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| <h3 class="section">20.6 Rounding Modes</h3> |
| |
| <p>Floating-point calculations are carried out internally with extra |
| precision, and then rounded to fit into the destination type. This |
| ensures that results are as precise as the input data. IEEE 754<!-- /@w --> |
| defines four possible rounding modes: |
| |
| <dl> |
| <dt>Round to nearest.<dd>This is the default mode. It should be used unless there is a specific |
| need for one of the others. In this mode results are rounded to the |
| nearest representable value. If the result is midway between two |
| representable values, the even representable is chosen. <dfn>Even</dfn> here |
| means the lowest-order bit is zero. This rounding mode prevents |
| statistical bias and guarantees numeric stability: round-off errors in a |
| lengthy calculation will remain smaller than half of <code>FLT_EPSILON</code>. |
| |
| <!-- @item Round toward @math{+@infinity{}} --> |
| <br><dt>Round toward plus Infinity.<dd>All results are rounded to the smallest representable value |
| which is greater than the result. |
| |
| <!-- @item Round toward @math{-@infinity{}} --> |
| <br><dt>Round toward minus Infinity.<dd>All results are rounded to the largest representable value which is less |
| than the result. |
| |
| <br><dt>Round toward zero.<dd>All results are rounded to the largest representable value whose |
| magnitude is less than that of the result. In other words, if the |
| result is negative it is rounded up; if it is positive, it is rounded |
| down. |
| </dl> |
| |
| <p class="noindent"><samp><span class="file">fenv.h</span></samp> defines constants which you can use to refer to the |
| various rounding modes. Each one will be defined if and only if the FPU |
| supports the corresponding rounding mode. |
| |
| <dl> |
| <!-- fenv.h --> |
| <!-- ISO --> |
| <a name="index-FE_005fTONEAREST-2389"></a><dt><code>FE_TONEAREST</code><dd>Round to nearest. |
| |
| <!-- fenv.h --> |
| <!-- ISO --> |
| <p><a name="index-FE_005fUPWARD-2390"></a><br><dt><code>FE_UPWARD</code><dd>Round toward +&infin;. |
| |
| <!-- fenv.h --> |
| <!-- ISO --> |
| <p><a name="index-FE_005fDOWNWARD-2391"></a><br><dt><code>FE_DOWNWARD</code><dd>Round toward -&infin;. |
| |
| <!-- fenv.h --> |
| <!-- ISO --> |
| <p><a name="index-FE_005fTOWARDZERO-2392"></a><br><dt><code>FE_TOWARDZERO</code><dd>Round toward zero. |
| </dl> |
| |
| <p>Underflow is an unusual case. Normally, IEEE 754<!-- /@w --> floating point |
| numbers are always normalized (see <a href="Floating-Point-Concepts.html#Floating-Point-Concepts">Floating Point Concepts</a>). |
| Numbers smaller than 2^r (where r is the minimum exponent, |
| <code>FLT_MIN_RADIX-1</code> for <var>float</var>) cannot be represented as |
| normalized numbers. Rounding all such numbers to zero or 2^r |
| would cause some algorithms to fail at 0. Therefore, they are left in |
| denormalized form. That produces loss of precision, since some bits of |
| the mantissa are stolen to indicate the decimal point. |
| |
| <p>If a result is too small to be represented as a denormalized number, it |
| is rounded to zero. However, the sign of the result is preserved; if |
| the calculation was negative, the result is <dfn>negative zero</dfn>. |
| Negative zero can also result from some operations on infinity, such as |
| 4/-&infin;. Negative zero behaves identically to zero except |
| when the <code>copysign</code> or <code>signbit</code> functions are used to check |
| the sign bit directly. |
| |
| <p>At any time one of the above four rounding modes is selected. You can |
| find out which one with this function: |
| |
| <!-- fenv.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: int <b>fegetround</b> (<var>void</var>)<var><a name="index-fegetround-2393"></a></var><br> |
| <blockquote><p>Returns the currently selected rounding mode, represented by one of the |
| values of the defined rounding mode macros. |
| </p></blockquote></div> |
| |
| <p class="noindent">To change the rounding mode, use this function: |
| |
| <!-- fenv.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: int <b>fesetround</b> (<var>int round</var>)<var><a name="index-fesetround-2394"></a></var><br> |
| <blockquote><p>Changes the currently selected rounding mode to <var>round</var>. If |
| <var>round</var> does not correspond to one of the supported rounding modes |
| nothing is changed. <code>fesetround</code> returns zero if it changed the |
| rounding mode, a nonzero value if the mode is not supported. |
| </p></blockquote></div> |
| |
| <p>You should avoid changing the rounding mode if possible. It can be an |
| expensive operation; also, some hardware requires you to compile your |
| program differently for it to work. The resulting code may run slower. |
| See your compiler documentation for details. |
| <!-- This section used to claim that functions existed to round one number --> |
| <!-- in a specific fashion. I can't find any functions in the library --> |
| <!-- that do that. -zw --> |
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