Algorithms for Triple-Word Arithmetic

Fabiano, Nicolas; Muller, Jean-Michel; Picot, Joris

doi:10.1109/tc.2019.2918451

Cited by 8 publications

(5 citation statements)

References 21 publications

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“…As seen in Section 5.1, we may rely on the available processor FP operations to build extended precision arithmetic, for instance using double-word algorithms, that roughly double the available precision. Triple-word (Fabiano, Muller and Picot 2019) and quad-word algorithms (Hida et al 2001) are available as well.…”

Section: Extended Precision Softwarementioning

confidence: 99%

Floating-point arithmetic

Boldo¹,

Jeannerod²,

Melquiond³

et al. 2023

Acta Numerica

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Floating-point numbers have an intuitive meaning when it comes to physics-based numerical computations, and they have thus become the most common way of approximating real numbers in computers. The IEEE-754 Standard has played a large part in making floating-point arithmetic ubiquitous today, by specifying its semantics in a strict yet useful way as early as 1985. In particular, floating-point operations should be performed as if their results were first computed with an infinite precision and then rounded to the target format. A consequence is that floating-point arithmetic satisfies the ‘standard model’ that is often used for analysing the accuracy of floating-point algorithms. But that is only scraping the surface, and floating-point arithmetic offers much more.In this survey we recall the history of floating-point arithmetic as well as its specification mandated by the IEEE-754 Standard. We also recall what properties it entails and what every programmer should know when designing a floating-point algorithm. We provide various basic blocks that can be implemented with floating-point arithmetic. In particular, one can actually compute the rounding error caused by some floating-point operations, which paves the way to designing more accurate algorithms. More generally, properties of floating-point arithmetic make it possible to extend the accuracy of computations beyond working precision.

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Section: Extended Precision Softwarementioning

confidence: 99%

Floating-point arithmetic

Boldo¹,

Jeannerod²,

Melquiond³

et al. 2023

Acta Numerica

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show abstract

“…As DD arithmetic is the simplest of the multi-component MPF arithmetic, its addition and multiplication needed for matrix multiplication, which directly adopt SIMDized EFT functions, can be easily implemented, as shown in Algorithm 1 and 2. One DD precision number is constructed with two binary64 as x[2] = (x[0], x [1]), and each array element is expressed as one _m256d variable. AVX2DDadd and AVX2DDmul simultaneously execute four calculations with x [2] and y [2], in which they have four DD numbers.…”

Section: Simdized Dd Addition and Multiplicationmentioning

confidence: 99%

“…Fabiano et.al. proposed the optimized triple word arithmetic [1]. In this arithmetic set, the renormalization of triple word number adopts VecSum and VSEB(k) (VecSum with Blanch).…”

Section: Simdized Td Addition and Multiplicationmentioning

confidence: 99%

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Acceleration of multiple precision matrix multiplication based on multi-component floating-point arithmetic using AVX2

Kouya

2021

Preprint

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In this paper, we report the results obtained from the acceleration of multi-binary64-type multiple precision matrix multiplication with AVX2. We target double-double (DD), triple-double (TD), and quad-double (QD) precision arithmetic designed by certain types of error-free transformation (EFT) arithmetic. Furthermore, we implement SIMDized EFT functions, which simultaneously compute with four bi-nary64 numbers on x86 64 computing environment, and by using help of them, we also develop SIMDized DD, TD, and QD additions and multiplications. In addition, AVX2 load/store functions were adopted to efficiently speed up reading and storing matrix elements from/to memory. Owing to these combined techniques, our implemented multiple precision matrix multiplications have been accelerated more than three times compared with non-accelerated ones. Our accelerated matrix multiplication modifies the performance of parallelization with OpenMP.

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“…A well-optimized triple-word arithmetic was proposed by Fabiano et al [ 1 ], which is one of the multi-component models of the multiple precision floating-point arithmetic, and is based on the error-free transformation techniques commonly used in DD and QD arithmetic. The triple-word arithmetic can be implemented by using the existing floating-point arithmetic such as IEEE single precision (binary32) or binary64.…”

Section: Triple-double Matrix Multiplicationmentioning

confidence: 99%

Performance Evaluation of Strassen Matrix Multiplication Supporting Triple-Double Precision Floating-Point Arithmetic

Kouya

2020

Computational Science and Its Applications – ICCSA 2020

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The Strassen matrix multiplication can be categorized into divide-and-conquer algorithms, and they are known as the most efficient algorithms. We previously implemented them supporting multiple precision floating-point arithmetic using MPFR and Bailey’s QD libraries and have shown their effectiveness in our papers and open-source codes. In preparation for a future release, we have introduced an optimized triple-word floating-point arithmetic proposed by Fabiano et al., and we found its utility in our implementation of multiple precision matrix multiplication. In this paper, we demonstrate the effectiveness of the Strassen triple-double precision matrix multiplication through performance evaluation compared to those based on QD and MPFR libraries.

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Algorithms for Triple-Word Arithmetic

Cited by 8 publications

References 21 publications

Floating-point arithmetic

Floating-point arithmetic

Acceleration of multiple precision matrix multiplication based on multi-component floating-point arithmetic using AVX2

Performance Evaluation of Strassen Matrix Multiplication Supporting Triple-Double Precision Floating-Point Arithmetic

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