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racket/collects/math/private/functions/log-gamma-zeros.rkt
Neil Toronto f2dc2027f6 Initial math library commit. The history for these changes is preserved 
in the original GitHub fork:

  https://github.com/ntoronto/racket

Some things about this are known to be broken (most egregious is that the
array tests DO NOT RUN because of a problem in typed/rackunit), about half
has no coverage in the tests, and half has no documentation. Fixes and
docs are coming. This is committed now to allow others to find errors and
inconsistency in the things that appear to be working, and to give the
author a (rather incomplete) sense of closure.
2012-11-16 11:39:51 -07:00

1092 lines
54 KiB
Racket
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#lang typed/racket/base
#|
Compute f(x) = log(abs(gamma(x))) near the roots of `f', accurately. This is done more or less by
brute force: a huge, nested `cond' determines whether `x' is near a root (or a tricky spot, as around
-2.625), and applies one of these many Taylor series expansions.
The Taylor series coefficients were computed by Maxima using
fpprec:45;
terms(x0) := cons(log(abs(gamma(x0))),map(lambda([n],psi[n](x0)/(n+1)!),[0,1,2,3,4,5,6,7,8,9,...]));
terms(bfloat(x0))
where x0 = -16.000000000000046, etc.
The last few digits in each are always wrong, but are fine after rounding to 53 bits, and are
probably fine for larger flonum precisions as well (though we'd need more terms to be more correct).
|#
(require "../../flonum.rkt")
(provide fllog-gamma-special-negative)
(: fllog-gamma-taylor-neg-16-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-16-eps x)
((make-flpolyfun
(3.42552033350358311733334120197391285997329364e-2
2.16519212854378804364052503881457935471686475e13
2.34402847675338276903976186040675803105186722e26
3.38351467129883525133298714499292391230168593e39
5.49446949983078390448145147509511783209827688e52
9.51726568924479766428829792505066627072141892e65
1.71722572963416977329253566951727611382059435e79
3.18696311377395951980006026131098541415051276e92
6.03783901691414892971142619565256231858278562e105
1.16205169002981209467014755360005441193795642e119
2.26445865499190633858041606972575513957782768e132
4.45726186836426745991677397375017774085015343e145
8.84659261955282023171874967614556357794891805e158
1.76811440347263289710978356338608287606666092e172
3.55485686099453407775024429998387503068981764e185
7.18381822062263098616415005664454304133158426e198
1.45821999976683265621312998703694106904655702e212
2.97160137899281100204818274837937667034221446e225
6.07663858635514120494896403934157760213721947e238
1.24646116122756945135868661223357906051452277e252
2.56388650008387675084781633626915713290845764e265
5.28695892234340315776889816230237875306469639e278
1.09269508497457661879972605257434704162622542e292
2.26302980571621375226727382723173851714553023e305))
(fl+ x 16.000000000000046)))
(: fllog-gamma-taylor-neg-16+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-16+eps x)
((make-flpolyfun
(+3.42552033352947932572398727165416329469955525e-2
-2.16519212854322734097485954674089084927480270e13
+2.34402847675338276903976186040675803105186553e26
-3.38351467129883525133298714499292391230168716e39
+5.49446949983078390448145147509511783209827688e52
-9.51726568924479766428829792505066627072141892e65
+1.71722572963416977329253566951727611382059435e79
-3.18696311377395951980006026131098541415051276e92
+6.03783901691414892971142619565256231858278562e105
-1.16205169002981209467014755360005441193795642e119
+2.26445865499190633858041606972575513957782768e132
-4.45726186836426745991677397375017774085015343e145
+8.84659261955282023171874967614556357794891805e158
-1.76811440347263289710978356338608287606666092e172
+3.55485686099453407775024429998387503068981764e185
-7.18381822062263098616415005664454304133158426e198
+1.45821999976683265621312998703694106904655702e212
-2.97160137899281100204818274837937667034221446e225
+6.07663858635514120494896403934157760213721947e238
-1.24646116122756945135868661223357906051452277e252
+2.56388650008387675084781633626915713290845764e265
-5.28695892234340315776889816230237875306469639e278
+1.09269508497457661879972605257434704162622542e292
-2.26302980571621375226727382723173851714553023e305))
(fl+ x 15.999999999999954)))
(: fllog-gamma-taylor-neg-15-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-15-eps x)
((make-flpolyfun
(1.15525490672681022185711739708783012180346524e-3
1.30918593819183868774692964788238733025798397e12
8.56983910376033938275220112806158380788981778e23
7.47967523145739725777963978774141027614667188e35
7.34421422643398169572992335407374271401272603e47
7.69195359383655244163976623081075588379362466e59
8.39183123521159038610388997065510730021596055e71
9.41697209896719459325296135717142749205948794e83
1.07874965207510576326789906427969304722014057e96
1.25536344473165336499588602372903828497531979e108
1.47915375224337636920269163322857941330755097e120
1.76044299350606257350766868342715948489863137e132
2.11268494440818660541928314338955356827359956e144
2.55313608096278164214923128293506360284438946e156
3.10377772294543137103271301928750979038863189e168
3.79252734013508678551325226736139002262904078e180
4.65480324740864694393008748960697082816914839e192
5.73553219439468045888102489477691980653676664e204
7.09171820265835310726774563051367548937070073e216
8.79572628807155765623735540489237622206442855e228
1.09394791138782982168391460633220354706621609e241
1.36398211685740940272776847469329107382146839e253
1.70453774336152266979708967262322846846670140e265
2.13453263408141799832562100864983139919557550e277
2.67806260459741351643411574808075407119390293e289
3.36583862738784320386893917339053511137912179e301))
(fl+ x 15.000000000000764)))
(: fllog-gamma-taylor-neg-15+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-15+eps x)
((make-flpolyfun
(+1.15525491091416550634620392019108538425965918e-3
-1.30918593818635666109027472714561810093410983e12
+8.56983910376033938275220112806158380785805751e23
-7.47967523145739725777963978774141027616053193e35
+7.34421422643398169572992335407374271401272603e47
-7.69195359383655244163976623081075588379362466e59
+8.39183123521159038610388997065510730021596055e71
-9.41697209896719459325296135717142749205948794e83
+1.07874965207510576326789906427969304722014057e96
-1.25536344473165336499588602372903828497531979e108
+1.47915375224337636920269163322857941330755097e120
-1.76044299350606257350766868342715948489863137e132
+2.11268494440818660541928314338955356827359956e144
-2.55313608096278164214923128293506360284438946e156
+3.10377772294543137103271301928750979038863189e168
-3.79252734013508678551325226736139002262904078e180
+4.65480324740864694393008748960697082816914839e192
-5.73553219439468045888102489477691980653676664e204
+7.09171820265835310726774563051367548937070073e216
-8.79572628807155765623735540489237622206442855e228
+1.09394791138782982168391460633220354706621609e241
-1.36398211685740940272776847469329107382146839e253
+1.70453774336152266979708967262322846846670140e265
-2.13453263408141799832562100864983139919557550e277
+2.67806260459741351643411574808075407119390293e289
-3.36583862738784320386893917339053511137912179e301))
(fl+ x 14.999999999999236)))
(: fllog-gamma-taylor-neg-14-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-14-eps x)
((make-flpolyfun
(7.05720414970836382974343795932367836228204258e-5
8.71844437724299607238818643993229044314817323e10
3.80056361772083925440439547664641650199223408e21
2.20900016681707654152682369867627068687209827e32
1.44442838123433135774968727885183741983569213e43
1.00745347986530949975061019345278118790493350e54
7.31952260549680798260379944355600071336462531e64
5.46984434588907051495462936676216116146823788e75
4.17274669703046177687747276824264410147395985e86
3.23376533131276231243373110151708319509140853e97
2.53740628523180393404941068473096530601602486e108
2.01111232359833350081224410957694968261608159e119
1.60726233517136341629818195230703984100819403e130
1.29349174785225078248774509860609135667443638e141
1.04717190088891217889982283945450481764946424e152
8.52106263962962986047317245298885003374543513e162
6.96472599903145134668055154185376490901542828e173
5.71497187985235353695785444163409490053443581e184
4.70575719810015273300645000279164929924814153e195
3.88675727840733627688644182717269723570884573e206
3.21921532816693137167156670390289667070550200e217
2.67300474058385899419220218361791442066160170e228
2.22451502797099844556885973986663205320032769e239
1.85510796441934987891353225082737600968833191e250
1.54997532843630249255321082937107619733383802e261
1.29728387391900772334015429402303245587856606e272
1.08752858614521596590096294535581746089179482e283))
(fl+ x 14.00000000001147)))
(: fllog-gamma-taylor-neg-14+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-14+eps x)
((make-flpolyfun
(+7.05721028462543395166741252987437638446269345e-5
-8.71844437670812674005602769958828165617820307e10
+3.80056361772083925440439547664641644750312443e21
-2.20900016681707654152682369867627070270744066e32
+1.44442838123433135774968727885183741983569213e43
-1.00745347986530949975061019345278118790493350e54
+7.31952260549680798260379944355600071336462531e64
-5.46984434588907051495462936676216116146823788e75
+4.17274669703046177687747276824264410147395985e86
-3.23376533131276231243373110151708319509140853e97
+2.53740628523180393404941068473096530601602486e108
-2.01111232359833350081224410957694968261608159e119
+1.60726233517136341629818195230703984100819403e130
-1.29349174785225078248774509860609135667443638e141
+1.04717190088891217889982283945450481764946424e152
-8.52106263962962986047317245298885003374543513e162
+6.96472599903145134668055154185376490901542828e173
-5.71497187985235353695785444163409490053443581e184
+4.70575719810015273300645000279164929924814153e195
-3.88675727840733627688644182717269723570884573e206
+3.21921532816693137167156670390289667070550200e217
-2.67300474058385899419220218361791442066160170e228
+2.22451502797099844556885973986663205320032769e239
-1.85510796441934987891353225082737600968833191e250
+1.54997532843630249255321082937107619733383802e261
-1.29728387391900772334015429402303245587856606e272
+1.08752858614521596590096294535581746089179482e283))
(fl+ x 13.99999999998853)))
(: fllog-gamma-taylor-neg-13-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-13-eps x)
((make-flpolyfun
(4.20070692126859694083520385628430681145952866e-6
6.22704696315015051305989305476581988406584914e9
1.93880569244302627999559554043966627136075792e19
8.04868939614601273767623947637595966084723446e28
3.75896751304948261029487848517086715088665702e38
1.87258137815444559355579202413716800486894348e48
9.71721014934499007442790146433391077060891507e57
5.18653062218378047648854673573809085536221324e67
2.82596735283228244362324473616029571252961981e77
1.56421612578310868828508606287502751413734642e87
8.76640254452700692209329834562293888490992251e96
4.96261821089367480005194792833420777195398124e106
2.83272519259599852918567471322345942565749047e116
1.62826272005727565118236773772165623741858943e126
9.41503496319752555773534041626705004304107636e135
5.47193405276234040824249545461113755887770115e145
3.19443659170955917253473379654127143885350501e155
1.87217945120619961652450055643490432162160360e165
1.10104743967196314558478384285610834792427926e175
6.49541758038405192149141424551847761237213585e184
3.84249067863447907906831685703870486213896574e194
2.27879713345922908440402121716008913743249147e204
1.35451687289014457756507563651717649104370105e214
8.06791669041266820186883279414411645394575841e223
4.81459921006082217623617815487345538030488875e233
2.87815059621718549831528798396536979929342541e243
1.72330566559340670986129762640372053529825002e253))
(fl+ x 13.00000000016059)))
(: fllog-gamma-taylor-neg-13+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-13+eps x)
((make-flpolyfun
(+4.20154292527072621914889857511622485654985997e-6
-6.22704695794431433259544850846871775302229610e9
+1.93880569244302627999559554043957827675071196e19
-8.04868939614601273767623947637614230970357822e28
+3.75896751304948261029487848517086715088665702e38
-1.87258137815444559355579202413716800486894348e48
+9.71721014934499007442790146433391077060891507e57
-5.18653062218378047648854673573809085536221324e67
+2.82596735283228244362324473616029571252961981e77
-1.56421612578310868828508606287502751413734642e87
+8.76640254452700692209329834562293888490992251e96
-4.96261821089367480005194792833420777195398124e106
+2.83272519259599852918567471322345942565749047e116
-1.62826272005727565118236773772165623741858943e126
+9.41503496319752555773534041626705004304107636e135
-5.47193405276234040824249545461113755887770115e145
+3.19443659170955917253473379654127143885350501e155
-1.87217945120619961652450055643490432162160360e165
+1.10104743967196314558478384285610834792427926e175
-6.49541758038405192149141424551847761237213585e184
+3.84249067863447907906831685703870486213896574e194
-2.27879713345922908440402121716008913743249147e204
+1.35451687289014457756507563651717649104370105e214
-8.06791669041266820186883279414411645394575841e223
+4.81459921006082217623617815487345538030488875e233
-2.87815059621718549831528798396536979929342541e243
+1.72330566559340670986129762640372053529825002e253))
(fl+ x 12.99999999983941)))
(: fllog-gamma-taylor-neg-12-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-12-eps x)
((make-flpolyfun
(7.92331889477662838176704627169679553918772480e-7
4.79001984580383156934295092578649219589398137e8
1.14721449406016195410092484708147166014958633e17
3.66345344330893113284755742851916848671782064e25
1.31610109538171333064294605008518749624155058e34
5.04332026617433628923319069504402651198760215e42
2.01313366969380958404839816353804766505096440e51
8.26538586791793206180756921553598759148456961e59
3.46424418652989138003252353575322225544849778e68
1.47500429481618622832411424312125875871473290e77
6.35876982680119455930209240151498201553943053e85
2.76896668224128240282212673231364014487122016e94
1.21581215161645932855804092279695053285167943e103
5.37578243473933196559200253766562886942293722e111
2.39108112409521134748162268919338033426509823e120
1.06897709118147796032098934601791965876435654e129
4.80039511356245954450366286956611957635723875e137
2.16414002262646382463492539807218520945450358e146
9.79036951376787668833331620803049384838472992e154
4.44278501255337092574540257286295318903639058e163
2.02169768551186469267664691423449813562994708e172
9.22283046166621209468650365753657254785284645e180
4.21694706804926224603612832242519303537955950e189
1.93210313408299233443396459331600576138416068e198
8.86919512810946571537814217489880825607931245e206
4.07842756376789482418778552477199675182815872e215
1.87843739106737609633367311897011914701215515e224))
(fl+ x 12.000000002087674)))
(: fllog-gamma-taylor-neg-12+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-12+eps x)
((make-flpolyfun
(+8.02878797874473321229715558287816009952298183e-7
-4.79001979528393130316004392463101229275295420e8
+1.14721449406016195410092484694807350730298753e17
-3.66345344330893113284755742873216175210420413e25
+1.31610109538171333064294605008518749624155058e34
-5.04332026617433628923319069504402651198760215e42
+2.01313366969380958404839816353804766505096440e51
-8.26538586791793206180756921553598759148456961e59
+3.46424418652989138003252353575322225544849778e68
-1.47500429481618622832411424312125875871473290e77
+6.35876982680119455930209240151498201553943053e85
-2.76896668224128240282212673231364014487122016e94
+1.21581215161645932855804092279695053285167943e103
-5.37578243473933196559200253766562886942293722e111
+2.39108112409521134748162268919338033426509823e120
-1.06897709118147796032098934601791965876435654e129
+4.80039511356245954450366286956611957635723875e137
-2.16414002262646382463492539807218520945450358e146
+9.79036951376787668833331620803049384838472992e154
-4.44278501255337092574540257286295318903639058e163
+2.02169768551186469267664691423449813562994708e172
-9.22283046166621209468650365753657254785284645e180
+4.21694706804926224603612832242519303537955950e189
-1.93210313408299233443396459331600576138416068e198
+8.86919512810946571537814217489880825607931245e206
-4.07842756376789482418778552477199675182815872e215
+1.87843739106737609633367311897011914701215515e224))
(fl+ x 11.999999997912326)))
(: fllog-gamma-taylor-neg-11-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-11-eps x)
((make-flpolyfun
(2.73464038994808618853990996713900546500363245e-8
3.99168059769041149523568978415501579131928981e7
7.96675602195659478355932730588437298457319315e14
2.12004956622456136373539147270355509119210030e22
6.34692015133814117725645182781078049260408379e29
2.02679011782791482635892447400196186445606399e37
6.74191524487342619925839700081682702106088296e44
2.30670605377823204002172584808694044342738686e52
8.05667908149243458094760269141474835194991556e59
2.85863889808333292404959945241460300717711546e67
1.02696954543122321631238034925453188849766691e75
3.72666741642077092523775278109077602761444219e82
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4.27517506575444098042953315198093408442090338e188
1.64087810744434949149168426042942529202007384e196))
(fl+ x 11.000000025052106)))
(: fllog-gamma-taylor-neg-11+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-11+eps x)
((make-flpolyfun
(+7.92108817813104976338185676938186326377669157e-9
-3.99167954308609387899529490984926221844918726e7
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(fl+ x 10.99999997494789)))
(: fllog-gamma-taylor-neg-10-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-10-eps x)
((make-flpolyfun
(1.26680513875652366050062659798555748836557448e-9
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1.37680628240094496354203214974016406530882430e169))
(fl+ x 10.000000275573013)))
(: fllog-gamma-taylor-neg-10+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-10+eps x)
((make-flpolyfun
(+1.77195439588259352534680065934485472482130717e-9
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(fl+ x 9.99999972442663)))
(: fllog-gamma-taylor-neg-9-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-9-eps x)
((make-flpolyfun
(3.44426332839150908016964631288898671467297262e-11
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1.37700517982992993725897300416472670257692228e143))
(fl+ x 9.000002755714823)))
(: fllog-gamma-taylor-neg-9+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-9+eps x)
((make-flpolyfun
(+5.64090189601985554680682097784018867907670734e-10
-3.62875496679368153944184754133409853882919443e5
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(fl+ x 8.999997244250979)))
(: fllog-gamma-taylor-neg-8-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-8-eps x)
((make-flpolyfun
(1.75595561986039004670754217836682458096818294e-11
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2.13382946385035377749021639708399303746666443e118))
(fl+ x 8.000024800270682)))
(: fllog-gamma-taylor-neg-8+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-8+eps x)
((make-flpolyfun
(+3.36862440532578048192178695865761670703028502e-11
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(fl+ x 7.9999751970958215)))
(: fllog-gamma-taylor-neg-7-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-7-eps x)
((make-flpolyfun
(3.21679479852632759745367246513718980001635171e-12
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5.63705852508412233807453056382118699556756662e72
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1.46947826096328253735160422033263210654528334e91
7.12416473540339088510115562375712909673904392e94))
(fl+ x 7.000198333407324)))
(: fllog-gamma-taylor-neg-7+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-7+eps x)
((make-flpolyfun
(+5.31301106573590200897337106433467557164151869e-14
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(fl+ x 6.999801507890638)))
(: fllog-gamma-taylor-neg-6-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-6-eps x)
((make-flpolyfun
(5.96373419303467500413792265481786840068918471e-13
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1.07633283427581150041381904802190294929609939e53
7.38121913528046491626428063066656081632724397e55
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3.49664243034293515162189387270168949733615312e61
2.41437081551384922759964267834853643796077387e64
1.67023842928535310631607098959951607194524987e67
1.15746431270476848578187718611073050993194347e70
8.03400643083259808484932459059707879728773282e72))
(fl+ x 6.001385294453154)))
(: fllog-gamma-taylor-neg-6+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-6+eps x)
((make-flpolyfun
(+3.98943097843441932465652026666091000412175563e-13
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-3.81964423882096250884715503265367483472767970e13
+2.28581065302601343849063472135311113229589014e16
-1.40699335579708310084132549548594488557761312e19
+8.84094489001693642835942147140025472914301438e21
-5.64345082040892431617497604635292443260529529e24
+3.64742052778954305530816853872536028608224385e27
-2.38117731596356223311008014087648517813605274e30
+1.56747910244616295308347662889941379734448921e33
-1.03905428353497913012761193552317328229344863e36
+6.92870606139998047359222823627264086936374153e38
-4.64394960184288117953293901427853908990631646e41
+3.12649226193266320638056018879596813548202725e44
-2.11313383169320236892244849825165874991284698e47
+1.43318417230683724207672956565653826555288418e50
-9.75033396470380191538409915247952968713605747e52
+6.65183825327027441169299935628486491306778882e55
-4.54936691657501621666687834880905567118335622e58
+3.11850297553268885844035989960765352437005977e61
-2.14209926096916557565934538015232402947359683e64
+1.47419444196325716857338412118878929175682948e67
-1.01630622646690800002319571389560738233771459e70
+7.01761972686052721453154108904317582064920521e72))
(fl+ x 5.998607480080876)))
(: fllog-gamma-taylor-neg-5-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-5-eps x)
((make-flpolyfun
(5.41885092655381025719595516331601539815956969e-15
1.23362184563354043865731028461414057744606020e2
7.40476243222876130900574826280080918143125990e3
6.00556168452820552068729377561100555529370641e5
5.48074863566570265324417981653566617471824549e7
5.33525074338452723153487397127655583797976263e9
5.41001609074791125558485600315244634161655653e11
5.64256713562444548684611921421374417220809909e13
6.00772100286369220122905461643344677542613570e15
6.49803732093028846484598058316207969435715638e17
7.11622512374105590783954671230253655863195756e19
7.87194337362483304708962106716775948612245357e21
8.78048223063437069160511062733533974970161719e23
9.86236859897728368260206634391870540840187331e25
1.11434973580101773101371794731069019794053142e28
1.26556151272168476205402111916756928204819325e30
1.44370846592998098137993770587584944505535249e32
1.65339085935578868895584777638475475524988199e34
1.90010200608912963279940507532952645389745432e36
2.19038667338894880563303719195863734661098943e38
2.53203300058817032335469415689374134458741212e40
2.93430349185055042449367970919755908893213651e42
3.40821208626557548218602650412971833405190776e44
3.96685590281426661708411918644940947372395895e46
4.62581209214915604420872188953110757048353555e48
5.40361237948187342779585787258489671029005435e50
6.32231043483605471166429476809359358293138224e52))
(fl+ x 5.0082181683225935)))
(: fllog-gamma-taylor-neg-5+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-5+eps x)
((make-flpolyfun
(+1.76836193508496134514687044485233022507412166e-14
-1.16535781616245753768503617985375944578579193e2
+6.99523595489431486546890813466674970256552674e3
-5.51419997851318044632166207860447589187985901e5
+4.89115827090625556320989173735148990068192623e7
-4.62774717173966484754236449207884450962286344e9
+4.56096042262246697956773952789936663634595578e11
-4.62357002765258798875348281864575148589999927e13
+4.78468574036034408626909392503392949786418539e15
-5.03000968987250701086043900444438734899839871e17
+5.35401097142582048519214039246283264279679805e19
-5.75644682625551268823564997647690621683299800e21
+6.24070799300926007561084330972156813164729147e23
-6.81302027775316489238836963643326917419501011e25
+7.48208993021257914055045142198078739956243209e27
-8.25900304352361522965771494862295861316385777e29
+9.15728705360304464559850646551668374267131592e31
-1.01930889756736968113544397731888110415266705e34
+1.13854488660950913720734782365262882725974297e36
-1.27566601891368590344217106037433472891809611e38
+1.43327167410740289698783159819255907081451586e40
-1.61438511479149974470063382576325819998775478e42
+1.82251740349289405753525333412414171829955385e44
-2.06174264827513679839882304004797713214906927e46
+2.33678617524058001428564879357591130591224694e48
-2.65312757448388074721204114962473788958612848e50
+3.01712094482992778631151481754365566073619428e52))
(fl+ x 4.991544640560048)))
(: fllog-gamma-taylor-neg-4-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-4-eps x)
((make-flpolyfun
(1.81301489023985618481731357730667339196237550e-14
2.67904808861410321327214397331379787908541540e1
3.24255322937858253736006013570778920615832578e2
5.46569182077741616725319810542178832035822966e3
1.04145555201462573673149727720949548371162345e5
2.11666941757771701328409351874486396871812932e6
4.48122159501603318236465855420526550067324334e7
9.75830163538331950363973953929618675079580747e8
2.16923649607162932956165726741988370482884479e10
4.89867910498275971952758694178019464785226189e11
1.12007244162039286479579040474081852339397694e13
2.58689045256111345612494171926626369243546123e14
6.02440400096176701173680787572491817237764037e15
1.41278668508301263198302773720333978605091028e17
3.33285577879941677496169133681524910584979240e18
7.90274391149113501309340654162886498293963454e19
1.88223479030962949513233495444370666816866525e21
4.50059018694097294452319570025381466521375657e22
1.07986756381211853649283994199009213197263987e24
2.59904627328961985271105601748597429270721239e25
6.27281137238709819146504016485952916588925394e26
1.51774055997381403059397905691577190791272758e28
3.68060121745351564615021740987651034473640274e29
8.94413244088949726258433997879455026504805354e30
2.17760661400824862848456267793034288891939313e32
5.31098740106663851246556066853754013568800558e33
1.29737804140610088127865888231535370941492990e35))
(fl+ x 4.039361839740536)))
(: fllog-gamma-taylor-neg-4+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-4+eps x)
((make-flpolyfun
(+8.78939314434977824362286615570184975741305041e-15
-2.07250608458039191801398354912074701126554797e1
+2.51714682586894146537732187837038486323386563e2
-3.73060471568071868389371231096324364995674596e3
+6.25880819187684354729315259842094937424442594e4
-1.11999150166557525866438224883417960801815493e6
+2.08771147265045175361955519775825227765181884e7
-4.00276998187490238282584216189014567996179702e8
+7.83439818373891296219323306813420731726214027e9
-1.55772242416585467557455053761730286461842180e11
+3.13595292529358977965704401186423364249660428e12
-6.37696161822735582434913575368842994406753210e13
+1.30756171365397339075006701778228546488926118e15
-2.69983387930444407360331692215175599784681891e16
+5.60775863730578162721388295019809496365914994e17
-1.17074697159654892886605119268916507675086169e19
+2.45511179588604297371791674939611648546057739e20
-5.16867574457309810939753320374439139711959337e21
+1.09192461783841750528498906123439438876072262e23
-2.31392093346042541326971161202647385124381260e24
+4.91710037482801295013628404667352873797929314e25
-1.04750650406095202238738008578691313752209369e27
+2.23661017141897372233705223490312596918865879e28
-4.78544240743225380874419144451265275847103823e29
+1.02583058100822899833668001151690176951567802e31
-2.20284443421392884416725580252156642000839635e32
+4.73791693429605430177700526875053064928423719e33))
(fl+ x 3.9552942848585984)))
(: fllog-gamma-taylor-neg-3-eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-3-eps x)
((make-flpolyfun
(1.69786559061210843022272847964297037675859473e-15
7.78188465813136214399528129997442792941073176e0
2.58313383723880324208953213398090230714514980e1
1.12268986297176516519245402454125678034401274e2
5.88890742238004921853852404638860526426305979e2
3.27719378549536664789687328052979137659712232e3
1.90230619843565318084065380848899905923057322e4
1.13559664237136370180852663772361533676044070e5
6.92049602289783024945417378060367661802816637e5
4.28437744364603821541980538763698670942244785e6
2.68555250555762961389414458921497932348535306e7
1.70037345821745965876919924552878737697863407e8
1.08557322157002487766499595339125057568363179e9
6.97911539667148383495178196114992142539256805e9
4.51355833575162388564796789592490869277550350e10
2.93399385882337138148673434081036371240625868e11
1.91572797345398693597903943399980424905205048e12
1.25576468663909262494300353917876036735521610e13
8.26015213762303044035070558862351046563406452e13
5.45017333309697979666936223400225945240521114e14
3.60609599644037084765420160822868349591590907e15
2.39194587729231083416912378422842318565585423e16
1.59019845254247279515221145045196949680406309e17
1.05937454968300158726334055013820581387376515e18
7.07081530914871512696425122433809109739803684e18
4.72763664773893258007091476870784841957917544e19
3.16602336231497561963030757035805419348902704e20))
(fl+ x 3.14358088834998)))
(: fllog-gamma-taylor-neg-3+eps (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-3+eps x)
((make-flpolyfun
(+1.73350924402450086109664880252280930803796840e-16
-1.91435018561160055062671998360109773008757615e0
+9.57518947570967212969871531897887298425867401e0
-2.00951349168426252669134955591785433284090043e1
+6.26272827135138013766227814092855203458008016e1
-1.94766155303446568125118695588276590924110440e2
+6.46905999712929665878051101089935247949317056e2
-2.19311277648062925608544541675088274405618955e3
+7.61039798665835057465104404450865651687943724e3
-2.68045784206181952947864995136383095044210192e4
+9.56174516433500463141401730092716465102711170e4
-3.44497593434834251677126191694547877536433278e5
+1.25156744652510559191680924102071419509151639e6
-4.57871659458173884379082476598177277475632514e6
+1.68504842706974508404657029185505612957061835e7
-6.23306848324077503176646009404048417739453040e7
+2.31593359803530705843535585519480903266158016e8
-8.63873252929326222756433185277849724061407159e8
+3.23354812607607746932238396579196197173278373e9
-1.21409064008644206975493902628524125704746969e10
+4.57117235417958792595283855057138618743523254e10
-1.72540549493287770025997384282337316967575029e11
+6.52740666207530857195855094792769006968850608e11
-2.47450533666850553797231795437602547002229320e12
+9.39848534763072962083260627797430059451185761e12
-3.57587213746677906598735906878666200860291196e13
+1.36270397145886520070135314755598407978744695e14))
(fl+ x 2.7476826467274127)))
(: fllog-gamma-taylor-neg-2.5 (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-2.5 x)
((make-flpolyfun
(5.61919235895009645091256851695098111749959486e-17
1.51560344802165768188749164398339627893778105e0
4.85832095163399627746734377392871693605230490e0
1.41129114307798124401510783653592177347873738e0
8.72178258381534729322013445636550188290755750e0
5.80041456659987799413315121154253330844366082e0
2.48249421218940773331265335049735451396738220e1
2.40388230622929786781070091918937520366699213e1
8.22291568270425660106801850355547569796638822e1
1.00648123545511701102453344855177421851423816e2
2.96436413507613369083757276470488051933710099e2
4.25179845684382619948469561167761367915934946e2
1.13045785817751386159338509375334334152608605e3
1.81120541880187359111104929471617512361784951e3
4.48721845318900263604051887722090475536449478e3
7.77615810234425735666130539513148262342615631e3
1.83471706999770464528633009895266966505869873e4
3.36304570557943903790064207274422028127017659e4
7.67210360239843918816147110553574167448750938e4
1.46431998292134176729885025076314444052562342e5
3.26433702221563796981145197985984661873939623e5
6.41568327741749042873610986617199799626504025e5
1.40797959337829879108442601929312734834777980e6
2.82702369687106235551822865228378508072204174e6
6.13938581529136603068173225483911055935198571e6
1.25222596563403475911690432815529823250889260e7
2.70076153104356837368592373996661893611674396e7))
(fl+ x 2.4570247382208006)))
(: fllog-gamma-taylor-neg-2.625 (Flonum -> Flonum))
(define (fllog-gamma-taylor-neg-2.625 x)
((make-flpolyfun
(-1.17511100308904859994666567899607070603472748e-1
-1.57662052808143238767682222467571768255158395e-1
+5.62280028708343342127100665301732994258009008e0
-5.03224286025454307606105505974515135386153910e0
+1.44026056764382911633999836745275241447715481e1
-2.48971464377258885456665463953421569146427274e1
+6.27638301132273807125391353825831515701418911e1
-1.33164187794833814803033486866170142306775977e2
+3.25020330318424079519181342335613727175845720e2
-7.50032453555325748153334099067598984903044042e2
+1.82939126670926077976630411313170194810520239e3
-4.39223051811670135063825899001404441576172064e3
+1.07991099207708003425000890908369348850196235e4
-2.64900388706763510403708190506232237093229157e4
+6.57316327437471829125602547425833118008518082e4
-1.63393764818489768773926609932691194453735854e5
+4.08791856900606820819408084967296134162542699e5
-1.02552442766763569800382316377975918319805950e6
+2.58350187169077781879301640113211587992619404e6))
(fl+ x 2.625)))
(: fllog-gamma-special-negative (Flonum -> Flonum))
;; For x < 0, either computes log(abs(gamma(x))) to good precision (so far, largest observed error is
;; 11 ulps; usually <= 5) or returns 0.0. Having 0.0 as the "I won't compute this" value works fine,
;; since log(abs(gamma(x))) has no zeros at any negative, rational x.
(define (fllog-gamma-special-negative x)
(cond
[(x . fl>= . -10.0)
(cond
[(x . fl>= . -6.0)
(cond
[(x . fl>= . -4.0)
(cond
[(and (x . fl>= . -2.5775) (x . fl<= . -2.3359))
(fllog-gamma-taylor-neg-2.5 x)]
[(and (x . fl>= . (fl- -2.7476826467274127 6.9e-2))
(x . fl<= . (fl+ -2.7476826467274127 6.9e-2)))
(fllog-gamma-taylor-neg-3+eps x)]
[(and (x . fl>= . -2.7) (x . fl<= . -2.575))
(fllog-gamma-taylor-neg-2.625 x)]
[(and (x . fl>= . (fl- -3.14358088834998 4.1e-2))
(x . fl<= . (fl+ -3.14358088834998 4.1e-2)))
(fllog-gamma-taylor-neg-3-eps x)]
[(and (x . fl>= . (fl- -3.9552942848585984 1.25e-2))
(x . fl<= . (fl+ -3.9552942848585984 1.25e-2)))
(fllog-gamma-taylor-neg-4+eps x)]
[else
0.0])]
[(and (x . fl>= . (fl- -4.039361839740536 1.1e-2))
(x . fl<= . (fl+ -4.039361839740536 1.1e-2)))
(fllog-gamma-taylor-neg-4-eps x)]
[(and (x . fl>= . (fl- -4.991544640560048 2.3e-3))
(x . fl<= . (fl+ -4.991544640560048 2.4e-3)))
(fllog-gamma-taylor-neg-5+eps x)]
[(and (x . fl>= . (fl- -5.0082181683225935 2.3e-3))
(x . fl<= . (fl+ -5.0082181683225935 2.3e-3)))
(fllog-gamma-taylor-neg-5-eps x)]
[(and (x . fl>= . (fl- -5.998607480080876 4.0e-4))
(x . fl<= . (fl+ -5.998607480080876 4.0e-4)))
(fllog-gamma-taylor-neg-6+eps x)]
[else
0.0])]
[(x . fl>= . -8.0)
(cond
[(and (x . fl>= . (fl- -6.001385294453154 4.0e-4))
(x . fl<= . (fl+ -6.001385294453154 4.0e-4)))
(fllog-gamma-taylor-neg-6-eps x)]
[(and (x . fl>= . (fl- -6.999801507890638 5.5e-5))
(x . fl<= . (fl+ -6.999801507890638 5.5e-5)))
(fllog-gamma-taylor-neg-7+eps x)]
[(and (x . fl>= . (fl- -7.000198333407324 5.5e-5))
(x . fl<= . (fl+ -7.000198333407324 5.5e-5)))
(fllog-gamma-taylor-neg-7-eps x)]
[(and (x . fl>= . (fl- -7.9999751970958215 7.1e-6))
(x . fl<= . (fl+ -7.9999751970958215 7.1e-6)))
(fllog-gamma-taylor-neg-8+eps x)]
[else
0.0])]
[(and (x . fl>= . (fl- -8.000024800270682 7.1e-6))
(x . fl<= . (fl+ -8.000024800270682 7.1e-6)))
(fllog-gamma-taylor-neg-8-eps x)]
[(and (x . fl>= . (fl- -8.999997244250979 7.8e-7))
(x . fl<= . (fl+ -8.999997244250979 7.8e-7)))
(fllog-gamma-taylor-neg-9+eps x)]
[(and (x . fl>= . (fl- -9.000002755714823 7.8e-7))
(x . fl<= . (fl+ -9.000002755714823 7.8e-7)))
(fllog-gamma-taylor-neg-9-eps x)]
[(and (x . fl>= . (fl- -9.99999972442663 8.0e-8))
(x . fl<= . (fl+ -9.99999972442663 8.0e-8)))
(fllog-gamma-taylor-neg-10+eps x)]
[else
0.0])]
[(x . fl>= . -13.0)
(cond
[(x . fl>= . -11.5)
(cond
[(and (x . fl>= . (fl- -10.000000275573013 8.0e-8))
(x . fl<= . (fl+ -10.000000275573013 8.0e-8)))
(fllog-gamma-taylor-neg-10-eps x)]
[(and (x . fl>= . (fl- -10.99999997494789 7.2e-9))
(x . fl<= . (fl+ -10.99999997494789 7.2e-9)))
(fllog-gamma-taylor-neg-11+eps x)]
[(and (x . fl>= . (fl- -11.000000025052106 7.2e-9))
(x . fl<= . (fl+ -11.000000025052106 7.2e-9)))
(fllog-gamma-taylor-neg-11-eps x)]
[else
0.0])]
[(and (x . fl>= . (fl- -11.999999997912326 6.0e-10))
(x . fl<= . (fl+ -11.999999997912326 6.0e-10)))
(fllog-gamma-taylor-neg-12+eps x)]
[(and (x . fl>= . (fl- -12.000000002087674 6.0e-10))
(x . fl<= . (fl+ -12.000000002087674 6.0e-10)))
(fllog-gamma-taylor-neg-12-eps x)]
[(and (x . fl>= . (fl- -12.99999999983941 4.5e-11))
(x . fl<= . (fl+ -12.99999999983941 4.5e-11)))
(fllog-gamma-taylor-neg-13+eps x)]
[else
0.0])]
[(x . fl>= . -15.0)
(cond
[(and (x . fl>= . (fl- -13.00000000016059 4.5e-11))
(x . fl<= . (fl+ -13.00000000016059 4.5e-11)))
(fllog-gamma-taylor-neg-13-eps x)]
[(and (x . fl>= . (fl- -13.99999999998853 3.4e-12))
(x . fl<= . (fl+ -13.99999999998853 3.4e-12)))
(fllog-gamma-taylor-neg-14+eps x)]
[(and (x . fl>= . (fl- -14.00000000001147 3.4e-12))
(x . fl<= . (fl+ -14.00000000001147 3.4e-12)))
(fllog-gamma-taylor-neg-14-eps x)]
[(and (x . fl>= . (fl- -14.999999999999236 2.1e-13))
(x . fl<= . (fl+ -14.999999999999236 2.1e-13)))
(fllog-gamma-taylor-neg-15+eps x)]
[else
0.0])]
[(and (x . fl>= . (fl- -15.000000000000764 2.1e-13))
(x . fl<= . (fl+ -15.000000000000764 2.1e-13)))
(fllog-gamma-taylor-neg-15-eps x)]
[(and (x . fl>= . (fl- -15.999999999999954 1e-14))
(x . fl<= . (fl+ -15.999999999999954 1e-14)))
(fllog-gamma-taylor-neg-16+eps x)]
[(and (x . fl>= . (fl- -16.000000000000046 1e-14))
(x . fl<= . (fl+ -16.000000000000046 1e-14)))
(fllog-gamma-taylor-neg-16-eps x)]
[else
0.0]))
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