r5rs-z-h-9.html

scheme 标准（r5rs）。Scheme是MIT发布的基于Lambda运算的语言

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_306"></a>truncate<i> x</i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_308"></a>round<i> x</i>)</tt>&nbsp;</div>
<p>

These procedures return integers.
<tt>Floor</tt> returns the largest integer not larger than <em>x</em>.
<tt>Ceiling</tt> returns the smallest integer not smaller than&nbsp;<em>x</em>.
<tt>Truncate</tt> returns the integer closest to <em>x</em> whose absolute
value is not larger than the absolute value of <em>x</em>.  <tt>Round</tt> returns the
closest integer to <em>x</em>, rounding to even when <em>x</em> is halfway between two
integers.<p>

<blockquote><em>Rationale:&nbsp;&nbsp;</em>
<tt>Round</tt> rounds to even for consistency with the default rounding
mode specified by the IEEE floating point standard.
</blockquote><p>

<blockquote><em>Note:&nbsp;&nbsp;</em>
If the argument to one of these procedures is inexact, then the result
will also be inexact.  If an exact value is needed, the
result should be passed to the <tt>inexact-&gt;exact</tt> procedure.
</blockquote><p>

<tt><p>(floor&nbsp;-4.3)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;&nbsp;-5.0<br>
(ceiling&nbsp;-4.3)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;&nbsp;-4.0<br>
(truncate&nbsp;-4.3)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;&nbsp;-4.0<br>
(round&nbsp;-4.3)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;&nbsp;-4.0<br>
<br>
(floor&nbsp;3.5)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;&nbsp;3.0<br>
(ceiling&nbsp;3.5)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;&nbsp;4.0<br>
(truncate&nbsp;3.5)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;&nbsp;3.0<br>
(round&nbsp;3.5)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;&nbsp;4.0&nbsp;&nbsp;;&nbsp;inexact<br>
<br>
(round&nbsp;7/2)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;&nbsp;4&nbsp;&nbsp;&nbsp;&nbsp;;&nbsp;exact<br>
(round&nbsp;7)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;&nbsp;7<p></tt><p>

<p><p>

<p><div align=left><u>library procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_310"></a>rationalize<i> x y</i>)</tt>&nbsp;</div>

<p>

<tt>Rationalize</tt> returns the <em>simplest</em> rational number
differing from <em>x</em> by no more than <em>y</em>.  A rational number <em>r</em><sub>1</sub> is
<em>simpler</em> <a name="%_idx_312"></a> than another rational number
<em>r</em><sub>2</sub> if <em>r</em><sub>1</sub>  =  <em>p</em><sub>1</sub>/<em>q</em><sub>1</sub> and <em>r</em><sub>2</sub>  =  <em>p</em><sub>2</sub>/<em>q</em><sub>2</sub> (in lowest terms) and |<em>p</em><sub>1</sub>|
<u>&lt;</u> |<em>p</em><sub>2</sub>| and |<em>q</em><sub>1</sub>| <u>&lt;</u> |<em>q</em><sub>2</sub>|.  Thus 3/5 is simpler than 4/7.
Although not all rationals are comparable in this ordering (consider 2/7
and 3/5) any interval contains a rational number that is simpler than
every other rational number in that interval (the simpler 2/5 lies
between 2/7 and 3/5).  Note that 0  =  0/1 is the simplest rational of
all.<p>

<tt><p>(rationalize<br>
&nbsp;&nbsp;(inexact-&gt;exact&nbsp;.3)&nbsp;1/10)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;1/3&nbsp;&nbsp;&nbsp;&nbsp;;&nbsp;exact<br>
(rationalize&nbsp;.3&nbsp;1/10)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;#i1/3&nbsp;&nbsp;;&nbsp;inexact<p></tt><p>

<p><p>

<p><div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_314"></a>exp<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_316"></a>log<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_318"></a>sin<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_320"></a>cos<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_322"></a>tan<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_324"></a>asin<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_326"></a>acos<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_328"></a>atan<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(atan<i> <em>y</em> <em>x</em></i>)</tt>&nbsp;</div>
<p>

These procedures are part of every implementation that supports
general
real numbers; they compute the usual transcendental functions.  <tt>Log</tt>
computes the natural logarithm of <em>z</em> (not the base ten logarithm).
<tt>Asin</tt>, <tt>acos</tt>, and <tt>atan</tt> compute arcsine (<tt>sin</tt><sup>-1</sup>),
arccosine (<tt>cos</tt><sup>-1</sup>), and arctangent (<tt>tan</tt><sup>-1</sup>), respectively.
The two-argument variant of <tt>atan</tt> computes <tt>(angle
(make-rectangular <em>x</em> <em>y</em>))</tt> (see below), even in implementations
that don't support general complex numbers.<p>

In general, the mathematical functions log, arcsine, arccosine, and
arctangent are multiply defined.
The value of <tt>log</tt> <em>z</em> is defined to be the one whose imaginary
part lies in the range from  - <img src="r5rs-Z-G-D-3.gif" border="0"> (exclusive) to <img src="r5rs-Z-G-D-3.gif" border="0"> (inclusive).
<tt>log</tt> 0 is undefined.
With <tt>log</tt> defined this way, the values of <tt>sin</tt><sup>-1</sup> <em>z</em>, <tt>cos</tt><sup>-1</sup> <em>z</em>,
and <tt>tan</tt><sup>-1</sup> <em>z</em> are according to the following formul&aelig;:
<p><div align=left><table><tr><td><tt>sin</tt><sup>-1</sup> <em>z</em>  =   - <em>i</em> <tt>log</tt> (<em>i</em> <em>z</em>  +  (1  -  <em>z</em><sup>2</sup>)<sup>1/2</sup>)</td></tr></table></div><p>
<p><div align=left><table><tr><td><tt>cos</tt><sup>-1</sup> <em>z</em>  =  <img src="r5rs-Z-G-D-3.gif" border="0"> / 2  -  <tt>sin</tt><sup>-1</sup> <em>z</em></td></tr></table></div><p>
<p><div align=left><table><tr><td><tt>tan</tt><sup>-1</sup> <em>z</em>  =  (<tt>log</tt> (1  +  <em>i</em> <em>z</em>)  -  <tt>log</tt> (1  -  <em>i</em> <em>z</em>)) / (2 <em>i</em>)</td></tr></table></div><p><p>

The above specification follows&nbsp;[<a href="r5rs-Z-H-14.html#%_sec_7.3">27</a>], which in turn
cites&nbsp;[<a href="r5rs-Z-H-14.html#%_sec_7.3">19</a>]; refer to these sources for more detailed
discussion of branch cuts, boundary conditions, and implementation of
these functions.  When it is possible these procedures produce a real
result from a real argument.<p>

<p>

<p><p>

<p><div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_330"></a>sqrt<i> <em>z</em></i>)</tt>&nbsp;</div>
<p>

Returns the principal square root of <em>z</em>.  The result will have
either positive real part, or zero real part and non-negative imaginary
part.
<p><p>

<p><div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_332"></a>expt<i> <em>z</em><sub>1</sub> <em>z</em><sub>2</sub></i>)</tt>&nbsp;</div>
<p>

Returns <em>z</em><sub>1</sub> raised to the power <em>z</em><sub>2</sub>.  For <em>z</em><sub>1</sub> <img src="r5rs-Z-G-D-8.gif" border="0"> 0
<p><div align=left><table><tr><td><em>z</em><sub>1</sub><sup><em>z</em><sub>2</sub></sup>  =  <em>e</em><sup><em>z</em><sub>2</sub> <tt>log</tt> <em>z</em><sub>1</sub></sup></td></tr></table></div><p>
0<sup><em>z</em></sup> is 1 if <em>z</em>  =  0 and 0 otherwise.
<p><p>

<p>

<p><div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_334"></a>make-rectangular<i> <em>x</em><sub>1</sub> <em>x</em><sub>2</sub></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_336"></a>make-polar<i> <em>x</em><sub>3</sub> <em>x</em><sub>4</sub></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_338"></a>real-part<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_340"></a>imag-part<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_342"></a>magnitude<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_344"></a>angle<i> <em>z</em></i>)</tt>&nbsp;</div>
<p>

These procedures are part of every implementation that supports
general
complex numbers.  Suppose <em>x</em><sub>1</sub>, <em>x</em><sub>2</sub>, <em>x</em><sub>3</sub>, and <em>x</em><sub>4</sub> are
real numbers and <em>z</em> is a complex number such that
<p><div align=left><table><tr><td> <em>z</em>  =  <em>x</em><sub>1</sub>  +  <em>x</em><sub>2</sub><em>i</em>
 =  <em>x</em><sub>3</sub>  &middot;  <em>e</em><sup><em>i</em> <em>x</em><sub>4</sub></sup></td></tr></table></div><p>
Then
<tt><p>(make-rectangular&nbsp;<em>x</em><sub>1</sub>&nbsp;<em>x</em><sub>2</sub>)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;<em>z</em><br>
(make-polar&nbsp;<em>x</em><sub>3</sub>&nbsp;<em>x</em><sub>4</sub>)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;<em>z</em><br>
(real-part&nbsp;<em>z</em>)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;<em>x</em><sub>1</sub><br>
(imag-part&nbsp;<em>z</em>)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;<em>x</em><sub>2</sub><br>
(magnitude&nbsp;<em>z</em>)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;|<em>x</em><sub>3</sub>|<br>
(angle&nbsp;<em>z</em>)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;===&gt;&nbsp;<em>x</em><sub><em>a</em><em>n</em><em>g</em><em>l</em><em>e</em></sub><br>
<p></tt>
where  - <img src="r5rs-Z-G-D-3.gif" border="0"> &lt; <em>x</em><sub><em>a</em><em>n</em><em>g</em><em>l</em><em>e</em></sub> <u>&lt;</u> <img src="r5rs-Z-G-D-3.gif" border="0"> with <em>x</em><sub><em>a</em><em>n</em><em>g</em><em>l</em><em>e</em></sub>  =  <em>x</em><sub>4</sub>  +  2<img src="r5rs-Z-G-D-3.gif" border="0"> <em>n</em>
for some integer <em>n</em>.<p>

<blockquote><em>Rationale:&nbsp;&nbsp;</em>
<tt>Magnitude</tt> is the same as <tt>abs</tt> for a real argument,
but <tt>abs</tt> must be present in all implementations, whereas
<tt>magnitude</tt> need only be present in implementations that support
general complex numbers.
</blockquote><p>

<p><p>

<p><div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_346"></a>exact-&gt;inexact<i> <em>z</em></i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_348"></a>inexact-&gt;exact<i> <em>z</em></i>)</tt>&nbsp;</div>
<p>

<tt>Exact-&gt;inexact</tt> returns an inexact representation of <em>z</em>.
The value returned is the
inexact number that is numerically closest to the argument.  
If an exact argument has no reasonably close inexact equivalent,
then a violation of an implementation restriction may be reported.<p>

<tt>Inexact-&gt;exact</tt> returns an exact representation of
<em>z</em>.  The value returned is the exact number that is numerically
closest to the argument.
If an inexact argument has no reasonably close exact equivalent,
then a violation of an implementation restriction may be reported.<p>


These procedures implement the natural one-to-one correspondence between
exact and inexact integers throughout an
implementation-dependent range.  See section&nbsp;<a href="#%_sec_6.2.3">6.2.3</a>.<p>

<p><p>

<p><p><p>

<a name="%_sec_6.2.6"></a>
<h3><a href="r5rs-Z-H-2.html#%_toc_%_sec_6.2.6">6.2.6&nbsp;&nbsp;Numerical input and output</a></h3><p>

<p><div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(<a name="%_idx_350"></a>number-&gt;string<i> z</i>)</tt>&nbsp;</div>

<div align=left><u>procedure:</u>&nbsp;&nbsp;<tt>(number-&gt;string<i> z radix</i>)</tt>&nbsp;</div>
<p>

<em>R</em><em>a</em><em>d</em><em>i</em><em>x</em> must be an exact integer, either 2, 8, 10, or 16.  If omitted,
<em>r</em><em>a</em><em>d</em><em>i</em><em>x</em> defaults to 10.
The procedure <tt>number-&gt;string</tt> takes a
number and a radix and returns as a