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+// Copyright 2011 John Maddock. Distributed under the Boost
+// 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 file has no include guards or namespaces - it's expanded inline inside default_ops.hpp
+//
+
+template <class T>
+void calc_log2(T& num, unsigned digits)
+{
+ typedef typename boost::multiprecision::detail::canonical<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);
+}
+
+template <class T>
+void calc_e(T& result, unsigned digits)
+{
+ typedef typename mpl::front<typename T::unsigned_types>::type ui_type;
+ //
+ // 1100 digits in string form:
+ //
+ const char* string_val = "2."
+ "7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274"
+ "2746639193200305992181741359662904357290033429526059563073813232862794349076323382988075319525101901"
+ "1573834187930702154089149934884167509244761460668082264800168477411853742345442437107539077744992069"
+ "5517027618386062613313845830007520449338265602976067371132007093287091274437470472306969772093101416"
+ "9283681902551510865746377211125238978442505695369677078544996996794686445490598793163688923009879312"
+ "7736178215424999229576351482208269895193668033182528869398496465105820939239829488793320362509443117"
+ "3012381970684161403970198376793206832823764648042953118023287825098194558153017567173613320698112509"
+ "9618188159304169035159888851934580727386673858942287922849989208680582574927961048419844436346324496"
+ "8487560233624827041978623209002160990235304369941849146314093431738143640546253152096183690888707016"
+ "7683964243781405927145635490613031072085103837505101157477041718986106873969655212671546889570350354"
+ "0212340784981933432106817012100562788023519303322474501585390473041995777709350366041699732972508869";
+ //
+ // Check if we can just construct from string:
+ //
+ if(digits < 3640) // 3640 binary digits ~ 1100 decimal digits
+ {
+ result = string_val;
+ return;
+ }
+
+ T lim;
+ lim = ui_type(1);
+ eval_ldexp(lim, lim, digits);
+
+ //
+ // Standard evaluation from the definition of e: http://functions.wolfram.com/Constants/E/02/
+ //
+ result = ui_type(2);
+ T denom;
+ denom = ui_type(1);
+ ui_type i = 2;
+ do{
+ eval_multiply(denom, i);
+ eval_multiply(result, i);
+ eval_add(result, ui_type(1));
+ ++i;
+ }while(denom.compare(lim) <= 0);
+ eval_divide(result, denom);
+}
+
+template <class T>
+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);
+}
+
+template <class T, const T& (*F)(void)>
+struct constant_initializer
+{
+ static void do_nothing()
+ {
+ init.do_nothing();
+ }
+private:
+ struct initializer
+ {
+ initializer()
+ {
+ F();
+ }
+ void do_nothing()const{}
+ };
+ static const initializer init;
+};
+
+template <class T, const T& (*F)(void)>
+typename constant_initializer<T, F>::initializer const constant_initializer<T, F>::init;
+
+template <class T>
+const T& get_constant_ln2()
+{
+ static T result;
+ static bool b = false;
+ if(!b)
+ {
+ calc_log2(result, boost::multiprecision::detail::digits2<number<T, et_on> >::value);
+ b = true;
+ }
+
+ constant_initializer<T, &get_constant_ln2<T> >::do_nothing();
+
+ return result;
+}
+
+template <class T>
+const T& get_constant_e()
+{
+ static T result;
+ static bool b = false;
+ if(!b)
+ {
+ calc_e(result, boost::multiprecision::detail::digits2<number<T, et_on> >::value);
+ b = true;
+ }
+
+ constant_initializer<T, &get_constant_e<T> >::do_nothing();
+
+ return result;
+}
+
+template <class T>
+const T& get_constant_pi()
+{
+ static T result;
+ static bool b = false;
+ if(!b)
+ {
+ calc_pi(result, boost::multiprecision::detail::digits2<number<T, et_on> >::value);
+ b = true;
+ }
+
+ constant_initializer<T, &get_constant_pi<T> >::do_nothing();
+
+ return result;
+}
+
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