f2dc2027f6
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.
158 lines
7.0 KiB
Racket
158 lines
7.0 KiB
Racket
#lang typed/racket/base
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;; Gautschi's algorithm (as presented by Temme) for the regularized upper gamma for x < 1
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(require "../../../flonum.rkt"
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"../../../base.rkt"
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"../../polynomial/chebyshev.rkt"
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"../gamma.rkt"
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"../log-gamma.rkt"
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"../stirling-error.rkt"
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"gamma-utils.rkt")
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(provide flgamma-upper-gautschi flgamma-lower-gautschi)
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(: fl1-1/gamma1p-taylor-0 (Float -> Float))
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(define (fl1-1/gamma1p-taylor-0 x)
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(fl* x ((make-flpolyfun
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(-5.7721566490153286060651209008240243104215933593992e-1
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+6.5587807152025388107701951514539048127976638047858e-1
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+4.2002635034095235529003934875429818711394500401104e-2
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-1.6653861138229148950170079510210523571778150224717e-1
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+4.219773455554433674820830128918739130165268418982e-2
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+9.6219715278769735621149216723481989753629422521117e-3
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-7.2189432466630995423950103404465727099048008802391e-3
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+1.1651675918590651121139710840183886668093337953828e-3
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+2.1524167411495097281572996305364780647824192337778e-4
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-1.2805028238811618615319862632816432339489209969465e-4
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+2.0134854780788238655689391421021818382294833296229e-5
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+1.2504934821426706573453594738330922423226556201567e-6
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-1.1330272319816958823741296203307449433240048389228e-6
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+2.0563384169776071034501541300205728365125790182239e-7
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-6.1160951044814158178624986828553428672758673305935e-9))
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x)))
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(: fl1-1/gamma1p-taylor-1 (Float -> Float))
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(define (fl1-1/gamma1p-taylor-1 x)
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(fl* (fl- x 1.0)
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((make-flpolyfun
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(+4.2278433509846713939348790991759756895784066406008e-1
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+2.3309373642178674168353160522779291232192571641851e-1
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-1.910911013876915061545276703523630936105312160174e-1
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+2.4552490005400016652826875250257857892749713770226e-2
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+1.7645244550144320095381426038929533408902970419595e-2
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-8.0232730222673465332665043665813344335400281674833e-3
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+8.0432977560424699087149402613476172363522728724358e-4
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+3.6083781625481812124247705788362694317410650813711e-4
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-1.4559614213986714842674709482997913669586458476042e-4
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+1.7545859751750962273548468501814813300972485066968e-5
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+2.5889950290372763821409229192070050813223482271545e-6
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-1.3385015468946057247955634453739128389996926090334e-6
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+2.0547431491290984242143382504316789567568776464973e-7
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+1.5952678485086792358158795888938797557013266594503e-10
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-6.2756218893322837414440866417447308428460020476605e-9))
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(fl- x 1.0))))
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(define fl1-1/gamma1p-0.3-0.7
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(inline-chebyshev-flpoly-fun
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0.3 0.7
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(-0.2357535799644732185145165985112978133459
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0.007192709283280881736611459461064511206541
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0.01049211807832747501559721918849629071077
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-3.459974677274507288096516749252814460585e-4
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-1.027210542517651843057394564412174832553e-5
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8.401234394534847347740864823467611880071e-7
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-1.304830795670152306426264416729164660901e-8
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-4.268734425630906391429002996659143191633e-10
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2.216361071246008579047775791962841508693e-11
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-3.016588637335897973295510846778736973194e-13
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-5.122627533431577404793896156393660398889e-15
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2.777819095314707020162876992918284024147e-16
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-4.289212074552049664300497789553082042299e-18)))
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(: fl1-1/gamma1p (Float -> Float))
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;; Computes 1-1/gamma(x+1); relative error <= 2*eps
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(define (fl1-1/gamma1p x)
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(cond [(x . fl< . 0.3) (fl1-1/gamma1p-taylor-0 x)]
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[(x . fl< . 0.7) (fl1-1/gamma1p-0.3-0.7 x)]
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[else (fl1-1/gamma1p-taylor-1 x)]))
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(: flgamma-gautschi-iter (Float Float -> Float))
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;; Calculates the series part of Gautschi's algorithm
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(define (flgamma-gautschi-iter k x)
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(let loop ([p (fl* k x)] [q (fl+ k 1.0)] [r (fl+ k 3.0)] [t 1.0] [v 1.0])
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(cond [((flabs t) . fl<= . (flabs (fl* (fl* 0.5 epsilon.0) v))) v]
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[else (let* ([p (fl+ p x)]
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[q (fl+ q r)]
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[r (fl+ r 2.0)]
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[t (fl/ (fl* (- p) t) q)]
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[v (fl+ v t)])
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(loop p q r t v))])))
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(: lg1-prod (Float Float -> Float))
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;; Calculates (lg1- (* k A)) in a way that maintains precision when k or A is very small
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(define (lg1-prod k A)
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(define kA (fl* k A))
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(cond [((flabs kA) . fl< . 1e-300)
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(let-values ([(k A) (if ((flabs k) . fl< . (flabs A)) (values k A) (values A k))])
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(fl- (lg1- (fl* (fl* (flexpt 2.0 80.0) k) A))
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;; Approximates sum_i=1^80 (fllog1p (exp (* (flexpt 2.0 i) k A)))
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;; (hint: (exp (* (flexpt 2.0 i) k A)) ~= 1 here)
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(fllog (flexpt 2.0 80.0))))]
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[else
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(lg1- kA)]))
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(: flgamma-upper-gautschi (Float Float Any -> Float))
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;; Temme's implementation of Gautschi's series for upper gamma, extended to compute logs
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(define (flgamma-upper-gautschi k x log?)
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(cond [log?
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(define p (flgamma-upper-gautschi k x #f))
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(cond
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[(and (p . > . +max-subnormal.0) (p . < . 0.8)) (fllog p)]
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[(k . < . 1e-32)
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(define y (flgamma-gautschi-iter k x))
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(+ (fllog k) (fllog (- (* x y) (+ (fllog x) gamma.0))))]
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[(k . < . 1e-16)
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(define y (flgamma-gautschi-iter k x))
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(lg1- (+ (fllog1p (* k (- 1.0 (* x y))))
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(* k (+ (fllog x) gamma.0 -1.0))))]
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[else
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(fllog1p (- (flgamma-lower-gautschi k x #f)))])]
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[(k . < . 1e-16)
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(define y (flgamma-gautschi-iter k x))
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;; Here, log(gamma(k)) ~ -log(k)-gamma*k, and log1p(k) ~ k
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(- (flexpm1 (+ (fllog1p (* k (- 1.0 (* x y))))
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(* k (+ (fllog x) gamma.0 -1.0)))))]
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[else
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(define y (flgamma-gautschi-iter k x))
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(define s (fl1-1/gamma1p k))
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(define u (fl- s (fl* (flexpm1 (fl* k (fllog x))) (fl- 1.0 s))))
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(define v (fl/ (fl* (fl* (fl* k (fl- 1.0 s))
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(flexp (fl* (fl+ k 1.0) (fllog x))))
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y)
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(fl+ k 1.0)))
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(fl+ u v)]))
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(: flgamma-lower-gautschi (Float Float Any -> Float))
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;; Gautschi's series for upper gamma, altered to compute lower
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(define (flgamma-lower-gautschi k x log?)
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(cond [log?
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(define p (flgamma-lower-gautschi k x #f))
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(cond
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[(and (p . > . +max-subnormal.0) (p . < . 0.8)) (fllog p)]
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[else (fllog1p (- (flgamma-upper-gautschi k x #f)))])]
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[(k . < . 1e-16)
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(define y (flgamma-gautschi-iter k x))
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;; Here, log(gamma(k)) ~ -log(k)-gamma*k, and log1p(k) ~ k
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(flexp (+ (fllog1p (* k (- 1.0 (* x y))))
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(* k (+ (fllog x) gamma.0 -1.0))))]
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[else
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(define y (flgamma-gautschi-iter k x))
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(define 1-s (fl/ 1.0 (fl* k (flgamma k))))
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(define 1-u (fl* 1-s (flexpt x k)))
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(define v (fl/ (fl* (fl* (fl* k 1-s)
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(flexp (fl* (fl+ k 1.0) (fllog x))))
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y)
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(fl+ k 1.0)))
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(fl- 1-u v)]))
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