How does IEEE 754 actually store a floating point number?

It splits the number into three fields: a sign bit, a biased exponent and a mantissa (fraction). The value is reconstructed as (−1)^sign × 1.mantissa × 2^(exponent − bias). For single precision the layout is 1 sign bit, 8 exponent bits and 23 mantissa bits; the bias is 127. So 1.0 becomes sign 0, exponent 127, mantissa 0, which is the bit word 0x3F800000.

What do the sign, exponent and mantissa parts mean?

The sign bit is 0 for positive and 1 for negative. The exponent is stored with a bias added (127 for single, 1023 for double) so it can hold negative powers without a separate sign; subtract the bias to get the real exponent. The mantissa holds the fraction after an implied leading 1. Example: in 0x3E200000 the sign is 0, the raw exponent is 124 so the real exponent is −3, and the value works out to 1.25 × 2^−3 = 0.15625.

Why can a computer not store 0.1 exactly?

0.1 in binary is 0.0001100110011… repeating forever, just as 1/3 repeats in decimal. IEEE 754 has a fixed 52-bit mantissa for doubles, so the value is rounded to the nearest representable number, which is about 0.1000000000000000055511151231257827. That tiny error is why 0.1 + 0.2 prints 0.30000000000000004. Paste 0.1 into the tool and you will see the nonzero repeating mantissa for yourself.

What is the difference between single and double precision?

Single precision uses 32 bits (8 exponent, 23 mantissa) and gives about 7 significant decimal digits, with a range up to roughly 3.4 × 10^38. Double precision uses 64 bits (11 exponent, 52 mantissa), about 15 to 17 significant digits, range up to roughly 1.8 × 10^308. Doubles cost twice the memory but lose far less to rounding, which is why most languages default to them for `number` and `double`.

How are Infinity and NaN encoded as bit patterns?

Both use the all-ones exponent field. If the mantissa is all zero, the value is infinity (sign bit decides positive or negative), so single-precision +Infinity is 0x7F800000. If any mantissa bit is set, it is NaN, for example 0x7FC00000 is a common quiet NaN. The tool flags these with a class badge, so you can paste a suspicious hex word and instantly see whether it is a real number, an infinity or a NaN.

What are subnormal (denormal) numbers?

When the exponent field is all zeros but the mantissa is not, the implied leading 1 is dropped and the number is subnormal. These fill the gap between the smallest normal value and zero so underflow is gradual rather than a sudden jump. For single precision anything below about 1.18 × 10^−38 becomes subnormal; type 1e-40 in single precision and the class badge reads subnormal.

Debug a floating point mismatch in code

Your test asserts 0.1 + 0.2 === 0.3 and it fails. Paste 0.1, then 0.2, then 0.3 in double precision and read the mantissas side by side. You will see that the stored bits for 0.1 + 0.2 differ from the stored bits for 0.3 by one unit in the last place, which is exactly why the equality breaks and why you should compare with an epsilon instead.

Understand a raw value from a memory dump or packet

A binary protocol hands you four bytes, 0x42 0x28 0x00 0x00, that are supposed to be a float. Concatenate them to 0x42280000, drop it into the bit-pattern field in single precision, and the tool decodes it to 42.0. No more squinting at a hex editor and doing the exponent math by hand.

Teach or learn how floats really work

A CS lecturer can show 1.0 as 0x3F800000, then 2.0, then 0.5, and let students watch only the exponent change while the mantissa stays zero. Toggle to double precision and the same numbers reappear with an 11-bit exponent. The colored bit strip makes the three fields obvious without drawing them on a whiteboard.

Verify a hand-written float encoder or decoder

Writing serialization for an embedded target without a standard library? Encode a handful of values in your own code, then check each one against this tool. If your encoder emits 0x3E200000 for 0.15625 and the tool agrees, your sign, exponent and mantissa packing is correct.

IEEE 754 Floating Point Converter (Single + Double)

Decimal to single and double precision bits, sign exponent mantissa split, full hex and binary, all in your browser

Runs locally
Category Developer & DevOps
Best for Formatting, validating, shrinking, or inspecting code-adjacent text.

IEEE 754 Floating Point

Precision:

Decimal valuee.g. 0.15625, -2, 3.14159, 1e-40

Bit pattern (binary or hex)binary digits or hex, 0x / 0b optional

Try:

00111110001000000000000000000000

Sign0 (1 bit)0 → +

Exponent01111100 (8 bits)Raw 124 · E = exp − bias = -3

Mantissa23 bits2097152

Hex0x3e200000

Binary00111110001000000000000000000000

Decoded value0.15625

Class: normalbias = 127

value = (−1)^sign × 1.mantissa × 2^(exp − bias)

Everything runs on ArrayBuffer in your browser — the same encoding the CPU uses. Nothing is uploaded.

What this tool does

A precise IEEE 754 floating point converter for developers and CS students. Type a decimal value and see exactly how the CPU stores it: the single (32-bit) or double (64-bit) layout broken into its sign bit, exponent field and mantissa, plus the complete hex word and the full binary string. Go the other way too. Paste a hex or binary bit pattern and the tool decodes it back to the decimal value it represents. Every conversion runs on a real ArrayBuffer and DataView, the same encoding path the hardware uses, so 0.15625 lands on 0x3E200000 and 0.1 shows its true repeating mantissa rather than a rounded approximation. Special values are labelled clearly: positive and negative zero, infinity, NaN and subnormal (denormal) numbers each get their own class badge, with the biased exponent and the real exponent both shown. One-click copy for hex, binary and the decoded value. The decimal input and precision are saved in the URL so a shared link reopens the exact same number.

Tool details

Input: Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Developer & DevOps · Developer; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How IEEE 754 Floating Point Converter fits into your work

Use it in the small gaps between coding, reviewing, debugging, and shipping.

Developer jobs

Formatting, validating, shrinking, or inspecting code-adjacent text.
Preparing snippets for documentation, tickets, commits, or handoff.
Checking a small payload quickly without switching tools.

Developer checks

Run irreversible transforms like minify or obfuscate on a copy.
Keep secrets out of pasted snippets unless the tool explicitly stays local.
Use your normal tests or linter before shipping transformed code.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Debug a floating point mismatch in code
Your test asserts 0.1 + 0.2 === 0.3 and it fails. Paste 0.1, then 0.2, then 0.3 in double precision and read the mantissas side by side. You will see that the stored bits for 0.1 + 0.2 differ from the stored bits for 0.3 by one unit in the last place, which is exactly why the equality breaks and why you should compare with an epsilon instead.
Understand a raw value from a memory dump or packet
A binary protocol hands you four bytes, 0x42 0x28 0x00 0x00, that are supposed to be a float. Concatenate them to 0x42280000, drop it into the bit-pattern field in single precision, and the tool decodes it to 42.0. No more squinting at a hex editor and doing the exponent math by hand.
Teach or learn how floats really work
A CS lecturer can show 1.0 as 0x3F800000, then 2.0, then 0.5, and let students watch only the exponent change while the mantissa stays zero. Toggle to double precision and the same numbers reappear with an 11-bit exponent. The colored bit strip makes the three fields obvious without drawing them on a whiteboard.
Verify a hand-written float encoder or decoder
Writing serialization for an embedded target without a standard library? Encode a handful of values in your own code, then check each one against this tool. If your encoder emits 0x3E200000 for 0.15625 and the tool agrees, your sign, exponent and mantissa packing is correct.

Common pitfalls

Reading the stored exponent as the real one. The exponent field is biased — subtract 127 for single or 1023 for double to get the true power of two. A raw exponent of 124 means 2^−3, not 2^124. The tool shows both the raw value and E = exp − bias so you do not have to.
Expecting decimal fractions to be exact. Values like 0.1, 0.2 and 0.3 have no finite binary form, so they are stored rounded. Comparing them with === fails by design. Paste them in and the nonzero repeating mantissa explains why an epsilon comparison is the right fix.
Pasting a bit pattern of the wrong width. A 64-bit hex word will not decode as a single-precision float and vice versa. Match the precision toggle to the number of bits you have — 8 hex digits for single, 16 for double — or the tool rejects the input.

Privacy

Every step runs as plain JavaScript inside your browser tab using ArrayBuffer and DataView. The decimal value you type, the bit patterns you paste and the decoded results never leave the page, and nothing is logged. The one thing to know: the decimal input and the precision are encoded in the page URL so a shared link reopens the same number, which means a link pasted into chat records that value in the recipient server's access log. For anything sensitive, use the copy buttons and paste the text rather than sharing the URL.

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IEEE 754 Floating Point Converter (Single + Double)

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How IEEE 754 Floating Point Converter fits into your work

Developer jobs

Developer checks

Good next steps

Real-world use cases

Debug a floating point mismatch in code

Understand a raw value from a memory dump or packet

Teach or learn how floats really work

Verify a hand-written float encoder or decoder

Common pitfalls

Privacy

FAQ

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