Current File : //usr/include/boost/multiprecision/detail/functions/trig.hpp
// Copyright Christopher Kormanyos 2002 - 2011.
// Copyright 2011 John Maddock.
// Distributed under the Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt or copy at
// http://www.boost.org/LICENSE_1_0.txt)
// This work is based on an earlier work:
// "Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations",
// in ACM TOMS, {VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. http://doi.acm.org/10.1145/1916461.1916469
//
// This file has no include guards or namespaces - it's expanded inline inside default_ops.hpp
//
#ifdef BOOST_MSVC
#pragma warning(push)
#pragma warning(disable : 6326) // comparison of two constants
#endif
template <class T>
void hyp0F1(T& result, const T& b, const T& x)
{
typedef typename boost::multiprecision::detail::canonical<boost::int32_t, T>::type si_type;
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
// Compute the series representation of Hypergeometric0F1 taken from
// http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric0F1/06/01/01/
// There are no checks on input range or parameter boundaries.
T x_pow_n_div_n_fact(x);
T pochham_b(b);
T bp(b);
eval_divide(result, x_pow_n_div_n_fact, pochham_b);
eval_add(result, ui_type(1));
si_type n;
T tol;
tol = ui_type(1);
eval_ldexp(tol, tol, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value());
eval_multiply(tol, result);
if (eval_get_sign(tol) < 0)
tol.negate();
T term;
const int series_limit =
boost::multiprecision::detail::digits2<number<T, et_on> >::value() < 100
? 100
: boost::multiprecision::detail::digits2<number<T, et_on> >::value();
// Series expansion of hyperg_0f1(; b; x).
for (n = 2; n < series_limit; ++n)
{
eval_multiply(x_pow_n_div_n_fact, x);
eval_divide(x_pow_n_div_n_fact, n);
eval_increment(bp);
eval_multiply(pochham_b, bp);
eval_divide(term, x_pow_n_div_n_fact, pochham_b);
eval_add(result, term);
bool neg_term = eval_get_sign(term) < 0;
if (neg_term)
term.negate();
if (term.compare(tol) <= 0)
break;
}
if (n >= series_limit)
BOOST_THROW_EXCEPTION(std::runtime_error("H0F1 Failed to Converge"));
}
template <class T>
void eval_sin(T& result, const T& x)
{
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The sin function is only valid for floating point types.");
if (&result == &x)
{
T temp;
eval_sin(temp, x);
result = temp;
return;
}
typedef typename boost::multiprecision::detail::canonical<boost::int32_t, T>::type si_type;
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
typedef typename mpl::front<typename T::float_types>::type fp_type;
switch (eval_fpclassify(x))
{
case FP_INFINITE:
case FP_NAN:
if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
{
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
errno = EDOM;
}
else
BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
return;
case FP_ZERO:
result = x;
return;
default:;
}
// Local copy of the argument
T xx = x;
// Analyze and prepare the phase of the argument.
// Make a local, positive copy of the argument, xx.
// The argument xx will be reduced to 0 <= xx <= pi/2.
bool b_negate_sin = false;
if (eval_get_sign(x) < 0)
{
xx.negate();
b_negate_sin = !b_negate_sin;
}
T n_pi, t;
// Remove even multiples of pi.
if (xx.compare(get_constant_pi<T>()) > 0)
{
eval_divide(n_pi, xx, get_constant_pi<T>());
eval_trunc(n_pi, n_pi);
t = ui_type(2);
eval_fmod(t, n_pi, t);
const bool b_n_pi_is_even = eval_get_sign(t) == 0;
eval_multiply(n_pi, get_constant_pi<T>());
if (n_pi.compare(get_constant_one_over_epsilon<T>()) > 0)
{
result = ui_type(0);
return;
}
else
eval_subtract(xx, n_pi);
BOOST_MATH_INSTRUMENT_CODE(xx.str(0, std::ios_base::scientific));
BOOST_MATH_INSTRUMENT_CODE(n_pi.str(0, std::ios_base::scientific));
// Adjust signs if the multiple of pi is not even.
if (!b_n_pi_is_even)
{
b_negate_sin = !b_negate_sin;
}
}
// Reduce the argument to 0 <= xx <= pi/2.
eval_ldexp(t, get_constant_pi<T>(), -1);
if (xx.compare(t) > 0)
{
eval_subtract(xx, get_constant_pi<T>(), xx);
BOOST_MATH_INSTRUMENT_CODE(xx.str(0, std::ios_base::scientific));
}
eval_subtract(t, xx);
const bool b_zero = eval_get_sign(xx) == 0;
const bool b_pi_half = eval_get_sign(t) == 0;
// Check if the reduced argument is very close to 0 or pi/2.
const bool b_near_zero = xx.compare(fp_type(1e-1)) < 0;
const bool b_near_pi_half = t.compare(fp_type(1e-1)) < 0;
;
if (b_zero)
{
result = ui_type(0);
}
else if (b_pi_half)
{
result = ui_type(1);
}
else if (b_near_zero)
{
eval_multiply(t, xx, xx);
eval_divide(t, si_type(-4));
T t2;
t2 = fp_type(1.5);
hyp0F1(result, t2, t);
BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
eval_multiply(result, xx);
}
else if (b_near_pi_half)
{
eval_multiply(t, t);
eval_divide(t, si_type(-4));
T t2;
t2 = fp_type(0.5);
hyp0F1(result, t2, t);
BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
}
else
{
// Scale to a small argument for an efficient Taylor series,
// implemented as a hypergeometric function. Use a standard
// divide by three identity a certain number of times.
// Here we use division by 3^9 --> (19683 = 3^9).
static const si_type n_scale = 9;
static const si_type n_three_pow_scale = static_cast<si_type>(19683L);
eval_divide(xx, n_three_pow_scale);
// Now with small arguments, we are ready for a series expansion.
eval_multiply(t, xx, xx);
eval_divide(t, si_type(-4));
T t2;
t2 = fp_type(1.5);
hyp0F1(result, t2, t);
BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
eval_multiply(result, xx);
// Convert back using multiple angle identity.
for (boost::int32_t k = static_cast<boost::int32_t>(0); k < n_scale; k++)
{
// Rescale the cosine value using the multiple angle identity.
eval_multiply(t2, result, ui_type(3));
eval_multiply(t, result, result);
eval_multiply(t, result);
eval_multiply(t, ui_type(4));
eval_subtract(result, t2, t);
}
}
if (b_negate_sin)
result.negate();
}
template <class T>
void eval_cos(T& result, const T& x)
{
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The cos function is only valid for floating point types.");
if (&result == &x)
{
T temp;
eval_cos(temp, x);
result = temp;
return;
}
typedef typename boost::multiprecision::detail::canonical<boost::int32_t, T>::type si_type;
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
typedef typename mpl::front<typename T::float_types>::type fp_type;
switch (eval_fpclassify(x))
{
case FP_INFINITE:
case FP_NAN:
if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
{
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
errno = EDOM;
}
else
BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
return;
case FP_ZERO:
result = ui_type(1);
return;
default:;
}
// Local copy of the argument
T xx = x;
// Analyze and prepare the phase of the argument.
// Make a local, positive copy of the argument, xx.
// The argument xx will be reduced to 0 <= xx <= pi/2.
bool b_negate_cos = false;
if (eval_get_sign(x) < 0)
{
xx.negate();
}
T n_pi, t;
// Remove even multiples of pi.
if (xx.compare(get_constant_pi<T>()) > 0)
{
eval_divide(t, xx, get_constant_pi<T>());
eval_trunc(n_pi, t);
BOOST_MATH_INSTRUMENT_CODE(n_pi.str(0, std::ios_base::scientific));
eval_multiply(t, n_pi, get_constant_pi<T>());
BOOST_MATH_INSTRUMENT_CODE(t.str(0, std::ios_base::scientific));
//
// If t is so large that all digits cancel the result of this subtraction
// is completely meaningless, just assume the result is zero for now...
//
// TODO We should of course do much better, see:
// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS" K C Ng 1992
//
if (n_pi.compare(get_constant_one_over_epsilon<T>()) > 0)
{
result = ui_type(1);
return;
}
else
eval_subtract(xx, t);
BOOST_MATH_INSTRUMENT_CODE(xx.str(0, std::ios_base::scientific));
// Adjust signs if the multiple of pi is not even.
t = ui_type(2);
eval_fmod(t, n_pi, t);
const bool b_n_pi_is_even = eval_get_sign(t) == 0;
if (!b_n_pi_is_even)
{
b_negate_cos = !b_negate_cos;
}
}
// Reduce the argument to 0 <= xx <= pi/2.
eval_ldexp(t, get_constant_pi<T>(), -1);
int com = xx.compare(t);
if (com > 0)
{
eval_subtract(xx, get_constant_pi<T>(), xx);
b_negate_cos = !b_negate_cos;
BOOST_MATH_INSTRUMENT_CODE(xx.str(0, std::ios_base::scientific));
}
const bool b_zero = eval_get_sign(xx) == 0;
const bool b_pi_half = com == 0;
// Check if the reduced argument is very close to 0.
const bool b_near_zero = xx.compare(fp_type(1e-1)) < 0;
if (b_zero)
{
result = si_type(1);
}
else if (b_pi_half)
{
result = si_type(0);
}
else if (b_near_zero)
{
eval_multiply(t, xx, xx);
eval_divide(t, si_type(-4));
n_pi = fp_type(0.5f);
hyp0F1(result, n_pi, t);
BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
}
else
{
eval_subtract(t, xx);
eval_sin(result, t);
}
if (b_negate_cos)
result.negate();
}
template <class T>
void eval_tan(T& result, const T& x)
{
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The tan function is only valid for floating point types.");
if (&result == &x)
{
T temp;
eval_tan(temp, x);
result = temp;
return;
}
T t;
eval_sin(result, x);
eval_cos(t, x);
eval_divide(result, t);
}
template <class T>
void hyp2F1(T& result, const T& a, const T& b, const T& c, const T& x)
{
// Compute the series representation of hyperg_2f1 taken from
// Abramowitz and Stegun 15.1.1.
// There are no checks on input range or parameter boundaries.
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
T x_pow_n_div_n_fact(x);
T pochham_a(a);
T pochham_b(b);
T pochham_c(c);
T ap(a);
T bp(b);
T cp(c);
eval_multiply(result, pochham_a, pochham_b);
eval_divide(result, pochham_c);
eval_multiply(result, x_pow_n_div_n_fact);
eval_add(result, ui_type(1));
T lim;
eval_ldexp(lim, result, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value());
if (eval_get_sign(lim) < 0)
lim.negate();
ui_type n;
T term;
const unsigned series_limit =
boost::multiprecision::detail::digits2<number<T, et_on> >::value() < 100
? 100
: boost::multiprecision::detail::digits2<number<T, et_on> >::value();
// Series expansion of hyperg_2f1(a, b; c; x).
for (n = 2; n < series_limit; ++n)
{
eval_multiply(x_pow_n_div_n_fact, x);
eval_divide(x_pow_n_div_n_fact, n);
eval_increment(ap);
eval_multiply(pochham_a, ap);
eval_increment(bp);
eval_multiply(pochham_b, bp);
eval_increment(cp);
eval_multiply(pochham_c, cp);
eval_multiply(term, pochham_a, pochham_b);
eval_divide(term, pochham_c);
eval_multiply(term, x_pow_n_div_n_fact);
eval_add(result, term);
if (eval_get_sign(term) < 0)
term.negate();
if (lim.compare(term) >= 0)
break;
}
if (n > series_limit)
BOOST_THROW_EXCEPTION(std::runtime_error("H2F1 failed to converge."));
}
template <class T>
void eval_asin(T& result, const T& x)
{
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The asin function is only valid for floating point types.");
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
typedef typename mpl::front<typename T::float_types>::type fp_type;
if (&result == &x)
{
T t(x);
eval_asin(result, t);
return;
}
switch (eval_fpclassify(x))
{
case FP_NAN:
case FP_INFINITE:
if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
{
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
errno = EDOM;
}
else
BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
return;
case FP_ZERO:
result = x;
return;
default:;
}
const bool b_neg = eval_get_sign(x) < 0;
T xx(x);
if (b_neg)
xx.negate();
int c = xx.compare(ui_type(1));
if (c > 0)
{
if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
{
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
errno = EDOM;
}
else
BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
return;
}
else if (c == 0)
{
result = get_constant_pi<T>();
eval_ldexp(result, result, -1);
if (b_neg)
result.negate();
return;
}
if (xx.compare(fp_type(1e-4)) < 0)
{
// http://functions.wolfram.com/ElementaryFunctions/ArcSin/26/01/01/
eval_multiply(xx, xx);
T t1, t2;
t1 = fp_type(0.5f);
t2 = fp_type(1.5f);
hyp2F1(result, t1, t1, t2, xx);
eval_multiply(result, x);
return;
}
else if (xx.compare(fp_type(1 - 1e-4f)) > 0)
{
T dx1;
T t1, t2;
eval_subtract(dx1, ui_type(1), xx);
t1 = fp_type(0.5f);
t2 = fp_type(1.5f);
eval_ldexp(dx1, dx1, -1);
hyp2F1(result, t1, t1, t2, dx1);
eval_ldexp(dx1, dx1, 2);
eval_sqrt(t1, dx1);
eval_multiply(result, t1);
eval_ldexp(t1, get_constant_pi<T>(), -1);
result.negate();
eval_add(result, t1);
if (b_neg)
result.negate();
return;
}
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
typedef typename boost::multiprecision::detail::canonical<long double, T>::type guess_type;
#else
typedef fp_type guess_type;
#endif
// Get initial estimate using standard math function asin.
guess_type dd;
eval_convert_to(&dd, xx);
result = (guess_type)(std::asin(dd));
// Newton-Raphson iteration, we should double our precision with each iteration,
// in practice this seems to not quite work in all cases... so terminate when we
// have at least 2/3 of the digits correct on the assumption that the correction
// we've just added will finish the job...
boost::intmax_t current_precision = eval_ilogb(result);
boost::intmax_t target_precision = current_precision - 1 - (std::numeric_limits<number<T> >::digits * 2) / 3;
// Newton-Raphson iteration
while (current_precision > target_precision)
{
T sine, cosine;
eval_sin(sine, result);
eval_cos(cosine, result);
eval_subtract(sine, xx);
eval_divide(sine, cosine);
eval_subtract(result, sine);
current_precision = eval_ilogb(sine);
if (current_precision <= (std::numeric_limits<typename T::exponent_type>::min)() + 1)
break;
}
if (b_neg)
result.negate();
}
template <class T>
inline void eval_acos(T& result, const T& x)
{
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The acos function is only valid for floating point types.");
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
switch (eval_fpclassify(x))
{
case FP_NAN:
case FP_INFINITE:
if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
{
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
errno = EDOM;
}
else
BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
return;
case FP_ZERO:
result = get_constant_pi<T>();
eval_ldexp(result, result, -1); // divide by two.
return;
}
eval_abs(result, x);
int c = result.compare(ui_type(1));
if (c > 0)
{
if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
{
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
errno = EDOM;
}
else
BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
return;
}
else if (c == 0)
{
if (eval_get_sign(x) < 0)
result = get_constant_pi<T>();
else
result = ui_type(0);
return;
}
eval_asin(result, x);
T t;
eval_ldexp(t, get_constant_pi<T>(), -1);
eval_subtract(result, t);
result.negate();
}
template <class T>
void eval_atan(T& result, const T& x)
{
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The atan function is only valid for floating point types.");
typedef typename boost::multiprecision::detail::canonical<boost::int32_t, T>::type si_type;
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
typedef typename mpl::front<typename T::float_types>::type fp_type;
switch (eval_fpclassify(x))
{
case FP_NAN:
result = x;
errno = EDOM;
return;
case FP_ZERO:
result = x;
return;
case FP_INFINITE:
if (eval_get_sign(x) < 0)
{
eval_ldexp(result, get_constant_pi<T>(), -1);
result.negate();
}
else
eval_ldexp(result, get_constant_pi<T>(), -1);
return;
default:;
}
const bool b_neg = eval_get_sign(x) < 0;
T xx(x);
if (b_neg)
xx.negate();
if (xx.compare(fp_type(0.1)) < 0)
{
T t1, t2, t3;
t1 = ui_type(1);
t2 = fp_type(0.5f);
t3 = fp_type(1.5f);
eval_multiply(xx, xx);
xx.negate();
hyp2F1(result, t1, t2, t3, xx);
eval_multiply(result, x);
return;
}
if (xx.compare(fp_type(10)) > 0)
{
T t1, t2, t3;
t1 = fp_type(0.5f);
t2 = ui_type(1u);
t3 = fp_type(1.5f);
eval_multiply(xx, xx);
eval_divide(xx, si_type(-1), xx);
hyp2F1(result, t1, t2, t3, xx);
eval_divide(result, x);
if (!b_neg)
result.negate();
eval_ldexp(t1, get_constant_pi<T>(), -1);
eval_add(result, t1);
if (b_neg)
result.negate();
return;
}
// Get initial estimate using standard math function atan.
fp_type d;
eval_convert_to(&d, xx);
result = fp_type(std::atan(d));
// Newton-Raphson iteration, we should double our precision with each iteration,
// in practice this seems to not quite work in all cases... so terminate when we
// have at least 2/3 of the digits correct on the assumption that the correction
// we've just added will finish the job...
boost::intmax_t current_precision = eval_ilogb(result);
boost::intmax_t target_precision = current_precision - 1 - (std::numeric_limits<number<T> >::digits * 2) / 3;
T s, c, t;
while (current_precision > target_precision)
{
eval_sin(s, result);
eval_cos(c, result);
eval_multiply(t, xx, c);
eval_subtract(t, s);
eval_multiply(s, t, c);
eval_add(result, s);
current_precision = eval_ilogb(s);
if (current_precision <= (std::numeric_limits<typename T::exponent_type>::min)() + 1)
break;
}
if (b_neg)
result.negate();
}
template <class T>
void eval_atan2(T& result, const T& y, const T& x)
{
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The atan2 function is only valid for floating point types.");
if (&result == &y)
{
T temp(y);
eval_atan2(result, temp, x);
return;
}
else if (&result == &x)
{
T temp(x);
eval_atan2(result, y, temp);
return;
}
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
switch (eval_fpclassify(y))
{
case FP_NAN:
result = y;
errno = EDOM;
return;
case FP_ZERO:
{
if (eval_signbit(x))
{
result = get_constant_pi<T>();
if (eval_signbit(y))
result.negate();
}
else
{
result = y; // Note we allow atan2(0,0) to be +-zero, even though it's mathematically undefined
}
return;
}
case FP_INFINITE:
{
if (eval_fpclassify(x) == FP_INFINITE)
{
if (eval_signbit(x))
{
// 3Pi/4
eval_ldexp(result, get_constant_pi<T>(), -2);
eval_subtract(result, get_constant_pi<T>());
if (eval_get_sign(y) >= 0)
result.negate();
}
else
{
// Pi/4
eval_ldexp(result, get_constant_pi<T>(), -2);
if (eval_get_sign(y) < 0)
result.negate();
}
}
else
{
eval_ldexp(result, get_constant_pi<T>(), -1);
if (eval_get_sign(y) < 0)
result.negate();
}
return;
}
}
switch (eval_fpclassify(x))
{
case FP_NAN:
result = x;
errno = EDOM;
return;
case FP_ZERO:
{
eval_ldexp(result, get_constant_pi<T>(), -1);
if (eval_get_sign(y) < 0)
result.negate();
return;
}
case FP_INFINITE:
if (eval_get_sign(x) > 0)
result = ui_type(0);
else
result = get_constant_pi<T>();
if (eval_get_sign(y) < 0)
result.negate();
return;
}
T xx;
eval_divide(xx, y, x);
if (eval_get_sign(xx) < 0)
xx.negate();
eval_atan(result, xx);
// Determine quadrant (sign) based on signs of x, y
const bool y_neg = eval_get_sign(y) < 0;
const bool x_neg = eval_get_sign(x) < 0;
if (y_neg != x_neg)
result.negate();
if (x_neg)
{
if (y_neg)
eval_subtract(result, get_constant_pi<T>());
else
eval_add(result, get_constant_pi<T>());
}
}
template <class T, class A>
inline typename enable_if<is_arithmetic<A>, void>::type eval_atan2(T& result, const T& x, const A& a)
{
typedef typename boost::multiprecision::detail::canonical<A, T>::type canonical_type;
typedef typename mpl::if_<is_same<A, canonical_type>, T, canonical_type>::type cast_type;
cast_type c;
c = a;
eval_atan2(result, x, c);
}
template <class T, class A>
inline typename enable_if<is_arithmetic<A>, void>::type eval_atan2(T& result, const A& x, const T& a)
{
typedef typename boost::multiprecision::detail::canonical<A, T>::type canonical_type;
typedef typename mpl::if_<is_same<A, canonical_type>, T, canonical_type>::type cast_type;
cast_type c;
c = x;
eval_atan2(result, c, a);
}
#ifdef BOOST_MSVC
#pragma warning(pop)
#endif
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