1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
|
;;; "bytenumb.scm" Byte integer and IEEE floating-point conversions.
; Copyright (C) 2003 Aubrey Jaffer
;
;Permission to copy this software, to modify it, to redistribute it,
;to distribute modified versions, and to use it for any purpose is
;granted, subject to the following restrictions and understandings.
;
;1. Any copy made of this software must include this copyright notice
;in full.
;
;2. I have made no warranty or representation that the operation of
;this software will be error-free, and I am under no obligation to
;provide any services, by way of maintenance, update, or otherwise.
;
;3. In conjunction with products arising from the use of this
;material, there shall be no use of my name in any advertising,
;promotional, or sales literature without prior written consent in
;each case.
(require 'byte)
(require 'logical)
;;@code{(require 'byte-number)}
;;@ftindex byte-number
;;@noindent
;;The multi-byte sequences produced and used by numeric conversion
;;routines are always big-endian. Endianness can be changed during
;;reading and writing bytes using @code{read-bytes} and
;;@code{write-bytes} @xref{Byte, read-bytes}.
;;
;;@noindent
;;The sign of the length argument to bytes/integer conversion
;;procedures determines the signedness of the number.
;;@body
;;Converts the first @code{(abs @var{n})} bytes of big-endian @1 array
;;to an integer. If @2 is negative then the integer coded by the
;;bytes are treated as two's-complement (can be negative).
;;
;;@example
;;(bytes->integer (bytes 0 0 0 15) -4) @result{} 15
;;(bytes->integer (bytes 0 0 0 15) 4) @result{} 15
;;(bytes->integer (bytes 255 255 255 255) -4) @result{} -1
;;(bytes->integer (bytes 255 255 255 255) 4) @result{} 4294967295
;;(bytes->integer (bytes 128 0 0 0) -4) @result{} -2147483648
;;(bytes->integer (bytes 128 0 0 0) 4) @result{} 2147483648
;;@end example
(define (bytes->integer bytes n)
(define cnt (abs n))
(cond ((zero? n) 0)
((and (negative? n) (> (byte-ref bytes 0) 127))
(do ((lng (- 255 (byte-ref bytes 0))
(+ (- 255 (byte-ref bytes idx)) (* 256 lng)))
(idx 1 (+ 1 idx)))
((>= idx cnt) (- -1 lng))))
(else
(do ((lng (byte-ref bytes 0)
(+ (byte-ref bytes idx) (* 256 lng)))
(idx 1 (+ 1 idx)))
((>= idx cnt) lng)))))
;;@body
;;Converts the integer @1 to a byte-array of @code{(abs @var{n})}
;;bytes. If @1 and @2 are both negative, then the bytes in the
;;returned array are coded two's-complement.
;;
;;@example
;;(bytes->list (integer->bytes 15 -4)) @result{} (0 0 0 15)
;;(bytes->list (integer->bytes 15 4)) @result{} (0 0 0 15)
;;(bytes->list (integer->bytes -1 -4)) @result{} (255 255 255 255)
;;(bytes->list (integer->bytes 4294967295 4)) @result{} (255 255 255 255)
;;(bytes->list (integer->bytes -2147483648 -4)) @result{} (128 0 0 0)
;;(bytes->list (integer->bytes 2147483648 4)) @result{} (128 0 0 0)
;;@end example
(define (integer->bytes n len)
(define bytes (make-bytes (abs len)))
(cond ((and (negative? n) (negative? len))
(do ((idx (+ -1 (abs len)) (+ -1 idx))
(res (- -1 n) (quotient res 256)))
((negative? idx) bytes)
(byte-set! bytes idx (- 255 (modulo res 256)))))
(else
(do ((idx (+ -1 (abs len)) (+ -1 idx))
(res n (quotient res 256)))
((negative? idx) bytes)
(byte-set! bytes idx (modulo res 256))))))
;;@body
;;@1 must be a 4-element byte-array. @0 calculates and returns the
;;value of @1 interpreted as a big-endian IEEE 4-byte (32-bit) number.
(define (bytes->ieee-float bytes)
(define zero (or (string->number "0.0") 0))
(define one (or (string->number "1.0") 1))
(define len (bytes-length bytes))
(define S (logbit? 7 (byte-ref bytes 0)))
(define E (+ (ash (logand #x7F (byte-ref bytes 0)) 1)
(ash (logand #x80 (byte-ref bytes 1)) -7)))
(if (not (eqv? 4 len))
(slib:error 'bytes->ieee-float 'wrong 'length len))
(do ((F (byte-ref bytes (+ -1 len))
(+ (byte-ref bytes idx) (/ F 256)))
(idx (+ -2 len) (+ -1 idx)))
((<= idx 1)
(set! F (/ (+ (logand #x7F (byte-ref bytes 1)) (/ F 256)) 128))
(cond ((< 0 E 255) (* (if S (- one) one) (expt 2 (- E 127)) (+ 1 F)))
((zero? E)
(if (zero? F)
(if S (- zero) zero)
(* (if S (- one) one) (expt 2 -126) F)))
;; E must be 255
((not (zero? F)) (/ zero zero))
(else (/ (if S (- one) one) zero))))))
;; S EEEEEEE E FFFFFFF FFFFFFFF FFFFFFFF
;; ========= ========= ======== ========
;; 0 1 8 9 31
;;@example
;;(bytes->ieee-float (bytes 0 0 0 0)) @result{} 0.0
;;(bytes->ieee-float (bytes #x80 0 0 0)) @result{} -0.0
;;(bytes->ieee-float (bytes #x40 0 0 0)) @result{} 2.0
;;(bytes->ieee-float (bytes #x40 #xd0 0 0)) @result{} 6.5
;;(bytes->ieee-float (bytes #xc0 #xd0 0 0)) @result{} -6.5
;;
;;(bytes->ieee-float (bytes 0 #x80 0 0)) @result{} 11.754943508222875e-39
;;(bytes->ieee-float (bytes 0 #x40 0 0)) @result{} 5.877471754111437e-39
;;(bytes->ieee-float (bytes 0 0 0 1)) @result{} 1.401298464324817e-45
;;
;;(bytes->ieee-float (bytes #xff #x80 0 0)) @result{} -inf.0
;;(bytes->ieee-float (bytes #x7f #x80 0 0)) @result{} +inf.0
;;(bytes->ieee-float (bytes #x7f #x80 0 1)) @result{} 0/0
;;(bytes->ieee-float (bytes #x7f #xc0 0 0)) @result{} 0/0
;;@end example
;;@body
;;@1 must be a 8-element byte-array. @0 calculates and returns the
;;value of @1 interpreted as a big-endian IEEE 8-byte (64-bit) number.
(define (bytes->ieee-double bytes)
(define zero (or (string->number "0.0") 0))
(define one (or (string->number "1.0") 1))
(define len (bytes-length bytes))
(define S (logbit? 7 (byte-ref bytes 0)))
(define E (+ (ash (logand #x7F (byte-ref bytes 0)) 4)
(ash (logand #xF0 (byte-ref bytes 1)) -4)))
(if (not (eqv? 8 len))
(slib:error 'bytes->ieee-double 'wrong 'length len))
(do ((F (byte-ref bytes (+ -1 len))
(+ (byte-ref bytes idx) (/ F 256)))
(idx (+ -2 len) (+ -1 idx)))
((<= idx 1)
(set! F (/ (+ (logand #x0F (byte-ref bytes 1)) (/ F 256)) 16))
(cond ((< 0 E 2047) (* (if S (- one) one) (expt 2 (- E 1023)) (+ 1 F)))
((zero? E)
(if (zero? F)
(if S (- zero) zero)
(* (if S (- one) one) (expt 2 -1022) F)))
;; E must be 2047
((not (zero? F)) (/ zero zero))
(else (/ (if S (- one) one) zero))))))
;; S EEEEEEE EEEE FFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF
;; ========= ========= ======== ======== ======== ======== ======== ========
;; 0 1 11 12 63
;;@example
;;(bytes->ieee-double (bytes 0 0 0 0 0 0 0 0)) @result{} 0.0
;;(bytes->ieee-double (bytes #x80 0 0 0 0 0 0 0)) @result{} -0.0
;;(bytes->ieee-double (bytes #x40 0 0 0 0 0 0 0)) @result{} 2.0
;;(bytes->ieee-double (bytes #x40 #x1A 0 0 0 0 0 0)) @result{} 6.5
;;(bytes->ieee-double (bytes #xC0 #x1A 0 0 0 0 0 0)) @result{} -6.5
;;
;;(bytes->ieee-double (bytes 0 8 0 0 0 0 0 0)) @result{} 11.125369292536006e-309
;;(bytes->ieee-double (bytes 0 4 0 0 0 0 0 0)) @result{} 5.562684646268003e-309
;;(bytes->ieee-double (bytes 0 0 0 0 0 0 0 1)) @result{} 4.0e-324
;;
;;(bytes->ieee-double (list->bytes '(127 239 255 255 255 255 255 255))) 179.76931348623157e306
;;(bytes->ieee-double (bytes #xFF #xF0 0 0 0 0 0 0)) @result{} -inf.0
;;(bytes->ieee-double (bytes #x7F #xF0 0 0 0 0 0 0)) @result{} +inf.0
;;(bytes->ieee-double (bytes #x7F #xF8 0 0 0 0 0 0)) @result{} 0/0
;;@end example
;;@args x
;;Returns a 4-element byte-array encoding the IEEE single-precision
;;floating-point of @1.
(define ieee-float->bytes
(let ((exactify (if (provided? 'inexact) inexact->exact identity)))
(lambda (flt)
(define byts (make-bytes 4 0))
(define S (and (real? flt) (negative? (if (zero? flt) (/ flt) flt))))
(define (scale flt scl)
(cond ((zero? scl) (out (/ flt 2) scl))
((>= flt 16)
(let ((flt/16 (/ flt 16)))
(cond ((= flt/16 flt)
(byte-set! byts 0 (if S #xFF #x7F))
(byte-set! byts 1 #x80)
byts)
(else (scale flt/16 (+ scl 4))))))
((>= flt 2) (scale (/ flt 2) (+ scl 1)))
((and (>= scl 4)
(< (* 16 flt) 1)) (scale (* flt 16) (+ scl -4)))
((< flt 1) (scale (* flt 2) (+ scl -1)))
(else (out (+ -1 flt) scl))))
(define (out flt scl)
(do ((flt (* 128 flt) (* 256 (- flt val)))
(val (exactify (floor (* 128 flt)))
(exactify (floor (* 256 (- flt val)))))
(idx 1 (+ 1 idx)))
((> idx 3)
(byte-set! byts 1 (bitwise-if #x80 (ash scl 7) (byte-ref byts 1)))
(byte-set! byts 0 (+ (if S 128 0) (ash scl -1)))
byts)
(byte-set! byts idx val)))
(set! flt (magnitude flt))
(cond ((zero? flt) (if S (byte-set! byts 0 #x80)) byts)
((or (not (real? flt))
(not (= flt flt)))
(byte-set! byts 0 (if S #xFF #x7F))
(byte-set! byts 1 #xC0)
byts)
(else (scale flt 127))))))
;;@example
;;(bytes->list (ieee-float->bytes 0.0)) @result{} (0 0 0 0)
;;(bytes->list (ieee-float->bytes -0.0)) @result{} (128 0 0 0)
;;(bytes->list (ieee-float->bytes 2.0)) @result{} (64 0 0 0)
;;(bytes->list (ieee-float->bytes 6.5)) @result{} (64 208 0 0)
;;(bytes->list (ieee-float->bytes -6.5)) @result{} (192 208 0 0)
;;
;;(bytes->list (ieee-float->bytes 11.754943508222875e-39)) @result{} ( 0 128 0 0)
;;(bytes->list (ieee-float->bytes 5.877471754111438e-39)) @result{} ( 0 64 0 0)
;;(bytes->list (ieee-float->bytes 1.401298464324817e-45)) @result{} ( 0 0 0 1)
;;
;;(bytes->list (ieee-float->bytes -inf.0)) @result{} (255 128 0 0)
;;(bytes->list (ieee-float->bytes +inf.0)) @result{} (127 128 0 0)
;;(bytes->list (ieee-float->bytes 0/0)) @result{} (127 192 0 0)
;;@end example
;;@args x
;;Returns a 8-element byte-array encoding the IEEE double-precision
;;floating-point of @1.
(define ieee-double->bytes
(let ((exactify (if (provided? 'inexact) inexact->exact identity)))
(lambda (flt)
(define byts (make-bytes 8 0))
(define S (and (real? flt) (negative? (if (zero? flt) (/ flt) flt))))
(define (scale flt scl)
(cond ((zero? scl) (out (/ flt 2) scl))
((>= flt 16)
(let ((flt/16 (/ flt 16)))
(cond ((= flt/16 flt)
(byte-set! byts 0 (if S #xFF #x7F))
(byte-set! byts 1 #xF0)
byts)
(else (scale flt/16 (+ scl 4))))))
((>= flt 2) (scale (/ flt 2) (+ scl 1)))
((and (>= scl 4)
(< (* 16 flt) 1)) (scale (* flt 16) (+ scl -4)))
((< flt 1) (scale (* flt 2) (+ scl -1)))
(else (out (+ -1 flt) scl))))
(define (out flt scl)
(do ((flt (* 16 flt) (* 256 (- flt val)))
(val (exactify (floor (* 16 flt)))
(exactify (floor (* 256 (- flt val)))))
(idx 1 (+ 1 idx)))
((> idx 7)
(byte-set! byts 1 (bitwise-if #xF0 (ash scl 4) (byte-ref byts 1)))
(byte-set! byts 0 (+ (if S 128 0) (ash scl -4)))
byts)
(byte-set! byts idx val)))
(set! flt (magnitude flt))
(cond ((zero? flt) (if S (byte-set! byts 0 #x80)) byts)
((or (not (real? flt))
(not (= flt flt)))
(byte-set! byts 0 #x7F)
(byte-set! byts 1 #xF8)
byts)
(else (scale flt 1023))))))
;;@example
;;(bytes->list (ieee-double->bytes 0.0)) @result{} (0 0 0 0 0 0 0 0)
;;(bytes->list (ieee-double->bytes -0.0)) @result{} (128 0 0 0 0 0 0 0)
;;(bytes->list (ieee-double->bytes 2.0)) @result{} (64 0 0 0 0 0 0 0)
;;(bytes->list (ieee-double->bytes 6.5)) @result{} (64 26 0 0 0 0 0 0)
;;(bytes->list (ieee-double->bytes -6.5)) @result{} (192 26 0 0 0 0 0 0)
;;
;;(bytes->list (ieee-double->bytes 11.125369292536006e-309))
;; @result{} ( 0 8 0 0 0 0 0 0)
;;(bytes->list (ieee-double->bytes 5.562684646268003e-309))
;; @result{} ( 0 4 0 0 0 0 0 0)
;;(bytes->list (ieee-double->bytes 4.0e-324))
;; @result{} ( 0 0 0 0 0 0 0 1)
;;
;;(bytes->list (ieee-double->bytes -inf.0)) @result{} (255 240 0 0 0 0 0 0)
;;(bytes->list (ieee-double->bytes +inf.0)) @result{} (127 240 0 0 0 0 0 0)
;;(bytes->list (ieee-double->bytes 0/0)) @result{} (127 248 0 0 0 0 0 0)
;;@end example
;;@subsubheading Byte Collation Order
;;
;;@noindent
;;The @code{string<?} ordering of big-endian byte-array
;;representations of fixed and IEEE floating-point numbers agrees with
;;the numerical ordering only when those numbers are non-negative.
;;
;;@noindent
;;Straighforward modification of these formats can extend the
;;byte-collating order to work for their entire ranges. This
;;agreement enables the full range of numbers as keys in
;;@dfn{indexed-sequential-access-method} databases.
;;@body
;;Modifies sign bit of @1 so that @code{string<?} ordering of
;;two's-complement byte-vectors matches numerical order. @0 returns
;;@1 and is its own functional inverse.
(define (integer-byte-collate! byte-vector)
(byte-set! byte-vector 0 (logxor #x80 (byte-ref byte-vector 0)))
byte-vector)
;;@body
;;Returns copy of @1 with sign bit modified so that @code{string<?}
;;ordering of two's-complement byte-vectors matches numerical order.
;;@0 is its own functional inverse.
(define (integer-byte-collate byte-vector)
(integer-byte-collate! (bytes-copy byte-vector)))
;;@body
;;Modifies @1 so that @code{string<?} ordering of IEEE floating-point
;;byte-vectors matches numerical order. @0 returns @1.
(define (ieee-byte-collate! byte-vector)
(cond ((logtest #x80 (byte-ref byte-vector 0))
(do ((idx (+ -1 (bytes-length byte-vector)) (+ -1 idx)))
((negative? idx))
(byte-set! byte-vector idx
(logxor #xFF (byte-ref byte-vector idx)))))
(else
(byte-set! byte-vector 0 (logxor #x80 (byte-ref byte-vector 0)))))
byte-vector)
;;@body
;;Given @1 modified by @code{ieee-byte-collate!}, reverses the @1
;;modifications.
(define (ieee-byte-decollate! byte-vector)
(cond ((not (logtest #x80 (byte-ref byte-vector 0)))
(do ((idx (+ -1 (bytes-length byte-vector)) (+ -1 idx)))
((negative? idx))
(byte-set! byte-vector idx
(logxor #xFF (byte-ref byte-vector idx)))))
(else
(byte-set! byte-vector 0 (logxor #x80 (byte-ref byte-vector 0)))))
byte-vector)
;;@body
;;Returns copy of @1 encoded so that @code{string<?} ordering of IEEE
;;floating-point byte-vectors matches numerical order.
(define (ieee-byte-collate byte-vector)
(ieee-byte-collate! (bytes-copy byte-vector)))
;;@body
;;Given @1 returned by @code{ieee-byte-collate}, reverses the @1
;;modifications.
(define (ieee-byte-decollate byte-vector)
(ieee-byte-decollate! (bytes-copy byte-vector)))
|
