1 ;; Copyright (C) 1986 Free Software Foundation, Inc.
2 ;; Author Bill Rosenblatt
4 ;; This file is part of GNU Emacs.
6 ;; GNU Emacs is free software; you can redistribute it and/or modify
7 ;; it under the terms of the GNU General Public License as published by
8 ;; the Free Software Foundation; either version 1, or (at your option)
11 ;; GNU Emacs is distributed in the hope that it will be useful,
12 ;; but WITHOUT ANY WARRANTY; without even the implied warranty of
13 ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14 ;; GNU General Public License for more details.
16 ;; You should have received a copy of the GNU General Public License
17 ;; along with GNU Emacs; see the file COPYING. If not, write to
18 ;; the Free Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
20 ;; Floating point arithmetic package.
22 ;; Floating point numbers are represented by dot-pairs (mant . exp)
23 ;; where mant is the 24-bit signed integral mantissa and exp is the
26 ;; Emacs LISP supports a 24-bit signed integer data type, which has a
27 ;; range of -(2**23) to +(2**23)-1, or -8388608 to 8388607 decimal.
28 ;; This gives six significant decimal digit accuracy. Exponents can
29 ;; be anything in the range -(2**23) to +(2**23)-1.
32 ;; function f converts from integer to floating point
33 ;; function string-to-float converts from string to floating point
34 ;; function fint converts a floating point to integer (with truncation)
35 ;; function float-to-string converts from floating point to string
38 ;; - Exponents outside of the range of +/-100 or so will cause certain
39 ;; functions (especially conversion routines) to take forever.
40 ;; - Very little checking is done for fixed point overflow/underflow.
41 ;; - No checking is done for over/underflow of the exponent
42 ;; (hardly necessary when exponent can be 2**23).
49 ;; fundamental implementation constants
51 "Base of exponent in this floating point representation.")
53 (defconst mantissa-bits
24
54 "Number of significant bits in this floating point representation.")
56 (defconst decimal-digits
6
57 "Number of decimal digits expected to be accurate.")
59 (defconst expt-digits
2
60 "Maximum permitted digits in a scientific notation exponent.")
63 (defconst maxbit
(1- mantissa-bits
)
64 "Number of highest bit")
66 (defconst mantissa-maxval
(1- (ash 1 maxbit
))
67 "Maximum permissable value of mantissa")
69 (defconst mantissa-minval
(ash 1 maxbit
)
70 "Minimum permissable value of mantissa")
72 (defconst floating-point-regexp
73 "^[ \t]*\\(-?\\)\\([0-9]*\\)\
74 \\(\\.\\([0-9]*\\)\\|\\)\
75 \\(\\(\\([Ee]\\)\\(-?\\)\\([0-9][0-9]*\\)\\)\\|\\)[ \t]*$"
76 "Regular expression to match floating point numbers. Extract matches:
80 8 - minus sign for power of ten
84 (defconst high-bit-mask
(ash 1 maxbit
)
85 "Masks all bits except the high-order (sign) bit.")
87 (defconst second-bit-mask
(ash 1 (1- maxbit
))
88 "Masks all bits except the highest-order magnitude bit")
90 ;; various useful floating point constants
93 (setq _f1
/2 '(4194304 . -
23))
95 (setq _f1
'(4194304 . -
22))
97 (setq _f10
'(5242880 . -
19))
99 ;; support for decimal conversion routines
100 (setq powers-of-10
(make-vector (1+ decimal-digits
) _f1
))
101 (aset powers-of-10
1 _f10
)
102 (aset powers-of-10
2 '(6553600 . -
16))
103 (aset powers-of-10
3 '(8192000 . -
13))
104 (aset powers-of-10
4 '(5120000 . -
9))
105 (aset powers-of-10
5 '(6400000 . -
6))
106 (aset powers-of-10
6 '(8000000 . -
3))
108 (setq all-decimal-digs-minval
(aref powers-of-10
(1- decimal-digits
))
109 highest-power-of-10
(aref powers-of-10 decimal-digits
))
111 (defun fashl (fnum) ; floating-point arithmetic shift left
112 (cons (ash (car fnum
) 1) (1- (cdr fnum
))))
114 (defun fashr (fnum) ; floating point arithmetic shift right
115 (cons (ash (car fnum
) -
1) (1+ (cdr fnum
))))
117 (defun normalize (fnum)
118 (if (> (car fnum
) 0) ; make sure next-to-highest bit is set
119 (while (zerop (logand (car fnum
) second-bit-mask
))
120 (setq fnum
(fashl fnum
)))
121 (if (< (car fnum
) 0) ; make sure highest bit is set
122 (while (zerop (logand (car fnum
) high-bit-mask
))
123 (setq fnum
(fashl fnum
)))
124 (setq fnum _f0
))) ; "standard 0"
127 (defun abs (n) ; integer absolute value
128 (if (>= n
0) n
(- n
)))
130 (defun fabs (fnum) ; re-normalize after taking abs value
131 (normalize (cons (abs (car fnum
)) (cdr fnum
))))
133 (defun xor (a b
) ; logical exclusive or
134 (and (or a b
) (not (and a b
))))
136 (defun same-sign (a b
) ; two f-p numbers have same sign?
137 (not (xor (natnump (car a
)) (natnump (car b
)))))
139 (defun extract-match (str i
) ; used after string-match
141 (substring str
(match-beginning i
) (match-end i
))
144 ;; support for the multiplication function
145 (setq halfword-bits
(/ mantissa-bits
2) ; bits in a halfword
146 masklo
(1- (ash 1 halfword-bits
)) ; isolate the lower halfword
147 maskhi
(lognot masklo
) ; isolate the upper halfword
148 round-limit
(ash 1 (/ halfword-bits
2)))
150 (defun hihalf (n) ; return high halfword, shifted down
151 (ash (logand n maskhi
) (- halfword-bits
)))
153 (defun lohalf (n) ; return low halfword
158 ;; Arithmetic functions
160 "Returns the sum of two floating point numbers."
161 (let ((f1 (fmax a1 a2
))
163 (if (same-sign a1 a2
)
164 (setq f1
(fashr f1
) ; shift right to avoid overflow
167 (cons (+ (car f1
) (ash (car f2
) (- (cdr f2
) (cdr f1
))))
170 (defun f- (a1 &optional a2
) ; unary or binary minus
171 "Returns the difference of two floating point numbers."
174 (normalize (cons (- (car a1
)) (cdr a1
)))))
176 (defun f* (a1 a2
) ; multiply in halfword chunks
177 "Returns the product of two floating point numbers."
178 (let* ((i1 (car (fabs a1
)))
180 (sign (not (same-sign a1 a2
)))
181 (prodlo (+ (hihalf (* (lohalf i1
) (lohalf i2
)))
182 (lohalf (* (hihalf i1
) (lohalf i2
)))
183 (lohalf (* (lohalf i1
) (hihalf i2
)))))
184 (prodhi (+ (* (hihalf i1
) (hihalf i2
))
185 (hihalf (* (hihalf i1
) (lohalf i2
)))
186 (hihalf (* (lohalf i1
) (hihalf i2
)))
188 (if (> (lohalf prodlo
) round-limit
)
189 (setq prodhi
(1+ prodhi
))) ; round off truncated bits
191 (cons (if sign
(- prodhi
) prodhi
)
192 (+ (cdr (fabs a1
)) (cdr (fabs a2
)) mantissa-bits
)))))
194 (defun f/ (a1 a2
) ; SLOW subtract-and-shift algorithm
195 "Returns the quotient of two floating point numbers."
196 (if (zerop (car a2
)) ; if divide by 0
197 (signal 'arith-error
(list "attempt to divide by zero" a1 a2
))
198 (let ((bits (1- maxbit
))
200 (dividend (car (fabs a1
)))
201 (divisor (car (fabs a2
)))
202 (sign (not (same-sign a1 a2
))))
203 (while (natnump bits
)
204 (if (< (- dividend divisor
) 0)
205 (setq quotient
(ash quotient
1))
206 (setq quotient
(1+ (ash quotient
1))
207 dividend
(- dividend divisor
)))
208 (setq dividend
(ash dividend
1)
211 (cons (if sign
(- quotient
) quotient
)
212 (- (cdr (fabs a1
)) (cdr (fabs a2
)) (1- maxbit
)))))))
215 "Returns the remainder of first floating point number divided by second."
216 (f- a1
(f* (ftrunc (f/ a1 a2
)) a2
)))
219 ;; Comparison functions
221 "Returns t if two floating point numbers are equal, nil otherwise."
225 "Returns t if first floating point number is greater than second,
227 (cond ((and (natnump (car a1
)) (< (car a2
) 0))
228 t
) ; a1 nonnegative, a2 negative
229 ((and (> (car a1
) 0) (<= (car a2
) 0))
230 t
) ; a1 positive, a2 nonpositive
231 ((and (<= (car a1
) 0) (natnump (car a2
)))
232 nil
) ; a1 nonpos, a2 nonneg
233 ((/= (cdr a1
) (cdr a2
)) ; same signs. exponents differ
234 (> (cdr a1
) (cdr a2
))) ; compare the mantissas.
236 (> (car a1
) (car a2
))))) ; same exponents.
239 "Returns t if first floating point number is greater than or equal to
240 second, nil otherwise."
241 (or (f> a1 a2
) (f= a1 a2
)))
244 "Returns t if first floating point number is less than second,
249 "Returns t if first floating point number is less than or equal to
250 second, nil otherwise."
254 "Returns t if first floating point number is not equal to second,
259 "Returns the minimum of two floating point numbers."
260 (if (f< a1 a2
) a1 a2
))
263 "Returns the maximum of two floating point numbers."
264 (if (f> a1 a2
) a1 a2
))
267 "Returns t if the floating point number is zero, nil otherwise."
271 "Returns t if the arg is a floating point number, nil otherwise."
272 (and (consp fnum
) (integerp (car fnum
)) (integerp (cdr fnum
))))
274 ;; Conversion routines
276 "Convert the integer argument to floating point, like a C cast operator."
277 (normalize (cons int
'0)))
279 (defun int-to-hex-string (int)
280 "Convert the integer argument to a C-style hexadecimal string."
283 (hex-chars "0123456789ABCDEF"))
284 (while (<= shiftval
0)
285 (setq str
(concat str
(char-to-string
287 (logand (lsh int shiftval
) 15))))
288 shiftval
(+ shiftval
4)))
291 (defun ftrunc (fnum) ; truncate fractional part
292 "Truncate the fractional part of a floating point number."
293 (cond ((natnump (cdr fnum
)) ; it's all integer, return number as is
295 ((<= (cdr fnum
) (- maxbit
)) ; it's all fractional, return 0
297 (t ; otherwise mask out fractional bits
298 (let ((mant (car fnum
)) (exp (cdr fnum
)))
300 (cons (if (natnump mant
) ; if negative, use absolute value
301 (ash (ash mant exp
) (- exp
))
302 (- (ash (ash (- mant
) exp
) (- exp
))))
305 (defun fint (fnum) ; truncate and convert to integer
306 "Convert the floating point number to integer, with truncation,
307 like a C cast operator."
308 (let* ((tf (ftrunc fnum
)) (tint (car tf
)) (texp (cdr tf
)))
309 (cond ((>= texp mantissa-bits
) ; too high, return "maxint"
311 ((<= texp
(- mantissa-bits
)) ; too low, return "minint"
314 (ash tint texp
))))) ; shift so that exponent is 0
316 (defun float-to-string (fnum &optional sci
)
317 "Convert the floating point number to a decimal string.
318 Optional second argument non-nil means use scientific notation."
319 (let* ((value (fabs fnum
)) (sign (< (car fnum
) 0))
320 (power 0) (result 0) (str "")
321 (temp 0) (pow10 _f1
))
325 (if (f>= value _f1
) ; find largest power of 10 <= value
326 (progn ; value >= 1, power is positive
327 (while (f<= (setq temp
(f* pow10 highest-power-of-10
)) value
)
329 power
(+ power decimal-digits
)))
330 (while (f<= (setq temp
(f* pow10 _f10
)) value
)
333 (progn ; value < 1, power is negative
334 (while (f> (setq temp
(f/ pow10 highest-power-of-10
)) value
)
336 power
(- power decimal-digits
)))
337 (while (f> pow10 value
)
338 (setq pow10
(f/ pow10 _f10
)
340 ; get value in range 100000 to 999999
341 (setq value
(f* (f/ value pow10
) all-decimal-digs-minval
)
342 result
(ftrunc value
))
344 (if (f> (f- value result
) _f1
/2) ; round up if remainder > 0.5
345 (setq int
(1+ (fint result
)))
346 (setq int
(fint result
)))
347 (setq str
(int-to-string int
))
349 (setq power
(1+ power
))))
351 (if sci
; scientific notation
352 (setq str
(concat (substring str
0 1) "." (substring str
1)
353 "E" (int-to-string power
)))
355 ; regular decimal string
356 (cond ((>= power
(1- decimal-digits
))
357 ; large power, append zeroes
358 (let ((zeroes (- power decimal-digits
)))
359 (while (natnump zeroes
)
360 (setq str
(concat str
"0")
361 zeroes
(1- zeroes
)))))
363 ; negative power, prepend decimal
364 ((< power
0) ; point and zeroes
365 (let ((zeroes (- (- power
) 2)))
366 (while (natnump zeroes
)
367 (setq str
(concat "0" str
)
369 (setq str
(concat "0." str
))))
371 (t ; in range, insert decimal point
373 (substring str
0 (1+ power
))
375 (substring str
(1+ power
)))))))
377 (if sign
; if negative, prepend minus sign
382 ;; string to float conversion.
383 ;; accepts scientific notation, but ignores anything after the first two
384 ;; digits of the exponent.
385 (defun string-to-float (str)
386 "Convert the string to a floating point number.
387 Accepts a decimal string in scientific notation, with exponent preceded
388 by either E or e. Only the six most significant digits of the integer
389 and fractional parts are used; only the first two digits of the exponent
390 are used. Negative signs preceding both the decimal number and the exponent
393 (if (string-match floating-point-regexp str
0)
396 ; calculate the mantissa
397 (let* ((int-subst (extract-match str
2))
398 (fract-subst (extract-match str
4))
399 (digit-string (concat int-subst fract-subst
))
400 (mant-sign (equal (extract-match str
1) "-"))
401 (leading-0s 0) (round-up nil
))
403 ; get rid of leading 0's
404 (setq power
(- (length int-subst
) decimal-digits
))
405 (while (and (< leading-0s
(length digit-string
))
406 (= (aref digit-string leading-0s
) ?
0))
407 (setq leading-0s
(1+ leading-0s
)))
408 (setq power
(- power leading-0s
)
409 digit-string
(substring digit-string leading-0s
))
411 ; if more than 6 digits, round off
412 (if (> (length digit-string
) decimal-digits
)
413 (setq round-up
(>= (aref digit-string decimal-digits
) ?
5)
414 digit-string
(substring digit-string
0 decimal-digits
))
415 (setq power
(+ power
(- decimal-digits
(length digit-string
)))))
417 ; round up and add minus sign, if necessary
418 (f (* (+ (string-to-int digit-string
)
420 (if mant-sign -
1 1))))
422 ; calculate the exponent (power of ten)
423 (let* ((expt-subst (extract-match str
9))
424 (expt-sign (equal (extract-match str
8) "-"))
425 (expt 0) (chunks 0) (tens 0) (exponent _f1
)
428 (setq expt
(+ (* (string-to-int
429 (substring expt-subst
0
430 (min expt-digits
(length expt-subst
))))
433 (if (< expt
0) ; if power of 10 negative
434 (setq expt
(- expt
) ; take abs val of exponent
435 func
'f
/)) ; and set up to divide, not multiply
437 (setq chunks
(/ expt decimal-digits
)
438 tens
(% expt decimal-digits
))
439 ; divide or multiply by "chunks" of 10**6
441 (setq exponent
(funcall func exponent highest-power-of-10
)
443 ; divide or multiply by remaining power of ten
444 (funcall func exponent
(aref powers-of-10 tens
)))))
446 _f0
)) ; if invalid, return 0