Machine Epsilon Calculator

Estimate machine epsilon for a simulated floating-point precision. This educational calculator shows the formula, result, and step-by-step interpretation.

Computer ScienceNumerical Methods

Calculator

What this calculator teaches

Machine epsilon measures spacing near 1 and helps explain floating-point roundoff.

Use the result as a learning aid. For classwork, still show the formula and intermediate reasoning so the final answer is not just a black-box number.

What Machine Epsilon Measures Near 1

Machine epsilon is the spacing from 1 to the next larger representable floating-point number under a common definition. For a normalized binary format with p bits of significand precision, that gap is ε=21-p. The precision count p includes the leading significand bit when the format represents it implicitly.

This value is about relative resolution near 1, not the smallest positive floating-point number. Very small magnitudes involve exponent limits and subnormal numbers, which are different properties of a format. Also note that some numerical-analysis texts use "machine epsilon" for unit roundoff, often half the spacing used here. Always check the convention.

Precision pε=21-pCommon interpretation
112-10=0.0009765625Binary16 spacing near 1
242-23≈1.19209×10-7Binary32 spacing near 1
532-52≈2.22045×10-16Binary64 spacing near 1

Floating-point spacing scales with exponent. Numbers around 2 have a larger absolute gap than numbers around 1, while the relative precision remains roughly comparable within the normalized range. That is why ε is most useful as a reference for relative rounding behavior rather than as one universal absolute error bound.

One practical experiment is to compare 1+ε with 1. Under the spacing convention used here, 1+ε is the next representable value above 1. A smaller increment may round back to 1 depending on the rounding mode, which demonstrates why tiny arithmetic changes can disappear in finite precision.

Formula & Symbols

ConceptFormula or rule
Machine epsilonε = 2^(1-p) for p precision bits

Worked example

Example: Double precision has p=53, so machine epsilon is about 2.22e-16.
Example 2: With p=24, ε=2-23≈1.1920929×10-7, the standard spacing near 1 for IEEE binary32 precision.
Example 3: With p=11, ε=2-10=0.0009765625, illustrating the much coarser spacing of a lower-precision significand.
Example 4: With p=4, ε=2-3=0.125. The next representable value above 1 in this simplified binary precision is 1.125.
Example 5: Increasing precision from p=24 to p=25 halves epsilon, because 21-25 is one-half of 21-24.

Common mistakes

⚠️
Using the wrong input format

Keep lists comma separated, matrices as rows separated by semicolons, and modular inputs as integers.

⚠️
Ignoring assumptions

Some methods require positive probabilities, valid moduli, independent trials, or small educational input sizes.

⚠️
Confusing epsilon with the smallest positive number

Epsilon describes spacing near 1. Minimum positive normal or subnormal values are controlled by the exponent range and are much smaller quantities.

⚠️
Mixing precision bits with decimal digits

The formula uses binary significand bits p. A format advertised with a certain number of decimal significant digits cannot be inserted directly as p.

FAQ

Can I use decimals?⌄
Most numerical calculators allow decimals, but modular arithmetic and coding-theory tools usually require integers or binary strings.
Is this for homework checking?⌄
Yes. The page is designed to show both the answer and the reasoning pattern.
Why does the result sometimes say approximate?⌄
Some probability, floating-point, and numerical methods naturally produce approximations.
Why is binary64 epsilon 2^-52 when the precision is 53 bits?⌄
The spacing formula is 21-p. Substituting p=53 gives 2-52. The 53-bit precision includes the leading significand bit used for normalized binary64 values.
Is machine epsilon an absolute error bound for every number?⌄
No. It describes local spacing near 1 under the stated convention. Absolute spacing changes with the exponent, so floating-point error is usually discussed in relative terms for normalized values.
What is unit roundoff?⌄
For rounding to nearest, unit roundoff is commonly half the gap from 1 to the next larger representable number. Some sources call that quantity machine epsilon, which is why definitions should be checked.
Does a smaller epsilon always prevent numerical error?⌄
More precision reduces individual rounding increments, but algorithms can still amplify error through cancellation, ill-conditioning, overflow, underflow, or repeated operations. Epsilon is one component of numerical-error analysis.

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Formula Explorer connections

Interpretation: This relationship converts, summarizes or checks numerical quantities using standard arithmetic and measurement rules. Assumption: Use consistent units, preserve enough significant digits, and round only the final result unless the method states otherwise.

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