Initial import of Calc 2.02f.

1 files changed, 1034 insertions, 0 deletions

diff --git a/lisp/calc/calc-funcs.el b/lisp/calc/calc-funcs.el
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+;; Calculator for GNU Emacs, part II [calc-funcs.el]
+;; Copyright (C) 1990, 1991, 1992, 1993 Free Software Foundation, Inc.
+;; Written by Dave Gillespie, daveg@synaptics.com.
+
+;; This file is part of GNU Emacs.
+
+;; GNU Emacs is distributed in the hope that it will be useful,
+;; but WITHOUT ANY WARRANTY.  No author or distributor
+;; accepts responsibility to anyone for the consequences of using it
+;; or for whether it serves any particular purpose or works at all,
+;; unless he says so in writing.  Refer to the GNU Emacs General Public
+;; License for full details.
+
+;; Everyone is granted permission to copy, modify and redistribute
+;; GNU Emacs, but only under the conditions described in the
+;; GNU Emacs General Public License.   A copy of this license is
+;; supposed to have been given to you along with GNU Emacs so you
+;; can know your rights and responsibilities.  It should be in a
+;; file named COPYING.  Among other things, the copyright notice
+;; and this notice must be preserved on all copies.
+
+
+
+;; This file is autoloaded from calc-ext.el.
+(require 'calc-ext)
+
+(require 'calc-macs)
+
+(defun calc-Need-calc-funcs () nil)
+
+
+(defun calc-inc-gamma (arg)
+  (interactive "P")
+  (calc-slow-wrapper
+   (if (calc-is-inverse)
+       (if (calc-is-hyperbolic)
+	   (calc-binary-op "gamG" 'calcFunc-gammaG arg)
+	 (calc-binary-op "gamQ" 'calcFunc-gammaQ arg))
+       (if (calc-is-hyperbolic)
+	   (calc-binary-op "gamg" 'calcFunc-gammag arg)
+	 (calc-binary-op "gamP" 'calcFunc-gammaP arg))))
+)
+
+(defun calc-erf (arg)
+  (interactive "P")
+  (calc-slow-wrapper
+   (if (calc-is-inverse)
+       (calc-unary-op "erfc" 'calcFunc-erfc arg)
+     (calc-unary-op "erf" 'calcFunc-erf arg)))
+)
+
+(defun calc-erfc (arg)
+  (interactive "P")
+  (calc-invert-func)
+  (calc-erf arg)
+)
+
+(defun calc-beta (arg)
+  (interactive "P")
+  (calc-slow-wrapper
+   (calc-binary-op "beta" 'calcFunc-beta arg))
+)
+
+(defun calc-inc-beta ()
+  (interactive)
+  (calc-slow-wrapper
+   (if (calc-is-hyperbolic)
+       (calc-enter-result 3 "betB" (cons 'calcFunc-betaB (calc-top-list-n 3)))
+     (calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3)))))
+)
+
+(defun calc-bessel-J (arg)
+  (interactive "P")
+  (calc-slow-wrapper
+   (calc-binary-op "besJ" 'calcFunc-besJ arg))
+)
+
+(defun calc-bessel-Y (arg)
+  (interactive "P")
+  (calc-slow-wrapper
+   (calc-binary-op "besY" 'calcFunc-besY arg))
+)
+
+(defun calc-bernoulli-number (arg)
+  (interactive "P")
+  (calc-slow-wrapper
+   (if (calc-is-hyperbolic)
+       (calc-binary-op "bern" 'calcFunc-bern arg)
+     (calc-unary-op "bern" 'calcFunc-bern arg)))
+)
+
+(defun calc-euler-number (arg)
+  (interactive "P")
+  (calc-slow-wrapper
+   (if (calc-is-hyperbolic)
+       (calc-binary-op "eulr" 'calcFunc-euler arg)
+     (calc-unary-op "eulr" 'calcFunc-euler arg)))
+)
+
+(defun calc-stirling-number (arg)
+  (interactive "P")
+  (calc-slow-wrapper
+   (if (calc-is-hyperbolic)
+       (calc-binary-op "str2" 'calcFunc-stir2 arg)
+     (calc-binary-op "str1" 'calcFunc-stir1 arg)))
+)
+
+(defun calc-utpb ()
+  (interactive)
+  (calc-prob-dist "b" 3)
+)
+
+(defun calc-utpc ()
+  (interactive)
+  (calc-prob-dist "c" 2)
+)
+
+(defun calc-utpf ()
+  (interactive)
+  (calc-prob-dist "f" 3)
+)
+
+(defun calc-utpn ()
+  (interactive)
+  (calc-prob-dist "n" 3)
+)
+
+(defun calc-utpp ()
+  (interactive)
+  (calc-prob-dist "p" 2)
+)
+
+(defun calc-utpt ()
+  (interactive)
+  (calc-prob-dist "t" 2)
+)
+
+(defun calc-prob-dist (letter nargs)
+  (calc-slow-wrapper
+   (if (calc-is-inverse)
+       (calc-enter-result nargs (concat "ltp" letter)
+			  (append (list (intern (concat "calcFunc-ltp" letter))
+					(calc-top-n 1))
+				  (calc-top-list-n (1- nargs) 2)))
+     (calc-enter-result nargs (concat "utp" letter)
+			(append (list (intern (concat "calcFunc-utp" letter))
+				      (calc-top-n 1))
+				(calc-top-list-n (1- nargs) 2)))))
+)
+
+
+
+
+;;; Sources:  Numerical Recipes, Press et al;
+;;;           Handbook of Mathematical Functions, Abramowitz & Stegun.
+
+
+;;; Gamma function.
+
+(defun calcFunc-gamma (x)
+  (or (math-numberp x) (math-reject-arg x 'numberp))
+  (calcFunc-fact (math-add x -1))
+)
+
+(defun math-gammap1-raw (x &optional fprec nfprec)   ; compute gamma(1 + x)
+  (or fprec
+      (setq fprec (math-float calc-internal-prec)
+	    nfprec (math-float (- calc-internal-prec))))
+  (cond ((math-lessp-float (calcFunc-re x) fprec)
+	 (if (math-lessp-float (calcFunc-re x) nfprec)
+	     (math-neg (math-div
+			(math-pi)
+			(math-mul (math-gammap1-raw
+				   (math-add (math-neg x)
+					     '(float -1 0))
+				   fprec nfprec)
+				  (math-sin-raw
+				   (math-mul (math-pi) x)))))
+	   (let ((xplus1 (math-add x '(float 1 0))))
+	     (math-div (math-gammap1-raw xplus1 fprec nfprec) xplus1))))
+	((and (math-realp x)
+	      (math-lessp-float '(float 736276 0) x))
+	 (math-overflow))
+	(t   ; re(x) now >= 10.0
+	 (let ((xinv (math-div 1 x))
+	       (lnx (math-ln-raw x)))
+	   (math-mul (math-sqrt-two-pi)
+		     (math-exp-raw
+		      (math-gamma-series
+		       (math-sub (math-mul (math-add x '(float 5 -1))
+					   lnx)
+				 x)
+		       xinv
+		       (math-sqr xinv)
+		       '(float 0 0)
+		       2))))))
+)
+
+(defun math-gamma-series (sum x xinvsqr oterm n)
+  (math-working "gamma" sum)
+  (let* ((bn (math-bernoulli-number n))
+	 (term (math-mul (math-div-float (math-float (nth 1 bn))
+					 (math-float (* (nth 2 bn)
+							(* n (1- n)))))
+			 x))
+	 (next (math-add sum term)))
+    (if (math-nearly-equal sum next)
+	next
+      (if (> n (* 2 calc-internal-prec))
+	  (progn
+	    ;; Need this because series eventually diverges for large enough n.
+	    (calc-record-why
+	     "*Gamma computation stopped early, not all digits may be valid")
+	    next)
+	(math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2)))))
+)
+
+
+;;; Incomplete gamma function.
+
+(defun calcFunc-gammaP (a x)
+  (if (equal x '(var inf var-inf))
+      '(float 1 0)
+    (math-inexact-result)
+    (or (Math-numberp a) (math-reject-arg a 'numberp))
+    (or (math-numberp x) (math-reject-arg x 'numberp))
+    (if (and (math-num-integerp a)
+	     (integerp (setq a (math-trunc a)))
+	     (> a 0) (< a 20))
+	(math-sub 1 (calcFunc-gammaQ a x))
+      (let ((math-current-gamma-value (calcFunc-gamma a)))
+	(math-div (calcFunc-gammag a x) math-current-gamma-value))))
+)
+
+(defun calcFunc-gammaQ (a x)
+  (if (equal x '(var inf var-inf))
+      '(float 0 0)
+    (math-inexact-result)
+    (or (Math-numberp a) (math-reject-arg a 'numberp))
+    (or (math-numberp x) (math-reject-arg x 'numberp))
+    (if (and (math-num-integerp a)
+	     (integerp (setq a (math-trunc a)))
+	     (> a 0) (< a 20))
+	(let ((n 0)
+	      (sum '(float 1 0))
+	      (term '(float 1 0)))
+	  (math-with-extra-prec 1
+	    (while (< (setq n (1+ n)) a)
+	      (setq term (math-div (math-mul term x) n)
+		    sum (math-add sum term))
+	      (math-working "gamma" sum))
+	    (math-mul sum (calcFunc-exp (math-neg x)))))
+      (let ((math-current-gamma-value (calcFunc-gamma a)))
+	(math-div (calcFunc-gammaG a x) math-current-gamma-value))))
+)
+
+(defun calcFunc-gammag (a x)
+  (if (equal x '(var inf var-inf))
+      (calcFunc-gamma a)
+    (math-inexact-result)
+    (or (Math-numberp a) (math-reject-arg a 'numberp))
+    (or (Math-numberp x) (math-reject-arg x 'numberp))
+    (math-with-extra-prec 2
+      (setq a (math-float a))
+      (setq x (math-float x))
+      (if (or (math-negp (calcFunc-re a))
+	      (math-lessp-float (calcFunc-re x)
+				(math-add-float (calcFunc-re a)
+						'(float 1 0))))
+	  (math-inc-gamma-series a x)
+	(math-sub (or math-current-gamma-value (calcFunc-gamma a))
+		  (math-inc-gamma-cfrac a x)))))
+)
+(setq math-current-gamma-value nil)
+
+(defun calcFunc-gammaG (a x)
+  (if (equal x '(var inf var-inf))
+      '(float 0 0)
+    (math-inexact-result)
+    (or (Math-numberp a) (math-reject-arg a 'numberp))
+    (or (Math-numberp x) (math-reject-arg x 'numberp))
+    (math-with-extra-prec 2
+      (setq a (math-float a))
+      (setq x (math-float x))
+      (if (or (math-negp (calcFunc-re a))
+	      (math-lessp-float (calcFunc-re x)
+				(math-add-float (math-abs-approx a)
+						'(float 1 0))))
+	  (math-sub (or math-current-gamma-value (calcFunc-gamma a))
+		    (math-inc-gamma-series a x))
+	(math-inc-gamma-cfrac a x))))
+)
+
+(defun math-inc-gamma-series (a x)
+  (if (Math-zerop x)
+      '(float 0 0)
+    (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x))
+	      (math-with-extra-prec 2
+		(let ((start (math-div '(float 1 0) a)))
+		  (math-inc-gamma-series-step start start a x)))))
+)
+
+(defun math-inc-gamma-series-step (sum term a x)
+  (math-working "gamma" sum)
+  (setq a (math-add a '(float 1 0))
+	term (math-div (math-mul term x) a))
+  (let ((next (math-add sum term)))
+    (if (math-nearly-equal sum next)
+	next
+      (math-inc-gamma-series-step next term a x)))
+)
+
+(defun math-inc-gamma-cfrac (a x)
+  (if (Math-zerop x)
+      (or math-current-gamma-value (calcFunc-gamma a))
+    (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x))
+	      (math-inc-gamma-cfrac-step '(float 1 0) x
+					 '(float 0 0) '(float 1 0)
+					 '(float 1 0) '(float 1 0) '(float 0 0)
+					 a x)))
+)
+
+(defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x)
+  (let ((ana (math-sub n a))
+	(anf (math-mul n fac)))
+    (setq n (math-add n '(float 1 0))
+	  a0 (math-mul (math-add a1 (math-mul a0 ana)) fac)
+	  b0 (math-mul (math-add b1 (math-mul b0 ana)) fac)
+	  a1 (math-add (math-mul x a0) (math-mul anf a1))
+	  b1 (math-add (math-mul x b0) (math-mul anf b1)))
+    (if (math-zerop a1)
+	(math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x)
+      (setq fac (math-div '(float 1 0) a1))
+      (let ((next (math-mul b1 fac)))
+	(math-working "gamma" next)
+	(if (math-nearly-equal next g)
+	    next
+	  (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x)))))
+)
+
+
+;;; Error function.
+
+(defun calcFunc-erf (x)
+  (if (equal x '(var inf var-inf))
+      '(float 1 0)
+    (if (equal x '(neg (var inf var-inf)))
+	'(float -1 0)
+      (if (Math-zerop x)
+	  x
+	(let ((math-current-gamma-value (math-sqrt-pi)))
+	  (math-to-same-complex-quad
+	   (math-div (calcFunc-gammag '(float 5 -1)
+				      (math-sqr (math-to-complex-quad-one x)))
+		     math-current-gamma-value)
+	   x)))))
+)
+
+(defun calcFunc-erfc (x)
+  (if (equal x '(var inf var-inf))
+      '(float 0 0)
+    (if (math-posp x)
+	(let ((math-current-gamma-value (math-sqrt-pi)))
+	  (math-div (calcFunc-gammaG '(float 5 -1) (math-sqr x))
+		    math-current-gamma-value))
+      (math-sub 1 (calcFunc-erf x))))
+)
+
+(defun math-to-complex-quad-one (x)
+  (if (eq (car-safe x) 'polar) (setq x (math-complex x)))
+  (if (eq (car-safe x) 'cplx)
+      (list 'cplx (math-abs (nth 1 x)) (math-abs (nth 2 x)))
+    x)
+)
+
+(defun math-to-same-complex-quad (x y)
+  (if (eq (car-safe y) 'cplx)
+      (if (eq (car-safe x) 'cplx)
+	  (list 'cplx
+		(if (math-negp (nth 1 y)) (math-neg (nth 1 x)) (nth 1 x))
+		(if (math-negp (nth 2 y)) (math-neg (nth 2 x)) (nth 2 x)))
+	(if (math-negp (nth 1 y)) (math-neg x) x))
+    (if (math-negp y)
+	(if (eq (car-safe x) 'cplx)
+	    (list 'cplx (math-neg (nth 1 x)) (nth 2 x))
+	  (math-neg x))
+      x))
+)
+
+
+;;; Beta function.
+
+(defun calcFunc-beta (a b)
+  (if (math-num-integerp a)
+      (let ((am (math-add a -1)))
+	(or (math-numberp b) (math-reject-arg b 'numberp))
+	(math-div 1 (math-mul b (calcFunc-choose (math-add b am) am))))
+    (if (math-num-integerp b)
+	(calcFunc-beta b a)
+      (math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b))
+		(calcFunc-gamma (math-add a b)))))
+)
+
+
+;;; Incomplete beta function.
+
+(defun calcFunc-betaI (x a b)
+  (cond ((math-zerop x)
+	 '(float 0 0))
+	((math-equal-int x 1)
+	 '(float 1 0))
+	((or (math-zerop a)
+	     (and (math-num-integerp a)
+		  (math-negp a)))
+	 (if (or (math-zerop b)
+		 (and (math-num-integerp b)
+		      (math-negp b)))
+	     (math-reject-arg b 'range)
+	   '(float 1 0)))
+	((or (math-zerop b)
+	     (and (math-num-integerp b)
+		  (math-negp b)))
+	 '(float 0 0))
+	((not (math-numberp a)) (math-reject-arg a 'numberp))
+	((not (math-numberp b)) (math-reject-arg b 'numberp))
+	((math-inexact-result))
+	(t (let ((math-current-beta-value (calcFunc-beta a b)))
+	     (math-div (calcFunc-betaB x a b) math-current-beta-value))))
+)
+
+(defun calcFunc-betaB (x a b)
+  (cond
+   ((math-zerop x)
+    '(float 0 0))
+   ((math-equal-int x 1)
+    (calcFunc-beta a b))
+   ((not (math-numberp x)) (math-reject-arg x 'numberp))
+   ((not (math-numberp a)) (math-reject-arg a 'numberp))
+   ((not (math-numberp b)) (math-reject-arg b 'numberp))
+   ((math-zerop a) (math-reject-arg a 'nonzerop))
+   ((math-zerop b) (math-reject-arg b 'nonzerop))
+   ((and (math-num-integerp b)
+	 (if (math-negp b)
+	     (math-reject-arg b 'range)
+	   (Math-natnum-lessp (setq b (math-trunc b)) 20)))
+    (and calc-symbolic-mode (or (math-floatp a) (math-floatp b))
+	 (math-inexact-result))
+    (math-mul
+     (math-with-extra-prec 2
+       (let* ((i 0)
+	      (term 1)
+	      (sum (math-div term a)))
+	 (while (< (setq i (1+ i)) b)
+	   (setq term (math-mul (math-div (math-mul term (- i b)) i) x)
+		 sum (math-add sum (math-div term (math-add a i))))
+	   (math-working "beta" sum))
+	 sum))
+     (math-pow x a)))
+   ((and (math-num-integerp a)
+	 (if (math-negp a)
+	     (math-reject-arg a 'range)
+	   (Math-natnum-lessp (setq a (math-trunc a)) 20)))
+    (math-sub (or math-current-beta-value (calcFunc-beta a b))
+	      (calcFunc-betaB (math-sub 1 x) b a)))
+   (t
+    (math-inexact-result)
+    (math-with-extra-prec 2
+      (setq x (math-float x))
+      (setq a (math-float a))
+      (setq b (math-float b))
+      (let ((bt (math-exp-raw (math-add (math-mul a (math-ln-raw x))
+					(math-mul b (math-ln-raw
+						     (math-sub '(float 1 0)
+							       x)))))))
+	(if (Math-lessp x (math-div (math-add a '(float 1 0))
+				    (math-add (math-add a b) '(float 2 0))))
+	    (math-div (math-mul bt (math-beta-cfrac a b x)) a)
+	  (math-sub (or math-current-beta-value (calcFunc-beta a b))
+		    (math-div (math-mul bt
+					(math-beta-cfrac b a (math-sub 1 x)))
+			      b)))))))
+)
+(setq math-current-beta-value nil)
+
+(defun math-beta-cfrac (a b x)
+  (let ((qab (math-add a b))
+	(qap (math-add a '(float 1 0)))
+	(qam (math-add a '(float -1 0))))
+    (math-beta-cfrac-step '(float 1 0)
+			  (math-sub '(float 1 0)
+				    (math-div (math-mul qab x) qap))
+			  '(float 1 0) '(float 1 0)
+			  '(float 1 0)
+			  qab qap qam a b x))
+)
+
+(defun math-beta-cfrac-step (az bz am bm m qab qap qam a b x)
+  (let* ((two-m (math-mul m '(float 2 0)))
+	 (d (math-div (math-mul (math-mul (math-sub b m) m) x)
+		      (math-mul (math-add qam two-m) (math-add a two-m))))
+	 (ap (math-add az (math-mul d am)))
+	 (bp (math-add bz (math-mul d bm)))
+	 (d2 (math-neg
+	      (math-div (math-mul (math-mul (math-add a m) (math-add qab m)) x)
+			(math-mul (math-add qap two-m) (math-add a two-m)))))
+	 (app (math-add ap (math-mul d2 az)))
+	 (bpp (math-add bp (math-mul d2 bz)))
+	 (next (math-div app bpp)))
+    (math-working "beta" next)
+    (if (math-nearly-equal next az)
+	next
+      (math-beta-cfrac-step next '(float 1 0)
+			    (math-div ap bpp) (math-div bp bpp)
+			    (math-add m '(float 1 0))
+			    qab qap qam a b x)))
+)
+
+
+;;; Bessel functions.
+
+;;; Should generalize this to handle arbitrary precision!
+
+(defun calcFunc-besJ (v x)
+  (or (math-numberp v) (math-reject-arg v 'numberp))
+  (or (math-numberp x) (math-reject-arg x 'numberp))
+  (let ((calc-internal-prec (min 8 calc-internal-prec)))
+    (math-with-extra-prec 3
+      (setq x (math-float (math-normalize x)))
+      (setq v (math-float (math-normalize v)))
+      (cond ((math-zerop x)
+	     (if (math-zerop v)
+		 '(float 1 0)
+	       '(float 0 0)))
+	    ((math-inexact-result))
+	    ((not (math-num-integerp v))
+	     (let ((start (math-div 1 (calcFunc-fact v))))
+	       (math-mul (math-besJ-series start start
+					   0
+					   (math-mul '(float -25 -2)
+						     (math-sqr x))
+					   v)
+			 (math-pow (math-div x 2) v))))
+	    ((math-negp (setq v (math-trunc v)))
+	     (if (math-oddp v)
+		 (math-neg (calcFunc-besJ (math-neg v) x))
+	       (calcFunc-besJ (math-neg v) x)))
+	    ((eq v 0)
+	     (math-besJ0 x))
+	    ((eq v 1)
+	     (math-besJ1 x))
+	    ((Math-lessp v (math-abs-approx x))
+	     (let ((j 0)
+		   (bjm (math-besJ0 x))
+		   (bj (math-besJ1 x))
+		   (two-over-x (math-div 2 x))
+		   bjp)
+	       (while (< (setq j (1+ j)) v)
+		 (setq bjp (math-sub (math-mul (math-mul j two-over-x) bj)
+				     bjm)
+		       bjm bj
+		       bj bjp))
+	       bj))
+	    (t
+	     (if (Math-lessp 100 v) (math-reject-arg v 'range))
+	     (let* ((j (logior (+ v (math-isqrt-small (* 40 v))) 1))
+		    (two-over-x (math-div 2 x))
+		    (jsum nil)
+		    (bjp '(float 0 0))
+		    (sum '(float 0 0))
+		    (bj '(float 1 0))
+		    bjm ans)
+	       (while (> (setq j (1- j)) 0)
+		 (setq bjm (math-sub (math-mul (math-mul j two-over-x) bj)
+				     bjp)
+		       bjp bj
+		       bj bjm)
+		 (if (> (nth 2 (math-abs-approx bj)) 10)
+		     (setq bj (math-mul bj '(float 1 -10))
+			   bjp (math-mul bjp '(float 1 -10))
+			   ans (and ans (math-mul ans '(float 1 -10)))
+			   sum (math-mul sum '(float 1 -10))))
+		 (or (setq jsum (not jsum))
+		     (setq sum (math-add sum bj)))
+		 (if (= j v)
+		     (setq ans bjp)))
+	       (math-div ans (math-sub (math-mul 2 sum) bj)))))))
+)
+
+(defun math-besJ-series (sum term k zz vk)
+  (math-working "besJ" sum)
+  (setq k (1+ k)
+	vk (math-add 1 vk)
+	term (math-div (math-mul term zz) (math-mul k vk)))
+  (let ((next (math-add sum term)))
+    (if (math-nearly-equal next sum)
+	next
+      (math-besJ-series next term k zz vk)))
+)
+
+(defun math-besJ0 (x &optional yflag)
+  (cond ((and (not yflag) (math-negp (calcFunc-re x)))
+	 (math-besJ0 (math-neg x)))
+	((Math-lessp '(float 8 0) (math-abs-approx x))
+	 (let* ((z (math-div '(float 8 0) x))
+		(y (math-sqr z))
+		(xx (math-add x '(float (bigneg 164 398 785) -9)))
+		(a1 (math-poly-eval y
+				    '((float (bigpos 211 887 093 2) -16)
+				      (float (bigneg 639 370 073 2) -15)
+				      (float (bigpos 407 510 734 2) -14)
+				      (float (bigneg 627 628 098 1) -12)
+				      (float 1 0))))
+		(a2 (math-poly-eval y
+				    '((float (bigneg 152 935 934) -16)
+				      (float (bigpos 161 095 621 7) -16)
+				      (float (bigneg 651 147 911 6) -15)
+				      (float (bigpos 765 488 430 1) -13)
+				      (float (bigneg 995 499 562 1) -11))))
+		(sc (math-sin-cos-raw xx)))
+	       (if yflag
+		   (setq sc (cons (math-neg (cdr sc)) (car sc))))
+	       (math-mul (math-sqrt
+			  (math-div '(float (bigpos 722 619 636) -9) x))
+			 (math-sub (math-mul (cdr sc) a1)
+				   (math-mul (car sc) (math-mul z a2))))))
+	 (t
+	  (let ((y (math-sqr x)))
+	    (math-div (math-poly-eval y
+				      '((float (bigneg 456 052 849 1) -7)
+					(float (bigpos 017 233 739 7) -5)
+					(float (bigneg 418 442 121 1) -2)
+					(float (bigpos 407 196 516 6) -1)
+					(float (bigneg 354 590 362 13) 0)
+					(float (bigpos 574 490 568 57) 0)))
+		      (math-poly-eval y
+				      '((float 1 0)
+					(float (bigpos 712 532 678 2) -7)
+					(float (bigpos 853 264 927 5) -5)
+					(float (bigpos 718 680 494 9) -3)
+					(float (bigpos 985 532 029 1) 0)
+					(float (bigpos 411 490 568 57) 0)))))))
+)
+
+(defun math-besJ1 (x &optional yflag)
+  (cond ((and (math-negp (calcFunc-re x)) (not yflag))
+	 (math-neg (math-besJ1 (math-neg x))))
+	((Math-lessp '(float 8 0) (math-abs-approx x))
+	 (let* ((z (math-div '(float 8 0) x))
+		(y (math-sqr z))
+		(xx (math-add x '(float (bigneg 491 194 356 2) -9)))
+		(a1 (math-poly-eval y
+				    '((float (bigneg 019 337 240) -15)
+				      (float (bigpos 174 520 457 2) -15)
+				      (float (bigneg 496 396 516 3) -14)
+				      (float 183105 -8)
+				      (float 1 0))))
+		(a2 (math-poly-eval y
+				    '((float (bigpos 412 787 105) -15)
+				      (float (bigneg 987 228 88) -14)
+				      (float (bigpos 096 199 449 8) -15)
+				      (float (bigneg 873 690 002 2) -13)
+				      (float (bigpos 995 499 687 4) -11))))
+		(sc (math-sin-cos-raw xx)))
+	   (if yflag
+	       (setq sc (cons (math-neg (cdr sc)) (car sc)))
+	     (if (math-negp x)
+		 (setq sc (cons (math-neg (car sc)) (math-neg (cdr sc))))))
+	   (math-mul (math-sqrt (math-div '(float (bigpos 722 619 636) -9) x))
+		     (math-sub (math-mul (cdr sc) a1)
+			       (math-mul (car sc) (math-mul z a2))))))
+	(t
+	 (let ((y (math-sqr x)))
+	   (math-mul
+	    x
+	    (math-div (math-poly-eval y
+				      '((float (bigneg 606 036 016 3) -8)
+					(float (bigpos 826 044 157) -4)
+					(float (bigneg 439 611 972 2) -3)
+					(float (bigpos 531 968 423 2) -1)
+					(float (bigneg 235 059 895 7) 0)
+					(float (bigpos 232 614 362 72) 0)))
+		      (math-poly-eval y
+				      '((float 1 0)
+					(float (bigpos 397 991 769 3) -7)
+					(float (bigpos 394 743 944 9) -5)
+					(float (bigpos 474 330 858 1) -2)
+					(float (bigpos 178 535 300 2) 0)
+					(float (bigpos 442 228 725 144)
+					       0))))))))
+)
+
+(defun calcFunc-besY (v x)
+  (math-inexact-result)
+  (or (math-numberp v) (math-reject-arg v 'numberp))
+  (or (math-numberp x) (math-reject-arg x 'numberp))
+  (let ((calc-internal-prec (min 8 calc-internal-prec)))
+    (math-with-extra-prec 3
+      (setq x (math-float (math-normalize x)))
+      (setq v (math-float (math-normalize v)))
+      (cond ((not (math-num-integerp v))
+	     (let ((sc (math-sin-cos-raw (math-mul v (math-pi)))))
+	       (math-div (math-sub (math-mul (calcFunc-besJ v x) (cdr sc))
+				   (calcFunc-besJ (math-neg v) x))
+			 (car sc))))
+	    ((math-negp (setq v (math-trunc v)))
+	     (if (math-oddp v)
+		 (math-neg (calcFunc-besY (math-neg v) x))
+	       (calcFunc-besY (math-neg v) x)))
+	    ((eq v 0)
+	     (math-besY0 x))
+	    ((eq v 1)
+	     (math-besY1 x))
+	    (t
+	     (let ((j 0)
+		   (bym (math-besY0 x))
+		   (by (math-besY1 x))
+		   (two-over-x (math-div 2 x))
+		   byp)
+	       (while (< (setq j (1+ j)) v)
+		 (setq byp (math-sub (math-mul (math-mul j two-over-x) by)
+				     bym)
+		       bym by
+		       by byp))
+	       by)))))
+)
+
+(defun math-besY0 (x)
+  (cond ((Math-lessp (math-abs-approx x) '(float 8 0))
+	 (let ((y (math-sqr x)))
+	   (math-add
+	    (math-div (math-poly-eval y
+				      '((float (bigpos 733 622 284 2) -7)
+					(float (bigneg 757 792 632 8) -5)
+					(float (bigpos 129 988 087 1) -2)
+					(float (bigneg 036 598 123 5) -1)
+					(float (bigpos 065 834 062 7) 0)
+					(float (bigneg 389 821 957 2) 0)))
+		      (math-poly-eval y
+				      '((float 1 0)
+					(float (bigpos 244 030 261 2) -7)
+					(float (bigpos 647 472 474) -4)
+					(float (bigpos 438 466 189 7) -3)
+					(float (bigpos 648 499 452 7) -1)
+					(float (bigpos 269 544 076 40) 0))))
+	    (math-mul '(float (bigpos 772 619 636) -9)
+		      (math-mul (math-besJ0 x) (math-ln-raw x))))))
+	((math-negp (calcFunc-re x))
+	 (math-add (math-besJ0 (math-neg x) t)
+		   (math-mul '(cplx 0 2)
+			     (math-besJ0 (math-neg x)))))
+	(t
+	 (math-besJ0 x t)))
+)
+
+(defun math-besY1 (x)
+  (cond ((Math-lessp (math-abs-approx x) '(float 8 0))
+	 (let ((y (math-sqr x)))
+	   (math-add
+	    (math-mul
+	     x
+	     (math-div (math-poly-eval y
+				       '((float (bigpos 935 937 511 8) -6)
+					 (float (bigneg 726 922 237 4) -3)
+					 (float (bigpos 551 264 349 7) -1)
+					 (float (bigneg 139 438 153 5) 1)
+					 (float (bigpos 439 527 127) 4)
+					 (float (bigneg 943 604 900 4) 3)))
+		       (math-poly-eval y
+				       '((float 1 0)
+					 (float (bigpos 885 632 549 3) -7)
+					 (float (bigpos 605 042 102) -3)
+					 (float (bigpos 002 904 245 2) -2)
+					 (float (bigpos 367 650 733 3) 0)
+					 (float (bigpos 664 419 244 4) 2)
+					 (float (bigpos 057 958 249) 5)))))
+	    (math-mul '(float (bigpos 772 619 636) -9)
+		      (math-sub (math-mul (math-besJ1 x) (math-ln-raw x))
+				(math-div 1 x))))))
+	((math-negp (calcFunc-re x))
+	 (math-neg
+	  (math-add (math-besJ1 (math-neg x) t)
+		    (math-mul '(cplx 0 2)
+			      (math-besJ1 (math-neg x))))))
+	(t
+	 (math-besJ1 x t)))
+)
+
+(defun math-poly-eval (x coefs)
+  (let ((accum (car coefs)))
+    (while (setq coefs (cdr coefs))
+      (setq accum (math-add (car coefs) (math-mul accum x))))
+    accum)
+)
+
+
+;;;; Bernoulli and Euler polynomials and numbers.
+
+(defun calcFunc-bern (n &optional x)
+  (if (and x (not (math-zerop x)))
+      (if (and calc-symbolic-mode (math-floatp x))
+	  (math-inexact-result)
+	(math-build-polynomial-expr (math-bernoulli-coefs n) x))
+    (or (math-num-natnump n) (math-reject-arg n 'natnump))
+    (if (consp n)
+	(progn
+	  (math-inexact-result)
+	  (math-float (math-bernoulli-number (math-trunc n))))
+      (math-bernoulli-number n)))
+)
+
+(defun calcFunc-euler (n &optional x)
+  (or (math-num-natnump n) (math-reject-arg n 'natnump))
+  (if x
+      (let* ((n1 (math-add n 1))
+	     (coefs (math-bernoulli-coefs n1))
+	     (fac (math-div (math-pow 2 n1) n1))
+	     (k -1)
+	     (x1 (math-div (math-add x 1) 2))
+	     (x2 (math-div x 2)))
+	(if (math-numberp x)
+	    (if (and calc-symbolic-mode (math-floatp x))
+		(math-inexact-result)
+	      (math-mul fac
+			(math-sub (math-build-polynomial-expr coefs x1)
+				  (math-build-polynomial-expr coefs x2))))
+	  (calcFunc-collect
+	   (math-reduce-vec
+	    'math-add
+	    (cons 'vec
+		  (mapcar (function
+			   (lambda (c)
+			     (setq k (1+ k))
+			     (math-mul (math-mul fac c)
+				       (math-sub (math-pow x1 k)
+						 (math-pow x2 k)))))
+			  coefs)))
+	   x)))
+    (math-mul (math-pow 2 n)
+	      (if (consp n)
+		  (progn
+		    (math-inexact-result)
+		    (calcFunc-euler n '(float 5 -1)))
+		(calcFunc-euler n '(frac 1 2)))))
+)
+
+(defun math-bernoulli-coefs (n)
+  (let* ((coefs (list (calcFunc-bern n)))
+	 (nn (math-trunc n))
+	 (k nn)
+	 (term nn)
+	 coef
+	 (calc-prefer-frac (or (integerp n) calc-prefer-frac)))
+    (while (>= (setq k (1- k)) 0)
+      (setq term (math-div term (- nn k))
+	    coef (math-mul term (math-bernoulli-number k))
+	    coefs (cons (if (consp n) (math-float coef) coef) coefs)
+	    term (math-mul term k)))
+    (nreverse coefs))
+)
+
+(defun math-bernoulli-number (n)
+  (if (= (% n 2) 1)
+      (if (= n 1)
+	  '(frac -1 2)
+	0)
+    (setq n (/ n 2))
+    (while (>= n math-bernoulli-cache-size)
+      (let* ((sum 0)
+	     (nk 1)     ; nk = n-k+1
+	     (fact 1)   ; fact = (n-k+1)!
+	     ofact
+	     (p math-bernoulli-b-cache)
+	     (calc-prefer-frac t))
+	(math-working "bernoulli B" (* 2 math-bernoulli-cache-size))
+	(while p
+	  (setq nk (+ nk 2)
+		ofact fact
+		fact (math-mul fact (* nk (1- nk)))
+		sum (math-add sum (math-div (car p) fact))
+		p (cdr p)))
+	(setq ofact (math-mul ofact (1- nk))
+	      sum (math-sub (math-div '(frac 1 2) ofact) sum)
+	      math-bernoulli-b-cache (cons sum math-bernoulli-b-cache)
+	      math-bernoulli-B-cache (cons (math-mul sum ofact)
+					   math-bernoulli-B-cache)
+	      math-bernoulli-cache-size (1+ math-bernoulli-cache-size))))
+    (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache))
+)
+
+;;;   Bn = n! bn
+;;;   bn = - sum_k=0^n-1 bk / (n-k+1)!
+
+;;; A faster method would be to use "tangent numbers", c.f., Concrete
+;;; Mathematics pg. 273.
+
+(setq math-bernoulli-b-cache '( (frac -174611
+				      (bigpos 0 200 291 698 662 857 802))
+				(frac 43867 (bigpos 0 944 170 217 94 109 5))
+				(frac -3617 (bigpos 0 880 842 622 670 10))
+				(frac 1 (bigpos 600 249 724 74))
+				(frac -691 (bigpos 0 368 674 307 1))
+				(frac 1 (bigpos 160 900 47))
+				(frac -1 (bigpos 600 209 1))
+				(frac 1 30240) (frac -1 720)
+				(frac 1 12) 1 ))
+
+(setq math-bernoulli-B-cache '( (frac -174611 330) (frac 43867 798)
+				(frac -3617 510) (frac 7 6) (frac -691 2730)
+				(frac 5 66) (frac -1 30) (frac 1 42)
+				(frac -1 30) (frac 1 6) 1 ))
+
+(setq math-bernoulli-cache-size 11)
+
+
+
+;;; Probability distributions.
+
+;;; Binomial.
+(defun calcFunc-utpb (x n p)
+  (if math-expand-formulas
+      (math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1)))
+    (calcFunc-betaI p x (math-add (math-sub n x) 1)))
+)
+(put 'calcFunc-utpb 'math-expandable t)
+
+(defun calcFunc-ltpb (x n p)
+  (math-sub 1 (calcFunc-utpb x n p))
+)
+(put 'calcFunc-ltpb 'math-expandable t)
+
+;;; Chi-square.
+(defun calcFunc-utpc (chisq v)
+  (if math-expand-formulas
+      (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2)))
+    (calcFunc-gammaQ (math-div v 2) (math-div chisq 2)))
+)
+(put 'calcFunc-utpc 'math-expandable t)
+
+(defun calcFunc-ltpc (chisq v)
+  (if math-expand-formulas
+      (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2)))
+    (calcFunc-gammaP (math-div v 2) (math-div chisq 2)))
+)
+(put 'calcFunc-ltpc 'math-expandable t)
+
+;;; F-distribution.
+(defun calcFunc-utpf (f v1 v2)
+  (if math-expand-formulas
+      (math-normalize (list 'calcFunc-betaI
+			    (list '/ v2 (list '+ v2 (list '* v1 f)))
+			    (list '/ v2 2)
+			    (list '/ v1 2)))
+    (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f)))
+		    (math-div v2 2)
+		    (math-div v1 2)))
+)
+(put 'calcFunc-utpf 'math-expandable t)
+
+(defun calcFunc-ltpf (f v1 v2)
+  (math-sub 1 (calcFunc-utpf f v1 v2))
+)
+(put 'calcFunc-ltpf 'math-expandable t)
+
+;;; Normal.
+(defun calcFunc-utpn (x mean sdev)
+  (if math-expand-formulas
+      (math-normalize
+       (list '/
+	     (list '+ 1
+		   (list 'calcFunc-erf
+			 (list '/ (list '- mean x)
+			       (list '* sdev (list 'calcFunc-sqrt 2)))))
+	     2))
+    (math-mul (math-add '(float 1 0)
+			(calcFunc-erf
+			 (math-div (math-sub mean x)
+				   (math-mul sdev (math-sqrt-2)))))
+	      '(float 5 -1)))
+)
+(put 'calcFunc-utpn 'math-expandable t)
+
+(defun calcFunc-ltpn (x mean sdev)
+  (if math-expand-formulas
+      (math-normalize
+       (list '/
+	     (list '+ 1
+		   (list 'calcFunc-erf
+			 (list '/ (list '- x mean)
+			       (list '* sdev (list 'calcFunc-sqrt 2)))))
+	     2))
+    (math-mul (math-add '(float 1 0)
+			(calcFunc-erf
+			 (math-div (math-sub x mean)
+				   (math-mul sdev (math-sqrt-2)))))
+	      '(float 5 -1)))
+)
+(put 'calcFunc-ltpn 'math-expandable t)
+
+;;; Poisson.
+(defun calcFunc-utpp (n x)
+  (if math-expand-formulas
+      (math-normalize (list 'calcFunc-gammaP x n))
+    (calcFunc-gammaP x n))
+)
+(put 'calcFunc-utpp 'math-expandable t)
+
+(defun calcFunc-ltpp (n x)
+  (if math-expand-formulas
+      (math-normalize (list 'calcFunc-gammaQ x n))
+    (calcFunc-gammaQ x n))
+)
+(put 'calcFunc-ltpp 'math-expandable t)
+
+;;; Student's t.  (As defined in Abramowitz & Stegun and Numerical Recipes.)
+(defun calcFunc-utpt (tt v)
+  (if math-expand-formulas
+      (math-normalize (list 'calcFunc-betaI
+			    (list '/ v (list '+ v (list '^ tt 2)))
+			    (list '/ v 2)
+			    '(float 5 -1)))
+    (calcFunc-betaI (math-div v (math-add v (math-sqr tt)))
+		    (math-div v 2)
+		    '(float 5 -1)))
+)
+(put 'calcFunc-utpt 'math-expandable t)
+
+(defun calcFunc-ltpt (tt v)
+  (math-sub 1 (calcFunc-utpt tt v))
+)
+(put 'calcFunc-ltpt 'math-expandable t)
+
+
+
+