About

Every floating-point number stored in a computer follows the IEEE 754 standard. A 64-bit double-precision float is partitioned into three fields: 1 sign bit, 11 exponent bits, and 52 mantissa (significand) bits. Misunderstanding this layout causes silent precision errors in financial calculations, physics simulations, and hash-dependent systems. This tool performs real binary decomposition using ArrayBuffer and DataView, letting you inspect and replace the mantissa fraction bits of any double-precision number. The result is a genuinely reassembled IEEE 754 value, not a textual approximation.

Note: the mantissa field stores only the fractional part. IEEE 754 normal numbers assume an implicit leading 1 bit (the "hidden bit"), so the true significand is 1.f where f is the stored 52-bit fraction. Subnormal numbers (exponent all zeros) lack this hidden bit. This tool preserves exponent and sign while swapping only the fraction bits. Values outside the 52-bit range are truncated from the right.

Formulas

An IEEE 754 double-precision number v is composed from three binary fields:

v = (−1)^s × 2^{e − 1023} × (1 + m2⁵²)

For subnormal numbers (exponent field = 0):

v = (−1)^s × 2⁻¹⁰²² × m2⁵²

Where s = sign bit (0 or 1), e = biased exponent (unsigned 11-bit integer, 0 - 2047), m = mantissa fraction (unsigned 52-bit integer, 0 - 2⁵²−1). This tool keeps s and e intact while replacing m with a user-supplied 52-bit binary string.

Reference Data

Field	Bits	Position	Range / Purpose
Sign	1	Bit 63	0 = positive, 1 = negative
Exponent	11	Bits 62 - 52	Biased by 1023; range 0 - 2047
Mantissa (fraction)	52	Bits 51 - 0	Fractional significand bits
Exponent 0	Subnormal number (no hidden bit) or ±0
Exponent 2047	±∞ (mantissa 0) or NaN (mantissa ≠ 0)
Smallest subnormal	5 × 10⁻³²⁴
Largest subnormal	≈ 2.225 × 10⁻³⁰⁸
Smallest normal	2.2250738585072014 × 10⁻³⁰⁸
Largest finite	1.7976931348623157 × 10³⁰⁸
Machine epsilon	2⁻⁵² ≈ 2.22 × 10⁻¹⁶
Precision (decimal)	15 - 17 significant digits
Bias	1023
Hidden bit	Implicit 1 for normal numbers
NaN payload	Any non-zero mantissa with exponent 2047
Quiet NaN	Bit 51 of mantissa = 1
Signaling NaN	Bit 51 = 0, remaining ≠ 0

Frequently Asked Questions

In IEEE 754 terminology, the stored 52-bit field is the "trailing significand" or "fraction." The full significand includes the implicit leading 1 for normal numbers, making it 53 bits effective. "Mantissa" is technically a logarithm term but is widely used colloquially to mean the stored fraction bits. This tool modifies the 52-bit fraction field only.

It does for normal numbers. When the mantissa fraction is all zeros, the significand equals exactly 1.0, so the value becomes (−1)^s × 2^(e−1023). However, if the exponent field is 0 (subnormal), mantissa all-zeros yields ±0. If the exponent is 2047, mantissa all-zeros yields ±∞.

Yes. Set the exponent to 2047 (all 11 bits = 1) and provide any non-zero mantissa. If bit 51 of the mantissa is 1, you get a quiet NaN. If bit 51 is 0 and at least one other mantissa bit is 1, the standard defines it as a signaling NaN, though JavaScript's DataView will read both as NaN.

Strings shorter than 52 characters are right-padded with zeros (the missing least-significant bits default to 0). Strings longer than 52 characters are truncated from the right to exactly 52 bits, discarding the excess low-order bits. A toast notification warns you in either case.

The sign is always preserved. The exponent is also preserved, so the magnitude stays within the same power-of-two interval [2^(e−1023), 2^(e−1022)). Within that interval, increasing the mantissa increases the absolute value linearly. The mantissa all-zeros gives the lower bound, all-ones gives the upper bound minus one ULP.

None at the binary level. The tool performs exact bit manipulation via ArrayBuffer. The decimal display uses JavaScript's native Number-to-string conversion, which guarantees a round-trip-safe representation (15-17 significant digits). The displayed decimal is the shortest string that uniquely identifies the underlying 64-bit pattern.