IEEE Floating Point Calculator

Calculator Form

Conversion mode

Precision

Decimal value

Stored hex or full bits

Input byte order for decode

Display digits

Report label

Rounding note

Bias note

Formula Used

Normal numbers use this form:

Value = (-1)^sign × 1.fraction₂ × 2^(exponent - bias)

Subnormal numbers use this form:

Value = (-1)^sign × 0.fraction₂ × 2^(1 - bias)

Single precision uses a bias of 127. Double precision uses a bias of 1023. Exponent fields filled with ones create infinity or NaN, depending on the fraction field.

How to Use This Calculator

Select decimal encoding or stored pattern decoding.
Choose single precision, double precision, or both.
Enter a decimal number, hexadecimal pattern, or full bit string.
Select the byte order when decoding stored bytes.
Press Calculate to place the result below the header.
Download a CSV or PDF report when needed.

Example Data Table

Decimal	Single hex	Double hex	Note
12.375	41460000	4028C00000000000	Exact binary fraction.
-0.15625	BE200000	BFC4000000000000	Negative exact fraction.
0.1	3DCCCCCD	3FB999999999999A	Rounded stored value.
Infinity	7F800000	7FF0000000000000	Special exponent pattern.

IEEE Floating Point Basics

IEEE floating point numbers store real values in fixed size binary fields. A value uses a sign bit, an exponent field, and a fraction field. This calculator separates those fields so each stored part becomes easy to inspect. It helps students, programmers, and testers see how a decimal value becomes machine data.

Why This Calculator Matters

Decimal numbers are familiar to humans. Computers usually store approximations in binary. Some values, such as 0.1, cannot be represented exactly. That can create small rounding differences in software, spreadsheets, sensors, and finance tools. The calculator shows the stored value, the bit pattern, and the hexadecimal form. These details make hidden rounding behavior visible.

Single And Double Precision

Single precision uses 32 bits. It has one sign bit, eight exponent bits, and twenty three fraction bits. Double precision uses 64 bits. It has one sign bit, eleven exponent bits, and fifty two fraction bits. Double precision normally gives more range and accuracy. Single precision is common in graphics, games, embedded work, and large numeric arrays.

Reading The Output

The sign bit tells whether the value is positive or negative. The exponent field stores a biased exponent. The fraction field stores the significand detail. Normal numbers use an implied leading one. Subnormal numbers do not. Special exponent patterns can represent zero, infinity, or NaN. The calculator marks these classes when the submitted value produces them.

Practical Uses

Use this tool when debugging numeric code. It is also useful when checking binary files, network packets, or hardware register values. Engineers can compare single and double precision output before choosing a data format. Teachers can use the table and export files for classroom examples. Developers can paste hexadecimal values and decode stored numbers during testing.

Workflow Tips

Start with a simple value, such as 1.5 or -13.25. Then try a difficult value, such as 0.1. Compare the encoded value against the original input. Notice the error field clearly.

Accuracy Notes

The conversion follows common IEEE 754 layout rules. The stored result depends on the selected precision. Rounding is handled by the runtime format. Always compare results with a tolerance when programming. Exact equality is unsafe for many decimal fractions.

FAQs

What is an IEEE floating point number?

It is a binary format for storing real numbers. It divides a stored value into a sign bit, exponent field, and fraction field. The layout helps computers represent very small and very large values.

What does single precision mean?

Single precision uses 32 bits. It has one sign bit, eight exponent bits, and twenty three fraction bits. It is smaller than double precision and is often used where memory matters.

What does double precision mean?

Double precision uses 64 bits. It has more exponent and fraction bits than single precision. It usually provides better accuracy and a wider numeric range for scientific and general calculations.

Why is 0.1 not exact?

The value 0.1 has a repeating binary expansion. The format must round it to a limited number of fraction bits. This creates a tiny difference between the typed value and stored value.

What is the exponent bias?

The exponent bias is a stored offset. Single precision uses 127. Double precision uses 1023. Subtracting the bias from the raw exponent gives the actual exponent for normal values.

What is a subnormal number?

A subnormal number is very close to zero. It has a raw exponent of zero and a nonzero fraction. It does not use the implied leading one used by normal values.

Can I decode hexadecimal values?

Yes. Choose the decode mode and paste a hex pattern. You may also paste a full binary bit pattern. Select the correct precision and byte order for accurate interpretation.

Why are CSV and PDF exports useful?

Exports help save results for lessons, bug reports, and documentation. CSV works well in spreadsheets. PDF is useful when you need a simple printable report.