lisp/emacs-lisp/float.el

   1 ;;; float.el --- floating point arithmetic package.
   2
   3 ;; Copyright (C) 1986 Free Software Foundation, Inc.
   4 ;; Author Bill Rosenblatt
   5
   6 ;; This file is part of GNU Emacs.
   7
   8 ;; GNU Emacs is free software; you can redistribute it and/or modify
   9 ;; it under the terms of the GNU General Public License as published by
  10 ;; the Free Software Foundation; either version 1, or (at your option)
  11 ;; any later version.
  12
  13 ;; GNU Emacs is distributed in the hope that it will be useful,
  14 ;; but WITHOUT ANY WARRANTY; without even the implied warranty of
  15 ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16 ;; GNU General Public License for more details.
  17
  18 ;; You should have received a copy of the GNU General Public License
  19 ;; along with GNU Emacs; see the file COPYING.  If not, write to
  20 ;; the Free Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
  21
  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
  24 ;; base 2 exponent.
  25 ;;
  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.
  30 ;;
  31 ;; User interface:
  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
  36 ;;
  37 ;; Caveats:
  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).
  43 ;;
  44 ;;
  45 ;; Bill Rosenblatt
  46 ;; June 20, 1986
  47 ;;
  48
  49 ;; fundamental implementation constants
  50 (defconst exp-base 2
  51   "Base of exponent in this floating point representation.")
  52
  53 (defconst mantissa-bits 24
  54   "Number of significant bits in this floating point representation.")
  55
  56 (defconst decimal-digits 6
  57   "Number of decimal digits expected to be accurate.")
  58
  59 (defconst expt-digits 2
  60   "Maximum permitted digits in a scientific notation exponent.")
  61
  62 ;; other constants
  63 (defconst maxbit (1- mantissa-bits)
  64   "Number of highest bit")
  65
  66 (defconst mantissa-maxval (1- (ash 1 maxbit))
  67   "Maximum permissable value of mantissa")
  68
  69 (defconst mantissa-minval (ash 1 maxbit)
  70   "Minimum permissable value of mantissa")
  71
  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:
  77 1 - minus sign
  78 2 - integer part
  79 4 - fractional part
  80 8 - minus sign for power of ten
  81 9 - power of ten
  82 ")
  83
  84 (defconst high-bit-mask (ash 1 maxbit)
  85   "Masks all bits except the high-order (sign) bit.")
  86
  87 (defconst second-bit-mask (ash 1 (1- maxbit))
  88   "Masks all bits except the highest-order magnitude bit")
  89
  90 ;; various useful floating point constants
  91 (setq _f0 '(0 . 1))
  92
  93 (setq _f1/2 '(4194304 . -23))
  94
  95 (setq _f1 '(4194304 . -22))
  96
  97 (setq _f10 '(5242880 . -19))
  98
  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))
 107
 108 (setq all-decimal-digs-minval (aref powers-of-10 (1- decimal-digits))
 109       highest-power-of-10 (aref powers-of-10 decimal-digits))
 110
 111 (defun fashl (fnum)                     ; floating-point arithmetic shift left
 112   (cons (ash (car fnum) 1) (1- (cdr fnum))))
 113
 114 (defun fashr (fnum)                     ; floating point arithmetic shift right
 115   (cons (ash (car fnum) -1) (1+ (cdr fnum))))
 116
 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"
 125   fnum)
 126
 127 (defun abs (n)                          ; integer absolute value
 128   (if (>= n 0) n (- n)))
 129
 130 (defun fabs (fnum)                      ; re-normalize after taking abs value
 131   (normalize (cons (abs (car fnum)) (cdr fnum))))
 132
 133 (defun xor (a b)                        ; logical exclusive or
 134   (and (or a b) (not (and a b))))
 135
 136 (defun same-sign (a b)                  ; two f-p numbers have same sign?
 137   (not (xor (natnump (car a)) (natnump (car b)))))
 138
 139 (defun extract-match (str i)            ; used after string-match
 140   (condition-case ()
 141       (substring str (match-beginning i) (match-end i))
 142     (error "")))
 143
 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)))
 149
 150 (defun hihalf (n)                       ; return high halfword, shifted down
 151   (ash (logand n maskhi) (- halfword-bits)))
 152
 153 (defun lohalf (n)                       ; return low halfword
 154   (logand n masklo))
 155
 156 ;; Visible functions
 157
 158 ;; Arithmetic functions
 159 (defun f+ (a1 a2)
 160   "Returns the sum of two floating point numbers."
 161   (let ((f1 (fmax a1 a2))
 162         (f2 (fmin a1 a2)))
 163     (if (same-sign a1 a2)
 164         (setq f1 (fashr f1)             ; shift right to avoid overflow
 165               f2 (fashr f2)))
 166     (normalize
 167      (cons (+ (car f1) (ash (car f2) (- (cdr f2) (cdr f1))))
 168            (cdr f1)))))
 169
 170 (defun f- (a1 &optional a2)             ; unary or binary minus
 171   "Returns the difference of two floating point numbers."
 172   (if a2
 173       (f+ a1 (f- a2))
 174     (normalize (cons (- (car a1)) (cdr a1)))))
 175
 176 (defun f* (a1 a2)                       ; multiply in halfword chunks
 177   "Returns the product of two floating point numbers."
 178   (let* ((i1 (car (fabs a1)))
 179          (i2 (car (fabs a2)))
 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)))
 187                     (hihalf prodlo))))
 188     (if (> (lohalf prodlo) round-limit)
 189         (setq prodhi (1+ prodhi)))      ; round off truncated bits
 190     (normalize
 191      (cons (if sign (- prodhi) prodhi)
 192            (+ (cdr (fabs a1)) (cdr (fabs a2)) mantissa-bits)))))
 193
 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))
 199           (quotient 0)
 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)
 209               bits (1- bits)))
 210       (normalize
 211        (cons (if sign (- quotient) quotient)
 212              (- (cdr (fabs a1)) (cdr (fabs a2)) (1- maxbit)))))))
 213
 214 (defun f% (a1 a2)
 215   "Returns the remainder of first floating point number divided by second."
 216   (f- a1 (f* (ftrunc (f/ a1 a2)) a2)))
 217
 218
 219 ;; Comparison functions
 220 (defun f= (a1 a2)
 221   "Returns t if two floating point numbers are equal, nil otherwise."
 222   (equal a1 a2))
 223
 224 (defun f> (a1 a2)
 225   "Returns t if first floating point number is greater than second,
 226 nil otherwise."
 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.
 235         (t
 236          (> (car a1) (car a2)))))       ; same exponents.
 237
 238 (defun f>= (a1 a2)
 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)))
 242
 243 (defun f< (a1 a2)
 244   "Returns t if first floating point number is less than second,
 245 nil otherwise."
 246   (not (f>= a1 a2)))
 247
 248 (defun f<= (a1 a2)
 249   "Returns t if first floating point number is less than or equal to
 250 second, nil otherwise."
 251   (not (f> a1 a2)))
 252
 253 (defun f/= (a1 a2)
 254   "Returns t if first floating point number is not equal to second,
 255 nil otherwise."
 256   (not (f= a1 a2)))
 257
 258 (defun fmin (a1 a2)
 259   "Returns the minimum of two floating point numbers."
 260   (if (f< a1 a2) a1 a2))
 261
 262 (defun fmax (a1 a2)
 263   "Returns the maximum of two floating point numbers."
 264   (if (f> a1 a2) a1 a2))
 265
 266 (defun fzerop (fnum)
 267   "Returns t if the floating point number is zero, nil otherwise."
 268   (= (car fnum) 0))
 269
 270 (defun floatp (fnum)
 271   "Returns t if the arg is a floating point number, nil otherwise."
 272   (and (consp fnum) (integerp (car fnum)) (integerp (cdr fnum))))
 273
 274 ;; Conversion routines
 275 (defun f (int)
 276   "Convert the integer argument to floating point, like a C cast operator."
 277   (normalize (cons int '0)))
 278
 279 (defun int-to-hex-string (int)
 280   "Convert the integer argument to a C-style hexadecimal string."
 281   (let ((shiftval -20)
 282         (str "0x")
 283         (hex-chars "0123456789ABCDEF"))
 284     (while (<= shiftval 0)
 285       (setq str (concat str (char-to-string
 286                         (aref hex-chars
 287                               (logand (lsh int shiftval) 15))))
 288             shiftval (+ shiftval 4)))
 289     str))
 290
 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
 294          fnum)
 295         ((<= (cdr fnum) (- maxbit))     ; it's all fractional, return 0
 296          '(0 . 1))
 297         (t                              ; otherwise mask out fractional bits
 298          (let ((mant (car fnum)) (exp (cdr fnum)))
 299            (normalize
 300             (cons (if (natnump mant)    ; if negative, use absolute value
 301                       (ash (ash mant exp) (- exp))
 302                     (- (ash (ash (- mant) exp) (- exp))))
 303                   exp))))))
 304
 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"
 310            mantissa-maxval)
 311           ((<= texp (- mantissa-bits))  ; too low, return "minint"
 312            mantissa-minval)
 313           (t                            ; in range
 314            (ash tint texp)))))          ; shift so that exponent is 0
 315
 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))
 322
 323     (if (f= fnum _f0)
 324         "0"
 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)
 328               (setq pow10 temp
 329                     power (+ power decimal-digits)))
 330             (while (f<= (setq temp (f* pow10 _f10)) value)
 331               (setq pow10 temp
 332                     power (1+ power))))
 333         (progn                          ; value < 1, power is negative
 334           (while (f> (setq temp (f/ pow10 highest-power-of-10)) value)
 335             (setq pow10 temp
 336                   power (- power decimal-digits)))
 337           (while (f> pow10 value)
 338             (setq pow10 (f/ pow10 _f10)
 339                   power (1- power)))))
 340                                           ; get value in range 100000 to 999999
 341       (setq value (f* (f/ value pow10) all-decimal-digs-minval)
 342             result (ftrunc value))
 343       (let (int)
 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))
 348         (if (>= int 1000000)
 349             (setq power (1+ power))))
 350
 351       (if sci                           ; scientific notation
 352           (setq str (concat (substring str 0 1) "." (substring str 1)
 353                             "E" (int-to-string power)))
 354
 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)))))
 362
 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)
 368                          zeroes (1- zeroes)))
 369                  (setq str (concat "0." str))))
 370
 371               (t                                ; in range, insert decimal point
 372                (setq str (concat
 373                           (substring str 0 (1+ power))
 374                           "."
 375                           (substring str (1+ power)))))))
 376
 377       (if sign                          ; if negative, prepend minus sign
 378           (concat "-" str)
 379         str))))
 380
 381
 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
 391 are recognized."
 392
 393   (if (string-match floating-point-regexp str 0)
 394       (let (power)
 395         (f*
 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))
 402
 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))
 410
 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)))))
 416
 417            ; round up and add minus sign, if necessary
 418            (f (* (+ (string-to-int digit-string)
 419                     (if round-up 1 0))
 420                  (if mant-sign -1 1))))
 421
 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)
 426                 (func 'f*))
 427
 428            (setq expt (+ (* (string-to-int
 429                              (substring expt-subst 0
 430                                         (min expt-digits (length expt-subst))))
 431                             (if expt-sign -1 1))
 432                          power))
 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
 436
 437            (setq chunks (/ expt decimal-digits)
 438                  tens (% expt decimal-digits))
 439            ; divide or multiply by "chunks" of 10**6
 440            (while (> chunks 0)
 441              (setq exponent (funcall func exponent highest-power-of-10)
 442                    chunks (1- chunks)))
 443            ; divide or multiply by remaining power of ten
 444            (funcall func exponent (aref powers-of-10 tens)))))
 445
 446     _f0))                               ; if invalid, return 0
 447
 448 (provide 'float)
 449
 450 ;;; float.el ends here