{"id":"https://openalex.org/W7118944198","doi":"https://doi.org/10.3390/mi17010084","title":"Hardware-Oriented Approximations of Softmax and RMSNorm for Efficient Transformer Inference","display_name":"Hardware-Oriented Approximations of Softmax and RMSNorm for Efficient Transformer Inference","publication_year":2026,"publication_date":"2026-01-07","ids":{"openalex":"https://openalex.org/W7118944198","doi":"https://doi.org/10.3390/mi17010084","pmid":"https://pubmed.ncbi.nlm.nih.gov/41597864"},"language":"en","primary_location":{"id":"doi:10.3390/mi17010084","doi":"https://doi.org/10.3390/mi17010084","is_oa":true,"landing_page_url":"https://doi.org/10.3390/mi17010084","pdf_url":"https://www.mdpi.com/2072-666X/17/1/84/pdf?version=1767780965","source":{"id":"https://openalex.org/S96702057","display_name":"Micromachines","issn_l":"2072-666X","issn":["2072-666X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","norway-1"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Micromachines","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2072-666X/17/1/84/pdf?version=1767780965","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111335736","display_name":"Yiwen Kang","orcid":"https://orcid.org/0009-0006-0807-5877","observed_orcids":["https://orcid.org/0009-0006-0807-5877"]},"institutions":[{"id":"https://openalex.org/I21193070","display_name":"Beijing Jiaotong University","ror":"https://ror.org/01yj56c84","country_code":"CN","type":"education","lineage":["https://openalex.org/I21193070"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yiwen Kang","raw_affiliation_strings":["Beijing Key Laboratory of Advanced Information Science and Network Technology, Beijing 100044, China","Institute of Information Science, Beijing Jiaotong University, Beijing 100044, China"],"raw_orcid":"https://orcid.org/0009-0006-0807-5877","affiliations":[{"raw_affiliation_string":"Beijing Key Laboratory of Advanced Information Science and Network Technology, Beijing 100044, China","institution_ids":[]},{"raw_affiliation_string":"Institute of Information Science, Beijing Jiaotong University, Beijing 100044, China","institution_ids":["https://openalex.org/I21193070"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101589049","display_name":"Dong Wang","orcid":"https://orcid.org/0000-0002-0068-8824","observed_orcids":["https://orcid.org/0000-0002-0068-8824"]},"institutions":[{"id":"https://openalex.org/I21193070","display_name":"Beijing Jiaotong University","ror":"https://ror.org/01yj56c84","country_code":"CN","type":"education","lineage":["https://openalex.org/I21193070"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Dong Wang","raw_affiliation_strings":["Beijing Key Laboratory of Advanced Information Science and Network Technology, Beijing 100044, China","Institute of Information Science, Beijing Jiaotong University, Beijing 100044, China"],"raw_orcid":"https://orcid.org/0000-0002-0068-8824","affiliations":[{"raw_affiliation_string":"Beijing Key Laboratory of Advanced Information Science and Network Technology, Beijing 100044, China","institution_ids":[]},{"raw_affiliation_string":"Institute of Information Science, Beijing Jiaotong University, Beijing 100044, China","institution_ids":["https://openalex.org/I21193070"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5101589049"],"corresponding_institution_ids":["https://openalex.org/I21193070"],"apc_list":{"value":2100,"currency":"CHF","value_usd":2534},"apc_paid":{"value":2100,"currency":"CHF","value_usd":2534},"fwci":7.3797,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.95457561,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":97},"biblio":{"volume":"17","issue":"1","first_page":"84","last_page":"84"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.7865999937057495,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.7865999937057495,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10036","display_name":"Advanced Neural Network Applications","score":0.0934000015258789,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10363","display_name":"Low-power high-performance VLSI design","score":0.060600001364946365,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/softmax-approximation","display_name":"softmax approximation","score":0.6639999747276306},{"id":"https://openalex.org/keywords/rmsnorm","display_name":"RMSNorm","score":0.8830000162124634},{"id":"https://openalex.org/keywords/transformer-inference","display_name":"transformer inference","score":0.7990000247955322},{"id":"https://openalex.org/keywords/hardware-acceleration","display_name":"hardware acceleration","score":0.7129999995231628},{"id":"https://openalex.org/keywords/range-reduction","display_name":"range reduction","score":0.718999981880188},{"id":"https://openalex.org/keywords/bit-width-optimization","display_name":"bit-width optimization","score":0.8460000157356262},{"id":"https://openalex.org/keywords/error-compensation","display_name":"error compensation","score":0.7509999871253967},{"id":"https://openalex.org/keywords/fpga-accelerator","display_name":"FPGA accelerator","score":0.7649999856948853},{"id":"https://openalex.org/keywords/large-language-models","display_name":"large language models","score":0.8410000205039978},{"id":"https://openalex.org/keywords/code-generation","display_name":"code generation","score":0.8080000281333923}],"concepts":[{"id":"https://openalex.org/C188441871","wikidata":"https://www.wikidata.org/wiki/Q7554146","display_name":"Softmax function","level":3,"score":0.781000018119812},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6636000275611877},{"id":"https://openalex.org/C134835016","wikidata":"https://www.wikidata.org/wiki/Q690265","display_name":"Lookup table","level":2,"score":0.6579999923706055},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.5264000296592712},{"id":"https://openalex.org/C66322947","wikidata":"https://www.wikidata.org/wiki/Q11658","display_name":"Transformer","level":3,"score":0.4731000065803528},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4641000032424927},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.43230000138282776},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.33390000462532043},{"id":"https://openalex.org/C203005215","wikidata":"https://www.wikidata.org/wiki/Q79798","display_name":"Machine translation","level":2,"score":0.3285999894142151},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.3167000114917755},{"id":"https://openalex.org/C32610155","wikidata":"https://www.wikidata.org/wiki/Q1798621","display_name":"Approximate string matching","level":3,"score":0.3125999867916107},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.3061000108718872},{"id":"https://openalex.org/C2777904410","wikidata":"https://www.wikidata.org/wiki/Q7397","display_name":"Software","level":2,"score":0.2994000017642975},{"id":"https://openalex.org/C36941000","wikidata":"https://www.wikidata.org/wiki/Q209455","display_name":"VHDL","level":3,"score":0.29499998688697815},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.2948000133037567},{"id":"https://openalex.org/C158946198","wikidata":"https://www.wikidata.org/wiki/Q131187","display_name":"Taylor series","level":2,"score":0.29330000281333923},{"id":"https://openalex.org/C113775141","wikidata":"https://www.wikidata.org/wiki/Q428691","display_name":"Computer engineering","level":1,"score":0.29269999265670776},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.28349998593330383},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.2777000069618225},{"id":"https://openalex.org/C82876162","wikidata":"https://www.wikidata.org/wiki/Q17096504","display_name":"Latency (audio)","level":2,"score":0.2727999985218048},{"id":"https://openalex.org/C42935608","wikidata":"https://www.wikidata.org/wiki/Q190411","display_name":"Field-programmable gate array","level":2,"score":0.2702000141143799},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2694000005722046},{"id":"https://openalex.org/C192328126","wikidata":"https://www.wikidata.org/wiki/Q4514647","display_name":"Schematic","level":2,"score":0.26589998602867126},{"id":"https://openalex.org/C157922185","wikidata":"https://www.wikidata.org/wiki/Q173198","display_name":"Logic synthesis","level":3,"score":0.2655999958515167},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.2630999982357025},{"id":"https://openalex.org/C13164978","wikidata":"https://www.wikidata.org/wiki/Q600158","display_name":"Hardware acceleration","level":3,"score":0.262800008058548},{"id":"https://openalex.org/C81843906","wikidata":"https://www.wikidata.org/wiki/Q173156","display_name":"Digital electronics","level":3,"score":0.26179999113082886},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.25440001487731934},{"id":"https://openalex.org/C2777742833","wikidata":"https://www.wikidata.org/wiki/Q1964083","display_name":"Reciprocal","level":2,"score":0.2540999948978424},{"id":"https://openalex.org/C148764684","wikidata":"https://www.wikidata.org/wiki/Q621751","display_name":"Approximation algorithm","level":2,"score":0.25360000133514404}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/mi17010084","doi":"https://doi.org/10.3390/mi17010084","is_oa":true,"landing_page_url":"https://doi.org/10.3390/mi17010084","pdf_url":"https://www.mdpi.com/2072-666X/17/1/84/pdf?version=1767780965","source":{"id":"https://openalex.org/S96702057","display_name":"Micromachines","issn_l":"2072-666X","issn":["2072-666X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","norway-1"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Micromachines","raw_type":"journal-article"},{"id":"pmid:41597864","doi":null,"is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/41597864","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Micromachines","raw_type":"Journal Article"},{"id":"pmh:oai:doaj.org/article:8a81f2e0fde5475683f50310c127135b","doi":null,"is_oa":true,"landing_page_url":"https://doaj.org/article/8a81f2e0fde5475683f50310c127135b","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Micromachines, Vol 17, Iss 1, p 84 (2026)","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:12843610","doi":null,"is_oa":true,"landing_page_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC12843610/","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Micromachines (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/mi17010084","doi":"https://doi.org/10.3390/mi17010084","is_oa":true,"landing_page_url":"https://doi.org/10.3390/mi17010084","pdf_url":"https://www.mdpi.com/2072-666X/17/1/84/pdf?version=1767780965","source":{"id":"https://openalex.org/S96702057","display_name":"Micromachines","issn_l":"2072-666X","issn":["2072-666X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","norway-1"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Micromachines","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Industry, innovation and infrastructure","score":0.428600013256073,"id":"https://metadata.un.org/sdg/9"}],"sustainable_development_goals_aurora":[{"display_name":"Decent work and economic growth","score":0.5460644364356995,"id":"https://metadata.un.org/sdg/8"}],"study_designs":[],"awards":[{"id":"https://openalex.org/G1172879264","display_name":null,"funder_award_id":"Grant L244050 and L252018","funder_id":"https://openalex.org/F4320322919","funder_display_name":"Natural Science Foundation of Beijing Municipality"}],"funders":[{"id":"https://openalex.org/F4320322919","display_name":"Natural Science Foundation of Beijing Municipality","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7118944198.pdf","grobid_xml":"https://content.openalex.org/works/W7118944198.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"With":[0],"the":[1,57,75,84,88,152],"rapid":[2],"advancement":[3],"of":[4,90],"Transformer-based":[5],"large":[6],"language":[7],"models":[8,11],"(LLMs),":[9],"these":[10,171],"have":[12],"found":[13],"widespread":[14],"applications":[15],"in":[16,28],"industrial":[17],"domains":[18],"such":[19],"as":[20,130],"code":[21],"generation":[22],"and":[23,53,59,66,97,111,114,158,165,183],"non-functional":[24],"requirement":[25],"(NFR)":[26],"classification":[27],"software":[29],"engineering.":[30],"However,":[31],"recent":[32],"research":[33],"has":[34],"primarily":[35],"focused":[36],"on":[37,83,170],"optimizing":[38],"linear":[39],"matrix":[40],"operations,":[41],"while":[42,70,192],"nonlinear":[43],"operators":[44,61],"remain":[45],"relatively":[46],"underexplored.":[47],"This":[48],"paper":[49],"proposes":[50],"hardware-efficient":[51],"approximation":[52,96],"acceleration":[54],"methods":[55],"for":[56],"Softmax":[58,76],"RMSNorm":[60],"to":[62,108,122,150],"reduce":[63],"resource":[64,190],"cost":[65],"accelerate":[67],"Transformer":[68],"inference":[69],"maintaining":[71],"model":[72,194],"accuracy.":[73,125,195],"For":[74,144],"operator,":[77],"an":[78,115,173],"additional":[79],"range":[80],"reduction":[81],"based":[82],"SafeSoftmax":[85],"technique":[86],"enables":[87],"adoption":[89],"a":[91,131,136,166],"bipartite":[92],"lookup":[93,139,164],"table":[94,163],"(LUT)":[95],"acceleration.":[98],"The":[99,126],"bit-width":[100],"configuration":[101],"is":[102,119,128,147,177],"optimized":[103],"through":[104],"Pareto":[105],"frontier":[106],"analysis":[107],"balance":[109],"precision":[110],"hardware":[112,189],"cost,":[113],"error":[116],"compensation":[117],"mechanism":[118],"further":[120,148],"applied":[121],"preserve":[123],"numerical":[124],"division":[127],"reformulated":[129],"logarithmic":[132],"subtraction":[133],"implemented":[134],"with":[135,186],"small":[137],"LOD-driven":[138],"table,":[140],"eliminating":[141],"expensive":[142],"dividers.":[143],"RMSNorm,":[145],"LOD":[146],"leveraged":[149],"decompose":[151],"reciprocal":[153],"square":[154],"root":[155],"into":[156],"mantissa":[157],"exponent":[159],"parts,":[160],"enabling":[161],"parallel":[162],"single":[167],"multiplication.":[168],"Based":[169],"optimizations,":[172],"FPGA-based":[174],"pipelined":[175],"accelerator":[176],"implemented,":[178],"achieving":[179],"low":[180],"operator-level":[181],"latency":[182],"power":[184],"consumption":[185],"significantly":[187],"reduced":[188],"usage":[191],"preserving":[193]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-09-21T07:43:02.447757","created_date":"2026-01-08T00:00:00"}