🔢IEEE-754 Float Converter

About this tool

Type a decimal number such as 0.1, −2.5 or 1e-10. Both single-precision (32-bit) and double-precision (64-bit) layouts appear, colour-coded into sign, exponent and mantissa, with the hex value, the exact value stored and the rounding error. To decode raw bits, paste 8 or 16 hex digits.

How floating-point numbers are stored

IEEE-754 is the standard almost every processor uses for non-integer numbers. A number is stored like scientific notation in base 2: a sign bit, an exponent and a fraction (the mantissa). A 32-bit float has 1 sign bit, 8 exponent bits with a bias of 127, and 23 fraction bits. A 64-bit double has 1, 11 and 52 bits, with a bias of 1023. Normal numbers have an implied leading 1 that isn't stored, which buys an extra bit of precision. The catch is that most decimal fractions repeat forever in binary, just as 1/3 does in decimal, so they have to be rounded. 0.1 as a double is really 0.1000000000000000055511151231257827…, stored as hex 3FB999999999999A. As a float, it's 0.100000001490116…, stored as 3DCCCCCD. These tiny errors are why 0.1 + 0.2 prints 0.30000000000000004 in JavaScript and Python, and why money should be handled in whole cents or with decimal types. Special bit patterns represent zero (including −0), infinity, NaN and subnormal numbers near zero. Single precision keeps about 7 significant decimal digits, and double about 15 to 17. Honest limits: this shows the standard binary formats only, not half precision, bfloat16 or decimal floating point. The binary calculator, the number base converter and the scientific notation converter are related.

Frequently asked questions