void eval_log(T& result, const T& arg)
{
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The log function is only valid for floating point types.");
//
// We use a variation of http://dlmf.nist.gov/4.45#i
// using frexp to reduce the argument to x * 2^n,
// then let y = x - 1 and compute:
// log(x) = log(2) * n + log1p(1 + y)
//
typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
typedef typename boost::multiprecision::detail::canonical<unsigned, 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;
exp_type e;
T t;
eval_frexp(t, arg, &e);
bool alternate = false;
if(t.compare(fp_type(2) / fp_type(3)) <= 0)
{
alternate = true;
eval_ldexp(t, t, 1);
--e;
}
eval_multiply(result, get_constant_ln2<T>(), canonical_exp_type(e));
INSTRUMENT_BACKEND(result);
eval_subtract(t, ui_type(1)); /* -0.3 <= t <= 0.3 */
if(!alternate)
t.negate(); /* 0 <= t <= 0.33333 */
T pow = t;
T lim;
T t2;
if(alternate)
eval_add(result, t);
else
eval_subtract(result, t);
eval_multiply(lim, result, std::numeric_limits<number<T, et_on> >::epsilon().backend());
if(eval_get_sign(lim) < 0)
lim.negate();
INSTRUMENT_BACKEND(lim);
ui_type k = 1;
do
{
++k;
eval_multiply(pow, t);
eval_divide(t2, pow, k);
INSTRUMENT_BACKEND(t2);
if(alternate && ((k & 1) != 0))
eval_add(result, t2);
else
eval_subtract(result, t2);
INSTRUMENT_BACKEND(result);
}while(lim.compare(t2) < 0);
}
void calc_log2(T& num, unsigned digits)
{
typedef typename geofeatures_boost::multiprecision::detail::canonical<geofeatures_boost::uint32_t, T>::type ui_type;
typedef typename mpl::front<typename T::signed_types>::type si_type;
//
// String value with 1100 digits:
//
static const char* string_val = "0."
"6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875"
"4200148102057068573368552023575813055703267075163507596193072757082837143519030703862389167347112335"
"0115364497955239120475172681574932065155524734139525882950453007095326366642654104239157814952043740"
"4303855008019441706416715186447128399681717845469570262716310645461502572074024816377733896385506952"
"6066834113727387372292895649354702576265209885969320196505855476470330679365443254763274495125040606"
"9438147104689946506220167720424524529612687946546193165174681392672504103802546259656869144192871608"
"2938031727143677826548775664850856740776484514644399404614226031930967354025744460703080960850474866"
"3852313818167675143866747664789088143714198549423151997354880375165861275352916610007105355824987941"
"4729509293113897155998205654392871700072180857610252368892132449713893203784393530887748259701715591"
"0708823683627589842589185353024363421436706118923678919237231467232172053401649256872747782344535347"
"6481149418642386776774406069562657379600867076257199184734022651462837904883062033061144630073719489";
//
// Check if we can just construct from string:
//
if(digits < 3640) // 3640 binary digits ~ 1100 decimal digits
{
num = string_val;
return;
}
//
// We calculate log2 from using the formula:
//
// ln(2) = 3/4 SUM[n>=0] ((-1)^n * N!^2 / (2^n(2n+1)!))
//
// Numerator and denominator are calculated separately and then
// divided at the end, we also precalculate the terms up to n = 5
// since these fit in a 32-bit integer anyway.
//
// See Gourdon, X., and Sebah, P. The logarithmic constant: log 2, Jan. 2004.
// Also http://www.mpfr.org/algorithms.pdf.
//
num = static_cast<ui_type>(1180509120uL);
T denom, next_term, temp;
denom = static_cast<ui_type>(1277337600uL);
next_term = static_cast<ui_type>(120uL);
si_type sign = -1;
ui_type limit = digits / 3 + 1;
for(ui_type n = 6; n < limit; ++n)
{
temp = static_cast<ui_type>(2);
eval_multiply(temp, ui_type(2 * n));
eval_multiply(temp, ui_type(2 * n + 1));
eval_multiply(num, temp);
eval_multiply(denom, temp);
sign = -sign;
eval_multiply(next_term, n);
eval_multiply(temp, next_term, next_term);
if(sign < 0)
temp.negate();
eval_add(num, temp);
}
eval_multiply(denom, ui_type(4));
eval_multiply(num, ui_type(3));
INSTRUMENT_BACKEND(denom);
INSTRUMENT_BACKEND(num);
eval_divide(num, denom);
INSTRUMENT_BACKEND(num);
}
