Double Precision Floating Point Numbers Research Articles

Optimizing the representation of intervals

A representation of intervals is proposed that, instead of both endpoints, uses the lower endpoint and the width of the interval. This proposed representation is more efficient since both endpoints have many corresponding bits that are equal. Consequently, the width of the interval can be represented with a smaller number of bits than an endpoint, resulting in a better utilization of the number of bits available. Considering the case in which the total number of bits is the same as for the traditional representation (say, two double-precision floating-point numbers) we determine the number of bits of the lower endpoint and of the width so that the rounding error is minimized. The use of this combination of bits would result in a variable representation, which is difficult to implement. Alternatively, it is possible to use a fixed partition that gives good results for a variety of applications.The proposed representation is evaluated for several computations. Using a total number of 128 bits to represent the interval we conclude that it produces intervals that are substantially narrower than those obtained with the traditional representation.Alternative implementations of the calculation of the width are proposed. The interval widths produced are shown to be very similar, so that the choice of the alternative can be made by considerations such as area and delay.

Double precision rational approximation algorithms for the standard normal first and second order loss functions

We present double precision algorithms based upon piecewise rational approximations for the standard normal first and second order loss functions. These functions are used frequently in inventory management. No direct approximation or closed formulation exists for the standard first and second order loss functions. Current state-of-the-art algorithms require intermediate computations of the cumulative normal distribution or tabulations and they do not compute to full double precision. We deal with both these issues and present direct double precision accurate algorithms which are valid in the full range of double precision floating point numbers.

Double precision rational approximation algorithm for the inverse standard normal first order loss function

We present a double precision algorithm based upon rational approximations for the inverse standard normal first order loss function. This function is used frequently in inventory management. No direct approximation or closed formulation exists for the inverse standard normal first order loss function. Calculations are currently based on root-finding methods and intermediate computations of the cumulative normal distribution or tabulations. Results then depend on the accuracy and valid range of that underlying function. We deal with these issues and present a direct, double precision accurate algorithm valid in the full range of double precision floating point numbers.

IEEE-754 compliant Algorithms for Fast Multiplication of Double Precision Floating Point Numbers

In image and signal processing applications floating-point multiplication is major concern in calculations. Performing multiplication on floating point data is a long course of action and requires huge quantity of processing time. By improving the speed of multiplication task overall speed of the system can be enhanced. The Bottleneck in the floating point multiplication process is the multiplication of mantissas which needs 53*53 bit integer multiplier for double precision floating point numbers. By improving the speed of multiplication task the overall speed of the system can be improved. In this paper a comparison on the existing trends in mantissa multiplication is made. Vedic and Canonic Signed digit algorithms were compared on parameters like speed, complexity of routing, pipelining, resource required on FPGA. The comparison showed that Canonic Signed Digit Algorithm is better than Vedic algorithm in terms of speed and resources required on spartan3 FPGA. .

Simulations of pulse propagation in optical fibers using graphics processor units

Several simple cases of pulse propagation in optical fibers have been simulated using a graphics processor unit. Comparisons with simulations in a computer processor unit are also reported. Speedups from 4 to 36 have been obtained by different numerical methods, reaching a similar accuracy both in computer processor unit and in graphics processor unit, working with double-precision floating point numbers. Best results are achieved in simulations when the number of points in t grid rises. Therefore, the results indicate that graphics processor units could be a good tool to improve numerical simulations of pulse propagation in fibers.

Fast finite Hilbert transform via DE quadrature scheme

The finite Hilbert transform (FHT) or inverse finite Hilbert transform (IFHT) is recently found to have some important applications in computerized tomography (CT) arena, where they are used to filter the derivatives of back-projected data in the chord-line based CT reconstruction algorithms. In this paper, we implemented, improved and validated a fast numerical solution to the FHT via a double exponential (DE) integration scheme. A same strategy can be used to compute IFHT. To overcome the underflow of floating-point numbers, we first determined the range of variable transformation from the minimum positive value of single or double precision floating point number, the integration step can be further determined by the range of variable transformation and the integration level. Two functions with their known analytical FHTs are used to validate the implementation of the FHT via DE scheme. The surface map and 2D contour of the FHT transformation error with respect to integration level and the range of the variable transformation are used to numerically determine the optimal numbers for a fast FHT. Given a specific precision, the lowest integration level and the optimal range of variable transformation, which are used to transform a signal with a certain degree of fluctuation, can be numerically determined by the surface map and 2D contour of the standard deviation of transformation error. These two numbers can then be taken to efficiently compute the FHT for other signals with the same or less degree of fluctuation. The FHT via DE scheme and the numerical method to determine the integration level and the range of transformation can be used for fast FHT in certain applications, such as data filtering in chord-line based CT reconstruction algorithms.

Integer Representation of Decimal Numbers for Exact Computations.

A scheme is presented and software is documented for representing as integers input decimal numbers that have been stored in a computer as double precision floating point numbers and for carrying out multiplications, additions and subtractions based on these numbers in an exact manner. The input decimal numbers must not have more than nine digits to the left of the decimal point. The decimal fractions of their floating point representations are all first rounded off at a prespecified location, a location no more than nine digits away from the decimal point. The number of digits to the left of the decimal point for each input number besides not being allowed to exceed nine must then be such that the total number of digits from the leftmost digit of the number to the location where round-off is to occur does not exceed fourteen.

Efficient and reliable methods for rounded-interval arithmetic

We present an efficient and reliable method for computing the unit-in-the-last-place ( ulp) of a double-precision floating-point number, taking advantage of the standard binary representation for floatingpoint numbers defined by IEEE Std 754-1985. The ulp is necessary to perform software rounding for robust rounded-interval arithmetic (RIA) operations. Hardware rounding, using two of the standard rounding modes defined by IEEE-754, may be more efficient. RIA has been used to produce robust software systems for the solution of systems of nonlinear equations, interrogation of geometric and differential properties of curves and surfaces, curve and surface intersections, and solid modeling.

An integrated prolog architecture for symbolic and numeric executions

Numerically intensive calculations are not well supported by Prolog, yet there are important applications that require tightly coupled symbolic and numeric calculations. The Aquarius Numeric Processor (ANP) is an extended numeric Instruction Set Architecture based on the Berkeley Programmed Logic Machine (PLM) to support integrated symbolic and numeric calculations. This extension expands the existing numeric data type to include 32- and 64-bit integers, and single and double precision floating-point numbers conforming to the IEEE Standard P754. A new class of data structure, numeric arrays, is added to represent matrices and arrays found in most scientific programming languages. Powerful numeric instructions are included to manipulate the new data types. Dynamic type checking and coercing of operands are done. The ANP and PLM together provide for the efficient execution for symbolic and numeric operations written in AI languages such as Prolog and Lisp. Simulated performance results indicate the system will achieve about 10 MFLOPs on the Prolog version of some Whetstone and Linpack benchmarks and close to 20 MFLOPS on some matrix operations (all in double precision).

Some data conversions for managing the internal and output form of formac constants

Three PL/I functions for managing Formac constants have been throughly tested and documented. These functions are: FLT&CHR for converting double precision floating point numbers to rational form as a PL/I character strings, TRUNCATE for truncating the numerator and denominator of a Formac rational constant to prescribed length, and EXT&FLT for constructing a PL/I character string containing an arbitrary length (precision) floating point equivalent of a Formac rational constant.