Posit number format

Research article: Beating Floating Point at its Own Game: Posit Arithmetic
Most of the images below are taken from above article

Also see the float-posit converters I made using the universal c++ library.

An n-bit posit number contains:

1. 1 compulsory sign bit
2. 1 or more regime bits
3. 0 or more exponent bits, length (es) is pre-determined by the standard
4. 0 or more fraction bits
   So it possible that the posit number only contains 1 sign and rest regime bits

Sign bit

0 : positive number
1 : negative number

Unlike IEEE754, ==negative numbers in posit are in 2’s complement form==
So to get equivalent decimal value, first convert all n bits to 2’s complement form.

Because it does not use signed magnitude form, there’s no negative zero,
also both +inf and - inf are represented by same bit pattern

Regime bits

Run of 1s (positives) or 0s (negatives) terminated by a bit of opposite length

The numerical value of regime bits “k” determines the power of useed
Where useed = $2^{(2^{es})}$

It is possible that whole posit number contains only regime bits with 1 sign bit
e.g. if n = 4, 0111 contains 3 regime bits, whose k = 2
while 0110 also contains 3 regime bits, last 0 is terminator, thus k = 1
Fractions will be present only if regime bits are terminated at least within n-1 bits
Exponent length is pre-determined at least in standard posit formats.

Exponent bits

These are optional
es = number of exponent bits
Standard posit formats (posit8, posit16, posit32 etc) have pre-defined number of exponent bits (es)

Unlike IEEE754 the numerical value of exponent bits e is unsigned integer. So no negative exponent.
Similar to IEEE754, the e determines the power of 2.

Fraction bits

There are optional
The numerical value of fraction is always preceded by 1. There are no denormal numbers
The calculation is similar to IEEE754
Number of fraction bits is not fixed by standard.

Overall Value

Note: the last statement in following diagram is ambiguous, as posit uses 2’s complement format, not signed magnitude format.

Explaination from Beating Floating Point at its Own Game: Posit Arithmetic

The sign bit 0 means the value is positive. The regime bits 0001 have a run of three 0s, which means k is −3; hence, the scale factor contributed by the regime is 256−3. The exponent bits, 101, represent 5 as an unsigned binary integer, and contribute another scale factor of 25. Lastly, the fraction bits 11011101 represent 221 as an unsigned binary integer, so the fraction is 1 + 221/256. The expression shown underneath the bit fields in fig. 5 works out to 477/134217728 ≈ 3.55393 × 10−6.

Differences with IEEE754

1. No NaNs
2. Fixed representation for 0, no -ve / +ve zero
3. Fixed represnetation for both +inf and -inf
4. No denormals
5. 2’s complement representation of negative numbers
6. No fixed length of fraction bits or exponent bits

Standard posit formats

• Posit8 is 8 bits with es=0;
• Posit16 is 16 bits with es=1;
• Posit32 is 32 bits with es=2.
• Posit64 is 64 bits with es=3
• Posit128 is 128 bits with es=4

Further Read

Details of posit (long read)
https://posithub.org/docs/Posits4.pdf

Other resources

1. Unum: https://en.wikipedia.org/wiki/Unum_(number_format)#

2. PPT Standford lecture by John (short read), not really clear what happens as no explaination: https://web.stanford.edu/class/ee380/Abstracts/170201-slides.pdf