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    <title>Arithmetic operations</title>
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    <section class="sect1" title="Arithmetic operations" epub:type="subchapter" id="Arithmetic-operations">
      <div class="titlepage">
        <div>
          <div>
            <h2 class="title" style="clear: both">Arithmetic operations</h2>
          </div>
        </div>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877111776" class="indexterm"/> <code class="function">real-valued?</code> <em class="replaceable"><code><em class="replaceable"><code>obj</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877108848" class="indexterm"/> <code class="function">rational-valued?</code> <em class="replaceable"><code><em class="replaceable"><code>obj</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877105920" class="indexterm"/> <code class="function">integer-valued?</code> <em class="replaceable"><code><em class="replaceable"><code>obj</code></em></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>These numerical type predicates can be applied to any kind of argument.
The <code class="literal">real-valued?</code> procedure returns <code class="literal">#t</code> if the object is a
number object and is equal in the sense of <code class="literal">=</code> to some real number
object, or if the object is a NaN, or a complex number object whose real
part is a NaN and whose imaginary part is zero in the sense of
<code class="literal">zero?</code>.  The <code class="literal">rational-valued?</code> and <code class="literal">integer-valued?</code>
procedures return <code class="literal">#t</code> if the object is a number object and is equal
in the sense of <code class="literal">=</code> to some object of the named type, and otherwise
they return <code class="literal">#f</code>.
</p>
          <pre class="screen">(real-valued? +nan.0)                  ⇒ #t
(real-valued? +nan.0+0i)               ⇒ #t
(real-valued? -inf.0)                  ⇒ #t
(real-valued? 3)                       ⇒ #t
(real-valued? -2.5+0.0i)               ⇒ #t

(real-valued? -2.5+0i)                 ⇒ #t
(real-valued? -2.5)                    ⇒ #t
(real-valued? #e1e10)                  ⇒ #t

(rational-valued? +nan.0)              ⇒ #f
(rational-valued? -inf.0)              ⇒ #f
(rational-valued? 6/10)                ⇒ #t
(rational-valued? 6/10+0.0i)           ⇒ #t
(rational-valued? 6/10+0i)             ⇒ #t
(rational-valued? 6/3)                 ⇒ #t

(integer-valued? 3+0i)                 ⇒ #t
(integer-valued? 3+0.0i)               ⇒ #t
(integer-valued? 3.0)                  ⇒ #t
(integer-valued? 3.0+0.0i)             ⇒ #t
(integer-valued? 8/4)                  ⇒ #t
</pre>
          <div class="blockquote">
            <blockquote class="blockquote">
              <p><span class="emphasis"><em>Note:</em></span> These procedures test whether a given number object can be
coerced to the specified type without loss of numerical accuracy.
Specifically, the behavior of these predicates differs from the behavior
of <code class="literal">real?</code>, <code class="literal">rational?</code>, and <code class="literal">integer?</code> on complex number
objects whose imaginary part is inexact zero.
</p>
            </blockquote>
          </div>
          <div class="blockquote">
            <blockquote class="blockquote">
              <p><span class="emphasis"><em>Note:</em></span> The behavior of these type predicates on inexact number
objects is unreliable, because any inaccuracy may affect the result.
</p>
            </blockquote>
          </div>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877092816" class="indexterm"/> <code class="function">exact-integer?</code> <em class="replaceable"><code>z</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>Returns <code class="literal">#t</code> if <em class="replaceable"><code>z</code></em> is both exact and an integer; otherwise
returns <code class="literal">#f</code>.
</p>
          <pre class="screen">(exact-integer? 32)                    ⇒ #t
(exact-integer? 32.0)                  ⇒ #t
(exact-integer? 32/5)                  ⇒ #f
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877087776" class="indexterm"/> <code class="function">finite?</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>Returns <code class="literal">#t</code> if <em class="replaceable"><code>z</code></em> is finite real number
(i.e. an infinity and not a NaN),
or if <em class="replaceable"><code>z</code></em> is a complex number
whose real and imaginary parts are both finite.
</p>
          <pre class="screen">(finite? 3)             ⇒ #t
(finite? +inf.0)        ⇒ #f
(finite? 3.0+inf.0i)    ⇒ #f
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877082576" class="indexterm"/> <code class="function">infinite?</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>Return <code class="literal">#t</code> if <em class="replaceable"><code>z</code></em> is
an infinite real number (<code class="literal">+int.0</code> or <code class="literal">-inf.0</code>),
or if <em class="replaceable"><code>z</code></em> is a complex number where either
real or imaginary parts or both are infinite.
</p>
          <pre class="screen">(infinite? 5.0)         ⇒ #f
(infinite? +inf.0)      ⇒ #t
(infinite? +nan.0)      ⇒ #f
(infinite? 3.0+inf.0i)  ⇒ #t
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877076368" class="indexterm"/> <code class="function">nan?</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>For a real numer returns whether its is a NaN;
for a complex number if the real or imaginary parts or both is a NaN.
</p>
          <pre class="screen">(nan? +nan.0)           ⇒ #t
(nan? 32)               ⇒ #f
(nan? +nan.0+5.0i)      ⇒ #t
(nan? 1+2i)             ⇒ #f
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877072288" class="indexterm"/> <code class="function">+</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em> <em class="replaceable"><code>…</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877069008" class="indexterm"/> <code class="function">*</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em> <em class="replaceable"><code>…</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>These procedures return the sum or product of their arguments.
</p>
          <pre class="screen">(+ 3 4)                          ⇒  7
(+ 3)                            ⇒  3
(+)                              ⇒  0
(+ +inf.0 +inf.0)                ⇒  +inf.0
(+ +inf.0 -inf.0)                ⇒  +nan.0

(* 4)                            ⇒  4
(*)                              ⇒  1
(* 5 +inf.0)                     ⇒  +inf.0
(* -5 +inf.0)                    ⇒  -inf.0
(* +inf.0 +inf.0)                ⇒  +inf.0
(* +inf.0 -inf.0)                ⇒  -inf.0
(* 0 +inf.0)                     ⇒  +nan.0
(* 0 +nan.0)                     ⇒  +nan.0
(* 1.0 0)                        ⇒  0.0
</pre>
          <p>For any real number object <em class="replaceable"><code>x</code></em> that is neither infinite nor NaN:
</p>
          <pre class="screen">(+ +inf.0 <em class="replaceable"><code>x</code></em>)                   ⇒  +inf.0
(+ -inf.0 <em class="replaceable"><code>x</code></em>)                   ⇒  -inf.0
</pre>
          <p>For any real number object <em class="replaceable"><code>x</code></em>:
</p>
          <pre class="screen">(+ +nan.0 <em class="replaceable"><code>x</code></em>)                   ⇒  +nan.0
</pre>
          <p>For any real number object <em class="replaceable"><code>x</code></em> that is not an exact 0:
</p>
          <pre class="screen">(* +nan.0 <em class="replaceable"><code>x</code></em>)                   ⇒  +nan.0
</pre>
          <p>The behavior of <code class="literal">-0.0</code> is illustrated by the following examples:
</p>
          <pre class="screen">(+  0.0 -0.0)  ⇒  0.0
(+ -0.0  0.0)  ⇒  0.0
(+  0.0  0.0)  ⇒  0.0
(+ -0.0 -0.0)  ⇒ -0.0
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877055440" class="indexterm"/> <code class="function">-</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877052688" class="indexterm"/> <code class="function">-</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em><em class="replaceable"><code><sub>1</sub></code></em> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em><em class="replaceable"><code><sub>2</sub></code></em> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em><em class="replaceable"><code><sub>3</sub></code></em> <em class="replaceable"><code>…</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>With two or more arguments, this procedures returns the difference of
its arguments, associating to the left.  With one argument, however, it
returns the negation (additive inverse) of its argument.
</p>
          <pre class="screen">(- 3 4)                               ⇒  -1
(- 3 4 5)                             ⇒  -6
(- 3)                                 ⇒  -3
(- +inf.0 +inf.0)                     ⇒  +nan.0
</pre>
          <p>The behavior of <code class="literal">-0.0</code> is illustrated by the following examples:
</p>
          <pre class="screen">(-  0.0)       ⇒ -0.0
(- -0.0)       ⇒  0.0
(-  0.0 -0.0)  ⇒  0.0
(- -0.0  0.0)  ⇒ -0.0
(-  0.0  0.0)  ⇒  0.0
(- -0.0 -0.0)  ⇒  0.0
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877042880" class="indexterm"/> <code class="function">/</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877040016" class="indexterm"/> <code class="function">/</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em><em class="replaceable"><code><sub>1</sub></code></em> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em><em class="replaceable"><code><sub>2</sub></code></em> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em><em class="replaceable"><code><sub>3</sub></code></em> <em class="replaceable"><code>…</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>If all of the arguments are exact, then the divisors must all be
nonzero.  With two or more arguments, this procedure returns the
quotient of its arguments, associating to the left.  With one argument,
however, it returns the multiplicative inverse of its argument.
</p>
          <pre class="screen">(/ 3 4 5)                         ⇒  3/20
(/ 3)                             ⇒  1/3
(/ 0.0)                           ⇒  +inf.0
(/ 1.0 0)                         ⇒  +inf.0
(/ -1 0.0)                        ⇒  -inf.0
(/ +inf.0)                        ⇒  0.0
(/ 0 0)                           ⇒  exception &amp;assertion
(/ 3 0)                           ⇒  exception &amp;assertion
(/ 0 3.5)                         ⇒  0.0
(/ 0 0.0)                         ⇒  +nan.0
(/ 0.0 0)                         ⇒  +nan.0
(/ 0.0 0.0)                       ⇒  +nan.0
</pre>
          <p>If this procedure is applied to mixed non–rational real and non–real
complex arguments, it either raises an exception with condition type
<code class="literal">&amp;implementation-restriction</code> or returns an unspecified number
object.
</p>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877030928" class="indexterm"/> <code class="function">floor/</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877027744" class="indexterm"/> <code class="function">truncate/</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877024560" class="indexterm"/> <code class="function">div-and-mod</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667877021376" class="indexterm"/> <code class="function">div0-and-mod0</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>These procedures implement number–theoretic integer division.
They accept two real numbers <em class="replaceable"><code>x</code></em> and <em class="replaceable"><code>y</code></em> as operands,
where <em class="replaceable"><code>y</code></em> must be nonzero.
In all cases the result is two values <em class="replaceable"><code>q</code></em> (an integer) and <em class="replaceable"><code>r</code></em> (a real)
that satisfy the equations:
</p>
          <pre class="screen"><em class="replaceable"><code>x</code></em> = <em class="replaceable"><code>q</code></em> * <em class="replaceable"><code>y</code></em> + <em class="replaceable"><code>r</code></em>
<em class="replaceable"><code>q</code></em> = <em class="replaceable"><code>rounding-op</code></em>(<em class="replaceable"><code>x</code></em>/<em class="replaceable"><code>y</code></em>)
</pre>
          <p>The result is inexact if either argument is inexact.
</p>
          <p>For <code class="literal">floor/</code> the <em class="replaceable"><code>rounding-op</code></em> is the <code class="literal">floor</code> function (below).
</p>
          <pre class="screen">(floor/ 123 10)         ⇒  12 3
(floor/ 123 -10)        ⇒  -13 -7
(floor/ -123 10)        ⇒  -13 7
(floor/ -123 -10)       ⇒  12 -3
</pre>
          <p>For <code class="literal">truncate/</code> the <em class="replaceable"><code>rounding-op</code></em> is the <code class="literal">truncate</code> function.
</p>
          <pre class="screen">(truncate/ 123 10)      ⇒  12 3
(truncate/ 123 -10)     ⇒  -12 3
(truncate/ -123 10)     ⇒  -12 -3
(truncate/ -123 -10)    ⇒  12 -3
</pre>
          <p>For <code class="literal">div-and-mod</code> the <em class="replaceable"><code>rounding-op</code></em> is either <code class="literal">floor</code>
(if <em class="replaceable"><code>y</code></em> is positive) or <code class="literal">ceiling</code> (if <em class="replaceable"><code>y</code></em> is negative).
We have:
</p>
          <pre class="screen">0  &lt;= <em class="replaceable"><code>r</code></em> &lt; |<em class="replaceable"><code>y</code></em>|
</pre>
          <pre class="screen">(div-and-mod 123 10)    ⇒  12 3
(div-and-mod 123 -10)   ⇒  -12 3
(div-and-mod -123 10)   ⇒  -13 7
(div-and-mod -123 -10)  ⇒  13 7
</pre>
          <p>For <code class="literal">div0-and-mod0</code> the <em class="replaceable"><code>rounding-op</code></em> is the <code class="literal">round</code> function,
and <code class="literal">r</code> lies within a half–open interval centered on zero.
</p>
          <pre class="screen">-|<em class="replaceable"><code>y</code></em>/2| &lt;= <em class="replaceable"><code>r</code></em> &lt; |<em class="replaceable"><code>y</code></em>/2|
</pre>
          <pre class="screen">(div0-and-mod0 123 10)   ⇒  12 3
(div0-and-mod0 123 -10)  ⇒  -12 3
(div0-and-mod0 -123 10)  ⇒  -12 -3
(div0-and-mod0 -123 -10) ⇒  12 -3
(div0-and-mod0 127 10)   ⇒  13 -3
(div0-and-mod0 127 -10)  ⇒  -13 -3
(div0-and-mod0 -127 10)  ⇒  -13 3
(div0-and-mod0 -127 -10) ⇒  13 3
</pre>
          <p>The inconsistent naming is for historical reasons: <code class="literal">div-and-mod</code>
and <code class="literal">div0-and-mod0</code> are from R6RS, while <code class="literal">floor/</code> and
<code class="literal">truncate/</code> are from R7RS.
</p>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876995616" class="indexterm"/> <code class="function">floor-quotient</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876992432" class="indexterm"/> <code class="function">truncate-quotient</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876989248" class="indexterm"/> <code class="function">div</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876986128" class="indexterm"/> <code class="function">div0</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>These procedures return the quotient part (first value)
of respectively <code class="literal">floor/</code>, <code class="literal">truncate/</code>,
<code class="literal">div-and-mod</code>, and <code class="literal">div0-and-mod0</code>.
</p>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876980640" class="indexterm"/> <code class="function">floor-remainder</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876977456" class="indexterm"/> <code class="function">truncate-remainder</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876974272" class="indexterm"/> <code class="function">mod</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876971152" class="indexterm"/> <code class="function">mod0</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>These procedures return the remainder part (second value)
of respectively <code class="literal">floor/</code>, <code class="literal">truncate/</code>,
<code class="literal">div-and-mod</code>, and <code class="literal">div0-and-mod0</code>.
</p>
          <p>As a Kawa extension <em class="replaceable"><code>y</code></em> may be zero, in which case the result is <em class="replaceable"><code>x</code></em>:
</p>
          <pre class="screen">(mod 123 0)     ⇒  123 ;; Kawa extension
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876964016" class="indexterm"/> <code class="function">quotient</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876960832" class="indexterm"/> <code class="function">remainder</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876957648" class="indexterm"/> <code class="function">modulo</code> <em class="replaceable"><code>x</code></em> <em class="replaceable"><code>y</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>These are equivalent to <code class="literal">truncate-quotient</code>,
<code class="literal">truncate-remainder</code>, and <code class="literal">floor-remainder</code>, respectively.
These are provided for backward compatibility.
</p>
          <pre class="screen">(remainder 13 4)     ⇒ 1
(remainder -13 4)    ⇒ -1
(remainder 13 -4)    ⇒ 1
(remainder -13 -4)   ⇒ -1
(remainder -13 -4.0) ⇒ -1.0
(modulo 13 4)   ⇒ 1
(modulo -13 4)  ⇒ 3
(modulo 13 -4)  ⇒ -4
(modulo -13 -4) ⇒ -1
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876951792" class="indexterm"/> <code class="function">abs</code> <em class="replaceable"><code><em class="replaceable"><code>x</code></em></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>Returns the absolute value of its argument.
</p>
          <pre class="screen">(abs -7)                         ⇒  7
(abs -inf.0)                     ⇒  +inf.0
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876947792" class="indexterm"/> <code class="function">gcd</code> <em class="replaceable"><code><em class="replaceable"><code>n</code></em></code></em><em class="replaceable"><code><sub>1</sub></code></em> <em class="replaceable"><code>…</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876944128" class="indexterm"/> <code class="function">lcm</code> <em class="replaceable"><code><em class="replaceable"><code>n</code></em></code></em><em class="replaceable"><code><sub>1</sub></code></em> <em class="replaceable"><code>…</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>These procedures return the greatest common divisor or least common
multiple of their arguments.  The result is always non–negative.
The arguments must be integers; if an argument is inexact, so is the result.
</p>
          <pre class="screen">(gcd 32 -36)                     ⇒  4
(gcd)                            ⇒  0
(lcm 32 -36)                     ⇒  288
(lcm 32.0 -36)                   ⇒  288.0 ; inexact
(lcm)                            ⇒  1
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876938896" class="indexterm"/> <code class="function">numerator</code> <em class="replaceable"><code><em class="replaceable"><code>q</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876935968" class="indexterm"/> <code class="function">denominator</code> <em class="replaceable"><code><em class="replaceable"><code>q</code></em></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>These procedures return the numerator or denominator of their argument;
the result is computed as if the argument was represented as a fraction
in lowest terms.  The denominator is always positive.  The denominator
of <code class="literal">0</code> is defined to be <code class="literal">1</code>.
The arguments must be integers; if an argument is inexact, so is the result.
</p>
          <pre class="screen">(numerator   (/ 6 4))            ⇒  3
(denominator (/ 6 4))            ⇒  2
(denominator (inexact (/ 6 4)))        ⇒  2.0
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876930832" class="indexterm"/> <code class="function">floor</code> <em class="replaceable"><code><em class="replaceable"><code>x</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876927904" class="indexterm"/> <code class="function">ceiling</code> <em class="replaceable"><code><em class="replaceable"><code>x</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876924976" class="indexterm"/> <code class="function">truncate</code> <em class="replaceable"><code><em class="replaceable"><code>x</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876922048" class="indexterm"/> <code class="function">round</code> <em class="replaceable"><code><em class="replaceable"><code>x</code></em></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>These procedures return inexact integer objects for inexact arguments
that are not infinities or NaNs, and exact integer objects for exact
rational arguments.
</p>
          <div class="variablelist" epub:type="list">
            <dl class="variablelist">
              <dt class="term"><code class="literal">floor</code>
</dt>
              <dd>
                <p>Returns the largest integer object not larger than <em class="replaceable"><code>x</code></em>.
</p>
              </dd>
              <dt class="term"><code class="literal">ceiling</code>
</dt>
              <dd>
                <p>Returns the smallest integer object not smaller than <em class="replaceable"><code>x</code></em>.
</p>
              </dd>
              <dt class="term"><code class="literal">truncate</code>
</dt>
              <dd>
                <p>Returns the integer object closest to <em class="replaceable"><code>x</code></em> whose absolute value is
not larger than the absolute value of <em class="replaceable"><code>x</code></em>.
</p>
              </dd>
              <dt class="term"><code class="literal">round</code>
</dt>
              <dd>
                <p>Returns the closest integer object to <em class="replaceable"><code>x</code></em>, rounding to even when
<em class="replaceable"><code>x</code></em> represents a number halfway between two integers.
</p>
              </dd>
            </dl>
          </div>
          <p>If the argument to one of these procedures is inexact, then the result
is also inexact.  If an exact value is needed, the result should be
passed to the <code class="literal">exact</code> procedure.
</p>
          <p>Although infinities and NaNs are not integer objects, these procedures
return an infinity when given an infinity as an argument, and a NaN when
given a NaN.
</p>
          <pre class="screen">(floor -4.3)                     ⇒  -5.0
(ceiling -4.3)                   ⇒  -4.0
(truncate -4.3)                  ⇒  -4.0
(round -4.3)                     ⇒  -4.0

(floor 3.5)                      ⇒  3.0
(ceiling 3.5)                    ⇒  4.0
(truncate 3.5)                   ⇒  3.0
(round 3.5)                      ⇒  4.0

(round 7/2)                      ⇒  4
(round 7)                        ⇒  7

(floor +inf.0)                   ⇒  +inf.0
(ceiling -inf.0)                 ⇒  -inf.0
(round +nan.0)                   ⇒  +nan.0
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876907856" class="indexterm"/> <code class="function">rationalize</code> <em class="replaceable"><code><em class="replaceable"><code>x</code></em></code></em><em class="replaceable"><code><sub>1</sub></code></em> <em class="replaceable"><code><em class="replaceable"><code>x</code></em></code></em><em class="replaceable"><code><sub>2</sub></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>The <code class="literal">rationalize</code> procedure returns a number object representing
the <span class="emphasis"><em>simplest</em></span> rational number differing from <em class="replaceable"><code>x</code></em><sub>1</sub> by no more
than <em class="replaceable"><code>x</code></em><sub>2</sub>.
</p>
          <p>A rational number <span class="emphasis"><em>r_1</em></span> is <span class="emphasis"><em>simpler</em></span> than another rational
number <span class="emphasis"><em>r_2</em></span> if <code class="literal">r_1 = p_1/q_1</code> and <code class="literal">r_2 = p_2/q_2</code> (in
lowest terms) and <code class="literal">|p_1| &lt;= |p_2|</code> and <code class="literal">|q_1| &lt;= |q_2|</code>.  Thus
<code class="literal">3/5</code> is simpler than <code class="literal">4/7</code>.
</p>
          <p>Although not all rationals are comparable in this ordering (consider
<code class="literal">2/7</code> and <code class="literal">3/5</code>) any interval contains a rational number that
is simpler than every other rational number in that interval (the
simpler <code class="literal">2/5</code> lies between <code class="literal">2/7</code> and <code class="literal">3/5</code>).
</p>
          <p>Note that <code class="literal">0 = 0/1</code> is the simplest rational of all.
</p>
          <pre class="screen">(rationalize (exact .3) 1/10)          ⇒ 1/3
(rationalize .3 1/10)                  ⇒ #i1/3  ; approximately

(rationalize +inf.0 3)                 ⇒  +inf.0
(rationalize +inf.0 +inf.0)            ⇒  +nan.0

</pre>
          <p>The first two examples hold only in implementations whose inexact real
number objects have sufficient precision.
</p>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876891376" class="indexterm"/> <code class="function">exp</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876888512" class="indexterm"/> <code class="function">log</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876885648" class="indexterm"/> <code class="function">log</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em><em class="replaceable"><code><sub>1</sub></code></em> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em><em class="replaceable"><code><sub>2</sub></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876881504" class="indexterm"/> <code class="function">sin</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876878592" class="indexterm"/> <code class="function">cos</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876875728" class="indexterm"/> <code class="function">tan</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876872816" class="indexterm"/> <code class="function">asin</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876869888" class="indexterm"/> <code class="function">acos</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876866960" class="indexterm"/> <code class="function">atan</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876864032" class="indexterm"/> <code class="function">atan</code> <em class="replaceable"><code><em class="replaceable"><code>x</code></em></code></em><em class="replaceable"><code><sub>1</sub></code></em> <em class="replaceable"><code><em class="replaceable"><code>x</code></em></code></em><em class="replaceable"><code><sub>2</sub></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>These procedures compute the usual transcendental functions.
</p>
          <p>The <code class="literal">exp</code> procedure computes the base–<em class="replaceable"><code>e</code></em> exponential of
<em class="replaceable"><code>z</code></em>.
The <code class="literal">log</code> procedure with a single argument computes the natural
logarithm of <em class="replaceable"><code>z</code></em> (<span class="bold"><strong>not</strong></span> the base–10 logarithm); <code class="literal">(log
<em class="replaceable"><code>z</code></em><sub>1</sub> <em class="replaceable"><code>z</code></em><sub>2</sub>)</code> computes the base–<em class="replaceable"><code>z</code></em><sub>2</sub> logarithm of <em class="replaceable"><code>z</code></em><sub>1</sub>.
</p>
          <p>The <code class="literal">asin</code>, <code class="literal">acos</code>, and <code class="literal">atan</code> procedures compute
arcsine, arccosine, and arctangent, respectively.  The two–argument
variant of <code class="literal">atan</code> computes:
</p>
          <pre class="screen">(angle (make-rectangular <em class="replaceable"><code>x</code></em><sub>2</sub> <em class="replaceable"><code>x</code></em><sub>1</sub>))
</pre>
          <p>These procedures may return inexact results even when given exact
arguments.
</p>
          <pre class="screen">(exp +inf.0)    ⇒ +inf.0
(exp -inf.0)    ⇒ 0.0
(log +inf.0)    ⇒ +inf.0
(log 0.0)       ⇒ -inf.0
(log 0)         ⇒ exception &amp;assertion
(log -inf.0)    ⇒ +inf.0+3.141592653589793i    ; approximately
(atan -inf.0)   ⇒ -1.5707963267948965          ; approximately
(atan +inf.0)   ⇒ 1.5707963267948965           ; approximately
(log -1.0+0.0i) ⇒ 0.0+3.141592653589793i       ; approximately
(log -1.0-0.0i) ⇒ 0.0-3.141592653589793i       ; approximately
                                                ; if -0.0 is distinguished
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876847392" class="indexterm"/> <code class="function">sinh</code> <em class="replaceable"><code>z</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876844592" class="indexterm"/> <code class="function">cosh</code> <em class="replaceable"><code>z</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876841792" class="indexterm"/> <code class="function">tanh</code> <em class="replaceable"><code>z</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876838992" class="indexterm"/> <code class="function">asinh</code> <em class="replaceable"><code>z</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876836192" class="indexterm"/> <code class="function">acosh</code> <em class="replaceable"><code>z</code></em></p>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876833392" class="indexterm"/> <code class="function">atanh</code> <em class="replaceable"><code>z</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>The hyperbolic functions.
</p>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876830032" class="indexterm"/> <code class="function">square</code> <em class="replaceable"><code>z</code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>Returns the square of <em class="replaceable"><code>z</code></em>.
This is equivalent to <code class="literal">(* <em class="replaceable"><code>z</code></em> <em class="replaceable"><code>z</code></em>)</code>.
</p>
          <pre class="screen">(square 42)    ⇒ 1764
(square 2.0)   ⇒ 4.0
</pre>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876824672" class="indexterm"/> <code class="function">sqrt</code> <em class="replaceable"><code><em class="replaceable"><code>z</code></em></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>Returns the principal square root of <em class="replaceable"><code>z</code></em>.  For rational <em class="replaceable"><code>z</code></em>, the
result has either positive real part, or zero real part and
non–negative imaginary part.  The value of <code class="literal">(sqrt <em class="replaceable"><code>z</code></em>)</code> could be
expressed as:
</p>
          <pre class="screen">e^((log z)/2)
</pre>
          <p>The <code class="literal">sqrt</code> procedure may return an inexact result even when given
an exact argument.
</p>
          <pre class="screen">(sqrt -5)                   ⇒  0.0+2.23606797749979i ; approximately
(sqrt +inf.0)               ⇒  +inf.0
(sqrt -inf.0)               ⇒  +inf.0i
</pre>
          <p>Note that if the argument is a primitive number (such as <code class="literal">double</code>) or an
instance of the corresponding boxed class (such as <code class="literal">java.lang.Double</code>)
then we use the real-number version of <code class="literal">sqrt</code>:
</p>
          <pre class="screen">(sqrt (-&gt;double -5))        ⇒  NaN
</pre>
          <p>That is, we get different a result for <code class="literal">java.lang.Double</code>
and <code class="literal">gnu.math.DFloNum</code>, even for arguments that are numerically equal
in the sense of <code class="literal">=</code>.
This is so that the compiler can use the <code class="literal">java.lang.Math.sqrt</code>
method without object allocation when the argument is a <code class="literal">double</code>
(and because we want <code class="literal">double</code> and <code class="literal">java.lang.Double</code> to behave
consistently).
</p>
        </blockquote>
      </div>
      <p class="synopsis" kind="Procedure"><span class="kind">Procedure</span><span class="ignore">: </span><a id="idm139667876811680" class="indexterm"/> <code class="function">exact-integer-sqrt</code> <em class="replaceable"><code><em class="replaceable"><code>k</code></em></code></em></p>
      <div class="blockquote">
        <blockquote class="blockquote">
          <p>The <code class="literal">exact-integer-sqrt</code> procedure returns two non–negative exact
integer objects <span class="emphasis"><em>s</em></span> and <span class="emphasis"><em>r</em></span> where <code class="literal"><em class="replaceable"><code>k</code></em> = s^2 + r</code> and
<code class="literal"><em class="replaceable"><code>k</code></em> &lt; (s+1)^2</code>.
</p>
          <pre class="screen">(exact-integer-sqrt 4)  ⇒ 2 0 ; two return values
(exact-integer-sqrt 5)  ⇒ 2 1 ; two return values
</pre>
        </blockquote>
      </div>
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