How is bfloat16 different from IEEE-754 half-precision (float16)?

bfloat16 uses an 8-bit exponent and 7-bit mantissa, while float16 uses a 5-bit exponent and 10-bit mantissa. bfloat16 keeps float32's full dynamic range but with fewer significant digits, which suits deep-learning training where range matters more than precision.

Why is bfloat16 popular in machine learning?

Because it is literally the top 16 bits of a float32, converting between them is trivial and overflow behaves the same. Halving memory and bandwidth speeds up training and inference on TPUs and modern GPUs with little loss in model quality.

What is the exponent bias?

The bias is 127, the same as float32. A stored exponent of 0 means subnormal or zero, 255 means infinity or NaN, and any value in between represents an unbiased exponent of stored minus 127.

How much precision does bfloat16 have?

With a 7-bit mantissa plus the implicit leading 1, bfloat16 carries about 8 bits of significand, roughly 2 to 3 decimal significant digits. That is much coarser than float32's 7 digits but acceptable for gradient updates.

What does the tool do when my number is not exactly representable?

It rounds to the nearest bfloat16 using round-to-nearest-even, the IEEE default, then shows the value that bit pattern actually represents. Comparing your input to the decoded value reveals the rounding error.

bfloat16 Inspector — Gera Tools

bfloat16 (brain floating point) is a 16-bit format that keeps the full exponent range of a 32-bit float while throwing away most of its precision. This inspector lets you encode a decimal number into its nearest bfloat16 and decode any 16-bit pattern back into the real number it represents, showing every field.

How it works

A bfloat16 packs three fields into 16 bits:

[ sign : 1 ][ exponent : 8 ][ mantissa : 7 ]

The exponent uses a bias of 127. For a normal value the decoded number is:

value = (-1)^sign × (1 + mantissa/128) × 2^(exponent − 127)

When the stored exponent is 0 the value is subnormal (no implicit leading 1) or zero; when it is 255 the value is infinity (mantissa 0) or NaN. Because bfloat16 is exactly the high 16 bits of an IEEE-754 single, encoding a number just truncates the low mantissa bits — this tool applies round-to-nearest-even so the result matches real hardware.

Example and notes

The hex value 0x3F80 decodes to exactly 1.0: sign 0, exponent 127, mantissa 0. Entering 0.15625 encodes to 0x3E20 and decodes back to exactly 0.15625 because that value is representable. Try a value like 0.1 to see rounding error appear between what you typed and what bfloat16 can store. Remember that bfloat16 trades precision for range — it is designed for neural-network weights and activations, not financial arithmetic.