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| |
| <h3 class="section">19.4 Exponentiation and Logarithms</h3> |
| |
| <p><a name="index-exponentiation-functions-2131"></a><a name="index-power-functions-2132"></a><a name="index-logarithm-functions-2133"></a> |
| <!-- math.h --> |
| <!-- ISO --> |
| |
| <div class="defun"> |
| — Function: double <b>exp</b> (<var>double x</var>)<var><a name="index-exp-2134"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>expf</b> (<var>float x</var>)<var><a name="index-expf-2135"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>expl</b> (<var>long double x</var>)<var><a name="index-expl-2136"></a></var><br> |
| <blockquote><p>These functions compute <code>e</code> (the base of natural logarithms) raised |
| to the power <var>x</var>. |
| |
| <p>If the magnitude of the result is too large to be representable, |
| <code>exp</code> signals overflow. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>exp2</b> (<var>double x</var>)<var><a name="index-exp2-2137"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>exp2f</b> (<var>float x</var>)<var><a name="index-exp2f-2138"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>exp2l</b> (<var>long double x</var>)<var><a name="index-exp2l-2139"></a></var><br> |
| <blockquote><p>These functions compute <code>2</code> raised to the power <var>x</var>. |
| Mathematically, <code>exp2 (x)</code> is the same as <code>exp (x * log (2))</code>. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- GNU --> |
| <div class="defun"> |
| — Function: double <b>exp10</b> (<var>double x</var>)<var><a name="index-exp10-2140"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- GNU --> |
| — Function: float <b>exp10f</b> (<var>float x</var>)<var><a name="index-exp10f-2141"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- GNU --> |
| — Function: long double <b>exp10l</b> (<var>long double x</var>)<var><a name="index-exp10l-2142"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- GNU --> |
| — Function: double <b>pow10</b> (<var>double x</var>)<var><a name="index-pow10-2143"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- GNU --> |
| — Function: float <b>pow10f</b> (<var>float x</var>)<var><a name="index-pow10f-2144"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- GNU --> |
| — Function: long double <b>pow10l</b> (<var>long double x</var>)<var><a name="index-pow10l-2145"></a></var><br> |
| <blockquote><p>These functions compute <code>10</code> raised to the power <var>x</var>. |
| Mathematically, <code>exp10 (x)</code> is the same as <code>exp (x * log (10))</code>. |
| |
| <p>These functions are GNU extensions. The name <code>exp10</code> is |
| preferred, since it is analogous to <code>exp</code> and <code>exp2</code>. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>log</b> (<var>double x</var>)<var><a name="index-log-2146"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>logf</b> (<var>float x</var>)<var><a name="index-logf-2147"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>logl</b> (<var>long double x</var>)<var><a name="index-logl-2148"></a></var><br> |
| <blockquote><p>These functions compute the natural logarithm of <var>x</var>. <code>exp (log |
| (</code><var>x</var><code>))</code> equals <var>x</var>, exactly in mathematics and approximately in |
| C. |
| |
| <p>If <var>x</var> is negative, <code>log</code> signals a domain error. If <var>x</var> |
| is zero, it returns negative infinity; if <var>x</var> is too close to zero, |
| it may signal overflow. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>log10</b> (<var>double x</var>)<var><a name="index-log10-2149"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>log10f</b> (<var>float x</var>)<var><a name="index-log10f-2150"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>log10l</b> (<var>long double x</var>)<var><a name="index-log10l-2151"></a></var><br> |
| <blockquote><p>These functions return the base-10 logarithm of <var>x</var>. |
| <code>log10 (</code><var>x</var><code>)</code> equals <code>log (</code><var>x</var><code>) / log (10)</code>. |
| |
| </blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>log2</b> (<var>double x</var>)<var><a name="index-log2-2152"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>log2f</b> (<var>float x</var>)<var><a name="index-log2f-2153"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>log2l</b> (<var>long double x</var>)<var><a name="index-log2l-2154"></a></var><br> |
| <blockquote><p>These functions return the base-2 logarithm of <var>x</var>. |
| <code>log2 (</code><var>x</var><code>)</code> equals <code>log (</code><var>x</var><code>) / log (2)</code>. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>logb</b> (<var>double x</var>)<var><a name="index-logb-2155"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>logbf</b> (<var>float x</var>)<var><a name="index-logbf-2156"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>logbl</b> (<var>long double x</var>)<var><a name="index-logbl-2157"></a></var><br> |
| <blockquote><p>These functions extract the exponent of <var>x</var> and return it as a |
| floating-point value. If <code>FLT_RADIX</code> is two, <code>logb</code> is equal |
| to <code>floor (log2 (x))</code>, except it's probably faster. |
| |
| <p>If <var>x</var> is de-normalized, <code>logb</code> returns the exponent <var>x</var> |
| would have if it were normalized. If <var>x</var> is infinity (positive or |
| negative), <code>logb</code> returns &infin;. If <var>x</var> is zero, |
| <code>logb</code> returns &infin;. It does not signal. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: int <b>ilogb</b> (<var>double x</var>)<var><a name="index-ilogb-2158"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: int <b>ilogbf</b> (<var>float x</var>)<var><a name="index-ilogbf-2159"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: int <b>ilogbl</b> (<var>long double x</var>)<var><a name="index-ilogbl-2160"></a></var><br> |
| <blockquote><p>These functions are equivalent to the corresponding <code>logb</code> |
| functions except that they return signed integer values. |
| </p></blockquote></div> |
| |
| <p class="noindent">Since integers cannot represent infinity and NaN, <code>ilogb</code> instead |
| returns an integer that can't be the exponent of a normal floating-point |
| number. <samp><span class="file">math.h</span></samp> defines constants so you can check for this. |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Macro: int <b>FP_ILOGB0</b><var><a name="index-FP_005fILOGB0-2161"></a></var><br> |
| <blockquote><p><code>ilogb</code> returns this value if its argument is <code>0</code>. The |
| numeric value is either <code>INT_MIN</code> or <code>-INT_MAX</code>. |
| |
| <p>This macro is defined in ISO C99<!-- /@w -->. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Macro: int <b>FP_ILOGBNAN</b><var><a name="index-FP_005fILOGBNAN-2162"></a></var><br> |
| <blockquote><p><code>ilogb</code> returns this value if its argument is <code>NaN</code>. The |
| numeric value is either <code>INT_MIN</code> or <code>INT_MAX</code>. |
| |
| <p>This macro is defined in ISO C99<!-- /@w -->. |
| </p></blockquote></div> |
| |
| <p>These values are system specific. They might even be the same. The |
| proper way to test the result of <code>ilogb</code> is as follows: |
| |
| <pre class="smallexample"> i = ilogb (f); |
| if (i == FP_ILOGB0 || i == FP_ILOGBNAN) |
| { |
| if (isnan (f)) |
| { |
| /* <span class="roman">Handle NaN.</span> */ |
| } |
| else if (f == 0.0) |
| { |
| /* <span class="roman">Handle 0.0.</span> */ |
| } |
| else |
| { |
| /* <span class="roman">Some other value with large exponent,</span> |
| <span class="roman">perhaps +Inf.</span> */ |
| } |
| } |
| </pre> |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>pow</b> (<var>double base, double power</var>)<var><a name="index-pow-2163"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>powf</b> (<var>float base, float power</var>)<var><a name="index-powf-2164"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>powl</b> (<var>long double base, long double power</var>)<var><a name="index-powl-2165"></a></var><br> |
| <blockquote><p>These are general exponentiation functions, returning <var>base</var> raised |
| to <var>power</var>. |
| |
| <p>Mathematically, <code>pow</code> would return a complex number when <var>base</var> |
| is negative and <var>power</var> is not an integral value. <code>pow</code> can't |
| do that, so instead it signals a domain error. <code>pow</code> may also |
| underflow or overflow the destination type. |
| </p></blockquote></div> |
| |
| <p><a name="index-square-root-function-2166"></a><!-- math.h --> |
| <!-- ISO --> |
| |
| <div class="defun"> |
| — Function: double <b>sqrt</b> (<var>double x</var>)<var><a name="index-sqrt-2167"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>sqrtf</b> (<var>float x</var>)<var><a name="index-sqrtf-2168"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>sqrtl</b> (<var>long double x</var>)<var><a name="index-sqrtl-2169"></a></var><br> |
| <blockquote><p>These functions return the nonnegative square root of <var>x</var>. |
| |
| <p>If <var>x</var> is negative, <code>sqrt</code> signals a domain error. |
| Mathematically, it should return a complex number. |
| </p></blockquote></div> |
| |
| <p><a name="index-cube-root-function-2170"></a><!-- math.h --> |
| <!-- BSD --> |
| |
| <div class="defun"> |
| — Function: double <b>cbrt</b> (<var>double x</var>)<var><a name="index-cbrt-2171"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- BSD --> |
| — Function: float <b>cbrtf</b> (<var>float x</var>)<var><a name="index-cbrtf-2172"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- BSD --> |
| — Function: long double <b>cbrtl</b> (<var>long double x</var>)<var><a name="index-cbrtl-2173"></a></var><br> |
| <blockquote><p>These functions return the cube root of <var>x</var>. They cannot |
| fail; every representable real value has a representable real cube root. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>hypot</b> (<var>double x, double y</var>)<var><a name="index-hypot-2174"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>hypotf</b> (<var>float x, float y</var>)<var><a name="index-hypotf-2175"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>hypotl</b> (<var>long double x, long double y</var>)<var><a name="index-hypotl-2176"></a></var><br> |
| <blockquote><p>These functions return <code>sqrt (</code><var>x</var><code>*</code><var>x</var><code> + |
| </code><var>y</var><code>*</code><var>y</var><code>)</code>. This is the length of the hypotenuse of a right |
| triangle with sides of length <var>x</var> and <var>y</var>, or the distance |
| of the point (<var>x</var>, <var>y</var>) from the origin. Using this function |
| instead of the direct formula is wise, since the error is |
| much smaller. See also the function <code>cabs</code> in <a href="Absolute-Value.html#Absolute-Value">Absolute Value</a>. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>expm1</b> (<var>double x</var>)<var><a name="index-expm1-2177"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>expm1f</b> (<var>float x</var>)<var><a name="index-expm1f-2178"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>expm1l</b> (<var>long double x</var>)<var><a name="index-expm1l-2179"></a></var><br> |
| <blockquote><p>These functions return a value equivalent to <code>exp (</code><var>x</var><code>) - 1</code>. |
| They are computed in a way that is accurate even if <var>x</var> is |
| near zero—a case where <code>exp (</code><var>x</var><code>) - 1</code> would be inaccurate owing |
| to subtraction of two numbers that are nearly equal. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>log1p</b> (<var>double x</var>)<var><a name="index-log1p-2180"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>log1pf</b> (<var>float x</var>)<var><a name="index-log1pf-2181"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>log1pl</b> (<var>long double x</var>)<var><a name="index-log1pl-2182"></a></var><br> |
| <blockquote><p>These functions returns a value equivalent to <code>log (1 + </code><var>x</var><code>)</code><!-- /@w -->. |
| They are computed in a way that is accurate even if <var>x</var> is |
| near zero. |
| </p></blockquote></div> |
| |
| <p><a name="index-complex-exponentiation-functions-2183"></a><a name="index-complex-logarithm-functions-2184"></a> |
| ISO C99<!-- /@w --> defines complex variants of some of the exponentiation and |
| logarithm functions. |
| |
| <!-- complex.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: complex double <b>cexp</b> (<var>complex double z</var>)<var><a name="index-cexp-2185"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex float <b>cexpf</b> (<var>complex float z</var>)<var><a name="index-cexpf-2186"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex long double <b>cexpl</b> (<var>complex long double z</var>)<var><a name="index-cexpl-2187"></a></var><br> |
| <blockquote><p>These functions return <code>e</code> (the base of natural |
| logarithms) raised to the power of <var>z</var>. |
| Mathematically, this corresponds to the value |
| |
| <p>exp (z) = exp (creal (z)) * (cos (cimag (z)) + I * sin (cimag (z))) |
| </p></blockquote></div> |
| |
| <!-- complex.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: complex double <b>clog</b> (<var>complex double z</var>)<var><a name="index-clog-2188"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex float <b>clogf</b> (<var>complex float z</var>)<var><a name="index-clogf-2189"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex long double <b>clogl</b> (<var>complex long double z</var>)<var><a name="index-clogl-2190"></a></var><br> |
| <blockquote><p>These functions return the natural logarithm of <var>z</var>. |
| Mathematically, this corresponds to the value |
| |
| <p>log (z) = log (cabs (z)) + I * carg (z) |
| |
| <p class="noindent"><code>clog</code> has a pole at 0, and will signal overflow if <var>z</var> equals |
| or is very close to 0. It is well-defined for all other values of |
| <var>z</var>. |
| </p></blockquote></div> |
| |
| <!-- complex.h --> |
| <!-- GNU --> |
| <div class="defun"> |
| — Function: complex double <b>clog10</b> (<var>complex double z</var>)<var><a name="index-clog10-2191"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- GNU --> |
| — Function: complex float <b>clog10f</b> (<var>complex float z</var>)<var><a name="index-clog10f-2192"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- GNU --> |
| — Function: complex long double <b>clog10l</b> (<var>complex long double z</var>)<var><a name="index-clog10l-2193"></a></var><br> |
| <blockquote><p>These functions return the base 10 logarithm of the complex value |
| <var>z</var>. Mathematically, this corresponds to the value |
| |
| <p>log (z) = log10 (cabs (z)) + I * carg (z) |
| |
| <p>These functions are GNU extensions. |
| </p></blockquote></div> |
| |
| <!-- complex.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: complex double <b>csqrt</b> (<var>complex double z</var>)<var><a name="index-csqrt-2194"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex float <b>csqrtf</b> (<var>complex float z</var>)<var><a name="index-csqrtf-2195"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex long double <b>csqrtl</b> (<var>complex long double z</var>)<var><a name="index-csqrtl-2196"></a></var><br> |
| <blockquote><p>These functions return the complex square root of the argument <var>z</var>. Unlike |
| the real-valued functions, they are defined for all values of <var>z</var>. |
| </p></blockquote></div> |
| |
| <!-- complex.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: complex double <b>cpow</b> (<var>complex double base, complex double power</var>)<var><a name="index-cpow-2197"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex float <b>cpowf</b> (<var>complex float base, complex float power</var>)<var><a name="index-cpowf-2198"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex long double <b>cpowl</b> (<var>complex long double base, complex long double power</var>)<var><a name="index-cpowl-2199"></a></var><br> |
| <blockquote><p>These functions return <var>base</var> raised to the power of |
| <var>power</var>. This is equivalent to <code>cexp (y * clog (x))</code><!-- /@w --> |
| </p></blockquote></div> |
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