125:
2288:(MIT license) The implementations where ES is constant the code is generated. The generator should be able to generate for all sizes {8,16,32} and ES below the size. However, the ones not included into the library by default are not tested, fuzzed, or supported. For some operations on 32/ES, mixing and matching ES is possible. However, this is not tested.
251: − 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
172:
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
167:
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
2876:
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
1505:
Fully integrated with C/C++ types and automatic conversions. Supports full C++ math library (native and conversion to/from IEEE). Runtime integrations: MTL4/MTL5, Eigen, Trilinos, HPR-BLAS. Application integrations: G+SMO, FDBB, FEniCS, ODEintV2, TVM.ai. Hardware accelerator integration (Xilinx,
2080:
Application-level posit-capable RISC-V core based on CVA6 that can execute all posit instructions, including the quire fused operations. PERCIVAL is the first work that integrates the complete posit ISA and quire in hardware. It allows the native execution of posit instructions as well as the
2195:
Fully integrated with C/C++ types and automatic conversions. Supports full C++ math library (native and conversion to/from IEEE). Runtime integrations: GNU Utils, OpenBLAS, CBLAS. Application integrations: in progress. Compiler support extended: C/C++, G++, GFortran & LLVM (in progress).
1302:
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.
2356:
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
3540:
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:
403:
1493:
posit<8,0> 130 MPOPS posit<16,1> 115 MPOPS posit<32,2> 105 MPOPS posit<64,3> 50 MPOPS posit<128,4> 1 MPOPS posit<256,5> 800 KPOPS
1039:
900:
1104:
761:
965:
826:
3894:
1863:
Intended for Deep
Learning applications Addition, Subtraction and Multiplication only. A proof of concept matrix multiplier has been built, but is off-spec in its precision
1431:
Open source (MIT license). Original definition and prototype. Most complete environment for comparing IEEE floats and posits. Many examples of use, including linear solvers
3531:
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.
3284:
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.
1257:
1217:
3454:
David Mallasén; Alberto A. Del Barrio; Manuel Prieto-Matias (2023-05-11). "Big-PERCIVAL: Exploring the Native Use of 64-Bit Posit
Arithmetic in Scientific Computing".
1177:
632:
687:
4210:
2299:
SoftPosit is a software implementation of posits based on
Berkeley SoftFloat. It allows software comparison between posits and floats. It currently supports
1147:
4038:
3975:
3115:
1660:
Leverages Julia's templated mathematics standard library, can natively do matrix and tensor operations, complex numbers, FFT, DiffEQ. Support for valids
3989:
3404:
1287:
Several software and hardware solutions implement posits. The first complete parameterized posit arithmetic hardware generator was proposed in 2018.
168:
for very large or very small values), and
Gustafson claims that they offer better accuracy. Studies confirm that for some applications, posits with
3867:
2897:
Solving differential equations and evaluating integrals with unums guarantee correct answers but may not be as fast as methods that usually work.
4143:
1691:
1401:
Open source license. Fastest and most comprehensive C library for posits presently. Designed for plug-in comparison of IEEE floats and posits.
4072:
4068:
3239:
3194:
3026:
3553:
Rohit
Chaurasiya, John Gustafson, Rahul Shrestha, Jonathan Neudorfer, Sangeeth Nambiar, Kaustav Niyogi, Farhad Merchant, Rainer Leupers, "
3232:
3861:"Right-Sizing Precision: Unleashed Computing: The need to right-size precision to save energy, bandwidth, storage, and electrical power"
2970:
3946:
2163:
Fully software stack with compilers, debugger, IDE environment and math libraries for applications. C, C++, Python languages supported
3961:
4174:
4034:
3920:
3715:"Reducing Memory Footprints in Explicit Model Predictive Control using Universal Numbers. Submitted to the IFAC World Congress 2017"
4005:
3828:
3202:
3085:
2887:
Unum computation may deliver overly loose bounds from the selection of an algebraically correct but numerically unstable algorithm.
4101:
3886:
2923:
160:
In
February 2017, Gustafson officially introduced Type III unums (posits), for fixed floating-point-like values and valids for
3379:
1735:
This project was developed due to the fact that
SigmoidNumbers and FastSigmoid by Isaac Yonemoto is not maintained currently.
3522:
J. Chen, Z. Al-Ars, and H. Hofstee, "A Matrix-Multiply Unit for Posits in Reconfigurable Logic Using (Open)CAPI." ACM, 2018.
3330:
John L. Gustafson and I. Yonemoto. (February 2017) Beyond Floating Point: Next Generation Computer Arithmetic. . Available:
3307:
306:
4298:
3004:
2907:
239:
identical bits, followed by a bit of opposite value if any remaining bits are available, to represent an unsigned integer
3120:
4064:
1291:
1997:
Vinay Saxena, Research and Technology Centre, Robert Bosch, India (RTC-IN) and Farhad Merchant, RWTH Aachen University
992:
3994:
3967:
3938:
3909:
3856:
3735:
3711:
3354:
3228:
3141:
3000:
1625:
853:
31:
4122:
The Unum Number Format: Mathematical Foundations, Implementation and Comparison to IEEE 754 Floating-Point Numbers
1738:
Supports basic linear algebra functions in Julia (Matrix multiplication, Matrix solve, Elgen decomposition, etc.)
1063:
714:
173:
bits (since it has more fraction bits for accuracy) reducing network and memory bandwidth and power requirements.
4027:
3014:
2044:
Verified against SoftPosit for (es=2) and tested with several applications for (es=2) and (es=3). No known bugs.
1932:
Supports +,-,/,*. Implements both logarithmic posit and normal, "linear" posits License: CC-BY-NC 4.0 at present
924:
4129:
785:
3971:
3987:
2985:
JG: The word "unum" is short for "universal number," the same way the word "bit" is short for "binary digit."
2285:
Fuzzing against C softposit with a lot of iterations for 16/1 and 32/2. Explicitly testing edge cases found.
3822:"Prof. W. Kahan's Commentary on "THE END of ERROR — Unum Computing" by John L. Gustafson, (2015) CRC Press"
4185:
2933:
1259:
except for the special value with a leading sign bit of 1 and all other bits equal to 0 (which represents
78:
54:
as a superset of the IEEE-754 floating-point format. The defining features of the Type I unum format are:
2866:
criticizes type I unums on the following grounds (some are addressed in type II and type III standards):
4204:
3860:
4274:
2201:
1769:
Open source (MIT license). Easy-to-use interface. Neural net example. Comprehensive functions support.
4120:
3343:
3059:
2282:
80 MPOPS for div32/2 and similar linear functions. Much higher for truncate and much lower for exp.
4229:
4165:. Prentice-Hall Series in Automatic Computation (1st ed.). Englewood Cliffs, New Jersey, USA:
4057:
3479:
2918:
1268:
1220:
161:
90:
4133:
3455:
2928:
2047:
First complete posit-capable RISC-V core. Supports dynamic switching between (es=2) and (es=3).
117:
Type II Unums were introduced in 2016 as a redesign of Unums that broke IEEE-754 compatibility.
1226:
1186:
295:
If the sign bit is 0, the posit value is 0 (which is unsigned and the only value for which the
4170:
4023:
3475:
3022:
3010:
2913:
1264:
50:
The first version of unums, formally known as Type I unum, was introduced in Gustafson's book
1988:
Multi-million random tests are performed for up to 32-bit posit with various ES combinations
3496:
3436:
3075:
2974:
1179:, used to accumulate exact sums of products of those posits without rounding or overflow in
1156:
100:
3942:
598:
3959:
3265:
2954:
1499:
Randoms for large posit configs. Uses induction to prove nbits+1 is correct no known bugs
1397:
16: Exhaustive except FMA, quire 32: Exhaustive test is still in progress. No known bugs.
656:
221:
4186:"J Programming Language Implementation of 3-bit, 4-bit, 8-bit and 16-bit Precision Unums"
3913:
1632:
1117:: 32-bit posit is expected to be sufficient to solve almost all classes of applications.
4028:"Up-to-date Interval Arithmetic from closed intervals to connected sets of real numbers"
3998:
3821:
3063:
4094:
3568:
3179:
2191:
Applications tested and being made available for seamless adoption www.calligotech.com
2157:
World's first posit-enabled ASIC with octa-core RISC-V processor and Quire implemented.
3513:
S. Chung, "Provably Correct Posit Arithmetic with Fixed-Point Big Integer." ACM, 2018.
3254:
124:
4292:
4166:
4090:
3817:
3175:
3145:
3041:
2863:
2145:
2140:
No divide or square root. First full processor design to replace floats with posits.
1838:
184:(e.g., the 16-bit posit format is "posit16") and consists of four sequential fields:
3542:
2261:
85: = 1). In this way, the unums cover the entire extended real number line .
4067:(LSBU), UK: Institute of Informatics & Automation (IIA), Faculty EEE & CS,
3554:
2110:
Ops: arithmetic, comparisons, sqrt, sin, cos, tan, acos, asin, atan, pow, exp, log
1279:
Valids are described as a Type III Unum mode that bounds results in a given range.
1132:
232:
164:. In March 2022, a standard was ratified and published by the Posit Working Group.
97:, providing the guarantee that the resulting interval contains the exact solution.
3799:
3217:
3300:
3269:
3039:
2189:
Exhaustive tests completed for 32 bits and 64 bits with Quire support completed.
2014:- ES=2 | N=7,8,9,10,11,12 Selective (20000*65536) combinations for - ES=1 | N=16
1699:
284:
The two encodings in which all non-sign bits are 0 have special interpretations:
1904:
1600:
1180:
220:: all remaining bits available after exponent, representing a non-negative real
217:
59:
27:
2870:
The description of unums sidesteps using calculus for solving physics problems.
2231:
3331:
3149:
3107:"Program: Special Session: The Great Debate: John Gustafson and William Kahan"
3106:
2362:
1774:
1729:
Posit (8,0), Posit (16,1), Posit (32,2) Other formats lack full functionality
1665:
1542:
1436:
3758:
3588:
1744:
1468:
3484:
3440:
3405:"A New Number Format for Computers Could Nuke Approximation Errors for Good"
3074:(2). Publishing Center of South Ural State University, Chelyabinsk, Russia.
3018:
2891:
2166:
Applications tested successfully - image and video compression, more to come
2055:
211:: up to 2 bits as available after regime, representing an unsigned integer
3614:
2115:
2086:
3943:"A Radical Approach to Computation with Real Numbers – Unums version 2.0"
3243:
3198:
2880:
Unums provide only two kinds of numerical exception, quiet and signaling
2074:
Speed of design is based on the underlying hardware platform (ASIC/FPGA)
1983:
Speed of design is based on the underlying hardware platform (ASIC/FPGA)
1976:
Able to generate any combination of word-size (N) and exponent-size (ES)
1449:
Convert decimal to posit 6, 8, 16, 32; generate tables 2–17 with es 1–4.
1382:
C++ wrapper to override operators Python wrapper using SWIG of SoftPosit
208:
188:
180:-bit posit is given a label of "posit" followed by the decimal digits of
63:
35:
4197:
3666:
3482:(March 1986). "The Arithmetic of the Digital Computer: A New Approach".
3347:
3080:
1868:
1406:
4244:
3972:"An Energy-Efficient and massively parallel approach to valid numerics"
3781:
3738:
3714:
3640:
3111:
1963:
1370:
105:
4259:
2028:(Sugandha Tiwari, Neel Gala, Chester Rebeiro, V.Kamakoti, IIT MADRAS)
4138:
3763:
3695:
3645:
3619:
3593:
3151:
The Great Debate @ARITH23: John Gustafson and William Kahan (1:34:41)
1295:
3887:"Slidecast: John Gustafson Explains Energy Efficient Unum Computing"
3690:
3500:
2979:
2958:
1512:
73:, which determines whether the unum corresponds to an exact number (
3460:
1299:
3433:
Universal Coding of the Reals: Alternatives to IEEE Floating Point
2226:
Does 128-by-128 matrix-matrix multiplication (SGEMM) using quire.
3431:
Lindstrom, Peter; Lloyd, Scott; Hittinger, Jeffrey (March 2018).
2959:"The End of (Numeric) Error: An interview with John L. Gustafson"
2125:
FPGA version of the "Neo" VLIW processor with posit numeric unit
2018:
To be used in commercial products. To the best of our knowledge.
1571:
26:) are a family of number formats and arithmetic for implementing
4033:(preprint). Institut für Angewandte und Numerische Mathematik –
3492:
2366:
1799:
4 levels of operations working with posits. Special support for
1714:
8-bit (es=0..2) 16-bit (es=0..2) 24-bit (es=1..2) 32-bit (es=2)
1223:
signed integer, interpreted as a multiple of units of magnitude
3759:"GitHub - stillwater-sc/Universal: Universal Number Arithmetic"
3435:. Conference for Next Generation Arithmetic. Art. 5. ACM.
2160:
PCIe accelerator card with this silicon will be ready June 2024
1808:
417:
distributes values uniformly between adjacent combinations of (
34:
in 2015. They are designed as an alternative to the ubiquitous
2881:
2873:
Unums can be expensive in terms of time and power consumption.
2358:
1800:
480:
anything not mathematically definable as a unique real number
119:
4206:
Embedded Implementation of Explicit Model Predictive Control
1340:
1183:
for vectors of up to 2 or more elements (the exact limit is
1787:
4 to 64 (any es value); "Template version is 2 to 63 bits"
3064:"Beating Floating Point at its Own Game: Posit Arithmetic"
1488:
Arbitrary quire configurations with programmable capacity
89:
For computation with the format, Gustafson proposed using
2552:// To print result in binary (warning: non-portable code)
1149:, the standard defines a corresponding "quire" type quire
3426:
3424:
4225:
3054:
3052:
3050:
3048:
3017:. Vol. 24 (2nd corrected printing, 1st ed.).
398:{\textstyle ((1-3s)+f)\times 2^{(1-2s)\times (4r+e+s)}}
136:
1159:
1135:
1066:
995:
927:
856:
788:
717:
659:
601:
309:
3999:"A Radical Approach to Computation with Real Numbers"
3734:
Ingole, Deepak; Kvasnica, Michal; De Silva, Himeshi;
3710:
Ingole, Deepak; Kvasnica, Michal; De Silva, Himeshi;
3641:"Unum (Universal Number) types and operations: Unums"
3294:
3292:
2894:
for problems requiring low precision are not obvious.
1229:
1189:
3380:"New Approach Could Sink Floating Point Computation"
3233:""The Great Debate": Unum arithmetic position paper"
2275:
16/1 32/2 (included is a generic 32/ES for ES<32)
2065:
Open-Source Posit RISC-V Core with Quire Capability
4226:"Conference for Next Generation Arithmetic (CoNGA)"
3186:
A Radical Approach to Computation with Real Numbers
3555:Parameterized Posit Arithmetic Hardware Generator.
1251:
1211:
1171:
1141:
1098:
1033:
959:
894:
820:
755:
681:
626:
397:
201: − 1), representing an unsigned integer
3543:https://sites.google.com/view/asplos-emc2/program
2077:Functionality testing of each posit instruction.
2020:***First ever integration of posits in RISC-V***
1992:It supports rounding-to-nearest rounding method.
3270:"Unums 2.0: An Interview with John L. Gustafson"
2995:
2993:
1860:Comprehensively tested for 8-bit, no known bugs
1833:Supports + – × ÷ √ reciprocal, negate, compare
1733:Open source. Issues and suggestions on GitHub.
1590:~20–45 MPOPS on 4.9 GHz Skylake core
1497:Complete validation suite for arbitrary posits
1034:{\textstyle 2^{-120}\approx 7.5\times 10^{-37}}
281:is 1), and an absent fraction is treated as 0.
4095:"Prof. W. Kahan's Comments on SORN Arithmetic"
1428:Exhaustive for low precisions. No known bugs.
1392:~60 to 110 MPOPS on x86 core (Broadwell)
895:{\textstyle 2^{-56}\approx 1.4\times 10^{-17}}
16:Variant of floating-point numbers in computers
4211:Slovak University of Technology in Bratislava
2862:William M. Kahan, the principal architect of
2504:// To check answer by converting it to double
2081:standard floating-point ones simultaneously.
2032:BSV (Bluespec System Verilog) Implementation
1099:{\textstyle 2^{120}\approx 1.3\times 10^{36}}
756:{\textstyle 2^{-24}\approx 6.0\times 10^{-8}}
58:a variable-width storage format for both the
8:
3776:
3774:
3170:
3168:
2333:Less than, equal, less than equal comparison
2327:Convert signed and unsigned integer to posit
2324:Convert posit to signed and unsigned integer
960:{\textstyle 2^{56}\approx 7.2\times 10^{16}}
4058:"Unum – an expedient extension of IEEE 754"
3332:https://www.youtube.com/watch?v=aP0Y1uAA-2Y
1290:Unum implementations have been explored in
821:{\textstyle 2^{24}\approx 1.7\times 10^{7}}
3116:23rd IEEE Symposium on Computer Arithmetic
2175:<32, 2> with Quire 512 bits support.
1878:C# with Hastlayer for hardware generation
1482:Arbitrary precision posit float valid (p)
191:: 1 bit, representing an unsigned integer
4137:
3459:
3079:
2978:
1930:Does not strictly conform to posit spec.
1234:
1228:
1194:
1188:
1158:
1134:
1090:
1071:
1065:
1022:
1000:
994:
951:
932:
926:
883:
861:
855:
812:
793:
787:
744:
722:
716:
664:
658:
606:
600:
347:
308:
288:If the sign bit is 1, the posit value is
3068:Supercomputing Frontiers and Innovations
2744:// Accumulate products without roundings
1478:C library Python wrapper Golang library
1305:
436:
429:adjusts the sign symmetrically about 0.
3615:"Julia Implementation of Unums: README"
2946:
1964:PACoGen: Posit Arthmetic Core Generator
303:Otherwise, the posit value is equal to
93:with a pair of unums, what he called a
3691:"GitHub - ityonemo/Unum2: Pivot Unums"
3301:"Standard for Posit Arithmetic (2022)"
1380:C library based on Berkeley SoftFloat
258:(representing 0), a one-bit exponent E
4069:Bremen University of Applied Sciences
3240:IEEE Symposium on Computer Arithmetic
3195:IEEE Symposium on Computer Arithmetic
2068:Posit<32,2> with 512-bit quire
7:
4260:"Community Source Code Contribution"
3782:"Cerlane Leong / SoftPosit · GitLab"
3589:"Unum arithmetic in Julia: Unums.jl"
2035:32-bit posit with (es=2) and (es=3)
1830:Bugs found; status of fixes unknown
3182:THE END of ERROR — Unum Computation
3180:"A Critique of John L. Gustafson's
3119:. Silicon Valley, USA. 2016-07-12.
2971:Association for Computing Machinery
2330:Convert posit to another posit size
2009:Similar to floats of same bit size
1723:A*STAR "SoftPosit" (Cerlane Leong)
1687:Known bug in 32-bit multiplication
1263:). Quires are based on the work of
197:regime: at least 2 bits and up to (
103:and Gustafson debated unums at the
3739:"MATLAB Prototype of unum (munum)"
3299:Posit Working Group (2022-03-02).
2369:version 9.1.0 (clang-902.0.39.2).
1986:Exhaustive tests for 8-bit posit.
1915:Any (parameterized SystemVerilog)
1386:8, 16, 32 published and complete;
14:
4035:Karlsruhe Institute of Technology
2963:Ubiquity – Information Everywhere
2000:Verilog generator for VLSI, FPGA
1971:Verilog HDL for Posit Arithmetic
1952:"2 GHz", not translated to MPOPS
1779:Federico Rossi, Emanuele Ruffaldi
81:between consecutive exact unums (
38:. The latest version is known as
3006:The End of Error: Unum Computing
2621:Fused dot product with quire16_t
1484:Unum type 1 (p) Unum type 2 (p)
273:(representing the integer 0 if E
123:
36:IEEE 754 floating-point standard
4232:from the original on 2017-11-04
4161:Sterbenz, Pat H. (1974-05-01).
4149:from the original on 2017-01-07
4107:from the original on 2016-08-01
4078:from the original on 2016-07-12
4044:from the original on 2016-07-12
4011:from the original on 2016-07-10
3978:from the original on 2016-07-10
3949:from the original on 2016-07-10
3926:from the original on 2016-06-06
3897:from the original on 2016-07-10
3873:from the original on 2016-06-06
3834:from the original on 2016-08-01
3378:Feldman, Michael (2019-07-08).
3360:from the original on 2017-11-05
3313:from the original on 2022-09-26
3208:from the original on 2016-07-25
3123:from the original on 2016-05-30
3088:from the original on 2017-11-04
2924:Floating-point error mitigation
1622:Limited Testing; no known bugs
1472:Stillwater Supercomputing, Inc
1463:Table generator and conversion
1056:1111111111111111111111111111111
4196:(Roger Stokes' download link:
3958:(NB. PDFs come without notes:
3885:Brueckner, Rich (2015-03-02).
2892:short precision floating point
2352:cast unsigned integer to posit
2318:Fused-dot-product (with quire)
1581:SoftPosit bindings for Python
1552:SoftPosit bindings for Racket
1491:posit<4,0> 1 GPOPS
982:000000000000000000000000000000
390:
369:
363:
348:
337:
328:
313:
310:
1:
4203:Ingole, Deepak (2017-09-28).
4119:Hunhold, Laslo (2016-11-08).
3569:"Implementing Unums in Julia"
3403:Byrne, Michael (2016-04-24).
2908:Karlsruhe Accurate Arithmetic
2256:A DNN framework using posits
1955:Partial; known rounding bugs
1899:Requires Microsoft .Net APIs
4065:London South Bank University
4056:Risse, Thomas (2016-03-10).
3914:"The end of numerical error"
2096:Single file C# MIT Licensed
1923:(RTL for FPGA/ASIC designs)
4275:"Anatomy of a posit number"
4132:, Mathematisches Institut.
3567:Byrne, Simon (2016-03-29).
2223:Only one known case tested
1365:RacEr GP-GPU has 512 cores
1362:Exhaustive. No known bugs.
30:on a computer, proposed by
4315:
4163:Floating-Point Computation
4037:(KIT), Germany. ID 15/02.
3062:; Yonemoto, Isaac (2017).
2890:The benefits of unum over
2627:// Convert double to posit
2241:Python library (software)
2026:Posit-enabled RISC-V core
1537:Open source (MIT license)
1503:Open source. MIT license.
169:
4184:Cave, Skip (2016-08-17).
3015:CRC Computational Science
2137:Extensive; no known bugs
2107:Extensive; no known bugs
1809:bfp:Beyond Floating Point
1534:Extensive; no known bugs
1350:World's first FPGA GPGPU
1252:{\displaystyle 2^{16-8n}}
1219:). The quire format is a
1212:{\displaystyle 2^{23+4n}}
247:if the first bit is 0 or
3238:. Santa Clara, CA, USA:
3193:. Santa Clara, CA, USA:
2807:// Convert back to posit
2624:
2381:
2336:Round to nearest integer
1675:Julia and C/C++ library
1654:No known bugs (posits).
1043:smallest positive value
904:smallest positive value
765:smallest positive value
636:smallest positive value
409:scales by powers of 16,
3441:10.1145/3190339.3190344
2387:"softposit.h"
2349:convert posit to double
2346:convert double to posit
2183:Yes - Fully supported.
2146:PNU: Posit Numeric Unit
1656:Division bugs (valids)
1172:{\textstyle 16\times n}
1108:largest positive value
969:largest positive value
830:largest positive value
691:largest positive value
413:scales by powers of 2,
77: = 0), or an
3974:(PPT). OCRAR Seminar.
2934:Tapered floating point
2211:Specific-purpose FPGA
1958:Yet to be open-source
1908:Jeff Johnson, Facebook
1610:Octave implementation
1253:
1213:
1173:
1143:
1100:
1035:
961:
896:
822:
757:
683:
628:
627:{\textstyle 2^{-4n+8}}
399:
231:The regime field uses
3557:" ICCD 2018: 334-341.
3344:Gustafson, John Leroy
3060:Gustafson, John Leroy
1883:(64bits in progress)
1803:types (non-standard)
1476:C++ template library
1416:Mathematica notebook
1254:
1214:
1174:
1144:
1101:
1036:
962:
897:
823:
758:
684:
682:{\textstyle 2^{4n-8}}
629:
400:
156:Posit (Type III Unum)
4299:Floating point types
3800:"Berkeley SoftFloat"
3667:"jwmerrill/Pnums.jl"
3480:Miranker, Willard L.
3384:www.nextplatform.com
2929:Elias gamma (γ) code
2534:"dZ: %.15f
1967:Manish Kumar Jaiswal
1892:Click here for more
1712:Based on softposit;
1227:
1187:
1157:
1133:
1064:
993:
925:
854:
786:
715:
657:
599:
307:
4130:Universität zu Köln
4128:(Bachelor thesis).
3081:10.14529/jsfi170206
2919:Significant figures
1946:16, 32, extendable
1873:Lombiq Technologies
1763:< 20 MPOPS
1645:All <32, all ES
1457:interactive widget
1425:< 80 KPOPS
1269:Willard L. Miranker
438:
299:function returns 0)
162:interval arithmetic
91:interval arithmetic
4024:Kulisch, Ulrich W.
3995:Gustafson, John L.
3968:Gustafson, John L.
3939:Gustafson, John L.
3910:Gustafson, John L.
3857:Gustafson, John L.
3736:Gustafson, John L.
3712:Gustafson, John L.
3476:Kulisch, Ulrich W.
3348:"Posit Arithmetic"
3268:(September 2016).
3229:Gustafson, John L.
3142:Gustafson, John L.
3011:Chapman & Hall
3001:Gustafson, John L.
2843:convertP16ToDouble
2831:// To check answer
2702:convertDoubleToP16
2681:convertDoubleToP16
2660:convertDoubleToP16
2639:convertDoubleToP16
2315:Fused-multiply-add
2186:500 MHz * 8 Cores
1869:Lombiq Arithmetics
1848:Julia and Verilog
1796:A few basic tests
1678:8, 16, 32, all ES
1506:Intel, Achronix).
1446:JavaScript widget
1249:
1209:
1169:
1139:
1129:type of precision
1096:
1031:
957:
892:
818:
753:
679:
624:
437:
395:
205:as described below
135:. You can help by
4091:Kahan, William M.
3818:Kahan, William M.
3176:Kahan, William M.
3146:Kahan, William M.
3028:978-1-4822-3986-7
2914:Q (number format)
2723:// Set quire to 0
2516:convertP8ToDouble
2378:Add with posit8_t
2292:
2291:
2220:16–64 GPOPS
1694:in shock studies
1265:Ulrich W. Kulisch
1112:
1111:
176:The format of an
153:
152:
32:John L. Gustafson
24:universal numbers
4306:
4285:
4283:
4282:
4270:
4268:
4267:
4255:
4253:
4252:
4240:
4238:
4237:
4214:
4195:
4193:
4192:
4180:
4157:
4155:
4154:
4148:
4141:
4127:
4115:
4113:
4112:
4106:
4099:
4086:
4084:
4083:
4077:
4063:(presentation).
4062:
4052:
4050:
4049:
4043:
4032:
4019:
4017:
4016:
4010:
4004:. SuperFri.org.
4003:
3986:
3984:
3983:
3957:
3955:
3954:
3934:
3932:
3931:
3925:
3918:
3905:
3903:
3902:
3881:
3879:
3878:
3872:
3865:
3843:
3842:
3840:
3839:
3833:
3826:
3814:
3808:
3807:
3796:
3790:
3789:
3778:
3769:
3768:
3755:
3749:
3748:
3746:
3745:
3731:
3725:
3724:
3722:
3721:
3707:
3701:
3700:
3687:
3681:
3680:
3678:
3677:
3663:
3657:
3656:
3654:
3653:
3637:
3631:
3630:
3628:
3627:
3611:
3605:
3604:
3602:
3601:
3585:
3579:
3578:
3576:
3575:
3564:
3558:
3551:
3545:
3538:
3532:
3529:
3523:
3520:
3514:
3511:
3505:
3504:
3472:
3466:
3465:
3463:
3451:
3445:
3444:
3428:
3419:
3418:
3416:
3415:
3400:
3394:
3393:
3391:
3390:
3375:
3369:
3368:
3366:
3365:
3359:
3352:
3340:
3334:
3328:
3322:
3321:
3319:
3318:
3312:
3305:
3296:
3287:
3286:
3281:
3280:
3274:Ubiquity.ACM.org
3266:Tichy, Walter F.
3262:
3256:
3253:
3251:
3250:
3237:
3225:
3219:
3216:
3214:
3213:
3207:
3192:
3172:
3163:
3162:
3160:
3159:
3138:
3132:
3131:
3129:
3128:
3103:
3097:
3096:
3094:
3093:
3083:
3056:
3043:
3038:
3036:
3035:
2997:
2988:
2987:
2982:
2955:Tichy, Walter F.
2951:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2341:Helper functions
2276:
2271:Pure Go library
2134:~1.2 GPOPS
2049:More info here.
1851:8, 16, 32, ES=0
1437:posit-javascript
1359:~3.2 TPOPS
1306:
1262:
1258:
1256:
1255:
1250:
1248:
1247:
1221:two's complement
1218:
1216:
1215:
1210:
1208:
1207:
1178:
1176:
1175:
1170:
1148:
1146:
1145:
1140:
1105:
1103:
1102:
1097:
1095:
1094:
1076:
1075:
1058:
1057:
1054:
1040:
1038:
1037:
1032:
1030:
1029:
1008:
1007:
987:
986:
983:
980:
966:
964:
963:
958:
956:
955:
937:
936:
919:
918:
915:
901:
899:
898:
893:
891:
890:
869:
868:
848:
847:
844:
841:
827:
825:
824:
819:
817:
816:
798:
797:
780:
779:
776:
762:
760:
759:
754:
752:
751:
730:
729:
709:
708:
705:
702:
688:
686:
685:
680:
678:
677:
651:
650:
647:
633:
631:
630:
625:
623:
622:
593:
592:
589:
586:
573:
568:
566:
563:
560:
557:
544:
539:
538:
535:
522:
517:
516:
513:
500:
495:
494:
491:
477:
472:
471:
468:
439:
404:
402:
401:
396:
394:
393:
298:
291:
269:
254:
148:
145:
127:
120:
101:William M. Kahan
52:The End of Error
4314:
4313:
4309:
4308:
4307:
4305:
4304:
4303:
4289:
4288:
4280:
4278:
4273:
4265:
4263:
4258:
4250:
4248:
4243:
4235:
4233:
4224:
4221:
4202:
4190:
4188:
4183:
4177:
4160:
4152:
4150:
4146:
4125:
4118:
4110:
4108:
4104:
4097:
4089:
4081:
4079:
4075:
4060:
4055:
4047:
4045:
4041:
4030:
4022:
4014:
4012:
4008:
4001:
3993:
3981:
3979:
3966:
3952:
3950:
3941:(2016-06-03) .
3937:
3929:
3927:
3923:
3916:
3908:
3900:
3898:
3891:The Rich Report
3884:
3876:
3874:
3870:
3863:
3855:
3852:
3850:Further reading
3847:
3846:
3837:
3835:
3831:
3824:
3816:
3815:
3811:
3798:
3797:
3793:
3780:
3779:
3772:
3757:
3756:
3752:
3743:
3741:
3733:
3732:
3728:
3719:
3717:
3709:
3708:
3704:
3689:
3688:
3684:
3675:
3673:
3665:
3664:
3660:
3651:
3649:
3639:
3638:
3634:
3625:
3623:
3613:
3612:
3608:
3599:
3597:
3587:
3586:
3582:
3573:
3571:
3566:
3565:
3561:
3552:
3548:
3539:
3535:
3530:
3526:
3521:
3517:
3512:
3508:
3501:10.1137/1028001
3474:
3473:
3469:
3453:
3452:
3448:
3430:
3429:
3422:
3413:
3411:
3402:
3401:
3397:
3388:
3386:
3377:
3376:
3372:
3363:
3361:
3357:
3350:
3342:
3341:
3337:
3329:
3325:
3316:
3314:
3310:
3303:
3298:
3297:
3290:
3278:
3276:
3264:
3263:
3259:
3248:
3246:
3235:
3227:
3226:
3222:
3211:
3209:
3205:
3190:
3178:(2016-07-16) .
3174:
3173:
3166:
3157:
3155:
3140:
3139:
3135:
3126:
3124:
3105:
3104:
3100:
3091:
3089:
3058:
3057:
3046:
3033:
3031:
3029:
3003:(2016-02-04) .
2999:
2998:
2991:
2980:10.1145/2913029
2953:
2952:
2948:
2943:
2904:
2884:(Not-a-Number).
2860:
2855:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2708:0.8797607421875
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2687:0.4998779296875
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2618:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2375:
2343:
2297:
2274:
1974:Any precision.
1754:Python library
1633:Sigmoid Numbers
1531:~20 MPOPS
1522:Python library
1456:
1395:8: Exhaustive;
1285:
1283:Implementations
1277:
1260:
1230:
1225:
1224:
1190:
1185:
1184:
1155:
1154:
1131:
1130:
1123:
1086:
1067:
1062:
1061:
1055:
1052:
1051:
1018:
996:
991:
990:
984:
981:
978:
977:
947:
928:
923:
922:
917:111111111111111
916:
913:
912:
879:
857:
852:
851:
845:
842:
839:
838:
808:
789:
784:
783:
777:
774:
773:
740:
718:
713:
712:
706:
703:
700:
699:
660:
655:
654:
648:
645:
644:
602:
597:
596:
590:
587:
584:
583:
571:
564:
561:
558:
555:
554:
542:
536:
533:
532:
520:
514:
511:
510:
498:
492:
489:
488:
475:
469:
466:
465:
443:
435:
343:
305:
304:
296:
289:
280:
276:
272:
267:
263:
261:
257:
252:
222:dyadic rational
158:
149:
143:
140:
133:needs expansion
115:
48:
17:
12:
11:
5:
4312:
4310:
4302:
4301:
4291:
4290:
4287:
4286:
4271:
4256:
4241:
4220:
4219:External links
4217:
4216:
4215:
4209:(PhD Thesis).
4200:
4181:
4175:
4158:
4116:
4093:(2016-07-15).
4087:
4053:
4020:
3991:
3970:(2016-06-06).
3964:
3935:
3906:
3893:. Inside HPC.
3882:
3859:(March 2013).
3851:
3848:
3845:
3844:
3820:(2016-07-15).
3809:
3804:www.jhauser.us
3791:
3770:
3750:
3726:
3702:
3682:
3658:
3632:
3606:
3580:
3559:
3546:
3533:
3524:
3515:
3506:
3467:
3446:
3420:
3395:
3370:
3346:(2017-10-10).
3335:
3323:
3288:
3257:
3231:(2016-07-12).
3220:
3164:
3148:(2016-07-12).
3133:
3098:
3044:
3027:
2989:
2957:(April 2016).
2945:
2944:
2942:
2939:
2938:
2937:
2931:
2926:
2921:
2916:
2911:
2903:
2900:
2899:
2898:
2895:
2888:
2885:
2878:
2874:
2871:
2859:
2856:
2666:0.987060546875
2625:
2382:
2374:
2371:
2365:) 4.8.5 Apple
2354:
2353:
2350:
2347:
2342:
2339:
2338:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2296:
2293:
2290:
2289:
2286:
2283:
2280:
2277:
2272:
2269:
2258:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2228:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2198:
2197:
2193:
2187:
2184:
2181:
2180:
2179:
2176:
2171:
2170:
2169:
2167:
2164:
2161:
2158:
2153:
2142:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2112:
2111:
2108:
2105:
2103:
2100:
2097:
2094:
2083:
2082:
2078:
2075:
2072:
2069:
2066:
2063:
2060:David Mallasén
2052:
2051:
2045:
2042:
2039:
2036:
2033:
2030:
2023:
2022:
2016:
2010:
2007:
2004:
2001:
1998:
1994:
1993:
1990:
1984:
1981:
1978:
1972:
1969:
1960:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1935:
1934:
1928:
1925:
1919:
1916:
1913:
1912:SystemVerilog
1910:
1901:
1900:
1897:
1894:
1890:10 MPOPS
1888:
1885:
1879:
1876:
1865:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:Isaac Yonemoto
1835:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:Clément Guérin
1805:
1804:
1797:
1794:
1791:
1788:
1785:
1782:
1771:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1741:
1740:
1731:
1725:
1719:
1716:
1710:
1709:Julia library
1707:
1696:
1695:
1688:
1685:
1682:
1679:
1676:
1673:
1670:Isaac Yonemoto
1662:
1661:
1658:
1652:
1649:
1646:
1643:
1642:Julia library
1640:
1637:Isaac Yonemoto
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1142:{\textstyle n}
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1125:For each posit
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843:00000000000000
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292:("not a real")
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277:is 0 or 2 if E
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262:is treated as
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15:
13:
10:
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4178:
4176:0-13-322495-3
4172:
4168:
4167:Prentice Hall
4164:
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3795:
3792:
3787:
3783:
3777:
3775:
3771:
3767:. 2019-06-16.
3766:
3765:
3760:
3754:
3751:
3740:
3737:
3730:
3727:
3716:
3713:
3706:
3703:
3699:. 2019-04-29.
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3012:
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2976:
2973:(ACM): 1–14.
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2864:IEEE 754-1985
2857:
2645:1.02783203125
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2240:
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2233:
2232:Deep PeNSieve
2230:
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2222:
2219:
2216:
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2200:
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2178:<64, 3>
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2120:REX Computing
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2011:
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1570:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1549:
1548:
1544:
1543:softposit-rkt
1541:
1540:
1536:
1533:
1530:
1527:
1524:
1521:
1519:
1518:
1514:
1511:
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1494:
1490:
1487:
1485:
1481:
1479:
1475:
1473:
1470:
1467:
1466:
1462:
1460:Fully tested
1459:
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1445:
1443:
1442:
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1435:
1434:
1430:
1427:
1424:
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1308:
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1244:
1241:
1238:
1235:
1231:
1222:
1204:
1201:
1198:
1195:
1191:
1182:
1166:
1163:
1160:
1153:of precision
1152:
1136:
1128:
1120:
1118:
1116:
1107:
1091:
1087:
1083:
1080:
1077:
1072:
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1042:
1026:
1023:
1019:
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903:
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834:
833:
829:
813:
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802:
799:
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790:
782:
772:
769:
768:
764:
748:
745:
741:
737:
734:
731:
726:
723:
719:
711:
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695:
694:
690:
674:
671:
668:
665:
661:
653:
643:
640:
639:
635:
619:
616:
613:
610:
607:
603:
595:
582:
579:
578:
575:
570:
553:
550:
549:
546:
541:
531:
528:
527:
524:
519:
509:
506:
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502:
497:
487:
484:
483:
479:
474:
464:
461:
460:
456:
453:
450:
447:
441:
440:
432:
430:
428:
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412:
408:
387:
384:
381:
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366:
360:
357:
354:
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344:
340:
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331:
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316:
294:
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194:
190:
187:
186:
185:
183:
179:
174:
171:
165:
163:
155:
147:
138:
134:
131:This section
129:
126:
122:
121:
118:
112:
110:
108:
107:
102:
98:
96:
92:
84:
80:
76:
72:
68:
65:
61:
57:
56:
55:
53:
45:
43:
41:
37:
33:
29:
25:
21:
4279:. Retrieved
4277:. 2018-04-11
4264:. Retrieved
4249:. Retrieved
4234:. Retrieved
4205:
4189:. Retrieved
4162:
4151:. Retrieved
4139:1701.00722v1
4121:
4109:. Retrieved
4080:. Retrieved
4046:. Retrieved
4013:. Retrieved
3980:. Retrieved
3951:. Retrieved
3928:. Retrieved
3899:. Retrieved
3890:
3875:. Retrieved
3836:. Retrieved
3812:
3803:
3794:
3785:
3762:
3753:
3742:. Retrieved
3729:
3718:. Retrieved
3705:
3694:
3685:
3674:. Retrieved
3670:
3661:
3650:. Retrieved
3644:
3635:
3624:. Retrieved
3618:
3609:
3598:. Retrieved
3592:
3583:
3572:. Retrieved
3562:
3549:
3536:
3527:
3518:
3509:
3488:
3483:
3470:
3449:
3432:
3412:. Retrieved
3408:
3398:
3387:. Retrieved
3383:
3373:
3362:. Retrieved
3338:
3326:
3315:. Retrieved
3283:
3277:. Retrieved
3273:
3260:
3247:. Retrieved
3223:
3210:. Retrieved
3185:
3181:
3156:. Retrieved
3150:
3136:
3125:. Retrieved
3110:
3101:
3090:. Retrieved
3071:
3067:
3032:. Retrieved
3005:
2984:
2966:
2962:
2949:
2861:
2620:
2619:
2377:
2376:
2355:
2298:
2265:
2264:
2236:Raul Murillo
2235:
2234:
2205:
2204:
2190:
2150:Calligo Tech
2149:
2148:
2119:
2118:
2091:Chris Lomont
2090:
2089:
2059:
2058:
2048:
2027:
2019:
2013:
1987:
1975:
1966:
1938:
1931:
1922:
1907:
1891:
1882:
1872:
1871:
1842:
1841:
1818:C++ library
1812:
1811:
1784:C++ library
1778:
1777:
1748:
1747:
1737:
1734:
1728:
1722:
1713:
1704:Milan Klöwer
1703:
1702:
1700:SoftPosit.jl
1669:
1668:
1655:
1636:
1635:
1605:Diego Coelho
1604:
1603:
1601:positsoctave
1575:
1574:
1546:
1545:
1516:
1515:
1504:
1498:
1492:
1483:
1477:
1471:
1440:
1439:
1410:
1409:
1396:
1381:
1374:
1373:
1344:
1343:
1324:
1311:
1310:
1289:
1286:
1278:
1181:dot products
1150:
1126:
1124:
1114:
1113:
445:
426:
422:
418:
414:
410:
406:
302:
283:
248:
244:
240:
236:
233:unary coding
230:
224:
212:
202:
198:
192:
181:
177:
175:
166:
159:
141:
137:adding to it
132:
116:
113:Type II Unum
109:conference.
104:
99:
94:
88:
82:
74:
70:
51:
49:
39:
28:real numbers
23:
19:
18:
4245:"SoftPosit"
4213:, Slovakia.
4071:, Germany.
2783:q16_fdp_add
2753:q16_fdp_add
2576:printBinary
2321:Square root
2206:Jianyu Chen
1881:8, 16, 32.
1749:Ken Mercado
1721:Similar to
1666:FastSigmoid
1547:David Thien
1345:VividSparks
1320:Precisions
405:, in which
227:less than 1
60:significand
46:Type I Unum
4281:2019-08-09
4266:2018-06-13
4251:2018-06-13
4236:2017-11-04
4191:2017-05-03
4153:2016-10-23
4111:2016-08-01
4082:2016-07-12
4048:2016-07-12
4015:2016-07-10
3982:2016-07-10
3953:2016-07-10
3930:2016-06-06
3901:2016-06-10
3877:2016-06-06
3838:2016-08-01
3744:2016-11-15
3720:2016-11-15
3676:2017-01-30
3671:Github.com
3652:2016-05-30
3626:2016-05-30
3600:2016-05-30
3574:2016-05-30
3461:2305.06946
3414:2019-07-09
3389:2019-07-09
3364:2017-11-04
3317:2022-12-21
3279:2017-01-30
3249:2016-07-20
3212:2016-07-25
3158:2016-07-20
3127:2016-05-30
3092:2017-11-04
3034:2016-05-30
2941:References
2819:q16_to_p16
2363:SUSE Linux
2266:Jaap Aarts
2244:8, 16, 32
1939:Tokyo Tech
1839:Verilog.jl
3485:SIAM Rev.
3019:CRC Press
2969:(April).
2810:posit16_t
2714:quire16_t
2693:posit16_t
2672:posit16_t
2651:posit16_t
2630:posit16_t
2295:SoftPosit
2253:Unknown
2250:Unknown
2099:Any size
1905:Deepfloat
1857:Unknown
1827:Unknown
1793:Unknown
1766:Unknown
1745:PySigmoid
1684:Unknown
1651:Unknown
1619:Unknown
1593:Unknown
1576:Bill Zorn
1564:Unknown
1561:Unknown
1469:Universal
1407:posit4.nb
1371:SoftPosit
1325:Support?
1239:−
1164:×
1084:×
1078:≈
1024:−
1016:×
1010:≈
1002:−
945:×
939:≈
885:−
877:×
871:≈
863:−
806:×
800:≈
746:−
738:×
732:≈
724:−
672:−
608:−
367:×
355:−
341:×
320:−
243:that is −
144:June 2021
4293:Category
4230:Archived
4228:. 2017.
4144:Archived
4102:Archived
4073:Archived
4039:Archived
4026:(2015).
4006:Archived
3997:(2016).
3976:Archived
3947:Archived
3921:Archived
3912:(2015).
3895:Archived
3868:Archived
3829:Archived
3495:: 1–40.
3355:Archived
3308:Archived
3244:ARITH 23
3203:Archived
3199:ARITH 23
3184:and his
3121:Archived
3086:Archived
2902:See also
2877:storage.
2858:Critique
2582:uint64_t
2423:posit8_t
2384:#include
2373:Examples
2309:Multiply
2306:Subtract
2202:IBM-TACC
2087:LibPosit
2056:PERCIVAL
1927:Limited
1896:Partial
1775:cppPosit
1690:Used by
1332:Testing
1309:Project
1048:posit32
974:posit32
909:posit16
835:posit16
433:Examples
218:fraction
209:exponent
79:interval
64:exponent
3945:(PPT).
3154:(video)
3112:Arith23
2732:q16_clr
2564:castUI8
2555:uint8_t
1626:GNU GPL
1513:Speedgo
778:1111111
770:posit8
696:posit8
451:Binary
425:), and
106:Arith23
4262:. 2018
4247:. 2018
4173:
3786:GitLab
3764:GitHub
3696:GitHub
3646:GitHub
3620:GitHub
3594:GitHub
3025:
2834:double
2606:return
2540:"
2528:printf
2507:double
2486:p8_add
2468:castP8
2450:castP8
2312:Divide
2116:unumjl
1441:A*STAR
1411:A*STAR
1375:A*STAR
1341:GP-GPU
1335:Notes
1329:Speed
1323:Quire
1312:author
1296:MATLAB
704:000000
457:Notes
454:Value
444:(posit
95:ubound
40:posits
4147:(PDF)
4134:arXiv
4126:(PDF)
4105:(PDF)
4098:(PDF)
4076:(PDF)
4061:(PDF)
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4031:(PDF)
4009:(PDF)
4002:(PDF)
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3917:(PDF)
3871:(PDF)
3864:(PDF)
3832:(PDF)
3825:(PDF)
3491:(1).
3456:arXiv
3358:(PDF)
3351:(PDF)
3311:(PDF)
3304:(PDF)
3236:(PDF)
3206:(PDF)
3191:(PDF)
2936:(TFP)
2910:(KAA)
2591:&
2361:gcc (
2262:Gosit
1943:FPGA
1727:Yes:
1317:Type
1292:Julia
1275:Valid
1121:Quire
170:quire
71:u-bit
66:, and
20:Unums
4171:ISBN
3493:SIAM
3409:Vice
3023:ISBN
2967:2016
2474:0x23
2456:0xF2
2414:argv
2408:char
2402:argc
2393:main
2367:LLVM
2247:Yes
2217:Yes
2071:Yes
2012:N=8
2003:All
1921:N/A
1918:Yes
1887:Yes
1821:Any
1760:Yes
1757:All
1718:Yes
1692:LLNL
1648:Yes
1613:All
1587:Yes
1584:All
1572:sfpy
1558:Yes
1555:All
1525:All
1455:N/A;
1452:N/A
1422:Yes
1419:All
1389:Yes
1356:Yes
1298:. A
1294:and
1267:and
1115:Note
641:Any
580:Any
551:Any
529:Any
507:Any
485:Any
462:Any
442:type
297:sign
189:sign
62:and
3497:doi
3437:doi
3076:doi
2975:doi
2882:NaN
2594:uiZ
2558:uiZ
2399:int
2390:int
2359:GNU
2303:Add
2279:No
2214:32
2131:No
2128:32
2102:No
2038:No
2006:No
1980:No
1949:No
1854:No
1824:No
1801:NaN
1790:No
1681:No
1616:No
1528:No
1353:32
1300:C++
1261:NaR
1081:1.3
1073:120
1013:7.5
1005:120
942:7.2
874:1.4
803:1.7
735:6.0
572:0.5
537:10…
515:10…
476:NaR
290:NaR
235:of
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4100:.
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