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;
return;
case FP_ZERO:
result = ui_type(0);
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
static const boost::int32_t double_digits10_minus_a_few = std::numeric_limits<double>::digits10 - 3;
T s, c, t;
for(boost::int32_t digits = double_digits10_minus_a_few; digits <= std::numeric_limits<number<T, et_on> >::digits10; digits *= 2)
{
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);
}
if(b_neg)
result.negate();
}
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();
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(0);
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();
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));
unsigned current_digits = std::numeric_limits<guess_type>::digits - 5;
unsigned target_precision = boost::multiprecision::detail::digits2<number<T, et_on> >::value;
// Newton-Raphson iteration
while(current_digits < 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_digits *= 2;
/*
T lim;
eval_ldexp(lim, result, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value);
if(eval_get_sign(s) < 0)
s.negate();
if(eval_get_sign(lim) < 0)
lim.negate();
if(lim.compare(s) >= 0)
break;
*/
}
if(b_neg)
result.negate();
}
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();
}
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();
}
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::int32_t, T>::type si_type;
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
typedef typename T::exponent_type exp_type;
typedef typename boost::multiprecision::detail::canonical<exp_type, T>::type canonical_exp_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:
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
return;
case FP_ZERO:
result = ui_type(0);
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)
{
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
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;
}
// Get initial estimate using standard math function asin.
double dd;
eval_convert_to(&dd, xx);
result = fp_type(std::asin(dd));
// Newton-Raphson iteration
while(true)
{
T s, c;
eval_sin(s, result);
eval_cos(c, result);
eval_subtract(s, xx);
eval_divide(s, c);
eval_subtract(result, s);
T lim;
eval_ldexp(lim, result, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value);
if(eval_get_sign(s) < 0)
s.negate();
if(eval_get_sign(lim) < 0)
lim.negate();
if(lim.compare(s) >= 0)
break;
}
if(b_neg)
result.negate();
}
