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Source for wiki FixnumsCowan version 2
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127.11.51.1
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FixnumsCowan
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{{{
#!html
<p>
Fixnums are an implementation-defined subset of the exact integers.
Every implementation must define its fixnum range as a closed
interval [-2<sup>w-1</sup>, 2<sup>w-1</sup>-1],
such that <em>w</em> is an integer <em>w</em> greater than or equal to 24. Every
mathematical integer within an implementation's fixnum range must
correspond to an exact integer that is representable within the
implementation.
A fixnum is an exact integer whose value lies within this
fixnum range.</p>
<p>
Fixnum operations perform integer arithmetic on their fixnum
arguments. If any argument is not a fixnum, or if the mathematical result
is not representable as a fixnum, it is an error. In particular, this means
that fixnum operations may return a mathematically incorrect fixnum in these
situations without raising an error.</p>
<p>
This section uses <i>fx</i>, <i>fx<sub>1</sub></i>, <i>fx<sub>2</sub></i>, etc., as parameter
names for fixnum arguments.</p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_856"></a>fixnum?<i> obj</i>)</tt></div>
<p>
Returns <tt>#t</tt> if <i>obj</i> is an exact
integer within the fixnum range, <tt>#f</tt> otherwise.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_858"></a>fixnum-width<i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_860"></a>least-fixnum<i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_862"></a>greatest-fixnum<i></i>)</tt></div>
<p>
These procedures return <em>w</em>,
−2<sup><em>w</em>−1</sup> and 2<sup><em>w</em>−1</sup> − 1: the
width, minimum and the maximum value of the fixnum range, respectively.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_864"></a>fx=?<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i> <tt>...</tt></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_866"></a>fx>?<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i> <tt>...</tt></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_868"></a>fx<?<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i> <tt>...</tt></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_870"></a>fx>=?<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i> <tt>...</tt></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_872"></a>fx<=?<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i> <tt>...</tt></i>)</tt></div>
<p>
These procedures return <tt>#t</tt> if their arguments are (respectively):
equal, monotonically increasing, monotonically decreasing,
monotonically nondecreasing, or monotonically nonincreasing,
<tt>#f</tt> otherwise.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_874"></a>fxzero?<i> fx</i>)</tt></div>
<div align=left><tt>(<a name="node_idx_876"></a>fxpositive?<i> fx</i>)</tt></div>
<div align=left><tt>(<a name="node_idx_878"></a>fxnegative?<i> fx</i>)</tt></div>
<div align=left><tt>(<a name="node_idx_880"></a>fxodd?<i> fx</i>)</tt></div>
<div align=left><tt>(<a name="node_idx_882"></a>fxeven?<i> fx</i>)</tt></div>
<p>
These numerical predicates test a fixnum for a particular property,
returning <tt>#t</tt> or <tt>#f</tt>. The five properties tested by
these procedures are: whether the number object is zero, greater than zero,
less than zero, odd, or even.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_884"></a>fxmax<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <tt>...</tt></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_886"></a>fxmin<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <tt>...</tt></i>)</tt></div>
<p>
These procedures return the maximum or minimum of their arguments.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_888"></a>fx+<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_890"></a>fx*<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<p>
These procedures return the sum or product of their arguments.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_892"></a>fx-<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_894"></a>fx-<i> fx</i>)</tt></div>
<p>
With two arguments, this procedure returns the difference
<i>fx<sub>1</sub></i>−<i>fx<sub>2</sub></i>.</p>
<p>
With one argument, this procedure returns the additive
inverse of its argument.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_896a"></a>fxfloor/<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_898a"></a>fxfloor-quotient<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_900a"></a>fxfloor-remainder<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_896b"></a>fxceiling/<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_898b"></a>fxceiling-quotient<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_900b"></a>fxceiling-remainder<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_896c"></a>fxtruncate/<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_898c"></a>fxtruncate-quotient<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_900c"></a>fxtruncate-remainder<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_896d"></a>fxround/<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_898d"></a>fxround-quotient<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_900d"></a>fxround-remainder<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_896e"></a>fxeuclidean/<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_898e"></a>fxeuclidean-quotient<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_900e"></a>fxeuclidean-remainder<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_896f"></a>fxbalanced/<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_898f"></a>fxbalanced-quotient<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_900f"></a>fxbalanced-remainder<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<p>It is an error if <i>fx<sub>2</sub></i> is zero.
These procedures implement integer division and
return the results of the corresponding operations
specified in <a href="DivisionRiastradh">DivisionRiastradh</a>.</p>
<p>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_908"></a>fx+/carry<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i></i>)</tt></div>
<p>
Returns the two fixnum results of the following computation:
</p>
<tt>(let* ((s (+ <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i>))<br/>
(s0 (mod0 s (expt 2 (fixnum-width))))<br/>
(s1 (div0 s (expt 2 (fixnum-width)))))<br/>
(values s0 s1))</p>
<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_910"></a>fx-/carry<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i></i>)</tt></div>
<p>
Returns the two fixnum results of the following computation:
</p>
<tt>(let* ((d (- <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i>))<br/>
(d0 (mod0 d (expt 2 (fixnum-width))))<br/>
(d1 (div0 d (expt 2 (fixnum-width)))))<br/>
(values d0 d1))</p>
<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_912"></a>fx*/carry<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i></i>)</tt></div>
<p>
Returns the two fixnum results of the following computation:
</p>
<tt>(let* ((s (+ (* <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i>) <i>fx<sub>3</sub></i>))<br/>
(s0 (mod0 s (expt 2 (fixnum-width))))<br/>
(s1 (div0 s (expt 2 (fixnum-width)))))<br/>
= (values s0 s1))</p>
<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_914"></a>fxnot<i> <i>fx</i></i>)</tt></div>
<p>
Returns the unique fixnum that is congruent
mod 2<sup><em>w</em></sup> to the one's-complement of <i>fx</i>.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_916"></a>fxand<i> <i>fx<sub>1</sub></i> <tt>...</tt></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_918"></a>fxior<i> <i>fx<sub>1</sub></i> <tt>...</tt></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_920"></a>fxxor<i> <i>fx<sub>1</sub></i> <tt>...</tt></i>)</tt></div>
<p>
These procedures return the fixnum that is the bit-wise “and”,
“inclusive or”, or “exclusive or” of the two's complement
representations of their arguments. If they are passed only one
argument, they return that argument. If they are passed no arguments,
they return the fixnum (either −1 or 0) that acts as identity for the
operation.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_922"></a>fxif<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i></i>)</tt></div>
<p>
Returns the fixnum that is the bit-wise “if” of the two's complement
representations of its arguments, i.e. for each bit, if it is 1 in
<i>fx<sub>1</sub></i>, the corresponding bit in <i>fx<sub>2</sub></i> becomes the value of
the corresponding bit in the result, and if it is 0, the corresponding
bit in <i>fx<sub>3</sub></i> becomes the corresponding bit in the value of the
result. This is the fixnum result of the following computation:
</p>
<tt>(fxior (fxand <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i>)<br/>
(fxand (fxnot <i>fx<sub>1</sub></i>) <i>fx<sub>3</sub></i>))</p>
<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_924"></a>fxbit-count<i> <i>fx</i></i>)</tt></div>
<p>
If <i>fx</i> is non-negative, this procedure returns the
number of 1 bits in the two's complement representation of <i>fx</i>.
Otherwise it returns the result of the following computation:
</p>
<tt>(fxnot (fxbit-count (fxnot <i>ei</i>)))<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_926"></a>fxlength<i> <i>fx</i></i>)</tt></div>
<p>
Returns the number of bits needed to represent <i>fx</i> if it is
positive, and the number of bits needed to represent <tt>(fxnot
<i>fx</i>)</tt> if it is negative, which is the fixnum result of the
following computation:
</p>
<tt>(do ((result 0 (+ result 1))<br/>
(bits (if (fxnegative? <i>fx</i>)<br/>
(fxnot <i>fx</i>)<br/>
<i>fx</i>)<br/>
(fxarithmetic-shift-right bits 1)))<br/>
((fxzero? bits)<br/>
result))</p>
<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_928"></a>fxfirst-bit-set<i> <i>fx</i></i>)</tt></div>
<p>
Returns the index of the least significant 1 bit in
the two's complement representation of <i>fx</i>. If
<i>fx</i> is 0, then −1 is returned.
</p>
<tt>(fxfirst-bit-set 0) <span style="margin-left: 2em">‌</span>⇒ -1<br/>
(fxfirst-bit-set 1) <span style="margin-left: 2em">‌</span>⇒ 0<br/>
(fxfirst-bit-set -4) <span style="margin-left: 2em">‌</span>⇒ 2</p>
<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_930"></a>fxbit-set?<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<p>
It is an error if <i>fx<sub>2</sub></i> is negative, or is greater than or equal to <tt>(fixnum-width)</tt>. The <tt>fxbit-set?</tt> procedure returns
<tt>#t</tt> if the <i>fx<sub>2</sub></i>th bit is 1 in the two's complement
representation of <i>fx<sub>1</sub></i>, and <tt>#f</tt> otherwise. This is the
fixnum result of the following computation:
</p>
<tt>(not
<br/> (fxzero?</p>
<br/> (fxand <i>fx<sub>1</sub></i><br/>
(fxarithmetic-shift-left 1 <i>fx<sub>2</sub></i>))))</p>
<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_932"></a>fxcopy-bit<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i></i>)</tt></div>
<p>
It is an error if <i>fx<sub>2</sub></i> is negative, or is greater than or equal to <tt>(fixnum-width)</tt>. It is also an error if <i>Fx<sub>3</sub></i> is neither 0 nor
1. The <tt>fxcopy-bit</tt> procedure returns the result of replacing
the <i>fx<sub>2</sub></i>th bit of <i>fx<sub>1</sub></i> by <i>fx<sub>3</sub></i>, which is
the result of the following computation:
</p>
<tt>(let* ((mask (fxarithmetic-shift-left 1 <i>fx<sub>2</sub></i>)))<br/>
(fxif mask<br/>
(fxarithmetic-shift-left <i>fx<sub>3</sub></i> <i>fx<sub>2</sub></i>)<br/>
<i>fx<sub>1</sub></i>))</p>
<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_934"></a>fxbit-field<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i></i>)</tt></div>
<p>
It is an error if either of <i>Fx<sub>2</sub></i> and <i>fx<sub>3</sub></i> is negative, or if either is greater than or equal to
<tt>(fixnum-width)</tt>. It is also an error if <i>fx<sub>2</sub></i> is greater than
<i>fx<sub>3</sub></i>. The <tt>fxbit-field</tt> procedure returns the
number represented by the bits at the positions from <i>fx<sub>2</sub></i> (inclusive) to
<i>fx<sub>3</sub></i> (exclusive), which is
the fixnum result of the following computation:
</p>
<tt>(let* ((mask (fxnot<br/>
(fxarithmetic-shift-left -1 <i>fx<sub>3</sub></i>))))<br/>
(fxarithmetic-shift-right (fxand <i>fx<sub>1</sub></i> mask)<br/>
<i>fx<sub>2</sub></i>))</p>
<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_936"></a>fxcopy-bit-field<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i> <i>fx<sub>4</sub></i></i>)</tt></div>
<p>
It is an error if either <i>fx<sub>2</sub></i> or <i>fx<sub>3</sub></i> or if either is greater than
<tt>(fixnum-width)</tt>. It is also an error if <i>fx<sub>2</sub></i> is greater than
<i>fx<sub>3</sub></i>. The <tt>fxcopy-bit-field</tt> procedure returns
the result of replacing in <i>fx<sub>1</sub></i> the bits at positions from
<i>fx<sub>2</sub></i> (inclusive) to <i>fx<sub>3</sub></i> (exclusive) by the corresponding bits in <i>fx<sub>4</sub></i>, which
is the fixnum result of the following computation:
</p>
<tt>(let* ((to <i>fx<sub>1</sub></i>)<br/>
(start <i>fx<sub>2</sub></i>)<br/>
(end <i>fx<sub>3</sub></i>)<br/>
(from <i>fx<sub>4</sub></i>)<br/>
(mask1 (fxarithmetic-shift-left -1 start))<br/>
(mask2 (fxnot<br/>
(fxarithmetic-shift-left -1 end)))<br/>
(mask (fxand mask1 mask2)))<br/>
(fxif mask<br/>
(fxarithmetic-shift-left from start)<br/>
to))</p>
<p></tt>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_938"></a>fxarithmetic-shift<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<p>
It is an error if the absolute value of <i>fx<sub>2</sub></i> is greater than or equal to
<tt>(fixnum-width)</tt>. If
</p>
<tt>(floor (* <i>fx<sub>1</sub></i> (expt 2 <i>fx<sub>2</sub></i>)))<p></tt>
is a fixnum, then that fixnum is returned; otherwise it is an error.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_940"></a>fxarithmetic-shift-left<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<div align=left><tt>(<a name="node_idx_942"></a>fxarithmetic-shift-right<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i></i>)</tt></div>
<p>
It is an error if <i>fx<sub>2</sub></i> is negative, or is greater than or equal to <tt>(fixnum-width)</tt>.
The <tt>fxarithmetic-shift-left</tt> procedure behaves the same as <tt>fxarithmetic-shift</tt>, and <tt>(fxarithmetic-shift-right <i>fx<sub>1</sub></i>
<i>fx<sub>2</sub></i>)</tt> behaves the same as <tt>(fxarithmetic-shift <i>fx<sub>1</sub></i>
(fx- <i>fx<sub>2</sub></i>))</tt>.
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_944"></a>fxrotate-bit-field<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i> <i>fx<sub>4</sub></i></i>)</tt></div>
<p>
It is an error if any of <i>Fx<sub>2</sub></i>, <i>fx<sub>3</sub></i>, and <i>fx<sub>4</sub></i>
are negative, or are greater than or equal to
<tt>(fixnum-width)</tt>. It is also an error if <i>Fx<sub>2</sub></i> is greater than
<i>fx<sub>3</sub></i>, or if <i>Fx<sub>4</sub></i> is greater than or equal to the difference
between <i>fx<sub>3</sub></i> and <i>fx<sub>2</sub></i>. The <tt>fxrotate-bit-field</tt>
procedure returns the result of cyclically permuting in <i>fx<sub>1</sub></i> the
bits at positions from <i>fx<sub>2</sub></i> (inclusive) to <i>fx<sub>3</sub></i>
(exclusive) by <i>fx<sub>4</sub></i> bits
towards the more significant bits, which is the result of the
following computation:
</p>
<tt>(let* ((n <i>fx<sub>1</sub></i>)<br/>
(start <i>fx<sub>2</sub></i>)<br/>
(end <i>fx<sub>3</sub></i>)<br/>
(count <i>fx<sub>4</sub></i>)<br/>
(width (fx- end start)))<br/>
(if (fxpositive? width)<br/>
(let* ((count (fxmod count width))<br/>
(field0<br/>
(fxbit-field n start end))<br/>
(field1<br/>
(fxarithmetic-shift-left<br/>
field0 count))<br/>
(field2<br/>
(fxarithmetic-shift-right<br/>
field0 (fx- width count)))<br/>
(field (fxior field1 field2)))<br/>
(fxcopy-bit-field n start end field))<br/>
n))</p>
<p></tt></p>
<p>
</p>
<p></p>
<p>
</p>
<p></p>
<div align=left><tt>(<a name="node_idx_946"></a>fxreverse-bit-field<i> <i>fx<sub>1</sub></i> <i>fx<sub>2</sub></i> <i>fx<sub>3</sub></i></i>)</tt></div>
<p>
It is an error if either of <i>Fx<sub>2</sub></i> and <i>fx<sub>3</sub></i> is negative, or if either is
greater than
<tt>(fixnum-width)</tt>. It is also an error if <i>fx<sub>2</sub></i> is greater than
<i>fx<sub>3</sub></i>. The <tt>fxreverse-bit-field</tt> procedure
returns
the fixnum obtained from <i>fx<sub>1</sub></i> by reversing the
order of the bits at positions from <i>fx<sub>2</sub></i> (inclusive) to
<i>fx<sub>3</sub></i> (exclusive).
</p>
<tt>(fxreverse-bit-field <tt>#</tt>b1010010 1 4) <br><span style="margin-left: 2em">‌</span><span style="margin-left: 2em">‌</span>⇒ 88 ; <tt>#</tt>b1011000<p></tt></p>
<p>
</p>
<p></p>
<p>
</p>
}}}
time
2016-07-11 08:23:52
version
2