<h3><a name="3.18" href="#3.18">3.18</a></h3>
<p><!--para 1 -->
-<b> ??? x???</b><br>
+<b> [^ x ^]</b><br>
ceiling of x: the least integer greater than or equal to x
<p><!--para 2 -->
- EXAMPLE ???2.4??? is 3, ???-2.4??? is -2.
+ EXAMPLE [^2.4^] is 3, [^-2.4^] is -2.
<h3><a name="3.19" href="#3.19">3.19</a></h3>
<p><!--para 1 -->
-<b> ??? x???</b><br>
+<b> [_ x _]</b><br>
floor of x: the greatest integer less than or equal to x
<p><!--para 2 -->
- EXAMPLE ???2.4??? is 2, ???-2.4??? is -3.
+ EXAMPLE [_2.4_] is 2, [_-2.4_] is -3.
<!--page 19 -->
<h2><a name="4" href="#4">4. Conformance</a></h2>
b base or radix of exponent representation (an integer > 1)
e exponent (an integer between a minimum emin and a maximum emax )
p precision (the number of base-b digits in the significand)
- fk nonnegative integers less than b (the significand digits)</pre>
+ f<sub>k</sub> nonnegative integers less than b (the significand digits)</pre>
A floating-point number (x) is defined by the following model:
<pre>
p
- x = sb e (Sum) f k b-k ,
- k=1
- emin <= e <= emax</pre>
+ x = s b<sup>e</sup> (Sum) f<sub>k</sub> b<sup>-k</sup> , emin <= e <= emax
+ k=1</pre>
<p><!--para 3 -->
- In addition to normalized floating-point numbers ( f 1 > 0 if x != 0), floating types may be
+ In addition to normalized floating-point numbers ( f<sub>1</sub> > 0 if x != 0), floating types may be
able to contain other kinds of floating-point numbers, such as subnormal floating-point
- numbers (x != 0, e = emin , f 1 = 0) and unnormalized floating-point numbers (x != 0,
- e > emin , f 1 = 0), and values that are not floating-point numbers, such as infinities and
+ numbers (x != 0, e = emin , f<sub>1</sub> = 0) and unnormalized floating-point numbers (x != 0,
+ e > emin , f<sub>1</sub> = 0), and values that are not floating-point numbers, such as infinities and
NaNs. A NaN is an encoding signifying Not-a-Number. A quiet NaN propagates
through almost every arithmetic operation without raising a floating-point exception; a
signaling NaN generally raises a floating-point exception when occurring as an
All integer values in the <a href="#7.7"><float.h></a> header, except FLT_ROUNDS, shall be constant
expressions suitable for use in #if preprocessing directives; all floating values shall be
constant expressions. All except DECIMAL_DIG, FLT_EVAL_METHOD, FLT_RADIX,
- and FLT_ROUNDS have separate names for all three floating-point types. The floating-
- point model representation is provided for all values except FLT_EVAL_METHOD and
+ and FLT_ROUNDS have separate names for all three floating-point types. The floating-point
+ model representation is provided for all values except FLT_EVAL_METHOD and
FLT_ROUNDS.
<p><!--para 7 -->
The rounding mode for floating-point addition is characterized by the implementation-
those shown, with the same sign:
<ul>
<li> radix of exponent representation, b
- FLT_RADIX 2
+<pre> FLT_RADIX 2</pre>
<li> number of base-FLT_RADIX digits in the floating-point significand, p
- FLT_MANT_DIG
+<pre> FLT_MANT_DIG
DBL_MANT_DIG
- LDBL_MANT_DIG
+ LDBL_MANT_DIG</pre>
<li> number of decimal digits, n, such that any floating-point number in the widest
supported floating type with pmax radix b digits can be rounded to a floating-point
number with n decimal digits and back again without change to the value,
<pre>
- ??? pmax log10 b if b is a power of 10
- ???
- ??? ???1 + pmax log10 b??? otherwise</pre>
- DECIMAL_DIG 10
+ { pmax log10 b if b is a power of 10
+ {
+ { [^1 + pmax log10 b^] otherwise</pre>
+<pre> DECIMAL_DIG 10</pre>
<li> number of decimal digits, q, such that any floating-point number with q decimal digits
can be rounded into a floating-point number with p radix b digits and back again
without change to the q decimal digits,
<!--page 38 -->
<pre>
- ??? p log10 b if b is a power of 10
- ???
- ??? ???( p - 1) log10 b??? otherwise</pre>
- FLT_DIG 6
+ { p log10 b if b is a power of 10
+ {
+ { [_( p - 1) log10 b_] otherwise</pre>
+<pre> FLT_DIG 6
DBL_DIG 10
- LDBL_DIG 10
+ LDBL_DIG 10</pre>
<li> minimum negative integer such that FLT_RADIX raised to one less than that power is
a normalized floating-point number, emin
- FLT_MIN_EXP
+<pre> FLT_MIN_EXP
DBL_MIN_EXP
- LDBL_MIN_EXP
+ LDBL_MIN_EXP</pre>
<li> minimum negative integer such that 10 raised to that power is in the range of
- normalized floating-point numbers, ???log10 b emin -1 ???
-<pre>
- ??? ???</pre>
- FLT_MIN_10_EXP -37
+ normalized floating-point numbers, [^log10 b<sup>emin -1</sup>^]
+<pre> FLT_MIN_10_EXP -37
DBL_MIN_10_EXP -37
- LDBL_MIN_10_EXP -37
+ LDBL_MIN_10_EXP -37</pre>
<li> maximum integer such that FLT_RADIX raised to one less than that power is a
representable finite floating-point number, emax
- FLT_MAX_EXP
+<pre> FLT_MAX_EXP
DBL_MAX_EXP
- LDBL_MAX_EXP
+ LDBL_MAX_EXP</pre>
<li> maximum integer such that 10 raised to that power is in the range of representable
- finite floating-point numbers, ???log10 ((1 - b- p )b emax )???
- FLT_MAX_10_EXP +37
+ finite floating-point numbers, [_log10 ((1 - b<sup>-p</sup>)b<sup>emax</sup>)_]
+<pre> FLT_MAX_10_EXP +37
DBL_MAX_10_EXP +37
- LDBL_MAX_10_EXP +37
+ LDBL_MAX_10_EXP +37</pre>
</ul>
<p><!--para 10 -->
The values given in the following list shall be replaced by constant expressions with
implementation-defined values that are greater than or equal to those shown:
<ul>
-<li> maximum representable finite floating-point number, (1 - b- p )b emax
- FLT_MAX 1E+37
+<li> maximum representable finite floating-point number, (1 - b<sup>-p</sup>)b<sup>emax</sup>
+<pre> FLT_MAX 1E+37
DBL_MAX 1E+37
- LDBL_MAX 1E+37
+ LDBL_MAX 1E+37</pre>
</ul>
<p><!--para 11 -->
The values given in the following list shall be replaced by constant expressions with
implementation-defined (positive) values that are less than or equal to those shown:
<ul>
<li> the difference between 1 and the least value greater than 1 that is representable in the
- given floating point type, b1- p
+ given floating point type, b<sup>1-p</sup>
<!--page 39 -->
- FLT_EPSILON 1E-5
+<pre> FLT_EPSILON 1E-5
DBL_EPSILON 1E-9
- LDBL_EPSILON 1E-9
-<li> minimum normalized positive floating-point number, b emin -1
- FLT_MIN 1E-37
+ LDBL_EPSILON 1E-9</pre>
+<li> minimum normalized positive floating-point number, b<sup>emin -1</sup>
+<pre> FLT_MIN 1E-37
DBL_MIN 1E-37
- LDBL_MIN 1E-37
+ LDBL_MIN 1E-37</pre>
</ul>
Recommended practice
<p><!--para 12 -->
float:
<pre>
6
- x = s16e (Sum) f k 16-k ,
- k=1
- -31 <= e <= +32</pre>
+ x = s 16<sup>e</sup> (Sum) f<sub>k</sub> 16<sup>-k</sup> , -31 <= e <= +32
+ k=1</pre>
<pre>
FLT_RADIX 16
<a href="#7.7"><float.h></a> header for types float and double:
<pre>
24
- x f = s2e (Sum) f k 2-k ,
- k=1
- -125 <= e <= +128</pre>
+ xf = s 2<sup>e</sup> (Sum) f<sub>k</sub> 2<sup>-k</sup> , -125 <= e <= +128
+ k=1</pre>
<pre>
53
- x d = s2e (Sum) f k 2-k ,
- k=1
- -1021 <= e <= +1024</pre>
+ xd = s 2<sup>e</sup> (Sum) f<sub>k</sub> 2<sup>-k</sup> , -1021 <= e <= +1024
+ k=1</pre>
+
<pre>
FLT_RADIX 2
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