lisp/emacs-lisp/float.el

   1 ;;; float.el --- obsolete floating point arithmetic package
   2
   3 ;; Copyright (C) 1986 Free Software Foundation, Inc.
   4
   5 ;; Author: Bill Rosenblatt
   6 ;; Maintainer: FSF
   7 ;; Keywords: extensions
   8
   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
  13 ;; the Free Software Foundation; either version 2, or (at your option)
  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
  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.
  25
  26 ;;; Commentary:
  27
  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
  55 ;;; Code:
  56
  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))
  75   "Maximum permissible value of mantissa")
  76
  77 (defconst mantissa-minval (ash 1 maxbit)
  78   "Minimum permissible value of mantissa")
  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 (defconst _f0 '(0 . 1))
 100
 101 (defconst _f1/2 '(4194304 . -23))
 102
 103 (defconst _f1 '(4194304 . -22))
 104
 105 (defconst _f10 '(5242880 . -19))
 106
 107 ;; support for decimal conversion routines
 108 (defvar 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 (defconst all-decimal-digs-minval (aref powers-of-10 (1- decimal-digits)))
 117 (defconst 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 (defconst halfword-bits (/ mantissa-bits 2)) ; bits in a halfword
 154 (defconst masklo (1- (ash 1 halfword-bits))) ; isolate the lower halfword
 155 (defconst maskhi (lognot masklo))       ; isolate the upper halfword
 156 (defconst 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.
 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
 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
 455
 456 (provide 'float)
 457
 458 ;;; float.el ends here