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.
1021 lines
48 KiB
Racket
1021 lines
48 KiB
Racket
#lang typed/racket/base
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#|
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The constants used for argument reduction in flexp/error and flexpm1/error
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|#
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(require "../../../base.rkt"
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"../flonum-constants.rkt"
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"../flonum-functions.rkt")
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(provide expm1-min expm1-max exp-min flexpm1-reduction)
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;; Exponent base used for argument reduction
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(define base (fl+ 1.0 #i1/32))
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;; Bounds for which flexpm1-small/error is accurate
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(define expm1-min 0.0005)
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(define expm1-max (- (fllog +min.0)))
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(define exp-min 0.00118)
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;; Least and greatest integer exponents used in argument reduction
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(define kmin (exact-floor (fl/ (fllog expm1-min) (fllog base))))
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(define kmax (exact-floor (fl/ (fllog expm1-max) (fllog base))))
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#|
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Argument reduction for exponential:
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exp(x) = exp(base^k) * exp(x - base^k)
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where base = 1 + ε is chosen to minimize *both* of the following:
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1. The number of reductions necessary to get arguments near zero
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2. The size of the table of exp(base^k)
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These two goals are at odds: the closer `base' is to 1, the larger the table is, but fewer
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reductions are necessary. Currently, the table has 462 entries, and it takes <= 4 reductions to get
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`x' to where (+ 1 (flexpm1-small/error x)) is accurate.
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The natural number `k' is chosen to reduce positive `x' as close to zero as possible without going
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negative.
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|#
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;; Generated by (parameterize ([bf-precision 1070]) (bigfloat->fl2 (bfexpm1 (bf -650.0))))
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(define-values (expm1-neg-650-hi expm1-neg-650-lo)
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(values -1.0 5.111951948651156e-283))
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(: flexpm1-reduction (Flonum -> (Values Flonum Flonum Flonum)))
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(define (flexpm1-reduction x)
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(cond [(x . fl> . expm1-max) (values x 0.0 0.0)]
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[(x . fl>= . expm1-min)
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(define k (flfloor (fl/ (fllog x) (fllog base))))
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(define expm1-b^k (vector-ref positive-ds (- (fl->exact-integer k) kmin)))
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(let*-values ([(d2 d1) (values (car expm1-b^k) (cdr expm1-b^k))])
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(values (fl- x (flexpt base k)) d2 d1))]
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[(x . fl> . (- expm1-min)) (values x 0.0 0.0)]
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[(x . fl> . -650.0)
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(define k (flfloor (fl/ (fllog (- x)) (fllog base))))
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(define expm1-b^k (vector-ref negative-ds (- (fl->exact-integer k) kmin)))
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(let*-values ([(d2 d1) (values (car expm1-b^k) (cdr expm1-b^k))])
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(values (fl+ x (flexpt base k)) d2 d1))]
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[(x . fl> . (fllog +min.0))
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;; This special case keeps flexpm1-reduction from returning subnormals
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(values (fl+ x 650.0) expm1-neg-650-hi expm1-neg-650-lo)]
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[else
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(values x 0.0 0.0)]))
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;; The tables below were generated by the following code:
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#|
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(require "../../../bigfloat.rkt"
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"expansion-base.rkt")
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(: positive-ds (Vectorof (Pair Flonum Flonum)))
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;; exp(base^k) - 1 for each kmin <= k <= kmax
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(define positive-ds
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(build-vector (- (+ kmax 1) kmin)
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(λ: ([k : Index])
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(parameterize ([bf-precision 256])
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(define x (flexpt base (fl (+ k kmin))))
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(define-values (y2 y1) (bigfloat->fl2 (bfexpm1 (bf x))))
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(cons y2 y1)))))
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(: negative-ds (Vectorof (Pair Flonum Flonum)))
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;; exp(-base^k) - 1 for each kmin <= k <= kmax
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(define negative-ds
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(build-vector (- (+ kmax 1) kmin)
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(λ: ([k : Index])
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(parameterize ([bf-precision 1070])
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(define x (flexpt base (fl (+ k kmin))))
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(define-values (y2 y1) (bigfloat->fl2 (bfexpm1 (bf (- x)))))
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(cons y2 y1)))))
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|#
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(define positive-ds
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#((0.00048511293333186416 . 7.287053259455084e-21)
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(0.0005002765039165264 . 5.174812552711603e-20)
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(0.0005159141767895225 . 2.1220112778569672e-20)
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(0.0005320407829306539 . 2.7353612783220783e-20)
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(0.0005486716177583588 . 1.436172748406251e-20)
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(0.000565822455705062 . 2.1992133061039514e-20)
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(0.0005835095652519211 . -2.010504872218286e-21)
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(0.000601749724437574 . 4.982650386129834e-20)
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(0.000620560236855965 . -7.430039714114664e-21)
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(0.0006399589481588143 . 3.8122182693055644e-20)
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(0.000659964263078802 . 2.138888295545083e-20)
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(0.0006805951629900561 . -5.314495618174954e-20)
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(0.0007018712240230741 . 4.6179617103884916e-20)
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(0.0007238126357517673 . -4.1596964367213793e-20)
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(0.0007464402204708848 . -1.8880161157866444e-21)
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(0.0007697754530826812 . 4.73133055222851e-20)
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(0.0007938404816122941 . 2.04622269272477e-20)
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(0.0008186581483719438 . -4.693486485344378e-20)
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(0.0008442520117947219 . -1.5493768351251446e-20)
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(0.0008706463689594116 . 2.853861232957886e-22)
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(0.0008978662788284951 . -5.4068667279405054e-20)
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(0.0009259375862222202 . -4.194952457490763e-20)
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(0.0009548869465523614 . -3.386529900189101e-20)
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(0.0009847418513400793 . -5.669583993860675e-20)
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(0.0010155306545430959 . -4.166473178227564e-20)
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(0.001047282599718231 . 9.521065082803394e-20)
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(0.00108002784804621 . 6.154424315085578e-20)
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(0.0011137975072465416 . 8.454888707177185e-20)
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(0.0011486236614111957 . 8.953086830150095e-20)
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(0.0011845394017867567 . 7.217159457790775e-20)
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(0.0012215788585357235 . 7.477344191447046e-20)
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(0.0012597772335086511 . 6.566772750951499e-21)
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(0.0012991708340598853 . 4.3179037768796526e-20)
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(0.0013397971079407453 . -1.0797335953851804e-19)
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(0.0013816946793051388 . 1.0047591559543256e-19)
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(0.0014249033858637816 . -2.5695478448084455e-20)
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(0.0014694643172244001 . 8.06630571658122e-20)
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(0.0015154198544565735 . 8.668353541071613e-20)
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(0.0015628137109211689 . -5.774014185861802e-20)
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(0.0016116909744056864 . 7.387283203139758e-20)
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(0.0016620981506082358 . 3.6190462953392175e-20)
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(0.0017140832080143208 . -1.1589495995584767e-20)
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(0.0017676956242121183 . 7.435250652996099e-20)
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(0.0018229864336935118 . 1.6584560557414573e-20)
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(0.0018800082771897484 . -5.314279891721045e-20)
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(0.00193881545259229 . 7.057409849045952e-20)
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(0.001999463967511161 . -9.116330016191191e-20)
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(0.0020620115935249127 . -2.0864709589013526e-19)
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(0.002126517922178207 . -4.876090730004397e-20)
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(0.00219304442278497 . 2.137019248779804e-20)
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(0.0022616545020970828 . 5.096078611740868e-20)
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(0.0023324135659006976 . 2.917296418544145e-20)
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(0.002405389082604425 . 1.2440609575139753e-19)
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(0.002480650648885927 . 2.726808826095052e-20)
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(0.002558270057465778 . 7.769966232307108e-20)
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(0.0026383213670799297 . 8.849414494444815e-20)
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(0.002720880974724625 . -1.8166863250830977e-19)
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(0.0028060276902502623 . 4.170398395508256e-20)
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(0.002893842813383458 . 1.3342105820505055e-19)
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(0.0029844102132593774 . 9.195101569892898e-20)
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(0.0030778164105494097 . -1.6343723184574783e-19)
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(0.003174150662272302 . -1.2879846095787289e-20)
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(0.003273505049380108 . -1.9626022851944244e-19)
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(0.0033759745672136167 . 1.0910988961422856e-20)
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(0.0034816572189254108 . 5.0829634028060636e-20)
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(0.003590654111972309 . -1.9843487184941522e-19)
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(0.003703069557782712 . 1.873310694823313e-19)
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(0.0038190111747082828 . 1.3562826460964407e-19)
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(0.0039385899943734605 . 3.46645056046899e-19)
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(0.004061920571540579 . 2.210205307217469e-19)
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(0.004189121097612757 . 6.473176348368055e-20)
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(0.004320313517901348 . 3.9553759423126633e-19)
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(0.004455623652789569 . 1.3778848742222007e-19)
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(0.004595181322928888 . -1.0442938740181728e-19)
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(0.004739120478610037 . -3.316247561247527e-19)
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(0.004887579333455926 . -2.7453099482491593e-19)
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(0.00504070050258943 . 3.7024133899284865e-19)
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(0.005198631145435001 . -2.1647643968448829e-19)
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(0.005361523113319176 . 1.888029366421676e-19)
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(0.005529533102041619 . -1.372189645824532e-19)
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(0.005702822809595014 . 2.8864400392230173e-19)
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(0.005881559099219254 . -8.575078874790418e-20)
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(0.006065914167982702 . -3.382932389054231e-19)
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(0.0062560657210910635 . -1.4881102497593667e-20)
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(0.006452197152132442 . 3.798115355624615e-19)
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(0.006654497729475611 . 2.7200995607274075e-19)
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(0.006863162789047346 . -2.7878483350076027e-19)
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(0.0070783939337239355 . 3.7048964867346634e-20)
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(0.007300399239581616 . -9.480859418132428e-20)
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(0.0075293934692608295 . 1.6517761136833034e-19)
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(0.007765598292709817 . -3.9745416880549755e-19)
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(0.008009242515584148 . -2.4044687013794144e-19)
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(0.008260562315590457 . -6.587400191683076e-19)
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(0.008519801487074855 . 8.387020028532755e-19)
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(0.008787211694169282 . -8.196097050138885e-19)
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(0.009063052732822494 . -3.2301461454261893e-19)
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(0.009347592802056508 . -7.025623380613278e-19)
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(0.00964110878480403 . 8.655823851246648e-19)
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(0.009943886538698017 . -2.5356392849550307e-19)
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(0.010256221197200734 . -8.086026309433652e-19)
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(0.010578417481476882 . 5.640352871418658e-19)
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(0.01091079002343331 . 5.970706595372109e-19)
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(0.011253663700366743 . -3.695457946677901e-19)
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(0.01160737398168092 . 3.289555843193904e-19)
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(0.011972267288155378 . -8.730097316112283e-20)
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(0.012348701364270177 . 3.8872528489959077e-19)
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(0.012737045664114032 . -4.7729496603624285e-19)
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(0.013137681751427724 . 6.478204344064299e-19)
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(0.013551003714360316 . 1.3298832303228675e-20)
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(0.013977418595542815 . 2.0257438569694155e-19)
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(0.014417346838112431 . -7.13005949816736e-19)
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(0.014871222748350642 . 5.681340686254808e-19)
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(0.015339494975630027 . -6.612062080438212e-19)
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(0.015822627010398216 . 8.296530982748908e-19)
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(0.016321097700962718 . -1.659895381856688e-18)
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(0.016835401789877486 . -1.3733254647069333e-18)
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(0.01736605047077166 . -9.783029760164461e-19)
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(0.017913571966502208 . -1.6404265725425483e-18)
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(0.01847851212955637 . 7.250935299760642e-19)
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(0.019061435065676088 . -1.4091769087450886e-18)
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(0.019662923781725627 . -6.952696577288123e-20)
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(0.02028358085887576 . -7.566567727966786e-19)
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(0.020924029152232494 . -9.59612072439542e-19)
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(0.021584912518096692 . -6.41488169822601e-20)
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(0.02226689657010246 . -7.336613651912944e-19)
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(0.022970669465547214 . -3.657425923384842e-19)
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(0.023696942723295693 . 1.7301455690443072e-18)
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(0.02444645207471311 . 1.242011812585554e-18)
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(0.025219958349160486 . -1.8439819571806356e-19)
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(0.02601824839566741 . 7.648227504707697e-19)
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(0.02684213604248493 . -3.024364317498657e-20)
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(0.027692463096313864 . -1.4810746531228193e-18)
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(0.028570100383102384 . -5.174668144972875e-19)
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(0.029475948832411267 . 4.448286316523317e-20)
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(0.030410940607456194 . -5.81533876508634e-19)
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(0.031376040283054726 . -1.4158564427160428e-18)
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(0.032372246073831004 . 2.2334054291587723e-18)
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(0.033400591115164815 . -2.5221828976912458e-18)
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(0.0344621447995138 . -8.510898707950959e-20)
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(0.035558014170888795 . -2.2354500799952555e-18)
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(0.03668934538042334 . 1.1684089702155055e-18)
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(0.037857325206150046 . -1.0125447739512992e-18)
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(0.039063182640279324 . -2.0920943476429972e-18)
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(0.040308190547470785 . -7.985899169864652e-19)
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(0.041593667397795686 . 2.07644114717057e-18)
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(0.04292097907831042 . -2.9718456774634723e-18)
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(0.04429154078739765 . -6.340767355681303e-19)
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(0.04570681901628487 . 2.1704004890048954e-18)
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(0.04716833362241929 . 2.9906195295324253e-18)
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(0.04867765999966722 . 1.9020569655558005e-19)
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(0.050236431350613966 . 2.9557106476059737e-18)
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(0.05184634106656999 . -1.515543925210912e-18)
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(0.053509145221242116 . -2.31880578111979e-18)
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(0.05522666518440587 . -1.4087074223799722e-18)
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(0.05700079036231938 . 3.193530579389267e-18)
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(0.05883348107205249 . -2.145260640991126e-18)
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(0.06072677155736839 . 3.0535299407780733e-18)
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(0.06268277315429321 . -1.9158317755446028e-18)
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(0.06470367761504209 . -6.452572296512067e-18)
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(0.06679176059954327 . 5.706584124310573e-18)
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(0.06894938534441653 . -4.893311618639166e-18)
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(0.07117900651992193 . 1.3836556609129577e-18)
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(0.07348317428610542 . 1.0150560143811903e-18)
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(0.0758645385601292 . -1.2725691451908302e-19)
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(0.07832585350759605 . 1.5354906427541542e-18)
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(0.08086998227155874 . 4.412576875593593e-18)
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(0.08349990195385616 . -9.270840538531389e-19)
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(0.08621870886444084 . 2.4601285608902732e-18)
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(0.0890296240554657 . -3.6117807566340314e-18)
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(0.09193599915808591 . -6.5483276811005975e-19)
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(0.0949413225412153 . -1.2057826985840237e-18)
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(0.09804922581285948 . 3.613052734932594e-18)
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(0.10126349068614282 . 3.63332655592265e-18)
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(0.10458805623376095 . -5.446963683297344e-18)
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(0.10802702655633457 . 2.4068800861136225e-19)
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(0.11158467889202853 . 5.547091525555895e-18)
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(0.11526547219684118 . -6.242140070908552e-18)
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(0.11907405622718036 . 1.9047348542199263e-18)
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(0.12301528115873718 . 4.260716267093159e-18)
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(0.12709420777826422 . -6.998082090230668e-19)
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(0.13131611828768022 . -2.022280195305526e-18)
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(0.135686527762978 . 1.3778664949460334e-17)
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(0.1402111963137291 . -7.957031399343018e-18)
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(0.14489614199258216 . 6.079724335094065e-18)
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(0.1497476545080694 . -1.0403594901115149e-19)
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(0.15477230979829656 . 9.750626447544526e-18)
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(0.15997698552772965 . -7.415235340924643e-18)
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(0.1653688775743427 . 1.233525816475006e-17)
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(0.17095551757989683 . 5.162148117868752e-18)
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(0.17674479164212215 . -5.735071010010956e-18)
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(0.18274496023412792 . -7.236699239167251e-18)
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(0.18896467944351802 . 1.2126334623173915e-18)
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