logarithm.h

/*  This file is part of the Vc library. {{{
Copyright © 2009-2015 Matthias Kretz <kretz@kde.org>
 
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
    * Redistributions of source code must retain the above copyright
      notice, this list of conditions and the following disclaimer.
    * Redistributions in binary form must reproduce the above copyright
      notice, this list of conditions and the following disclaimer in the
      documentation and/or other materials provided with the distribution.
    * Neither the names of contributing organizations nor the
      names of its contributors may be used to endorse or promote products
      derived from this software without specific prior written permission.
 
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 
}}}*/
 
/* The log implementations are based on code from Julien Pommier which carries the following
   copyright information:
 */
/*
   Inspired by Intel Approximate Math library, and based on the
   corresponding algorithms of the cephes math library
*/
/* Copyright (C) 2007  Julien Pommier
 
  This software is provided 'as-is', without any express or implied
  warranty.  In no event will the authors be held liable for any damages
  arising from the use of this software.
 
  Permission is granted to anyone to use this software for any purpose,
  including commercial applications, and to alter it and redistribute it
  freely, subject to the following restrictions:
 
  1. The origin of this software must not be misrepresented; you must not
     claim that you wrote the original software. If you use this software
     in a product, an acknowledgment in the product documentation would be
     appreciated but is not required.
  2. Altered source versions must be plainly marked as such, and must not be
     misrepresented as being the original software.
  3. This notice may not be removed or altered from any source distribution.
 
  (this is the zlib license)
*/
 
#ifdef Vc_COMMON_MATH_H_INTERNAL
 
enum LogarithmBase {
    BaseE, Base10, Base2
};
 
namespace Detail
{
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
                                        AVX::Const<T>, SSE::Const<T>>::type;
 
template<LogarithmBase Base>
struct LogImpl
{
    template<typename T, typename Abi> static Vc_ALWAYS_INLINE void log_series(Vector<T, Abi> &Vc_RESTRICT x, typename Vector<T, Abi>::AsArg exponent) {
        typedef Vector<T, Abi> V;
        typedef Detail::Const<T, Abi> C;
        // Taylor series around x = 2^exponent
        //   f(x) = ln(x)   → exponent * ln(2) → C::ln2_small + C::ln2_large
        //  f'(x) =    x⁻¹  →  x               → 1
        // f''(x) = -  x⁻²  → -x² / 2          → C::_1_2()
        //        =  2!x⁻³  →  x³ / 3          → C::P(8)
        //        = -3!x⁻⁴  → -x⁴ / 4          → C::P(7)
        //        =  4!x⁻⁵  →  x⁵ / 5          → C::P(6)
        // ...
        // The high order coefficients are adjusted to reduce the error that occurs from ommission
        // of higher order terms.
        // P(0) is the smallest term and |x| < 1 ⇒ |xⁿ| > |xⁿ⁺¹|
        // The order of additions must go from smallest to largest terms
        const V x2 = x * x; // 0 → 4
#ifdef Vc_LOG_ILP
        V y2 = (C::P(6) * /*4 →  8*/ x2 + /* 8 → 11*/ C::P(7) * /*1 → 5*/ x) + /*11 → 14*/ C::P(8);
        V y0 = (C::P(0) * /*5 →  9*/ x2 + /* 9 → 12*/ C::P(1) * /*2 → 6*/ x) + /*12 → 15*/ C::P(2);
        V y1 = (C::P(3) * /*6 → 10*/ x2 + /*10 → 13*/ C::P(4) * /*3 → 7*/ x) + /*13 → 16*/ C::P(5);
        const V x3 = x2 * x;  // 7 → 11
        const V x6 = x3 * x3; // 11 → 15
        const V x9 = x6 * x3; // 15 → 19
        V y = (y0 * /*19 → 23*/ x9 + /*23 → 26*/ y1 * /*16 → 20*/ x6) + /*26 → 29*/ y2 * /*14 → 18*/ x3;
#elif defined Vc_LOG_ILP2
        /*
         *                            name start done
         *  movaps %xmm0, %xmm1     ; x     0     1
         *  movaps %xmm0, %xmm2     ; x     0     1
         *  mulps %xmm1, %xmm1      ; x2    1     5 *xmm1
         *  movaps <P8>, %xmm15     ; y8    1     2
         *  mulps %xmm1, %xmm2      ; x3    5     9 *xmm2
         *  movaps %xmm1, %xmm3     ; x2    5     6
         *  movaps %xmm1, %xmm4     ; x2    5     6
         *  mulps %xmm3, %xmm3      ; x4    6    10 *xmm3
         *  movaps %xmm2, %xmm5     ; x3    9    10
         *  movaps %xmm2, %xmm6     ; x3    9    10
         *  mulps %xmm2, %xmm4      ; x5    9    13 *xmm4
         *  movaps %xmm3, %xmm7     ; x4   10    11
         *  movaps %xmm3, %xmm8     ; x4   10    11
         *  movaps %xmm3, %xmm9     ; x4   10    11
         *  mulps %xmm5, %xmm5      ; x6   10    14 *xmm5
         *  mulps %xmm3, %xmm6      ; x7   11    15 *xmm6
         *  mulps %xmm7, %xmm7      ; x8   12    16 *xmm7
         *  movaps %xmm4, %xmm10    ; x5   13    14
         *  mulps %xmm4, %xmm8      ; x9   13    17 *xmm8
         *  mulps %xmm5, %xmm10     ; x11  14    18 *xmm10
         *  mulps %xmm5, %xmm9      ; x10  15    19 *xmm9
         *  mulps <P0>, %xmm10      ; y0   18    22
         *  mulps <P1>, %xmm9       ; y1   19    23
         *  mulps <P2>, %xmm8       ; y2   20    24
         *  mulps <P3>, %xmm7       ; y3   21    25
         *  addps %xmm10, %xmm9     ; y    23    26
         *  addps %xmm9, %xmm8      ; y    26    29
         *  addps %xmm8, %xmm7      ; y    29    32
         */
        const V x3 = x2 * x;  // 4  → 8
        const V x4 = x2 * x2; // 5  → 9
        const V x5 = x2 * x3; // 8  → 12
        const V x6 = x3 * x3; // 9  → 13
        const V x7 = x4 * x3; // 
        const V x8 = x4 * x4;
        const V x9 = x5 * x4;
        const V x10 = x5 * x5;
        const V x11 = x5 * x6; // 13 → 17
        V y = C::P(0) * x11 + C::P(1) * x10 + C::P(2) * x9 + C::P(3) * x8 + C::P(4) * x7
            + C::P(5) * x6  + C::P(6) * x5  + C::P(7) * x4 + C::P(8) * x3;
#else
        V y = C::P(0);
        Vc::Common::unrolled_loop<int, 1, 9>([&](int i) { y = y * x + C::P(i); });
        y *= x * x2;
#endif
        switch (Base) {
        case BaseE:
            // ln(2) is split in two parts to increase precision (i.e. ln2_small + ln2_large = ln(2))
            y += exponent * C::ln2_small();
            y -= x2 * C::_1_2(); // [0, 0.25[
            x += y;
            x += exponent * C::ln2_large();
            break;
        case Base10:
            y += exponent * C::ln2_small();
            y -= x2 * C::_1_2(); // [0, 0.25[
            x += y;
            x += exponent * C::ln2_large();
            x *= C::log10_e();
            break;
        case Base2:
            {
                const V x_ = x;
                x *= C::log2_e();
                y *= C::log2_e();
                y -= x_ * x * C::_1_2(); // [0, 0.25[
                x += y;
                x += exponent;
                break;
            }
        }
    }
 
template <typename Abi>
static Vc_ALWAYS_INLINE void log_series(Vector<double, Abi> &Vc_RESTRICT x,
                                        typename Vector<double, Abi>::AsArg exponent)
{
    typedef Vector<double, Abi> V;
    typedef Detail::Const<double, Abi> C;
        const V x2 = x * x;
        V y = C::P(0);
        V y2 = C::Q(0) + x;
        Vc::Common::unrolled_loop<int, 1, 5>([&](int i) {
            y = y * x + C::P(i);
            y2 = y2 * x + C::Q(i);
        });
        y2 = x / y2;
        y = y * x + C::P(5);
        y = x2 * y * y2;
        // TODO: refactor the following with the float implementation:
        switch (Base) {
        case BaseE:
            // ln(2) is split in two parts to increase precision (i.e. ln2_small + ln2_large = ln(2))
            y += exponent * C::ln2_small();
            y -= x2 * C::_1_2(); // [0, 0.25[
            x += y;
            x += exponent * C::ln2_large();
            break;
        case Base10:
            y += exponent * C::ln2_small();
            y -= x2 * C::_1_2(); // [0, 0.25[
            x += y;
            x += exponent * C::ln2_large();
            x *= C::log10_e();
            break;
        case Base2:
            {
                const V x_ = x;
                x *= C::log2_e();
                y *= C::log2_e();
                y -= x_ * x * C::_1_2(); // [0, 0.25[
                x += y;
                x += exponent;
                break;
            }
        }
    }
 
template <typename T, typename Abi, typename V = Vector<T, Abi>>
static inline Vector<T, Abi> calc(V _x)
{
        typedef typename V::Mask M;
    typedef Detail::Const<T, Abi> C;
 
        V x(_x);
 
        const M invalidMask = x < V::Zero();
        const M infinityMask = x == V::Zero();
        const M denormal = x <= C::min();
 
        x(denormal) *= V(Vc::Detail::doubleConstant<1, 0, 54>()); // 2²⁵
        V exponent = Detail::exponent(x.data());                    // = ⎣log₂(x)⎦
        exponent(denormal) -= 54;
 
        x.setZero(C::exponentMask()); // keep only the fractional part ⇒ x ∈ [1, 2[
        x = Detail::operator|(x,
                              C::_1_2());  // and set the exponent to 2⁻¹   ⇒ x ∈ [½, 1[
 
        // split calculation in two cases:
        // A: x ∈ [½, √½[
        // B: x ∈ [√½, 1[
        // √½ defines the point where Δe(x) := log₂(x) - ⎣log₂(x)⎦ = ½, i.e.
        // log₂(√½) - ⎣log₂(√½)⎦ = ½ * -1 - ⎣½ * -1⎦ = -½ + 1 = ½
 
        const M smallX = x < C::_1_sqrt2();
        x(smallX) += x; // => x ∈ [√½,     1[ ∪ [1.5, 1 + √½[
        x -= V::One();  // => x ∈ [√½ - 1, 0[ ∪ [0.5, √½[
        exponent(!smallX) += V::One();
 
        log_series(x, exponent); // A: (ˣ⁄₂ᵉ - 1, e)  B: (ˣ⁄₂ᵉ⁺¹ - 1, e + 1)
 
        x.setQnan(invalidMask);        // x < 0 → NaN
        x(infinityMask) = C::neginf(); // x = 0 → -∞
 
        return x;
    }
};
}  // namespace Detail
 
template <typename T, typename Abi>
Vc_INTRINSIC Vc_CONST Vector<T, detail::not_fixed_size_abi<Abi>> log(
    const Vector<T, Abi> &x)
{
    return Detail::LogImpl<BaseE>::calc<T, Abi>(x);
}
template <typename T, typename Abi>
Vc_INTRINSIC Vc_CONST Vector<T, detail::not_fixed_size_abi<Abi>> log10(
    const Vector<T, Abi> &x)
{
    return Detail::LogImpl<Base10>::calc<T, Abi>(x);
}
template <typename T, typename Abi>
Vc_INTRINSIC Vc_CONST Vector<T, detail::not_fixed_size_abi<Abi>> log2(
    const Vector<T, Abi> &x)
{
    return Detail::LogImpl<Base2>::calc<T, Abi>(x);
}
 
#endif // Vc_COMMON_MATH_H_INTERNAL