01
Select the meaning of the text
Start with decimal 0.1 and binary32. Selecting hex changes how the current text is read; it does not convert the text. Paste a complete word, with its most significant bit on the left.
Numbers, Bitwise & Units
IEEE 754 converterSee the value behind the bits
Compare the exact decimal input with its stored value, or decode a complete hexadecimal or binary word.
IEEE 754 converter
Input
Your input is processed in this browser and is not sent to a server.
Decimal uses a dot and optional e/E exponent. Hex requires 8/16 digits; binary requires 32/64 bits. Raw words allow 0x/0b and spaces or tabs.
Results
Decimal display · 9 / 17 significant digits
0.100000001
Normal
Sign bit · Bits: 1
0
Stored exponent · Bits: 8
01111011
Fraction field · unsigned integer · Bits: 23
10011001100110011001101
- Precision
- 32-bit · binary32
- Input representation
- Decimal value
- Hexadecimal word
- 0x3DCCCCCD
- Binary word
- 00111101110011001100110011001101
- Sign bit
- 0
- Stored exponent
- 123
- Effective exponent
- -4
- Fraction field · unsigned integer
- 5033165
- Implicit leading bit
- 1
- Exact value formula
- 13421773 × 2^-27
- Conversion outcome
- Rounded to nearest, ties to even
Exact stored decimal
0.100000001490116119384765625
Signed error · stored − input
0.000000001490116119384765625
Options
Decimal rounds directly to the selected format: nearest, ties to even. Changing precision or representation reinterprets the current text. Limit: 1024 characters; explicit decimal exponent −4096…4096.
Inspection ready.
An IEEE 754 converter is useful when a debugger shows 0.1 but a comparison fails. The short decimal display conceals the stored fraction; open the exact expansion and compare its signed error with the original input.
This floating point converter keeps the mathematical value separate from its encoded word. Use decimal to IEEE 754 for a new value, or read a captured hex pattern without normalizing its NaN payload.
Why use this IEEE 754 converter?
Spot a rounding difference
The IEEE 754 converter rounds the exact decimal target separately for each precision. For 0.1, binary32 stores 0.100000001490116119384765625; binary64 is closer but still not exact. The signed difference tells you whether the stored value lies above or below your target.
Keep different NaNs different
A register dump may contain 7FC00001 or 7FC00002. Both are quiet NaNs, yet their payloads are 1 and 2. Reading the original word preserves that distinction, together with its sign and quiet bit, instead of replacing every NaN with one encoding.
Check meaning before changing precision
02
Compare the two decimal displays
The compact display uses 9 significant digits for binary32 or 17 for binary64. Expand the exact stored decimal to inspect every digit. Error is stored minus input, so a negative error means the stored value is smaller.
03
Review a replacement before applying it
Example opens a preview with five choices, including negative zero and a payload-bearing NaN. Apply changes the input; Undo restores it and its result. Automatic mode waits 350 ms; switch it off for Inspect or Ctrl/Cmd + Enter.
Read the complete stored word
Decode sign, exponent and fraction
The bit diagram groups 1/8/23 bits for binary32 and 1/11/52 for binary64. Stored exponent, effective exponent and implicit leading bit are shown separately. Normal values use a leading 1, while subnormal values use 0.
Use exact decimal arithmetic for conversion
The engine rounds a decimal rational directly to the target significand, avoiding a decimal → double → float detour. A value just above the binary32 midpoint 1.000000059604644775390625 therefore selects the next float even when Number would first collapse it to the midpoint.
Recover words and test vectors
Float to hex and hex to float share this page. Binary and hex outputs preserve every bit, including signaling NaNs. Copy either word or export a report with the input, precision, fields, exact decimal and rounding error for a reproducible check.
Rounding, syntax and format boundaries
- Two binary formats and one rounding rule
- Only binary32 and binary64 are supported. Decimal conversion uses round to nearest, ties to even. At a halfway point the retained significand with an even low bit wins. There are no half, bfloat16, extended precision or selectable rounding modes.
- A raw word has an exact width
- Hex must contain 8 or 16 digits; binary must contain 32 or 64 bits. Matching prefixes and ASCII spaces or tabs are allowed. Other separators, incomplete words and excess bits are rejected. The tool never pads, truncates or swaps bytes to make input fit.
- Overflow and tiny values remain visible
- Out-of-range finite input may round to signed Infinity; very small input may become subnormal or signed zero. These are conversion outcomes, not syntax failures. Signed error is available only for a finite decimal input and a finite result, not for raw words or special literals.
- Payload meaning is explicit
- The fraction’s highest bit is interpreted as the quiet flag: 1 means quiet. Payload excludes that bit. Decimal NaN text creates a canonical quiet NaN with payload zero; use hex or binary to supply a different pattern. Hardware exceptions and legacy architecture conventions are not simulated.
- One bounded value at a time
- Enter at most 1024 characters, with an explicit decimal exponent from −4096 through 4096. Decimal syntax uses ASCII digits and a dot, without commas, expressions or line breaks. No files, URLs, batches, byte-order conversion, bit-toggle buttons, neighboring-value navigation, ULP distance or relative error are provided.
IEEE 754 converter questions
Why does 0.1 need binary rounding?
A finite binary fraction has a denominator that is a power of two. One tenth also needs a factor of five, so its binary expansion repeats. The selected format keeps a finite significand and rounds the rest. View the full exact decimal to see the approximation instead of relying on a short display.
How do float32 and float64 differ?
Binary32 has 23 stored fraction bits and an implicit bit for normal values; binary64 has 52 plus the implicit bit. Their exponent fields are 8 and 11 bits. Double precision offers finer spacing and a wider range, but values such as 0.1 remain inexact in either format.
How can a hex word recover a decimal?
Choose hexadecimal input and the matching precision, then paste the full word. For example, 3F800000 in binary32 gives exactly 1. This reinterprets the word as floating point; it does not convert the hexadecimal integer 1065353216 into a floating-point number.
Which bit fields identify subnormal values?
A stored exponent of zero with a nonzero fraction identifies a subnormal. The leading bit is zero and the effective exponent is −126 for binary32 or −1022 for binary64. This fills the gap toward zero gradually; a zero fraction instead represents signed zero.
How are Infinity, NaN and signed zero represented?
An all-ones exponent with a zero fraction is Infinity; a nonzero fraction is NaN. Signed zero has a zero exponent and fraction but retains the sign bit. For NaN, the separate quiet bit and payload fields explain why different words can belong to the same value class.
Something looks wrong?
Inspect another value
Use the IEEE 754 converter to check a captured word or a difficult decimal boundary. Change one input at a time, then copy the report when the current result is ready.