Commit d017bf4e authored by Richard M. Stallman's avatar Richard M. Stallman
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Initial revision

parent 1a20f48d
;; 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
;; 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]*\\)\
\\(\\(\\([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"
(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)))
(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
(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)))
(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.
(> (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)))
(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
((<= (cdr fnum) (- maxbit)) ; it's all fractional, return 0
'(0 . 1))
(t ; otherwise mask out fractional bits
(let ((mant (car fnum)) (exp (cdr fnum)))
(cons (if (natnump mant) ; if negative, use absolute value
(ash (ash mant exp) (- exp))
(- (ash (ash (- mant) 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"
((<= texp (- mantissa-bits)) ; too low, return "minint"
(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)
(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)
;; 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)
; 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))
(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
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