IEEE 754 float converter

Almost every computer stores fractional numbers in the IEEE 754 floating-point format. Understanding its bit layout explains surprising behaviour, like why 0.1 + 0.2 is not exactly 0.3 in JavaScript, Python and most other languages. This converter shows the exact bits of single-precision (32-bit) and double-precision (64-bit) values.

How to use it

1. Enter a decimal number, or a hex or binary bit pattern.
2. See the sign, exponent and mantissa fields, the hex representation and the exact stored value.

The layout

• Single (float32): 1 sign bit, 8 exponent bits (bias 127), 23 fraction bits. About 7 significant decimal digits.
• Double (float64): 1 sign bit, 11 exponent bits (bias 1023), 52 fraction bits. About 15 to 17 significant digits.

Value = (-1)^sign × 1.fraction × 2^(exponent - bias) for normal numbers.

Worked example

-6.25 = -1.5625 × 2². Sign = 1, exponent = 2 + 127 = 129 = 10000001, fraction = .5625 = 1001000.... As float32 the bits are 1 10000001 10010000000000000000000, or 0xC0C80000.

Why 0.1 is inexact

0.1 has no finite binary representation, just as 1/3 has none in decimal. The nearest double is 0.1000000000000000055511151231257827. Adding two such approximations gives 0.30000000000000004.

Special values

• Exponent all ones with zero fraction: positive or negative infinity.
• Exponent all ones with non-zero fraction: NaN.
• Exponent all zeros: zero (positive and negative) and subnormal numbers.

For money, use integers or decimal types instead of floats.