root/trunk/lisp/calc/calc-bin.el

;;; calc-bin.el --- binary functions for Calc

;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
;;   2005, 2006, 2007, 2008 Free Software Foundation, Inc.

;; Author: David Gillespie <daveg@synaptics.com>
;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>

;; This file is part of GNU Emacs.

;; GNU Emacs is free software; you can redistribute it and/or modify
;; it under the terms of the GNU General Public License as published by
;; the Free Software Foundation; either version 3, or (at your option)
;; any later version.

;; GNU Emacs is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
;; GNU General Public License for more details.

;; You should have received a copy of the GNU General Public License
;; along with GNU Emacs; see the file COPYING.  If not, write to the
;; Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
;; Boston, MA 02110-1301, USA.

;;; Commentary:

;;; Code:

;; This file is autoloaded from calc-ext.el.

(require 'calc-ext)
(require 'calc-macs)

;;; b-prefix binary commands.

(defun calc-and (n)
  (interactive "P")
  (calc-slow-wrapper
   (calc-enter-result 2 "and"
                      (append '(calcFunc-and)
                              (calc-top-list-n 2)
                              (and n (list (prefix-numeric-value n)))))))

(defun calc-or (n)
  (interactive "P")
  (calc-slow-wrapper
   (calc-enter-result 2 "or"
                      (append '(calcFunc-or)
                              (calc-top-list-n 2)
                              (and n (list (prefix-numeric-value n)))))))

(defun calc-xor (n)
  (interactive "P")
  (calc-slow-wrapper
   (calc-enter-result 2 "xor"
                      (append '(calcFunc-xor)
                              (calc-top-list-n 2)
                              (and n (list (prefix-numeric-value n)))))))

(defun calc-diff (n)
  (interactive "P")
  (calc-slow-wrapper
   (calc-enter-result 2 "diff"
                      (append '(calcFunc-diff)
                              (calc-top-list-n 2)
                              (and n (list (prefix-numeric-value n)))))))

(defun calc-not (n)
  (interactive "P")
  (calc-slow-wrapper
   (calc-enter-result 1 "not"
                      (append '(calcFunc-not)
                              (calc-top-list-n 1)
                              (and n (list (prefix-numeric-value n)))))))

(defun calc-lshift-binary (n)
  (interactive "P")
  (calc-slow-wrapper
   (let ((hyp (if (calc-is-hyperbolic) 2 1)))
     (calc-enter-result hyp "lsh"
                        (append '(calcFunc-lsh)
                                (calc-top-list-n hyp)
                                (and n (list (prefix-numeric-value n))))))))

(defun calc-rshift-binary (n)
  (interactive "P")
  (calc-slow-wrapper
   (let ((hyp (if (calc-is-hyperbolic) 2 1)))
     (calc-enter-result hyp "rsh"
                        (append '(calcFunc-rsh)
                                (calc-top-list-n hyp)
                                (and n (list (prefix-numeric-value n))))))))

(defun calc-lshift-arith (n)
  (interactive "P")
  (calc-slow-wrapper
   (let ((hyp (if (calc-is-hyperbolic) 2 1)))
     (calc-enter-result hyp "ash"
                        (append '(calcFunc-ash)
                                (calc-top-list-n hyp)
                                (and n (list (prefix-numeric-value n))))))))

(defun calc-rshift-arith (n)
  (interactive "P")
  (calc-slow-wrapper
   (let ((hyp (if (calc-is-hyperbolic) 2 1)))
     (calc-enter-result hyp "rash"
                        (append '(calcFunc-rash)
                                (calc-top-list-n hyp)
                                (and n (list (prefix-numeric-value n))))))))

(defun calc-rotate-binary (n)
  (interactive "P")
  (calc-slow-wrapper
   (let ((hyp (if (calc-is-hyperbolic) 2 1)))
     (calc-enter-result hyp "rot"
                        (append '(calcFunc-rot)
                                (calc-top-list-n hyp)
                                (and n (list (prefix-numeric-value n))))))))

(defun calc-clip (n)
  (interactive "P")
  (calc-slow-wrapper
   (calc-enter-result 1 "clip"
                      (append '(calcFunc-clip)
                              (calc-top-list-n 1)
                              (and n (list (prefix-numeric-value n)))))))

(defun calc-word-size (n)
  (interactive "P")
  (calc-wrapper
   (or n (setq n (read-string (format "Binary word size: (default %d) "
                                      calc-word-size))))
   (setq n (if (stringp n)
               (if (equal n "")
                   calc-word-size
                 (if (string-match "\\`[-+]?[0-9]+\\'" n)
                     (string-to-number n)
                   (error "Expected an integer")))
             (prefix-numeric-value n)))
   (or (= n calc-word-size)
       (if (> (math-abs n) 100)
           (calc-change-mode 'calc-word-size n calc-leading-zeros)
         (calc-change-mode '(calc-word-size calc-previous-modulo)
                           (list n (math-power-of-2 (math-abs n)))
                           calc-leading-zeros)))
   (if (< n 0)
       (message "Binary word size is %d bits (2's complement)" (- n))
     (message "Binary word size is %d bits" n))))





;;; d-prefix mode commands.

(defun calc-radix (n)
  (interactive "NDisplay radix (2-36): ")
  (calc-wrapper
   (if (and (>= n 2) (<= n 36))
       (progn
         (calc-change-mode 'calc-number-radix n t)
         ;; also change global value so minibuffer sees it
         (setq-default calc-number-radix calc-number-radix))
     (setq n calc-number-radix))
   (message "Number radix is %d" n)))

(defun calc-decimal-radix ()
  (interactive)
  (calc-radix 10))

(defun calc-binary-radix ()
  (interactive)
  (calc-radix 2))

(defun calc-octal-radix ()
  (interactive)
  (calc-radix 8))

(defun calc-hex-radix ()
  (interactive)
  (calc-radix 16))

(defun calc-leading-zeros (n)
  (interactive "P")
  (calc-wrapper
   (if (calc-change-mode 'calc-leading-zeros n t t)
       (message "Zero-padding integers to %d digits (assuming radix %d)"
                (let* ((calc-internal-prec 6))
                  (math-compute-max-digits (math-abs calc-word-size)
                                           calc-number-radix))
                calc-number-radix)
     (message "Omitting leading zeros on integers"))))


(defvar math-power-of-2-cache (list 1 2 4 8 16 32 64 128 256 512 1024))
(defvar math-big-power-of-2-cache nil)
(defun math-power-of-2 (n)    ;  [I I] [Public]
  (if (and (natnump n) (<= n 100))
      (or (nth n math-power-of-2-cache)
          (let* ((i (length math-power-of-2-cache))
                 (val (nth (1- i) math-power-of-2-cache)))
            (while (<= i n)
              (setq val (math-mul val 2)
                    math-power-of-2-cache (nconc math-power-of-2-cache
                                                 (list val))
                    i (1+ i)))
            val))
    (let ((found (assq n math-big-power-of-2-cache)))
      (if found
          (cdr found)
        (let ((po2 (math-ipow 2 n)))
          (setq math-big-power-of-2-cache
                (cons (cons n po2) math-big-power-of-2-cache))
          po2)))))

(defun math-integer-log2 (n)    ; [I I] [Public]
  (let ((i 0)
        (p math-power-of-2-cache)
        val)
    (while (and p (Math-natnum-lessp (setq val (car p)) n))
      (setq p (cdr p)
            i (1+ i)))
    (if p
        (and (equal val n)
             i)
      (while (Math-natnum-lessp
              (prog1
                  (setq val (math-mul val 2))
                (setq math-power-of-2-cache (nconc math-power-of-2-cache
                                                   (list val))))
              n)
        (setq i (1+ i)))
      (and (equal val n)
           i))))




;;; Bitwise operations.

(defun calcFunc-and (a b &optional w)   ; [I I I] [Public]
  (cond ((Math-messy-integerp w)
         (calcFunc-and a b (math-trunc w)))
        ((and w (not (integerp w)))
         (math-reject-arg w 'fixnump))
        ((and (integerp a) (integerp b))
         (math-clip (logand a b) w))
        ((or (eq (car-safe a) 'mod) (eq (car-safe b) 'mod))
         (math-binary-modulo-args 'calcFunc-and a b w))
        ((not (Math-num-integerp a))
         (math-reject-arg a 'integerp))
        ((not (Math-num-integerp b))
         (math-reject-arg b 'integerp))
        (t (math-clip (cons 'bigpos
                            (math-and-bignum (math-binary-arg a w)
                                             (math-binary-arg b w)))
                      w))))

(defun math-binary-arg (a w)
  (if (not (Math-integerp a))
      (setq a (math-trunc a)))
  (if (Math-integer-negp a)
      (math-not-bignum (cdr (math-bignum-test (math-sub -1 a)))
                       (math-abs (if w (math-trunc w) calc-word-size)))
    (cdr (Math-bignum-test a))))

(defun math-binary-modulo-args (f a b w)
  (let (mod)
    (if (eq (car-safe a) 'mod)
        (progn
          (setq mod (nth 2 a)
                a (nth 1 a))
          (if (eq (car-safe b) 'mod)
              (if (equal mod (nth 2 b))
                  (setq b (nth 1 b))
                (math-reject-arg b "*Inconsistent modulos"))))
      (setq mod (nth 2 b)
            b (nth 1 b)))
    (if (Math-messy-integerp mod)
        (setq mod (math-trunc mod))
      (or (Math-integerp mod)
          (math-reject-arg mod 'integerp)))
    (let ((bits (math-integer-log2 mod)))
      (if bits
          (if w
              (if (/= w bits)
                  (calc-record-why
                   "*Warning: Modulo inconsistent with word size"))
            (setq w bits))
        (calc-record-why "*Warning: Modulo is not a power of 2"))
      (math-make-mod (if b
                         (funcall f a b w)
                       (funcall f a w))
                     mod))))

(defun math-and-bignum (a b)   ; [l l l]
  (and a b
       (let ((qa (math-div-bignum-digit a 512))
             (qb (math-div-bignum-digit b 512)))
         (math-mul-bignum-digit (math-and-bignum (math-norm-bignum (car qa))
                                                  (math-norm-bignum (car qb)))
                                 512
                                 (logand (cdr qa) (cdr qb))))))

(defun calcFunc-or (a b &optional w)   ; [I I I] [Public]
  (cond ((Math-messy-integerp w)
         (calcFunc-or a b (math-trunc w)))
        ((and w (not (integerp w)))
         (math-reject-arg w 'fixnump))
        ((and (integerp a) (integerp b))
         (math-clip (logior a b) w))
        ((or (eq (car-safe a) 'mod) (eq (car-safe b) 'mod))
         (math-binary-modulo-args 'calcFunc-or a b w))
        ((not (Math-num-integerp a))
         (math-reject-arg a 'integerp))
        ((not (Math-num-integerp b))
         (math-reject-arg b 'integerp))
        (t (math-clip (cons 'bigpos
                            (math-or-bignum (math-binary-arg a w)
                                            (math-binary-arg b w)))
                      w))))

(defun math-or-bignum (a b)   ; [l l l]
  (and (or a b)
       (let ((qa (math-div-bignum-digit a 512))
             (qb (math-div-bignum-digit b 512)))
         (math-mul-bignum-digit (math-or-bignum (math-norm-bignum (car qa))
                                                 (math-norm-bignum (car qb)))
                                 512
                                 (logior (cdr qa) (cdr qb))))))

(defun calcFunc-xor (a b &optional w)   ; [I I I] [Public]
  (cond ((Math-messy-integerp w)
         (calcFunc-xor a b (math-trunc w)))
        ((and w (not (integerp w)))
         (math-reject-arg w 'fixnump))
        ((and (integerp a) (integerp b))
         (math-clip (logxor a b) w))
        ((or (eq (car-safe a) 'mod) (eq (car-safe b) 'mod))
         (math-binary-modulo-args 'calcFunc-xor a b w))
        ((not (Math-num-integerp a))
         (math-reject-arg a 'integerp))
        ((not (Math-num-integerp b))
         (math-reject-arg b 'integerp))
        (t (math-clip (cons 'bigpos
                            (math-xor-bignum (math-binary-arg a w)
                                             (math-binary-arg b w)))
                      w))))

(defun math-xor-bignum (a b)   ; [l l l]
  (and (or a b)
       (let ((qa (math-div-bignum-digit a 512))
             (qb (math-div-bignum-digit b 512)))
         (math-mul-bignum-digit (math-xor-bignum (math-norm-bignum (car qa))
                                                  (math-norm-bignum (car qb)))
                                 512
                                 (logxor (cdr qa) (cdr qb))))))

(defun calcFunc-diff (a b &optional w)   ; [I I I] [Public]
  (cond ((Math-messy-integerp w)
         (calcFunc-diff a b (math-trunc w)))
        ((and w (not (integerp w)))
         (math-reject-arg w 'fixnump))
        ((and (integerp a) (integerp b))
         (math-clip (logand a (lognot b)) w))
        ((or (eq (car-safe a) 'mod) (eq (car-safe b) 'mod))
         (math-binary-modulo-args 'calcFunc-diff a b w))
        ((not (Math-num-integerp a))
         (math-reject-arg a 'integerp))
        ((not (Math-num-integerp b))
         (math-reject-arg b 'integerp))
        (t (math-clip (cons 'bigpos
                            (math-diff-bignum (math-binary-arg a w)
                                              (math-binary-arg b w)))
                      w))))

(defun math-diff-bignum (a b)   ; [l l l]
  (and a
       (let ((qa (math-div-bignum-digit a 512))
             (qb (math-div-bignum-digit b 512)))
         (math-mul-bignum-digit (math-diff-bignum (math-norm-bignum (car qa))
                                                   (math-norm-bignum (car qb)))
                                 512
                                 (logand (cdr qa) (lognot (cdr qb)))))))

(defun calcFunc-not (a &optional w)   ; [I I] [Public]
  (cond ((Math-messy-integerp w)
         (calcFunc-not a (math-trunc w)))
        ((eq (car-safe a) 'mod)
         (math-binary-modulo-args 'calcFunc-not a nil w))
        ((and w (not (integerp w)))
         (math-reject-arg w 'fixnump))
        ((not (Math-num-integerp a))
         (math-reject-arg a 'integerp))
        ((< (or w (setq w calc-word-size)) 0)
         (math-clip (calcFunc-not a (- w)) w))
        (t (math-normalize
            (cons 'bigpos
                  (math-not-bignum (math-binary-arg a w)
                                   w))))))

(defun math-not-bignum (a w)   ; [l l]
  (let ((q (math-div-bignum-digit a 512)))
    (if (<= w 9)
        (list (logand (lognot (cdr q))
                      (1- (lsh 1 w))))
      (math-mul-bignum-digit (math-not-bignum (math-norm-bignum (car q))
                                               (- w 9))
                              512
                              (logxor (cdr q) 511)))))

(defun calcFunc-lsh (a &optional n w)   ; [I I] [Public]
  (setq a (math-trunc a)
        n (if n (math-trunc n) 1))
  (if (eq (car-safe a) 'mod)
      (math-binary-modulo-args 'calcFunc-lsh a n w)
    (setq w (if w (math-trunc w) calc-word-size))
    (or (integerp w)
        (math-reject-arg w 'fixnump))
    (or (Math-integerp a)
        (math-reject-arg a 'integerp))
    (or (Math-integerp n)
        (math-reject-arg n 'integerp))
    (if (< w 0)
        (math-clip (calcFunc-lsh a n (- w)) w)
      (if (Math-integer-negp a)
          (setq a (math-clip a w)))
      (cond ((or (Math-lessp n (- w))
                 (Math-lessp w n))
             0)
            ((< n 0)
             (math-quotient (math-clip a w) (math-power-of-2 (- n))))
            (t
             (math-clip (math-mul a (math-power-of-2 n)) w))))))

(defun calcFunc-rsh (a &optional n w)   ; [I I] [Public]
  (calcFunc-lsh a (math-neg (or n 1)) w))

(defun calcFunc-ash (a &optional n w)   ; [I I] [Public]
  (if (or (null n)
          (not (Math-negp n)))
      (calcFunc-lsh a n w)
    (setq a (math-trunc a)
          n (if n (math-trunc n) 1))
    (if (eq (car-safe a) 'mod)
        (math-binary-modulo-args 'calcFunc-ash a n w)
      (setq w (if w (math-trunc w) calc-word-size))
      (or (integerp w)
          (math-reject-arg w 'fixnump))
      (or (Math-integerp a)
          (math-reject-arg a 'integerp))
      (or (Math-integerp n)
          (math-reject-arg n 'integerp))
      (if (< w 0)
          (math-clip (calcFunc-ash a n (- w)) w)
        (if (Math-integer-negp a)
            (setq a (math-clip a w)))
        (let ((two-to-sizem1 (math-power-of-2 (1- w)))
              (sh (calcFunc-lsh a n w)))
          (cond ((Math-natnum-lessp a two-to-sizem1)
                 sh)
                ((Math-lessp n (- 1 w))
                 (math-add (math-mul two-to-sizem1 2) -1))
                (t (let ((two-to-n (math-power-of-2 (- n))))
                     (math-add (calcFunc-lsh (math-add two-to-n -1)
                                             (+ w n) w)
                               sh)))))))))

(defun calcFunc-rash (a &optional n w)   ; [I I] [Public]
  (calcFunc-ash a (math-neg (or n 1)) w))

(defun calcFunc-rot (a &optional n w)   ; [I I] [Public]
  (setq a (math-trunc a)
        n (if n (math-trunc n) 1))
  (if (eq (car-safe a) 'mod)
      (math-binary-modulo-args 'calcFunc-rot a n w)
    (setq w (if w (math-trunc w) calc-word-size))
    (or (integerp w)
        (math-reject-arg w 'fixnump))
    (or (Math-integerp a)
        (math-reject-arg a 'integerp))
    (or (Math-integerp n)
        (math-reject-arg n 'integerp))
    (if (< w 0)
        (math-clip (calcFunc-rot a n (- w)) w)
      (if (Math-integer-negp a)
          (setq a (math-clip a w)))
      (cond ((or (Math-integer-negp n)
                 (not (Math-natnum-lessp n w)))
             (calcFunc-rot a (math-mod n w) w))
            (t
             (math-add (calcFunc-lsh a (- n w) w)
                       (calcFunc-lsh a n w)))))))

(defun math-clip (a &optional w)   ; [I I] [Public]
  (cond ((Math-messy-integerp w)
         (math-clip a (math-trunc w)))
        ((eq (car-safe a) 'mod)
         (math-binary-modulo-args 'math-clip a nil w))
        ((and w (not (integerp w)))
         (math-reject-arg w 'fixnump))
        ((not (Math-num-integerp a))
         (math-reject-arg a 'integerp))
        ((< (or w (setq w calc-word-size)) 0)
         (setq a (math-clip a (- w)))
         (if (Math-natnum-lessp a (math-power-of-2 (- -1 w)))
             a
           (math-sub a (math-power-of-2 (- w)))))
        ((Math-negp a)
         (math-normalize (cons 'bigpos (math-binary-arg a w))))
        ((and (integerp a) (< a 1000000))
         (if (>= w 20)
             a
           (logand a (1- (lsh 1 w)))))
        (t
         (math-normalize
          (cons 'bigpos
                (math-clip-bignum (cdr (math-bignum-test (math-trunc a)))
                                  w))))))

(defalias 'calcFunc-clip 'math-clip)

(defun math-clip-bignum (a w)   ; [l l]
  (let ((q (math-div-bignum-digit a 512)))
    (if (<= w 9)
        (list (logand (cdr q)
                      (1- (lsh 1 w))))
      (math-mul-bignum-digit (math-clip-bignum (math-norm-bignum (car q))
                                                (- w 9))
                              512
                              (cdr q)))))

(defvar math-max-digits-cache nil)
(defun math-compute-max-digits (w r)
  (let* ((pair (+ (* r 100000) w))
         (res (assq pair math-max-digits-cache)))
    (if res
        (cdr res)
      (let* ((calc-command-flags nil)
             (digs (math-ceiling (math-div w (math-real-log2 r)))))
        (setq math-max-digits-cache (cons (cons pair digs)
                                          math-max-digits-cache))
        digs))))

(defvar math-log2-cache (list '(2 . 1)
                              '(4 . 2)
                              '(8 . 3)
                              '(10 . (float 332193 -5))
                              '(16 . 4)
                              '(32 . 5)))
(defun math-real-log2 (x)   ;;; calc-internal-prec must be 6
  (let ((res (assq x math-log2-cache)))
    (if res
        (cdr res)
      (let* ((calc-symbolic-mode nil)
             (calc-display-working-message nil)
             (log (calcFunc-log x 2)))
        (setq math-log2-cache (cons (cons x log) math-log2-cache))
        log))))

(defconst math-radix-digits ["0" "1" "2" "3" "4" "5" "6" "7" "8" "9"
                             "A" "B" "C" "D" "E" "F" "G" "H" "I" "J"
                             "K" "L" "M" "N" "O" "P" "Q" "R" "S" "T"
                             "U" "V" "W" "X" "Y" "Z"])

(defsubst math-format-radix-digit (a)   ; [X D]
  (aref math-radix-digits a))

(defun math-format-radix (a)   ; [X S]
  (if (< a calc-number-radix)
      (if (< a 0)
          (concat "-" (math-format-radix (- a)))
        (math-format-radix-digit a))
    (let ((s ""))
      (while (> a 0)
        (setq s (concat (math-format-radix-digit (% a calc-number-radix)) s)
              a (/ a calc-number-radix)))
      s)))

(defconst math-binary-digits ["000" "001" "010" "011"
                              "100" "101" "110" "111"])
(defun math-format-binary (a)   ; [X S]
  (if (< a 8)
      (if (< a 0)
          (concat "-" (math-format-binary (- a)))
        (math-format-radix a))
    (let ((s ""))
      (while (> a 7)
        (setq s (concat (aref math-binary-digits (% a 8)) s)
              a (/ a 8)))
      (concat (math-format-radix a) s))))

(defun math-format-bignum-radix (a)   ; [X L]
  (cond ((null a) "0")
        ((and (null (cdr a))
              (< (car a) calc-number-radix))
         (math-format-radix-digit (car a)))
        (t
         (let ((q (math-div-bignum-digit a calc-number-radix)))
           (concat (math-format-bignum-radix (math-norm-bignum (car q)))
                   (math-format-radix-digit (cdr q)))))))

(defun math-format-bignum-binary (a)   ; [X L]
  (cond ((null a) "0")
        ((null (cdr a))
         (math-format-binary (car a)))
        (t
         (let ((q (math-div-bignum-digit a 512)))
           (concat (math-format-bignum-binary (math-norm-bignum (car q)))
                   (aref math-binary-digits (/ (cdr q) 64))
                   (aref math-binary-digits (% (/ (cdr q) 8) 8))
                   (aref math-binary-digits (% (cdr q) 8)))))))

(defun math-format-bignum-octal (a)   ; [X L]
  (cond ((null a) "0")
        ((null (cdr a))
         (math-format-radix (car a)))
        (t
         (let ((q (math-div-bignum-digit a 512)))
           (concat (math-format-bignum-octal (math-norm-bignum (car q)))
                   (math-format-radix-digit (/ (cdr q) 64))
                   (math-format-radix-digit (% (/ (cdr q) 8) 8))
                   (math-format-radix-digit (% (cdr q) 8)))))))

(defun math-format-bignum-hex (a)   ; [X L]
  (cond ((null a) "0")
        ((null (cdr a))
         (math-format-radix (car a)))
        (t
         (let ((q (math-div-bignum-digit a 256)))
           (concat (math-format-bignum-hex (math-norm-bignum (car q)))
                   (math-format-radix-digit (/ (cdr q) 16))
                   (math-format-radix-digit (% (cdr q) 16)))))))

;;; Decompose into integer and fractional parts, without depending
;;; on calc-internal-prec.
(defun math-float-parts (a need-frac)    ; returns ( int frac fracdigs )
  (if (>= (nth 2 a) 0)
      (list (math-scale-rounding (nth 1 a) (nth 2 a)) '(float 0 0) 0)
    (let* ((d (math-numdigs (nth 1 a)))
           (n (- (nth 2 a))))
      (if need-frac
          (if (>= n d)
              (list 0 a n)
            (let ((qr (math-idivmod (nth 1 a) (math-scale-int 1 n))))
              (list (car qr) (math-make-float (cdr qr) (- n)) n)))
        (list (math-scale-rounding (nth 1 a) (nth 2 a))
              '(float 0 0) 0)))))

(defun math-format-radix-float (a prec)
  (let ((fmt (car calc-float-format))
        (figs (nth 1 calc-float-format))
        (point calc-point-char)
        (str nil)
        pos)
    (if (eq fmt 'fix)
        (let* ((afigs (math-abs figs))
               (fp (math-float-parts a (> afigs 0)))
               (calc-internal-prec (+ 3 (max (nth 2 fp)
                                             (math-convert-radix-digits
                                              afigs t))))
               (int (car fp))
               (frac (math-round (math-mul (math-normalize (nth 1 fp))
                                           (math-radix-float-power afigs)))))
          (if (not (and (math-zerop frac) (math-zerop int) (< figs 0)))
              (let ((math-radix-explicit-format nil))
                (let ((calc-group-digits nil))
                  (setq str (if (> afigs 0) (math-format-number frac) ""))
                  (if (< (length str) afigs)
                      (setq str (concat (make-string (- afigs (length str)) ?0)
                                        str))
                    (if (> (length str) afigs)
                        (setq str (substring str 1)
                              int (math-add int 1))))
                  (setq str (concat (math-format-number int) point str)))
                (when calc-group-digits
                  (setq str (math-group-float str))))
            (setq figs 0))))
    (or str
        (let* ((prec calc-internal-prec)
               (afigs (if (> figs 0)
                          figs
                        (max 1 (+ figs
                                  (1- (math-convert-radix-digits
                                       (max prec
                                            (math-numdigs (nth 1 a)))))))))
               (calc-internal-prec (+ 3 (math-convert-radix-digits afigs t)))
               (explo -1) (vlo (math-radix-float-power explo))
               (exphi 1) (vhi (math-radix-float-power exphi))
               expmid vmid eadj)
          (setq a (math-normalize a))
          (if (Math-zerop a)
              (setq explo 0)
            (if (math-lessp-float '(float 1 0) a)
                (while (not (math-lessp-float a vhi))
                  (setq explo exphi vlo vhi
                        exphi (math-mul exphi 2)
                        vhi (math-radix-float-power exphi)))
              (while (math-lessp-float a vlo)
                (setq exphi explo vhi vlo
                      explo (math-mul explo 2)
                      vlo (math-radix-float-power explo))))
            (while (not (eq (math-sub exphi explo) 1))
              (setq expmid (math-div2 (math-add explo exphi))
                    vmid (math-radix-float-power expmid))
              (if (math-lessp-float a vmid)
                  (setq exphi expmid vhi vmid)
                (setq explo expmid vlo vmid)))
            (setq a (math-div-float a vlo)))
          (let* ((sc (math-round (math-mul a (math-radix-float-power
                                              (1- afigs)))))
                 (math-radix-explicit-format nil))
            (let ((calc-group-digits nil))
              (setq str (math-format-number sc))))
          (if (> (length str) afigs)
              (setq str (substring str 0 -1)
                    explo (1+ explo)))
          (if (and (eq fmt 'float)
                   (math-lessp explo (+ (if (= figs 0)
                                            (1- (math-convert-radix-digits
                                                 prec))
                                          afigs)
                                        calc-display-sci-high 1))
                   (math-lessp calc-display-sci-low explo))
              (let ((dpos (1+ explo)))
                (cond ((<= dpos 0)
                       (setq str (concat "0" point (make-string (- dpos) ?0)
                                         str)))
                      ((> dpos (length str))
                       (setq str (concat str (make-string (- dpos (length str))
                                                          ?0) point)))
                      (t
                       (setq str (concat (substring str 0 dpos) point
                                         (substring str dpos)))))
                (setq explo nil))
            (setq eadj (if (eq fmt 'eng)
                           (min (math-mod explo 3) (length str))
                         0)
                  str (concat (substring str 0 (1+ eadj)) point
                              (substring str (1+ eadj)))))
          (setq pos (length str))
          (while (eq (aref str (1- pos)) ?0) (setq pos (1- pos)))
          (and explo (eq (aref str (1- pos)) ?.) (setq pos (1- pos)))
          (setq str (substring str 0 pos))
          (when calc-group-digits
            (setq str (math-group-float str)))
          (if explo
              (let ((estr (let ((calc-number-radix 10)
                                (calc-group-digits nil))
                            (math-format-number
                             (math-sub explo eadj)))))
                (setq str (if (or (memq calc-language '(math maple))
                                  (> calc-number-radix 14))
                              (format "%s*%d.^%s" str calc-number-radix estr)
                            (format "%se%s" str estr)))))))
    str))

(defvar math-radix-digits-cache nil)

(defun math-convert-radix-digits (n &optional to-dec)
  (let ((key (cons n (cons to-dec calc-number-radix))))
    (or (cdr (assoc key math-radix-digits-cache))
        (let* ((calc-internal-prec 6)
               (log (math-div (math-real-log2 calc-number-radix)
                              '(float 332193 -5))))
          (cdr (car (setq math-radix-digits-cache
                          (cons (cons key (math-ceiling (if to-dec
                                                            (math-mul n log)
                                                          (math-div n log))))
                                math-radix-digits-cache))))))))

(defvar math-radix-float-cache-tag nil)
(defvar math-radix-float-cache)

(defun math-radix-float-power (n)
  (if (eq n 0)
      '(float 1 0)
    (or (and (eq calc-number-radix (car math-radix-float-cache-tag))
             (<= calc-internal-prec (cdr math-radix-float-cache-tag)))
        (setq math-radix-float-cache-tag (cons calc-number-radix
                                               calc-internal-prec)
              math-radix-float-cache nil))
    (math-normalize
     (or (cdr (assoc n math-radix-float-cache))
         (cdr (car (setq math-radix-float-cache
                         (cons (cons
                                n
                                (let ((calc-internal-prec
                                       (cdr math-radix-float-cache-tag)))
                                  (if (math-negp n)
                                      (math-div-float '(float 1 0)
                                                      (math-radix-float-power
                                                       (math-neg n)))
                                    (math-mul-float (math-sqr-float
                                                     (math-radix-float-power
                                                      (math-div2 n)))