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 calc_pi(T& result, unsigned digits)
{
typedef typename mpl::front<typename T::unsigned_types>::type ui_type;
typedef typename mpl::front<typename T::float_types>::type real_type;
//
// 1100 digits in string form:
//
const char* string_val = "3."
"1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679"
"8214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196"
"4428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273"
"7245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094"
"3305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912"
"9833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132"
"0005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235"
"4201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859"
"5024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303"
"5982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989"
"3809525720106548586327886593615338182796823030195203530185296899577362259941389124972177528347913152";
//
// Check if we can just construct from string:
//
if(digits < 3640) // 3640 binary digits ~ 1100 decimal digits
{
result = string_val;
return;
}
T a;
a = ui_type(1);
T b;
T A(a);
T B;
B = real_type(0.5f);
T D;
D = real_type(0.25f);
T lim;
lim = ui_type(1);
eval_ldexp(lim, lim, -(int)digits);
//
// This algorithm is from:
// Schonhage, A., Grotefeld, A. F. W., and Vetter, E. Fast Algorithms: A Multitape Turing
// Machine Implementation. BI Wissenschaftverlag, 1994.
// Also described in MPFR's algorithm guide: http://www.mpfr.org/algorithms.pdf.
//
// Let:
// a[0] = A[0] = 1
// B[0] = 1/2
// D[0] = 1/4
// Then:
// S[k+1] = (A[k]+B[k]) / 4
// b[k] = sqrt(B[k])
// a[k+1] = a[k]^2
// B[k+1] = 2(A[k+1]-S[k+1])
// D[k+1] = D[k] - 2^k(A[k+1]-B[k+1])
// Stop when |A[k]-B[k]| <= 2^(k-p)
// and PI = B[k]/D[k]
unsigned k = 1;
do
{
eval_add(result, A, B);
eval_ldexp(result, result, -2);
eval_sqrt(b, B);
eval_add(a, b);
eval_ldexp(a, a, -1);
eval_multiply(A, a, a);
eval_subtract(B, A, result);
eval_ldexp(B, B, 1);
eval_subtract(result, A, B);
bool neg = eval_get_sign(result) < 0;
if(neg)
result.negate();
if(result.compare(lim) <= 0)
break;
if(neg)
result.negate();
eval_ldexp(result, result, k - 1);
eval_subtract(D, result);
++k;
eval_ldexp(lim, lim, 1);
}
while(true);
eval_divide(result, B, D);
}
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();
}
