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<h1>Rounding Policies</h1>
<p>In order to be as general as possible, the library uses a class to
compute all the necessary functions rounded upward or downward. This class
is the first parameter of <code>policies</code>, it is also the type named
<code>rounding</code> in the policy definition of
<code>interval</code>.</p>
<p>By default, it is <code>interval_lib::rounded_math<T></code>. The
class <code>interval_lib::rounded_math</code> is already specialized for
the standard floating types (<code>float</code> , <code>double</code> and
<code>long double</code>). So if the base type of your intervals is not one
of these, a good solution would probably be to provide a specialization of
this class. But if the default specialization of
<code>rounded_math<T></code> for <code>float</code>,
<code>double</code>, or <code>long double</code> is not what you seek, or
you do not want to specialize
<code>interval_lib::rounded_math<T></code> (say because you prefer to
work in your own namespace) you can also define your own rounding policy
and pass it directly to <code>interval_lib::policies</code>.</p>
<h2>Requirements</h2>
<p>Here comes what the class is supposed to provide. The domains are
written next to their respective functions (as you can see, the functions
do not have to worry about invalid values, but they have to handle infinite
arguments).</p>
<pre>
/* Rounding requirements */
struct rounding {
// defaut constructor, destructor
rounding();
~rounding();
// mathematical operations
T add_down(T, T); // [-∞;+∞][-∞;+∞]
T add_up (T, T); // [-∞;+∞][-∞;+∞]
T sub_down(T, T); // [-∞;+∞][-∞;+∞]
T sub_up (T, T); // [-∞;+∞][-∞;+∞]
T mul_down(T, T); // [-∞;+∞][-∞;+∞]
T mul_up (T, T); // [-∞;+∞][-∞;+∞]
T div_down(T, T); // [-∞;+∞]([-∞;+∞]-{0})
T div_up (T, T); // [-∞;+∞]([-∞;+∞]-{0})
T sqrt_down(T); // ]0;+∞]
T sqrt_up (T); // ]0;+∞]
T exp_down(T); // [-∞;+∞]
T exp_up (T); // [-∞;+∞]
T log_down(T); // ]0;+∞]
T log_up (T); // ]0;+∞]
T cos_down(T); // [0;2π]
T cos_up (T); // [0;2π]
T tan_down(T); // ]-π/2;π/2[
T tan_up (T); // ]-π/2;π/2[
T asin_down(T); // [-1;1]
T asin_up (T); // [-1;1]
T acos_down(T); // [-1;1]
T acos_up (T); // [-1;1]
T atan_down(T); // [-∞;+∞]
T atan_up (T); // [-∞;+∞]
T sinh_down(T); // [-∞;+∞]
T sinh_up (T); // [-∞;+∞]
T cosh_down(T); // [-∞;+∞]
T cosh_up (T); // [-∞;+∞]
T tanh_down(T); // [-∞;+∞]
T tanh_up (T); // [-∞;+∞]
T asinh_down(T); // [-∞;+∞]
T asinh_up (T); // [-∞;+∞]
T acosh_down(T); // [1;+∞]
T acosh_up (T); // [1;+∞]
T atanh_down(T); // [-1;1]
T atanh_up (T); // [-1;1]
T median(T, T); // [-∞;+∞][-∞;+∞]
T int_down(T); // [-∞;+∞]
T int_up (T); // [-∞;+∞]
// conversion functions
T conv_down(U);
T conv_up (U);
// unprotected rounding class
typedef ... unprotected_rounding;
};
</pre>
<p>The constructor and destructor of the rounding class have a very
important semantic requirement: they are responsible for setting and
resetting the rounding modes of the computation on T. For instance, if T is
a standard floating point type and floating point computation is performed
according to the Standard IEEE 754, the constructor can save the current
rounding state, each <code>_up</code> (resp. <code>_down</code>) function
will round up (resp. down), and the destructor will restore the saved
rounding state. Indeed this is the behavior of the default rounding
policy.</p>
<p>The meaning of all the mathematical functions up until
<code>atanh_up</code> is clear: each function returns number representable
in the type <code>T</code> which is a lower bound (for <code>_down</code>)
or upper bound (for <code>_up</code>) on the true mathematical result of
the corresponding function. The function <code>median</code> computes the
average of its two arguments rounded to its nearest representable number.
The functions <code>int_down</code> and <code>int_up</code> compute the
nearest integer smaller or bigger than their argument. Finally,
<code>conv_down</code> and <code>conv_up</code> are responsible of the
conversions of values of other types to the base number type: the first one
must round down the value and the second one must round it up.</p>
<p>The type <code>unprotected_rounding</code> allows to remove all
controls. For reasons why one might to do this, see the <a href=
"#Protection">protection</a> paragraph below.</p>
<h2>Overview of the provided classes</h2>
<p>A lot of classes are provided. The classes are organized by level. At
the bottom is the class <code>rounding_control</code>. At the next level
come <code>rounded_arith_exact</code>, <code>rounded_arith_std</code> and
<code>rounded_arith_opp</code>. Then there are
<code>rounded_transc_dummy</code>, <code>rounded_transc_exact</code>,
<code>rounded_transc_std</code> and <code>rounded_transc_opp</code>. And
finally are <code>save_state</code> and <code>save_state_nothing</code>.
Each of these classes provide a set of members that are required by the
classes of the next level. For example, a <code>rounded_transc_...</code>
class needs the members of a <code>rounded_arith_...</code> class.</p>
<p>When they exist in two versions <code>_std</code> and <code>_opp</code>,
the first one does switch the rounding mode each time, and the second one
tries to keep it oriented toward plus infinity. The main purpose of the
<code>_opp</code> version is to speed up the computations through the use
of the "opposite trick" (see the <a href="#perf">performance notes</a>).
This version requires the rounding mode to be upward before entering any
computation functions of the class. It guarantees that the rounding mode
will still be upward at the exit of the functions.</p>
<p>Please note that it is really a very bad idea to mix the
<code>_opp</code> version with the <code>_std</code> since they do not have
compatible properties.</p>
<p>There is a third version named <code>_exact</code> which computes the
functions without changing the rounding mode. It is an "exact" version
because it is intended for a base type that produces exact results.</p>
<p>The last version is the <code>_dummy</code> version. It does not do any
computations but still produces compatible results.</p>
<p>Please note that it is possible to use the "exact" version for an
inexact base type, e.g. <code>float</code> or <code>double</code>. In that
case, the inclusion property is no longer guaranteed, but this can be
useful to speed up the computation when the inclusion property is not
desired strictly. For instance, in computer graphics, a small error due to
floating-point roundoff is acceptable as long as an approximate version of
the inclusion property holds.</p>
<p>Here comes what each class defines. Later, when they will be described
more thoroughly, these members will not be repeated. Please come back here
in order to see them. Inheritance is also used to avoid repetitions.</p>
<pre>
template <class T>
struct rounding_control
{
typedef ... rounding_mode;
void set_rounding_mode(rounding_mode);
void get_rounding_mode(rounding_mode&);
void downward ();
void upward ();
void to_nearest();
T to_int(T);
T force_rounding(T);
};
template <class T, class Rounding>
struct rounded_arith_... : Rounding
{
void init();
T add_down(T, T);
T add_up (T, T);
T sub_down(T, T);
T sub_up (T, T);
T mul_down(T, T);
T mul_up (T, T);
T div_down(T, T);
T div_up (T, T);
T sqrt_down(T);
T sqrt_up (T);
T median(T, T);
T int_down(T);
T int_up (T);
};
template <class T, class Rounding>
struct rounded_transc_... : Rounding
{
T exp_down(T);
T exp_up (T);
T log_down(T);
T log_up (T);
T cos_down(T);
T cos_up (T);
T tan_down(T);
T tan_up (T);
T asin_down(T);
T asin_up (T);
T acos_down(T);
T acos_up (T);
T atan_down(T);
T atan_up (T);
T sinh_down(T);
T sinh_up (T);
T cosh_down(T);
T cosh_up (T);
T tanh_down(T);
T tanh_up (T);
T asinh_down(T);
T asinh_up (T);
T acosh_down(T);
T acosh_up (T);
T atanh_down(T);
T atanh_up (T);
};
template <class Rounding>
struct save_state_... : Rounding
{
save_state_...();
~save_state_...();
typedef ... unprotected_rounding;
};
</pre>
<h2>Synopsis.</h2>
<pre>
namespace boost {
namespace numeric {
namespace interval_lib {
<span style="color: #FF0000">/* basic rounding control */</span>
template <class T> struct rounding_control;
<span style="color: #FF0000">/* arithmetic functions rounding */</span>
template <class T, class Rounding = rounding_control<T> > struct rounded_arith_exact;
template <class T, class Rounding = rounding_control<T> > struct rounded_arith_std;
template <class T, class Rounding = rounding_control<T> > struct rounded_arith_opp;
<span style="color: #FF0000">/* transcendental functions rounding */</span>
template <class T, class Rounding> struct rounded_transc_dummy;
template <class T, class Rounding = rounded_arith_exact<T> > struct rounded_transc_exact;
template <class T, class Rounding = rounded_arith_std<T> > struct rounded_transc_std;
template <class T, class Rounding = rounded_arith_opp<T> > struct rounded_transc_opp;
<span style="color: #FF0000">/* rounding-state-saving classes */</span>
template <class Rounding> struct save_state;
template <class Rounding> struct save_state_nothing;
<span style="color: #FF0000">/* default policy for type T */</span>
template <class T> struct rounded_math;
template <> struct rounded_math<float>;
template <> struct rounded_math<double>;
<span style=
"color: #FF0000">/* some metaprogramming to convert a protected to unprotected rounding */</span>
template <class I> struct unprotect;
} // namespace interval_lib
} // namespace numeric
} // namespace boost
</pre>
<h2>Description of the provided classes</h2>
<p>We now describe each class in the order they appear in the definition of
a rounding policy (this outermost-to-innermost order is the reverse order
from the synopsis).</p>
<h3 id="Protection">Protection control</h3>
<p>Protection refers to the fact that the interval operations will be
surrounded by rounding mode controls. Unprotecting a class means to remove
all the rounding controls. Each rounding policy provides a type
<code>unprotected_rounding</code>. The required type
<code>unprotected_rounding</code> gives another rounding class that enables
to work when nested inside rounding. For example, the first three lines
below should all produce the same result (because the first operation is
the rounding constructor, and the last is its destructor, which take care
of setting the rounding modes); and the last line is allowed to have an
undefined behavior (since no rounding constructor or destructor is ever
called).</p>
<pre>
T c; { rounding rnd; c = rnd.add_down(a, b); }
T c; { rounding rnd1; { rounding rnd2; c = rnd2.add_down(a, b); } }
T c; { rounding rnd1; { rounding::unprotected_rounding rnd2; c = rnd2.add_down(a, b); } }
T d; { rounding::unprotected_rounding rnd; d = rnd.add_down(a, b); }
</pre>
<p>Naturally <code>rounding::unprotected_rounding</code> may simply be
<code>rounding</code> itself. But it can improve performance if it is a
simplified version with empty constructor and destructor. In order to avoid
undefined behaviors, in the library, an object of type
<code>rounding::unprotected_rounding</code> is guaranteed to be created
only when an object of type <code>rounding</code> is already alive. See the
<a href="#perf">performance notes</a> for some additional details.</p>
<p>The support library defines a metaprogramming class template
<code>unprotect</code> which takes an interval type <code>I</code> and
returns an interval type <code>unprotect<I>::type</code> where the
rounding policy has been unprotected. Some information about the types:
<code>interval<T, interval_lib::policies<Rounding, _>
>::traits_type::rounding</code> <b>is</b> the same type as
<code>Rounding</code>, and <code>unprotect<interval<T,
interval_lib::policies<Rounding, _> > >::type</code> <b>is</b>
the same type as <code>interval<T,
interval_lib::policies<Rounding::unprotected, _> ></code>.</p>
<h3>State saving</h3>
<p>First comes <code>save_state</code>. This class is responsible for
saving the current rounding mode and calling init in its constructor, and
for restoring the saved rounding mode in its destructor. This class also
defines the <code>unprotected_rounding</code> type.</p>
<p>If the rounding mode does not require any state-saving or
initialization, <code>save_state_nothing</code> can be used instead of
<code>save_state</code>.</p>
<h3>Transcendental functions</h3>
<p>The classes <code>rounded_transc_exact</code>,
<code>rounded_transc_std</code> and <code>rounded_transc_opp</code> expect
the std namespace to provide the functions exp log cos tan acos asin atan
cosh sinh tanh acosh asinh atanh. For the <code>_std</code> and
<code>_opp</code> versions, all these functions should respect the current
rounding mode fixed by a call to downward or upward.</p>
<p><strong>Please note:</strong> Unfortunately, the latter is rarely the
case. It is the reason why a class <code>rounded_transc_dummy</code> is
provided which does not depend on the functions from the std namespace.
There is no magic, however. The functions of
<code>rounded_transc_dummy</code> do not compute anything. They only return
valid values. For example, <code>cos_down</code> always returns -1. In this
way, we do verify the inclusion property for the default implementation,
even if this has strictly no value for the user. In order to have useful
values, another policy should be used explicitely, which will most likely
lead to a violation of the inclusion property. In this way, we ensure that
the violation is clearly pointed out to the user who then knows what he
stands against. This class could have been used as the default
transcendental rounding class, but it was decided it would be better for
the compilation to fail due to missing declarations rather than succeed
thanks to valid but unusable functions.</p>
<h3>Basic arithmetic functions</h3>
<p>The classes <code>rounded_arith_std</code> and
<code>rounded_arith_opp</code> expect the operators + - * / and the
function <code>std::sqrt</code> to respect the current rounding mode.</p>
<p>The class <code>rounded_arith_exact</code> requires
<code>std::floor</code> and <code>std::ceil</code> to be defined since it
can not rely on <code>to_int</code>.</p>
<h3>Rounding control</h3>
<p>The functions defined by each of the previous classes did not need any
explanation. For example, the behavior of <code>add_down</code> is to
compute the sum of two numbers rounded downward. For
<code>rounding_control</code>, the situation is a bit more complex.</p>
<p>The basic function is <code>force_rounding</code> which returns its
argument correctly rounded accordingly to the current rounding mode if it
was not already the case. This function is necessary to handle delayed
rounding. Indeed, depending on the way the computations are done, the
intermediate results may be internaly stored in a more precise format and
it can lead to a wrong rounding. So the function enforces the rounding.
<a href="#extreg">Here</a> is an example of what happens when the rounding
is not enforced.</p>
<p>The function <code>get_rounding_mode</code> returns the current rounding
mode, <code>set_rounding_mode</code> sets the rounding mode back to a
previous value returned by <code>get_rounding_mode</code>.
<code>downward</code>, <code>upward</code> and <code>to_nearest</code> sets
the rounding mode in one of the three directions. This rounding mode should
be global to all the functions that use the type <code>T</code>. For
example, after a call to <code>downward</code>,
<code>force_rounding(x+y)</code> is expected to return the sum rounded
toward -∞.</p>
<p>The function <code>to_int</code> computes the nearest integer
accordingly to the current rounding mode.</p>
<p>The non-specialized version of <code>rounding_control</code> does not do
anything. The functions for the rounding mode are empty, and
<code>to_int</code> and <code>force_rounding</code> are identity functions.
The <code>pi_</code> constant functions return suitable integers (for
example, <code>pi_up</code> returns <code>T(4)</code>).</p>
<p>The class template <code>rounding_control</code> is specialized for
<code>float</code>, <code>double</code> and <code>long double</code> in
order to best use the floating point unit of the computer.</p>
<h2>Template class <tt>rounded_math</tt></h2>
<p>The default policy (aka <code>rounded_math<T></code>) is simply
defined as:</p>
<pre>
template <class T> struct rounded_math<T> : save_state_nothing<rounded_arith_exact<T> > {};
</pre>
<p>and the specializations for <code>float</code>, <code>double</code> and
<code>long double</code> use <code>rounded_arith_opp</code>, as in:</p>
<pre>
template <> struct rounded_math<float> : save_state<rounded_arith_opp<float> > {};
template <> struct rounded_math<double> : save_state<rounded_arith_opp<double> > {};
template <> struct rounded_math<long double> : save_state<rounded_arith_opp<long double> > {};
</pre>
<h2 id="perf">Performance Issues</h2>
<p>This paragraph deals mostly with the performance of the library with
intervals using the floating-point unit (FPU) of the computer. Let's
consider the sum of [<i>a</i>,<i>b</i>] and [<i>c</i>,<i>d</i>] as an
example. The result is [<code>down</code>(<i>a</i>+<i>c</i>),
<code>up</code>(<i>b</i>+<i>d</i>)], where <code>down</code> and
<code>up</code> indicate the rounding mode needed.</p>
<h3>Rounding Mode Switch</h3>
<p>If the FPU is able to use a different rounding mode for each operation,
there is no problem. For example, it's the case for the Alpha processor:
each floating-point instruction can specify a different rounding mode.
However, the IEEE-754 Standard does not require such a behavior. So most of
the FPUs only provide some instructions to set the rounding mode for all
subsequent operations. And generally, these instructions need to flush the
pipeline of the FPU.</p>
<p>In this situation, the time needed to sum [<i>a</i>,<i>b</i>] and
[<i>c</i>,<i>d</i>] is far worse than the time needed to calculate
<i>a</i>+<i>b</i> and <i>c</i>+<i>d</i> since the two additions cannot be
parallelized. Consequently, the objective is to diminish the number of
rounding mode switches.</p>
<p>If this library is not used to provide exact computations, but only for
pair arithmetic, the solution is quite simple: do not use rounding. In that
case, doing the sum [<i>a</i>,<i>b</i>] and [<i>c</i>,<i>d</i>] will be as
fast as computing <i>a</i>+<i>b</i> and <i>c</i>+<i>d</i>. Everything is
perfect.</p>
<p>However, if exact computations are required, such a solution is totally
unthinkable. So, are we penniless? No, there is still a trick available.
Indeed, down(<i>a</i>+<i>c</i>) = -up(-<i>a</i>-<i>c</i>) if the unary
minus is an exact operation. It is now possible to calculate the whole sum
with the same rounding mode. Generally, the cost of the mode switching is
worse than the cost of the sign changes.</p>
<h4>Speeding up consecutive operations</h4>
<p>The interval addition is not the only operation; most of the interval
operations can be computed by setting the rounding direction of the FPU
only once. So the operations of the floating point rounding policy assume
that the direction is correctly set. This assumption is usually not true in
a program (the user and the standard library expect the rounding direction
to be to nearest), so these operations have to be enclosed in a shell that
sets the floating point environment. This protection is done by the
constructor and destructor of the rounding policy.</p>
<p>Les us now consider the case of two consecutive interval additions:
[<i>a</i>,<i>b</i>] + [<i>c</i>,<i>d</i>] + [<i>e</i>,<i>f</i>]. The
generated code should look like:</p>
<pre>
init_rounding_mode(); // rounding object construction during the first addition
t1 = -(-a - c);
t2 = b + d;
restore_rounding_mode(); // rounding object destruction
init_rounding_mode(); // rounding object construction during the second addition
x = -(-t1 - e);
y = t2 + f;
restore_rounding_mode(); // rounding object destruction
// the result is the interval [x,y]
</pre>
<p>Between the two operations, the rounding direction is restored, and then
initialized again. Ideally, compilers should be able to optimize this
useless code away. But unfortunately they are not, and this slows the code
down by an order of magnitude. In order to avoid this bottleneck, the user
can tell to the interval operations that they do not need to be protected
anymore. It will then be up to the user to protect the interval
computations. The compiler will then be able to generate such a code:</p>
<pre>
init_rounding_mode(); // done by the user
x = -(-a - c - e);
y = b + d + f;
restore_rounding_mode(); // done by the user
</pre>
<p>The user will have to create a rounding object. And as long as this
object is alive, unprotected versions of the interval operations can be
used. They are selected by using an interval type with a specific rounding
policy. If the initial interval type is <code>I</code>, then
<code>I::traits_type::rounding</code> is the type of the rounding object,
and <code>interval_lib::unprotect<I>::type</code> is the type of the
unprotected interval type.</p>
<p>Because the rounding mode of the FPU is changed during the life of the
rounding object, any arithmetic floating point operation that does not
involve the interval library can lead to unexpected results. And
reciprocally, using unprotected interval operation when no rounding object
is alive will produce intervals that are not guaranteed anymore to contain
the real result.</p>
<h4 id="perfexp">Example</h4>
<p>Here is an example of Horner's scheme to compute the value of a polynom.
The rounding mode switches are disabled for the whole computation.</p>
<pre>
// I is an interval class, the polynom is a simple array
template<class I>
I horner(const I& x, const I p[], int n) {
// save and initialize the rounding mode
typename I::traits_type::rounding rnd;
// define the unprotected version of the interval type
typedef typename boost::numeric::interval_lib::unprotect<I>::type R;
const R& a = x;
R y = p[n - 1];
for(int i = n - 2; i >= 0; i--) {
y = y * a + (const R&)(p[i]);
}
return y;
// restore the rounding mode with the destruction of rnd
}
</pre>
<p>Please note that a rounding object is specially created in order to
protect all the interval computations. Each interval of type I is converted
in an interval of type R before any operations. If this conversion is not
done, the result is still correct, but the interest of this whole
optimization has disappeared. Whenever possible, it is good to convert to
<code>const R&</code> instead of <code>R</code>: indeed, the function
could already be called inside an unprotection block so the types
<code>R</code> and <code>I</code> would be the same interval, no need for a
conversion.</p>
<h4>Uninteresting remark</h4>
<p>It was said at the beginning that the Alpha processors can use a
specific rounding mode for each operation. However, due to the instruction
format, the rounding toward plus infinity is not available. Only the
rounding toward minus infinity can be used. So the trick using the change
of sign becomes essential, but there is no need to save and restore the
rounding mode on both sides of an operation.</p>
<h3 id="extreg">Extended Registers</h3>
<p>There is another problem besides the cost of the rounding mode switch.
Some FPUs use extended registers (for example, float computations will be
done with double registers, or double computations with long double
registers). Consequently, many problems can arise.</p>
<p>The first one is due to to the extended precision of the mantissa. The
rounding is also done on this extended precision. And consequently, we
still have down(<i>a</i>+<i>b</i>) = -up(-<i>a</i>-<i>b</i>) in the
extended registers. But back to the standard precision, we now have
down(<i>a</i>+<i>b</i>) < -up(-<i>a</i>-<i>b</i>) instead of an
equality. A solution could be not to use this method. But there still are
other problems, with the comparisons between numbers for example.</p>
<p>Naturally, there is also a problem with the extended precision of the
exponent. To illustrate this problem, let <i>m</i> be the biggest number
before +<i>inf</i>. If we calculate 2*[<i>m</i>,<i>m</i>], the answer
should be [<i>m</i>,<i>inf</i>]. But due to the extended registers, the FPU
will first store [<i>2m</i>,<i>2m</i>] and then convert it to
[<i>inf</i>,<i>inf</i>] at the end of the calculus (when the rounding mode
is toward +<i>inf</i>). So the answer is no more accurate.</p>
<p>There is only one solution: to force the FPU to convert the extended
values back to standard precision after each operation. Some FPUs provide
an instruction able to do this conversion (for example the PowerPC
processors). But for the FPUs that do not provide it (the x86 processors),
the only solution is to write the values to memory and read them back. Such
an operation is obviously very expensive.</p>
<h2>Some Examples</h2>
<p>Here come several cases:</p>
<ul>
<li>if you need precise computations with the <code>float</code> or
<code>double</code> types, use the default
<code>rounded_math<T></code>;</li>
<li>for fast wide intervals without any rounding nor precision, use
<code>save_state_nothing<rounded_transc_exact<T>
></code>;</li>
<li>for an exact type (like int or rational with a little help for
infinite and NaN values) without support for transcendental functions,
the solution could be
<code>save_state_nothing<rounded_transc_dummy<T,
rounded_arith_exact<T> > ></code> or directly
<code>save_state_nothing<rounded_arith_exact<T>
></code>;</li>
<li>if it is a more complex case than the previous ones, the best thing
is probably to directly define your own policy.</li>
</ul>
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<p>Revised
<!--webbot bot="Timestamp" s-type="EDITED" s-format="%Y-%m-%d" startspan -->2006-12-24<!--webbot bot="Timestamp" endspan i-checksum="12172" --></p>
<p><i>Copyright © 2002 Guillaume Melquiond, Sylvain Pion, Hervé
Brönnimann, Polytechnic University<br>
Copyright © 2004-2005 Guillaume Melquiond, ENS Lyon</i></p>
<p><i>Distributed under the Boost Software License, Version 1.0. (See
accompanying file <a href="../../../../LICENSE_1_0.txt">LICENSE_1_0.txt</a>
or copy at <a href=
"http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p>
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