IEEE 754 Floating-Point Converter

Computers store most non-integer numbers in IEEE 754 binary floating point, which cannot represent 0.1 or 1/3 exactly. This converter shows the exact bits a number becomes in half (16-bit), single (32-bit float) and double (64-bit double) precision, splits them into sign, exponent and mantissa, and prints the exact decimal value that is really stored together with the rounding error. It also decodes a hex or binary bit pattern back into a number, including zeros, subnormals, infinities and NaN.

How to use this tool

1. Type a number — a decimal such as 0.1, a fraction such as 1/3, a C hex float, inf or nan — or paste a bit pattern.
2. Choose whether the value is a number or hex/binary bits.
3. Pick half, single or double precision.
4. Read the hex and binary encoding, the field breakdown, the exact stored value and the error; the table compares all three formats.

Formula

value = (−1)^sign × 1.mantissa₂ × 2^(exponent − bias) · subnormal: (−1)^sign × 0.mantissa₂ × 2^(1 − bias)

Bias = 15 (half), 127 (single), 1023 (double). An exponent field of all zeros means zero or a subnormal; all ones means infinity (mantissa 0) or NaN. Conversion uses exact integer arithmetic and rounds to nearest, ties to even, so the result matches a correctly rounding compiler or parser; the shortest round-trip decimal is the shortest number that converts back to the same bits.

Example

0.1 in single precision becomes 0x3DCCCCCD = 0 01111011 10011001100110011001101: sign 0, exponent 123 − 127 = −4, significand 1.60000002384185791015625. The stored value is exactly 0.100000001490116119384765625, so the rounding error is +0.000000001490116119384765625. In double precision 0.1 becomes 0x3FB999999999999A, about 0.1000000000000000055511.