Unum (number format)

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.[2] 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 [−∞,+∞].

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.[4][5][6][7]

Type II Unums were introduced in 2016[8] as a redesign of Unums that broke IEEE-754 compatibility.

In February 2017, Gustafson officially introduced Type III unums, posits for fixed floating-point-like values and valids for interval arithmetic.[3] In March 2021, a standard was ratified and published by the Posit Working Group.[9]

Posits[3][10][11] 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.[12][13] Studies[14][15] 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:

1. sign: 1 bit, representing an unsigned integer s
2. regime: at least 2 bits and up to (n − 1), representing an unsigned integer r as described below
3. exponent: up to 2 bits as available after regime, representing an unsigned integer e
4. 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 − 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.

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 ${\textstyle ((1-3s)+f)\times 2^{(1-2s)\times (4r+e+s)}}$, 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

type (positn)	Binary	Value	Notes
any	1 0…	NaR	anything not mathematically definable as a unique real number[9]
any	0 0…	${\textstyle 0}$
any	0 10…	${\textstyle 1}$
any	1 10…	${\textstyle -1}$
any	0 01 11 0…	${\textstyle 0.5}$
any	0 0…1	${\textstyle 2^{-4n+8}}$	smallest positive value
any	0 1…	${\textstyle 2^{4n-8}}$	largest positive value
posit8	0 0000001	${\textstyle 2^{-24}\approx 6.0\times 10^{-8}}$	smallest positive value
posit8	0 1111111	${\textstyle 2^{24}\approx 1.7\times 10^{7}}$	largest positive value
posit16	0 000000000000001	${\textstyle 2^{-56}\approx 1.4\times 10^{-17}}$	smallest positive value
posit16	0 111111111111111	${\textstyle 2^{56}\approx 7.2\times 10^{16}}$	largest positive value
posit32	0 0000000000000000000000000000001	${\textstyle 2^{-120}\approx 7.5\times 10^{-37}}$	smallest positive value
posit32	0 1111111111111111111111111111111	${\textstyle 2^{120}\approx 1.3\times 10^{36}}$	largest positive value

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

Valids are described as a Type III Unum mode that bounds results in a given range.[3]

Several software and hardware solutions implement posits.[14][17][18][19][20] The first complete parameterized posit arithmetic hardware generator was proposed in 2018.[21]

Unum implementations have been explored in Julia[22][23][24][25][26][27] and MATLAB.[28][29] A C++ version[30] with support for any posit sizes combined with any number of exponent bits is available. A fast implementation in C, SoftPosit,[31] provided by the NGA research team based on Berkeley SoftFloat adds to the available software implementations.

#include "softposit.h"

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

    posit8_t pA, pB, pZ;
    pA = castP8(0xF2);
    pB = castP8(0x23);

    pZ = p8_add(pA, pB);

    //To check answer by converting it to double
    double dZ = convertP8ToDouble(pZ);
    printf("dZ: %.15f\n", dZ);

    //To print result in binary (warning: non-portable code)
    uint8_t uiZ = castUI8(pZ);
    printBinary((uint64_t*)&uiZ, 8);

    return 0;
}

//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);

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):[6][33]

• 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.

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

• Gustafson, John L. (March 2013). "Right-Sizing Precision: Unleashed Computing: The need to right-size precision to save energy, bandwidth, storage, and electrical power" (PDF). Archived (PDF) from the original on 2016-06-06. Retrieved 2016-06-06.
• Brueckner, Rich (2015-03-02). "Slidecast: John Gustafson Explains Energy Efficient Unum Computing". The Rich Report. Inside HPC. Archived from the original on 2016-07-10. Retrieved 2016-06-10.
• Gustafson, John L. (2015). "The end of numerical error" (PDF). Archived (PDF) from the original on 2016-06-06. Retrieved 2016-06-06.
• Gustafson, John L. (2016-06-03) [2016-02-22]. "A Radical Approach to Computation with Real Numbers – Unums version 2.0" (PPT). Archived from the original on 2016-07-10. Retrieved 2016-07-10. (NB. PDFs come without notes: )
• Gustafson, John L. (2016-06-06). "An Energy-Efficient and massively parallel approach to valid numerics" (PPT). OCRAR Seminar. Archived from the original on 2016-07-10. Retrieved 2016-07-10.
• Gustafson, John L. (2016). "A Radical Approach to Computation with Real Numbers" (PDF). SuperFri.org. Archived (PDF) from the original on 2016-07-10. Retrieved 2016-07-10.
• Kulisch, Ulrich W. (2015). "Up-to-date Interval Arithmetic from closed intervals to connected sets of real numbers" (PDF) (preprint). Institut für Angewandte und Numerische Mathematik – Karlsruhe Institute of Technology (KIT), Germany. ID 15/02. Archived (PDF) from the original on 2016-07-12. Retrieved 2016-07-12.
• Risse, Thomas (2016-03-10). "Unum – an expedient extension of IEEE 754" (PDF) (presentation). London South Bank University (LSBU), UK: Institute of Informatics & Automation (IIA), Faculty EEE & CS, Bremen University of Applied Sciences, Germany. Archived (PDF) from the original on 2016-07-12. Retrieved 2016-07-12.
• Kahan, William M. (2016-07-15). "Prof. W. Kahan's Comments on SORN Arithmetic" (PDF). Archived (PDF) from the original on 2016-08-01. Retrieved 2016-08-01.
• Hunhold, Laslo (2016-11-08). The Unum Number Format: Mathematical Foundations, Implementation and Comparison to IEEE 754 Floating-Point Numbers (PDF) (Bachelor thesis). Universität zu Köln, Mathematisches Institut. arXiv:1701.00722v1. Archived (PDF) from the original on 2017-01-07. Retrieved 2016-10-23.
• Sterbenz, Pat H. (1974-05-01). Floating-Point Computation. Prentice-Hall Series in Automatic Computation (1st ed.). Englewood Cliffs, New Jersey, USA: Prentice Hall. ISBN 0-13-322495-3.
• Cave, Skip (2016-08-17). "J Programming Language Implementation of 3-bit, 4-bit, 8-bit and 16-bit Precision Unums". Retrieved 2017-05-03. (Roger Stokes' download link: )
• Ingole, Deepak (2017-09-28). Embedded Implementation of Explicit Model Predictive Control (PhD Thesis). Slovak University of Technology in Bratislava, Slovakia.

• "Conference for Next Generation Arithmetic (CoNGA)". 2017. Archived from the original on 2017-11-04. Retrieved 2017-11-04.
• "SoftPosit". 2018. Retrieved 2018-06-13.
• "Community Source Code Contribution". 2018. Retrieved 2018-06-13.
• "Anatomy of a posit number". 2018-04-11. Retrieved 2019-08-09.