IEEE 754 Floating Point Calculator

It converts a decimal number into its IEEE 754 binary representation and breaks the bits into the three fields the standard defines: sign, exponent, and mantissa (also called the fraction). You get both single precision (32-bit, binary32) and double precision (64-bit, binary64) at once, shown in binary and hex. It also works in reverse, decoding a hex bit pattern back to the decimal value it stores.

Single precision uses 32 bits: 1 sign bit, 8 exponent bits, and 23 mantissa bits, with an exponent bias of 127. Double precision uses 64 bits: 1 sign bit, 11 exponent bits, and 52 mantissa bits, with a bias of 1023. Double precision gives you roughly 15 to 17 significant decimal digits versus about 6 to 9 for single, so the same value usually stores more accurately as a double.

For a normal value the formula is (-1)^sign times 1.mantissa times 2 to the power of (stored exponent minus bias). The number is first written in binary scientific notation, the leading 1 is dropped because it is implied, and the remaining fraction bits become the mantissa. The exponent is stored with the bias added so it can represent both negative and positive powers without a separate sign.

0.1 has no finite representation in base 2, the same way 1/3 has no finite representation in base 10. The hardware rounds it to the nearest value it can store, which in single precision is the bit pattern 0x3DCCCCCD. Try entering 0.1 in encode mode to see the repeating mantissa pattern that causes the small error you sometimes notice in floating point math.

When the exponent field is all zeros and the mantissa is nonzero, the value is subnormal (denormalized), which lets the format represent numbers closer to zero by dropping the implied leading 1. When the exponent field is all ones, a zero mantissa means infinity and a nonzero mantissa means NaN (not a number). The calculator labels each of these special cases when it detects them.

The hex and binary output is shown in big-endian (most significant bit first), which is the standard way IEEE 754 layouts are written in textbooks and documentation. That means the sign bit is on the left, followed by the exponent field, then the mantissa. Stored byte order in memory depends on your CPU, but the logical bit pattern shown here is the same on every platform.