as well as on the device. We describe compiler options for OpenMP and SYCL programs, and describe source-level changes in SYCL.

OpenMP

In OpenMP, the following -fp-model options may be used to control the semantics of floating-point computations on the host as well as on the device.

-fp-model=precise

This option tells the compiler to strictly adhere to value-safe optimizations when implementing floating-point calculations. It disables optimizations (such as re-association, multiply by reciprocal, and zero folding) that can change the result of floating-point calculations. The increased accuracy that comes with -fp-model=precise may result in lower performance.

-fp-model=fast

This option is the default for both host and device compilations at -O2 and above. The option tells the compiler to use more aggressive optimizations when implementing floating-point calculations. These optimizations increase speed but may affect the accuracy or reproducibility of floating-point computations.

In C/C++, the -fp-model=fast option is equivalent to the -ffast-math option. With this option (at -O2 and above), all 7 fast-math flags (nnan, ninf, nsz, arcp, contract, afn, reassoc) are set by the C/C++ front-end. (See https://llvm.org/docs/LangRef.html#fast-math-flags for a description of the fast-math flags in LLVM.)

In Fortran, on the other hand, the language rules dictate that we cannot set the nnan flag (No NaNs) by default. So the -fp-model=fast option (at -O2 and above) only sets the other 6 fast-math flags (ninf, nsz, arcp, contract, afn, reassoc). To set all 7 fast-math flags in Fortran, use the -ffast-math option.

-Xopenmp-target-backend “-options -cl-fast-relaxed-math”

The -fp-model=fast (or -ffast-math) option does not enable native math instructions on the Intel GPU (Intel^® Data Center GPU Max Series). You need to compile with -Xopenmp-target-backend “-options -cl-fast-relaxed-math” to get native math instructions on the GPU. Native math instructions give even lower accuracy than what is allowed with -fp-model=fast.

-Xopenmp-target-backend “-options -cl-fast-relaxed-math” passes the -cl-fast-relaxed-math option to the backend in the compilation tool chain for the device. -cl-fast-relaxed-math relaxes the precision of commonly used math functions on the device. It offers a quick way to get performance gains for kernels with many math library function calls, as long as the accuracy requirements are satisfied by what is provided through native math instructions.

The -cl-fast-relaxed-math option affects the compilation of the entire program and does not permit fine control of the resulting numeric accuracy.

Note that Intel GPUs support native for single precision (float, real) only.

Notes (OpenMP):

When -fp-model is specified on the compilation command line (outside of the -fopenmp-targets=spir64=”…” set of options), the -fp-model option applies to both the host and the device compilations. For example, the following compilation command specifies -fp-model=precise for both the host and the device compilations:

  icx/icpx/ifx -O2 -fiopenmp -fopenmp-targets=spir64 -fp-model=precise

You can specify different ``-fp-model`` keywords for the host and
the device compilations as shown below.

To specify -fp-model=fast for the host compilation, and -fp-model=precise for the device compilation:

icx/icpx/ifx -O2 -fiopenmp -fopenmp-targets=spir64=”-fp-model=precise” -fp-model=fast

Or:

icx/icpx/ifx -O2 -fiopenmp -fopenmp-targets=spir64=”-fp-model=precise”
(No need to specify ``-fp-model=fast`` since it is the default at -O2 or higher.)

To specify -fp-model=precise for the host compilation, and -fp-model=fast for the device compilation:

icx/icpx/ifx -O3 -fiopenmp -fopenmp-targets=spir64=”-fp-model=fast” -fp-model=precise

To specify -fp-model=fast for the host compilation, and relaxed-math for the device compilation:

icx/icpx/ifx -O2 -fiopenmp -fopenmp-targets=spir64 -Xopenmp-target-backend “-options -cl-fast-relaxed-math” -fp-model=fast

Or:

icx/ifx -O2 -fiopenmp -fopenmp-targets=spir64 -Xopenmp-target-backend “-options -cl-fast-relaxed-math”
(No need to explicitly specify ``-fp-model=fast`` since it is the default at -O2 or higher.)

You can combine the fast, precise, and relaxed-math options with -fimf-precision=low (medium or high) option to fine-tune precision on the host side. The -fimf-precision option is not supported on the device side currently.

The following table shows a summary of how to set the various options (precise, fast-math, relaxed math) in OpenMP to (a) both the host and the device (second column); (b) to the host only (third column); and (c) to the device only (fourth column).

OpenMP - Summary of Options
Floating Point Semantics	Apply to Host and Device Compilations	Apply to Host Compilation Only	Apply to Device Compilation Only
Precise	`-fp-model=precise`	-fp-model=precise, and specify ``-fiopenmp -fopenmp-targets=spir64="-fp-model=fast"	`-fiopenmp -fopenmp-targets=spir64="-fp-model=precise"`
Fast-math	`-fp-model=fast` (default)	`-fp-model=fast`, and specify `-fiopenmp -fopenmp-targets=spir64="-fp-model= precise"`	`-fp-model=precise`, and specify `-fiopenmp -fopenmp-targets=spir64="-fp-model=fast"`
Relaxed-math (native instructions)	Applies to device only	Applies to device only	`-Xopenmp-target-backend "-options -cl-fast-relaxed-math"`

SYCL

In SYCL, as in OpenMP, the -fp-model=fast and -fp-model=precise options may be used for both host and device compilations.

In SYCL, -fp-model=fast option is equivalent to the -ffast-math option, and is the default for both host and device compilations at -O2 and above.

To specify relaxed-math for device compilation, use the compiler option -Xsycl-target-backend “-options -cl-fast-relaxed-math”. You need to compile with this option to get native math instructions on the GPU.

Note that SYCL (and Intel GPUs) support native for single precision (float, real) only.

Notes (SYCL):

When -fp-model is specified on the compilation command line (outside of any -Xsycl-target option), the -fp-model option applies to both the host and the device compilations. For example, the following compilation command specifies -fp-model=precise for both the host and the device compilations:

  icx/icpx/ifx -fsycl -fp-model=precise

You can specify different ``-fp-model`` keywords for the host and
device compilations as shown below.

To specify -fp-model=fast for the host compilation, and -fp-model=precise for the device compilation:

icx/icpx -fsycl -Xsycl-target-frontend "-fp-model=precise" -fp-model=fast

Or:

icx/icpx -fsycl -Xsycl-target-frontend "-fp-model=precise"
(No need to specify ``-fp-model=fast`` since it is the default at -O2 or higher.)

To specify -fp-model=precise for the host compilation, and -fp-model=fast for the device compilation:

icx/icpx -fsycl -Xsycl-target-frontend "-fp-model=fast" -fp-model=precise

To specify -fp-model=fast for the host compilation, and relaxed-math for the device compilation:

icx/icpx -fsycl -Xsycl-target-backend “-options -cl-fast-relaxed-math” -fp-model=fast

Or:

icx/icpx -fsycl -Xsycl-target-backend “-options -cl-fast-relaxed-math”
(No need to specify ``-fp-model=fast`` since it is the default at -O2 or higher.)

The following table shows a summary of how to set the various options (math, fast-math, relaxed math) in SYCL to (a) both the host and the device (second column); (b) to the host only (third column); and (c) to the device only (fourth column).

SYCL - Summary of Options
Floating Point Semantics	Apply to Host and Device Compilations	Apply to Host Compilation Only	Apply to Device Compilation Only
Precise	`-fp-model=precise`	`-fp-model=precise`, and specify `-Xsycl-target-frontend "-fp-mode=fast"`	`-Xsycl-target-frontend "-fp-model=precise"`
Fast-math	`-fp-model=fast` (default)	`-fp-model=fast`, and specify `-Xsycl-target-frontend "-fp-model=precise"`	`-fp-model=precise`, and specify `-Xsycl-target-frontend "-fp-model=fast"`
Relaxed-math (native instructions)	Applies to device only	Applies to device only	`-Xsycl-target-backend "-options -cl-fast-relaxed-math"`

Guidelines

In general, here are some guidelines for which options to use:

Do not specify inconsistent options. The result will be unpredictable.
The most commonly used option is -fp-model=fast for both host and device.
Use relaxed-math for best performance on the device.
Use -fp-model=precise for highest precision.

Example: The log Function

The math library (a component of a programming language’s standard library) contains functions (or subroutines) for the most common mathematical functions, such as exponential, logarithmic, power, and trigonometric functions.

Different implementations of the math library functions may not have the same accuracy or be rounded in the same way. The value returned by a math library function may vary between one compiler release and another, due to algorithmic and optimization changes.

The accuracy of a math library function can be controlled via compiler options or via the source code. We use the log (natural logarithm) math function as an example to illustrate this.

OpenMP / C++ (test_log_omp.cpp)

The following is an OpenMP C++ program that calls the std::log function on the device (from inside OpenMP target regions). The program includes the cmath header file which contains definitions for common math functions.

#include <iostream>
#include <assert.h>
#include <chrono>
#include <cmath>

#if FP_SIZE == 32
  typedef float FP_TYPE;
  static constexpr FP_TYPE VALIDATION_THRESHOLD = 1e-3;
#elif FP_SIZE == 64
  typedef double FP_TYPE;
  static constexpr FP_TYPE VALIDATION_THRESHOLD = 1e-6;
#endif

template<typename T>
void do_work (unsigned NELEMENTS, unsigned NREPETITIONS, T initial_value, T *res)
{
	#pragma omp target teams distribute parallel for map(present,alloc:res[0:NELEMENTS])
	for (unsigned j = 0; j < NELEMENTS; j++)
	{
		T tmp = initial_value;
		for (unsigned i = 0; i < NREPETITIONS; ++i)
			tmp += std::log(tmp);
		res[j] = tmp;
	}
}

int main (int argc, char *argv[])
{
	static constexpr unsigned NELEMENTS = 64*1024*1024;
	static constexpr unsigned NREPETITIONS = 1024;

	#pragma omp target
	{ }

	FP_TYPE initial_value = 2;
	FP_TYPE ref_res = initial_value;
	for (unsigned i = 0; i < NREPETITIONS; ++i)
		ref_res += std::log(ref_res);
	std::cout << "reference result = " << ref_res << std::endl;

	{
		FP_TYPE * std_res = new FP_TYPE[NELEMENTS];
		assert (std_res != nullptr);

		std::chrono::duration<float, std::micro> elapsed;
		#pragma omp target data map(std_res[0:NELEMENTS])
		{
			auto tbegin = std::chrono::system_clock::now();
			do_work<FP_TYPE> (NELEMENTS, NREPETITIONS, initial_value, std_res);
			auto tend = std::chrono::system_clock::now();
			elapsed = tend - tbegin;
		}
		std::cout << "std::log result[0] = " << std_res[0] << std::endl;

		bool allequal = true;
		for (auto i = 1; i < NELEMENTS; ++i)
			allequal = allequal and std_res[0] == std_res[i];
		if (allequal)
		{
			if (std::abs(ref_res - std_res[0])/std::abs(ref_res) < std::abs(VALIDATION_THRESHOLD))
				std::cout << "std::log validates. Total execution time is " << elapsed.count() << " us." << std::endl;
			else
				std::cout << "std::log does not validate (ref=" << ref_res << " std_res=" << std_res[0] << " mix=" << std::abs(ref_res - std_res[0])/std::abs(ref_res) << ")" << std::endl;
		}
		else
			std::cout << "std::log does not validate, results are not equal." << std::endl;

		delete [] std_res;
	}

	return 0;
}

Sample compilation and run commands for test_log_omp.cpp:

icpx -O2 -fiopenmp -fopenmp-targets=spir64 test_log_omp.cpp \
     -DREAL_ELEMENT -DFP_SIZE=64 -fp-model=fast -fopenmp-version=51

OMP_TARGET_OFFLOAD=MANDATORY ./a.out

OpenMP / Fortran (test_log_omp_f_mod.f90)

The following is a OpenMP Fortran module that calls the Fortran intrinsic math function, log, on the device (from inside OpenMP target regions):

MODULE test

    USE ISO_C_BINDING

CONTAINS

  SUBROUTINE log_real_sp (nelements, nrepetitions, initial_value, res) bind(C,NAME='log_real_sp')
    IMPLICIT NONE
    INTEGER(KIND=C_INT), VALUE :: nelements, nrepetitions
    REAL(C_FLOAT), VALUE :: initial_value
    REAL(C_FLOAT) :: res(0:nelements-1), tmp
    INTEGER :: i, j

    !$OMP TARGET TEAMS DISTRIBUTE PARALLEL DO PRIVATE(tmp)
    DO j = 0, nelements-1
      tmp = initial_value
      DO i = 0, nrepetitions-1
        tmp = tmp + log(tmp)
      END DO
      res(j) = tmp
    END DO
    RETURN
  END SUBROUTINE log_real_sp

  SUBROUTINE log_real_dp (nelements, nrepetitions, initial_value, res) bind(C,NAME='log_real_dp')
    IMPLICIT NONE
    INTEGER(KIND=C_INT), VALUE :: nelements, nrepetitions
    REAL(C_DOUBLE), VALUE :: initial_value
    REAL(C_DOUBLE) :: res(0:nelements-1), tmp
    INTEGER :: i, j

    !$OMP TARGET TEAMS DISTRIBUTE PARALLEL DO PRIVATE(tmp)
    DO j = 0, nelements-1
      tmp = initial_value
      DO i = 0, nrepetitions-1
        tmp = tmp + log(tmp)
      END DO
      res(j) = tmp
    END DO
    RETURN
  END SUBROUTINE log_real_dp

END MODULE test

Sample compilation command for test_log_omp_f.f90:

ifx -c -O2 -fiopenmp -fopenmp-targets=spir64 test_log_omp_f_mod.f90

SYCL (test_log_sycl.cpp)

In SYCL, you can control floating point semantics at the source-level by choosing which math function to call. For example, the SYCL program below calls the following three different versions of the log function:

std::log : Refers to the log function in the C++ standard library. The particular implementation chosen will be according to what the compiler options (-fp-model and -cl-fast-relaxed-math) prescribe. For example, to get the implementation that uses native math instructions, you need to compile with the -cl-fast-relaxed-math option.
sycl::log : Refers to the log function in the sycl namespace that is provided by SYCL. This function may use native instructions, even when the -cl-fast-relaxed-math option is not specified. Precision is spelled out by the SYCL specification.
sycl::native::log : Refers to the native log function in the sycl namespace that is provided by SYCL. This function uses native math instructions, and the -cl-fast-relaxed-math option is not needed. Note that SYCL (and Intel GPUs) support native for single precision (float, real) only. Precision is spelled out by the SYCL specification.

#include <CL/sycl.hpp>
#include <iostream>
#include <assert.h>
#include <chrono>
#include <cmath>

#if FP_SIZE == 32
  typedef float FP_TYPE;
  static constexpr FP_TYPE VALIDATION_THRESHOLD = 1e-3;
#elif FP_SIZE == 64
  typedef double FP_TYPE;
  static constexpr FP_TYPE VALIDATION_THRESHOLD = 1e-6;
#endif

template<typename T>
void do_work_std (sycl::queue &q, unsigned NELEMENTS, unsigned NREPETITIONS, T initial_value, T *res)
{
	q.submit([&](sycl::handler& h) {
		h.parallel_for(NELEMENTS, [=] (auto j)
		{
			FP_TYPE tmp = initial_value;
			for (unsigned i = 0; i < NREPETITIONS; ++i)
				tmp += std::log(tmp);
			res[j] = tmp;
			});
	}).wait();
}

template<typename T>
void do_work_sycl (sycl::queue &q, unsigned NELEMENTS, unsigned NREPETITIONS, T initial_value, T *res)
{
	q.submit([&](sycl::handler& h) {
		h.parallel_for(NELEMENTS, [=] (auto j)
		{
			FP_TYPE tmp = initial_value;
			for (unsigned i = 0; i < NREPETITIONS; ++i)
				tmp += sycl::log(tmp);
			res[j] = tmp;
			});
	}).wait();
}
# if FP_SIZE == 32
template<typename T>
void do_work_sycl_native (sycl::queue &q, unsigned NELEMENTS, unsigned NREPETITIONS, T initial_value, T *res)
{
	q.submit([&](sycl::handler& h) {
		h.parallel_for(NELEMENTS, [=] (auto j)
		{
			FP_TYPE tmp = initial_value;
			for (unsigned i = 0; i < NREPETITIONS; ++i)
				tmp += sycl::native::log(tmp);
			res[j] = tmp;
			});
	}).wait();
}
# endif

int main (int argc, char *argv[])
{
	static constexpr unsigned NELEMENTS = 64*1024*1024;
	static constexpr unsigned NREPETITIONS = 1024;

	sycl::device d (sycl::gpu_selector_v);
	sycl::queue q (d);

	q.submit([&](sycl::handler& h) {
		h.single_task ([=]() { });
	}).wait();

	FP_TYPE initial_value = 2;
	FP_TYPE ref_res = initial_value;
	for (unsigned i = 0; i < NREPETITIONS; ++i)
		ref_res += std::log(ref_res);
	std::cout << "reference result = " << ref_res << std::endl;

	{
		FP_TYPE * std_res = new FP_TYPE[NELEMENTS];
		assert (std_res != nullptr);

		std::chrono::duration<float, std::micro> elapsed;

		{
			auto * res = sycl::malloc_device<FP_TYPE>(NELEMENTS, q);
			auto tbegin = std::chrono::system_clock::now();
			do_work_std<FP_TYPE>(q, NELEMENTS, NREPETITIONS, initial_value, res);
			auto tend = std::chrono::system_clock::now();
			elapsed = tend - tbegin;
			q.memcpy (std_res, res, NELEMENTS*sizeof(FP_TYPE)).wait();
			sycl::free (res, q);
		}
		std::cout << "std::log result[0] = " << std_res[0] << std::endl;

		bool allequal = true;
		for (auto i = 1; i < NELEMENTS; ++i)
			allequal = allequal and std_res[0] == std_res[i];
		if (allequal)
		{
			if (std::abs(ref_res - std_res[0])/std::abs(ref_res) < std::abs(VALIDATION_THRESHOLD))
				std::cout << "std::log validates. Total execution time is " << elapsed.count() << " us." << std::endl;
			else
				std::cout << "std::log does not validate (ref=" << ref_res << " std_res=" << std_res[0] << " mix=" << std::abs(ref_res - std_res[0])/std::abs(ref_res) << ")" << std::endl;
		}
		else
			std::cout << "std::log does not validate, results are not equal." << std::endl;

		delete [] std_res;
	}

	{
		FP_TYPE * sycl_res = new FP_TYPE[NELEMENTS];
		assert (sycl_res != nullptr);

		std::chrono::duration<float, std::micro> elapsed;

		{
			auto * res = sycl::malloc_device<FP_TYPE>(NELEMENTS, q);
			auto tbegin = std::chrono::system_clock::now();
			do_work_sycl<FP_TYPE>(q, NELEMENTS, NREPETITIONS, initial_value, res);
			auto tend = std::chrono::system_clock::now();
			elapsed = tend - tbegin;
			q.memcpy (sycl_res, res, NELEMENTS*sizeof(FP_TYPE)).wait();
			sycl::free (res, q);
		}
		std::cout << "sycl::log result[0] = " << sycl_res[0] << std::endl;

		bool allequal = true;
		for (auto i = 1; i < NELEMENTS; ++i)
			allequal = allequal and sycl_res[0] == sycl_res[i];
		if (allequal)
		{
			if (std::abs(ref_res - sycl_res[0])/std::abs(ref_res) < std::abs(VALIDATION_THRESHOLD))
				std::cout << "sycl::log validates. Total execution time is " << elapsed.count() << " us." << std::endl;
			else
				std::cout << "sycl::log does not validate (ref=" << ref_res << " sycl_res=" << sycl_res[0] << " mix=" << std::abs(ref_res - sycl_res[0])/std::abs(ref_res) << ")" << std::endl;
		}
		else
			std::cout << "sycl::log does not validate, results are not equal." << std::endl;

		delete [] sycl_res;
	}
# if FP_SIZE == 32
	{
		FP_TYPE * sycl_native_res = new FP_TYPE[NELEMENTS];
		assert (sycl_native_res != nullptr);

		std::chrono::duration<float, std::micro> elapsed;

		{
			auto * res = sycl::malloc_device<FP_TYPE>(NELEMENTS, q);
			auto tbegin = std::chrono::system_clock::now();
			do_work_sycl_native<FP_TYPE>(q, NELEMENTS, NREPETITIONS, initial_value, res);
			auto tend = std::chrono::system_clock::now();
			elapsed = tend - tbegin;
			q.memcpy (sycl_native_res, res, NELEMENTS*sizeof(FP_TYPE)).wait();
			sycl::free (res, q);
		}
		std::cout << "sycl::native::log result[0] = " << sycl_native_res[0] << std::endl;

		bool allequal = true;
		for (auto i = 1; i < NELEMENTS; ++i)
			allequal = allequal and sycl_native_res[0] == sycl_native_res[i];
		if (allequal)
		{
			if (std::abs(ref_res - sycl_native_res[0])/std::abs(ref_res) < std::abs(VALIDATION_THRESHOLD))
				std::cout << "sycl::native::log validates. Total execution time is " << elapsed.count() << " us." << std::endl;
			else
				std::cout << "sycl::native::log does not validate (ref=" << ref_res << " sycl_native_res=" << sycl_native_res[0] << " mix=" << std::abs(ref_res - sycl_native_res[0])/std::abs(ref_res) << ")" << std::endl;
		}
		else
			std::cout << "sycl::native::log does not validate, results are not equal." << std::endl;

		delete [] sycl_native_res;
	}
# endif // FP_SIZE == 32

	return 0;
}

Sample compilation and run commands for test_log_sycl.cpp:

icpx -fsycl -O2 test_log_sycl.cpp -DREAL_ELEMENT -DFP_SIZE=64 -fp-model=fast

OMP_TARGET_OFFLOAD=MANDATORY ./a.out

Performance Experiments

We present performance results when running the different programs (OpenMP C++, OpenMP Fortran and SYCL) that call the log function. On the particular Intel^® Data Center GPU Max Series used (1-stack only), the performance of the log function, in single-precision was as follows.

Performance of log - Default Precision (Fast-math)
Version	Time (sec)
OpenMP/C++ (`std::log`)	93,118
OpenMP/Fortran (`log`)	94,342
SYCL (`std::log`)	31,835
SYCL (`sycl::log`)	31,644
SYCL (`sycl::native::log`)	31,684

Performance of log - Fast-math
Version	Time (sec)
OpenMP/C++ (`std::log`)	93,181
OpenMP/Fortran (`log`)	94,467
SYCL (`std::log`)	31,657
SYCL (`sycl::log`)	32,064
SYCL (`sycl::native::log`)	31,452

Performance of log - Precise
Version	Time (sec)
OpenMP/C++ (`std::log`)	92,971
OpenMP/Fortran (`log`)	94,444
SYCL (`std::log`)	94,592
SYCL (`sycl::log`)	94,852
SYCL (`sycl::native::log`)	40,778

Performance of log - Relaxed-math
Version	Time (sec)
OpenMP/C++ (`std::log`)	35,251
OpenMP/Fortran (`log`)	35,787
SYCL (`std::log`)	31,314
SYCL (`sycl::log`)	32,077
SYCL (`sycl::native::log`)	32,141

Observations:

In OpenMP (C and Fortran): std::log follows what the compiler options (-fp-model, -cl-relaxed-math) prescribe. The -cl-relaxed-math option is needed to use native instructions for std::log.
In SYCL: sycl::log may use native instructions, even with just -fp-model=fast.
In SYCL: sycl::native::log will always use native machine instructions. The -cl-relaxed-math option is not needed.
In OpenMP and SYCL: When the -cl-relaxed-math option is specified, native machine instructions will be used for the log function on the device.
-fp-model=precise produces more accurate results, but the performance will be lower.

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