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1a06eabd ER |
1 | ;;; float.el --- floating point arithmetic package. |
2 | ||
9750e079 ER |
3 | ;; Copyright (C) 1986 Free Software Foundation, Inc. |
4 | ||
e5167999 ER |
5 | ;; Author: Bill Rosenblatt |
6 | ;; Maintainer: FSF | |
fd7fa35a | 7 | ;; Keywords: extensions |
e5167999 | 8 | |
d017bf4e RS |
9 | ;; This file is part of GNU Emacs. |
10 | ||
11 | ;; GNU Emacs is free software; you can redistribute it and/or modify | |
12 | ;; it under the terms of the GNU General Public License as published by | |
e5167999 | 13 | ;; the Free Software Foundation; either version 2, or (at your option) |
d017bf4e RS |
14 | ;; any later version. |
15 | ||
16 | ;; GNU Emacs is distributed in the hope that it will be useful, | |
17 | ;; but WITHOUT ANY WARRANTY; without even the implied warranty of | |
18 | ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
19 | ;; GNU General Public License for more details. | |
20 | ||
21 | ;; You should have received a copy of the GNU General Public License | |
b578f267 EN |
22 | ;; along with GNU Emacs; see the file COPYING. If not, write to the |
23 | ;; Free Software Foundation, Inc., 59 Temple Place - Suite 330, | |
24 | ;; Boston, MA 02111-1307, USA. | |
d017bf4e | 25 | |
e5167999 ER |
26 | ;;; Commentary: |
27 | ||
d017bf4e RS |
28 | ;; Floating point numbers are represented by dot-pairs (mant . exp) |
29 | ;; where mant is the 24-bit signed integral mantissa and exp is the | |
30 | ;; base 2 exponent. | |
31 | ;; | |
32 | ;; Emacs LISP supports a 24-bit signed integer data type, which has a | |
33 | ;; range of -(2**23) to +(2**23)-1, or -8388608 to 8388607 decimal. | |
34 | ;; This gives six significant decimal digit accuracy. Exponents can | |
35 | ;; be anything in the range -(2**23) to +(2**23)-1. | |
36 | ;; | |
37 | ;; User interface: | |
38 | ;; function f converts from integer to floating point | |
39 | ;; function string-to-float converts from string to floating point | |
40 | ;; function fint converts a floating point to integer (with truncation) | |
41 | ;; function float-to-string converts from floating point to string | |
42 | ;; | |
43 | ;; Caveats: | |
44 | ;; - Exponents outside of the range of +/-100 or so will cause certain | |
45 | ;; functions (especially conversion routines) to take forever. | |
46 | ;; - Very little checking is done for fixed point overflow/underflow. | |
47 | ;; - No checking is done for over/underflow of the exponent | |
48 | ;; (hardly necessary when exponent can be 2**23). | |
49 | ;; | |
50 | ;; | |
51 | ;; Bill Rosenblatt | |
52 | ;; June 20, 1986 | |
53 | ;; | |
54 | ||
e5167999 ER |
55 | ;;; Code: |
56 | ||
d017bf4e RS |
57 | ;; fundamental implementation constants |
58 | (defconst exp-base 2 | |
59 | "Base of exponent in this floating point representation.") | |
60 | ||
61 | (defconst mantissa-bits 24 | |
62 | "Number of significant bits in this floating point representation.") | |
63 | ||
64 | (defconst decimal-digits 6 | |
65 | "Number of decimal digits expected to be accurate.") | |
66 | ||
67 | (defconst expt-digits 2 | |
68 | "Maximum permitted digits in a scientific notation exponent.") | |
69 | ||
70 | ;; other constants | |
71 | (defconst maxbit (1- mantissa-bits) | |
72 | "Number of highest bit") | |
73 | ||
74 | (defconst mantissa-maxval (1- (ash 1 maxbit)) | |
eb8c3be9 | 75 | "Maximum permissible value of mantissa") |
d017bf4e RS |
76 | |
77 | (defconst mantissa-minval (ash 1 maxbit) | |
eb8c3be9 | 78 | "Minimum permissible value of mantissa") |
d017bf4e RS |
79 | |
80 | (defconst floating-point-regexp | |
81 | "^[ \t]*\\(-?\\)\\([0-9]*\\)\ | |
82 | \\(\\.\\([0-9]*\\)\\|\\)\ | |
83 | \\(\\(\\([Ee]\\)\\(-?\\)\\([0-9][0-9]*\\)\\)\\|\\)[ \t]*$" | |
84 | "Regular expression to match floating point numbers. Extract matches: | |
85 | 1 - minus sign | |
86 | 2 - integer part | |
87 | 4 - fractional part | |
88 | 8 - minus sign for power of ten | |
89 | 9 - power of ten | |
90 | ") | |
91 | ||
92 | (defconst high-bit-mask (ash 1 maxbit) | |
93 | "Masks all bits except the high-order (sign) bit.") | |
94 | ||
95 | (defconst second-bit-mask (ash 1 (1- maxbit)) | |
96 | "Masks all bits except the highest-order magnitude bit") | |
97 | ||
98 | ;; various useful floating point constants | |
99 | (setq _f0 '(0 . 1)) | |
100 | ||
101 | (setq _f1/2 '(4194304 . -23)) | |
102 | ||
103 | (setq _f1 '(4194304 . -22)) | |
104 | ||
105 | (setq _f10 '(5242880 . -19)) | |
106 | ||
107 | ;; support for decimal conversion routines | |
108 | (setq powers-of-10 (make-vector (1+ decimal-digits) _f1)) | |
109 | (aset powers-of-10 1 _f10) | |
110 | (aset powers-of-10 2 '(6553600 . -16)) | |
111 | (aset powers-of-10 3 '(8192000 . -13)) | |
112 | (aset powers-of-10 4 '(5120000 . -9)) | |
113 | (aset powers-of-10 5 '(6400000 . -6)) | |
114 | (aset powers-of-10 6 '(8000000 . -3)) | |
115 | ||
116 | (setq all-decimal-digs-minval (aref powers-of-10 (1- decimal-digits)) | |
117 | highest-power-of-10 (aref powers-of-10 decimal-digits)) | |
118 | ||
119 | (defun fashl (fnum) ; floating-point arithmetic shift left | |
120 | (cons (ash (car fnum) 1) (1- (cdr fnum)))) | |
121 | ||
122 | (defun fashr (fnum) ; floating point arithmetic shift right | |
123 | (cons (ash (car fnum) -1) (1+ (cdr fnum)))) | |
124 | ||
125 | (defun normalize (fnum) | |
126 | (if (> (car fnum) 0) ; make sure next-to-highest bit is set | |
127 | (while (zerop (logand (car fnum) second-bit-mask)) | |
128 | (setq fnum (fashl fnum))) | |
129 | (if (< (car fnum) 0) ; make sure highest bit is set | |
130 | (while (zerop (logand (car fnum) high-bit-mask)) | |
131 | (setq fnum (fashl fnum))) | |
132 | (setq fnum _f0))) ; "standard 0" | |
133 | fnum) | |
134 | ||
135 | (defun abs (n) ; integer absolute value | |
136 | (if (>= n 0) n (- n))) | |
137 | ||
138 | (defun fabs (fnum) ; re-normalize after taking abs value | |
139 | (normalize (cons (abs (car fnum)) (cdr fnum)))) | |
140 | ||
141 | (defun xor (a b) ; logical exclusive or | |
142 | (and (or a b) (not (and a b)))) | |
143 | ||
144 | (defun same-sign (a b) ; two f-p numbers have same sign? | |
145 | (not (xor (natnump (car a)) (natnump (car b))))) | |
146 | ||
147 | (defun extract-match (str i) ; used after string-match | |
148 | (condition-case () | |
149 | (substring str (match-beginning i) (match-end i)) | |
150 | (error ""))) | |
151 | ||
152 | ;; support for the multiplication function | |
153 | (setq halfword-bits (/ mantissa-bits 2) ; bits in a halfword | |
154 | masklo (1- (ash 1 halfword-bits)) ; isolate the lower halfword | |
155 | maskhi (lognot masklo) ; isolate the upper halfword | |
156 | round-limit (ash 1 (/ halfword-bits 2))) | |
157 | ||
158 | (defun hihalf (n) ; return high halfword, shifted down | |
159 | (ash (logand n maskhi) (- halfword-bits))) | |
160 | ||
161 | (defun lohalf (n) ; return low halfword | |
162 | (logand n masklo)) | |
163 | ||
164 | ;; Visible functions | |
165 | ||
166 | ;; Arithmetic functions | |
167 | (defun f+ (a1 a2) | |
168 | "Returns the sum of two floating point numbers." | |
169 | (let ((f1 (fmax a1 a2)) | |
170 | (f2 (fmin a1 a2))) | |
171 | (if (same-sign a1 a2) | |
172 | (setq f1 (fashr f1) ; shift right to avoid overflow | |
173 | f2 (fashr f2))) | |
174 | (normalize | |
175 | (cons (+ (car f1) (ash (car f2) (- (cdr f2) (cdr f1)))) | |
176 | (cdr f1))))) | |
177 | ||
178 | (defun f- (a1 &optional a2) ; unary or binary minus | |
179 | "Returns the difference of two floating point numbers." | |
180 | (if a2 | |
181 | (f+ a1 (f- a2)) | |
182 | (normalize (cons (- (car a1)) (cdr a1))))) | |
183 | ||
184 | (defun f* (a1 a2) ; multiply in halfword chunks | |
185 | "Returns the product of two floating point numbers." | |
186 | (let* ((i1 (car (fabs a1))) | |
187 | (i2 (car (fabs a2))) | |
188 | (sign (not (same-sign a1 a2))) | |
189 | (prodlo (+ (hihalf (* (lohalf i1) (lohalf i2))) | |
190 | (lohalf (* (hihalf i1) (lohalf i2))) | |
191 | (lohalf (* (lohalf i1) (hihalf i2))))) | |
192 | (prodhi (+ (* (hihalf i1) (hihalf i2)) | |
193 | (hihalf (* (hihalf i1) (lohalf i2))) | |
194 | (hihalf (* (lohalf i1) (hihalf i2))) | |
195 | (hihalf prodlo)))) | |
196 | (if (> (lohalf prodlo) round-limit) | |
197 | (setq prodhi (1+ prodhi))) ; round off truncated bits | |
198 | (normalize | |
199 | (cons (if sign (- prodhi) prodhi) | |
200 | (+ (cdr (fabs a1)) (cdr (fabs a2)) mantissa-bits))))) | |
201 | ||
202 | (defun f/ (a1 a2) ; SLOW subtract-and-shift algorithm | |
203 | "Returns the quotient of two floating point numbers." | |
204 | (if (zerop (car a2)) ; if divide by 0 | |
205 | (signal 'arith-error (list "attempt to divide by zero" a1 a2)) | |
206 | (let ((bits (1- maxbit)) | |
207 | (quotient 0) | |
208 | (dividend (car (fabs a1))) | |
209 | (divisor (car (fabs a2))) | |
210 | (sign (not (same-sign a1 a2)))) | |
211 | (while (natnump bits) | |
212 | (if (< (- dividend divisor) 0) | |
213 | (setq quotient (ash quotient 1)) | |
214 | (setq quotient (1+ (ash quotient 1)) | |
215 | dividend (- dividend divisor))) | |
216 | (setq dividend (ash dividend 1) | |
217 | bits (1- bits))) | |
218 | (normalize | |
219 | (cons (if sign (- quotient) quotient) | |
220 | (- (cdr (fabs a1)) (cdr (fabs a2)) (1- maxbit))))))) | |
221 | ||
222 | (defun f% (a1 a2) | |
223 | "Returns the remainder of first floating point number divided by second." | |
224 | (f- a1 (f* (ftrunc (f/ a1 a2)) a2))) | |
225 | ||
226 | ||
227 | ;; Comparison functions | |
228 | (defun f= (a1 a2) | |
229 | "Returns t if two floating point numbers are equal, nil otherwise." | |
230 | (equal a1 a2)) | |
231 | ||
232 | (defun f> (a1 a2) | |
233 | "Returns t if first floating point number is greater than second, | |
234 | nil otherwise." | |
235 | (cond ((and (natnump (car a1)) (< (car a2) 0)) | |
236 | t) ; a1 nonnegative, a2 negative | |
237 | ((and (> (car a1) 0) (<= (car a2) 0)) | |
238 | t) ; a1 positive, a2 nonpositive | |
239 | ((and (<= (car a1) 0) (natnump (car a2))) | |
240 | nil) ; a1 nonpos, a2 nonneg | |
241 | ((/= (cdr a1) (cdr a2)) ; same signs. exponents differ | |
242 | (> (cdr a1) (cdr a2))) ; compare the mantissas. | |
243 | (t | |
244 | (> (car a1) (car a2))))) ; same exponents. | |
245 | ||
246 | (defun f>= (a1 a2) | |
247 | "Returns t if first floating point number is greater than or equal to | |
248 | second, nil otherwise." | |
249 | (or (f> a1 a2) (f= a1 a2))) | |
250 | ||
251 | (defun f< (a1 a2) | |
252 | "Returns t if first floating point number is less than second, | |
253 | nil otherwise." | |
254 | (not (f>= a1 a2))) | |
255 | ||
256 | (defun f<= (a1 a2) | |
257 | "Returns t if first floating point number is less than or equal to | |
258 | second, nil otherwise." | |
259 | (not (f> a1 a2))) | |
260 | ||
261 | (defun f/= (a1 a2) | |
262 | "Returns t if first floating point number is not equal to second, | |
263 | nil otherwise." | |
264 | (not (f= a1 a2))) | |
265 | ||
266 | (defun fmin (a1 a2) | |
267 | "Returns the minimum of two floating point numbers." | |
268 | (if (f< a1 a2) a1 a2)) | |
269 | ||
270 | (defun fmax (a1 a2) | |
271 | "Returns the maximum of two floating point numbers." | |
272 | (if (f> a1 a2) a1 a2)) | |
273 | ||
274 | (defun fzerop (fnum) | |
275 | "Returns t if the floating point number is zero, nil otherwise." | |
276 | (= (car fnum) 0)) | |
277 | ||
278 | (defun floatp (fnum) | |
279 | "Returns t if the arg is a floating point number, nil otherwise." | |
280 | (and (consp fnum) (integerp (car fnum)) (integerp (cdr fnum)))) | |
281 | ||
282 | ;; Conversion routines | |
283 | (defun f (int) | |
284 | "Convert the integer argument to floating point, like a C cast operator." | |
285 | (normalize (cons int '0))) | |
286 | ||
287 | (defun int-to-hex-string (int) | |
288 | "Convert the integer argument to a C-style hexadecimal string." | |
289 | (let ((shiftval -20) | |
290 | (str "0x") | |
291 | (hex-chars "0123456789ABCDEF")) | |
292 | (while (<= shiftval 0) | |
293 | (setq str (concat str (char-to-string | |
294 | (aref hex-chars | |
295 | (logand (lsh int shiftval) 15)))) | |
296 | shiftval (+ shiftval 4))) | |
297 | str)) | |
298 | ||
299 | (defun ftrunc (fnum) ; truncate fractional part | |
300 | "Truncate the fractional part of a floating point number." | |
301 | (cond ((natnump (cdr fnum)) ; it's all integer, return number as is | |
302 | fnum) | |
303 | ((<= (cdr fnum) (- maxbit)) ; it's all fractional, return 0 | |
304 | '(0 . 1)) | |
305 | (t ; otherwise mask out fractional bits | |
306 | (let ((mant (car fnum)) (exp (cdr fnum))) | |
307 | (normalize | |
308 | (cons (if (natnump mant) ; if negative, use absolute value | |
309 | (ash (ash mant exp) (- exp)) | |
310 | (- (ash (ash (- mant) exp) (- exp)))) | |
311 | exp)))))) | |
312 | ||
313 | (defun fint (fnum) ; truncate and convert to integer | |
314 | "Convert the floating point number to integer, with truncation, | |
315 | like a C cast operator." | |
316 | (let* ((tf (ftrunc fnum)) (tint (car tf)) (texp (cdr tf))) | |
317 | (cond ((>= texp mantissa-bits) ; too high, return "maxint" | |
318 | mantissa-maxval) | |
319 | ((<= texp (- mantissa-bits)) ; too low, return "minint" | |
320 | mantissa-minval) | |
321 | (t ; in range | |
322 | (ash tint texp))))) ; shift so that exponent is 0 | |
323 | ||
324 | (defun float-to-string (fnum &optional sci) | |
325 | "Convert the floating point number to a decimal string. | |
326 | Optional second argument non-nil means use scientific notation." | |
327 | (let* ((value (fabs fnum)) (sign (< (car fnum) 0)) | |
328 | (power 0) (result 0) (str "") | |
329 | (temp 0) (pow10 _f1)) | |
330 | ||
331 | (if (f= fnum _f0) | |
332 | "0" | |
333 | (if (f>= value _f1) ; find largest power of 10 <= value | |
334 | (progn ; value >= 1, power is positive | |
335 | (while (f<= (setq temp (f* pow10 highest-power-of-10)) value) | |
336 | (setq pow10 temp | |
337 | power (+ power decimal-digits))) | |
338 | (while (f<= (setq temp (f* pow10 _f10)) value) | |
339 | (setq pow10 temp | |
340 | power (1+ power)))) | |
341 | (progn ; value < 1, power is negative | |
342 | (while (f> (setq temp (f/ pow10 highest-power-of-10)) value) | |
343 | (setq pow10 temp | |
344 | power (- power decimal-digits))) | |
345 | (while (f> pow10 value) | |
346 | (setq pow10 (f/ pow10 _f10) | |
347 | power (1- power))))) | |
348 | ; get value in range 100000 to 999999 | |
349 | (setq value (f* (f/ value pow10) all-decimal-digs-minval) | |
350 | result (ftrunc value)) | |
351 | (let (int) | |
352 | (if (f> (f- value result) _f1/2) ; round up if remainder > 0.5 | |
353 | (setq int (1+ (fint result))) | |
354 | (setq int (fint result))) | |
355 | (setq str (int-to-string int)) | |
356 | (if (>= int 1000000) | |
357 | (setq power (1+ power)))) | |
358 | ||
359 | (if sci ; scientific notation | |
360 | (setq str (concat (substring str 0 1) "." (substring str 1) | |
361 | "E" (int-to-string power))) | |
362 | ||
363 | ; regular decimal string | |
364 | (cond ((>= power (1- decimal-digits)) | |
365 | ; large power, append zeroes | |
366 | (let ((zeroes (- power decimal-digits))) | |
367 | (while (natnump zeroes) | |
368 | (setq str (concat str "0") | |
369 | zeroes (1- zeroes))))) | |
370 | ||
371 | ; negative power, prepend decimal | |
372 | ((< power 0) ; point and zeroes | |
373 | (let ((zeroes (- (- power) 2))) | |
374 | (while (natnump zeroes) | |
375 | (setq str (concat "0" str) | |
376 | zeroes (1- zeroes))) | |
377 | (setq str (concat "0." str)))) | |
378 | ||
379 | (t ; in range, insert decimal point | |
380 | (setq str (concat | |
381 | (substring str 0 (1+ power)) | |
382 | "." | |
383 | (substring str (1+ power))))))) | |
384 | ||
385 | (if sign ; if negative, prepend minus sign | |
386 | (concat "-" str) | |
387 | str)))) | |
388 | ||
389 | ||
390 | ;; string to float conversion. | |
391 | ;; accepts scientific notation, but ignores anything after the first two | |
392 | ;; digits of the exponent. | |
393 | (defun string-to-float (str) | |
394 | "Convert the string to a floating point number. | |
3bfd7edb JB |
395 | Accepts a decimal string in scientific notation, with exponent preceded |
396 | by either E or e. Only the six most significant digits of the integer | |
397 | and fractional parts are used; only the first two digits of the exponent | |
398 | are used. Negative signs preceding both the decimal number and the exponent | |
d017bf4e RS |
399 | are recognized." |
400 | ||
401 | (if (string-match floating-point-regexp str 0) | |
402 | (let (power) | |
403 | (f* | |
404 | ; calculate the mantissa | |
405 | (let* ((int-subst (extract-match str 2)) | |
406 | (fract-subst (extract-match str 4)) | |
407 | (digit-string (concat int-subst fract-subst)) | |
408 | (mant-sign (equal (extract-match str 1) "-")) | |
409 | (leading-0s 0) (round-up nil)) | |
410 | ||
411 | ; get rid of leading 0's | |
412 | (setq power (- (length int-subst) decimal-digits)) | |
413 | (while (and (< leading-0s (length digit-string)) | |
414 | (= (aref digit-string leading-0s) ?0)) | |
415 | (setq leading-0s (1+ leading-0s))) | |
416 | (setq power (- power leading-0s) | |
417 | digit-string (substring digit-string leading-0s)) | |
418 | ||
419 | ; if more than 6 digits, round off | |
420 | (if (> (length digit-string) decimal-digits) | |
421 | (setq round-up (>= (aref digit-string decimal-digits) ?5) | |
422 | digit-string (substring digit-string 0 decimal-digits)) | |
423 | (setq power (+ power (- decimal-digits (length digit-string))))) | |
424 | ||
425 | ; round up and add minus sign, if necessary | |
426 | (f (* (+ (string-to-int digit-string) | |
427 | (if round-up 1 0)) | |
428 | (if mant-sign -1 1)))) | |
429 | ||
430 | ; calculate the exponent (power of ten) | |
431 | (let* ((expt-subst (extract-match str 9)) | |
432 | (expt-sign (equal (extract-match str 8) "-")) | |
433 | (expt 0) (chunks 0) (tens 0) (exponent _f1) | |
434 | (func 'f*)) | |
435 | ||
436 | (setq expt (+ (* (string-to-int | |
437 | (substring expt-subst 0 | |
438 | (min expt-digits (length expt-subst)))) | |
439 | (if expt-sign -1 1)) | |
440 | power)) | |
441 | (if (< expt 0) ; if power of 10 negative | |
442 | (setq expt (- expt) ; take abs val of exponent | |
443 | func 'f/)) ; and set up to divide, not multiply | |
444 | ||
445 | (setq chunks (/ expt decimal-digits) | |
446 | tens (% expt decimal-digits)) | |
447 | ; divide or multiply by "chunks" of 10**6 | |
448 | (while (> chunks 0) | |
449 | (setq exponent (funcall func exponent highest-power-of-10) | |
450 | chunks (1- chunks))) | |
451 | ; divide or multiply by remaining power of ten | |
452 | (funcall func exponent (aref powers-of-10 tens))))) | |
453 | ||
454 | _f0)) ; if invalid, return 0 | |
49116ac0 JB |
455 | |
456 | (provide 'float) | |
457 | ||
1a06eabd | 458 | ;;; float.el ends here |