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Unums (universal numbers) are a family of formats and arithmetic, similar to floating point, proposed by John L. Gustafson in 2015. They are designed as an alternative to the ubiquitous IEEE 754 floating-point standard. The latest version (known as posits) can be used as a drop-in replacement for programs that do not depend on specific features of IEEE 754.

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 .

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.

This section needs expansion. You can help by adding to it. (June 2021)

Type III Unum (posit and valid)

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

Posit

Posits are a hardware-friendly version of unum where difficulties faced in the original type I unum due to its variable size are resolved. Similar size posits when compared to floats offer a bigger dynamic range and more fraction bits for accuracy. In an independent study, Lindstrom, Lloyd and Hittinger from Lawrence Livermore National Laboratory confirmed that posits 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 minimise 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.

Posits have variable-sized index and mantissa bitfields, with the split being specified by a "regime" indicator. Gustafson claimed that they offer better precision than standard floating-point numbers while taking up fewer bits.

Posits have a different format than IEEE 754 floats. They consist of four parts: sign, regime, exponent, and fraction (also known as significand/mantissa). For a n-bit posit, regime can be of length 2 to (n − 1). The format of the regime is such that it is a repetition of a same-sign bit and terminated by a different-sign bit.

Example:

Example 1 shows a regime with 14 same-sign bits (bit 0), terminated by a different-sign bit (bit 1). As there are 14 same-sign bits, the runlength of the regime is 14.

Example 2 shows a regime with 3 same-sign bits (bit 1), terminated by a different-sign bit (bit 0). As there are 3 same-sign bits, the runlength of the regime is 3.

Sign, exponent and fraction bits are very similar to IEEE 754; however, posits may omit either or both of the exponent and fraction bits, leaving a posit that consists of only sign and regime bits. Example 3 shows the longest possible regime runlength for a 16-bit posit, where the regime terminating bit, exponent bit and fraction bits are beyond the length of the size of the posit. Example 4 illustrates the shortest possible runlength of 1 for a 16-bit posit with one exponent bit (bit value = 1) and 12 fraction bits (bit value = 100000000001).

The recommended posit sizes and corresponding exponent bits and quire sizes:

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

Quire

Quire is one of the most useful features of posits. It is a special data type that gives posits an "near-infinity" number of bits to accumulate dot products. It is 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){
    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;
}

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

1. Tichy, Walter F. (April 2016). "The End of (Numeric) Error: An interview with John L. Gustafson". Ubiquity – Information Everywhere. 2016 (April). Association for Computing Machinery (ACM): 1–14. doi:10.1145/2913029. Archived from the original on 2016-07-10. Retrieved 2016-07-10. JG: The word "unum" is short for "universal number," the same way the word "bit" is short for "binary digit."
2. ^ Gustafson, John L. (2016-02-04) . The End of Error: Unum Computing. Chapman & Hall / CRC Computational Science. Vol. 24 (2nd corrected printing, 1st ed.). CRC Press. ISBN 978-1-4822-3986-7. Retrieved 2016-05-30.
3. ^ Gustafson, John Leroy; Yonemoto, Isaac (2017). "Beating Floating Point at its Own Game: Posit Arithmetic". Supercomputing Frontiers and Innovations. 4 (2). Publishing Center of South Ural State University, Chelyabinsk, Russia. doi:10.14529/jsfi170206. Archived from the original on 2017-11-04. Retrieved 2017-11-04.
4. "Program: Special Session: The Great Debate: John Gustafson and William Kahan". Arith23: 23rd IEEE Symposium on Computer Arithmetic. Silicon Valley, USA. 2016-07-12. Archived from the original on 2016-05-30. Retrieved 2016-05-30.
5. Gustafson, John L.; Kahan, William M. (2016-07-12). The Great Debate @ARITH23: John Gustafson and William Kahan (1:34:41) (video). Retrieved 2016-07-20.
6. ^ Kahan, William M. (2016-07-16) . "A Critique of John L. Gustafson's THE END of ERROR — Unum Computation and his A Radical Approach to Computation with Real Numbers" (PDF). Santa Clara, CA, USA: IEEE Symposium on Computer Arithmetic, ARITH 23. Archived (PDF) from the original on 2016-07-25. Retrieved 2016-07-25.
7. Gustafson, John L. (2016-07-12). ""The Great Debate": Unum arithmetic position paper" (PDF). Santa Clara, CA, USA: IEEE Symposium on Computer Arithmetic, ARITH 23. Retrieved 2016-07-20.
8. Tichy, Walter F. (September 2016). "Unums 2.0: An Interview with John L. Gustafson". Ubiquity.ACM.org. Retrieved 2017-01-30. I started out calling them "unums 2.0," which seemed to be as good a name for the concept as any, but it is really not a "latest release" so much as it is an alternative.
9. "Standard for Posit Arithmetic (2022)" (PDF). 2022-03-02. Archived (PDF) from the original on 2022-09-26. Retrieved 2022-12-21. {{cite web}}: Unknown parameter |authors= ignored (help)
10. John L. Gustafson and I. Yonemoto. (February 2017) Beyond Floating Point: Next Generation Computer Arithmetic. . Available: https://www.youtube.com/watch?v=aP0Y1uAA-2Y
11. Gustafson, John Leroy (2017-10-10). "Posit Arithmetic" (PDF). Archived (PDF) from the original on 2017-11-05. Retrieved 2017-11-04.
12. ^ P. Lindstrom, S. Lloyd, and J. Hittinger, "Universal Coding of the Reals: Alternatives to IEEE Floating Point". In Conference for Next Generation Arithmetic. ACM, 2018.
13. Feldman, Michael (2019-07-08). "New Approach Could Sink Floating Point Computation". www.nextplatform.com. Retrieved 2019-07-09.
14. Byrne, Michael (2016-04-24). "A New Number Format for Computers Could Nuke Approximation Errors for Good". Vice. Retrieved 2019-07-09.
15. Kulisch, Ulrich W.; Miranker, Willard L. (March 1986). "The Arithmetic of the Digital Computer: A New Approach". SIAM Rev. 28 (1). SIAM: 1–40. doi:10.1137/1028001.
16. S. Chung, "Provably Correct Posit Arithmetic with Fixed-Point Big Integer." ACM, 2018.
17. J. Chen, Z. Al-Ars, and H. Hofstee, "A Matrix-Multiply Unit for Posits in Reconfigurable Logic Using (Open)CAPI." ACM, 2018.
18. Z. Lehoczky, A. Szabo, and B. Farkas, "High-level .NET Software Implementations of Unum Type I and Posit with Simultaneous FPGA Implementation Using Hastlayer." ACM, 2018.
19. S. Langroudi, T. Pandit, and D. Kudithipudi, "Deep Learning Inference on Embedded Devices: Fixed-Point vs Posit". In Energy Efficiency Machine Learning and Cognitive Computing for Embedded Applications (EMC), 2018. . Available: https://sites.google.com/view/asplos-emc2/program
20. Rohit Chaurasiya, John Gustafson, Rahul Shrestha, Jonathan Neudorfer, Sangeeth Nambiar, Kaustav Niyogi, Farhad Merchant, Rainer Leupers, "Parameterized Posit Arithmetic Hardware Generator." ICCD 2018: 334-341.
21. Byrne, Simon (2016-03-29). "Implementing Unums in Julia". Retrieved 2016-05-30.
22. "Unum arithmetic in Julia: Unums.jl". Retrieved 2016-05-30.
23. "Julia Implementation of Unums: README". Retrieved 2016-05-30.
24. "Unum (Universal Number) types and operations: Unums". Retrieved 2016-05-30.
25. "jwmerrill/Pnums.jl". Github.com. Retrieved 2017-01-30.
26. "GitHub - ityonemo/Unum2: Pivot Unums". 2019-04-29.
27. Ingole, Deepak; Kvasnica, Michal; De Silva, Himeshi; Gustafson, John L. "Reducing Memory Footprints in Explicit Model Predictive Control using Universal Numbers. Submitted to the IFAC World Congress 2017". Retrieved 2016-11-15.
28. Ingole, Deepak; Kvasnica, Michal; De Silva, Himeshi; Gustafson, John L. "MATLAB Prototype of unum (munum)". Retrieved 2016-11-15.
29. "GitHub - stillwater-sc/Universal: Universal Number Arithmetic". 2019-06-16.
30. ^ "Cerlane Leong / SoftPosit · GitLab". GitLab.
31. "Berkeley SoftFloat". www.jhauser.us.
32. Kahan, William M. (2016-07-15). "Prof. W. Kahan's Commentary on "THE END of ERROR — Unum Computing" by John L. Gustafson, (2015) CRC Press" (PDF). Archived (PDF) from the original on 2016-08-01. Retrieved 2016-08-01.

Further reading

• 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) . "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.

External links

• "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.

• Floating point types