## Algorithms for Arbitrary Precision Floating Point Arithmetic (1991)

Venue: | Proceedings of the 10th Symposium on Computer Arithmetic |

Citations: | 69 - 1 self |

### BibTeX

@INPROCEEDINGS{Priest91algorithmsfor,

author = {Douglas M. Priest},

title = {Algorithms for Arbitrary Precision Floating Point Arithmetic},

booktitle = {Proceedings of the 10th Symposium on Computer Arithmetic},

year = {1991},

pages = {132--145},

publisher = {IEEE Computer Society Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present techniques which may be used to perform computations of very high accuracy using only straightforward floating point arithmetic operations of limited precision, and we prove the validity of these techniques under very general hypotheses satisfied by most implementations of floating point arithmetic. To illustrate the application of these techniques, we present an algorithm which computes the intersection of a line and a line segment. The algorithm is guaranteed to correctly decide whether an intersection exists and, if so, to produce the coordinates of the intersection point accurate to full precision. Moreover, the algorithm is usually quite efficient; only in a few cases does guaranteed accuracy necessitate an expensive computation. 1. Introduction "How accurate is a computed result if each intermediate quantity is computed using floating point arithmetic of a given precision?" The casual reader of Wilkinson's famous treatise [21] and similar roundoff error analyses might...

### Citations

2321 |
The Art of Computer Programming
- Knuth
- 1968
(Show Context)
Citation Context ...articular, if both expansions have n components, the number of basic multiply-and-add steps is O(n 2 ). Algorithms which multiply n-digit numbers in fewer than O(n log 2 n) steps are known (see Knuth =-=[10]-=-), but we have elected not to use them for several reasons. Specifically, most of these algorithms improve upon the classical method only for very large n, while in practice we would expect to apply t... |

116 |
A Floating-Point Technique for Extending the Available Precision
- Dekker
- 1971
(Show Context)
Citation Context ... least, reasonable enough that we are obliged to consider more carefully the trade-offs between cost and accuracy. Our algorithms are based on an approach pioneered by M��ller [17], Kahan [7], Dek=-=ker [4]-=-, Pichat [19], Linnainmaa [13, 14], and several others [10, 12], but our hypotheses are slightly more general than theirs. Moreover, rather than simply extending the accuracy to approximately twice th... |

100 |
Rounding Errors
- Wilkinson
- 1963
(Show Context)
Citation Context ...n. 1. Introduction "How accurate is a computed result if each intermediate quantity is computed using floating point arithmetic of a given precision?" The casual reader of Wilkinson's famous=-= treatise [21]-=- and similar roundoff error analyses might conclude that the most one can hope to say about the accuracy of a computation carried out in fixed precision floating point arithmetic is that the computed ... |

83 |
Computer Arithmetic in Theory and Practice
- Kulisch, Miranker
- 1981
(Show Context)
Citation Context ...rected roundings, or interval arithmetic, which must be implemented in a combination of hardware and low-level software to be efficient. Other techniques, such as one proposed by Kulisch and Miranker =-=[11]-=-, require direct access to the exponent and significand fields of floating point numbers, and obtaining these quantities, even when the programming environment provides a convenient way to do so, is m... |

73 |
Verifiable implementations of geometric algorithms using finite precision arithmetic
- Milenkovic
- 1988
(Show Context)
Citation Context ...point. This problem arises in many computer graphics and computer-aided design applications; numerous authors have recognized both the importance and difficulty of solving the problem accurately (see =-=[6, 15, 16, 18]-=- and references therein). Milenkovic [16], in particular, has shown that if the single and double precision arithmetics satisfy t 2s2t 1 +1 then line and line segment intersection calculations may be ... |

61 |
Further remarks on reducing truncation errors
- Kahan
- 1965
(Show Context)
Citation Context ...roblems---at least, reasonable enough that we are obliged to consider more carefully the trade-offs between cost and accuracy. Our algorithms are based on an approach pioneered by M��ller [17], Ka=-=han [7]-=-, Dekker [4], Pichat [19], Linnainmaa [13, 14], and several others [10, 12], but our hypotheses are slightly more general than theirs. Moreover, rather than simply extending the accuracy to approximat... |

37 |
Underflow and the reliability of numerical software
- Demmel
- 1984
(Show Context)
Citation Context ....) Nevertheless, we believe that in most applications, intermediate results may be rounded or rescaled so that underflow does not pose a serious problem. Even when underflow cannot be avoided, Demmel =-=[5]-=- points out that the cumulative effect of the resulting errors can often be estimated by extending the error analysis to include an underflow term whose magnitude is bounded by the underflow threshhol... |

36 |
Correction d’une somme en arithmétique à virgule flottante
- Pichat
- 1972
(Show Context)
Citation Context ...onable enough that we are obliged to consider more carefully the trade-offs between cost and accuracy. Our algorithms are based on an approach pioneered by M��ller [17], Kahan [7], Dekker [4], Pic=-=hat [19]-=-, Linnainmaa [13, 14], and several others [10, 12], but our hypotheses are slightly more general than theirs. Moreover, rather than simply extending the accuracy to approximately twice the working pre... |

35 | Floating-Point Computation - Sterbenz - 1974 |

20 |
A Fortran multiple precision arithmetic package
- BRENT
- 1978
(Show Context)
Citation Context ...ysis can guarantee based on the widest precision of arithmetic supported in whatever computing environment is available, then he must instead resort to a subroutine library such as Brent's MP package =-=[3]-=- in order to compute with higher precision arithmetic. That both conclusions are wrong follows from the existence of techniques which allow a program to compute to arbitrarily high accuracy using only... |

12 |
Precision Geometry: A General Technique for Calculating Line and Segment Intersections using Rounded Arithmetic
- Double
- 1989
(Show Context)
Citation Context ...point. This problem arises in many computer graphics and computer-aided design applications; numerous authors have recognized both the importance and difficulty of solving the problem accurately (see =-=[6, 15, 16, 18]-=- and references therein). Milenkovic [16], in particular, has shown that if the single and double precision arithmetics satisfy t 2s2t 1 +1 then line and line segment intersection calculations may be ... |

11 |
Analysis of some known methods of improving the accuracy of floatingpoint sums
- Linnainmaa
- 1974
(Show Context)
Citation Context ...t we are obliged to consider more carefully the trade-offs between cost and accuracy. Our algorithms are based on an approach pioneered by M��ller [17], Kahan [7], Dekker [4], Pichat [19], Linnain=-=maa [13, 14]-=-, and several others [10, 12], but our hypotheses are slightly more general than theirs. Moreover, rather than simply extending the accuracy to approximately twice the working precision, as do most of... |

10 |
Floating-point computation of functions with maximum accuracy
- Bohlender
- 1977
(Show Context)
Citation Context ...oreover, rather than simply extending the accuracy to approximately twice the working precision, as do most of the aforementioned references, our algorithms expand upon methods developed by Bohlender =-=[1]-=- and Kahan [9] which compute to arbitrarily high accuracy. Below, we give algorithms for exact addition and multiplication and arbitrarily accurate division of extended precision numbers using only fi... |

8 |
What Do We Need Beyond
- Bohlender
- 1990
(Show Context)
Citation Context ...omponents of y are desired. Many other algorithms which evaluate a sum or an inner product to working precision (i.e., yielding the first component of an expansion) have been developed; see Bohlender =-=[2]-=- for a survey of several approaches. Unfortunately, many of these techniques rely on special features, such as an extra wide fixed point accumulator, directed roundings, or interval arithmetic, which ... |

3 |
Recipes for Geometry and Numerical Analysis
- Dobkin, Silver
- 1988
(Show Context)
Citation Context ...point. This problem arises in many computer graphics and computer-aided design applications; numerous authors have recognized both the importance and difficulty of solving the problem accurately (see =-=[6, 15, 16, 18]-=- and references therein). Milenkovic [16], in particular, has shown that if the single and double precision arithmetics satisfy t 2s2t 1 +1 then line and line segment intersection calculations may be ... |

3 |
for doubled-precision floating-point computations
- Software
- 1981
(Show Context)
Citation Context ...dding all the partial products and renormalizing to obtain a t-digit expansion. The method of splitting components was first proposed by Dekker [4], and we shall rely on the proof given by Linnainmaa =-=[14]-=-. To clarify the statements and proofs, we define the number of leading nonzero digits of a t-digit number x to be the smallest integer d such that x = mfi k with fi d\Gamma1sjmj ! fi d , or zero if x... |

3 |
Numerical Stability of Simple Geometric Algorithms
- Ottman, Thiemt, et al.
- 1987
(Show Context)
Citation Context |

2 |
Parallel Algorithms for the Rounding-Exact Summation of Floating-Point Numbers, Computing 28
- Leuprecht, Oberaigner
- 1982
(Show Context)
Citation Context ...more carefully the trade-offs between cost and accuracy. Our algorithms are based on an approach pioneered by M��ller [17], Kahan [7], Dekker [4], Pichat [19], Linnainmaa [13, 14], and several oth=-=ers [10, 12]-=-, but our hypotheses are slightly more general than theirs. Moreover, rather than simply extending the accuracy to approximately twice the working precision, as do most of the aforementioned reference... |

2 |
Quasi Double Precision
- M��ller
- 1965
(Show Context)
Citation Context ...e for many problems---at least, reasonable enough that we are obliged to consider more carefully the trade-offs between cost and accuracy. Our algorithms are based on an approach pioneered by M��l=-=ler [17]-=-, Kahan [7], Dekker [4], Pichat [19], Linnainmaa [13, 14], and several others [10, 12], but our hypotheses are slightly more general than theirs. Moreover, rather than simply extending the accuracy to... |