64 BIT Floating-Point Utility

IEEE 754 Double Precision Converter

Convert decimal values to IEEE 754 double-precision binary and hexadecimal, or decode a 64-bit IEEE 754 binary or hexadecimal pattern back to its numeric value. Inspect the sign bit, 11-bit exponent, 52-bit fraction, bias, classification and complete Binary64 representation.

✓ IEEE 754 Binary64 ✓ Decimal → Binary ✓ Hex → Decimal ✓ Binary → Decimal ✓ 11-bit Exponent ✓ 52-bit Fraction ✓ NaN & Infinity

754

IEEE 754 Binary64 Conversion

● Ready

Input Value Decimal example: 10.5, -3.25, 0.1, Infinity, -Infinity or NaN.

Conversion Options

Input Type

Binary Display

IEEE 754 double precision: Binary64 contains 1 sign bit, 11 exponent bits and 52 fraction bits. Normalized values use exponent bias 1023.

IEEE 754 Double Precision Result Converted

IEEE 754 Binary64 Representation —

Sign · 1 bit —

Exponent · 11 bits —

Fraction · 52 bits —

Decimal Value —

Hexadecimal —

Sign —

Biased Exponent —

Unbiased Exponent —

Exponent Hex —

Fraction Hex —

Classification —

Precision —

Exponent Bias 1023

Total Bits 64

Status —

IEEE 754 Field Breakdown —

Value Reconstruction —

Validation & Notes —

What Is IEEE 754 Double Precision?

IEEE 754 double precision is a 64-bit binary floating-point format commonly called Binary64 or simply a double. It is widely used by programming languages, processors, scientific applications, databases and numerical software when more precision or range is needed than the 32-bit single-precision format can provide.

A Binary64 value divides its 64 bits into three fields: one sign bit, an 11-bit exponent and a 52-bit stored fraction. For normal finite numbers there is also an implicit leading binary 1, giving approximately 53 bits of significand precision.

IEEE 754 Double Precision Bit Layout

The 64-bit representation has a fixed structure. The most significant bit is the sign, followed by the exponent field and then the fraction field.

Bit 63:
Sign

Bits 62–52:
Exponent
11 bits

Bits 51–0:
Fraction / Mantissa field
52 stored bits

Total:
1 + 11 + 52
= 64 bits

Field	Bits	Purpose
Sign	1	Determines positive or negative sign
Exponent	11	Stores the exponent using bias 1023
Fraction	52	Stores the fractional part of the significand

IEEE 754 Double Precision Formula

For a normalized finite Binary64 value, the numerical value can be described using the sign, exponent and fraction fields.

Value =
(-1)^Sign
×
(1 + Fraction)
×
2^(Exponent - 1023)

The stored fraction represents binary fractions such as 2^-1, 2^-2, 2^-3 and so on. The leading 1 in normalized values is implicit and is not physically stored in the 52-bit fraction field.

Example: Convert 10.5 to IEEE 754 Double Precision

The decimal value 10.5 has an exact finite binary representation, making it a useful example for understanding Binary64.

Decimal:
10.5

Binary:
1010.1

Normalized:
1.0101 × 2^3

Sign:
0

Unbiased exponent:
3

Biased exponent:
3 + 1023
= 1026

Exponent binary:
10000000010

IEEE 754 Hex:
4025000000000000

The remaining fraction bits after the significant binary digits are filled with zeros to produce the complete 52-bit fraction field.

Example: Decode 3FF0000000000000

The hexadecimal pattern 3FF0000000000000 is one of the most recognizable Binary64 values because it represents decimal 1 exactly.

Hex:
3FF0000000000000

Binary:
0011111111110000000000000000000000000000000000000000000000000000

Sign:
0

Exponent:
01111111111
= 1023

Unbiased exponent:
1023 - 1023
= 0

Fraction:
0

Value:
1 × 2^0
= 1

Exponent Bias 1023

The IEEE 754 Binary64 exponent field is unsigned, but floating-point numbers need both positive and negative exponents. A bias of 1023 solves this by storing the mathematical exponent after adding 1023.

Stored exponent =
Actual exponent + 1023

Example:

Actual exponent:
3

Stored:
3 + 1023
= 1026

Exponent values 0 and 2047 have special meanings, so they are not used as ordinary normalized exponents.

Normal and Subnormal Double-Precision Values

When the exponent field is between 1 and 2046, the number is normalized and uses an implicit leading 1 in the significand. When the exponent field is zero but the fraction is nonzero, the value is subnormal.

Exponent	Fraction	Classification
1–2046	Any	Normal finite number
0	0	Positive or negative zero
0	Nonzero	Subnormal number
2047	0	Positive or negative infinity
2047	Nonzero	NaN

Positive Zero and Negative Zero

IEEE 754 contains two zero encodings. Positive zero has a sign bit of zero, while negative zero has a sign bit of one. All exponent and fraction bits are zero in both cases.

+0:

0000000000000000


-0:

8000000000000000

Although positive and negative zero compare as equal in many calculations, the sign can matter in some floating-point operations and mathematical functions.

Infinity in IEEE 754 Binary64

Positive and negative infinity use an exponent field containing all ones and a fraction field containing all zeros. The sign bit distinguishes positive from negative infinity.

Positive Infinity:

7FF0000000000000


Negative Infinity:

FFF0000000000000

NaN in IEEE 754 Double Precision

NaN means Not a Number. It uses an all-ones exponent field together with a nonzero fraction. NaNs can arise from undefined floating-point operations and may also carry additional payload information in their fraction bits.

Typical NaN pattern:

7FF8000000000000

Exponent:
11111111111

Fraction:
Nonzero

Classification:
NaN

Double Precision vs Single Precision

IEEE 754 single precision uses 32 bits, whereas double precision uses 64 bits. Binary64 provides both a larger exponent range and significantly greater precision.

Property	Single / Binary32	Double / Binary64
Total bits	32	64
Sign bits	1	1
Exponent bits	8	11
Fraction bits	23	52
Exponent bias	127	1023
Approx. decimal precision	About 7 digits	About 15–17 digits

Why Decimal 0.1 Is Not Exact in Binary64

Many decimal fractions cannot be represented using a finite binary fraction. Decimal 0.1 is one of the best-known examples. Its binary fractional expansion repeats indefinitely, so IEEE 754 stores the closest representable Binary64 value.

Decimal requested:
0.1

IEEE 754 Binary64 hex:
3FB999999999999A

Stored value:
nearest representable double to 0.1

This behavior explains many small floating-point rounding effects seen in programming languages and numerical calculations.

Hexadecimal IEEE 754 Representation

Hexadecimal is particularly convenient for displaying floating-point bit patterns because every hexadecimal digit represents exactly four bits. Therefore a 64-bit double always fits into exactly 16 hexadecimal digits.

64 bits
÷ 4 bits per hex digit
= 16 hexadecimal digits

Example:

0100000000100101000000000000000000000000000000000000000000000000

Hex:

4025000000000000

Where IEEE 754 Double Precision Is Used

Binary64 is widely used in scientific computing, engineering calculations, JavaScript numbers, C and C++ double values, Java double values, Python floats, databases, spreadsheets, simulations, graphics, numerical algorithms and many hardware floating-point units.

A converter that exposes the raw sign, exponent and fraction fields is useful for debugging numerical software, studying floating-point representation, inspecting memory dumps and understanding precision or rounding problems.

IEEE 754 Double Precision Converter FAQs

How many bits are in IEEE 754 double precision?

Double precision contains 64 bits: one sign bit, 11 exponent bits and 52 stored fraction bits.

What is the exponent bias for Binary64?

The exponent bias is 1023. For ordinary normalized numbers, subtract 1023 from the stored exponent to obtain the unbiased exponent.

What is 1.0 in IEEE 754 double precision?

Decimal 1.0 has hexadecimal representation 3FF0000000000000 in IEEE 754 Binary64.

What is 10.5 in IEEE 754 double precision?

Decimal 10.5 is represented by hexadecimal 4025000000000000.

How many hexadecimal digits represent a double?

Exactly 16 hexadecimal digits represent all 64 bits of an IEEE 754 Binary64 value.

How precise is a double?

Binary64 provides 53 bits of significand precision for normalized numbers, which corresponds to roughly 15 to 17 significant decimal digits.

Can IEEE 754 represent negative zero?

Yes. Negative zero has sign bit 1 with all exponent and fraction bits set to zero. Its hexadecimal representation is 8000000000000000.

How is infinity represented?

Infinity uses an exponent containing all ones and a fraction containing all zeros. The sign bit determines positive or negative infinity.

How is NaN represented?

NaN uses an exponent containing all ones together with a nonzero fraction. Multiple different NaN bit patterns are possible.

What is a subnormal double?

A subnormal value has an exponent field of zero and a nonzero fraction. It does not use the implicit leading 1 used by normal floating-point numbers.

Why is 0.1 not represented exactly?

Decimal 0.1 has an infinitely repeating binary expansion, so Binary64 stores the nearest representable finite binary value.

Can this converter decode raw 64-bit hexadecimal?

Yes. Select IEEE 754 Hexadecimal and enter exactly 16 hexadecimal digits. The tool decodes the complete Binary64 pattern.