$2$ Google$$s bfloat$16$

The bfloat$16$ format, shown in Figure $1,$ has similar fields as the IEEE $754$ floating$-$point format. The

sign$-$bit field $$S$$ indicates whether the number is positive $($i$.$e$.,$ S $=0)$ or negative $($i$.$e$.,$ S $=1).$ The

exponent field $$E$$ encodes the value of the exponent as a biased signed$-$value, where the bias $$B$$ is

equal to $127$; assuming the unsigned value of the exponent field $$E$$ is val$($E$),$ the value encoded by the

exponent field is:

exp $=$ val$($E$)127$

The field $$F$$ represents the fraction$/$mantissa$.$ The mantissa is encoded as an unsigned binary number

with the weight of the most$-$significant bit $($i$.$e$.,$ bit number $6$ in the format$)$ is $21$

; subsequent bits have

the following weights: $22$

$,23$

$,...,27$

$.$ To convert the binary fraction to decimal, one can use the

following equation:

F raction $=$ F$6\backslash$times

$1$

$2$

$1$

$+$ F$5\backslash$times

$1$

$2$

$2$

$+$ F$4\backslash$times

$1$

$2$

$3$

$+$ F$3\backslash$times

$1$

$2$

$4$

$+$ F$2\backslash$times

$1$

$2$

$5$

$+$ F$1\backslash$times

$1$

$2$

$6$

$+$ F$0\backslash$times

$1$

$2$

$6$

$1$

$1514131211109876543210$

S E F

Figure $1$: bfloat$16$ Format and Encoding$$The bfloat$16$ format and encoding showing the different

fields; S: sign bit indicator, E: exponent value, F: fraction$/$mantissa value.

Much like IEEE$-754$ format, floating$-$point numbers can either be normalized or denormalized.

Normalized floating$-$point numbers have an implicit $1$ in the integer part, which is not represented by

the aforementioned fields. Denormalized floating$-$point numbers have an implicit $0$ in the integer part

of the number. The equation to calculate the value of the number encoded by the bfloat$16$ floatingpoint format is influenced by the value of the exponent field $$E$$ and the value of the fraction field $$F$.$

The following subsections outline the different equations based on the values of the exponent field $$E$$

and the fraction field $$F$$: