Valid (number Format)

Unums (''universal numbers'') are a family of number formats and arithmetic for implementing real numbers on a computer, proposed by John L. Gustafson in 2015. They are designed as an alternative to the ubiquitous IEEE 754 floating-point standard. The latest version is known as ''posits''.

Type I Unum

The first version of unums, formally known as Type I unum, was introduced in Gustafson's book ''The End of Error'' as a superset of the IEEE-754 floating-point format. The defining features of the Type I unum format are:
* a variable-width storage format for both the significand and exponent, and
* a ''u-bit'', which determines whether the unum corresponds to an exact number (''u'' = 0), or an interval between consecutive exact unums (''u'' = 1). In this way, the unums cover the entire extended real number line ˆ’∞,+∞

For computation with the format, Gustafson proposed using interval arithmetic with a pair of unums, what he called a ''ubound'', providing the guarantee that the resulting interval contains the exact solution.

William M. Kahan and Gustafson debated unums at the '' Arith23'' conference.

Type II Unum

Type II Unums were introduced in 2016 as a redesign of Unums that broke IEEE-754 compatibility. In addition to the sign bit and the interval bit mentioned earlier, the Type II Unum uses a bit to indicate inversion. These three operations make it possible, starting from a finite set of points between one and infinity, to quantify the entire projective line except for four points: the two exceptions, 0 and ∞, and then 1 and −1. This set of points is chosen arbitrarily, and arithmetic operations involving them are not performed logically but rather by using a lookup table. The size of such a table becomes prohibitive for an encoding format spanning multiple bytes. This challenge necessitated the development of the Type III Unum, known as the posit, discussed below.

Posit (Type III Unum)

In February 2017, Gustafson officially introduced Type III unums (posits), for fixed floating-point-like values and valids for interval arithmetic. In March 2022, a standard was ratified and published by the Posit Working Group.

Posits are a hardware-friendly version of unum where difficulties faced in the original type I unum due to its variable size are resolved. Compared to IEEE 754 floats of similar size, posits offer a bigger dynamic range and more fraction bits for values with magnitude near 1 (but fewer fraction bits for very large or very small values), and Gustafson claims that they offer better accuracy. Studies confirm that for some applications, posits with quire out-perform floats in accuracy. Posits have superior accuracy in the range near one, where most computations occur. This makes it very attractive to the current trend in deep learning to minimize the number of bits used. It potentially helps any application to accelerate by enabling the use of fewer bits (since it has more fraction bits for accuracy) reducing network and memory bandwidth and power requirements.

The format of an ''n''-bit posit is given a label of "posit" followed by the decimal digits of ''n'' (e.g., the 16-bit posit format is "posit16") and consists of four sequential fields:

# sign: 1 bit, representing an unsigned integer ''s''
# regime: at least 2 bits and up to (''n'' âˆ’ 1), representing an unsigned integer ''r'' as described below
# exponent: generally 2 bits as available after regime, representing an unsigned integer ''e''
# fraction: all remaining bits available after exponent, representing a non-negative real dyadic rational ''f'' less than 1

The regime field uses unary coding of ''k'' identical bits, followed by a bit of opposite value if any remaining bits are available, to represent an unsigned integer ''r'' that is −''k'' if the first bit is 0 or ''k'' âˆ’ 1 if the first bit is 1. The sign, exponent, and fraction fields are analogous to IEEE 754 sign, exponent, and significand fields (respectively), except that the posit exponent and fraction fields may be absent or truncated and implicitly extended with zeroes—an absent exponent is treated as 00₂ (representing 0), a one-bit exponent E₁ is treated as E10₂ (representing the integer 0 if E₁ is 0 or 2 if E₁ is 1), and an absent fraction is treated as 0. Negative numbers (''s'' is 1) are encoded as 2's complements.

The two encodings in which all non-sign bits are 0 have special interpretations:

* If the sign bit is 1, the posit value is NaR ("not a real")
* If the sign bit is 0, the posit value is 0 (which is unsigned and the only value for which the sign function returns 0)

Otherwise, the posit value is equal to $((1 - 3s) + f) \times 2^$, in which ''r'' scales by powers of 16, ''e'' scales by powers of 2, ''f'' distributes values uniformly between adjacent combinations of (''r'', ''e''), and ''s'' adjusts the sign symmetrically about 0.

Examples

Note: 32-bit posit is expected to be sufficient to solve almost all classes of applications.

Quire

For each posit''n'' type of precision $n$, the standard defines a corresponding "quire" type quire''n'' of precision $16 \times n$, used to accumulate exact sums of products of those posits without rounding or overflow in dot products for vectors of up to 2³¹ or more elements (the exact limit is $2^$). The quire format is a two's complement signed integer, interpreted as a multiple of units of magnitude $2^$ except for the special value with a leading sign bit of 1 and all other bits equal to 0 (which represents NaR). Quires are based on the work of Ulrich W. Kulisch and Willard L. Miranker.

Valid

Valids are described as a Type III Unum mode that bounds results in a given range.

Implementations

Several software and hardware solutions implement posits. The first complete parameterized posit arithmetic hardware generator was proposed in 2018.

Unum implementations have been explored in Julia and MATLAB. A C++ version with support for any posit sizes combined with any number of exponent bits is available. A fast implementation in C, SoftPosit, provided by the NGA research team based on Berkeley SoftFloat adds to the available software implementations.

SoftPosit

SoftPosit is a software implementation of posits based on Berkeley SoftFloat. It allows software comparison between posits and floats. It currently supports

* Add
* Subtract
* Multiply
* Divide
* Fused-multiply-add
* Fused-dot-product (with quire)
* Square root
* Convert posit to signed and unsigned integer
* Convert signed and unsigned integer to posit
* Convert posit to another posit size
* Less than, equal, less than equal comparison
* Round to nearest integer

Helper functions

* convert double to posit
* convert posit to double
* cast unsigned integer to posit

It works for 16-bit posits with one exponent bit and 8-bit posit with zero exponent bit. Support for 32-bit posits and flexible type (2-32 bits with two exponent bits) is pending validation. It supports x86_64 systems. It has been tested on GNU gcc (SUSE Linux) 4.8.5 Apple LLVM version 9.1.0 (clang-902.0.39.2).

Examples

''Add with posit8_t''

#include "softposit.h"

int main(int argc, char *argv[])

''Fused dot product with quire16_t''

// Convert double to posit
posit16_t pA = convertDoubleToP16(1.02783203125);
posit16_t pB = convertDoubleToP16(0.987060546875);
posit16_t pC = convertDoubleToP16(0.4998779296875);
posit16_t pD = convertDoubleToP16(0.8797607421875);

quire16_t qZ;

// Set quire to 0
qZ = q16_clr(qZ);

// Accumulate products without roundings
qZ = q16_fdp_add(qZ, pA, pB);
qZ = q16_fdp_add(qZ, pC, pD);

// Convert back to posit
posit16_t pZ = q16_to_p16(qZ);

// To check answer
double dZ = convertP16ToDouble(pZ);

Critique

William M. Kahan, the principal architect of IEEE 754-1985 criticizes type I unums on the following grounds (some are addressed in type II and type III standards):
* The description of unums sidesteps using calculus for solving physics problems.
* Unums can be expensive in terms of time and power consumption.
* Each computation in unum space is likely to change the bit length of the structure. This requires either unpacking them into a fixed-size space, or data allocation, deallocation, and garbage collection during unum operations, similar to the issues for dealing with variable-length records in mass storage.
* Unums provide only two kinds of numerical exception, quiet and signaling NaN (Not-a-Number).
* Unum computation may deliver overly loose bounds from the selection of an algebraically correct but numerically unstable algorithm.
* The benefits of unum over short precision floating point for problems requiring low precision are not obvious.
* Solving differential equations and evaluating integrals with unums guarantee correct answers but may not be as fast as methods that usually work.

See also

* Karlsruhe Accurate Arithmetic (KAA)
* Q (number format)
* Significant figures
* Floating-point error mitigation
* Elias gamma (γ) code
* Tapered floating point (TFP)

References

Further reading

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* (Roger Stokes' download link


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External links

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* {{cite web , url=https://www.johndcook.com/blog/2018/04/11/anatomy-of-a-posit-number/ , title=Anatomy of a posit number , date=2018-04-11 , access-date=2019-08-09

Floating point types
Articles with example C code