IEEE 754 double calculator
IEEE 754 double precision is the 64-bit floating-point format behind the default number type in most programming languages, spreadsheets and scientific software. This calculator converts any decimal number you enter into its exact IEEE 754 double-precision binary representation. It separates the 64 bits into the three fields the standard defines: one sign bit, 0 for positive and 1 for negative, an 11-bit biased exponent that encodes the power of two with a bias of 1023, and a 52-bit mantissa, the fraction that follows the implied leading 1. It also shows the full 64-bit string and the compact 16-digit hexadecimal form, which is how the value appears in memory or a binary file. Compared with single precision, the wider exponent gives a far larger range and the longer mantissa gives about 15 to 16 significant decimal digits, which is why double precision is the workhorse of numerical computing. Laying out the fields makes clear why certain decimals cannot be stored exactly. Enter your own value, positive or negative, to inspect its encoding or to check an exercise. Every field here is produced deterministically by the IEEE 754 double-precision encoding, explained in full below with a worked example that reconciles exactly to the calculator.
IEEE 754 double precision packs a number as 1 sign bit, 11 exponent bits (bias 1023), 52 mantissa bits. The value 12.375 has a biased exponent of 1026 (true exponent 3) and encodes to hexadecimal 4028C00000000000.
IEEE 754 double-precision layout
value = (-1)^sign x 1.mantissa x 2^(exponent - 1023)
bit 63 = sign (0 positive, 1 negative)
bits 62..52 = exponent, stored with a bias of 1023
bits 51..0 = mantissa (fraction after the implied leading 1)
total = 64 bits = 16 hexadecimal digits
The number is normalized to 1 point something times a power of two. The sign goes in the top bit, the true power of two plus 1023 goes in the 11-bit exponent field, and the fractional bits go in the 52-bit mantissa, with the leading 1 implied and not stored.
Worked example
Encode the decimal value 12.375 in IEEE 754 double precision.
- 12.375 in binary is 1100.011 = 1.100011 x 2^3, so the true exponent is 3
- Biased exponent = 3 + 1023 = 1026 = 10000000010 (11 bits)
- Mantissa = 100011 padded to 52 bits, sign bit = 0
- The 64-bit pattern groups into hex as 4028C00000000000
- So 12.375 encodes to 4028C00000000000
The value 12.375 encodes to 4028C00000000000 in hexadecimal, with a biased exponent of 1026. These are the calculator's default inputs, so the results above match the widget exactly.
IEEE 754 double calculator: frequently asked questions
What is IEEE 754 double precision?
IEEE 754 double precision is a 64-bit standard for storing floating-point numbers, the basis of the double type and of the default number in many languages. It uses a sign bit, an 11-bit biased exponent and a 52-bit mantissa, giving roughly 15 to 16 significant decimal digits of precision.
How does double precision differ from single?
Single precision uses 32 bits with an 8-bit exponent and a 23-bit mantissa; double precision uses 64 bits with an 11-bit exponent and a 52-bit mantissa. The wider exponent gives a much larger range and the longer mantissa gives far more precision, at the cost of twice the storage.
What is the exponent bias of 1023?
The 11-bit exponent field stores the true power of two plus a bias of 1023, so it can represent exponents from about minus 1022 to plus 1023 as an unsigned number. A stored value of 1026 means an actual exponent of 1026 minus 1023, which is 3.
Why is the hexadecimal 16 digits long?
Sixty-four bits group into sixteen blocks of four bits, and each four-bit block is one hexadecimal digit, so a double precision value is always sixteen hex digits. This compact form is how the value typically appears in a memory dump or a binary file.
Why can some decimals not be stored exactly?
Many decimal fractions have no finite binary form, so they are rounded to the nearest value the 52-bit mantissa can hold, which is the source of small floating-point errors. A value like 12.375, a sum of exact powers of two, stores perfectly. The encoding is deterministic and reconciles exactly to the worked example.
Official sources
- Numerical computing standards and reference data: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.