{"id":"https://openalex.org/W2295743822","doi":"https://doi.org/10.1145/1596550.1596579","title":"Beautiful differentiation","display_name":"Beautiful differentiation","publication_year":2009,"publication_date":"2009-08-31","ids":{"openalex":"https://openalex.org/W2295743822","doi":"https://doi.org/10.1145/1596550.1596579","mag":"2295743822"},"language":"en","primary_location":{"id":"doi:10.1145/1596550.1596579","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1596550.1596579","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 14th ACM SIGPLAN international conference on Functional programming","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058654790","display_name":"Conal Elliott","orcid":"https://orcid.org/0000-0002-9480-9518"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Conal M. Elliott","raw_affiliation_strings":["LambdaPix, San Andreas, USA"],"affiliations":[{"raw_affiliation_string":"LambdaPix, San Andreas, USA","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5058654790"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.7656,"has_fulltext":false,"cited_by_count":46,"citation_normalized_percentile":{"value":0.93491439,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":null,"issue":null,"first_page":"191","last_page":"202"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9944000244140625,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9944000244140625,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9866999983787537,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9825000166893005,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/automatic-differentiation","display_name":"Automatic differentiation","score":0.8746305704116821},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.7642416954040527},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6704397201538086},{"id":"https://openalex.org/keywords/chain-rule","display_name":"Chain rule (probability)","score":0.6043594479560852},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5993951559066772},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5985837578773499},{"id":"https://openalex.org/keywords/mode","display_name":"Mode (computer interface)","score":0.5175168514251709},{"id":"https://openalex.org/keywords/numerical-differentiation","display_name":"Numerical differentiation","score":0.5060867667198181},{"id":"https://openalex.org/keywords/partial-derivative","display_name":"Partial derivative","score":0.49865221977233887},{"id":"https://openalex.org/keywords/sampling","display_name":"Sampling (signal processing)","score":0.45240432024002075},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.4189852476119995},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.4060376286506653},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3883998692035675},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3463331460952759},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3444675803184509},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.33510178327560425},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.29772549867630005},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.22273743152618408},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.16451960802078247},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08183234930038452}],"concepts":[{"id":"https://openalex.org/C133512626","wikidata":"https://www.wikidata.org/wiki/Q787371","display_name":"Automatic differentiation","level":3,"score":0.8746305704116821},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.7642416954040527},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6704397201538086},{"id":"https://openalex.org/C33825631","wikidata":"https://www.wikidata.org/wiki/Q17004731","display_name":"Chain rule (probability)","level":5,"score":0.6043594479560852},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5993951559066772},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5985837578773499},{"id":"https://openalex.org/C48677424","wikidata":"https://www.wikidata.org/wiki/Q6888088","display_name":"Mode (computer interface)","level":2,"score":0.5175168514251709},{"id":"https://openalex.org/C154181440","wikidata":"https://www.wikidata.org/wiki/Q1683292","display_name":"Numerical differentiation","level":2,"score":0.5060867667198181},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.49865221977233887},{"id":"https://openalex.org/C140779682","wikidata":"https://www.wikidata.org/wiki/Q210868","display_name":"Sampling (signal processing)","level":3,"score":0.45240432024002075},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.4189852476119995},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.4060376286506653},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3883998692035675},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3463331460952759},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3444675803184509},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.33510178327560425},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.29772549867630005},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.22273743152618408},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.16451960802078247},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08183234930038452},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C106131492","wikidata":"https://www.wikidata.org/wiki/Q3072260","display_name":"Filter (signal processing)","level":2,"score":0.0},{"id":"https://openalex.org/C49698424","wikidata":"https://www.wikidata.org/wiki/Q1989192","display_name":"Law of total probability","level":4,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.0},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C57830394","wikidata":"https://www.wikidata.org/wiki/Q278079","display_name":"Posterior probability","level":3,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C107673813","wikidata":"https://www.wikidata.org/wiki/Q812534","display_name":"Bayesian probability","level":2,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C31972630","wikidata":"https://www.wikidata.org/wiki/Q844240","display_name":"Computer vision","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1596550.1596579","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1596550.1596579","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 14th ACM SIGPLAN international conference on Functional programming","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W24803201","https://openalex.org/W84930537","https://openalex.org/W1577258119","https://openalex.org/W1601890769","https://openalex.org/W1964298340","https://openalex.org/W2041337116","https://openalex.org/W2058760524","https://openalex.org/W2078944436","https://openalex.org/W2080124533","https://openalex.org/W2103009337","https://openalex.org/W2107214848","https://openalex.org/W2112615547","https://openalex.org/W2116542587","https://openalex.org/W2116813111","https://openalex.org/W2133221106","https://openalex.org/W4256418598","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W4230546540","https://openalex.org/W2553993726","https://openalex.org/W4386148205","https://openalex.org/W2009634354","https://openalex.org/W2370856440","https://openalex.org/W4252202378","https://openalex.org/W2295743822","https://openalex.org/W2127375336","https://openalex.org/W3148322078","https://openalex.org/W2486169089"],"abstract_inverted_index":{"Automatic":[0,111],"differentiation":[1,112],"(AD)":[2],"is":[3,142],"a":[4,71,106],"precise,":[5],"efficient,":[6],"and":[7,59,66,73,80,108,163],"convenient":[8],"method":[9],"for":[10],"computing":[11],"derivatives":[12,31,134],"of":[13,28,50,62,94,102,104,115,136,157],"functions.":[14],"Its":[15],"forward-mode":[16,53],"implementation":[17,49,69],"can":[18],"be":[19],"quite":[20],"simple":[21,72],"even":[22,168],"when":[23],"extended":[24],"to":[25,78,127,131,144,160],"compute":[26],"all":[27,133],"the":[29,56,82,84,87,100,121,140,170],"higher-order":[30,128],"as":[32],"well.":[33],"The":[34,155],"higher-dimensional":[35],"case":[36],"has":[37],"also":[38],"been":[39],"tackled,":[40],"though":[41],"with":[42,120],"extra":[43],"complexity.":[44],"This":[45],"paper":[46,85],"develops":[47],"an":[48],"higher-dimensional,":[51],"higher-order,":[52],"AD":[54,91,129,158],"in":[55,99,148],"extremely":[57],"general":[58,165],"elegant":[60,162],"setting":[61,141],"calculus":[63],"on":[64],"manifolds":[65],"derives":[67],"that":[68],"from":[70,125],"precise":[74],"specification.":[75],"In":[76],"order":[77],"motivate":[79],"discover":[81],"implementation,":[83],"poses":[86],"question":[88],"\"What":[89],"does":[90],"mean,":[92],"independently":[93],"implementation?\"":[95],"An":[96],"answer":[97],"arises":[98],"form":[101],"naturality":[103,117],"sampling":[105,132],"function":[107],"its":[109],"derivative.":[110],"flows":[113],"out":[114],"this":[116,161],"condition,":[118],"together":[119],"chain":[122],"rule.":[123],"Graduating":[124],"first-order":[126],"corresponds":[130],"instead":[135],"just":[137],"one.":[138],"Next,":[139],"expanded":[143],"arbitrary":[145],"vector":[146],"spaces,":[147],"which":[149,167],"derivative":[150],"values":[151],"are":[152],"linear":[153],"maps.":[154],"specification":[156],"adapts":[159],"very":[164],"setting,":[166],"simplifies":[169],"development.":[171]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":17},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
