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Computational error due to rounding numbers
In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic
Round-off_error
Details of data storage in a spreadsheet application
not the numbers that it displays. Moreover, the error in Excel's answer is not simply round-off error, it is an effect in floating point calculations
Numeric precision in Microsoft Excel
Numeric_precision_in_Microsoft_Excel
Replacing a number with a simpler value
123500". On the other hand, rounding of exact numbers will introduce some round-off error in the reported result. Rounding is almost unavoidable when reporting
Rounding
Logical error that can often be found in programming
An off-by-one error or off-by-one bug (known by acronyms OBOE, OBOB, OBO, and OB1) is a logic error that involves a number that differs from its intended
Off-by-one_error
Loss of precision in numerical analysis
floating-point difference is computed exactly, by the Sterbenz lemma—there is no rounding error introduced by the floating-point subtraction operation. Formally, catastrophic
Catastrophic_cancellation
Process of mapping a continuous set to a countable set
an input value and its quantized value (such as round-off error) is referred to as quantization error, noise or distortion. A device or algorithmic function
Quantization (signal processing)
Quantization_(signal_processing)
Mathematical optimization algorithm
stability if the conjugate gradient method misbehaves, e.g., due to round-off error. The formulas x k + 1 := x k + α k p k {\displaystyle \mathbf {x} _{k+1}:=\mathbf
Conjugate_gradient_method
Mathematical concept
accuracy, leading to unavoidable truncation or rounding. Another common source is inherent measurement error, stemming from the practical limitations of
Approximation_error
are round-off error and truncation error. It can be the combined effect of two kinds of error in a calculation. The first is referred to as round-off error
Numerical_error
Algorithm to compute rounding error
2Sum is a floating-point algorithm for computing the exact round-off error in a floating-point addition operation. 2Sum and its variant Fast2Sum were
2Sum
Computer approximation for real numbers
regained. The Kahan summation algorithm may be used to reduce the errors. Round-off error can affect the convergence and accuracy of iterative numerical
Floating-point_arithmetic
Discrete Fourier transform algorithm
are computed with infinite precision. However, in the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT
Fast_Fourier_transform
Non-zero element of a matrix selected by an algorithm
allow the algorithm to proceed successfully, and possibly to reduce round-off error. It is often used for verifying row echelon form. Pivoting might be
Pivot_element
Matrix decomposition method
(BFGS). Loss of the positive-definite condition through round-off error is avoided if rather than updating an approximation to the inverse
Cholesky_decomposition
Algorithmic technique
floating-point numbers that substantially reduces the accumulated round-off error compared to naively accumulating the sum in sequence. Although there
Pairwise_summation
Ability of numerical algorithms to remain accurate under small changes of inputs
algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large
Numerical_stability
Method for numerical integration
denote the round-off error when f ( x i ) {\displaystyle f(x_{i})} is calculated, and E ( h ) {\displaystyle E(h)} denote the total accumulated error in Composite
Simpson's_rule
Theorem in numerical analysis
show than convergence (and would be needed in any event to show that round-off error will not destroy the computation). Hence convergence is usually shown
Lax_equivalence_theorem
Polynomial equation of degree two
two very nearly equal numbers (the case of large b), which causes round-off error in a numerical evaluation. The figure shows the difference between[clarification
Quadratic_equation
Digit necessary to represent a quantity
algorithm Precision (computer science) Round-off error Lower, Stephen (2021-03-31). "Significant Figures and Rounding". Chemistry - LibreTexts. Chemistry
Significant_figures
Algorithm for determinants of integers
introduces division, which results in round-off errors when implemented using floating point numbers. Round-off errors can be avoided if all the numbers are
Bareiss_algorithm
Floating-point accuracy metric
requires an ulp function Least significant bit (LSB) Machine epsilon Round-off error Goldberg, David (March 1991). "What Every Computer Scientist Should
Unit_in_the_last_place
Approximation method in statistics
the inverse of the error variance-covariance matrix of the observations. In the absence of round-off error and of experimental error in the independent
Non-linear_least_squares
Numerical analysis procedure
x , t ) {\displaystyle u(x,t)} of the PDE on the grid. Define the round-off error ϵ j n {\displaystyle \epsilon _{j}^{n}} as ϵ j n = N j n − u j n {\displaystyle
Von Neumann stability analysis
Von_Neumann_stability_analysis
Quantization error in numerical analysis
resolution in the range (values), nor in round-off error arising from floating-point arithmetic. Discretization error would occur even if it were possible
Discretization_error
Measure of variation in statistics
consider round-off error, arithmetic overflow, and arithmetic underflow. The method below calculates the running sums method with reduced rounding errors. This
Standard_deviation
(i.e. if its condition number multiplied by the machine's relative round-off error is appreciably large). In that case, including the smallest singular
Numerical methods for linear least squares
Numerical_methods_for_linear_least_squares
Function's sensitivity to argument change
will be after approximation. Note that this is before the effects of round-off error are taken into account; conditioning is a property of the matrix, not
Condition_number
Idea that small causes can have large effects
original numbers, but were the rounded-off values that had appeared in the original printout. The initial round-off errors were the culprits; they were
Butterfly_effect
Error from taking a finite sum of an infinite series
by mistake, round-off error (the consequence of using finite precision floating point numbers on computers), is also called truncation error, especially
Truncation_error
Rendering issue in 3D modeling
caused by limited sub-pixel precision, floating point and fixed point round-off errors. The more z-buffer precision one uses, the less likely it is that z-fighting
Z-fighting
Algorithms for calculating square roots
{\displaystyle \ x_{n}\ } being calculated, to avoid significant round-off error. When in the program above the estimates — the highlighted lines 41
Square_root_algorithms
Digital signal sample rate converter
CIC filters are implemented with fixed-point arithmetic without any round-off error to ensure exact cancellation of each comb's zero at DC with each integrator's
Cascaded integrator–comb filter
Cascaded_integrator–comb_filter
Methods for numerical approximations
and additions. Generally, it is important to estimate and control round-off errors arising from the use of floating-point arithmetic. Interpolation solves
Numerical_analysis
Algorithm that estimates unknowns from a series of measurements over time
numerical stability. If the process noise covariance Qk is small, round-off error often causes a small positive eigenvalue of the state covariance matrix
Kalman_filter
and assist humans in their day-to-day lives and keep everyone safe. round-off error The difference between the result produced by a given algorithm using
Glossary_of_computer_science
Volunteer project using software to search for Mersenne prime numbers
floating-point numbers does not introduce errors into the LL calculation. The program verifies that the round-off error is no greater than 0.4 every 128 iterations
Great Internet Mersenne Prime Search
Great_Internet_Mersenne_Prime_Search
C programming language standard, 1999 revision
precision than required for intermediate computations can minimize round-off error (the typedef double_t can be used for code that is portable under all
C99
computation to reduce round-off error Truncation — rounding a floating-point number by discarding all digits after a certain digit Round-off error Numeric precision
List of numerical analysis topics
List_of_numerical_analysis_topics
Decimal representation of real numbers in computing
numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically occur when converting between decimal fractions
Decimal_floating_point
Coins with errors while minting
necessarily make an error coin valuable. Although no other coin may be the same as a coin with a particular off-center strike, off-center strikes of varying
Mint-made_errors
Error in numerical data
underflow Limit of detection Precision bias Propagation of uncertainty Round-off error Rounding Significant figures "Overprecision". Fallacy files. Pirie, Madsen
False_precision
Subset of x86 instruction set architecture for floating-point arithmetic
high-precision scalar unit for numerical calculations sensitive to round-off error and requiring the 64-bit mantissa precision and extended range available
X87
Class of numerical techniques
methods are round-off error, the loss of precision due to computer rounding of decimal quantities, and truncation error or discretization error, the difference
Finite_difference_method
right corner of the net in off the post to win the game for Brazil on a 2–1 scoreline. The victory advanced Brazil to the round of 16, Japan were eliminated
2026 FIFA World Cup round of 32
2026_FIFA_World_Cup_round_of_32
Type of cellular automaton
the boolean states mean there will be exact computing without any round-off error due to floating-point precision, and that the cellular automata system
Lattice_gas_automaton
Recursive algorighm in linear algebra
computationally, but more sensitive to computational inaccuracies like round-off errors. The Bareiss algorithm for Toeplitz matrices (not to be confused with
Levinson_recursion
128-bit computer number format
precision results more reliably and accurately by minimising overflow and round-off errors in intermediate calculations and scratch variables. William Kahan,
Quadruple-precision floating-point format
Quadruple-precision_floating-point_format
Projectile impact patterns
size of a dollar coin. This coincidentally can be rounded down to just 1 inch, and the round-off error of assuming 1 MOA equals "1 inch at 100 yards" is
Shot_grouping
Molecular modeling tool in chemistry
be close in Cartesian space (and thus small round-off errors can accumulate to large force-field errors.) The optimally fastest and most numerically
Z-matrix_(chemistry)
Type of numeric sequence used in statistics
uses only integer numbers internally, which makes it robust against round-off errors. def halton_sequence(b): """Generator function for Halton sequence
Halton_sequence
1970s synthesizer
inherent effect of low precision calculations in the digital circuitry - round-off error produced digital non-linearity and extra harmonics resulting in digital
Coupland Digital Music Synthesizer
Coupland_Digital_Music_Synthesizer
with dependent variable. restricted randomization robust statistics round-off error sample That part of a population which is actually observed or measured
Glossary of probability and statistics
Glossary_of_probability_and_statistics
Method of encoding digital data on multiple carrier frequencies
require fewer operations, which translates to fewer round-off errors in the equalizer. Those round-off errors can be viewed as numerical noise and are inevitable
Orthogonal frequency-division multiplexing
Orthogonal_frequency-division_multiplexing
Numerical linear algebra algorithm
that S must not be explicitly calculated which reduces the danger of round-off errors. Note that J S J T = J A T A J T = J A T J T J A J T = B T B {\displaystyle
Jacobi_eigenvalue_algorithm
Calculation of position based on satellite signal timing
t_{{\text{clock}},i}-\delta t_{\text{clock,rec}}} , whose right side is round-off-error resistive during calculation. The geometric range is calculated as
Satellite_navigation_solution
Espriella narrowly defeated Iván Cepeda in the run-off election held on 21 June 2026. In the first round, de la Espriella led with 43.7% of the vote, followed
2026 Colombian presidential election
2026_Colombian_presidential_election
Device for suppressing part of a discretely-sampled signal
provide advantages such as improved numerical stability and reduced round-off error. Some structures are better for fixed-point arithmetic and others may
Digital_filter
Technique for reducing low-resolution image distortion
fractals have unlimited detail and no noise other than arithmetic round-off error, they illustrate aliasing more clearly than do photographs or other
Spatial_anti-aliasing
English computer scientist (1912–1954)
17 February 2019. Retrieved 18 March 2019. Turing, A.M. (1948). "Rounding-Off Errors in Matrix Processes". The Quarterly Journal of Mechanics and Applied
Alan_Turing
Upper bound on rounding error in floating-point arithmetic
machine precision is an upper bound on the relative approximation error due to rounding in floating point number systems. This value characterizes computer
Machine_epsilon
BASIC programming language dialect
using a three-byte mantissa is only about 6.5 decimal digits, and round-off error is common. 6502 implementations of Microsoft BASIC utilized 40-bit
Commodore_BASIC
Regression models accounting for possible errors in independent variables
In statistics, an errors-in-variables model or a measurement error model is a regression model that accounts for measurement errors in the independent
Errors-in-variables_model
number of significant figures are retained to minimise propagation of round-off errors; they should not be taken to indicate the relative uncertainty of the
Hydrolysis_constant
Obsolete keyword in the Java programming language
compute intermediate results using such extended formats. This may avoid round-off errors, overflows and underflows that would otherwise occur, but can cause
Strictfp
Internal representation of numeric values in a digital computer
precision is a well-studied problem. Estimates of the magnitude of round-off errors and methods to limit their effect on large calculations are part of
Computer_number_format
Amateur mathematician and school owner (1812–1882)
incorrect because of round-off errors). He subsequently expanded his calculation to 607 decimal places in April 1853, but an error introduced at the start
William_Shanks
Algorithm for pseudo-random number sampling
ziggurat. At the end, we should have yn = f(0). There will be some round-off error, but it is a useful sanity test to see that it is acceptably small
Ziggurat_algorithm
File format and programming language
real number, and performing calculations may produce unacceptable round-off errors. Software which can be used to render PostScript documents: Ghostscript
PostScript
Frequency-division multiple access scheme
multi-tap FIR or IIR-filters. Less computations result in less compounded round-off error, which can be viewed as numerical noise. A related concept is the combination
Single-carrier_FDMA
Mechanical analog computer
"One effect was that users felt close to the numbers, aware of rounding-off errors and systematic inaccuracies, unlike users of today's computer-design
Slide_rule
Canadian stock exchange (1906–1999)
exchange's index provides an example of large errors arising from the accumulation of seemingly minor round-off errors. In January 1982 the index was initialized
Vancouver_Stock_Exchange
British television baking competition
replacing Leith. Periodically, accidents and other errors have influenced the results of a round of judging. Several have had a significant impact on
The_Great_British_Bake_Off
Golf course in Rio de Janeiro, Brazil
golfers. Any discrepancy between metre and yard distances is due to round off error when converting from SI units to USC units. "Olympic Games Facilities:
Olympic_Golf_Course
Method of data analysis
suffers from loss of orthogonality of PCs due to machine precision round-off errors accumulated in each iteration and matrix deflation by subtraction.
Principal_component_analysis
Computer algebra system
Because all numeric arithmetic is double precision floating point, and round-off error is not tracked, Mathomatic is not suitable for applications requiring
Mathomatic
Varying methods used to calculate pi
being incorrect due to round-off errors). He subsequently expanded his calculation to 607 decimal places in April 1853, but an error introduced right at
Approximations_of_pi
Computational method for solving partial differential equations
its shape parameter-A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation," Engineering
Kansa_method
Method of estimating a statistical model's parameters
{\displaystyle x_{i+r-1}} , let δ {\displaystyle \delta } represent the round-off error. All of the true values should then fall in the range x ± δ {\displaystyle
Maximum_spacing_estimation
new pipeline also supports floating point calculations to minimize round-off errors, which are inherent in integer processing. Once the color pipeline
Windows_Color_System
Misconception that people in the Middle Ages believed Earth to be flat
obliged to abjure his error. This error however at length prevailed, the Earth became a globe, and Descartes declared it was whirled round its axis by a vortex
Myth_of_the_flat_Earth
2025 EP by Yves
Soft Error is the third extended play by South Korean singer Yves. It was released by Paix Per Mil on August 7, 2025, and contains six tracks, including
Soft_Error
Italian-American applied mathematician
1985 and completing her Ph.D. in 1989. Her dissertation, A Stochastic Round Off Error Analysis for the Fast Fourier Transform, was supervised by John Tolle
Daniela_Calvetti
Image compression standard and coding system
because it has to be implemented in floating or fix-point and causes round-off errors. The ICT shall be used only with the 9/7 wavelet transform. Reversible
JPEG_2000
Algorithm in numerical analysis
accumulated error, based on the adjusted y. = 2.80000 - 2.75987 As expected, the low-order parts can be retained in c with no or minor round-off effects.
Kahan_summation_algorithm
First edition of the IEEE 754 floating-point standard
higher precision than that required for the final result, to minimise round-off errors: the standard only specifies minimum precision and exponent requirements
IEEE_754-1985
In the trading card collecting hobby, an error card is a card that shows incorrect information or some other unintended flaw. It can contain a mistake
Error_card
Class of numerical methods
algorithm suffers from numerical instabilities due to floating point round-off errors. To see why, consider the first function, φ 1 ( z ) = e z − 1 z , {\displaystyle
Exponential_integrator
European football tournament
and the play-off round. It was played from 1 July to 27 August 2014. A total of 55 teams competed in the qualifying phase and play-off round to decide 10
2014–15 UEFA Champions League qualifying
2014–15_UEFA_Champions_League_qualifying
130th season of the Australian Football League (AFL)
decisions since the 2025 finals series. There was a key error late in the Geelong v Adelaide match in round 3: the boundary umpire incorrectly paid a last disposal
2026_AFL_season
Aviation term
inherently slow and repetitive approach, which is also subject to round-off errors if the speed increment between the steps is not carefully selected
Balanced_field_takeoff
(2019), p. 194. sfnp error: no target: CITEREFDeAngelis2019 (help) "One Hit Wonder Jermaine Stewart We Don't Have to Take Our Clothes Off". Castbox.fm. Marx
List of one-hit wonders in the United States
List_of_one-hit_wonders_in_the_United_States
teams in the group. Group winners qualified for the World Cup. Second round (play-offs): Sixteen teams (twelve group runners-up and four best Nations League
2026 FIFA World Cup qualification (UEFA)
2026_FIFA_World_Cup_qualification_(UEFA)
NHL postseason tournament
on CBC this postseason, and any first-round games that appeared on that streaming service was due to an error. In the U.S., this marked the third year
2024_Stanley_Cup_playoffs
Sexual stimulation of one's own genitals
held that "the effort to achieve complete abstinence was a very serious error." The work of sexologist Alfred Kinsey during the 1940s and 1950s, most
Masturbation
Formula One motor race
June 2026 at the Red Bull Ring in Spielberg, Austria. It was the eighth round of the 2026 Formula One World Championship. George Russell (Mercedes) converted
2026_Austrian_Grand_Prix
European football tournament
qualifying phase (preliminary and first to third rounds) and the play-off round. It began on 25 June and ended on 28 August 2019. A total of 53 teams
2019–20 UEFA Champions League qualifying
2019–20_UEFA_Champions_League_qualifying
presidential election, no candidate secured an outright majority in the first round, a run-off was conducted on 13 April 2025. Incumbent President Daniel Noboa was
2025 Ecuadorian general election
2025_Ecuadorian_general_election
Method to improve accuracy of numerical solutions to systems of linear equations
A\mathbf {x} =\mathbf {b} \,,} due to the compounded accumulation of rounding errors, the computed solution x ^ {\displaystyle {\hat {\mathbf {x} }}} may
Iterative_refinement
Verónika Mendoza to finish in second and earn a place in the second round. The run-off was held on 5 June 2016. With support from those opposing Fujimori
2016 Peruvian general election
2016_Peruvian_general_election
ROUND OFF-ERROR
ROUND OFF-ERROR
Surname or Lastname
English
English : variant of Rolfe.North German : variant of Ruoff.
Boy/Male
Hindu, Indian
Round Face
Surname or Lastname
English
English : patronymic from Round.
Boy/Male
Tamil
Gundapa | கà¯à®¨à¯à®¤à®¾à®ªà®¾Â
Round
Gundapa | கà¯à®¨à¯à®¤à®¾à®ªà®¾Â
Boy/Male
Hindu
Round
Surname or Lastname
Welsh
Welsh : nickname for a red-haired person (see Gough).English (of Cornish and Breton origin) : occupational name from Cornish and Breton goff ‘smith’ (cognate with Gaelic gobha). The surname is common in East Anglia, where it is of Breton origin, introduced by followers of William the Conqueror.Irish : reduced form of McGoff.Edward Goffe was a farmer in Cambridge MA whose house was acquired by Harvard College some time before 1654 and used as a dormitory, known as Goffe’s College.
Surname or Lastname
English (chiefly West Midlands)
English (chiefly West Midlands) : nickname for a plump person, from Middle English, Old French rond, rund ‘fat’, ‘round’ (Latin rotundus).
Boy/Male
Arabic
Round
Surname or Lastname
English (also found in Ireland)
English (also found in Ireland) : from a pet form of Lamb 1 and 2.
Surname or Lastname
English
English : variant of Bond
Surname or Lastname
English
English : from Middle English p(o)und ‘enclosure (especially for confining animals)’; a topographic name for someone who lived near an enclosure in which animals were kept, or a metonymic occupational name for an official responsible for rounding up stray animals and placing them in a pound.Probably a translation of German Pfund or the North German cognate Pund.
Female
Portuguese
Portuguese form of Latin Ofelia, OFÉLIA means "help."
Surname or Lastname
English
English : from Old French boeuf ‘bull’, a nickname for a powerfully built man. In some cases it may have been originally a metonymic occupational name for a herdsman. Compare Bouvier.German (Böff) : from the short form of a Germanic personal name with bod- (Old Saxon bodo ‘messenger’), as in Bodo.
Girl/Female
Arabic
Tree of Good Scent
Boy/Male
Anglo, British, English
Name of a King
Surname or Lastname
English
English : unexplained. Compare Grounds.Perhaps an Americanized form of German Grund.
Surname or Lastname
English (found mainly in Wales)
English (found mainly in Wales) : variant of Glasscock 2.
Boy/Male
Indian, Sanskrit
Round Eye
Boy/Male
Greek
Round basket.
Surname or Lastname
English
English : presumably a variant of Mount.
ROUND OFF-ERROR
ROUND OFF-ERROR
Boy/Male
Muslim
Clear
Male
Finnish
Finnish form of Greek Paulos, PAULI means "small."
Girl/Female
American, Australian, Welsh
Fair; Good; Blend of Glenda and Linda
Girl/Female
Muslim/Islamic
Happy
Boy/Male
Arabic
Lover; Madly in Love
Boy/Male
German, Scandinavian
Warrior
Girl/Female
Latin
Brave.
Boy/Male
Hindu
Girl/Female
American, Australian
A Form of Shanice
Boy/Male
Indian, Tamil
Lord Krishna
ROUND OFF-ERROR
ROUND OFF-ERROR
ROUND OFF-ERROR
ROUND OFF-ERROR
ROUND OFF-ERROR
a.
Having a round back or shoulders; round-shouldered.
adv.
In circumference; as, a ball is ten inches round.
n.
That which goes round a whole circle or company; as, a round of applause.
prep.
On every side of, so as to encompass or encircle; around; about; as, the people atood round him; to go round the city; to wind a cable round a windlass.
a.
Having a curved outline or form; especially, one like the arc of a circle or an ellipse, or a portion of the surface of a sphere; rotund; bulging; protuberant; not angular or pointed; as, a round arch; round hills.
a.
Round; circular; spherical.
a.
Outspoken; plain and direct; unreserved; unqualified; not mincing; as, a round answer; a round oath.
a.
Uttered or emitted with a full tone; as, a round voice; a round note.
n.
A course ending where it began; a circuit; a beat; especially, one freguently or regulary traversed; also, the act of traversing a circuit; as, a watchman's round; the rounds of the postman.
v. t.
To go round wholly or in part; to go about (a corner or point); as, to round a corner; to round Cape Horn.
a.
Not inconsiderable; large; hence, generous; free; as, a round price.
n.
An assembly; a group; a circle; as, a round of politicians.
adv.
Circularly; in a circular form or manner; by revolving or reversing one's position; as, to turn one's head round; a wheel turns round.
v. t.
To make circular, spherical, or cylindrical; to give a round or convex figure to; as, to round a silver coin; to round the edges of anything.
v. i.
To go round, as a guard.
adv.
On all sides; around.
n.
Same as Round of beef, below.
a.
Round.
n.
Anything round, as a circle, a globe, a ring. "The golden round" [the crown].
v. i.
To go or turn round; to wheel about.