Trailing whitepace deleted.

[emacs.git] / lisp / calc / calc-funcs.el

;;; calc-funcs.el --- well-known functions for Calc

;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc.

;; Author: David Gillespie <daveg@synaptics.com>
;; Maintainers: D. Goel <deego@gnufans.org>
;;              Colin Walters <walters@debian.org>

;; 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.

;;; Commentary:

;;; Code:

;; 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.

(defvar math-current-gamma-value nil)
(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))))))

(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.

(defvar math-current-beta-value nil)
(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))))))))

(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))))))

(defvar 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 ))

(defvar 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 ))

(defvar math-bernoulli-cache-size 11)
(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.


;;; 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)


;;; calc-funcs.el ends here