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+;; Copyright (C) 1986 Free Software Foundation, Inc.
+;; Author Bill Rosenblatt
+
+;; 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 1, 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, 675 Mass Ave, Cambridge, MA 02139, USA.
+
+;; Floating point arithmetic package.
+;;
+;; Floating point numbers are represented by dot-pairs (mant . exp)
+;; where mant is the 24-bit signed integral mantissa and exp is the
+;; base 2 exponent.
+;;
+;; Emacs LISP supports a 24-bit signed integer data type, which has a
+;; range of -(2**23) to +(2**23)-1, or -8388608 to 8388607 decimal.
+;; This gives six significant decimal digit accuracy.  Exponents can
+;; be anything in the range -(2**23) to +(2**23)-1.
+;;
+;; User interface:
+;; function f converts from integer to floating point
+;; function string-to-float converts from string to floating point
+;; function fint converts a floating point to integer (with truncation)
+;; function float-to-string converts from floating point to string
+;;                   
+;; Caveats:
+;; -  Exponents outside of the range of +/-100 or so will cause certain 
+;;    functions (especially conversion routines) to take forever.
+;; -  Very little checking is done for fixed point overflow/underflow.
+;; -  No checking is done for over/underflow of the exponent
+;;    (hardly necessary when exponent can be 2**23).
+;; 
+;;
+;; Bill Rosenblatt
+;; June 20, 1986
+;;
+
+;; fundamental implementation constants
+(defconst exp-base 2
+  "Base of exponent in this floating point representation.")
+
+(defconst mantissa-bits 24
+  "Number of significant bits in this floating point representation.")
+
+(defconst decimal-digits 6
+  "Number of decimal digits expected to be accurate.")
+
+(defconst expt-digits 2
+  "Maximum permitted digits in a scientific notation exponent.")
+
+;; other constants
+(defconst maxbit (1- mantissa-bits)
+  "Number of highest bit")
+
+(defconst mantissa-maxval (1- (ash 1 maxbit))
+  "Maximum permissable value of mantissa")
+
+(defconst mantissa-minval (ash 1 maxbit)
+  "Minimum permissable value of mantissa")
+
+(defconst floating-point-regexp
+  "^[ \t]*\\(-?\\)\\([0-9]*\\)\
+\\(\\.\\([0-9]*\\)\\|\\)\
+\\(\\(\\([Ee]\\)\\(-?\\)\\([0-9][0-9]*\\)\\)\\|\\)[ \t]*$"
+  "Regular expression to match floating point numbers.  Extract matches:
+1 - minus sign
+2 - integer part
+4 - fractional part
+8 - minus sign for power of ten
+9 - power of ten
+")
+
+(defconst high-bit-mask (ash 1 maxbit)
+  "Masks all bits except the high-order (sign) bit.")
+
+(defconst second-bit-mask (ash 1 (1- maxbit))
+  "Masks all bits except the highest-order magnitude bit")
+
+;; various useful floating point constants
+(setq _f0 '(0 . 1))
+
+(setq _f1/2 '(4194304 . -23))
+
+(setq _f1 '(4194304 . -22))
+
+(setq _f10 '(5242880 . -19))
+
+;; support for decimal conversion routines
+(setq powers-of-10 (make-vector (1+ decimal-digits) _f1))
+(aset powers-of-10 1 _f10)
+(aset powers-of-10 2 '(6553600 . -16))
+(aset powers-of-10 3 '(8192000 . -13))
+(aset powers-of-10 4 '(5120000 . -9))
+(aset powers-of-10 5 '(6400000 . -6))
+(aset powers-of-10 6 '(8000000 . -3))
+
+(setq all-decimal-digs-minval (aref powers-of-10 (1- decimal-digits))
+      highest-power-of-10 (aref powers-of-10 decimal-digits))
+
+(defun fashl (fnum)                     ; floating-point arithmetic shift left
+  (cons (ash (car fnum) 1) (1- (cdr fnum))))
+
+(defun fashr (fnum)                     ; floating point arithmetic shift right
+  (cons (ash (car fnum) -1) (1+ (cdr fnum))))
+
+(defun normalize (fnum)
+  (if (> (car fnum) 0)                  ; make sure next-to-highest bit is set
+      (while (zerop (logand (car fnum) second-bit-mask))
+        (setq fnum (fashl fnum)))
+    (if (< (car fnum) 0)                ; make sure highest bit is set
+        (while (zerop (logand (car fnum) high-bit-mask))
+          (setq fnum (fashl fnum)))
+      (setq fnum _f0)))                 ; "standard 0"
+  fnum)
+      
+(defun abs (n)                          ; integer absolute value
+  (if (>= n 0) n (- n)))
+
+(defun fabs (fnum)                      ; re-normalize after taking abs value
+  (normalize (cons (abs (car fnum)) (cdr fnum))))
+
+(defun xor (a b)                        ; logical exclusive or
+  (and (or a b) (not (and a b))))
+
+(defun same-sign (a b)                  ; two f-p numbers have same sign?
+  (not (xor (natnump (car a)) (natnump (car b)))))
+
+(defun extract-match (str i)            ; used after string-match
+  (condition-case ()
+      (substring str (match-beginning i) (match-end i))
+    (error "")))
+
+;; support for the multiplication function
+(setq halfword-bits (/ mantissa-bits 2) ; bits in a halfword
+      masklo (1- (ash 1 halfword-bits)) ; isolate the lower halfword
+      maskhi (lognot masklo)            ; isolate the upper halfword
+      round-limit (ash 1 (/ halfword-bits 2)))
+
+(defun hihalf (n)                       ; return high halfword, shifted down
+  (ash (logand n maskhi) (- halfword-bits)))
+
+(defun lohalf (n)                       ; return low halfword
+  (logand n masklo))
+
+;; Visible functions
+
+;; Arithmetic functions
+(defun f+ (a1 a2)
+  "Returns the sum of two floating point numbers."
+  (let ((f1 (fmax a1 a2))
+        (f2 (fmin a1 a2)))
+    (if (same-sign a1 a2)
+        (setq f1 (fashr f1)             ; shift right to avoid overflow
+              f2 (fashr f2)))
+    (normalize
+     (cons (+ (car f1) (ash (car f2) (- (cdr f2) (cdr f1))))
+           (cdr f1)))))
+
+(defun f- (a1 &optional a2)             ; unary or binary minus
+  "Returns the difference of two floating point numbers."
+  (if a2
+      (f+ a1 (f- a2))
+    (normalize (cons (- (car a1)) (cdr a1)))))
+
+(defun f* (a1 a2)                       ; multiply in halfword chunks
+  "Returns the product of two floating point numbers."
+  (let* ((i1 (car (fabs a1)))
+         (i2 (car (fabs a2)))
+         (sign (not (same-sign a1 a2)))
+         (prodlo (+ (hihalf (* (lohalf i1) (lohalf i2)))
+                    (lohalf (* (hihalf i1) (lohalf i2)))
+                    (lohalf (* (lohalf i1) (hihalf i2)))))
+         (prodhi (+ (* (hihalf i1) (hihalf i2))
+                    (hihalf (* (hihalf i1) (lohalf i2)))
+                    (hihalf (* (lohalf i1) (hihalf i2)))
+                    (hihalf prodlo))))
+    (if (> (lohalf prodlo) round-limit)
+        (setq prodhi (1+ prodhi)))      ; round off truncated bits
+    (normalize
+     (cons (if sign (- prodhi) prodhi)
+           (+ (cdr (fabs a1)) (cdr (fabs a2)) mantissa-bits)))))
+
+(defun f/ (a1 a2)                       ; SLOW subtract-and-shift algorithm
+  "Returns the quotient of two floating point numbers."
+  (if (zerop (car a2))                  ; if divide by 0
+      (signal 'arith-error (list "attempt to divide by zero" a1 a2))
+    (let ((bits (1- maxbit))
+          (quotient 0) 
+          (dividend (car (fabs a1)))
+          (divisor (car (fabs a2)))
+          (sign (not (same-sign a1 a2))))
+      (while (natnump bits)
+        (if (< (- dividend divisor) 0)
+            (setq quotient (ash quotient 1))
+          (setq quotient (1+ (ash quotient 1))
+                dividend (- dividend divisor)))
+        (setq dividend (ash dividend 1)
+              bits (1- bits)))
+      (normalize
+       (cons (if sign (- quotient) quotient)
+             (- (cdr (fabs a1)) (cdr (fabs a2)) (1- maxbit)))))))
+  
+(defun f% (a1 a2)
+  "Returns the remainder of first floating point number divided by second."
+  (f- a1 (f* (ftrunc (f/ a1 a2)) a2)))
+          
+
+;; Comparison functions
+(defun f= (a1 a2)
+  "Returns t if two floating point numbers are equal, nil otherwise."
+  (equal a1 a2))
+
+(defun f> (a1 a2)
+  "Returns t if first floating point number is greater than second,
+nil otherwise."
+  (cond ((and (natnump (car a1)) (< (car a2) 0)) 
+         t)                             ; a1 nonnegative, a2 negative
+        ((and (> (car a1) 0) (<= (car a2) 0))
+         t)                             ; a1 positive, a2 nonpositive
+        ((and (<= (car a1) 0) (natnump (car a2)))
+         nil)                           ; a1 nonpos, a2 nonneg
+        ((/= (cdr a1) (cdr a2))         ; same signs.  exponents differ
+         (> (cdr a1) (cdr a2)))         ; compare the mantissas.
+        (t
+         (> (car a1) (car a2)))))       ; same exponents.
+
+(defun f>= (a1 a2)
+  "Returns t if first floating point number is greater than or equal to 
+second, nil otherwise."
+  (or (f> a1 a2) (f= a1 a2)))
+
+(defun f< (a1 a2)
+  "Returns t if first floating point number is less than second,
+nil otherwise."
+  (not (f>= a1 a2)))
+
+(defun f<= (a1 a2)
+  "Returns t if first floating point number is less than or equal to
+second, nil otherwise."
+  (not (f> a1 a2)))
+
+(defun f/= (a1 a2)
+  "Returns t if first floating point number is not equal to second,
+nil otherwise."
+  (not (f= a1 a2)))
+
+(defun fmin (a1 a2)
+  "Returns the minimum of two floating point numbers."
+  (if (f< a1 a2) a1 a2))
+
+(defun fmax (a1 a2)
+  "Returns the maximum of two floating point numbers."
+  (if (f> a1 a2) a1 a2))
+      
+(defun fzerop (fnum)
+  "Returns t if the floating point number is zero, nil otherwise."
+  (= (car fnum) 0))
+
+(defun floatp (fnum)
+  "Returns t if the arg is a floating point number, nil otherwise."
+  (and (consp fnum) (integerp (car fnum)) (integerp (cdr fnum))))
+
+;; Conversion routines
+(defun f (int)
+  "Convert the integer argument to floating point, like a C cast operator."
+  (normalize (cons int '0)))
+
+(defun int-to-hex-string (int)
+  "Convert the integer argument to a C-style hexadecimal string."
+  (let ((shiftval -20)
+        (str "0x")
+        (hex-chars "0123456789ABCDEF"))
+    (while (<= shiftval 0)
+      (setq str (concat str (char-to-string 
+                        (aref hex-chars
+                              (logand (lsh int shiftval) 15))))
+            shiftval (+ shiftval 4)))
+    str))
+
+(defun ftrunc (fnum)                    ; truncate fractional part
+  "Truncate the fractional part of a floating point number."
+  (cond ((natnump (cdr fnum))           ; it's all integer, return number as is
+         fnum)
+        ((<= (cdr fnum) (- maxbit))     ; it's all fractional, return 0
+         '(0 . 1))
+        (t                              ; otherwise mask out fractional bits
+         (let ((mant (car fnum)) (exp (cdr fnum)))
+           (normalize 
+            (cons (if (natnump mant)    ; if negative, use absolute value
+                      (ash (ash mant exp) (- exp))
+                    (- (ash (ash (- mant) exp) (- exp))))
+                  exp))))))
+
+(defun fint (fnum)                      ; truncate and convert to integer
+  "Convert the floating point number to integer, with truncation, 
+like a C cast operator."
+  (let* ((tf (ftrunc fnum)) (tint (car tf)) (texp (cdr tf)))
+    (cond ((>= texp mantissa-bits)      ; too high, return "maxint"
+           mantissa-maxval)
+          ((<= texp (- mantissa-bits))  ; too low, return "minint"
+           mantissa-minval)
+          (t                            ; in range
+           (ash tint texp)))))          ; shift so that exponent is 0
+
+(defun float-to-string (fnum &optional sci)
+  "Convert the floating point number to a decimal string.
+Optional second argument non-nil means use scientific notation."
+  (let* ((value (fabs fnum)) (sign (< (car fnum) 0))
+         (power 0) (result 0) (str "") 
+         (temp 0) (pow10 _f1))
+
+    (if (f= fnum _f0)
+        "0"
+      (if (f>= value _f1)                       ; find largest power of 10 <= value
+          (progn                                ; value >= 1, power is positive
+            (while (f<= (setq temp (f* pow10 highest-power-of-10)) value)
+              (setq pow10 temp
+                    power (+ power decimal-digits)))
+            (while (f<= (setq temp (f* pow10 _f10)) value)
+              (setq pow10 temp
+                    power (1+ power))))
+        (progn                          ; value < 1, power is negative
+          (while (f> (setq temp (f/ pow10 highest-power-of-10)) value)
+            (setq pow10 temp
+                  power (- power decimal-digits)))
+          (while (f> pow10 value)
+            (setq pow10 (f/ pow10 _f10)
+                  power (1- power)))))
+                                          ; get value in range 100000 to 999999
+      (setq value (f* (f/ value pow10) all-decimal-digs-minval)
+            result (ftrunc value))
+      (let (int)
+        (if (f> (f- value result) _f1/2)        ; round up if remainder > 0.5
+            (setq int (1+ (fint result)))
+          (setq int (fint result)))
+        (setq str (int-to-string int))
+        (if (>= int 1000000)
+            (setq power (1+ power))))
+
+      (if sci                           ; scientific notation
+          (setq str (concat (substring str 0 1) "." (substring str 1)
+                            "E" (int-to-string power)))
+
+                                          ; regular decimal string
+        (cond ((>= power (1- decimal-digits))
+                                          ; large power, append zeroes
+               (let ((zeroes (- power decimal-digits)))
+                 (while (natnump zeroes)
+                   (setq str (concat str "0")
+                         zeroes (1- zeroes)))))
+
+                                          ; negative power, prepend decimal
+              ((< power 0)              ; point and zeroes
+               (let ((zeroes (- (- power) 2)))
+                 (while (natnump zeroes)
+                   (setq str (concat "0" str)
+                         zeroes (1- zeroes)))
+                 (setq str (concat "0." str))))
+
+              (t                                ; in range, insert decimal point
+               (setq str (concat
+                          (substring str 0 (1+ power))
+                          "."
+                          (substring str (1+ power)))))))
+
+      (if sign                          ; if negative, prepend minus sign
+          (concat "-" str)
+        str))))
+
+    
+;; string to float conversion.
+;; accepts scientific notation, but ignores anything after the first two
+;; digits of the exponent.
+(defun string-to-float (str)
+  "Convert the string to a floating point number.
+Accepts a decimal string in scientific notation, 
+with exponent preceded by either E or e.
+Only the 6 most significant digits of the integer and fractional parts
+are used; only the first two digits of the exponent are used.
+Negative signs preceding both the decimal number and the exponent
+are recognized."
+
+  (if (string-match floating-point-regexp str 0)
+      (let (power)
+        (f*
+         ; calculate the mantissa
+         (let* ((int-subst (extract-match str 2))
+                (fract-subst (extract-match str 4))
+                (digit-string (concat int-subst fract-subst))
+                (mant-sign (equal (extract-match str 1) "-"))
+                (leading-0s 0) (round-up nil))
+
+           ; get rid of leading 0's
+           (setq power (- (length int-subst) decimal-digits))
+           (while (and (< leading-0s (length digit-string))
+                       (= (aref digit-string leading-0s) ?0))
+             (setq leading-0s (1+ leading-0s)))
+           (setq power (- power leading-0s)
+                 digit-string (substring digit-string leading-0s))
+           
+           ; if more than 6 digits, round off
+           (if (> (length digit-string) decimal-digits)
+               (setq round-up (>= (aref digit-string decimal-digits) ?5)
+                     digit-string (substring digit-string 0 decimal-digits))
+             (setq power (+ power (- decimal-digits (length digit-string)))))
+
+           ; round up and add minus sign, if necessary
+           (f (* (+ (string-to-int digit-string)
+                    (if round-up 1 0))
+                 (if mant-sign -1 1))))
+           
+         ; calculate the exponent (power of ten)
+         (let* ((expt-subst (extract-match str 9))
+                (expt-sign (equal (extract-match str 8) "-"))
+                (expt 0) (chunks 0) (tens 0) (exponent _f1)
+                (func 'f*))
+ 
+           (setq expt (+ (* (string-to-int
+                             (substring expt-subst 0
+                                        (min expt-digits (length expt-subst))))
+                            (if expt-sign -1 1))
+                         power))
+           (if (< expt 0)               ; if power of 10 negative
+               (setq expt (- expt)      ; take abs val of exponent
+                     func 'f/))         ; and set up to divide, not multiply
+
+           (setq chunks (/ expt decimal-digits)
+                 tens (% expt decimal-digits))
+           ; divide or multiply by "chunks" of 10**6
+           (while (> chunks 0)  
+             (setq exponent (funcall func exponent highest-power-of-10)
+                   chunks (1- chunks)))
+           ; divide or multiply by remaining power of ten
+           (funcall func exponent (aref powers-of-10 tens)))))
+                  
+    _f0))                               ; if invalid, return 0
+
+