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351 lines
8.3 KiB
Plaintext
351 lines
8.3 KiB
Plaintext
[section:powers Basic Functions]
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[section:sin_pi sin_pi]
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``
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#include <boost/math/special_functions/sin_pi.hpp>
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``
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namespace boost{ namespace math{
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template <class T>
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``__sf_result`` sin_pi(T x);
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template <class T, class ``__Policy``>
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``__sf_result`` sin_pi(T x, const ``__Policy``&);
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}} // namespaces
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Returns the sine of ['[pi]x].
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The return type of this function is computed using the __arg_promotion_rules:
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the return is `double` when /x/ is an integer type and T otherwise.
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[optional_policy]
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This function performs exact all-integer arithmetic argument reduction before computing the sine of ['[pi]x].
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[table_sin_pi]
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[endsect]
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[section:cos_pi cos_pi]
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``
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#include <boost/math/special_functions/cos_pi.hpp>
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``
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namespace boost{ namespace math{
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template <class T>
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``__sf_result`` cos_pi(T x);
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template <class T, class ``__Policy``>
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``__sf_result`` cos_pi(T x, const ``__Policy``&);
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}} // namespaces
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Returns the cosine of ['[pi]x].
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The return type of this function is computed using the __arg_promotion_rules:
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the return is `double` when /x/ is an integer type and T otherwise.
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[optional_policy]
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This function performs exact all-integer arithmetic argument reduction before computing the cosine of ['[pi]x].
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[table_cos_pi]
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[endsect]
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[section:log1p log1p]
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``
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#include <boost/math/special_functions/log1p.hpp>
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``
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namespace boost{ namespace math{
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template <class T>
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``__sf_result`` log1p(T x);
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template <class T, class ``__Policy``>
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``__sf_result`` log1p(T x, const ``__Policy``&);
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}} // namespaces
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Returns the natural logarithm of `x+1`.
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The return type of this function is computed using the __arg_promotion_rules:
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the return is `double` when /x/ is an integer type and T otherwise.
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[optional_policy]
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There are many situations where it is desirable to compute `log(x+1)`.
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However, for small `x` then `x+1` suffers from catastrophic cancellation errors
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so that `x+1 == 1` and `log(x+1) == 0`, when in fact for very small x, the
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best approximation to `log(x+1)` would be `x`. `log1p` calculates the best
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approximation to `log(1+x)` using a Taylor series expansion for accuracy
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(less than __te).
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Alternatively note that there are faster methods available,
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for example using the equivalence:
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log(1+x) == (log(1+x) * x) / ((1+x) - 1)
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However, experience has shown that these methods tend to fail quite spectacularly
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once the compiler's optimizations are turned on, consequently they are
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used only when known not to break with a particular compiler.
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In contrast, the series expansion method seems to be reasonably
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immune to optimizer-induced errors.
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Finally when BOOST_HAS_LOG1P is defined then the `float/double/long double`
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specializations of this template simply forward to the platform's
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native (POSIX) implementation of this function.
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The following graph illustrates the behaviour of log1p:
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[graph log1p]
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[h4 Accuracy]
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For built in floating point types `log1p`
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should have approximately 1 epsilon accuracy.
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[table_log1p]
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[h4 Testing]
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A mixture of spot test sanity checks, and random high precision test values
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calculated using NTL::RR at 1000-bit precision.
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[endsect]
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[section:expm1 expm1]
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``
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#include <boost/math/special_functions/expm1.hpp>
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``
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namespace boost{ namespace math{
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template <class T>
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``__sf_result`` expm1(T x);
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template <class T, class ``__Policy``>
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``__sf_result`` expm1(T x, const ``__Policy``&);
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}} // namespaces
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Returns e[super x] - 1.
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The return type of this function is computed using the __arg_promotion_rules:
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the return is `double` when /x/ is an integer type and T otherwise.
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[optional_policy]
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For small x, then __ex is very close to 1, as a result calculating __exm1 results
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in catastrophic cancellation errors when x is small. `expm1` calculates __exm1 using
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rational approximations (for up to 128-bit long doubles), otherwise via
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a series expansion when x is small (giving an accuracy of less than __te).
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Finally when BOOST_HAS_EXPM1 is defined then the `float/double/long double`
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specializations of this template simply forward to the platform's
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native (POSIX) implementation of this function.
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The following graph illustrates the behaviour of expm1:
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[graph expm1]
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[h4 Accuracy]
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For built in floating point types `expm1`
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should have approximately 1 epsilon accuracy.
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[table_expm1]
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[h4 Testing]
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A mixture of spot test sanity checks, and random high precision test values
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calculated using NTL::RR at 1000-bit precision.
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[endsect]
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[section:cbrt cbrt]
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``
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#include <boost/math/special_functions/cbrt.hpp>
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``
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namespace boost{ namespace math{
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template <class T>
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``__sf_result`` cbrt(T x);
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template <class T, class ``__Policy``>
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``__sf_result`` cbrt(T x, const ``__Policy``&);
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}} // namespaces
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Returns the cubed root of x: x[super 1/3].
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The return type of this function is computed using the __arg_promotion_rules:
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the return is `double` when /x/ is an integer type and T otherwise.
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[optional_policy]
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Implemented using Halley iteration.
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The following graph illustrates the behaviour of cbrt:
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[graph cbrt]
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[h4 Accuracy]
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For built in floating-point types `cbrt`
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should have approximately 2 epsilon accuracy.
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[table_cbrt]
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[h4 Testing]
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A mixture of spot test sanity checks, and random high precision test values
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calculated using NTL::RR at 1000-bit precision.
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[endsect]
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[section:sqrt1pm1 sqrt1pm1]
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``
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#include <boost/math/special_functions/sqrt1pm1.hpp>
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``
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namespace boost{ namespace math{
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template <class T>
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``__sf_result`` sqrt1pm1(T x);
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template <class T, class ``__Policy``>
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``__sf_result`` sqrt1pm1(T x, const ``__Policy``&);
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}} // namespaces
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Returns `sqrt(1+x) - 1`.
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The return type of this function is computed using the __arg_promotion_rules:
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the return is `double` when /x/ is an integer type and T otherwise.
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[optional_policy]
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This function is useful when you need the difference between sqrt(x) and 1, when
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x is itself close to 1.
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Implemented in terms of `log1p` and `expm1`.
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The following graph illustrates the behaviour of sqrt1pm1:
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[graph sqrt1pm1]
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[h4 Accuracy]
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For built in floating-point types `sqrt1pm1`
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should have approximately 3 epsilon accuracy.
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[table_sqrt1pm1]
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[h4 Testing]
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A selection of random high precision test values
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calculated using NTL::RR at 1000-bit precision.
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[endsect]
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[section:powm1 powm1]
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``
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#include <boost/math/special_functions/powm1.hpp>
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``
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namespace boost{ namespace math{
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template <class T1, class T2>
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``__sf_result`` powm1(T1 x, T2 y);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` powm1(T1 x, T2 y, const ``__Policy``&);
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}} // namespaces
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Returns x[super y ] - 1.
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The return type of this function is computed using the __arg_promotion_rules
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when T1 and T2 are dufferent types.
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[optional_policy]
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There are two domains where this is useful: when y is very small, or when
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x is close to 1.
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Implemented in terms of `expm1`.
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The following graph illustrates the behaviour of powm1:
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[graph powm1]
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[h4 Accuracy]
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Should have approximately 2-3 epsilon accuracy.
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[table_powm1]
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[h4 Testing]
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A selection of random high precision test values
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calculated using NTL::RR at 1000-bit precision.
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[endsect]
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[section:hypot hypot]
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template <class T1, class T2>
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``__sf_result`` hypot(T1 x, T2 y);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` hypot(T1 x, T2 y, const ``__Policy``&);
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__effects computes [equation hypot]
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in such a way as to avoid undue underflow and overflow.
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The return type of this function is computed using the __arg_promotion_rules
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when T1 and T2 are of different types.
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[optional_policy]
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When calculating [equation hypot] it's quite easy for the intermediate terms to either
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overflow or underflow, even though the result is in fact perfectly representable.
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[h4 Implementation]
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The function is even and symmetric in x and y, so first take assume
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['x,y > 0] and ['x > y] (we can permute the arguments if this is not the case).
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Then if ['x * [epsilon][space] >= y] we can simply return /x/.
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Otherwise the result is given by:
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[equation hypot2]
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[endsect]
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[include pow.qbk]
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[endsect][/section:powers Logs, Powers, Roots and Exponentials]
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[/
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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