Textbook
The notation $a\approx b$ says that one number approximates another; the symbol supplies no tolerance. If $b$ approximates an exact value $a$, the absolute error is $|a-b|$. A specified accuracy requirement is $|a-b|\le\varepsilon$, with a chosen $\varepsilon\ge0$.
For example, $\sqrt2\approx1.414$. Whether that rounding is acceptable depends on the tolerance of the task. The number of decimal places shown does not replace an error bound. The error and its tolerance must use the same units.
For $a\ne0$, relative error is $|a-b|/|a|$: the error divided by the size of the reference value. It is undefined at $a=0$, so a criterion of accuracy needs particular care near zero.
Informally, $u\ll v$ for positive quantities often says that $u/v$ is small on the scale of the task. There is no universal “one hundred times smaller” threshold. Some fields give this symbol a precise specialized meaning; the stated definition takes priority.
A tolerance applies to a difference in values, rather than to their printed representation. Converting both durations from seconds to milliseconds multiplies the absolute error and its tolerance by 1000 but leaves relative error unchanged. Comparing an error in one unit with a tolerance in another is invalid. Equality at the tolerance boundary satisfies the stated non-strict inequality; that boundary is part of the acceptance rule.
Definitions and notation: Deep Learning: Notation.
In plain language
Approximation error measures the difference between exact and approximate values, while a tolerance sets the permitted boundary. One millisecond is small beside ten seconds but large beside one microsecond, so comparison needs common units and a suitable scale. A tolerance resembles parking-space markings: the same result fits one requirement and fails another. Extra zeroes on a display look authoritative but cannot move the painted lines.
Example
In this constructed example, the exact duration is 10 s and a measurement reads 10.001 s. The same error is tested against two tolerances.
Approximation and error
Common units
A millisecond is 0.001 seconds.
Step 1 of 4: Common units
Read all steps
Common units
A millisecond is 0.001 seconds.
- Durations, s
- 1010.001current
Error calculation
Absolute error is 0.001 s; relative error is 0.0001.
- Absolute; relative
- 0.001current0.0001included
Tolerance check
A tolerance of 0.01 s accepts the error.
- Error; tolerance, s
- 0.001included0.01included
A stricter tolerance
With a tolerance of 0.0001 s, the same measurement fails.
- Error; tolerance, s
- 0.001excluded0.0001current
For the 10.001 s measurement, error is a positive distance even though exact value minus measured value is negative. Tightening the tolerance in the last step changes the requirement, not the measurement. The relative error 0.0001 equals 0.01%, since converting a ratio to percent multiplies it by 100. It differs from the absolute error 0.001 s: their meanings and units differ.
An approximation is acceptable relative to a chosen tolerance. An error that looks small is not enough to establish that.