138 lines
4.2 KiB
Plaintext
138 lines
4.2 KiB
Plaintext
///////////////////////////////////////////////////////////////
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// Copyright 2013 John Maddock. Distributed under the Boost
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// Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_
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#ifndef BOOST_MULTIPRECISION_CPP_BIN_FLOAT_TRANSCENDENTAL_HPP
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#define BOOST_MULTIPRECISION_CPP_BIN_FLOAT_TRANSCENDENTAL_HPP
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namespace boost{ namespace multiprecision{ namespace backends{
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template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
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void eval_exp_taylor(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
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{
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static const int bits = cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE>::bit_count;
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//
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// Taylor series for small argument, note returns exp(x) - 1:
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//
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res = limb_type(0);
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cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> num(arg), denom, t;
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denom = limb_type(1);
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eval_add(res, num);
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for(unsigned k = 2; ; ++k)
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{
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eval_multiply(denom, k);
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eval_multiply(num, arg);
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eval_divide(t, num, denom);
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eval_add(res, t);
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if(eval_is_zero(t) || (res.exponent() - bits > t.exponent()))
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break;
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}
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}
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template <unsigned Digits, digit_base_type DigitBase, class Allocator, class Exponent, Exponent MinE, Exponent MaxE>
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void eval_exp(cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &res, const cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> &arg)
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{
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//
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// This is based on MPFR's method, let:
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//
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// n = floor(x / ln(2))
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//
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// Then:
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//
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// r = x - n ln(2) : 0 <= r < ln(2)
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//
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// We can reduce r further by dividing by 2^k, with k ~ sqrt(n),
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// so if:
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//
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// e0 = exp(r / 2^k) - 1
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//
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// With e0 evaluated by taylor series for small arguments, then:
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//
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// exp(x) = 2^n (1 + e0)^2^k
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//
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// Note that to preserve precision we actually square (1 + e0) k times, calculating
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// the result less one each time, i.e.
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//
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// (1 + e0)^2 - 1 = e0^2 + 2e0
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//
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// Then add the final 1 at the end, given that e0 is small, this effectively wipes
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// out the error in the last step.
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//
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using default_ops::eval_multiply;
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using default_ops::eval_subtract;
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using default_ops::eval_add;
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using default_ops::eval_convert_to;
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int type = eval_fpclassify(arg);
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bool isneg = eval_get_sign(arg) < 0;
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if(type == (int)FP_NAN)
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{
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res = arg;
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return;
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}
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else if(type == (int)FP_INFINITE)
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{
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res = arg;
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if(isneg)
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res = limb_type(0u);
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else
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res = arg;
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return;
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}
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else if(type == (int)FP_ZERO)
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{
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res = limb_type(1);
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return;
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}
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cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> t, n;
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if(isneg)
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{
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t = arg;
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t.negate();
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eval_exp(res, t);
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t.swap(res);
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res = limb_type(1);
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eval_divide(res, t);
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return;
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}
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eval_divide(n, arg, default_ops::get_constant_ln2<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> >());
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eval_floor(n, n);
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eval_multiply(t, n, default_ops::get_constant_ln2<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> >());
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eval_subtract(t, arg);
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t.negate();
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if(eval_get_sign(t) < 0)
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{
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// There are some very rare cases where arg/ln2 is an integer, and the subsequent multiply
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// rounds up, in that situation t ends up negative at this point which breaks our invariants below:
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t = limb_type(0);
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}
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BOOST_ASSERT(t.compare(default_ops::get_constant_ln2<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinE, MaxE> >()) < 0);
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Exponent k, nn;
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eval_convert_to(&nn, n);
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k = nn ? Exponent(1) << (msb(nn) / 2) : 0;
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eval_ldexp(t, t, -k);
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eval_exp_taylor(res, t);
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//
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// Square 1 + res k times:
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//
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for(int s = 0; s < k; ++s)
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{
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t.swap(res);
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eval_multiply(res, t, t);
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eval_ldexp(t, t, 1);
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eval_add(res, t);
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}
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eval_add(res, limb_type(1));
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eval_ldexp(res, res, nn);
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}
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}}} // namespaces
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#endif
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