Computing Correctly Rounded Integer Powers in Floating-Point Arithmetic

23 pagesWe introduce several algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a fused multiply-add (fma) instruction is available. We aim at always obtaining correctly-rounded results in round-to-nearest mode, that is, our algorithms return the floating-point number that is nearest the exact value

Computing Integer Powers in Floating-Point Arithmetic

We introduce two algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a fused multiply-add (fma) instruction is available. We show that our log-time algorithm always produce faithfully-rounded results, discuss the possibility of getting correctly rounded results, and show that results correctly rounded in double precision can be obtained if extended-precision is available with the possibility to round into double precision (with a single rounding).Comment: Laboratoire LIP : CNRS/ENS Lyon/INRIA/Universit\'e Lyon

Parallel Algorithms for Summing Floating-Point Numbers

The problem of exactly summing n floating-point numbers is a fundamental problem that has many applications in large-scale simulations and computational geometry. Unfortunately, due to the round-off error in standard floating-point operations, this problem becomes very challenging. Moreover, all existing solutions rely on sequential algorithms which cannot scale to the huge datasets that need to be processed. In this paper, we provide several efficient parallel algorithms for summing n floating point numbers, so as to produce a faithfully rounded floating-point representation of the sum. We present algorithms in PRAM, external-memory, and MapReduce models, and we also provide an experimental analysis of our MapReduce algorithms, due to their simplicity and practical efficiency.Comment: Conference version appears in SPAA 201

Efficient implementation of the Hardy-Ramanujan-Rademacher formula

We describe how the Hardy-Ramanujan-Rademacher formula can be implemented to allow the partition function $p(n)$ to be computed with softly optimal complexity $O(n^{1/2+o(1)})$ and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate $p(10^{19})$, an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of $p(n)$, where our implementation of the Hardy-Ramanujan-Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver's tabulation of 76,065 congruences.Comment: updated version containing an unconditional complexity proof; accepted for publication in LMS Journal of Computation and Mathematic

Computing hypergeometric functions rigorously

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0F_1$, ${}_1F_1$, ${}_2F_1$ and ${}_2F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function ${}_pF_q$ and computation of high-order parameter derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case E-G in section 8.5 (table 6, figure 2); adjusted paper siz

Multiplication by rational constants: LIP research report 2011-3

International audienceMultiplications by simple rational constants often appear in fixed-point or floating-point application code, for instance in the form of division by an integer constant. The hardware implementation of such operations is of practical interest to FPGA-accelerated computing. It is well known that the binary representation of rational constants is eventually periodic. This article shows how this feature can be exploited to implement multiplication by a rational constant in a number of additions that is logarithmic in the precision. An open-source implementation of these techniques is provided, and is shown to be practically relevant for constants with small numerators and denominators, where it provides improvements of 20 to 40\% in area with respect to the state of the art. It is also shown that for such constants, the additional cost for a correctly rounded result is very small, and that correct rounding very often comes for free in practice

Error bounds on complex floating-point multiplication

Given floating-point arithmetic with t-digit base-β significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values z0 and z1 can be computed with maximum absolute error |z0||z1|1/2β 1-t√5. In particular, this provides relative error bounds of 2-24√5 and 2-53√5. for IEEE 754 single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for IEEE 754 single and double precision arithmetic