CR-LIBM: a correctly rounded elementary function library

Daramy, Catherine; Defour, David; Dinechin, Florent de; Muller, Jean-Michel

doi:10.1117/12.505591

Cited by 29 publications

(39 citation statements)

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“…Instead, we formalized theorems necessary for validating results produced by FPTaylor. The validity of results is checked against specifications of floating-point rounding operations given by (1) and (12). We chose HOL Light as the tool for our formalization because it is the only proof assistant for which there exists a tool for formal verification of nonlinear inequalities (including inequalities with transcendental functions) [53].…”

Section: Formal Verification Of Fptaylor Results In Hol Lightmentioning

confidence: 99%

“…As mentioned above, in our current formalization we define such a rounding operator as any operator satisfying (1) and (12). We also implemented a comprehensive formalization of floating-point arithmetic in HOL Light (our floating-point formalization is available in the HOL Light distribution).…”

Section: Formal Verification Of Fptaylor Results In Hol Lightmentioning

confidence: 99%

“…So we can still use (3) to model transcendental functions but we need to increase values of and δ appropriately. There exist software libraries that exactly compute rounded values of transcendental functions [12,17]. For such libraries, (3) can be applied without any changes.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Solovyev

Jacobsen

Rakamarić

et al. 2015

FM 2015: Formal Methods

123

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Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide overestimates. Also, there are no available rigorous approaches that handle transcendental functions. We have developed a new approach called Symbolic Taylor Expansions that avoids this difficulty, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. We release FPTaylor along with our benchmarks for evaluation. Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor ExpansionsAlexey Solovyev, Charles Jacobsen, Zvonimir Rakamarić, and Ganesh Gopalakrishnan School of Computing, University of Utah, Salt Lake City, UT 84112, USA {monad,charlesj,zvonimir,ganesh}@cs.utah.edu Abstract. Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide overestimates. Also, there are no available rigorous approaches that handle transcendental functions. We have developed a new approach called Symbolic Taylor Expansions that avoids this difficulty, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. We release FPTaylor along with our benchmarks for evaluation.

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Section: Formal Verification Of Fptaylor Results In Hol Lightmentioning

confidence: 99%

Section: Formal Verification Of Fptaylor Results In Hol Lightmentioning

confidence: 99%

See 1 more Smart Citation

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Solovyev

Jacobsen

Rakamarić

et al. 2015

FM 2015: Formal Methods

123

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“…• Gappa (Melquiond): a tool that computes error bounds on FP calculations, and generates formal proofs of these bounds [9]; These tools have been heavily used for building our CR-LIBM library of correctly-rounded elementary functions [4], [6], [7].…”

Section: Various Tools Designed By the Arenaire Teammentioning

confidence: 99%

Generating function approximations at compile time

Muller

2006

2006 Fortieth Asilomar Conference on Signals, Systems and Computers

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Abstract-Usually, the mathematical functions used in a numerical programs are decomposed into elementary functions (such as sine, cosine, exponential, logarithm...), and for each of these functions, we use a program from a library. This may have some drawbacks: first in frequent cases, it is a compound function (e.g. log(1 + exp(−x))) that is needed, so that directly building a polynomial or rational approximation for that function (instead of decomposing it) would result in a faster and/or more accurate calculation. Also, at compile-time, we might have some information (e.g., on the range of the input value) that could help to simplify the program. We investigate the possibility of directly building accurate approximations at compile-time.

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“…Dentro de las aproximaciones por software podremos utilizar las instrucciones en coma flotante disponibles por los microprocesadores de propósito general [10], [11] o bien utilizar una serie de librerías matemáticas sobre las cuales se han implementado mediante lenguaje C una serie de funciones para el cálculo matemático de dichas funciones elementales. Dentro de las librerías más importantes tenemos CRLIBM [12], LIBULTIM, etc. Estas librerías tienen la desventaja de estar orientadas para su utilización en microprocesadores de propósito general y en muchos casos suelen ser demasiado lentas para poder ser utilizadas en aplicaciones en tiempo real, ahora bien, permiten generar resultados con una precisión muy elevada.…”

Section: Introductionunclassified

Implementaciones de Funciones Elementales en Dispositivos FPGA

Mazon¹

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In this thesis we have designed several hardware architectures of some typical digital subsystems for high performance communications systems, aiming at optimized implementations for these systems. The work has focused on two areas: the approximation of elementary functions, specifically the logarithm and arctangent, and the design of an emulator of additive Gaussian noise channel. The architectures have been designed at all times with the objective to achieve an efficient implementation in Field Programmable Gate Arrays devices (FPGAs), due to its increasing use in digital high performance communications systems. For the approximation of logarithm we have proposed two different architectures, one based on the use of multipartite table methods and the other based on the Mitchell's approximation with two correction stages: a piecewise linear interpolation with power-of-two slopes and truncated mantissa, and a LUT-based correction stage that corrects the piecewise interpolation error. A first architecture for the approximation of atan(y/x) is based on the computation of the reciprocal of x and the arctangent approximation, using multipartite Look-up tables (LUTs). This proposal reduces the power consumption compared to the best techniques described in the literature, such based on CORDIC. A second strategy for the approximation of atan(y/x) is based on logarithmic transformations, which transforms the calculation of the division of two inputs in a simple subtraction and requires the computation of atan(2 w ). This second strategy has resulted in two architectures, a first in which both the logarithm and atan(2 w ) have been implemented using multipartite LUTs, also combined with the use of non-uniform segmentation techniques in the calculation of atan(2 w ), and a second architecture using a piecewise linear interpolation with power-of-two slopes and correction tables. The results obtained with this strategy improve the first architecture discussed. Two architectures for the approximation of arctangent and one for the logarithm have resulted in three publications in internationals journals. We also have proposed several architectures for a Gaussian white noise generator. The designs are based on the Inversion method, specifically approximating the inverse of cumulative distributions functions by polynomial interpolation and non-uniform segmentation. These architectures offer its output a standard deviation of ±13.1σ and 13 fractional bits, higher values than practically all hardware generators present in the literature, employing, in comparison, fewer FPGA resources. Compared to Gaussian channel implementations based on the inversion method presented by other authors, our architectures achieve a significant reduction in area partially or completely eliminating the barrel-shifter. The results for Gaussian channel emulator have been sent to an international journal, being currently under review process.

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CR-LIBM: a correctly rounded elementary function library

Cited by 29 publications

References 0 publications

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Generating function approximations at compile time

Implementaciones de Funciones Elementales en Dispositivos FPGA

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