What every computer scientist should know about floating-point arithmetic

By David Goldberg. 1991

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Citations in the corpus

Formal Security Analysis of Neural Networks using Symbolic Intervals (2018, background): “A large number of matrix multiplication in the DNN in floating point could lead to a severe precision drops after round-up [10].”
SymPy: Symbolic computing in Python (2017, background): “Float numbers do not track their accuracy, and should be used with caution within symbolic expressions since familiar dangers of ﬂoating-point arithmetic apply [16].”
Floating-Point Arithmetic (2020, background): “Several slightly different definitions of the ulp function appear in the literature (Goldberg 1991, Harrison 1999, Kahan 2004a, Overton 2001), they differ near powers of 2.”
Robustness Testing of Autonomy Software (2017, background): “numbers are a frequent source of unexpected software errors [16].”
Automatically improving accuracy for floating point expressions (2015, methodology): “The series expansion for large positive bmakes it more accurate than the form described in common surveys and textbooks [17, 19], which omit the 10(127) < b case.”
Beating Floating Point at its Own Game: Posit Arithmetic (2017, background): “Here’s a classic “thin triangle” problem [1]: Find the area of a triangle with sides a, b, c, when two of the sides b and c are just 3 Units in the Last Place (ULPs) longer than half the longest side a (fig.”
Precimonious: Tuning assistant for floating-point precision (2013, background): “Floating-point arithmetic [15, 26] is used in applications from a wide variety of domains such as high-performance computing, graphics and finance.”
Combining Secret Sharing and Garbled Circuits for Efficient Private IEEE 754 Floating-Point Computations (2015, unknown): “The main shortcomings are in not rounding inexact results to nearest representable floating-point numbers, lack of support for gradual underflow and missing error handling [15].”
Dual coordinate descent methods for logistic regression and maximum entropy models (2011, background): “That is, if z ≈ −αi , the relative numerical error of calculating αi + z can be large (Goldberg 1991).”
MIPLIB 2010 - Mixed Integer Programming Library version 5 (2011, unknown): “A survey of the issues that can arise in floating-point computation can be found in [60].”
Chain of Thought Empowers Transformers to Solve Inherently Serial Problems (2024, background): “Immediate rounding makes iterated addition over floating point numbers no longer associative (Goldberg, 1991), for example, round ( a The associativity of integer addition plays a crucial role in the fact that the iterated addition over n different n -bit binary integers are in TC 0 .”
O(1) Computation of Legendre Polynomials and Gauss-Legendre Nodes and Weights for Parallel Computing (2012, unknown): “Indeed, the floating-point representation fl(x) of the input x only limits the absolute error of the input to [9] δ = |fl(x)− x| ≤ |x| ǫ, (2.”
A Study on Sigmoid Kernels for SVM and the Training of non-PSD Kernels by SMO-type Methods (2005, methodology): “In this case, using the trick under IEEE floating point standard (Goldberg 1991), we can make sure that (6.”
On significance of the least significant bits for differential privacy (2012, background): “For a brief introduction to the subject see Goldberg [ Gol91 ]; Muller et al. serves as a current reference [MBdD + 10].”