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