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| 24 | </div> | |
| 25 | <div class="section"> | |
| 26 | <div class="titlepage"><div><div><h4 class="title"> | |
| 27 | <a name="boost_multiprecision.tut.limits.constants"></a><a class="link" href="constants.html" title="std::numeric_limits<> constants">std::numeric_limits<> | |
| 28 | constants</a> | |
| 29 | </h4></div></div></div> | |
| 30 | <h5> | |
| 31 | <a name="boost_multiprecision.tut.limits.constants.h0"></a> | |
| 32 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.is_specialized"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.is_specialized">is_specialized</a> | |
| 33 | </h5> | |
| 34 | <p> | |
| 35 | <code class="computeroutput"><span class="keyword">true</span></code> for all arithmetic types | |
| 36 | (integer, floating and fixed-point) for which <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">numeric_limits</span></code> | |
| 37 | is specialized. | |
| 38 | </p> | |
| 39 | <p> | |
| 40 | A typical test is | |
| 41 | </p> | |
| 42 | <pre class="programlisting"><span class="keyword">if</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">is_specialized</span> <span class="special">==</span> <span class="keyword">false</span><span class="special">)</span> | |
| 43 | <span class="special">{</span> | |
| 44 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"type "</span> <span class="special"><<</span> <span class="keyword">typeid</span><span class="special">(</span><span class="identifier">T</span><span class="special">).</span><span class="identifier">name</span><span class="special">()</span> <span class="special"><<</span> <span class="string">" is not specialized for std::numeric_limits!"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
| 45 | <span class="comment">// ...</span> | |
| 46 | <span class="special">}</span> | |
| 47 | </pre> | |
| 48 | <p> | |
| 49 | Typically <code class="computeroutput"><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">is_specialized</span></code> | |
| 50 | is <code class="computeroutput"><span class="keyword">true</span></code> for all <code class="computeroutput"><span class="identifier">T</span></code> where the compile-time constant members | |
| 51 | of <code class="computeroutput"><span class="identifier">numeric_limits</span></code> are indeed | |
| 52 | known at compile time, and don't vary at runtime. For example floating-point | |
| 53 | types with runtime-variable precision such as <code class="computeroutput"><span class="identifier">mpfr_float</span></code> | |
| 54 | have no <code class="computeroutput"><span class="identifier">numeric_limits</span></code> | |
| 55 | specialization as it would be impossible to define all the members at compile | |
| 56 | time. In contrast the precision of a type such as <code class="computeroutput"><span class="identifier">mpfr_float_50</span></code> | |
| 57 | is known at compile time, and so it <span class="emphasis"><em>does</em></span> have a <code class="computeroutput"><span class="identifier">numeric_limits</span></code> specialization. | |
| 58 | </p> | |
| 59 | <p> | |
| 60 | Note that not all the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code> | |
| 61 | member constants and functions are meaningful for all user-defined types | |
| 62 | (UDT), such as the decimal and binary multiprecision types provided here. | |
| 63 | More information on this is given in the sections below. | |
| 64 | </p> | |
| 65 | <h5> | |
| 66 | <a name="boost_multiprecision.tut.limits.constants.h1"></a> | |
| 67 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.infinity"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.infinity">infinity</a> | |
| 68 | </h5> | |
| 69 | <p> | |
| 70 | For floating-point types, ∞ is defined wherever possible, but clearly infinity | |
| 71 | is meaningless for __arbitrary_precision arithmetic backends, and there | |
| 72 | is one floating-point type (GMP's <code class="computeroutput"><span class="identifier">mpf_t</span></code>, | |
| 73 | see <a class="link" href="../floats/gmp_float.html" title="gmp_float">gmp_float</a>) | |
| 74 | which has no notion of infinity or NaN at all. | |
| 75 | </p> | |
| 76 | <p> | |
| 77 | A typical test whether infinity is implemented is | |
| 78 | </p> | |
| 79 | <pre class="programlisting"><span class="keyword">if</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">has_infinity</span><span class="special">)</span> | |
| 80 | <span class="special">{</span> | |
| 81 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
| 82 | <span class="special">}</span> | |
| 83 | </pre> | |
| 84 | <p> | |
| 85 | and using tests like this is strongly recommended to improve portability. | |
| 86 | </p> | |
| 87 | <p> | |
| 88 | If the backend is switched to a type that does not support infinity then, | |
| 89 | without checks like this, there will be trouble. | |
| 90 | </p> | |
| 91 | <h5> | |
| 92 | <a name="boost_multiprecision.tut.limits.constants.h2"></a> | |
| 93 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.is_signed"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.is_signed">is_signed</a> | |
| 94 | </h5> | |
| 95 | <p> | |
| 96 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">is_signed</span> <span class="special">==</span> | |
| 97 | <span class="keyword">true</span></code> if the type <code class="computeroutput"><span class="identifier">T</span></code> | |
| 98 | is signed. | |
| 99 | </p> | |
| 100 | <p> | |
| 101 | For built-in binary types, the sign is held in a single bit, but for other | |
| 102 | types (cpp_dec_float and cpp_bin_float) it may be a separate storage element, | |
| 103 | usually <code class="computeroutput"><span class="keyword">bool</span></code>. | |
| 104 | </p> | |
| 105 | <h5> | |
| 106 | <a name="boost_multiprecision.tut.limits.constants.h3"></a> | |
| 107 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.is_exact"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.is_exact">is_exact</a> | |
| 108 | </h5> | |
| 109 | <p> | |
| 110 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">is_exact</span> <span class="special">==</span> | |
| 111 | <span class="keyword">true</span></code> if type T uses exact representations. | |
| 112 | </p> | |
| 113 | <p> | |
| 114 | This is defined as <code class="computeroutput"><span class="keyword">true</span></code> for | |
| 115 | all integer types and <code class="computeroutput"><span class="keyword">false</span></code> | |
| 116 | for floating-point types. | |
| 117 | </p> | |
| 118 | <p> | |
| 119 | <a href="http://stackoverflow.com/questions/14203654/stdnumeric-limitsis-exact-what-is-a-usable-definition" target="_top">A | |
| 120 | usable definition</a> has been discussed. | |
| 121 | </p> | |
| 122 | <p> | |
| 123 | ISO/IEC 10967-1, Language independent arithmetic, noted by the C++ Standard | |
| 124 | defines | |
| 125 | </p> | |
| 126 | <pre class="programlisting"><span class="identifier">A</span> <span class="identifier">floating</span><span class="special">-</span><span class="identifier">point</span> <span class="identifier">type</span> <span class="identifier">F</span> <span class="identifier">shall</span> <span class="identifier">be</span> <span class="identifier">a</span> <span class="identifier">finite</span> <span class="identifier">subset</span> <span class="identifier">of</span> <span class="special">[</span><span class="identifier">real</span><span class="special">].</span> | |
| 127 | </pre> | |
| 128 | <p> | |
| 129 | The important practical distinction is that all integers (up to <code class="computeroutput"><span class="identifier">max</span><span class="special">()</span></code>) | |
| 130 | can be stored exactly. | |
| 131 | </p> | |
| 132 | <p> | |
| 133 | <a href="http://en.wikipedia.org/wiki/Rational_number" target="_top">Rational</a> | |
| 134 | types using two integer types are also exact. | |
| 135 | </p> | |
| 136 | <p> | |
| 137 | Floating-point types <span class="bold"><strong>cannot store all real values</strong></span> | |
| 138 | (those in the set of ℜ) <span class="bold"><strong>exactly</strong></span>. For example, | |
| 139 | 0.5 can be stored exactly in a binary floating-point, but 0.1 cannot. What | |
| 140 | is stored is the nearest representable real value, that is, rounded to | |
| 141 | nearest. | |
| 142 | </p> | |
| 143 | <p> | |
| 144 | Fixed-point types (usually decimal) are also defined as exact, in that | |
| 145 | they only store a <span class="bold"><strong>fixed precision</strong></span>, so | |
| 146 | half cents or pennies (or less) cannot be stored. The results of computations | |
| 147 | are rounded up or down, just like the result of integer division stored | |
| 148 | as an integer result. | |
| 149 | </p> | |
| 150 | <p> | |
| 151 | There are number of proposals to <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n3407.html" target="_top">add | |
| 152 | Decimal floating-point Support to C++</a>. | |
| 153 | </p> | |
| 154 | <p> | |
| 155 | <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2009/n2849.pdf" target="_top">Decimal | |
| 156 | TR</a>. | |
| 157 | </p> | |
| 158 | <p> | |
| 159 | And also <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n3352.html" target="_top">C++ | |
| 160 | Binary Fixed-Point Arithmetic</a>. | |
| 161 | </p> | |
| 162 | <h5> | |
| 163 | <a name="boost_multiprecision.tut.limits.constants.h4"></a> | |
| 164 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.is_bounded"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.is_bounded">is_bounded</a> | |
| 165 | </h5> | |
| 166 | <p> | |
| 167 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">is_bounded</span> <span class="special">==</span> | |
| 168 | <span class="keyword">true</span></code> if the set of values represented | |
| 169 | by the type <code class="computeroutput"><span class="identifier">T</span></code> is finite. | |
| 170 | </p> | |
| 171 | <p> | |
| 172 | This is <code class="computeroutput"><span class="keyword">true</span></code> for all built-in | |
| 173 | integer, fixed and floating-point types, and most multi-precision types. | |
| 174 | </p> | |
| 175 | <p> | |
| 176 | It is only <code class="computeroutput"><span class="keyword">false</span></code> for a few | |
| 177 | __arbitrary_precision types like <code class="computeroutput"><span class="identifier">cpp_int</span></code>. | |
| 178 | </p> | |
| 179 | <p> | |
| 180 | Rational and fixed-exponent representations are exact but not integer. | |
| 181 | </p> | |
| 182 | <h5> | |
| 183 | <a name="boost_multiprecision.tut.limits.constants.h5"></a> | |
| 184 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.is_modulo"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.is_modulo">is_modulo</a> | |
| 185 | </h5> | |
| 186 | <p> | |
| 187 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">is_modulo</span></code> is defined as <code class="computeroutput"><span class="keyword">true</span></code> if adding two positive values of type | |
| 188 | T can yield a result less than either value. | |
| 189 | </p> | |
| 190 | <p> | |
| 191 | <code class="computeroutput"><span class="identifier">is_modulo</span> <span class="special">==</span> | |
| 192 | <span class="keyword">true</span></code> means that the type does not | |
| 193 | overflow, but, for example, 'wraps around' to zero, when adding one to | |
| 194 | the <code class="computeroutput"><span class="identifier">max</span><span class="special">()</span></code> | |
| 195 | value. | |
| 196 | </p> | |
| 197 | <p> | |
| 198 | For most built-in integer types, <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><>::</span><span class="identifier">is_modulo</span></code> | |
| 199 | is <code class="computeroutput"><span class="keyword">true</span></code>. | |
| 200 | </p> | |
| 201 | <p> | |
| 202 | <code class="computeroutput"><span class="keyword">bool</span></code> is the only exception. | |
| 203 | </p> | |
| 204 | <p> | |
| 205 | The modulo behaviour is sometimes useful, but also can be unexpected, and | |
| 206 | sometimes undesired, behaviour. | |
| 207 | </p> | |
| 208 | <p> | |
| 209 | Overflow of signed integers can be especially unexpected, possibly causing | |
| 210 | change of sign. | |
| 211 | </p> | |
| 212 | <p> | |
| 213 | Boost.Multiprecision integer type <code class="computeroutput"><span class="identifier">cpp_int</span></code> | |
| 214 | is not modulo because as an __arbitrary_precision types, it expands to | |
| 215 | hold any value that the machine resources permit. | |
| 216 | </p> | |
| 217 | <p> | |
| 218 | However fixed precision <a class="link" href="../ints/cpp_int.html" title="cpp_int">cpp_int</a>'s | |
| 219 | may be modulo if they are unchecked (i.e. they behave just like built in | |
| 220 | integers), but not if they are checked (overflow causes an exception to | |
| 221 | be raised). | |
| 222 | </p> | |
| 223 | <p> | |
| 224 | Built-in and multi-precision floating-point types are normally not modulo. | |
| 225 | </p> | |
| 226 | <p> | |
| 227 | Where possible, overflow is to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><>::</span><span class="identifier">infinity</span><span class="special">()</span></code>, provided <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><>::</span><span class="identifier">has_infinity</span> | |
| 228 | <span class="special">==</span> <span class="keyword">true</span></code>. | |
| 229 | </p> | |
| 230 | <h5> | |
| 231 | <a name="boost_multiprecision.tut.limits.constants.h6"></a> | |
| 232 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.radix"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.radix">radix</a> | |
| 233 | </h5> | |
| 234 | <p> | |
| 235 | Constant <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">radix</span></code> returns either 2 (for built-in | |
| 236 | and binary types) or 10 (for decimal types). | |
| 237 | </p> | |
| 238 | <h5> | |
| 239 | <a name="boost_multiprecision.tut.limits.constants.h7"></a> | |
| 240 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.digits"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.digits">digits</a> | |
| 241 | </h5> | |
| 242 | <p> | |
| 243 | The number of <code class="computeroutput"><span class="identifier">radix</span></code> digits | |
| 244 | that be represented without change: | |
| 245 | </p> | |
| 246 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
| 247 | <li class="listitem"> | |
| 248 | for integer types, the number of <span class="bold"><strong>non-sign bits</strong></span> | |
| 249 | in the significand. | |
| 250 | </li> | |
| 251 | <li class="listitem"> | |
| 252 | for floating types, the number of <span class="bold"><strong>radix digits</strong></span> | |
| 253 | in the significand. | |
| 254 | </li> | |
| 255 | </ul></div> | |
| 256 | <p> | |
| 257 | The values include any implicit bit, so for example, for the ubiquious | |
| 258 | <code class="computeroutput"><span class="keyword">double</span></code> using 64 bits (<a href="http://en.wikipedia.org/wiki/Double_precision_floating-point_format" target="_top">IEEE | |
| 259 | binary64 </a>), <code class="computeroutput"><span class="identifier">digits</span></code> | |
| 260 | == 53, even though there are only 52 actual bits of the significand stored | |
| 261 | in the representation. The value of <code class="computeroutput"><span class="identifier">digits</span></code> | |
| 262 | reflects the fact that there is one implicit bit which is always set to | |
| 263 | 1. | |
| 264 | </p> | |
| 265 | <p> | |
| 266 | The Boost.Multiprecision binary types do not use an implicit bit, so the | |
| 267 | <code class="computeroutput"><span class="identifier">digits</span></code> member reflects | |
| 268 | exactly how many bits of precision were requested: | |
| 269 | </p> | |
| 270 | <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">53</span><span class="special">,</span> <span class="identifier">digit_base_2</span><span class="special">></span> <span class="special">></span> <span class="identifier">float64</span><span class="special">;</span> | |
| 271 | <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">113</span><span class="special">,</span> <span class="identifier">digit_base_2</span><span class="special">></span> <span class="special">></span> <span class="identifier">float128</span><span class="special">;</span> | |
| 272 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">float64</span><span class="special">>::</span><span class="identifier">digits</span> <span class="special">==</span> <span class="number">53.</span> | |
| 273 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">float128</span><span class="special">>::</span><span class="identifier">digits</span> <span class="special">==</span> <span class="number">113.</span> | |
| 274 | </pre> | |
| 275 | <p> | |
| 276 | For the most common case of <code class="computeroutput"><span class="identifier">radix</span> | |
| 277 | <span class="special">==</span> <span class="number">2</span></code>, | |
| 278 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span></code> is the number of bits in the representation, | |
| 279 | not counting any sign bit. | |
| 280 | </p> | |
| 281 | <p> | |
| 282 | For a decimal integer type, when <code class="computeroutput"><span class="identifier">radix</span> | |
| 283 | <span class="special">==</span> <span class="number">10</span></code>, | |
| 284 | it is the number of decimal digits. | |
| 285 | </p> | |
| 286 | <h5> | |
| 287 | <a name="boost_multiprecision.tut.limits.constants.h8"></a> | |
| 288 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.digits10"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.digits10">digits10</a> | |
| 289 | </h5> | |
| 290 | <p> | |
| 291 | Constant <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits10</span></code> returns the number of decimal | |
| 292 | digits that can be represented without change or loss. | |
| 293 | </p> | |
| 294 | <p> | |
| 295 | For example, <code class="computeroutput"><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">unsigned</span> <span class="keyword">char</span><span class="special">>::</span><span class="identifier">digits10</span></code> is 2. | |
| 296 | </p> | |
| 297 | <p> | |
| 298 | This somewhat inscrutable definition means that an <code class="computeroutput"><span class="keyword">unsigned</span> | |
| 299 | <span class="keyword">char</span></code> can hold decimal values <code class="computeroutput"><span class="number">0.</span><span class="special">.</span><span class="number">99</span></code> | |
| 300 | without loss of precision or accuracy, usually from truncation. | |
| 301 | </p> | |
| 302 | <p> | |
| 303 | Had the definition been 3 then that would imply it could hold 0..999, but | |
| 304 | as we all know, an 8-bit <code class="computeroutput"><span class="keyword">unsigned</span> | |
| 305 | <span class="keyword">char</span></code> can only hold 0..255, and an | |
| 306 | attempt to store 256 or more will involve loss or change. | |
| 307 | </p> | |
| 308 | <p> | |
| 309 | For bounded integers, it is thus <span class="bold"><strong>one less</strong></span> | |
| 310 | than number of decimal digits you need to display the biggest integer | |
| 311 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()</span></code>. | |
| 312 | This value can be used to predict the layout width required for | |
| 313 | </p> | |
| 314 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> | |
| 315 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">short</span><span class="special">>::</span><span class="identifier">digits10</span> <span class="special">+</span><span class="number">1</span> <span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="comment">// digits10+1, and +1 for sign.</span> | |
| 316 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpos</span> <span class="special"><<</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">short</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)()</span> <span class="comment">// +32767</span> | |
| 317 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span> | |
| 318 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">short</span><span class="special">>::</span><span class="identifier">digits10</span> <span class="special">+</span><span class="number">1</span> <span class="special">+</span><span class="number">1</span><span class="special">)</span> | |
| 319 | <span class="special"><<</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">short</span><span class="special">>::</span><span class="identifier">min</span><span class="special">)()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// -32767</span> | |
| 320 | </pre> | |
| 321 | <p> | |
| 322 | For example, <code class="computeroutput"><span class="keyword">unsigned</span> <span class="keyword">short</span></code> | |
| 323 | is often stored in 16 bits, so the maximum value is 0xFFFF or 65535. | |
| 324 | </p> | |
| 325 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> | |
| 326 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">unsigned</span> <span class="keyword">short</span><span class="special">>::</span><span class="identifier">digits10</span> <span class="special">+</span><span class="number">1</span> <span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="comment">// digits10+1, and +1 for sign.</span> | |
| 327 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpos</span> <span class="special"><<</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">unsigned</span> <span class="keyword">short</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)()</span> <span class="comment">// 65535</span> | |
| 328 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span> | |
| 329 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">unsigned</span> <span class="keyword">short</span><span class="special">>::</span><span class="identifier">digits10</span> <span class="special">+</span><span class="number">1</span> <span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="comment">// digits10+1, and +1 for sign.</span> | |
| 330 | <span class="special"><<</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">unsigned</span> <span class="keyword">short</span><span class="special">>::</span><span class="identifier">min</span><span class="special">)()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0</span> | |
| 331 | </pre> | |
| 332 | <p> | |
| 333 | For bounded floating-point types, if we create a <code class="computeroutput"><span class="keyword">double</span></code> | |
| 334 | with a value with <code class="computeroutput"><span class="identifier">digits10</span></code> | |
| 335 | (usually 15) decimal digits, <code class="computeroutput"><span class="number">1e15</span></code> | |
| 336 | or <code class="computeroutput"><span class="number">1000000000000000</span></code> : | |
| 337 | </p> | |
| 338 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> | |
| 339 | <span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="number">1e15</span><span class="special">;</span> | |
| 340 | <span class="keyword">double</span> <span class="identifier">dp1</span> <span class="special">=</span> <span class="identifier">d</span><span class="special">+</span><span class="number">1</span><span class="special">;</span> | |
| 341 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">d</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="special"><<</span> <span class="identifier">dp1</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
| 342 | <span class="comment">// 1000000000000000</span> | |
| 343 | <span class="comment">// 1000000000000001</span> | |
| 344 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">dp1</span> <span class="special">-</span> <span class="identifier">d</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1</span> | |
| 345 | </pre> | |
| 346 | <p> | |
| 347 | and we can increment this value to <code class="computeroutput"><span class="number">1000000000000001</span></code> | |
| 348 | as expected and show the difference too. | |
| 349 | </p> | |
| 350 | <p> | |
| 351 | But if we try to repeat this with more than <code class="computeroutput"><span class="identifier">digits10</span></code> | |
| 352 | digits, | |
| 353 | </p> | |
| 354 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> | |
| 355 | <span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="number">1e16</span><span class="special">;</span> | |
| 356 | <span class="keyword">double</span> <span class="identifier">dp1</span> <span class="special">=</span> <span class="identifier">d</span><span class="special">+</span><span class="number">1</span><span class="special">;</span> | |
| 357 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">d</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="special"><<</span> <span class="identifier">dp1</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
| 358 | <span class="comment">// 10000000000000000</span> | |
| 359 | <span class="comment">// 10000000000000000</span> | |
| 360 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">dp1</span> <span class="special">-</span> <span class="identifier">d</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0 !!!</span> | |
| 361 | </pre> | |
| 362 | <p> | |
| 363 | then we find that when we add one it has no effect, and display show that | |
| 364 | there is loss of precision. See <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">Loss | |
| 365 | of significance or cancellation error</a>. | |
| 366 | </p> | |
| 367 | <p> | |
| 368 | So <code class="computeroutput"><span class="identifier">digits10</span></code> is the number | |
| 369 | of decimal digits <span class="bold"><strong>guaranteed</strong></span> to be correct. | |
| 370 | </p> | |
| 371 | <p> | |
| 372 | For example, 'round-tripping' for <code class="computeroutput"><span class="keyword">double</span></code>: | |
| 373 | </p> | |
| 374 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
| 375 | <li class="listitem"> | |
| 376 | If a decimal string with at most <code class="computeroutput"><span class="identifier">digits10</span></code>( | |
| 377 | == 15) significant decimal digits is converted to <code class="computeroutput"><span class="keyword">double</span></code> | |
| 378 | and then converted back to the same number of significant decimal digits, | |
| 379 | then the final string will match the original 15 decimal digit string. | |
| 380 | </li> | |
| 381 | <li class="listitem"> | |
| 382 | If a <code class="computeroutput"><span class="keyword">double</span></code> floating-point | |
| 383 | number is converted to a decimal string with at least 17 decimal digits | |
| 384 | and then converted back to <code class="computeroutput"><span class="keyword">double</span></code>, | |
| 385 | then the result will be binary identical to the original <code class="computeroutput"><span class="keyword">double</span></code> value. | |
| 386 | </li> | |
| 387 | </ul></div> | |
| 388 | <p> | |
| 389 | For most purposes, you will much more likely want <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><>::</span><span class="identifier">max_digits10</span></code>, | |
| 390 | the number of decimal digits that ensure that a change of one least significant | |
| 391 | bit (ULP) produces a different decimal digits string. | |
| 392 | </p> | |
| 393 | <p> | |
| 394 | For the most common <code class="computeroutput"><span class="keyword">double</span></code> | |
| 395 | floating-point type,<code class="computeroutput"><span class="identifier">max_digits10</span></code> | |
| 396 | is <code class="computeroutput"><span class="identifier">digits10</span><span class="special">+</span><span class="number">2</span></code>, but you should use C++11 <code class="computeroutput"><span class="identifier">max_digits10</span></code> where possible (see <a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.max_digits10">below</a>). | |
| 397 | </p> | |
| 398 | <h5> | |
| 399 | <a name="boost_multiprecision.tut.limits.constants.h9"></a> | |
| 400 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.max_digits10"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.max_digits10">max_digits10</a> | |
| 401 | </h5> | |
| 402 | <p> | |
| 403 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max_digits10</span></code> was added for floating-point | |
| 404 | because <code class="computeroutput"><span class="identifier">digits10</span></code> decimal | |
| 405 | digits are insufficient to show a least significant bit (ULP) change giving | |
| 406 | puzzling displays like | |
| 407 | </p> | |
| 408 | <pre class="programlisting"><span class="number">0.666666666666667</span> <span class="special">!=</span> <span class="number">0.666666666666667</span> | |
| 409 | </pre> | |
| 410 | <p> | |
| 411 | from failure to 'round-trip', for example: | |
| 412 | </p> | |
| 413 | <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">write</span> <span class="special">=</span> <span class="number">2.</span><span class="special">/</span><span class="number">3</span><span class="special">;</span> <span class="comment">// Any arbitrary value that cannot be represented exactly.</span> | |
| 414 | <span class="keyword">double</span> <span class="identifier">read</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> | |
| 415 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">stringstream</span> <span class="identifier">s</span><span class="special">;</span> | |
| 416 | <span class="identifier">s</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">);</span> <span class="comment">// or `float64_t` for 64-bit IEE754 double.</span> | |
| 417 | <span class="identifier">s</span> <span class="special"><<</span> <span class="identifier">write</span><span class="special">;</span> | |
| 418 | <span class="identifier">s</span> <span class="special">>></span> <span class="identifier">read</span><span class="special">;</span> | |
| 419 | <span class="keyword">if</span><span class="special">(</span><span class="identifier">read</span> <span class="special">!=</span> <span class="identifier">write</span><span class="special">)</span> | |
| 420 | <span class="special">{</span> | |
| 421 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span> | |
| 422 | <span class="special"><<</span> <span class="identifier">read</span> <span class="special"><<</span> <span class="string">" != "</span> <span class="special"><<</span> <span class="identifier">write</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
| 423 | <span class="special">}</span> | |
| 424 | </pre> | |
| 425 | <p> | |
| 426 | If you wish to ensure that a change of one least significant bit (ULP) | |
| 427 | produces a different decimal digits string, then <code class="computeroutput"><span class="identifier">max_digits10</span></code> | |
| 428 | is the precision to use. | |
| 429 | </p> | |
| 430 | <p> | |
| 431 | For example: | |
| 432 | </p> | |
| 433 | <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">pi</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">double_constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">;</span> | |
| 434 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> | |
| 435 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pi</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 3.1415926535897931</span> | |
| 436 | </pre> | |
| 437 | <p> | |
| 438 | will display π to the maximum possible precision using a <code class="computeroutput"><span class="keyword">double</span></code>. | |
| 439 | </p> | |
| 440 | <p> | |
| 441 | and similarly for a much higher precision type: | |
| 442 | </p> | |
| 443 | <pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">;</span> | |
| 444 | ||
| 445 | <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">cpp_dec_float</span><span class="special"><</span><span class="number">50</span><span class="special">></span> <span class="special">></span> <span class="identifier">cpp_dec_float_50</span><span class="special">;</span> <span class="comment">// 50 decimal digits.</span> | |
| 446 | ||
| 447 | <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span><span class="special">;</span> | |
| 448 | ||
| 449 | <span class="identifier">cpp_dec_float_50</span> <span class="identifier">pi</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>();</span> | |
| 450 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> | |
| 451 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pi</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
| 452 | <span class="comment">// 3.141592653589793238462643383279502884197169399375105820974944592307816406</span> | |
| 453 | </pre> | |
| 454 | <p> | |
| 455 | For integer types, <code class="computeroutput"><span class="identifier">max_digits10</span></code> | |
| 456 | is implementation-dependent, but is usually <code class="computeroutput"><span class="identifier">digits10</span> | |
| 457 | <span class="special">+</span> <span class="number">2</span></code>. | |
| 458 | This is the output field-width required for the maximum value of the type | |
| 459 | T <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()</span></code> | |
| 460 | <span class="emphasis"><em>including a sign and a space</em></span>. | |
| 461 | </p> | |
| 462 | <p> | |
| 463 | So this will produce neat columns. | |
| 464 | </p> | |
| 465 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">int</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">)</span> <span class="special">...</span> | |
| 466 | </pre> | |
| 467 | <p> | |
| 468 | The extra two or three least-significant digits are 'noisy' and may be | |
| 469 | junk, but if you want to 'round-trip' - printing a value out as a decimal | |
| 470 | digit string and reading it back in - (most commonly during serialization | |
| 471 | and de-serialization) you must use <code class="computeroutput"><span class="identifier">os</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">)</span></code>. | |
| 472 | </p> | |
| 473 | <div class="note"><table border="0" summary="Note"> | |
| 474 | <tr> | |
| 475 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> | |
| 476 | <th align="left">Note</th> | |
| 477 | </tr> | |
| 478 | <tr><td align="left" valign="top"><p> | |
| 479 | For Microsoft Visual Studio 2010, <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">max_digits10</span></code> | |
| 480 | is wrongly defined as 8. It should be 9. | |
| 481 | </p></td></tr> | |
| 482 | </table></div> | |
| 483 | <div class="note"><table border="0" summary="Note"> | |
| 484 | <tr> | |
| 485 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> | |
| 486 | <th align="left">Note</th> | |
| 487 | </tr> | |
| 488 | <tr><td align="left" valign="top"> | |
| 489 | <p> | |
| 490 | For Microsoft Visual Studio before 2013, and default float format, a | |
| 491 | small range of values approximately 0.0001 to 0.004, with exponent values | |
| 492 | of 3F2 to 3F6, are wrongly input by one least significant bit, probably | |
| 493 | every third value of significand. | |
| 494 | </p> | |
| 495 | <p> | |
| 496 | A workaround is using scientific or exponential format <code class="computeroutput"><span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span></code>. | |
| 497 | </p> | |
| 498 | </td></tr> | |
| 499 | </table></div> | |
| 500 | <div class="note"><table border="0" summary="Note"> | |
| 501 | <tr> | |
| 502 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> | |
| 503 | <th align="left">Note</th> | |
| 504 | </tr> | |
| 505 | <tr><td align="left" valign="top"><p> | |
| 506 | BOOST_NO_CXX11_NUMERIC_LIMITS is a suitable feature-test macro to determine | |
| 507 | if <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">max_digits10</span></code> is implemented on any | |
| 508 | platform. | |
| 509 | </p></td></tr> | |
| 510 | </table></div> | |
| 511 | <p> | |
| 512 | If <code class="computeroutput"><span class="identifier">max_digits10</span></code> is not | |
| 513 | available, you should use the <a href="http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF" target="_top">Kahan | |
| 514 | formula for floating-point type T</a>. | |
| 515 | </p> | |
| 516 | <p> | |
| 517 | In C++, the equations for what Kahan (on page 4) describes as 'at least' | |
| 518 | and 'at most' are: | |
| 519 | </p> | |
| 520 | <pre class="programlisting"><span class="keyword">static</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="keyword">const</span> <span class="identifier">log10Two</span> <span class="special">=</span> <span class="number">0.30102999566398119521373889472449L</span><span class="special">;</span> <span class="comment">// log10(2.)</span> | |
| 521 | ||
| 522 | <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">int</span><span class="special">>(</span><span class="identifier">floor</span><span class="special">((</span><span class="identifier">significand_digits</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">log10Two</span><span class="special">));</span> <span class="comment">// == digits10 - 'at least' .</span> | |
| 523 | <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">int</span><span class="special">>(</span><span class="identifier">ceil</span><span class="special">(</span><span class="number">1</span> <span class="special">+</span> <span class="identifier">significand_digits</span> <span class="special">*</span> <span class="identifier">log10Two</span><span class="special">));</span> <span class="comment">// == max_digits10 - 'at most'.</span> | |
| 524 | </pre> | |
| 525 | <p> | |
| 526 | Unfortunately, these cannot be evaluated (at least by C++03) at <span class="bold"><strong>compile-time</strong></span>. So the following expression is often | |
| 527 | used instead. | |
| 528 | </p> | |
| 529 | <pre class="programlisting"><span class="identifier">max_digits10</span> <span class="special">=</span> <span class="number">2</span> <span class="special">+</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span> <span class="special">*</span> <span class="number">3010U</span><span class="special">/</span><span class="number">10000U</span><span class="special">;</span> | |
| 530 | ||
| 531 | <span class="comment">// == 2 + std::numeric_limits<T>::digits for double and 64-bit long double.</span> | |
| 532 | <span class="comment">// == 3 + std::numeric_limits<T>::digits for float, 80-bit long-double and __float128.</span> | |
| 533 | </pre> | |
| 534 | <p> | |
| 535 | often the actual values are computed for the C limits macros: | |
| 536 | </p> | |
| 537 | <pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">FLT_MAXDIG10</span> <span class="special">(</span><span class="number">2</span><span class="special">+</span><span class="identifier">FLT_MANT_DIG</span> <span class="special">*</span> <span class="number">3010U</span><span class="special">/</span><span class="number">10000U</span><span class="special">)</span> <span class="comment">// 9</span> | |
| 538 | <span class="preprocessor">#define</span> <span class="identifier">DBL_MAXDIG10</span> <span class="special">(</span><span class="number">2</span><span class="special">+</span> <span class="special">(</span><span class="identifier">DBL_MANT_DIG</span> <span class="special">*</span> <span class="number">3010U</span><span class="special">)/</span><span class="number">10000U</span><span class="special">)</span> <span class="comment">// 17</span> | |
| 539 | <span class="preprocessor">#define</span> <span class="identifier">LDBL_MAXDIG10</span> <span class="special">(</span><span class="number">2</span><span class="special">+</span> <span class="special">(</span><span class="identifier">LDBL_MANT_DIG</span> <span class="special">*</span> <span class="number">3010U</span><span class="special">)/</span><span class="number">10000U</span><span class="special">)</span> <span class="comment">// 17 for MSVC, 18 for others.</span> | |
| 540 | </pre> | |
| 541 | <p> | |
| 542 | The factor 3010U/10000U is <span class="emphasis"><em>log<sub>10</sub>(2) = 0.3010</em></span> that | |
| 543 | can be evaluated at compile-time using only <code class="computeroutput"><span class="keyword">short</span> | |
| 544 | <span class="keyword">unsigned</span> <span class="keyword">int</span></code>s | |
| 545 | to be a desirable <code class="computeroutput"><span class="keyword">const</span></code> or | |
| 546 | <code class="computeroutput"><span class="keyword">constexpr</span></code> (and usually also | |
| 547 | <code class="computeroutput"><span class="keyword">static</span></code>). | |
| 548 | </p> | |
| 549 | <p> | |
| 550 | Boost macros allow this to be done portably, see <a href="http://www.boost.org/doc/libs/1_58_0/libs/config/doc/html/boost_config/boost_macro_reference.html" target="_top">BOOST_CONSTEXPR_OR_CONST | |
| 551 | or BOOST_STATIC_CONSTEXPR</a>. | |
| 552 | </p> | |
| 553 | <p> | |
| 554 | (See also <a href="http://www.loria.fr/~zimmerma/mca/mca-cup-0.5.9.pdf" target="_top">Richard | |
| 555 | P. Brent and Paul Zimmerman, Modern Computer Arithmetic</a> Equation | |
| 556 | 3.8 on page 116). | |
| 557 | </p> | |
| 558 | <p> | |
| 559 | For example, to be portable (including obselete platforms) for type <code class="computeroutput"><span class="identifier">T</span></code> where <code class="computeroutput"><span class="identifier">T</span></code> | |
| 560 | may be: <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span> | |
| 561 | <span class="keyword">double</span></code>, <code class="computeroutput"><span class="number">128</span><span class="special">-</span><span class="identifier">bit</span> <span class="identifier">quad</span> <span class="identifier">type</span></code>, | |
| 562 | <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code> ... | |
| 563 | </p> | |
| 564 | <pre class="programlisting"> <span class="keyword">typedef</span> <span class="keyword">float</span> <span class="identifier">T</span><span class="special">;</span> | |
| 565 | ||
| 566 | <span class="preprocessor">#if</span> <span class="identifier">defined</span> <span class="identifier">BOOST_NO_CXX11_NUMERIC_LIMITS</span> | |
| 567 | <span class="comment">// No max_digits10 implemented.</span> | |
| 568 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">max_digits10</span><span class="special"><</span><span class="identifier">T</span><span class="special">>());</span> | |
| 569 | <span class="preprocessor">#else</span> | |
| 570 | <span class="preprocessor">#if</span><span class="special">(</span><span class="identifier">_MSC_VER</span> <span class="special"><=</span> <span class="number">1600</span><span class="special">)</span> | |
| 571 | <span class="comment">// Wrong value for std::numeric_limits<float>::max_digits10.</span> | |
| 572 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">max_digits10</span><span class="special"><</span><span class="identifier">T</span><span class="special">>());</span> | |
| 573 | <span class="preprocessor">#else</span> <span class="comment">// Use the C++11 max_digits10.</span> | |
| 574 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> | |
| 575 | <span class="preprocessor">#endif</span> | |
| 576 | <span class="preprocessor">#endif</span> | |
| 577 | ||
| 578 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"std::cout.precision(max_digits10) = "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 9</span> | |
| 579 | ||
| 580 | <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">1.2345678901234567889</span><span class="special">;</span> | |
| 581 | ||
| 582 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"x = "</span> <span class="special"><<</span> <span class="identifier">x</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">//</span> | |
| 583 | </pre> | |
| 584 | <p> | |
| 585 | which should output: | |
| 586 | </p> | |
| 587 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">max_digits10</span><span class="special">)</span> <span class="special">=</span> <span class="number">9</span> | |
| 588 | <span class="identifier">x</span> <span class="special">=</span> <span class="number">1.23456789</span> | |
| 589 | </pre> | |
| 590 | <h5> | |
| 591 | <a name="boost_multiprecision.tut.limits.constants.h10"></a> | |
| 592 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.round_style"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.round_style">round_style</a> | |
| 593 | </h5> | |
| 594 | <p> | |
| 595 | The rounding style determines how the result of floating-point operations | |
| 596 | is treated when the result cannot be <span class="bold"><strong>exactly represented</strong></span> | |
| 597 | in the significand. Various rounding modes may be provided: | |
| 598 | </p> | |
| 599 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
| 600 | <li class="listitem"> | |
| 601 | round to nearest up or down (default for floating-point types). | |
| 602 | </li> | |
| 603 | <li class="listitem"> | |
| 604 | round up (toward positive infinity). | |
| 605 | </li> | |
| 606 | <li class="listitem"> | |
| 607 | round down (toward negative infinity). | |
| 608 | </li> | |
| 609 | <li class="listitem"> | |
| 610 | round toward zero (integer types). | |
| 611 | </li> | |
| 612 | <li class="listitem"> | |
| 613 | no rounding (if decimal radix). | |
| 614 | </li> | |
| 615 | <li class="listitem"> | |
| 616 | rounding mode is not determinable. | |
| 617 | </li> | |
| 618 | </ul></div> | |
| 619 | <p> | |
| 620 | For integer types, <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">round_style</span></code> | |
| 621 | is always towards zero, so | |
| 622 | </p> | |
| 623 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">round_style</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">round_to_zero</span><span class="special">;</span> | |
| 624 | </pre> | |
| 625 | <p> | |
| 626 | A decimal type, <code class="computeroutput"><span class="identifier">cpp_dec_float</span></code> | |
| 627 | rounds in no particular direction, which is to say it doesn't round at | |
| 628 | all. And since there are several guard digits, it's not really the same | |
| 629 | as truncation (round toward zero) either. | |
| 630 | </p> | |
| 631 | <p> | |
| 632 | For floating-point types, it is normal to round to nearest. | |
| 633 | </p> | |
| 634 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">round_style</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">round_to_nearest</span><span class="special">;</span> | |
| 635 | </pre> | |
| 636 | <p> | |
| 637 | See function <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">round_error</span></code> for the maximum error (in | |
| 638 | ULP) that rounding can cause. | |
| 639 | </p> | |
| 640 | <h5> | |
| 641 | <a name="boost_multiprecision.tut.limits.constants.h11"></a> | |
| 642 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.has_denorm_loss"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.has_denorm_loss">has_denorm_loss</a> | |
| 643 | </h5> | |
| 644 | <p> | |
| 645 | <code class="computeroutput"><span class="keyword">true</span></code> if a loss of precision | |
| 646 | is detected as a <a href="http://en.wikipedia.org/wiki/Denormalization" target="_top">denormalization</a> | |
| 647 | loss, rather than an inexact result. | |
| 648 | </p> | |
| 649 | <p> | |
| 650 | Always <code class="computeroutput"><span class="keyword">false</span></code> for integer types. | |
| 651 | </p> | |
| 652 | <p> | |
| 653 | <code class="computeroutput"><span class="keyword">false</span></code> for all types which | |
| 654 | do not have <code class="computeroutput"><span class="identifier">has_denorm</span></code> | |
| 655 | == <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_present</span></code>. | |
| 656 | </p> | |
| 657 | <h5> | |
| 658 | <a name="boost_multiprecision.tut.limits.constants.h12"></a> | |
| 659 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.denorm_style"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.denorm_style">denorm_style</a> | |
| 660 | </h5> | |
| 661 | <p> | |
| 662 | <a href="http://en.wikipedia.org/wiki/Denormal_number" target="_top">Denormalized | |
| 663 | values</a> are representations with a variable number of exponent bits | |
| 664 | that can permit gradual underflow, so that, if type T is <code class="computeroutput"><span class="keyword">double</span></code>. | |
| 665 | </p> | |
| 666 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">denorm_min</span><span class="special">()</span> <span class="special"><</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">min</span><span class="special">()</span> | |
| 667 | </pre> | |
| 668 | <p> | |
| 669 | A type may have any of the following <code class="computeroutput"><span class="keyword">enum</span> | |
| 670 | <span class="identifier">float_denorm_style</span></code> values: | |
| 671 | </p> | |
| 672 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
| 673 | <li class="listitem"> | |
| 674 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_absent</span></code>, if it does not allow | |
| 675 | denormalized values. (Always used for all integer and exact types). | |
| 676 | </li> | |
| 677 | <li class="listitem"> | |
| 678 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_present</span></code>, if the floating-point | |
| 679 | type allows denormalized values. | |
| 680 | </li> | |
| 681 | <li class="listitem"> | |
| 682 | <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_indeterminate</span></code>, if indeterminate | |
| 683 | at compile time. | |
| 684 | </li> | |
| 685 | </ul></div> | |
| 686 | <h5> | |
| 687 | <a name="boost_multiprecision.tut.limits.constants.h13"></a> | |
| 688 | <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.tinyness_before_rounding"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.tinyness_before_rounding">Tinyness | |
| 689 | before rounding</a> | |
| 690 | </h5> | |
| 691 | <p> | |
| 692 | <code class="computeroutput"><span class="keyword">bool</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">tinyness_before</span></code> | |
| 693 | </p> | |
| 694 | <p> | |
| 695 | <code class="computeroutput"><span class="keyword">true</span></code> if a type can determine | |
| 696 | that a value is too small to be represent as a normalized value before | |
| 697 | rounding it. | |
| 698 | </p> | |
| 699 | <p> | |
| 700 | Generally true for <code class="computeroutput"><span class="identifier">is_iec559</span></code> | |
| 701 | floating-point built-in types, but false for integer types. | |
| 702 | </p> | |
| 703 | <p> | |
| 704 | Standard-compliant IEEE 754 floating-point implementations may detect the | |
| 705 | floating-point underflow at three predefined moments: | |
| 706 | </p> | |
| 707 | <div class="orderedlist"><ol class="orderedlist" type="1"> | |
| 708 | <li class="listitem"> | |
| 709 | After computation of a result with absolute value smaller than <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">min</span><span class="special">()</span></code>, | |
| 710 | such implementation detects <span class="emphasis"><em>tinyness before rounding</em></span> | |
| 711 | (e.g. UltraSparc). | |
| 712 | </li> | |
| 713 | <li class="listitem"> | |
| 714 | After rounding of the result to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span></code> | |
| 715 | bits, if the result is tiny, such implementation detects <span class="emphasis"><em>tinyness | |
| 716 | after rounding</em></span> (e.g. SuperSparc). | |
| 717 | </li> | |
| 718 | <li class="listitem"> | |
| 719 | If the conversion of the rounded tiny result to subnormal form resulted | |
| 720 | in the loss of precision, such implementation detects <span class="emphasis"><em>denorm | |
| 721 | loss</em></span>. | |
| 722 | </li> | |
| 723 | </ol></div> | |
| 724 | </div> | |
| 725 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 726 | <td align="left"></td> | |
| 727 | <td align="right"><div class="copyright-footer">Copyright © 2002-2013 John Maddock and Christopher Kormanyos<p> | |
| 728 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 729 | file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) | |
| 730 | </p> | |
| 731 | </div></td> | |
| 732 | </tr></table> | |
| 733 | <hr> | |
| 734 | <div class="spirit-nav"> | |
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