Gal's accurate tables is a method devised by Shmuel Gal to provide accurate values of

special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...

using a

lookup table In computer science, a lookup table (LUT) is an array data structure, array that replaces runtime (program lifecycle phase), runtime computation of a mathematical function (mathematics), function with a simpler array indexing operation, in a proc ...

and

interpolation In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...

. It is a fast and efficient method for generating values of functions like the

exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: * Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value * Ex ...

or the

trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

to within last-bit accuracy for almost all argument values without using extended precision arithmetic. The main idea in Gal's accurate tables is a different tabulation for the special function being computed. Commonly, the range is divided into several subranges, each with precomputed values and correction formulae. To compute the function, look up the closest point and compute a correction as a function of the distance. Gal's idea is to not precompute equally spaced values, but rather to perturb the points ''x'' so that both ''x'' and ''f''(''x'') are very nearly exactly representable in the chosen numeric format. By searching approximately 1000 values on either side of the desired value ''x'', a value can be found such that ''f''(''x'') can be represented with less than ±1/2000 bit of

rounding error In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Roun ...

. If the correction is also computed to ±1/2000 bit of accuracy (which does not require extra floating-point precision as long as the correction is less than 1/2000 the magnitude of the stored value ''f''(''x''), ''and'' the computed correction is more than ±1/1000 of a bit away from exactly half a bit (the difficult rounding case), then it is known whether the exact function value should be rounded up or down. The technique provides an efficient way to compute the function value to within ±1/1000 least-significant bit, i.e. 10 extra bits of precision. If this approximation is more than ±1/1000 of a bit away from exactly midway between two representable values (which happens 99.8% of the time), then the correctly rounded result is clear. Combined with an extended-precision fallback algorithm, this can compute the correctly rounded result in very reasonable ''average'' time. In 2/1000 (0.2%) of the time, such a higher-precision evaluation is required to resolve the rounding uncertainty, but this is infrequent enough that it has little effect on the average calculation time. The problem of generating function values which are accurate to the last bit is known as the

table-maker's dilemma Rounding or rounding off is the process of adjusting a number to an approximate, more convenient value, often with a shorter or simpler representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression √2 with ...

References

* * * * * {{cite book , author-first1=Damien , author-last1=Stehlé , author-first2=Paul , author-last2=Zimmermann , author-link2=Paul Zimmermann (mathematician) , chapter-url=http://hal.inria.fr/inria-00070644/PDF/RR-5359.pdf , pages=257–264 , date=2005 , access-date=2018-01-15 , url-status=live , archive-url=https://web.archive.org/web/20180115191657/https://hal.inria.fr/inria-00070644/PDF/RR-5359.pdf , archive-date=2018-01-15 , doi=10.1109/ARITH.2005.24, chapter=Gal's Accurate Tables Method Revisited , title=17th IEEE Symposium on Computer Arithmetic (ARITH'05) , isbn=0-7695-2366-8 Numerical analysis Computer arithmetic Interpolation

See also

References