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;;;; -*- Mode: lisp -*-
;;;;
;;;; Copyright (c) 2007 Raymond Toy
;;;;
;;;; Permission is hereby granted, free of charge, to any person
;;;; obtaining a copy of this software and associated documentation
;;;; files (the "Software"), to deal in the Software without
;;;; restriction, including without limitation the rights to use,
;;;; copy, modify, merge, publish, distribute, sublicense, and/or sell
;;;; copies of the Software, and to permit persons to whom the
;;;; Software is furnished to do so, subject to the following
;;;; conditions:
;;;;
;;;; The above copyright notice and this permission notice shall be
;;;; included in all copies or substantial portions of the Software.
;;;;
;;;; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
;;;; EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
;;;; OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
;;;; NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
;;;; HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
;;;; WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
;;;; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
;;;; OTHER DEALINGS IN THE SOFTWARE.
;;; This file contains various possible implementations of some of the
;;; core routines. These were experiments on faster and/or more
;;; accurate implementations. The routines inf qd-fun.lisp are the
;;; default, but you can select a different implementation from here
;;; if you want.
;;;
;;; The end of the file also includes some tests of the different
;;; implementations for speed.
(in-package #:octi)
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;; This works but seems rather slow, so we don't even compile it.
#+(or)
(defun exp-qd/newton (a)
(declare (type %quad-double a))
;; Newton iteration
;;
;; f(x) = log(x) - a
;;
;; x' = x - (log(x) - a)/(1/x)
;; = x - x*(log(x) - a)
;; = x*(1 + a - log(x))
(let ((a1 (add-qd-d a 1d0))
(x (make-qd-d (exp (qd-0 a)))))
(setf x (mul-qd x (sub-qd a1 (log-qd/agm x))))
(setf x (mul-qd x (sub-qd a1 (log-qd/agm x))))
(setf x (mul-qd x (sub-qd a1 (log-qd/agm x))))
x))
(defun expm1-qd/series (a)
(declare (type %quad-double a))
;; Compute exp(x) - 1.
;;
;; D(x) = exp(x) - 1
;;
;; First, write x = s*log(2) + r*k where s is an integer and |r*k| <
;; log(2)/2.
;;
;; Then D(x) = D(s*log(2)+r*k) = 2^s*exp(r*k) - 1
;; = 2^s*(exp(r*k)-1) - 1 + 2^s
;; = 2^s*D(r*k)+2^s-1
;; But
;; exp(r*k) = exp(r)^k
;; = (D(r) + 1)^k
;;
;; So
;; D(r*k) = (D(r) + 1)^k - 1
;;
;; For small r, D(r) can be computed using the Taylor series around
;; zero. To compute D(r*k) = (D(r) + 1)^k - 1, we use the binomial
;; theorem to expand out the power and to exactly cancel out the -1
;; term, which is the source of inaccuracy.
;;
;; We want to have small r so the Taylor series converges quickly,
;; but that means k is large, which means the binomial expansion is
;; long. We need to compromise. Let use choose k = 8. Then |r| <
;; log(2)/16 = 0.0433. For this range, the Taylor series converges
;; to 212 bits of accuracy with about 28 terms.
;;
;;
(flet ((taylor (x)
(declare (type %quad-double x))
;; Taylor series for exp(x)-1
;; = x+x^2/2!+x^3/3!+x^4/4!+...
;; = x*(1+x/2!+x^2/3!+x^3/4!+...)
(let ((sum +qd-one+)
(term +qd-one+))
(dotimes (k 28)
(setf term (div-qd-d (mul-qd term x) (float (cl:+ k 2) 1d0)))
(setf sum (add-qd sum term)))
(mul-qd x sum)))
(binom (x)
(declare (type %quad-double x))
;; (1+x)^8-1
;; = x*(8 + 28*x + 56*x^2 + 70*x^3 + 56*x^4 + 28*x^5 + 8*x^6 + x^7)
;; = x (x (x (x (x (x (x (x + 8) + 28) + 56) + 70) + 56) + 28) + 8)
(mul-qd
x
(add-qd-d
(mul-qd x
(add-qd-d
(mul-qd x
(add-qd-d
(mul-qd x
(add-qd-d
(mul-qd x
(add-qd-d
(mul-qd x
(add-qd-d
(mul-qd x
(add-qd-d x 8d0))
28d0))
56d0))
70d0))
56d0))
28d0))
8d0)))
(arg-reduce (x)
(declare (type %quad-double x))
;; Write x = s*log(2) + r*k where s is an integer and |r*k|
;; < log(2)/2, and k = 8.
(let* ((s (truncate (qd-0 (nint-qd (div-qd a +qd-log2+)))))
(r*k (sub-qd x (mul-qd-d +qd-log2+ (float s 1d0))))
(r (div-qd-d r*k 8d0)))
(values s r))))
(multiple-value-bind (s r)
(arg-reduce a)
(let* ((d (taylor r))
(dr (binom d)))
(add-qd-d (scale-float-qd dr s)
(cl:- (scale-float 1d0 s) 1))))))
(defun log-qd/newton (a)
(declare (type %quad-double a))
;; The Taylor series for log converges rather slowly. Hence, this
;; routine tries to determine the root of the function
;;
;; f(x) = exp(x) - a
;;
;; using Newton iteration. The iteration is
;;
;; x' = x - f(x) / f'(x)
;; = x - (1 - a * exp(-x))
;; = x + a * exp(-x) - 1
;;
;; Two iterations are needed.
(let ((x (make-qd-d (log (qd-0 a)))))
(dotimes (k 3)
(setf x (sub-qd-d (add-qd x (mul-qd a (exp-qd (neg-qd x))))
1d0)))
x))
;;(declaim (inline agm-qd))
(defun agm-qd (x y)
(declare (type %quad-double x y)
(optimize (speed 3)))
(let ((diff (qd-0 (abs-qd (sub-qd x y)))))
(cond ((< diff +qd-eps+)
x)
(t
(let ((a-mean (div-qd-d (add-qd x y) 2d0))
(g-mean (sqrt-qd (mul-qd x y))))
(agm-qd a-mean g-mean))))))
#+(or)
(defun agm-qd (x y)
(declare (type %quad-double x y)
(optimize (speed 3) (space 0) (safety 0)))
(let ((diff (qd-0 (abs-qd (sub-qd x y))))
(x x)
(y y))
(declare (double-float diff))
(loop while (> diff +qd-eps+)
do
(let ((a-mean (scale-float-qd (add-qd x y) -1))
(g-mean (sqrt-qd (mul-qd x y))))
(setf x a-mean)
(setf y g-mean)
(setf diff (qd-0 (abs-qd (sub-qd x y))))))
x))
(defun log-qd/agm (x)
(declare (type %quad-double x))
;; log(x) ~ pi/2/agm(1,4/x)*(1+O(1/x^2))
;;
;; Need to make x >= 2^(d/2) to get d bits of precision. We use
;;
;; log(2^k*x) = k*log(2)+log(x)
;;
;; to compute log(x). log(2^k*x) is computed using AGM.
;;
(multiple-value-bind (frac exp)
(decode-float (qd-0 x))
(declare (ignore frac))
(cond ((>= exp 106)
;; Big enough to use AGM
(div-qd +qd-pi/2+
(agm-qd +qd-one+
(div-qd (make-qd-d 4d0)
x))))
(t
;; log(x) = log(2^k*x) - k * log(2)
(let* ((k (cl:- 107 exp))
(big-x (scale-float-qd x k)))
;; Compute k*log(2) using extra precision by writing
;; log(2) = a + b, where a is the quad-double
;; approximation and b the rest.
(sub-qd (log-qd/agm big-x)
(add-qd (mul-qd-d +qd-log2+ (float k 1d0))
(mul-qd-d +qd-log2-extra+ (float k 1d0)))))))))
(defun log-qd/agm2 (x)
(declare (type %quad-double x))
;; log(x) ~ pi/4/agm(theta2(q^4)^2,theta3(q^4)^2)
;;
;; where q = 1/x
;;
;; Need to make x >= 2^(d/36) to get d bits of precision. We use
;;
;; log(2^k*x) = k*log(2)+log(x)
;;
;; to compute log(x). log(2^k*x) is computed using AGM.
;;
(multiple-value-bind (frac exp)
(decode-float (qd-0 x))
(declare (ignore frac))
(cond ((>= exp 7)
;; Big enough to use AGM (because d = 212 so x >= 2^5.8888)
(let* ((q (div-qd +qd-one+
x))
(q^4 (npow q 4))
(q^8 (sqr-qd q^4))
;; theta2(q^4) = 2*q*(1+q^8+q^24)
;; = 2*q*(1+q^8+(q^8)^3)
(theta2 (mul-qd-d
(mul-qd
q
(add-qd-d
(add-qd q^8
(npow q^8 3))
1d0))
2d0))
;; theta3(q^4) = 1+2*(q^4+q^16)
;; = 1+2*(q^4+(q^4)^4)
(theta3 (add-qd-d
(mul-qd-d
(add-qd q^4
(npow q^4 4))
2d0)
1d0)))
(div-qd +qd-pi/4+
(agm-qd (sqr-qd theta2)
(sqr-qd theta3)))))
(t
;; log(x) = log(2^k*x) - k * log(2)
(let* ((k (cl:- 7 exp))
(big-x (scale-float-qd x k)))
(sub-qd (log-qd/agm2 big-x)
(add-qd (mul-qd-d +qd-log2+ (float k 1d0))
(mul-qd-d +qd-log2-extra+ (float k 1d0)))))))))
(defun log-qd/agm3 (x)
(declare (type %quad-double x))
;; log(x) ~ pi/4/agm(theta2(q^4)^2,theta3(q^4)^2)
;;
;; where q = 1/x
;;
;; Need to make x >= 2^(d/36) to get d bits of precision. We use
;;
;; log(2^k*x) = k*log(2)+log(x)
;;
;; to compute log(x). log(2^k*x) is computed using AGM.
;;
(multiple-value-bind (frac exp)
(decode-float (qd-0 x))
(declare (ignore frac))
(cond ((>= exp 7)
;; Big enough to use AGM (because d = 212 so x >= 2^5.8888)
(let* ((q (div-qd +qd-one+
x))
(q^4 (npow q 4))
(q^8 (sqr-qd q^4))
;; theta2(q^4) = 2*q*(1+q^8+q^24)
;; = 2*q*(1+q^8+(q^8)^3)
(theta2 (mul-qd-d
(mul-qd
q
(add-qd-d
(add-qd q^8
(npow q^8 3))
1d0))
2d0))
;; theta3(q^4) = 1+2*(q^4+q^16)
;; = 1+2*(q^4+(q^4)^4)
(theta3 (add-qd-d
(mul-qd-d
(add-qd q^4
(npow q^4 4))
2d0)
1d0)))
;; Note that agm(theta2^2,theta3^2) = agm(2*theta2*theta3,theta2^2+theta3^2)/2
(div-qd +qd-pi/4+
(scale-float-qd
(agm-qd (scale-float-qd (mul-qd theta2 theta3) 1)
(add-qd (sqr-qd theta2)
(sqr-qd theta3)))
-1))))
(t
;; log(x) = log(2^k*x) - k * log(2)
(let* ((k (cl:- 7 exp))
(big-x (scale-float-qd x k)))
(sub-qd (log-qd/agm3 big-x)
(add-qd
(mul-qd-d +qd-log2+ (float k 1d0))
(mul-qd-d +qd-log2-extra+ (float k 1d0)))))))))
#+(or)
(defun atan-d (y x)
(let* ((r (abs (complex x y)))
(xx (cl:/ x r))
(yy (cl:/ y r)))
(let ((z (atan (float y 1f0) (float x 1f0)))
(sinz 0d0)
(cosz 0d0))
(format t "z = ~A~%" z)
(cond ((> xx yy)
(format t "xx > yy~%")
(dotimes (k 5)
(let* ((sinz (sin z))
(cosz (cos z))
(delta (cl:/ (cl:- yy sinz)
cosz)))
(format t "sz, dz = ~A ~A~%" sinz cosz)
(format t "delta = ~A~%" delta)
(setf z (cl:+ z delta))
(format t "z = ~A~%" z))))
(t
(dotimes (k 20)
(let ((sinz (sin z))
(cosz (cos z)))
(format t "sz, dz = ~A ~A~%" sinz cosz)
(setf z (cl:- z (cl:/ (cl:- xx cosz)
sinz)))
(format t "z = ~A~%" z)))))
z)))
#||
(defvar *table*)
(defvar *ttable*)
(defvar *cordic-scale*)
#+nil
(defun setup-cordic ()
(let ((table (make-array 34))
(ttable (make-array 34)))
(setf (aref table 0) 1d0)
(setf (aref table 1) 1d0)
(setf (aref table 2) 1d0)
(setf (aref ttable 0) (cl:/ pi 4))
(setf (aref ttable 1) (cl:/ pi 4))
(setf (aref ttable 2) (cl:/ pi 4))
(loop for k from 3 below 34 do
(setf (aref table k) (cl:* 0.5d0 (aref table (cl:1- k))))
(setf (aref ttable k) (atan (aref table k))))
(setf *table* table)
(setf *ttable* ttable)))
(defun setup-cordic ()
(let ((table (make-array 34))
(ttable (make-array 34)))
(setf (aref table 0) 4d0)
(setf (aref table 1) 2d0)
(setf (aref table 2) 1d0)
(setf (aref ttable 0) (atan 4d0))
(setf (aref ttable 1) (atan 2d0))
(setf (aref ttable 2) (cl:/ pi 4))
(loop for k from 3 below 34 do
(setf (aref table k) (cl:* 0.5d0 (aref table (cl:1- k))))
(setf (aref ttable k) (atan (aref table k))))
(setf *table* table)
(setf *ttable* ttable)))
(defun setup-cordic ()
(let ((table (make-array 34))
(ttable (make-array 34))
(scale 1d0))
(loop for k from 0 below 34 do
(setf (aref table k) (scale-float 1d0 (cl:- 2 k)))
(setf (aref ttable k) (atan (aref table k)))
(setf scale (cl:* scale (cos (aref ttable k)))))
(setf *table* table)
(setf *ttable* ttable)
(setf *cordic-scale* scale)))
(defun cordic-rot (x y)
(let ((z 0))
(dotimes (k (length *table*))
(cond ((plusp y)
(psetq x (cl:+ x (cl:* y (aref *table* k)))
y (cl:- y (cl:* x (aref *table* k))))
(incf z (aref *ttable* k)))
(t
(psetq x (cl:- x (cl:* y (aref *table* k)))
y (cl:+ y (cl:* x (aref *table* k))))
(decf z (aref *ttable* k)))
))
(values z x y)))
(defun cordic-vec (z)
(let ((x 1d0)
(y 0d0)
(scale 1d0))
(dotimes (k 12 (length *table*))
(setf scale (cl:* scale (cos (aref *ttable* k))))
(cond ((minusp z)
(psetq x (cl:+ x (cl:* y (aref *table* k)))
y (cl:- y (cl:* x (aref *table* k))))
(incf z (aref *ttable* k)))
(t
(psetq x (cl:- x (cl:* y (aref *table* k)))
y (cl:+ y (cl:* x (aref *table* k))))
(decf z (aref *ttable* k)))
))
(values x y z scale)))
(defun atan2-d (y x)
(multiple-value-bind (z dx dy)
(cordic-rot x y)
(let ((theta (cl:/ dy dx)))
(format t "theta = ~A~%" theta)
(let ((corr (cl:+ theta
(cl:- (cl:/ (expt theta 3)
3))
(cl:/ (expt theta 5)
5))))
(format t "corr = ~A~%" corr)
(cl:+ z corr)))))
(defun tan-d (r)
(multiple-value-bind (x y z)
(cordic-vec r)
(setf x (cl:* x *cordic-scale*))
(setf y (cl:* y *cordic-scale*))
(format t "x = ~A~%" x)
(format t "y = ~A~%" y)
(format t "z = ~A~%" z)
;; Need to finish of the rotation
(let ((st (sin z))
(ct (cos z)))
(format t "st, ct = ~A ~A~%" st ct)
(psetq x (cl:- (cl:* x ct) (cl:* y st))
y (cl:+ (cl:* y ct) (cl:* x st)))
(format t "x = ~A~%" x)
(format t "y = ~A~%" y)
(cl:/ y x)
)))
(defun sin-d (r)
(declare (type double-float r))
(multiple-value-bind (x y z s)
(cordic-vec r)
;; Need to finish the rotation
(let ((st (sin z))
(ct (cos z)))
(psetq x (cl:- (cl:* x ct) (cl:* y st))
y (cl:+ (cl:* y ct) (cl:* x st)))
(cl:* s y))))
||#
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#||
;; Here is a function for clisp that can be used to create the atan2 table
;; that we need.
(defun make-atan-table-data ()
(let ((scale 1l0))
(dotimes (k 67)
(let* ((x (scale-float 1L0 (- 2 k)))
(p (atan x)))
(setf scale (* scale (cos p)))
(multiple-value-bind (int exp sign)
(integer-decode-float p)
(let* ((len (integer-length int))
(wanted (ldb (byte 212 (- len 212)) int))
(bit (ldb (byte 1 (- len (* 4 53) 1)) int))
(roundp (not (zerop (ldb (byte (- len (* 4 53) 2) 0) int)))))
;;(format t "~&~v,'0b~%" len int)
;;(format t "~b~a~%" wanted (make-string (- len 212) :initial-element #\-))
;;(format t "~v,'-b~%" len (ash bit (- len 212 1)))
;;(format t "~v,'-b~%" len (ldb (byte (- len (* 4 53) 2) 0) int))
;; See if we need to round up the answer.
(when (= bit 1)
;; Round to even
(cond (roundp
(incf wanted))
(t
;; Round to even
(when (oddp wanted)
(incf wanted)))))
;;(format t "~b~a~%" wanted (make-string (- len 212) :initial-element #\-))
(let* ((i0 (ldb (byte 53 (* 3 53)) wanted))
(i1 (ldb (byte 53 (* 2 53)) wanted))
(i2 (ldb (byte 53 (* 1 53)) wanted))
(i3 (ldb (byte 53 0) wanted)))
(write `(make-qd-d
(scale-float (float ,i0 1d0) ,(+ exp (- len (* 1 53))))
(scale-float (float ,i1 1d0) ,(+ exp (- len (* 2 53))))
(scale-float (float ,i2 1d0) ,(+ exp (- len (* 3 53))))
(scale-float (float ,i3 1d0) ,(+ exp (- len (* 4 53)))))
:case :downcase))))))
scale))
||#
#+nil
(defconstant +atan-table+
(make-array 66
:initial-contents
(list
+qd-pi/4+
+qd-pi/4+
+qd-pi/4+
;; Do we need to make these values more accurate? (The
;; reader has quite a bit of roundoff.)
#.(qd-from-string "0.46364760900080611621425623146121440202853705428612026381093308872018q0")
#.(qd-from-string "0.24497866312686415417208248121127581091414409838118406712737591466738q0")
#.(qd-from-string "0.12435499454676143503135484916387102557317019176980408991511411911572q0")
#.(qd-from-string "0.062418809995957348473979112985505113606273887797499194607527816898697q0")
#.(qd-from-string "0.031239833430268276253711744892490977032495663725400040255315586255793q0")
#.(qd-from-string "0.0156237286204768308028015212565703189111141398009054178814105073966645q0")
#.(qd-from-string "0.0078123410601011112964633918421992816212228117250147235574539022483893q0")
#.(qd-from-string "0.003906230131966971827628665311424387140357490115202856215213095149011q0")
#.(qd-from-string "0.00195312251647881868512148262507671393161074677723351033905753396043094q0")
#.(qd-from-string "9.7656218955931943040343019971729085163419701581008759004900725226773q-4")
#.(qd-from-string "4.8828121119489827546923962564484866619236113313500303710940335348752q-4")
#.(qd-from-string "2.4414062014936176401672294325965998621241779097061761180790046091019q-4")
#.(qd-from-string "1.22070311893670204239058646117956300930829409015787498451939837846645q-4")
#.(qd-from-string "6.1035156174208775021662569173829153785143536833346179337671134316588q-5")
#.(qd-from-string "3.0517578115526096861825953438536019750949675119437837531021156883611q-5")
#.(qd-from-string "1.5258789061315762107231935812697885137429238144575874846241186407446q-5")
#.(qd-from-string "7.6293945311019702633884823401050905863507439184680771577638306965336q-6")
#.(qd-from-string "3.8146972656064962829230756163729937228052573039688663101874392503939q-6")
#.(qd-from-string "1.9073486328101870353653693059172441687143421654501533666700577234671q-6")
#.(qd-from-string "9.53674316405960879420670689923112390019634124498790160133611802076q-7")
#.(qd-from-string "4.7683715820308885992758382144924707587049404378664196740053215887142q-7")
#.(qd-from-string "2.3841857910155798249094797721893269783096898769063155913766911372218q-7")
#.(qd-from-string "1.19209289550780685311368497137922112645967587664586735576738225215437q-7")
#.(qd-from-string "5.9604644775390554413921062141788874250030195782366297314294565710003q-8")
#.(qd-from-string "2.9802322387695303676740132767709503349043907067445107249258477840843q-8")
#.(qd-from-string "1.4901161193847655147092516595963247108248930025964720012170057805491q-8")
#.(qd-from-string "7.4505805969238279871365645744953921132066925545665870075947601416172q-9")
#.(qd-from-string "3.725290298461914045267070571811923583671948328737040524231998269239q-9")
#.(qd-from-string "1.8626451492309570290958838214764904345065282835738863513491050124951q-9")
#.(qd-from-string "9.3132257461547851535573547768456130389292649614929067394376854242196q-10")
#.(qd-from-string "4.6566128730773925777884193471057016297347863891561617421323492554414q-10")
#.(qd-from-string "2.32830643653869628902042741838821270371274293204981860525486662280605q-10")
#.(qd-from-string "1.16415321826934814452599092729852658796396457380014290026584979170883q-10")
#.(qd-from-string "5.8207660913467407226496761591231582349549156257795272423976206167147q-11")
#.(qd-from-string "2.9103830456733703613273032698903947793693632003639830495829934525029q-11")
#.(qd-from-string "1.4551915228366851806639597837362993474211703608936710732067270213307q-11")
#.(qd-from-string "7.2759576141834259033201841046703741842764629388821429640111752890838q-12")
#.(qd-from-string "3.6379788070917129516601402005837967730345578669779258118296083646486q-12")
#.(qd-from-string "1.81898940354585647583007611882297459662931973336029253714520765350336q-12")
#.(qd-from-string "9.094947017729282379150388117278718245786649666696631862264792881855q-13")
#.(qd-from-string "4.5474735088646411895751949990348397807233312083369623012466392138249q-13")
#.(qd-from-string "2.2737367544323205947875976170668549725904164010421166413578155299654q-13")
#.(qd-from-string "1.1368683772161602973937988232271068715738020501302644662229139921281q-13")
#.(qd-from-string "5.6843418860808014869689941345026335894672525626628305471702634435609q-14")
#.(qd-from-string "2.8421709430404007434844970695472041986834065703328538172835210852389q-14")
#.(qd-from-string "1.42108547152020037174224853506058802483542582129160672712566632799217q-14")
#.(qd-from-string "7.1054273576010018587112426756616725310442822766145084088962160950957q-15")
#.(qd-from-string "3.5527136788005009293556213378756778163805352845768135511116874239215q-15")
#.(qd-from-string "1.7763568394002504646778106689434441020475669105721016938889503158663q-15")
#.(qd-from-string "8.881784197001252323389053344724227002559458638215127117361184578544q-16")
#.(qd-from-string "4.440892098500626161694526672362989312819932329776890889670147968684q-16")
#.(qd-from-string "2.22044604925031308084726333618160413285249154122211136120876849284695q-16")
#.(qd-from-string "1.11022302462515654042363166809081575098156144265276392015109606150467q-16")
#.(qd-from-string "5.5511151231257827021181583404540958606019518033159549001888700768492q-17")
#.(qd-from-string "2.7755575615628913510590791702270500685127439754144943625236087596052q-17")
#.(qd-from-string "1.3877787807814456755295395851135253015328429969268117953154510949506q-17")
#.(qd-from-string "6.9388939039072283776476979255676268417598037461585147441443138686883q-18")
#.(qd-from-string "3.4694469519536141888238489627838134626418504682698143430180392335861q-18")
#.(qd-from-string "1.7347234759768070944119244813919067365411688085337267928772549041983q-18")
#.(qd-from-string "8.673617379884035472059622406959533689231148510667158491096568630248q-19")
#.(qd-from-string "4.336808689942017736029811203479766845431237313833394811387071078781q-19")
#.(qd-from-string "2.16840434497100886801490560173988342281757653922917435142338388484765q-19")
#.(qd-from-string "1.08420217248550443400745280086994171142153300490364679392792298560597q-19")
))
"Table of atan(2^(-k)) for k = 1 to 64. But the first three entries are 1")
(defconstant +atan-table+
(make-array 67
:initial-contents
(list
(%make-qd-d (scale-float (float 5970951936056572 1.0d0) -52)
(scale-float (float 5427585433121543 1.0d0) -105)
(scale-float (float 5608515294538868 1.0d0) -158)
(scale-float (float 445395631680583 1.0d0) -211))
(%make-qd-d (scale-float (float 4986154552901188 1.0d0) -52)
(scale-float (float 3814906810089799 1.0d0) -105)
(scale-float (float 1896417689773139 1.0d0) -158)
(scale-float (float 3393132800284032 1.0d0) -211))
(%make-qd-d (scale-float (float 7074237752028440 1.0d0) -53)
(scale-float (float 2483878800010755 1.0d0) -106)
(scale-float (float 3956492004828932 1.0d0) -159)
(scale-float (float 2434854662709436 1.0d0) -212))
(%make-qd-d (scale-float (float 8352332796509007 1.0d0) -54)
(scale-float (float 3683087214424816 1.0d0) -107)
(scale-float (float 8240297260223171 1.0d0) -160)
(scale-float (float 5174086704442609 1.0d0) -213))
(%make-qd-d (scale-float (float 8826286527774941 1.0d0) -55)
(scale-float (float 3471944699336670 1.0d0) -108)
(scale-float (float 4798212191802497 1.0d0) -161)
(scale-float (float 6908472993489831 1.0d0) -214))
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(scale-float (float 6978747913895162 1.0d0) -109)
(scale-float (float 1204496828771308 1.0d0) -162)
(scale-float (float 6150314016033077 1.0d0) -215))
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(scale-float (float 6996384121843768 1.0d0) -110)
(scale-float (float 6481245652257127 1.0d0) -163)
(scale-float (float 6083920726820778 1.0d0) -216))
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(scale-float (float 5921825575778154 1.0d0) -111)
(scale-float (float 1742767809528138 1.0d0) -164)
(scale-float (float 3392785816514584 1.0d0) -217))
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(scale-float (float 4748718880757240 1.0d0) -218))
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(scale-float (float 5212528258452836 1.0d0) -222))
(%make-qd-d (scale-float (float 9007198538913211 1.0d0) -64)
(scale-float (float 6605122380416172 1.0d0) -117)
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(scale-float (float 2545695100421145 1.0d0) -223))
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(scale-float (float 6747877897971029 1.0d0) -224))
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(scale-float (float 8406719304424926 1.0d0) -250))
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(scale-float (float 3002399751580331 1.0d0) -272))
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(scale-float (float 9007199254740991 1.0d0) -168)
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(scale-float (float 3002399751580331 1.0d0) -274))
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"Table of atan(2^(-k)) for k = -2 to 64. But the first three entries are 1")
(defconstant +atan-power-table+
(make-array 67
:element-type 'double-float
:initial-contents
(loop for k from 0 below 67
collect (scale-float 1d0 (- 2 k)))
)
"Table of (2^(-k)) for k = -2 to 64. But the first three entries are 1")
(defconstant +cordic-scale+
#.(qd-from-string "0.065865828601599636584870082133151126045971796871364763285694473524426q0"))
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;; This is the basic CORDIC rotation. Based on code from
;; http://www.voidware.com/cordic.htm and
;; http://www.dspcsp.com/progs/cordic.c.txt.
;;
;; The only difference between this version and the typical CORDIC
;; implementation is that the first 3 rotations are all by pi/4. This
;; makes sense. If the angle is greater than pi/4, the rotations will
;; reduce it to at most pi/4. If the angle is less than pi/4, the 3
;; rotations by pi/4 will cause us to end back at the same place.
;; (Should we try to be smarter?)
(defun cordic-rot-qd (x y)
(declare (type %quad-double y x)
(optimize (speed 3)))
(let* ((zero +qd-zero+)
(z zero))
(declare (type %quad-double zero z))
(dotimes (k (length +atan-table+))
(declare (fixnum k))
(cond ((qd-> y zero)
(psetq x (add-qd x (mul-qd-d y (aref +atan-power-table+ k)))
y (sub-qd y (mul-qd-d x (aref +atan-power-table+ k))))
(setf z (add-qd z (aref +atan-table+ k))))
(t
(psetq x (sub-qd x (mul-qd-d y (aref +atan-power-table+ k)))
y (add-qd y (mul-qd-d x (aref +atan-power-table+ k))))
(setf z (sub-qd z (aref +atan-table+ k))))))
(values z x y)))
(defun atan2-qd/cordic (y x)
(declare (type %quad-double y x))
;; Use the CORDIC rotation to get us to a small angle. Then use the
;; Taylor series for atan to finish the computation.
(multiple-value-bind (z dx dy)
(cordic-rot-qd x y)
;; Use Taylor series to finish off the computation
(let* ((arg (div-qd dy dx))
(sq (neg-qd (sqr-qd arg)))
(sum +qd-one+))
;; atan(x) = x - x^3/3 + x^5/5 - ...
;; = x*(1-x^2/3+x^4/5-x^6/7+...)
(do ((k 3d0 (cl:+ k 2d0))
(term sq))
((< (abs (qd-0 term)) +qd-eps+))
(setf sum (add-qd sum (div-qd-d term k)))
(setf term (mul-qd term sq)))
(setf sum (mul-qd arg sum))
(add-qd z sum))))
(defun atan-qd/cordic (y)
(declare (type %quad-double y))
(atan2-qd/cordic y +qd-one+))
(defun atan-qd/duplication (y)
(declare (type %quad-double y)
(optimize (speed 3) (space 0)))
(cond ((< (abs (qd-0 y)) 1d-4)
;; Series
(let* ((arg y)
(sq (neg-qd (sqr-qd arg)))
(sum +qd-one+))
;; atan(x) = x - x^3/3 + x^5/5 - ...
;; = x*(1-x^2/3+x^4/5-x^6/7+...)
(do ((k 3d0 (cl:+ k 2d0))
(term sq))
((< (abs (qd-0 term)) +qd-eps+))
(setf sum (add-qd sum (div-qd-d term k)))
(setf term (mul-qd term sq)))
(mul-qd arg sum)))
(t
;; atan(x) = 2*atan(x/(1 + sqrt(1 + x^2)))
(let ((x (div-qd y
(add-qd-d (sqrt-qd (add-qd-d (sqr-qd y) 1d0))
1d0))))
(scale-float-qd (atan-qd/duplication x) 1)))))
(defun cordic-vec-qd (z)
(declare (type %quad-double z)
(optimize (speed 3)))
(let* ((x +qd-one+)
(y +qd-zero+)
(zero +qd-zero+))
(declare (type %quad-double zero x y))
(dotimes (k 30 (length +atan-table+))
(declare (fixnum k)
(inline mul-qd-d sub-qd add-qd))
(cond ((qd-> z zero)
(psetq x (sub-qd x (mul-qd-d y (aref +atan-power-table+ k)))
y (add-qd y (mul-qd-d x (aref +atan-power-table+ k))))
(setf z (sub-qd z (aref +atan-table+ k))))
(t
(psetq x (add-qd x (mul-qd-d y (aref +atan-power-table+ k)))
y (sub-qd y (mul-qd-d x (aref +atan-power-table+ k))))
(setf z (add-qd z (aref +atan-table+ k))))))
(values z x y)))
(defun tan-qd/cordic (r)
(declare (type %quad-double r))
(multiple-value-bind (z x y)
(cordic-vec-qd r)
;; Need to finish the rotation
(multiple-value-bind (st ct)
(sincos-taylor z)
(psetq x (sub-qd (mul-qd x ct) (mul-qd y st))
y (add-qd (mul-qd y ct) (mul-qd x st)))
(div-qd y x))))
(defun sin-qd/cordic (r)
(declare (type %quad-double r))
(multiple-value-bind (z x y)
(cordic-vec-qd r)