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.
216 lines
8.0 KiB
Racket
216 lines
8.0 KiB
Racket
#lang typed/racket/base
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erfc for large x implementation from:
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C Tellambura and A Annamalai. Efficient Computation of erfc(x) for Large Arguments.
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IEEE Transactions on Communications, 2000, vol 48, pp 529--532
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|#
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(require "../../flonum.rkt"
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"../../base.rkt"
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"../distributions/impl/normal-cdf.rkt"
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"../polynomial/chebyshev.rkt"
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"continued-fraction.rkt")
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(provide flerf flerfc*expsqr flerfc
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erf erfc)
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;; ===================================================================================================
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;; erf
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(: flerf-normal (Flonum -> Flonum))
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(define (flerf-normal x)
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(fl- (fl* 2.0 (standard-flnormal-cdf (fl* x (flsqrt 2.0)))) 1.0))
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(: flerf-taylor-0 (Flonum -> Flonum))
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;; Good for -0.5 <= x <= 0.5
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(define (flerf-taylor-0 x)
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(fl* x ((make-flpolyfun
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(+1.1283791670955125738961589031215451716881e0
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-3.7612638903183752463205296770718172389603e-1
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+1.1283791670955125738961589031215451716881e-1
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-2.6866170645131251759432354836227265992574e-2
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+5.2239776254421878421118467737108572763338e-3
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-8.5483270234508528325466583569814028158189e-4
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+1.2055332981789664251027338708563516791540e-4
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-1.4925650358406250977462419353459592218097e-5
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+1.6462114365889247401612962522198079652312e-6
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-1.6365844691234924317393003677039026554930e-7
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+1.4807192815879217239546050945892452597318e-8
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-1.2290555301717927352982888136906778835719e-9))
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(fl* x x))))
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(: flerf (Flonum -> Flonum))
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(define (flerf x)
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(let ([ax (flabs x)])
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(cond [(ax . fl> . 0.5) (flerf-normal x)]
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[(ax . fl< . 1e-13) (fl* x (fl/ 2.0 (flsqrt pi)))]
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[else (flerf-taylor-0 x)])))
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;; ===================================================================================================
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;; erfc(x)*exp(x^2)
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(define sqrtpi 1.772453850905516027298167483341145182798)
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(: flerfc*expsqr-huge (Flonum -> Flonum))
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;; Computed using erfc's asymptotic expansion
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;; Error < epsilon.0 for x > 1e8
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(define (flerfc*expsqr-huge x)
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(fl/ 1.0 (fl* x sqrtpi)))
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(: flerfc*expsqr-asym (Flonum -> Flonum))
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;; Computed using erfc's asymptotic expansion
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;; Error < epsilon.0 for x > 24
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(define (flerfc*expsqr-asym x)
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(define 1/2x^2 (fl/ 1.0 (fl* 2.0 (fl* x x))))
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(fl* (fl/ 1.0 (fl* x sqrtpi))
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((make-flpolyfun
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(1.0 -1.0 3.0 -15.0 105.0 -945.0 10395.0))
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1/2x^2)))
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(: foo (Flonum -> Flonum))
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(define (foo x)
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(define 2x^2 (fl* 2.0 (fl* x x)))
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(fl* (fl/ x sqrtpi)
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((make-flpolyfun
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(1.0 -1.0 3.0 -15.0 105.0 -945.0 10395.0))
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2x^2)))
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(: flerfc*expsqr-series (Flonum -> Flonum))
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;; Good for x > 5.0
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(define (flerfc*expsqr-series x)
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(define h (cond [(x . fl< . 5.4) 0.25]
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[(x . fl< . 8.0) 0.4]
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[(x . fl< . 16.0) 0.5]
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[else 0.51]))
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(define h^2 (* h h))
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(define z
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(fl+ (fl/ (fl/ 1.0 x) x)
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(let: loop : Flonum ([y : Flonum 0.0] [n : Flonum 1.0])
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(define n^2*h^2 (fl* (fl* n n) h^2))
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(define dy (fl/ (fl* 2.0 (flexp (- n^2*h^2))) (fl+ n^2*h^2 (fl* x x))))
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;(printf "~v~n" dy)
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(cond [(dy . fl< . (fl* (fl* 0.5 epsilon.0) y)) (fl+ dy y)]
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[else (loop (fl+ y dy) (fl+ n 1.0))]))))
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(fl* (fl/ (fl* x h) pi) z))
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(: flerfc*expsqr-3-5 (Flonum -> Flonum))
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(define (flerfc*expsqr-3-5 x)
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(fl/ ((inline-chebyshev-flpoly-fun
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3.0 5.0
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(76.91661738312371165176665510251236326338
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27.20465693840961735563948053921797067271
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3.386442685567191365542568636267208877102
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0.140993936422267459775165813598040765649
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-5.153134548269842201105342673544377759352e-6
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1.7508864589759466867096925622177881862e-6
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-3.043007141646851587889970096343295638291e-7
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4.243425771190431100938117624380917251699e-8
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-5.203867155244060760117564922569126026438e-9
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5.794849132093509459568321930026301391629e-10
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-5.921422840095836545874122164110445535066e-11
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5.533766942834863591670434965523217803471e-12
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-4.631550718317617310162582984555616243437e-13
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3.262025831452441605191178585599762596143e-14
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-1.513328953213075725641119234827526370187e-15
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-4.522283605960468274138123625976905984843e-17))
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x)
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(flexpt x 4.0)))
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(: flerfc*expsqr-1-3 (Flonum -> Flonum))
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(define (flerfc*expsqr-1-3 x)
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(fl/ ((inline-chebyshev-flpoly-fun
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1.0 3.0
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(0.9945093280638061048517513432019848175032
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0.05120904381630891334631711380945424146303
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-0.01422342070434339336273995600351825586225
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0.003357268323337835041126774361327403324835
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-7.114985025457904001609950417739639868647e-4
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1.390876463234288554054665709920181526987e-4
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-2.548042834519374090944060859469479045399e-5
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4.420149801027904714368125294893695251516e-6
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-7.314182887789212376555477464037660326951e-7
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1.160833279816164682586230015667162482652e-7
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-1.77457488690957160865337806653632235146e-8
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2.621904502367314860299110485351868396918e-9
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-3.754481078582135875876211640532994079565e-10
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5.222857161436770247649169925242741865874e-11
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-7.07220022924025262089289170578922817775e-12
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9.337553246718007012188722480344684913175e-13
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-1.203905834239132741551983752552150752983e-13
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1.517763413855051802253894051250284281458e-14
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-1.873156638647946872401895465528599118158e-15
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2.265073220841686859543609057755087846036e-16
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-2.651790674204626906157253423748608352303e-17))
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x)
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x))
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(define flerfc*expsqr-0.5-1
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(inline-chebyshev-flpoly-fun
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0.5 1.0
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(1.028508060089628120924659381992051204702
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-0.09353436157631646884296478224639544611431
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0.00734954356796596066014584525500838968569
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-5.170287683170954502817549118454885607861e-4
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3.327489181039831922712166420908522822156e-5
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-1.987760941931570403969164257181219771567e-6
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1.113719839925198075577451421375109070733e-7
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-5.898649609389129304942457516804088201602e-9
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2.971299014285280439436981007380165475227e-10
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-1.430488185776241939959484837699287692925e-11
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6.608616284222898801100004753695010173929e-13
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-2.939555278237892581547471127716213388842e-14
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1.262499691092374606344446613363361460578e-15
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-5.240060859272093179585053345170003668439e-17)))
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(: flerfc*expsqr-normal (Flonum -> Flonum))
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(define (flerfc*expsqr-normal x)
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(fl* (flexpsqr x) (fl* 2.0 (standard-flnormal-cdf (- (fl* x (flsqrt 2.0)))))))
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(: flerfc*expsqr (Flonum -> Flonum))
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(define (flerfc*expsqr x)
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(cond [(x . fl> . 1e8) (flerfc*expsqr-huge x)]
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[(x . fl> . 24.0) (flerfc*expsqr-asym x)]
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[(x . fl> . 5.0) (flerfc*expsqr-series x)]
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[(x . fl> . 3.0) (flerfc*expsqr-3-5 x)]
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[(x . fl> . 1.0) (flerfc*expsqr-1-3 x)]
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[(x . fl> . 0.5) (flerfc*expsqr-0.5-1 x)]
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[(x . fl> . -27.0) (flerfc*expsqr-normal x)]
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[else +inf.0]))
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;; ===================================================================================================
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;; erfc
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(define -erfc-max.0 -5.8635847487551676)
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(define +erfc-max.0 27.226017111108362)
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(: flerfc (Flonum -> Flonum))
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(define (flerfc x)
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(cond [(x . fl< . 0.0)
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(cond [(x . fl< . -erfc-max.0) 2.0]
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[else (- 2.0 (flerfc (- x)))])]
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[(x . fl> . 0.0)
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(cond [(x . fl> . +erfc-max.0) 0.0]
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[else (fl* (flerfc*expsqr x) (flgauss x))])]
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[(x . fl= . 0.0) 1.0]
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[else +nan.0]))
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;; ===================================================================================================
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(: erf (case-> (Zero -> Zero)
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(Flonum -> Flonum)
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(Real -> (U Zero Flonum))))
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(define (erf x)
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(cond [(flonum? x) (flerf x)]
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[(eqv? x 0) x]
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[else (flerf (fl x))]))
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(: erfc (case-> (Zero -> One)
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(Flonum -> Flonum)
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(Real -> (U One Flonum))))
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(define (erfc x)
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(cond [(flonum? x) (flerfc x)]
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[(eqv? x 0) 1]
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[else (flerfc (fl x))]))
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