[oct-git]OCT: A portable Lisp implementation for quad-double precision floats branch master updated. 0a607f3a6a6297597ae2a6c9896844803d42a6d9
Raymond Toy
rtoy at common-lisp.net
Tue Nov 26 19:54:08 UTC 2013
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commit 0a607f3a6a6297597ae2a6c9896844803d42a6d9
Author: Raymond Toy <toy.raymond at gmail.com>
Date: Tue Nov 26 09:55:46 2013 -0800
Switch to using atan-qd/taylor as the default atan implementation.
qd-fun.lisp:
o Use atan-qd/taylor and atan2-qd/taylor as the atan-qd and atan2-qd
implementation.
o Fix bug in atan-qd/taylor. We need a special case for x small where
we don't want to use the table. (Found by our rt-test. Yay!)
rt-tests.lisp:
o Add comment for oct.atan.4 for why it works.
diff --git a/qd-fun.lisp b/qd-fun.lisp
index 9564dcc..207a51b 100644
--- a/qd-fun.lisp
+++ b/qd-fun.lisp
@@ -1188,7 +1188,10 @@ is the cosine of A"
;; atan(x) = pi/2-atan(1/x), for x > 1
(sub-qd +qd-pi/2+
(atan-qd/taylor (div-qd +qd-one+ y))))
+ ((qd-<= y (aref octi::+qd-atan-partition+ 0))
+ (atan-taylor y))
(t
+ ;; The general case where the table is needed.
(let* ((i (find-atan-partition y))
(atan-xi (mul-qd +qd-pi/2+
(rational-to-qd (/ (1- (+ i 2))
@@ -1244,6 +1247,8 @@ is the cosine of A"
(neg-qd +qd-pi/4+))))
(let ((arg (atan-qd/taylor (div-qd y (abs-qd x)))))
+ ;; arg = atan2(y,|x|), so |arg| <= pi/2. Handle the case when x
+ ;; is negative.
(cond ((minusp-qd x)
(if (minusp-qd y)
(- (neg-qd +qd-pi+) arg)
@@ -1254,12 +1259,12 @@ is the cosine of A"
(defun atan2-qd (y x)
"atan2(y, x) = atan(y/x), but carefully handling the quadrant"
(declare (type %quad-double y x))
- (atan2-qd/newton y x))
+ (atan2-qd/taylor y x))
(defun atan-qd (y)
"Return the arc tangent of the %QUAD-DOUBLE number Y"
(declare (type %quad-double y))
- (atan-qd/newton y))
+ (atan-qd/taylor y))
(defun asin-qd (a)
"Return the arc sine of the %QUAD-DOUBLE number A"
diff --git a/rt-tests.lisp b/rt-tests.lisp
index 69f6ee7..289131e 100644
--- a/rt-tests.lisp
+++ b/rt-tests.lisp
@@ -222,6 +222,9 @@
nil)
(rt:deftest oct.atan.4
+ ;; We know atan(10^100) is pi/2 because atan(10^100) =
+ ;; pi/2-atan(10^-100) and atan(10^-100) is approx 10^-100, which
+ ;; is too small to affect pi/2.
(let* ((arg #q1q100)
(y (/ (atan arg) +pi+))
(true #q.5))
commit 793166a087afffde87c6d12a509069316884fd44
Author: Raymond Toy <toy.raymond at gmail.com>
Date: Tue Nov 26 09:36:02 2013 -0800
Reduce size of the atan partition and node tables by half.
qd-const.lisp:
o Update tables and generator to use tables with only half as many
entries as before. This is because atan-qd/taylor never needs to
compute directly for x > 1.
qd-fun.lisp:
o Use table length for binary search, so we're independent of other
constants.
o Remove FIXME.
diff --git a/qd-const.lisp b/qd-const.lisp
index 98c5ac9..f3a5bdc 100644
--- a/qd-const.lisp
+++ b/qd-const.lisp
@@ -3770,10 +3770,15 @@
;;
;; tan((2*k + 3)*pi/(4*s))
;;
-;; for k = 0, 1, ..., s, where s is the table size,
+;; for k = 0, 1, ..., s, where s is the partition size,
;; +qd-atan-partition-size+.
+;;
+;; This table, however only contains values up to k = s/2. The
+;; algorith for atan-qd/taylor uses the relationship atan(x) =
+;; pi/2-atan(1/x) for x > 1, so we never need to compute directly
+;; atan(x) for x > 1.
(defconstant +qd-atan-partition+
- (make-array +qd-atan-partition-size+ :initial-contents
+ (make-array (/ +qd-atan-partition-size+ 2) :initial-contents
(list
(multiple-value-bind (q0 q1 q2 q3)
(renorm-5 (scale-float (float 5305687677623182 1.0d0) -61)
@@ -7358,10766 +7363,3601 @@
(scale-float (float -8606500254519528 1.0d0) -165)
(scale-float (float -6855951617059037 1.0d0) -223)
(scale-float (float 7929633697693696 1.0d0) -277))
- (%make-qd-d q0 q1 q2 q3))
- (multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4524372768344304 1.0d0) -52)
- (scale-float (float 8317904890851914 1.0d0) -107)
- (scale-float (float 8594750619195149 1.0d0) -166)
- (scale-float (float -7306789725115964 1.0d0) -220)
- (scale-float (float -8574773586034688 1.0d0) -274))
- (%make-qd-d q0 q1 q2 q3))
- (multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4538274952087405 1.0d0) -52)
- (scale-float (float 8326257921126996 1.0d0) -106)
- (scale-float (float -6022210771707739 1.0d0) -162)
- (scale-float (float -4520561563519313 1.0d0) -216)
- (scale-float (float 7859974008373248 1.0d0) -270))
- (%make-qd-d q0 q1 q2 q3))
+ (%make-qd-d q0 q1 q2 q3)))))
+
+;; This is the set of evaluation nodes for the atan table. The table contains
+;;
+;; tan((2*k + 2)*pi/(4*s))
+;;
+;; for k = 0, 1, ..., s, where s is the partition size,
+;; +qd-atan-partition-size+.
+;;
+;; As explained in +qd-atan-partition+, we only store values up to k =
+;; s/2.
+(defconstant +qd-atan-nodes+
+ (make-array (/ +qd-atan-partition-size+ 2) :initial-contents
+ (list
(multiple-value-bind (q0 q1 q2 q3)
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+ (scale-float (float 5247049684764422 1.0d0) -226)
+ (scale-float (float -5541122207547392 1.0d0) -280))
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-
-;; This is the set of evaluation nodes for the atan table. The table contains
-;;
-;; tan((2*k + 2)*pi/(4*s))
-;;
-;; for k = 0, 1, ..., s, where s is the table size,
-;; +qd-atan-partition-size+.
-(defconstant +qd-atan-nodes+
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- (scale-float (float 6016821677785088 1.0d0) -268))
+ (renorm-5 (scale-float (float 8628403556015405 1.0d0) -53)
+ (scale-float (float -5689830371797970 1.0d0) -107)
+ (scale-float (float 8456917422509878 1.0d0) -161)
+ (scale-float (float 6498397382453532 1.0d0) -216)
+ (scale-float (float -7683261920575488 1.0d0) -271))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 7056493552344079 1.0d0) -47)
- (scale-float (float -6660391185023610 1.0d0) -101)
- (scale-float (float 8326117901300204 1.0d0) -156)
- (scale-float (float -5180116769562087 1.0d0) -211)
- (scale-float (float 6732836385259520 1.0d0) -265))
+ (renorm-5 (scale-float (float 8654938616431667 1.0d0) -53)
+ (scale-float (float 8182047312445503 1.0d0) -108)
+ (scale-float (float -6662523829600515 1.0d0) -162)
+ (scale-float (float 5290567255884442 1.0d0) -217)
+ (scale-float (float -6464888284348416 1.0d0) -271))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 7644684611709030 1.0d0) -47)
- (scale-float (float 7144281558445897 1.0d0) -105)
- (scale-float (float 5769916130091718 1.0d0) -159)
- (scale-float (float 6892393124855441 1.0d0) -213)
- (scale-float (float 8609597045211136 1.0d0) -267))
+ (renorm-5 (scale-float (float 8681552017138358 1.0d0) -53)
+ (scale-float (float -7006822305559368 1.0d0) -107)
+ (scale-float (float 7451129425931068 1.0d0) -161)
+ (scale-float (float -7173167161450929 1.0d0) -215)
+ (scale-float (float -5536285008453632 1.0d0) -269))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 8339806414116051 1.0d0) -47)
- (scale-float (float -4878300504338807 1.0d0) -101)
- (scale-float (float 5390699203123886 1.0d0) -156)
- (scale-float (float -7941956443494165 1.0d0) -210)
- (scale-float (float 8699025989009408 1.0d0) -265))
+ (renorm-5 (scale-float (float 8708244231382452 1.0d0) -53)
+ (scale-float (float -8087631037263965 1.0d0) -107)
+ (scale-float (float -8185307178763586 1.0d0) -161)
+ (scale-float (float -6511095932131732 1.0d0) -215)
+ (scale-float (float 8343018883710976 1.0d0) -272))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4586969091394693 1.0d0) -46)
- (scale-float (float -8978492069766178 1.0d0) -101)
- (scale-float (float 7053055367607091 1.0d0) -155)
- (scale-float (float -8712227340191945 1.0d0) -210)
- (scale-float (float -7846634958946304 1.0d0) -265))
+ (renorm-5 (scale-float (float 8735015735962084 1.0d0) -53)
+ (scale-float (float -8117625885586849 1.0d0) -107)
+ (scale-float (float 7067185933598292 1.0d0) -161)
+ (scale-float (float -5928047136609223 1.0d0) -217)
+ (scale-float (float -4613124548198400 1.0d0) -279))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 5096708286733636 1.0d0) -46)
- (scale-float (float -4572705006895138 1.0d0) -100)
- (scale-float (float 4718547123658870 1.0d0) -154)
- (scale-float (float 7157224158707570 1.0d0) -210)
- (scale-float (float -8590740628766720 1.0d0) -265))
+ (renorm-5 (scale-float (float 8761867011260840 1.0d0) -53)
+ (scale-float (float 8610737154062316 1.0d0) -107)
+ (scale-float (float 5316487286030695 1.0d0) -161)
+ (scale-float (float 8025315088065011 1.0d0) -216)
+ (scale-float (float -8789522686148608 1.0d0) -271))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 5733873284861593 1.0d0) -46)
- (scale-float (float -8604953363186775 1.0d0) -103)
- (scale-float (float 8626935765066666 1.0d0) -158)
- (scale-float (float -4858320692337793 1.0d0) -212)
- (scale-float (float -7449309035233280 1.0d0) -266))
+ (renorm-5 (scale-float (float 8788798541282450 1.0d0) -53)
+ (scale-float (float -5767028858726936 1.0d0) -110)
+ (scale-float (float 5809369621052884 1.0d0) -167)
+ (scale-float (float 7326787612774527 1.0d0) -221)
+ (scale-float (float -6442635476598784 1.0d0) -279))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 6553075144281224 1.0d0) -46)
- (scale-float (float 6620952143443553 1.0d0) -101)
- (scale-float (float -5268353655146108 1.0d0) -156)
- (scale-float (float 5893295495507489 1.0d0) -211)
- (scale-float (float -4800425745186816 1.0d0) -266))
+ (renorm-5 (scale-float (float 8815810813685869 1.0d0) -53)
+ (scale-float (float -8724536074899524 1.0d0) -109)
+ (scale-float (float 5172102500931788 1.0d0) -166)
+ (scale-float (float -8595123929132079 1.0d0) -220)
+ (scale-float (float -4972293416550400 1.0d0) -276))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 7645332295807713 1.0d0) -46)
- (scale-float (float -5080909471366464 1.0d0) -102)
- (scale-float (float 7938109188081450 1.0d0) -156)
- (scale-float (float 4928452855514440 1.0d0) -210)
- (scale-float (float 8885897390456832 1.0d0) -268))
+ (renorm-5 (scale-float (float 8842904319820777 1.0d0) -53)
+ (scale-float (float 7692158636243959 1.0d0) -107)
+ (scale-float (float -6981365851709947 1.0d0) -161)
+ (scale-float (float 5606403711887330 1.0d0) -216)
+ (scale-float (float 7207011875684352 1.0d0) -271))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4587238957440646 1.0d0) -45)
- (scale-float (float -7194218336984116 1.0d0) -99)
- (scale-float (float -8985364386653513 1.0d0) -153)
- (scale-float (float -8164942491672346 1.0d0) -207)
- (scale-float (float 6286694831915008 1.0d0) -262))
+ (renorm-5 (scale-float (float 8870079554763484 1.0d0) -53)
+ (scale-float (float 6730270946682744 1.0d0) -108)
+ (scale-float (float -6566124735755288 1.0d0) -163)
+ (scale-float (float 4704554665939634 1.0d0) -217)
+ (scale-float (float 5990258066063360 1.0d0) -271))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 5734089176174276 1.0d0) -45)
- (scale-float (float -7647633804412292 1.0d0) -101)
- (scale-float (float -7089963492055907 1.0d0) -155)
- (scale-float (float 8957378262525840 1.0d0) -210)
- (scale-float (float 8460149900050432 1.0d0) -265))
+ (renorm-5 (scale-float (float 8897337017353248 1.0d0) -53)
+ (scale-float (float -6246556062351795 1.0d0) -108)
+ (scale-float (float 7682958297604200 1.0d0) -163)
+ (scale-float (float -6314968768802122 1.0d0) -217)
+ (scale-float (float -6603959870652416 1.0d0) -271))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 7645494213403192 1.0d0) -45)
- (scale-float (float 4683660413726437 1.0d0) -99)
- (scale-float (float -7150841523991633 1.0d0) -153)
- (scale-float (float -4654257076122560 1.0d0) -207)
- (scale-float (float 5269682304483328 1.0d0) -261))
+ (renorm-5 (scale-float (float 8924677210229021 1.0d0) -53)
+ (scale-float (float 5469297920188961 1.0d0) -109)
+ (scale-float (float -8593195763191850 1.0d0) -163)
+ (scale-float (float -7114494562938148 1.0d0) -217)
+ (scale-float (float 8594557003857920 1.0d0) -271))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 5734143148494430 1.0d0) -44)
- (scale-float (float 8353770598111665 1.0d0) -99)
- (scale-float (float -8962023885528779 1.0d0) -158)
- (scale-float (float -5933825828214605 1.0d0) -212)
- (scale-float (float 5439908491034624 1.0d0) -266))
+ (renorm-5 (scale-float (float 8952100639866623 1.0d0) -53)
+ (scale-float (float -4764119273928549 1.0d0) -107)
+ (scale-float (float -4842944010947886 1.0d0) -161)
+ (scale-float (float 6684392706093202 1.0d0) -215)
+ (scale-float (float 5193091026345984 1.0d0) -269))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 5734156641542718 1.0d0) -43)
- (scale-float (float 8633141086175409 1.0d0) -98)
- (scale-float (float 5419442444201815 1.0d0) -153)
- (scale-float (float -8712131584355876 1.0d0) -211)
- (scale-float (float -5157997961543680 1.0d0) -266))
+ (renorm-5 (scale-float (float 8979607816616344 1.0d0) -53)
+ (scale-float (float -7789931843743237 1.0d0) -107)
+ (scale-float (float -8124365004371129 1.0d0) -164)
+ (scale-float (float 8996080488067980 1.0d0) -218)
+ (scale-float (float 6558996243415040 1.0d0) -272))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 7091480910882704 1.0d0) 204)
- (scale-float (float 8415670760171201 1.0d0) 148)
- (scale-float (float -8191062761010691 1.0d0) 94)
- (scale-float (float 8061527243862424 1.0d0) 39)
- (scale-float (float -8684710553714688 1.0d0) -17))
+ (renorm-5 (scale-float (float 4503599627370496 1.0d0) -52)
+ (scale-float (float -4503599627370496 1.0d0) -308) (scale-float (float 0 1.0d0) 0)
+ (scale-float (float 0 1.0d0) 0) (scale-float (float 0 1.0d0) 0))
(%make-qd-d q0 q1 q2 q3)))))
#||
@@ -18208,24 +11048,24 @@ something that we can use to initialize our tables.
;; Create atan table
(let (val)
- (dotimes (k +qd-atan-partition-size+)
+ (dotimes (k (/ +qd-atan-partition-size+ 2))
(push (convert-to-qd-parts (tan (* pi (/ (+ 3 (* 2 k))
(* 4 +qd-atan-partition-size+)))))
val))
(write `(defconstant +qd-atan-partition+
- (make-array +qd-atan-partition-size+
+ (make-array (/ +qd-atan-partition-size+ 2)
:initial-contents (list ,@(nreverse val))))
:case :downcase
:pretty t)
(values))
(let (val)
- (dotimes (k +qd-atan-partition-size+)
+ (dotimes (k (/ +qd-atan-partition-size+ 2))
(push (convert-to-qd-parts (tan (* pi (/ (+ 2 (* 2 k))
(* 4 +qd-atan-partition-size+)))))
val))
(write `(defconstant +qd-atan-nodes+
- (make-array +qd-atan-partition-size+
+ (make-array (/ +qd-atan-partition-size+ 2)
:initial-contents (list ,@(nreverse val))))
:case :downcase
:pretty t)
diff --git a/qd-fun.lisp b/qd-fun.lisp
index 35dbcd4..9564dcc 100644
--- a/qd-fun.lisp
+++ b/qd-fun.lisp
@@ -1117,10 +1117,8 @@ is the cosine of A"
;; the smallest entry greater than x. Return the index of the
;; entry.
(let* ((low 0)
- (high (1- octi::+qd-atan-partition-size+))
+ (high (length octi::+qd-atan-partition+))
(mid (ash (+ low high) -1)))
- ;; high starts one less than the end because the end actually has junk.
- ;; FIXME: Remove that junk last element.
(loop
(if (qd-< x (aref octi::+qd-atan-partition+ mid))
(setf high mid)
commit 4d09f5a39889e782bef99e2dda6d0d38327fa9f4
Author: Raymond Toy <toy.raymond at gmail.com>
Date: Tue Nov 26 09:11:45 2013 -0800
Fix atan/taylor bug and add atan2/taylor.
qd-fun.lisp:
o Fix issue with atan-qd/taylor for large args. We use the
relationship atan(x) = pi/2-atan(1/x) for x > 1.
o Add atan2-qd/taylor. This appears to be much faster than
atan2-qd/newton.
timing.lisp:
o Add timing test for atan2-qd.
diff --git a/qd-fun.lisp b/qd-fun.lisp
index e782a8a..35dbcd4 100644
--- a/qd-fun.lisp
+++ b/qd-fun.lisp
@@ -1182,21 +1182,76 @@ is the cosine of A"
;; atan(x) = (2*i-1)*pi/(4*s) + atan(e)
;;
;; Note that qd-atan-partition array starts with X[2].
- (if (minusp-qd y)
- (neg-qd (atan-qd/taylor (neg-qd y)))
- (let* ((i (find-atan-partition y))
- (atan-xi (mul-qd +qd-pi/2+
- (rational-to-qd (/ (1- (+ i 2))
- octi::+qd-atan-partition-size+))))
- (1/xi (div-qd +qd-one+ (aref octi::+qd-atan-nodes+ i)))
- (e1 (div-qd (add-qd-d (sqr-qd 1/xi) 1d0)
- (add-qd 1/xi y)))
- (e (sub-qd 1/xi
- e1)))
- ;;(format t "partition ~D~%" i)
- ;;(format t "atan(xi) = ~S~%" atan-xi)
- ;;(format t "e = ~S~%" e)
- (add-qd atan-xi (atan-taylor e)))))
+ (cond
+ ((minusp-qd y)
+ ;; atan(-x) = -atan(x)
+ (neg-qd (atan-qd/taylor (neg-qd y))))
+ ((qd-> y +qd-one+)
+ ;; atan(x) = pi/2-atan(1/x), for x > 1
+ (sub-qd +qd-pi/2+
+ (atan-qd/taylor (div-qd +qd-one+ y))))
+ (t
+ (let* ((i (find-atan-partition y))
+ (atan-xi (mul-qd +qd-pi/2+
+ (rational-to-qd (/ (1- (+ i 2))
+ octi::+qd-atan-partition-size+))))
+ (1/xi (div-qd +qd-one+ (aref octi::+qd-atan-nodes+ i)))
+ (e1 (div-qd (add-qd-d (sqr-qd 1/xi) 1d0)
+ (add-qd 1/xi y)))
+ (e (sub-qd 1/xi
+ e1)))
+ ;;(format t "partition ~D~%" i)
+ ;;(format t "atan(xi) = ~S~%" atan-xi)
+ ;;(format t "e = ~S~%" e)
+ (add-qd atan-xi (atan-taylor e))))))
+
+(defun atan2-qd/taylor (y x)
+ (declare (type %quad-double y x)
+ #+nil
+ (optimize (speed 3) (space 0)))
+ (cond ((zerop-qd x)
+ ;; x = 0
+ (cond ((zerop-qd y)
+ ;; Both x and y are zero. Use the signs of x and y to
+ ;; determine the result
+ (error "atan2(0,0)"))
+ (t
+ ;; x = 0, but y is not. Use the sign of y.
+ (return-from atan2-qd/taylor
+ (cond ((plusp (float-sign (qd-0 y)))
+ +qd-pi/2+)
+ (t
+ (neg-qd +qd-pi/2+)))))))
+ ((zerop-qd y)
+ ;; y = 0.
+ (return-from atan2-qd/taylor
+ ;; Use the sign of x and y to figure out the result.
+ (cond ((plusp (float-sign (qd-0 x)))
+ +qd-zero+)
+ ((plusp (float-sign (qd-0 y)))
+ +qd-pi+)
+ (t
+ (neg-qd +qd-pi+))))))
+
+ (when (qd-= x y)
+ (return-from atan2-qd/taylor
+ (if (plusp-qd y)
+ +qd-pi/4+
+ +qd-3pi/4+)))
+
+ (when (qd-= x (neg-qd y))
+ (return-from atan2-qd/taylor
+ (if (plusp-qd y)
+ +qd-3pi/4+
+ (neg-qd +qd-pi/4+))))
+
+ (let ((arg (atan-qd/taylor (div-qd y (abs-qd x)))))
+ (cond ((minusp-qd x)
+ (if (minusp-qd y)
+ (- (neg-qd +qd-pi+) arg)
+ (- +qd-pi+ arg)))
+ (t
+ arg))))
(defun atan2-qd (y x)
"atan2(y, x) = atan(y/x), but carefully handling the quadrant"
diff --git a/timing.lisp b/timing.lisp
index 1847c4c..9e2074a 100644
--- a/timing.lisp
+++ b/timing.lisp
@@ -231,3 +231,57 @@
;; We see that the taylor series is 21 times faster (!) and conses 11
;; times less. That's a pretty nice gain, at the expense of two
;; 1024-element tables.
+
+(defun time-atan2 (&optional (n 10000))
+ (declare (fixnum n))
+ (flet ((time-atan/newton ()
+ (let ((sum (octi::make-qd-d -1000d0)))
+ (declare (type octi::%quad-double sum)
+ (optimize (speed 3)))
+ (dotimes (k n)
+ (declare (fixnum k))
+ (setf sum (octi::atan2-qd/newton sum (mul-qd-d sum .1d0))))
+ sum))
+ (time-atan/taylor ()
+ (let ((sum (octi::make-qd-d -1000d0)))
+ (declare (type octi::%quad-double sum)
+ (optimize (speed 3)))
+ (dotimes (k n)
+ (declare (fixnum k))
+ (setf sum (octi::atan2-qd/taylor sum (mul-qd-d sum .1d0))))
+ sum)))
+ (format t "atan2-qd/newton ~d times~%" n)
+ #+cmu (ext:gc :full t)
+ (format t "sum = ~A~%" (time (time-atan/newton)))
+ (format t "atan2-qd/taylor ~d times~%" n)
+ #+cmu (ext:gc :full t)
+ (format t "sum = ~A~%" (time (time-atan/taylor)))))
+
+
+;; Some timing results on an iMac, 3.06 GHz Core i3:
+
+; atan2-qd/newton 10000 times
+
+; Evaluation took:
+; 6.37 seconds of real time
+; 6.321544 seconds of user run time
+; 0.026819 seconds of system run time
+; 19,474,336,804 CPU cycles
+; [Run times include 0.27 seconds GC run time]
+; 0 page faults and
+; 934,872,144 bytes consed.
+;
+; sum = #C(-1.670464979286058652105921398771024w0 1.39712341620073544419882362664172w-33)
+;
+; atan2-qd/taylor 10000 times
+
+; Evaluation took:
+; 0.32 seconds of real time
+; 0.320343 seconds of user run time
+; 0.001606 seconds of system run time
+; 989,558,772 CPU cycles
+; [Run times include 0.02 seconds GC run time]
+; 0 page faults and
+; 88,919,384 bytes consed.
+;
+; sum = #C(-1.670464979286058652105921398771027w0 4.47861132722031280908838833477765w-33)
commit b9a21f1f536aec53c186e4a8d40d75f9d6918bb1
Author: Raymond Toy <toy.raymond at gmail.com>
Date: Mon Nov 25 23:10:26 2013 -0800
Implement a new atan method that is 21 times faster.
The new method uses a table and the taylor series for atan to compute
atan for real args.
qd-const.lisp:
o Add the two tables used for the atan computation.
o Include clisp code that generates the tables.
qd-fun.lisp:
o Implement atan using a set of tables and a Taylor series.
timing.lisp:
o Add a simple timing program to evaluate how fast each atan method
is.
diff --git a/qd-const.lisp b/qd-const.lisp
index 56840b7..98c5ac9 100644
--- a/qd-const.lisp
+++ b/qd-const.lisp
@@ -3762,6 +3762,14363 @@
(%make-qd-d q0 q1 q2 q3))
"Quad-double value of Euler's constant")
+;; Table for atan computation. See atan-qd/taylor.
+(defconstant +qd-atan-partition-size+
+ 1024)
+
+;; This is the partition points for the atan table. The table contains
+;;
+;; tan((2*k + 3)*pi/(4*s))
+;;
+;; for k = 0, 1, ..., s, where s is the table size,
+;; +qd-atan-partition-size+.
+(defconstant +qd-atan-partition+
+ (make-array +qd-atan-partition-size+ :initial-contents
+ (list
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5305687677623182 1.0d0) -61)
+ (scale-float (float 5506082357036760 1.0d0) -115)
+ (scale-float (float -8811919955122674 1.0d0) -170)
+ (scale-float (float 6519548665950046 1.0d0) -226)
+ (scale-float (float -6440164966137856 1.0d0) -280))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 8842840540207337 1.0d0) -61)
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+ (renorm-5 (scale-float (float 5734160014802806 1.0d0) -42)
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+ (scale-float (float -6695998270734336 1.0d0) -264))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float -5734160014802806 1.0d0) -42)
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+
+;; This is the set of evaluation nodes for the atan table. The table contains
+;;
+;; tan((2*k + 2)*pi/(4*s))
+;;
+;; for k = 0, 1, ..., s, where s is the table size,
+;; +qd-atan-partition-size+.
+(defconstant +qd-atan-nodes+
+ (make-array +qd-atan-partition-size+ :initial-contents
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+ (scale-float (float -7038759865446038 1.0d0) -216)
+ (scale-float (float 7255865652412416 1.0d0) -273))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5091849743565917 1.0d0) -48)
+ (scale-float (float 6384455094568543 1.0d0) -102)
+ (scale-float (float 5702890449723630 1.0d0) -157)
+ (scale-float (float -7589384686425635 1.0d0) -218)
+ (scale-float (float 8058940122202112 1.0d0) -273))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5237623243959472 1.0d0) -48)
+ (scale-float (float -6250630418786862 1.0d0) -104)
+ (scale-float (float -7184429040914303 1.0d0) -159)
+ (scale-float (float 5513312508283453 1.0d0) -215)
+ (scale-float (float 4749731872374784 1.0d0) -270))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5391963180151410 1.0d0) -48)
+ (scale-float (float -7407081297448137 1.0d0) -102)
+ (scale-float (float 5185826904122421 1.0d0) -156)
+ (scale-float (float 7823667441420830 1.0d0) -210)
+ (scale-float (float 8936157189111808 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5555648319599754 1.0d0) -48)
+ (scale-float (float -5491336161082583 1.0d0) -102)
+ (scale-float (float -6554264313276709 1.0d0) -156)
+ (scale-float (float -7585496654613802 1.0d0) -210)
+ (scale-float (float 4713805106823168 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5729554775677781 1.0d0) -48)
+ (scale-float (float -7721231518321956 1.0d0) -108)
+ (scale-float (float 6182950135224119 1.0d0) -163)
+ (scale-float (float -5805908032198122 1.0d0) -217)
+ (scale-float (float 6138479531524096 1.0d0) -271))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5914671708628099 1.0d0) -48)
+ (scale-float (float -8127496465237598 1.0d0) -103)
+ (scale-float (float 5113636924531285 1.0d0) -158)
+ (scale-float (float 7610245412227071 1.0d0) -212)
+ (scale-float (float 5419049157689344 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6112120166707537 1.0d0) -48)
+ (scale-float (float 8554650084246937 1.0d0) -103)
+ (scale-float (float 6240992177685638 1.0d0) -161)
+ (scale-float (float 7196179540449182 1.0d0) -216)
+ (scale-float (float 6262113232224256 1.0d0) -272))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6323175825499951 1.0d0) -48)
+ (scale-float (float -8865111156616584 1.0d0) -102)
+ (scale-float (float 7085615420986638 1.0d0) -156)
+ (scale-float (float 7235233759408942 1.0d0) -211)
+ (scale-float (float -8649169230872576 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6549296599938903 1.0d0) -48)
+ (scale-float (float 8257698953810403 1.0d0) -102)
+ (scale-float (float 8532553234643244 1.0d0) -157)
+ (scale-float (float -8270427433372553 1.0d0) -211)
+ (scale-float (float -8578788487872512 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6792156392335400 1.0d0) -48)
+ (scale-float (float -6544379776982520 1.0d0) -104)
+ (scale-float (float 4901112354909220 1.0d0) -158)
+ (scale-float (float 6767333192271592 1.0d0) -216)
+ (scale-float (float -8842907181645824 1.0d0) -270))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7053686628412010 1.0d0) -48)
+ (scale-float (float -8951188254702623 1.0d0) -102)
+ (scale-float (float 8621287380014728 1.0d0) -158)
+ (scale-float (float -7857861379957792 1.0d0) -213)
+ (scale-float (float 4826892277383168 1.0d0) -269))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7336127761985185 1.0d0) -48)
+ (scale-float (float -8336340561589675 1.0d0) -104)
+ (scale-float (float -6723533502954940 1.0d0) -159)
+ (scale-float (float 7818179045611807 1.0d0) -214)
+ (scale-float (float -7201952627294208 1.0d0) -268))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7642093655818158 1.0d0) -48)
+ (scale-float (float 5443555368163400 1.0d0) -106)
+ (scale-float (float -7387394316037603 1.0d0) -162)
+ (scale-float (float 5304483945651657 1.0d0) -217)
+ (scale-float (float -4963064904417280 1.0d0) -271))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7974652757478940 1.0d0) -48)
+ (scale-float (float -4601485407011345 1.0d0) -102)
+ (scale-float (float -7916486289406152 1.0d0) -157)
+ (scale-float (float -7279545329272873 1.0d0) -211)
+ (scale-float (float 5134317856817152 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 8337431414068683 1.0d0) -48)
+ (scale-float (float -5987902372410459 1.0d0) -102)
+ (scale-float (float -5307411281694402 1.0d0) -158)
+ (scale-float (float -5679062825222479 1.0d0) -212)
+ (scale-float (float 8245649354752000 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 8734746705443862 1.0d0) -48)
+ (scale-float (float -5185035314729970 1.0d0) -103)
+ (scale-float (float 5370141748632671 1.0d0) -158)
+ (scale-float (float 8551587498597857 1.0d0) -213)
+ (scale-float (float 5905522198839296 1.0d0) -270))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4585889563702555 1.0d0) -47)
+ (scale-float (float -7954556241258531 1.0d0) -102)
+ (scale-float (float -5413256311151729 1.0d0) -156)
+ (scale-float (float 6878574492056157 1.0d0) -210)
+ (scale-float (float 4794200525701120 1.0d0) -264))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4827399903209683 1.0d0) -47)
+ (scale-float (float -6275711490776414 1.0d0) -105)
+ (scale-float (float 8099918832814676 1.0d0) -160)
+ (scale-float (float -6152065565819033 1.0d0) -214)
+ (scale-float (float -6790464220692480 1.0d0) -268))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5095736726291373 1.0d0) -47)
+ (scale-float (float 5702351686147640 1.0d0) -101)
+ (scale-float (float -8356071616588207 1.0d0) -155)
+ (scale-float (float 5407157322154023 1.0d0) -209)
+ (scale-float (float -5896165128568832 1.0d0) -264))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5395634118579270 1.0d0) -47)
+ (scale-float (float -8650595618423630 1.0d0) -102)
+ (scale-float (float -4637715105009220 1.0d0) -158)
+ (scale-float (float 6802436320551583 1.0d0) -213)
+ (scale-float (float 5674304881557504 1.0d0) -269))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5733009687095975 1.0d0) -47)
+ (scale-float (float 8484139752530675 1.0d0) -102)
+ (scale-float (float 5983330237824667 1.0d0) -156)
+ (scale-float (float 8366235526218494 1.0d0) -210)
+ (scale-float (float 5629484799655936 1.0d0) -264))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6115359067385373 1.0d0) -47)
+ (scale-float (float 7428268948615224 1.0d0) -101)
+ (scale-float (float 8097548842549957 1.0d0) -156)
+ (scale-float (float -7655306445395194 1.0d0) -210)
+ (scale-float (float -6488780694913024 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6552319505127374 1.0d0) -47)
+ (scale-float (float -8178523606996029 1.0d0) -105)
+ (scale-float (float -8699896243399604 1.0d0) -159)
+ (scale-float (float -7740332332082545 1.0d0) -214)
+ (scale-float (float 6016821677785088 1.0d0) -268))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7056493552344079 1.0d0) -47)
+ (scale-float (float -6660391185023610 1.0d0) -101)
+ (scale-float (float 8326117901300204 1.0d0) -156)
+ (scale-float (float -5180116769562087 1.0d0) -211)
+ (scale-float (float 6732836385259520 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7644684611709030 1.0d0) -47)
+ (scale-float (float 7144281558445897 1.0d0) -105)
+ (scale-float (float 5769916130091718 1.0d0) -159)
+ (scale-float (float 6892393124855441 1.0d0) -213)
+ (scale-float (float 8609597045211136 1.0d0) -267))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 8339806414116051 1.0d0) -47)
+ (scale-float (float -4878300504338807 1.0d0) -101)
+ (scale-float (float 5390699203123886 1.0d0) -156)
+ (scale-float (float -7941956443494165 1.0d0) -210)
+ (scale-float (float 8699025989009408 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4586969091394693 1.0d0) -46)
+ (scale-float (float -8978492069766178 1.0d0) -101)
+ (scale-float (float 7053055367607091 1.0d0) -155)
+ (scale-float (float -8712227340191945 1.0d0) -210)
+ (scale-float (float -7846634958946304 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5096708286733636 1.0d0) -46)
+ (scale-float (float -4572705006895138 1.0d0) -100)
+ (scale-float (float 4718547123658870 1.0d0) -154)
+ (scale-float (float 7157224158707570 1.0d0) -210)
+ (scale-float (float -8590740628766720 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5733873284861593 1.0d0) -46)
+ (scale-float (float -8604953363186775 1.0d0) -103)
+ (scale-float (float 8626935765066666 1.0d0) -158)
+ (scale-float (float -4858320692337793 1.0d0) -212)
+ (scale-float (float -7449309035233280 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6553075144281224 1.0d0) -46)
+ (scale-float (float 6620952143443553 1.0d0) -101)
+ (scale-float (float -5268353655146108 1.0d0) -156)
+ (scale-float (float 5893295495507489 1.0d0) -211)
+ (scale-float (float -4800425745186816 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7645332295807713 1.0d0) -46)
+ (scale-float (float -5080909471366464 1.0d0) -102)
+ (scale-float (float 7938109188081450 1.0d0) -156)
+ (scale-float (float 4928452855514440 1.0d0) -210)
+ (scale-float (float 8885897390456832 1.0d0) -268))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4587238957440646 1.0d0) -45)
+ (scale-float (float -7194218336984116 1.0d0) -99)
+ (scale-float (float -8985364386653513 1.0d0) -153)
+ (scale-float (float -8164942491672346 1.0d0) -207)
+ (scale-float (float 6286694831915008 1.0d0) -262))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5734089176174276 1.0d0) -45)
+ (scale-float (float -7647633804412292 1.0d0) -101)
+ (scale-float (float -7089963492055907 1.0d0) -155)
+ (scale-float (float 8957378262525840 1.0d0) -210)
+ (scale-float (float 8460149900050432 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7645494213403192 1.0d0) -45)
+ (scale-float (float 4683660413726437 1.0d0) -99)
+ (scale-float (float -7150841523991633 1.0d0) -153)
+ (scale-float (float -4654257076122560 1.0d0) -207)
+ (scale-float (float 5269682304483328 1.0d0) -261))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5734143148494430 1.0d0) -44)
+ (scale-float (float 8353770598111665 1.0d0) -99)
+ (scale-float (float -8962023885528779 1.0d0) -158)
+ (scale-float (float -5933825828214605 1.0d0) -212)
+ (scale-float (float 5439908491034624 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5734156641542718 1.0d0) -43)
+ (scale-float (float 8633141086175409 1.0d0) -98)
+ (scale-float (float 5419442444201815 1.0d0) -153)
+ (scale-float (float -8712131584355876 1.0d0) -211)
+ (scale-float (float -5157997961543680 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7091480910882704 1.0d0) 204)
+ (scale-float (float 8415670760171201 1.0d0) 148)
+ (scale-float (float -8191062761010691 1.0d0) 94)
+ (scale-float (float 8061527243862424 1.0d0) 39)
+ (scale-float (float -8684710553714688 1.0d0) -17))
+ (%make-qd-d q0 q1 q2 q3)))))
#||
Some sample test code to convert long-float data from clisp to
@@ -3848,4 +18205,30 @@ something that we can use to initialize our tables.
(dotimes (k 256)
(push (convert-to-qd-parts (cos (* pi (/ (1+ k) 1024)))) val))
`(make-array 256 :initial-contents (list ,@(nreverse val))))
+
+;; Create atan table
+(let (val)
+ (dotimes (k +qd-atan-partition-size+)
+ (push (convert-to-qd-parts (tan (* pi (/ (+ 3 (* 2 k))
+ (* 4 +qd-atan-partition-size+)))))
+ val))
+ (write `(defconstant +qd-atan-partition+
+ (make-array +qd-atan-partition-size+
+ :initial-contents (list ,@(nreverse val))))
+ :case :downcase
+ :pretty t)
+ (values))
+
+(let (val)
+ (dotimes (k +qd-atan-partition-size+)
+ (push (convert-to-qd-parts (tan (* pi (/ (+ 2 (* 2 k))
+ (* 4 +qd-atan-partition-size+)))))
+ val))
+ (write `(defconstant +qd-atan-nodes+
+ (make-array +qd-atan-partition-size+
+ :initial-contents (list ,@(nreverse val))))
+ :case :downcase
+ :pretty t)
+ (values))
+
||#
diff --git a/qd-fun.lisp b/qd-fun.lisp
index b3ae7b6..e782a8a 100644
--- a/qd-fun.lisp
+++ b/qd-fun.lisp
@@ -1102,6 +1102,102 @@ is the cosine of A"
#+nil (optimize (speed 3) (space 0)))
(atan2-qd/newton y +qd-one+))
+;; Computation of atan using a table and taylor series.
+
+(defun find-atan-partition-linear (x)
+ ;; Simple linear search. Search through the qd-atan-partition table
+ ;; to find the entry k such that x > X[k]. Return k.
+ (loop for k from 0 below octi::+qd-atan-partition-size+
+ ;; do (format t "k = ~D, atan = ~S~%" k (aref octi::+qd-atan-partition+ k))
+ when (qd-< x (aref octi::+qd-atan-partition+ k))
+ return k))
+
+(defun find-atan-partition (x)
+ ;; Perform binary search through the atan partition table to find
+ ;; the smallest entry greater than x. Return the index of the
+ ;; entry.
+ (let* ((low 0)
+ (high (1- octi::+qd-atan-partition-size+))
+ (mid (ash (+ low high) -1)))
+ ;; high starts one less than the end because the end actually has junk.
+ ;; FIXME: Remove that junk last element.
+ (loop
+ (if (qd-< x (aref octi::+qd-atan-partition+ mid))
+ (setf high mid)
+ (setf low mid))
+ (setf mid (ash (+ low high) -1))
+ ;;(format t "low, mid, high = ~A ~A ~A~%" low mid high)
+ (when (<= (- high low) 1)
+ (return)))
+ ;;(format t "mid = ~A~%" mid)
+ ;;(format t "X[mid] = ~A~%" (aref octi::+qd-atan-partition+ mid))
+ (if (qd-< x (aref octi::+qd-atan-partition+ mid))
+ mid
+ (1+ mid))))
+
+
+(defun atan-taylor (a)
+ ;; Taylor series for atan(x).
+ ;;
+ ;; atan(x) = sum((-1)^k*x^(2*k+1)/(2*k+1), k, 0, inf)
+ ;; = x - x^3/3 + x^5/5 - x^7/7 + x^9/9 - ...
+ ;; = x*(1 - x^2/3 + x^4/5 - x^6/7 + x^8/9 - ...)
+ ;;
+ ;; x should be small for this to be effective, but no check is made
+ ;; for this. The series on converges for |x| < 1.
+ (let* ((x2 (neg-qd (sqr-qd a)))
+ (term x2)
+ (s (make-qd-d 1d0))
+ (m 3))
+ (loop
+ (setf s (add-qd s (div-qd-d term (float m 1d0))))
+ (setf term (mul-qd term x2))
+ (setf m (+ m 2))
+ (when (<= (abs (qd-0 term)) +qd-eps+)
+ ;;(format t "terms = ~A~%" m)
+ (return)))
+ (mul-qd s a)))
+
+(defun atan-qd/taylor (y)
+ ;; To compute atan(x) for x in [0,inf], we partition the interval
+ ;; [0,inf] such that partition points X[i] and the evaluation nodes
+ ;; x[i] are defined by
+ ;;
+ ;; x[i] = tan((2*i-2)*pi/(4*s)), i = 2, ..., s + 1
+ ;; X[i] = tan((2*i-1)*pi/(4*s)), i = 1, ..., s
+ ;;
+ ;; where 0 = X[0] < X[1] < x[2] < X[2] < ... < x[s] < X[s] < x[s+1]
+ ;; = X[s+1] = inf.
+ ;;
+ ;; So if x is in the interval [X[i-1], X[i]}, the computation of
+ ;; atan(x) is given by
+ ;;
+ ;; atan(x) = atan(x[i]) + atan(e)
+ ;;
+ ;; where e = 1/x[i] - (1+1/x[i]^2)/(x+1/x[i])
+ ;;
+ ;; But since x[i] = tan((2*i-1)*pi/(4*s)), atan(x[i]) =
+ ;; (2*i-1)*pi/(4*s), so
+ ;;
+ ;; atan(x) = (2*i-1)*pi/(4*s) + atan(e)
+ ;;
+ ;; Note that qd-atan-partition array starts with X[2].
+ (if (minusp-qd y)
+ (neg-qd (atan-qd/taylor (neg-qd y)))
+ (let* ((i (find-atan-partition y))
+ (atan-xi (mul-qd +qd-pi/2+
+ (rational-to-qd (/ (1- (+ i 2))
+ octi::+qd-atan-partition-size+))))
+ (1/xi (div-qd +qd-one+ (aref octi::+qd-atan-nodes+ i)))
+ (e1 (div-qd (add-qd-d (sqr-qd 1/xi) 1d0)
+ (add-qd 1/xi y)))
+ (e (sub-qd 1/xi
+ e1)))
+ ;;(format t "partition ~D~%" i)
+ ;;(format t "atan(xi) = ~S~%" atan-xi)
+ ;;(format t "e = ~S~%" e)
+ (add-qd atan-xi (atan-taylor e)))))
+
(defun atan2-qd (y x)
"atan2(y, x) = atan(y/x), but carefully handling the quadrant"
(declare (type %quad-double y x))
diff --git a/timing.lisp b/timing.lisp
index 582671b..1847c4c 100644
--- a/timing.lisp
+++ b/timing.lisp
@@ -174,3 +174,60 @@
(format t "Sqrt QD-REAL (method) ~d times~%" n)
#+cmu (ext:gc :full t)
(time (sqrt-qd-real))))
+
+(defun time-atan (&optional (n 10000))
+ (declare (fixnum n))
+ (flet ((time-atan/newton ()
+ (let ((sum (octi::make-qd-d 0.01d0)))
+ (declare (type octi::%quad-double sum)
+ (optimize (speed 3)))
+ (dotimes (k n)
+ (declare (fixnum k))
+ (setf sum (octi::atan-qd/newton sum)))
+ sum))
+ (time-atan/taylor ()
+ (let ((sum (octi::make-qd-d 0.01d0)))
+ (declare (type octi::%quad-double sum)
+ (optimize (speed 3)))
+ (dotimes (k n)
+ (declare (fixnum k))
+ (setf sum (octi::atan-qd/taylor sum)))
+ sum)))
+ (format t "atan-qd/newton ~d times~%" n)
+ #+cmu (ext:gc :full t)
+ (format t "sum = ~A~%" (time (time-atan/newton)))
+ (format t "atan-qd/taylor ~d times~%" n)
+ #+cmu (ext:gc :full t)
+ (format t "sum = ~A~%" (time (time-atan/taylor)))))
+
+;; Some timing results on an iMac, 3.06 GHz Core i3:
+
+; atan-qd/newton 10000 times
+
+; Evaluation took:
+; 6.15 seconds of real time
+; 6.116626 seconds of user run time
+; 0.021624 seconds of system run time
+; 18,805,071,897 CPU cycles
+; [Run times include 0.24 seconds GC run time]
+; 0 page faults and
+; 893,037,112 bytes consed.
+;
+; sum = #C(0.0077459785628163552318041722744135w0 5.8166227464838760117515984152653w-36)
+;
+; atan-qd/taylor 10000 times
+
+; Evaluation took:
+; 0.29 seconds of real time
+; 0.28987 seconds of user run time
+; 0.001629 seconds of system run time
+; 892,698,785 CPU cycles
+; [Run times include 0.02 seconds GC run time]
+; 0 page faults and
+; 79,108,384 bytes consed.
+;
+; sum = #C(0.0077459785628163552318041722744135w0 5.81662274648387601175165266362535w-36)
+
+;; We see that the taylor series is 21 times faster (!) and conses 11
+;; times less. That's a pretty nice gain, at the expense of two
+;; 1024-element tables.
-----------------------------------------------------------------------
Summary of changes:
qd-const.lisp | 7223 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
qd-fun.lisp | 158 +-
rt-tests.lisp | 3 +
timing.lisp | 111 +
4 files changed, 7493 insertions(+), 2 deletions(-)
hooks/post-receive
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OCT: A portable Lisp implementation for quad-double precision floats
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