[oct-git]OCT: A portable Lisp implementation for quad-double precision floats branch master updated. 0a607f3a6a6297597ae2a6c9896844803d42a6d9

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commit 0a607f3a6a6297597ae2a6c9896844803d42a6d9
Author: Raymond Toy <toy.raymond@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@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)
-    (renorm-5 (scale-float (float 4552220182163332 1.0d0) -52)
-     (scale-float (float -7688418407588221 1.0d0) -113)
-     (scale-float (float 6056179036562030 1.0d0) -167)
-     (scale-float (float 4940221467301023 1.0d0) -222)
-     (scale-float (float -4749127669776384 1.0d0) -276))
+    (renorm-5 (scale-float (float 7074243300823009 1.0d0) -62)
+     (scale-float (float 6767281937885683 1.0d0) -116)
+     (scale-float (float 6495058557496386 1.0d0) -172)
+     (scale-float (float 5247049684764422 1.0d0) -226)
+     (scale-float (float -5541122207547392 1.0d0) -280))
     (%make-qd-d q0 q1 q2 q3))
    (multiple-value-bind (q0 q1 q2 q3)
-    (renorm-5 (scale-float (float 4566208724748110 1.0d0) -52)
-     (scale-float (float 6076277320393968 1.0d0) -108)
-     (scale-float (float 7876705740801642 1.0d0) -162)
-     (scale-float (float -6201281096292325 1.0d0) -218)
-     (scale-float (float 6268449461895168 1.0d0) -272))
+    (renorm-5 (scale-float (float 7074259947269390 1.0d0) -61)
+     (scale-float (float -7016692485563189 1.0d0) -116)
+     (scale-float (float 7360987505494805 1.0d0) -170)
+     (scale-float (float 7711403976890054 1.0d0) -224)
+     (scale-float (float 7755379282149376 1.0d0) -278))
     (%make-qd-d q0 q1 q2 q3))
    (multiple-value-bind (q0 q1 q2 q3)
-    (renorm-5 (scale-float (float 4580240848082718 1.0d0) -52)
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-     (scale-float (float -4889725374845191 1.0d0) -214)
-     (scale-float (float -8473860704370688 1.0d0) -271))
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     (%make-qd-d q0 q1 q2 q3))
    (multiple-value-bind (q0 q1 q2 q3)
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     (%make-qd-d q0 q1 q2 q3))
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     (%make-qd-d q0 q1 q2 q3))
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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 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@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@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)
+     (scale-float (float -5473160723154885 1.0d0) -115)
+     (scale-float (float -4535918610883809 1.0d0) -171)
+     (scale-float (float -4963578518408804 1.0d0) -225)
+     (scale-float (float -6712212661567488 1.0d0) -279))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6190017509796469 1.0d0) -60)
+     (scale-float (float 8647620749142769 1.0d0) -115)
+     (scale-float (float -6931221986832039 1.0d0) -171)
+     (scale-float (float -8517197728414346 1.0d0) -225)
+     (scale-float (float -4626619940470784 1.0d0) -283))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7958643881798693 1.0d0) -60)
+     (scale-float (float -5354492433078344 1.0d0) -114)
+     (scale-float (float -7327905104279995 1.0d0) -168)
+     (scale-float (float -6523976005184669 1.0d0) -223)
+     (scale-float (float -7854698297393152 1.0d0) -277))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 4863653855283737 1.0d0) -59)
+     (scale-float (float -6015787009420778 1.0d0) -113)
+     (scale-float (float 8594253106416801 1.0d0) -167)
+     (scale-float (float 7046824309216708 1.0d0) -222)
+     (scale-float (float -6520970423566336 1.0d0) -277))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5748008660632557 1.0d0) -59)
+     (scale-float (float 8797928203970591 1.0d0) -113)
+     (scale-float (float 8714055176617153 1.0d0) -167)
+     (scale-float (float -5612171940322240 1.0d0) -221)
+     (scale-float (float 4939191171620864 1.0d0) -275))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6632390519957926 1.0d0) -59)
+     (scale-float (float 8670207667135100 1.0d0) -114)
+     (scale-float (float 8149147453636330 1.0d0) -168)
+     (scale-float (float -8078741638624220 1.0d0) -222)
+     (scale-float (float 5251828477165568 1.0d0) -277))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7516803596781324 1.0d0) -59)
+     (scale-float (float -6201171919502881 1.0d0) -116)
+     (scale-float (float -7027033891914616 1.0d0) -173)
+     (scale-float (float 5952967054329307 1.0d0) -227)
+     (scale-float (float -8689209012387840 1.0d0) -282))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 8401252055212014 1.0d0) -59)
+     (scale-float (float -8018416525424526 1.0d0) -114)
+     (scale-float (float -8131489574139489 1.0d0) -168)
+     (scale-float (float 5356676700338153 1.0d0) -223)
+     (scale-float (float -8760547365355520 1.0d0) -279))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 4642870030012741 1.0d0) -58)
+     (scale-float (float 8661794336121368 1.0d0) -112)
+     (scale-float (float 5441429837488339 1.0d0) -167)
+     (scale-float (float -6919382284775521 1.0d0) -222)
+     (scale-float (float -5506479342911488 1.0d0) -276))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5085135888370956 1.0d0) -58)
+     (scale-float (float -7882846098076715 1.0d0) -113)
+     (scale-float (float -7762714374541670 1.0d0) -171)
+     (scale-float (float 8931845445931377 1.0d0) -225)
+     (scale-float (float 5556186022150144 1.0d0) -279))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5527425685852335 1.0d0) -58)
+     (scale-float (float 8127719015560079 1.0d0) -112)
+     (scale-float (float -6094331376989970 1.0d0) -166)
+     (scale-float (float 6973207316824032 1.0d0) -220)
+     (scale-float (float -8483611529240576 1.0d0) -274))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5969741506079427 1.0d0) -58)
+     (scale-float (float 8989322454423522 1.0d0) -112)
+     (scale-float (float 4820491773276110 1.0d0) -171)
+     (scale-float (float 5078156230612779 1.0d0) -225)
+     (scale-float (float 6795293461839872 1.0d0) -279))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6412085433164924 1.0d0) -58)
+     (scale-float (float 5647386557728108 1.0d0) -115)
+     (scale-float (float 5565185291589407 1.0d0) -169)
+     (scale-float (float -8187312701179060 1.0d0) -225)
+     (scale-float (float -8291776467828736 1.0d0) -279))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6854459551750966 1.0d0) -58)
+     (scale-float (float -5878879597999219 1.0d0) -112)
+     (scale-float (float -4586089337686870 1.0d0) -166)
+     (scale-float (float -8194332440371855 1.0d0) -224)
+     (scale-float (float 6231038519017472 1.0d0) -278))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7296865947048466 1.0d0) -58)
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+    (%make-qd-d q0 q1 q2 q3))
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+    (%make-qd-d q0 q1 q2 q3))
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+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5313675850830065 1.0d0) -48)
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+    (%make-qd-d q0 q1 q2 q3))
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+    (%make-qd-d q0 q1 q2 q3))
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+    (%make-qd-d q0 q1 q2 q3))
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+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6011778688436236 1.0d0) -48)
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+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6215860609535080 1.0d0) -48)
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+     (scale-float (float 4785333766536567 1.0d0) -210)
+     (scale-float (float -5015554907439104 1.0d0) -267))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6434253960626327 1.0d0) -48)
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+     (scale-float (float 5910480269657087 1.0d0) -216)
+     (scale-float (float 7260906967793664 1.0d0) -270))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6668519989190208 1.0d0) -48)
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+    (%make-qd-d q0 q1 q2 q3))
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+    (renorm-5 (scale-float (float 6920455602683467 1.0d0) -48)
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+    (%make-qd-d q0 q1 q2 q3))
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+    (renorm-5 (scale-float (float 7192139576377896 1.0d0) -48)
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+     (scale-float (float 7154798553989120 1.0d0) -271))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7485990077404228 1.0d0) -48)
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+     (scale-float (float 8559087960719360 1.0d0) -273))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7804836875785953 1.0d0) -48)
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+     (scale-float (float -5647019254008860 1.0d0) -210)
+     (scale-float (float 5168835195092992 1.0d0) -264))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 8152012811748831 1.0d0) -48)
+     (scale-float (float 4755270175277766 1.0d0) -102)
+     (scale-float (float 6561325893612995 1.0d0) -156)
+     (scale-float (float -6293819739494205 1.0d0) -214)
+     (scale-float (float -4611512610062336 1.0d0) -271))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 8531470788791185 1.0d0) -48)
+     (scale-float (float 4971209640929394 1.0d0) -102)
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+     (scale-float (float -8315187020880318 1.0d0) -210)
+     (scale-float (float -8323584824279040 1.0d0) -264))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 8947935008628436 1.0d0) -48)
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+     (scale-float (float 8685140401214535 1.0d0) -212)
+     (scale-float (float 7969202102599680 1.0d0) -269))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 4703549369983361 1.0d0) -47)
+     (scale-float (float 7852350636790280 1.0d0) -104)
+     (scale-float (float 8135489978399987 1.0d0) -158)
+     (scale-float (float -7174034823224577 1.0d0) -212)
+     (scale-float (float 8298739190988800 1.0d0) -267))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 4957943114250195 1.0d0) -47)
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+     (scale-float (float -7066653115289274 1.0d0) -211)
+     (scale-float (float -5251925558591488 1.0d0) -265))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5241402202311462 1.0d0) -47)
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+     (scale-float (float 5956295317296818 1.0d0) -210)
+     (scale-float (float -6370921775546368 1.0d0) -264))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5559211242425461 1.0d0) -47)
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+     (scale-float (float 8800956699377664 1.0d0) -272))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5918018612706102 1.0d0) -47)
+     (scale-float (float 6327539369914933 1.0d0) -102)
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+     (scale-float (float 5106528031699086 1.0d0) -217)
+     (scale-float (float 4966384115122176 1.0d0) -272))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6326306726590363 1.0d0) -47)
+     (scale-float (float 4932593287996622 1.0d0) -103)
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+     (scale-float (float -7780234803709193 1.0d0) -219)
+     (scale-float (float 6415528025391104 1.0d0) -273))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6795071305180165 1.0d0) -47)
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+     (scale-float (float -8270613876366648 1.0d0) -212)
+     (scale-float (float -6076525478084608 1.0d0) -266))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7338826700308355 1.0d0) -47)
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+     (scale-float (float 6242568037990400 1.0d0) -267))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7977135733986933 1.0d0) -47)
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+     (scale-float (float -5672208511336448 1.0d0) -266))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 8737013731825704 1.0d0) -47)
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+     (scale-float (float 4872662811672576 1.0d0) -271))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 4828425446673454 1.0d0) -46)
+     (scale-float (float -6916860673263500 1.0d0) -104)
+     (scale-float (float 6047527717683516 1.0d0) -158)
+     (scale-float (float 8454679980636421 1.0d0) -213)
+     (scale-float (float -7197223298793472 1.0d0) -267))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5396551697143330 1.0d0) -46)
+     (scale-float (float -5733671657124272 1.0d0) -104)
+     (scale-float (float -7168980379255407 1.0d0) -158)
+     (scale-float (float -5385528853569594 1.0d0) -213)
+     (scale-float (float 8970110362451968 1.0d0) -268))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6116168685368767 1.0d0) -46)
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+     (scale-float (float 7180988953329664 1.0d0) -264))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7057195213557535 1.0d0) -46)
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+     (scale-float (float -8681831307511366 1.0d0) -159)
+     (scale-float (float 8147107942114543 1.0d0) -213)
+     (scale-float (float 8684777780019200 1.0d0) -268))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 8340400121862057 1.0d0) -46)
+     (scale-float (float 6443522098103518 1.0d0) -103)
+     (scale-float (float -5063380033451047 1.0d0) -157)
+     (scale-float (float 7096754030320520 1.0d0) -211)
+     (scale-float (float -5649303769055232 1.0d0) -265))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5096951165270068 1.0d0) -45)
+     (scale-float (float -7367277215028988 1.0d0) -99)
+     (scale-float (float 5204165418458199 1.0d0) -155)
+     (scale-float (float -7328140217508302 1.0d0) -209)
+     (scale-float (float 4644105911926784 1.0d0) -265))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 6553264048624175 1.0d0) -45)
+     (scale-float (float -8728494544390632 1.0d0) -99)
+     (scale-float (float 6818471906941218 1.0d0) -155)
+     (scale-float (float -4839948684372485 1.0d0) -209)
+     (scale-float (float -6526652324249600 1.0d0) -266))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 4587306422959904 1.0d0) -44)
+     (scale-float (float -8132024147820829 1.0d0) -99)
+     (scale-float (float 8001039806955030 1.0d0) -154)
+     (scale-float (float 8285712093786516 1.0d0) -210)
+     (scale-float (float -5775142137495552 1.0d0) -264))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7645534692587744 1.0d0) -44)
+     (scale-float (float 7904996613277796 1.0d0) -98)
+     (scale-float (float -4949844840316895 1.0d0) -157)
+     (scale-float (float -5649413412498743 1.0d0) -213)
+     (scale-float (float -5510548699480064 1.0d0) -269))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 5734160014802806 1.0d0) -42)
+     (scale-float (float -5854173816779464 1.0d0) -97)
+     (scale-float (float -7171395319441940 1.0d0) -152)
+     (scale-float (float 6767673389735587 1.0d0) -207)
+     (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)
+     (scale-float (float 5854173816779464 1.0d0) -97)
+     (scale-float (float 7171395319441940 1.0d0) -152)
+     (scale-float (float -6767673389735587 1.0d0) -207)
+     (scale-float (float 6695997717610496 1.0d0) -264))
+    (%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 table size,
+;; +qd-atan-partition-size+.
+(defconstant +qd-atan-nodes+
+ (make-array +qd-atan-partition-size+ :initial-contents
+  (list
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7074243300823009 1.0d0) -62)
+     (scale-float (float 6767281937885683 1.0d0) -116)
+     (scale-float (float 6495058557496386 1.0d0) -172)
+     (scale-float (float 5247049684764422 1.0d0) -226)
+     (scale-float (float -5541122207547392 1.0d0) -280))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
+    (renorm-5 (scale-float (float 7074259947269390 1.0d0) -61)
+     (scale-float (float -7016692485563189 1.0d0) -116)
+     (scale-float (float 7360987505494805 1.0d0) -170)
+     (scale-float (float 7711403976890054 1.0d0) -224)
+     (scale-float (float 7755379282149376 1.0d0) -278))
+    (%make-qd-d q0 q1 q2 q3))
+   (multiple-value-bind (q0 q1 q2 q3)
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+    (%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(-)


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OCT:  A portable Lisp implementation for quad-double precision floats

    

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