NVIDIA Volta performance tweaks

[alexxy/gromacs.git] / src / gromacs / simd / simd_math.h

/*
 * This file is part of the GROMACS molecular simulation package.
 *
 * Copyright (c) 2012,2013,2014,2015,2017, by the GROMACS development team, led by
 * Mark Abraham, David van der Spoel, Berk Hess, and Erik Lindahl,
 * and including many others, as listed in the AUTHORS file in the
 * top-level source directory and at http://www.gromacs.org.
 *
 * GROMACS is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public License
 * as published by the Free Software Foundation; either version 2.1
 * of the License, or (at your option) any later version.
 *
 * GROMACS is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public
 * License along with GROMACS; if not, see
 * http://www.gnu.org/licenses, or write to the Free Software Foundation,
 * Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA.
 *
 * If you want to redistribute modifications to GROMACS, please
 * consider that scientific software is very special. Version
 * control is crucial - bugs must be traceable. We will be happy to
 * consider code for inclusion in the official distribution, but
 * derived work must not be called official GROMACS. Details are found
 * in the README & COPYING files - if they are missing, get the
 * official version at http://www.gromacs.org.
 *
 * To help us fund GROMACS development, we humbly ask that you cite
 * the research papers on the package. Check out http://www.gromacs.org.
 */
#ifndef GMX_SIMD_SIMD_MATH_H
#define GMX_SIMD_SIMD_MATH_H

/*! \libinternal \file
 *
 * \brief Math functions for SIMD datatypes.
 *
 * \attention This file is generic for all SIMD architectures, so you cannot
 * assume that any of the optional SIMD features (as defined in simd.h) are
 * present. In particular, this means you cannot assume support for integers,
 * logical operations (neither on floating-point nor integer values), shifts,
 * and the architecture might only have SIMD for either float or double.
 * Second, to keep this file clean and general, any additions to this file
 * must work for all possible SIMD architectures in both single and double
 * precision (if they support it), and you cannot make any assumptions about
 * SIMD width.
 *
 * \author Erik Lindahl <erik.lindahl@scilifelab.se>
 *
 * \inlibraryapi
 * \ingroup module_simd
 */

#include "config.h"

#include <cmath>

#include "gromacs/math/utilities.h"
#include "gromacs/simd/simd.h"
#include "gromacs/utility/basedefinitions.h"
#include "gromacs/utility/real.h"

namespace gmx
{

#if GMX_SIMD

/*! \cond libapi */
/*! \addtogroup module_simd */
/*! \{ */

/*! \name Implementation accuracy settings
 *  \{
 */

/*! \} */

#if GMX_SIMD_HAVE_FLOAT

/*! \name Single precision SIMD math functions
 *
 *  \note In most cases you should use the real-precision functions instead.
 *  \{
 */

/****************************************
 * SINGLE PRECISION SIMD MATH FUNCTIONS *
 ****************************************/

#if !GMX_SIMD_HAVE_NATIVE_COPYSIGN_FLOAT
/*! \brief Composes floating point value with the magnitude of x and the sign of y.
 *
 * \param x Values to set sign for
 * \param y Values used to set sign
 * \return  Magnitude of x, sign of y
 */
static inline SimdFloat gmx_simdcall
copysign(SimdFloat x, SimdFloat y)
{
#if GMX_SIMD_HAVE_LOGICAL
    return abs(x) | ( SimdFloat(GMX_FLOAT_NEGZERO) & y );
#else
    return blend(abs(x), -abs(x), y < setZero());
#endif
}
#endif

#if !GMX_SIMD_HAVE_NATIVE_RSQRT_ITER_FLOAT
/*! \brief Perform one Newton-Raphson iteration to improve 1/sqrt(x) for SIMD float.
 *
 * This is a low-level routine that should only be used by SIMD math routine
 * that evaluates the inverse square root.
 *
 *  \param lu Approximation of 1/sqrt(x), typically obtained from lookup.
 *  \param x  The reference (starting) value x for which we want 1/sqrt(x).
 *  \return   An improved approximation with roughly twice as many bits of accuracy.
 */
static inline SimdFloat gmx_simdcall
rsqrtIter(SimdFloat lu, SimdFloat x)
{
    SimdFloat tmp1 = x*lu;
    SimdFloat tmp2 = SimdFloat(-0.5f)*lu;
    tmp1 = fma(tmp1, lu, SimdFloat(-3.0f));
    return tmp1*tmp2;
}
#endif

/*! \brief Calculate 1/sqrt(x) for SIMD float.
 *
 *  \param x Argument that must be >0. This routine does not check arguments.
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid.
 */
static inline SimdFloat gmx_simdcall
invsqrt(SimdFloat x)
{
    SimdFloat lu = rsqrt(x);
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
    return lu;
}

/*! \brief Calculate 1/sqrt(x) for two SIMD floats.
 *
 * \param x0  First set of arguments, x0 must be positive - no argument checking.
 * \param x1  Second set of arguments, x1 must be positive - no argument checking.
 * \param[out] out0  Result 1/sqrt(x0)
 * \param[out] out1  Result 1/sqrt(x1)
 *
 *  In particular for double precision we can sometimes calculate square root
 *  pairs slightly faster by using single precision until the very last step.
 */
static inline void gmx_simdcall
invsqrtPair(SimdFloat x0,    SimdFloat x1,
            SimdFloat *out0, SimdFloat *out1)
{
    *out0 = invsqrt(x0);
    *out1 = invsqrt(x1);
}

#if !GMX_SIMD_HAVE_NATIVE_RCP_ITER_FLOAT
/*! \brief Perform one Newton-Raphson iteration to improve 1/x for SIMD float.
 *
 * This is a low-level routine that should only be used by SIMD math routine
 * that evaluates the reciprocal.
 *
 *  \param lu Approximation of 1/x, typically obtained from lookup.
 *  \param x  The reference (starting) value x for which we want 1/x.
 *  \return   An improved approximation with roughly twice as many bits of accuracy.
 */
static inline SimdFloat gmx_simdcall
rcpIter(SimdFloat lu, SimdFloat x)
{
    return lu*fnma(lu, x, SimdFloat(2.0f));
}
#endif

/*! \brief Calculate 1/x for SIMD float.
 *
 *  \param x Argument that must be nonzero. This routine does not check arguments.
 *  \return 1/x. Result is undefined if your argument was invalid.
 */
static inline SimdFloat gmx_simdcall
inv(SimdFloat x)
{
    SimdFloat lu = rcp(x);
#if (GMX_SIMD_RCP_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
    return lu;
}

/*! \brief Division for SIMD floats
 *
 * \param nom    Nominator
 * \param denom  Denominator
 *
 * \return nom/denom
 *
 * \note This function does not use any masking to avoid problems with
 *       zero values in the denominator.
 */
static inline SimdFloat gmx_simdcall
operator/(SimdFloat nom, SimdFloat denom)
{
    return nom*inv(denom);
}

/*! \brief Calculate 1/sqrt(x) for masked entries of SIMD float.
 *
 *  This routine only evaluates 1/sqrt(x) for elements for which mask is true.
 *  Illegal values in the masked-out elements will not lead to
 *  floating-point exceptions.
 *
 *  \param x Argument that must be >0 for masked-in entries
 *  \param m Mask
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid or
 *          entry was not masked, and 0.0 for masked-out entries.
 */
static inline SimdFloat
maskzInvsqrt(SimdFloat x, SimdFBool m)
{
    SimdFloat lu = maskzRsqrt(x, m);
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
    return lu;
}

/*! \brief Calculate 1/x for SIMD float, masked version.
 *
 *  \param x Argument that must be nonzero for non-masked entries.
 *  \param m Mask
 *  \return 1/x for elements where m is true, or 0.0 for masked-out entries.
 */
static inline SimdFloat gmx_simdcall
maskzInv(SimdFloat x, SimdFBool m)
{
    SimdFloat lu = maskzRcp(x, m);
#if (GMX_SIMD_RCP_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
    return lu;
}

/*! \brief Calculate sqrt(x) correctly for SIMD floats, including argument 0.0.
 *
 *  \param x Argument that must be >=0.
 *  \return sqrt(x). If x=0, the result will correctly be set to 0.
 *          The result is undefined if the input value is negative.
 */
static inline SimdFloat gmx_simdcall
sqrt(SimdFloat x)
{
    SimdFloat  res = maskzInvsqrt(x, setZero() < x);
    return res*x;
}

#if !GMX_SIMD_HAVE_NATIVE_LOG_FLOAT
/*! \brief SIMD float log(x). This is the natural logarithm.
 *
 * \param x Argument, should be >0.
 * \result The natural logarithm of x. Undefined if argument is invalid.
 */
static inline SimdFloat gmx_simdcall
log(SimdFloat x)
{
    const SimdFloat  one(1.0f);
    const SimdFloat  two(2.0f);
    const SimdFloat  invsqrt2(1.0f/std::sqrt(2.0f));
    const SimdFloat  corr(0.693147180559945286226764f);
    const SimdFloat  CL9(0.2371599674224853515625f);
    const SimdFloat  CL7(0.285279005765914916992188f);
    const SimdFloat  CL5(0.400005519390106201171875f);
    const SimdFloat  CL3(0.666666567325592041015625f);
    const SimdFloat  CL1(2.0f);
    SimdFloat        fExp, x2, p;
    SimdFBool        m;
    SimdFInt32       iExp;

    x     = frexp(x, &iExp);
    fExp  = cvtI2R(iExp);

    m     = x < invsqrt2;
    // Adjust to non-IEEE format for x<1/sqrt(2): exponent -= 1, mantissa *= 2.0
    fExp  = fExp - selectByMask(one, m);
    x     = x * blend(one, two, m);

    x     = (x-one) * inv( x+one );
    x2    = x * x;

    p     = fma(CL9, x2, CL7);
    p     = fma(p, x2, CL5);
    p     = fma(p, x2, CL3);
    p     = fma(p, x2, CL1);
    p     = fma(p, x, corr*fExp);

    return p;
}
#endif

#if !GMX_SIMD_HAVE_NATIVE_EXP2_FLOAT
/*! \brief SIMD float 2^x.
 *
 * \param x Argument.
 * \result 2^x. Undefined if input argument caused overflow.
 */
static inline SimdFloat gmx_simdcall
exp2(SimdFloat x)
{
    // Lower bound: Disallow numbers that would lead to an IEEE fp exponent reaching +-127.
    const SimdFloat  arglimit(126.0f);
    const SimdFloat  CC6(0.0001534581200287996416911311f);
    const SimdFloat  CC5(0.001339993121934088894618990f);
    const SimdFloat  CC4(0.009618488957115180159497841f);
    const SimdFloat  CC3(0.05550328776964726865751735f);
    const SimdFloat  CC2(0.2402264689063408646490722f);
    const SimdFloat  CC1(0.6931472057372680777553816f);
    const SimdFloat  one(1.0f);

    SimdFloat        intpart;
    SimdFloat        fexppart;
    SimdFloat        p;
    SimdFBool        m;

    fexppart  = ldexp(one, cvtR2I(x));
    intpart   = round(x);
    m         = abs(x) <= arglimit;
    fexppart  = selectByMask(fexppart, m);
    x         = x - intpart;

    p         = fma(CC6, x, CC5);
    p         = fma(p, x, CC4);
    p         = fma(p, x, CC3);
    p         = fma(p, x, CC2);
    p         = fma(p, x, CC1);
    p         = fma(p, x, one);
    x         = p * fexppart;
    return x;
}
#endif

#if !GMX_SIMD_HAVE_NATIVE_EXP_FLOAT
/*! \brief SIMD float exp(x).
 *
 * In addition to scaling the argument for 2^x this routine correctly does
 * extended precision arithmetics to improve accuracy.
 *
 * \param x Argument.
 * \result exp(x). Undefined if input argument caused overflow,
 * which can happen if abs(x) \> 7e13.
 */
static inline SimdFloat gmx_simdcall
exp(SimdFloat x)
{
    const SimdFloat  argscale(1.44269504088896341f);
    // Lower bound: Disallow numbers that would lead to an IEEE fp exponent reaching +-127.
    const SimdFloat  arglimit(126.0f);
    const SimdFloat  invargscale0(-0.693145751953125f);
    const SimdFloat  invargscale1(-1.428606765330187045e-06f);
    const SimdFloat  CC4(0.00136324646882712841033936f);
    const SimdFloat  CC3(0.00836596917361021041870117f);
    const SimdFloat  CC2(0.0416710823774337768554688f);
    const SimdFloat  CC1(0.166665524244308471679688f);
    const SimdFloat  CC0(0.499999850988388061523438f);
    const SimdFloat  one(1.0f);
    SimdFloat        fexppart;
    SimdFloat        intpart;
    SimdFloat        y, p;
    SimdFBool        m;

    y         = x * argscale;
    fexppart  = ldexp(one, cvtR2I(y));
    intpart   = round(y);
    m         = (abs(y) <= arglimit);
    fexppart  = selectByMask(fexppart, m);

    // Extended precision arithmetics
    x         = fma(invargscale0, intpart, x);
    x         = fma(invargscale1, intpart, x);

    p         = fma(CC4, x, CC3);
    p         = fma(p, x, CC2);
    p         = fma(p, x, CC1);
    p         = fma(p, x, CC0);
    p         = fma(x*x, p, x);
    x         = fma(p, fexppart, fexppart);
    return x;
}
#endif

/*! \brief SIMD float erf(x).
 *
 * \param x The value to calculate erf(x) for.
 * \result erf(x)
 *
 * This routine achieves very close to full precision, but we do not care about
 * the last bit or the subnormal result range.
 */
static inline SimdFloat gmx_simdcall
erf(SimdFloat x)
{
    // Coefficients for minimax approximation of erf(x)=x*P(x^2) in range [-1,1]
    const SimdFloat  CA6(7.853861353153693e-5f);
    const SimdFloat  CA5(-8.010193625184903e-4f);
    const SimdFloat  CA4(5.188327685732524e-3f);
    const SimdFloat  CA3(-2.685381193529856e-2f);
    const SimdFloat  CA2(1.128358514861418e-1f);
    const SimdFloat  CA1(-3.761262582423300e-1f);
    const SimdFloat  CA0(1.128379165726710f);
    // Coefficients for minimax approximation of erfc(x)=Exp(-x^2)*P((1/(x-1))^2) in range [0.67,2]
    const SimdFloat  CB9(-0.0018629930017603923f);
    const SimdFloat  CB8(0.003909821287598495f);
    const SimdFloat  CB7(-0.0052094582210355615f);
    const SimdFloat  CB6(0.005685614362160572f);
    const SimdFloat  CB5(-0.0025367682853477272f);
    const SimdFloat  CB4(-0.010199799682318782f);
    const SimdFloat  CB3(0.04369575504816542f);
    const SimdFloat  CB2(-0.11884063474674492f);
    const SimdFloat  CB1(0.2732120154030589f);
    const SimdFloat  CB0(0.42758357702025784f);
    // Coefficients for minimax approximation of erfc(x)=Exp(-x^2)*(1/x)*P((1/x)^2) in range [2,9.19]
    const SimdFloat  CC10(-0.0445555913112064f);
    const SimdFloat  CC9(0.21376355144663348f);
    const SimdFloat  CC8(-0.3473187200259257f);
    const SimdFloat  CC7(0.016690861551248114f);
    const SimdFloat  CC6(0.7560973182491192f);
    const SimdFloat  CC5(-1.2137903600145787f);
    const SimdFloat  CC4(0.8411872321232948f);
    const SimdFloat  CC3(-0.08670413896296343f);
    const SimdFloat  CC2(-0.27124782687240334f);
    const SimdFloat  CC1(-0.0007502488047806069f);
    const SimdFloat  CC0(0.5642114853803148f);
    const SimdFloat  one(1.0f);
    const SimdFloat  two(2.0f);

    SimdFloat        x2, x4, y;
    SimdFloat        t, t2, w, w2;
    SimdFloat        pA0, pA1, pB0, pB1, pC0, pC1;
    SimdFloat        expmx2;
    SimdFloat        res_erf, res_erfc, res;
    SimdFBool        m, maskErf;

    // Calculate erf()
    x2   = x * x;
    x4   = x2 * x2;

    pA0  = fma(CA6, x4, CA4);
    pA1  = fma(CA5, x4, CA3);
    pA0  = fma(pA0, x4, CA2);
    pA1  = fma(pA1, x4, CA1);
    pA0  = pA0*x4;
    pA0  = fma(pA1, x2, pA0);
    // Constant term must come last for precision reasons
    pA0  = pA0+CA0;

    res_erf = x*pA0;

    // Calculate erfc
    y       = abs(x);
    maskErf = SimdFloat(0.75f) <= y;
    t       = maskzInv(y, maskErf);
    w       = t-one;
    t2      = t*t;
    w2      = w*w;

    // No need for a floating-point sieve here (as in erfc), since erf()
    // will never return values that are extremely small for large args.
    expmx2  = exp( -y*y );

    pB1  = fma(CB9, w2, CB7);
    pB0  = fma(CB8, w2, CB6);
    pB1  = fma(pB1, w2, CB5);
    pB0  = fma(pB0, w2, CB4);
    pB1  = fma(pB1, w2, CB3);
    pB0  = fma(pB0, w2, CB2);
    pB1  = fma(pB1, w2, CB1);
    pB0  = fma(pB0, w2, CB0);
    pB0  = fma(pB1, w, pB0);

    pC0  = fma(CC10, t2, CC8);
    pC1  = fma(CC9, t2, CC7);
    pC0  = fma(pC0, t2, CC6);
    pC1  = fma(pC1, t2, CC5);
    pC0  = fma(pC0, t2, CC4);
    pC1  = fma(pC1, t2, CC3);
    pC0  = fma(pC0, t2, CC2);
    pC1  = fma(pC1, t2, CC1);

    pC0  = fma(pC0, t2, CC0);
    pC0  = fma(pC1, t, pC0);
    pC0  = pC0*t;

    // Select pB0 or pC0 for erfc()
    m        = two < y;
    res_erfc = blend(pB0, pC0, m);
    res_erfc = res_erfc * expmx2;

    // erfc(x<0) = 2-erfc(|x|)
    m        = x < setZero();
    res_erfc = blend(res_erfc, two-res_erfc, m);

    // Select erf() or erfc()
    res  = blend(res_erf, one-res_erfc, maskErf);

    return res;
}

/*! \brief SIMD float erfc(x).
 *
 * \param x The value to calculate erfc(x) for.
 * \result erfc(x)
 *
 * This routine achieves full precision (bar the last bit) over most of the
 * input range, but for large arguments where the result is getting close
 * to the minimum representable numbers we accept slightly larger errors
 * (think results that are in the ballpark of 10^-30 for single precision)
 * since that is not relevant for MD.
 */
static inline SimdFloat gmx_simdcall
erfc(SimdFloat x)
{
    // Coefficients for minimax approximation of erf(x)=x*P(x^2) in range [-1,1]
    const SimdFloat  CA6(7.853861353153693e-5f);
    const SimdFloat  CA5(-8.010193625184903e-4f);
    const SimdFloat  CA4(5.188327685732524e-3f);
    const SimdFloat  CA3(-2.685381193529856e-2f);
    const SimdFloat  CA2(1.128358514861418e-1f);
    const SimdFloat  CA1(-3.761262582423300e-1f);
    const SimdFloat  CA0(1.128379165726710f);
    // Coefficients for minimax approximation of erfc(x)=Exp(-x^2)*P((1/(x-1))^2) in range [0.67,2]
    const SimdFloat  CB9(-0.0018629930017603923f);
    const SimdFloat  CB8(0.003909821287598495f);
    const SimdFloat  CB7(-0.0052094582210355615f);
    const SimdFloat  CB6(0.005685614362160572f);
    const SimdFloat  CB5(-0.0025367682853477272f);
    const SimdFloat  CB4(-0.010199799682318782f);
    const SimdFloat  CB3(0.04369575504816542f);
    const SimdFloat  CB2(-0.11884063474674492f);
    const SimdFloat  CB1(0.2732120154030589f);
    const SimdFloat  CB0(0.42758357702025784f);
    // Coefficients for minimax approximation of erfc(x)=Exp(-x^2)*(1/x)*P((1/x)^2) in range [2,9.19]
    const SimdFloat  CC10(-0.0445555913112064f);
    const SimdFloat  CC9(0.21376355144663348f);
    const SimdFloat  CC8(-0.3473187200259257f);
    const SimdFloat  CC7(0.016690861551248114f);
    const SimdFloat  CC6(0.7560973182491192f);
    const SimdFloat  CC5(-1.2137903600145787f);
    const SimdFloat  CC4(0.8411872321232948f);
    const SimdFloat  CC3(-0.08670413896296343f);
    const SimdFloat  CC2(-0.27124782687240334f);
    const SimdFloat  CC1(-0.0007502488047806069f);
    const SimdFloat  CC0(0.5642114853803148f);
    // Coefficients for expansion of exp(x) in [0,0.1]
    // CD0 and CD1 are both 1.0, so no need to declare them separately
    const SimdFloat  CD2(0.5000066608081202f);
    const SimdFloat  CD3(0.1664795422874624f);
    const SimdFloat  CD4(0.04379839977652482f);
    const SimdFloat  one(1.0f);
    const SimdFloat  two(2.0f);

    /* We need to use a small trick here, since we cannot assume all SIMD
     * architectures support integers, and the flag we want (0xfffff000) would
     * evaluate to NaN (i.e., it cannot be expressed as a floating-point num).
     * Instead, we represent the flags 0xf0f0f000 and 0x0f0f0000 as valid
     * fp numbers, and perform a logical or. Since the expression is constant,
     * we can at least hope it is evaluated at compile-time.
     */
#if GMX_SIMD_HAVE_LOGICAL
    const SimdFloat         sieve(SimdFloat(-5.965323564e+29f) | SimdFloat(7.05044434e-30f));
#else
    const int               isieve   = 0xFFFFF000;
    GMX_ALIGNED(float, GMX_SIMD_FLOAT_WIDTH)  mem[GMX_SIMD_FLOAT_WIDTH];

    union {
        float f; int i;
    } conv;
    int                     i;
#endif

    SimdFloat        x2, x4, y;
    SimdFloat        q, z, t, t2, w, w2;
    SimdFloat        pA0, pA1, pB0, pB1, pC0, pC1;
    SimdFloat        expmx2, corr;
    SimdFloat        res_erf, res_erfc, res;
    SimdFBool        m, msk_erf;

    // Calculate erf()
    x2     = x * x;
    x4     = x2 * x2;

    pA0  = fma(CA6, x4, CA4);
    pA1  = fma(CA5, x4, CA3);
    pA0  = fma(pA0, x4, CA2);
    pA1  = fma(pA1, x4, CA1);
    pA1  = pA1 * x2;
    pA0  = fma(pA0, x4, pA1);
    // Constant term must come last for precision reasons
    pA0  = pA0 + CA0;

    res_erf = x * pA0;

    // Calculate erfc
    y       = abs(x);
    msk_erf = SimdFloat(0.75f) <= y;
    t       = maskzInv(y, msk_erf);
    w       = t - one;
    t2      = t * t;
    w2      = w * w;
    /*
     * We cannot simply calculate exp(-y2) directly in single precision, since
     * that will lose a couple of bits of precision due to the multiplication.
     * Instead, we introduce y=z+w, where the last 12 bits of precision are in w.
     * Then we get exp(-y2) = exp(-z2)*exp((z-y)*(z+y)).
     *
     * The only drawback with this is that it requires TWO separate exponential
     * evaluations, which would be horrible performance-wise. However, the argument
     * for the second exp() call is always small, so there we simply use a
     * low-order minimax expansion on [0,0.1].
     *
     * However, this neat idea requires support for logical ops (and) on
     * FP numbers, which some vendors decided isn't necessary in their SIMD
     * instruction sets (Hi, IBM VSX!). In principle we could use some tricks
     * in double, but we still need memory as a backup when that is not available,
     * and this case is rare enough that we go directly there...
     */
#if GMX_SIMD_HAVE_LOGICAL
    z       = y & sieve;
#else
    store(mem, y);
    for (i = 0; i < GMX_SIMD_FLOAT_WIDTH; i++)
    {
        conv.f  = mem[i];
        conv.i  = conv.i & isieve;
        mem[i]  = conv.f;
    }
    z = load(mem);
#endif
    q       = (z-y) * (z+y);
    corr    = fma(CD4, q, CD3);
    corr    = fma(corr, q, CD2);
    corr    = fma(corr, q, one);
    corr    = fma(corr, q, one);

    expmx2  = exp( -z*z );
    expmx2  = expmx2 * corr;

    pB1  = fma(CB9, w2, CB7);
    pB0  = fma(CB8, w2, CB6);
    pB1  = fma(pB1, w2, CB5);
    pB0  = fma(pB0, w2, CB4);
    pB1  = fma(pB1, w2, CB3);
    pB0  = fma(pB0, w2, CB2);
    pB1  = fma(pB1, w2, CB1);
    pB0  = fma(pB0, w2, CB0);
    pB0  = fma(pB1, w, pB0);

    pC0  = fma(CC10, t2, CC8);
    pC1  = fma(CC9, t2, CC7);
    pC0  = fma(pC0, t2, CC6);
    pC1  = fma(pC1, t2, CC5);
    pC0  = fma(pC0, t2, CC4);
    pC1  = fma(pC1, t2, CC3);
    pC0  = fma(pC0, t2, CC2);
    pC1  = fma(pC1, t2, CC1);

    pC0  = fma(pC0, t2, CC0);
    pC0  = fma(pC1, t, pC0);
    pC0  = pC0 * t;

    // Select pB0 or pC0 for erfc()
    m        = two < y;
    res_erfc = blend(pB0, pC0, m);
    res_erfc = res_erfc * expmx2;

    // erfc(x<0) = 2-erfc(|x|)
    m        = x < setZero();
    res_erfc = blend(res_erfc, two-res_erfc, m);

    // Select erf() or erfc()
    res  = blend(one-res_erf, res_erfc, msk_erf);

    return res;
}

/*! \brief SIMD float sin \& cos.
 *
 * \param x The argument to evaluate sin/cos for
 * \param[out] sinval Sin(x)
 * \param[out] cosval Cos(x)
 *
 * This version achieves close to machine precision, but for very large
 * magnitudes of the argument we inherently begin to lose accuracy due to the
 * argument reduction, despite using extended precision arithmetics internally.
 */
static inline void gmx_simdcall
sincos(SimdFloat x, SimdFloat *sinval, SimdFloat *cosval)
{
    // Constants to subtract Pi/4*x from y while minimizing precision loss
    const SimdFloat  argred0(-1.5703125);
    const SimdFloat  argred1(-4.83751296997070312500e-04f);
    const SimdFloat  argred2(-7.54953362047672271729e-08f);
    const SimdFloat  argred3(-2.56334406825708960298e-12f);
    const SimdFloat  two_over_pi(static_cast<float>(2.0f/M_PI));
    const SimdFloat  const_sin2(-1.9515295891e-4f);
    const SimdFloat  const_sin1( 8.3321608736e-3f);
    const SimdFloat  const_sin0(-1.6666654611e-1f);
    const SimdFloat  const_cos2( 2.443315711809948e-5f);
    const SimdFloat  const_cos1(-1.388731625493765e-3f);
    const SimdFloat  const_cos0( 4.166664568298827e-2f);
    const SimdFloat  half(0.5f);
    const SimdFloat  one(1.0f);
    SimdFloat        ssign, csign;
    SimdFloat        x2, y, z, psin, pcos, sss, ccc;
    SimdFBool        m;

#if GMX_SIMD_HAVE_FINT32_ARITHMETICS && GMX_SIMD_HAVE_LOGICAL
    const SimdFInt32 ione(1);
    const SimdFInt32 itwo(2);
    SimdFInt32       iy;

    z       = x * two_over_pi;
    iy      = cvtR2I(z);
    y       = round(z);

    m       = cvtIB2B((iy & ione) == SimdFInt32(0));
    ssign   = selectByMask(SimdFloat(GMX_FLOAT_NEGZERO), cvtIB2B((iy & itwo) == itwo));
    csign   = selectByMask(SimdFloat(GMX_FLOAT_NEGZERO), cvtIB2B(((iy+ione) & itwo) == itwo));
#else
    const SimdFloat  quarter(0.25f);
    const SimdFloat  minusquarter(-0.25f);
    SimdFloat        q;
    SimdFBool        m1, m2, m3;

    /* The most obvious way to find the arguments quadrant in the unit circle
     * to calculate the sign is to use integer arithmetic, but that is not
     * present in all SIMD implementations. As an alternative, we have devised a
     * pure floating-point algorithm that uses truncation for argument reduction
     * so that we get a new value 0<=q<1 over the unit circle, and then
     * do floating-point comparisons with fractions. This is likely to be
     * slightly slower (~10%) due to the longer latencies of floating-point, so
     * we only use it when integer SIMD arithmetic is not present.
     */
    ssign   = x;
    x       = abs(x);
    // It is critical that half-way cases are rounded down
    z       = fma(x, two_over_pi, half);
    y       = trunc(z);
    q       = z * quarter;
    q       = q - trunc(q);
    /* z now starts at 0.0 for x=-pi/4 (although neg. values cannot occur), and
     * then increased by 1.0 as x increases by 2*Pi, when it resets to 0.0.
     * This removes the 2*Pi periodicity without using any integer arithmetic.
     * First check if y had the value 2 or 3, set csign if true.
     */
    q       = q - half;
    /* If we have logical operations we can work directly on the signbit, which
     * saves instructions. Otherwise we need to represent signs as +1.0/-1.0.
     * Thus, if you are altering defines to debug alternative code paths, the
     * two GMX_SIMD_HAVE_LOGICAL sections in this routine must either both be
     * active or inactive - you will get errors if only one is used.
     */
#    if GMX_SIMD_HAVE_LOGICAL
    ssign   = ssign & SimdFloat(GMX_FLOAT_NEGZERO);
    csign   = andNot(q, SimdFloat(GMX_FLOAT_NEGZERO));
    ssign   = ssign ^ csign;
#    else
    ssign   = copysign(SimdFloat(1.0f), ssign);
    csign   = copysign(SimdFloat(1.0f), q);
    csign   = -csign;
    ssign   = ssign * csign;    // swap ssign if csign was set.
#    endif
    // Check if y had value 1 or 3 (remember we subtracted 0.5 from q)
    m1      = (q < minusquarter);
    m2      = (setZero() <= q);
    m3      = (q < quarter);
    m2      = m2 && m3;
    m       = m1 || m2;
    // where mask is FALSE, swap sign.
    csign   = csign * blend(SimdFloat(-1.0f), one, m);
#endif
    x       = fma(y, argred0, x);
    x       = fma(y, argred1, x);
    x       = fma(y, argred2, x);
    x       = fma(y, argred3, x);
    x2      = x * x;

    psin    = fma(const_sin2, x2, const_sin1);
    psin    = fma(psin, x2, const_sin0);
    psin    = fma(psin, x * x2, x);
    pcos    = fma(const_cos2, x2, const_cos1);
    pcos    = fma(pcos, x2, const_cos0);
    pcos    = fms(pcos, x2, half);
    pcos    = fma(pcos, x2, one);

    sss     = blend(pcos, psin, m);
    ccc     = blend(psin, pcos, m);
    // See comment for GMX_SIMD_HAVE_LOGICAL section above.
#if GMX_SIMD_HAVE_LOGICAL
    *sinval = sss ^ ssign;
    *cosval = ccc ^ csign;
#else
    *sinval = sss * ssign;
    *cosval = ccc * csign;
#endif
}

/*! \brief SIMD float sin(x).
 *
 * \param x The argument to evaluate sin for
 * \result Sin(x)
 *
 * \attention Do NOT call both sin & cos if you need both results, since each of them
 * will then call \ref sincos and waste a factor 2 in performance.
 */
static inline SimdFloat gmx_simdcall
sin(SimdFloat x)
{
    SimdFloat s, c;
    sincos(x, &s, &c);
    return s;
}

/*! \brief SIMD float cos(x).
 *
 * \param x The argument to evaluate cos for
 * \result Cos(x)
 *
 * \attention Do NOT call both sin & cos if you need both results, since each of them
 * will then call \ref sincos and waste a factor 2 in performance.
 */
static inline SimdFloat gmx_simdcall
cos(SimdFloat x)
{
    SimdFloat s, c;
    sincos(x, &s, &c);
    return c;
}

/*! \brief SIMD float tan(x).
 *
 * \param x The argument to evaluate tan for
 * \result Tan(x)
 */
static inline SimdFloat gmx_simdcall
tan(SimdFloat x)
{
    const SimdFloat  argred0(-1.5703125);
    const SimdFloat  argred1(-4.83751296997070312500e-04f);
    const SimdFloat  argred2(-7.54953362047672271729e-08f);
    const SimdFloat  argred3(-2.56334406825708960298e-12f);
    const SimdFloat  two_over_pi(static_cast<float>(2.0f/M_PI));
    const SimdFloat  CT6(0.009498288995810566122993911);
    const SimdFloat  CT5(0.002895755790837379295226923);
    const SimdFloat  CT4(0.02460087336161924491836265);
    const SimdFloat  CT3(0.05334912882656359828045988);
    const SimdFloat  CT2(0.1333989091464957704418495);
    const SimdFloat  CT1(0.3333307599244198227797507);

    SimdFloat        x2, p, y, z;
    SimdFBool        m;

#if GMX_SIMD_HAVE_FINT32_ARITHMETICS && GMX_SIMD_HAVE_LOGICAL
    SimdFInt32  iy;
    SimdFInt32  ione(1);

    z       = x * two_over_pi;
    iy      = cvtR2I(z);
    y       = round(z);
    m       = cvtIB2B((iy & ione) == ione);

    x       = fma(y, argred0, x);
    x       = fma(y, argred1, x);
    x       = fma(y, argred2, x);
    x       = fma(y, argred3, x);
    x       = selectByMask(SimdFloat(GMX_FLOAT_NEGZERO), m) ^ x;
#else
    const SimdFloat  quarter(0.25f);
    const SimdFloat  half(0.5f);
    const SimdFloat  threequarter(0.75f);
    SimdFloat        w, q;
    SimdFBool        m1, m2, m3;

    w       = abs(x);
    z       = fma(w, two_over_pi, half);
    y       = trunc(z);
    q       = z * quarter;
    q       = q - trunc(q);
    m1      = quarter <= q;
    m2      = q < half;
    m3      = threequarter <= q;
    m1      = m1 && m2;
    m       = m1 || m3;
    w       = fma(y, argred0, w);
    w       = fma(y, argred1, w);
    w       = fma(y, argred2, w);
    w       = fma(y, argred3, w);
    w       = blend(w, -w, m);
    x       = w * copysign( SimdFloat(1.0), x );
#endif
    x2      = x * x;
    p       = fma(CT6, x2, CT5);
    p       = fma(p, x2, CT4);
    p       = fma(p, x2, CT3);
    p       = fma(p, x2, CT2);
    p       = fma(p, x2, CT1);
    p       = fma(x2 * p, x, x);

    p       = blend( p, maskzInv(p, m), m);
    return p;
}

/*! \brief SIMD float asin(x).
 *
 * \param x The argument to evaluate asin for
 * \result Asin(x)
 */
static inline SimdFloat gmx_simdcall
asin(SimdFloat x)
{
    const SimdFloat limitlow(1e-4f);
    const SimdFloat half(0.5f);
    const SimdFloat one(1.0f);
    const SimdFloat halfpi(static_cast<float>(M_PI/2.0f));
    const SimdFloat CC5(4.2163199048E-2f);
    const SimdFloat CC4(2.4181311049E-2f);
    const SimdFloat CC3(4.5470025998E-2f);
    const SimdFloat CC2(7.4953002686E-2f);
    const SimdFloat CC1(1.6666752422E-1f);
    SimdFloat       xabs;
    SimdFloat       z, z1, z2, q, q1, q2;
    SimdFloat       pA, pB;
    SimdFBool       m, m2;

    xabs  = abs(x);
    m     = half < xabs;
    z1    = half * (one-xabs);
    m2    = xabs < one;
    q1    = z1 * maskzInvsqrt(z1, m2);
    q2    = xabs;
    z2    = q2 * q2;
    z     = blend(z2, z1, m);
    q     = blend(q2, q1, m);

    z2    = z * z;
    pA    = fma(CC5, z2, CC3);
    pB    = fma(CC4, z2, CC2);
    pA    = fma(pA, z2, CC1);
    pA    = pA * z;
    z     = fma(pB, z2, pA);
    z     = fma(z, q, q);
    q2    = halfpi - z;
    q2    = q2 - z;
    z     = blend(z, q2, m);

    m     = limitlow < xabs;
    z     = blend( xabs, z, m );
    z     = copysign(z, x);

    return z;
}

/*! \brief SIMD float acos(x).
 *
 * \param x The argument to evaluate acos for
 * \result Acos(x)
 */
static inline SimdFloat gmx_simdcall
acos(SimdFloat x)
{
    const SimdFloat one(1.0f);
    const SimdFloat half(0.5f);
    const SimdFloat pi(static_cast<float>(M_PI));
    const SimdFloat halfpi(static_cast<float>(M_PI/2.0f));
    SimdFloat       xabs;
    SimdFloat       z, z1, z2, z3;
    SimdFBool       m1, m2, m3;

    xabs  = abs(x);
    m1    = half < xabs;
    m2    = setZero() < x;

    z     = fnma(half, xabs, half);
    m3    = xabs <  one;
    z     = z * maskzInvsqrt(z, m3);
    z     = blend(x, z, m1);
    z     = asin(z);

    z2    = z + z;
    z1    = pi - z2;
    z3    = halfpi - z;
    z     = blend(z1, z2, m2);
    z     = blend(z3, z, m1);

    return z;
}

/*! \brief SIMD float asin(x).
 *
 * \param x The argument to evaluate atan for
 * \result Atan(x), same argument/value range as standard math library.
 */
static inline SimdFloat gmx_simdcall
atan(SimdFloat x)
{
    const SimdFloat halfpi(static_cast<float>(M_PI/2.0f));
    const SimdFloat CA17(0.002823638962581753730774f);
    const SimdFloat CA15(-0.01595690287649631500244f);
    const SimdFloat CA13(0.04250498861074447631836f);
    const SimdFloat CA11(-0.07489009201526641845703f);
    const SimdFloat CA9 (0.1063479334115982055664f);
    const SimdFloat CA7 (-0.1420273631811141967773f);
    const SimdFloat CA5 (0.1999269574880599975585f);
    const SimdFloat CA3 (-0.3333310186862945556640f);
    const SimdFloat one (1.0f);
    SimdFloat       x2, x3, x4, pA, pB;
    SimdFBool       m, m2;

    m     = x < setZero();
    x     = abs(x);
    m2    = one < x;
    x     = blend(x, maskzInv(x, m2), m2);

    x2    = x * x;
    x3    = x2 * x;
    x4    = x2 * x2;
    pA    = fma(CA17, x4, CA13);
    pB    = fma(CA15, x4, CA11);
    pA    = fma(pA, x4, CA9);
    pB    = fma(pB, x4, CA7);
    pA    = fma(pA, x4, CA5);
    pB    = fma(pB, x4, CA3);
    pA    = fma(pA, x2, pB);
    pA    = fma(pA, x3, x);

    pA    = blend(pA, halfpi-pA, m2);
    pA    = blend(pA, -pA, m);

    return pA;
}

/*! \brief SIMD float atan2(y,x).
 *
 * \param y Y component of vector, any quartile
 * \param x X component of vector, any quartile
 * \result Atan(y,x), same argument/value range as standard math library.
 *
 * \note This routine should provide correct results for all finite
 * non-zero or positive-zero arguments. However, negative zero arguments will
 * be treated as positive zero, which means the return value will deviate from
 * the standard math library atan2(y,x) for those cases. That should not be
 * of any concern in Gromacs, and in particular it will not affect calculations
 * of angles from vectors.
 */
static inline SimdFloat gmx_simdcall
atan2(SimdFloat y, SimdFloat x)
{
    const SimdFloat pi(static_cast<float>(M_PI));
    const SimdFloat halfpi(static_cast<float>(M_PI/2.0));
    SimdFloat       xinv, p, aoffset;
    SimdFBool       mask_xnz, mask_ynz, mask_xlt0, mask_ylt0;

    mask_xnz  = x != setZero();
    mask_ynz  = y != setZero();
    mask_xlt0 = x < setZero();
    mask_ylt0 = y < setZero();

    aoffset   = selectByNotMask(halfpi, mask_xnz);
    aoffset   = selectByMask(aoffset, mask_ynz);

    aoffset   = blend(aoffset, pi, mask_xlt0);
    aoffset   = blend(aoffset, -aoffset, mask_ylt0);

    xinv      = maskzInv(x, mask_xnz);
    p         = y * xinv;
    p         = atan(p);
    p         = p + aoffset;

    return p;
}

/*! \brief Calculate the force correction due to PME analytically in SIMD float.
 *
 * \param z2 \f$(r \beta)^2\f$ - see below for details.
 * \result Correction factor to coulomb force - see below for details.
 *
 * This routine is meant to enable analytical evaluation of the
 * direct-space PME electrostatic force to avoid tables.
 *
 * The direct-space potential should be \f$ \mbox{erfc}(\beta r)/r\f$, but there
 * are some problems evaluating that:
 *
 * First, the error function is difficult (read: expensive) to
 * approxmiate accurately for intermediate to large arguments, and
 * this happens already in ranges of \f$(\beta r)\f$ that occur in simulations.
 * Second, we now try to avoid calculating potentials in Gromacs but
 * use forces directly.
 *
 * We can simply things slight by noting that the PME part is really
 * a correction to the normal Coulomb force since \f$\mbox{erfc}(z)=1-\mbox{erf}(z)\f$, i.e.
 * \f[
 * V = \frac{1}{r} - \frac{\mbox{erf}(\beta r)}{r}
 * \f]
 * The first term we already have from the inverse square root, so
 * that we can leave out of this routine.
 *
 * For pme tolerances of 1e-3 to 1e-8 and cutoffs of 0.5nm to 1.8nm,
 * the argument \f$beta r\f$ will be in the range 0.15 to ~4, which is
 * the range used for the minimax fit. Use your favorite plotting program
 * to realize how well-behaved \f$\frac{\mbox{erf}(z)}{z}\f$ is in this range!
 *
 * We approximate \f$f(z)=\mbox{erf}(z)/z\f$ with a rational minimax polynomial.
 * However, it turns out it is more efficient to approximate \f$f(z)/z\f$ and
 * then only use even powers. This is another minor optimization, since
 * we actually \a want \f$f(z)/z\f$, because it is going to be multiplied by
 * the vector between the two atoms to get the vectorial force. The
 * fastest flops are the ones we can avoid calculating!
 *
 * So, here's how it should be used:
 *
 * 1. Calculate \f$r^2\f$.
 * 2. Multiply by \f$\beta^2\f$, so you get \f$z^2=(\beta r)^2\f$.
 * 3. Evaluate this routine with \f$z^2\f$ as the argument.
 * 4. The return value is the expression:
 *
 * \f[
 *    \frac{2 \exp{-z^2}}{\sqrt{\pi} z^2}-\frac{\mbox{erf}(z)}{z^3}
 * \f]
 *
 * 5. Multiply the entire expression by \f$\beta^3\f$. This will get you
 *
 *  \f[
 *    \frac{2 \beta^3 \exp(-z^2)}{\sqrt{\pi} z^2} - \frac{\beta^3 \mbox{erf}(z)}{z^3}
 *  \f]
 *
 *    or, switching back to \f$r\f$ (since \f$z=r \beta\f$):
 *
 *  \f[
 *    \frac{2 \beta \exp(-r^2 \beta^2)}{\sqrt{\pi} r^2} - \frac{\mbox{erf}(r \beta)}{r^3}
 *  \f]
 *
 *    With a bit of math exercise you should be able to confirm that
 *    this is exactly
 *
 *  \f[
 *   \frac{\frac{d}{dr}\left( \frac{\mbox{erf}(\beta r)}{r} \right)}{r}
 *  \f]
 *
 * 6. Add the result to \f$r^{-3}\f$, multiply by the product of the charges,
 *    and you have your force (divided by \f$r\f$). A final multiplication
 *    with the vector connecting the two particles and you have your
 *    vectorial force to add to the particles.
 *
 * This approximation achieves an error slightly lower than 1e-6
 * in single precision and 1e-11 in double precision
 * for arguments smaller than 16 (\f$\beta r \leq 4 \f$);
 * when added to \f$1/r\f$ the error will be insignificant.
 * For \f$\beta r \geq 7206\f$ the return value can be inf or NaN.
 *
 */
static inline SimdFloat gmx_simdcall
pmeForceCorrection(SimdFloat z2)
{
    const SimdFloat  FN6(-1.7357322914161492954e-8f);
    const SimdFloat  FN5(1.4703624142580877519e-6f);
    const SimdFloat  FN4(-0.000053401640219807709149f);
    const SimdFloat  FN3(0.0010054721316683106153f);
    const SimdFloat  FN2(-0.019278317264888380590f);
    const SimdFloat  FN1(0.069670166153766424023f);
    const SimdFloat  FN0(-0.75225204789749321333f);

    const SimdFloat  FD4(0.0011193462567257629232f);
    const SimdFloat  FD3(0.014866955030185295499f);
    const SimdFloat  FD2(0.11583842382862377919f);
    const SimdFloat  FD1(0.50736591960530292870f);
    const SimdFloat  FD0(1.0f);

    SimdFloat        z4;
    SimdFloat        polyFN0, polyFN1, polyFD0, polyFD1;

    z4             = z2 * z2;

    polyFD0        = fma(FD4, z4, FD2);
    polyFD1        = fma(FD3, z4, FD1);
    polyFD0        = fma(polyFD0, z4, FD0);
    polyFD0        = fma(polyFD1, z2, polyFD0);

    polyFD0        = inv(polyFD0);

    polyFN0        = fma(FN6, z4, FN4);
    polyFN1        = fma(FN5, z4, FN3);
    polyFN0        = fma(polyFN0, z4, FN2);
    polyFN1        = fma(polyFN1, z4, FN1);
    polyFN0        = fma(polyFN0, z4, FN0);
    polyFN0        = fma(polyFN1, z2, polyFN0);

    return polyFN0 * polyFD0;
}



/*! \brief Calculate the potential correction due to PME analytically in SIMD float.
 *
 * \param z2 \f$(r \beta)^2\f$ - see below for details.
 * \result Correction factor to coulomb potential - see below for details.
 *
 * See \ref pmeForceCorrection for details about the approximation.
 *
 * This routine calculates \f$\mbox{erf}(z)/z\f$, although you should provide \f$z^2\f$
 * as the input argument.
 *
 * Here's how it should be used:
 *
 * 1. Calculate \f$r^2\f$.
 * 2. Multiply by \f$\beta^2\f$, so you get \f$z^2=\beta^2*r^2\f$.
 * 3. Evaluate this routine with z^2 as the argument.
 * 4. The return value is the expression:
 *
 *  \f[
 *   \frac{\mbox{erf}(z)}{z}
 *  \f]
 *
 * 5. Multiply the entire expression by beta and switching back to \f$r\f$ (since \f$z=r \beta\f$):
 *
 *  \f[
 *    \frac{\mbox{erf}(r \beta)}{r}
 *  \f]
 *
 * 6. Subtract the result from \f$1/r\f$, multiply by the product of the charges,
 *    and you have your potential.
 *
 * This approximation achieves an error slightly lower than 1e-6
 * in single precision and 4e-11 in double precision
 * for arguments smaller than 16 (\f$ 0.15 \leq \beta r \leq 4 \f$);
 * for \f$ \beta r \leq 0.15\f$ the error can be twice as high;
 * when added to \f$1/r\f$ the error will be insignificant.
 * For \f$\beta r \geq 7142\f$ the return value can be inf or NaN.
 */
static inline SimdFloat gmx_simdcall
pmePotentialCorrection(SimdFloat z2)
{
    const SimdFloat  VN6(1.9296833005951166339e-8f);
    const SimdFloat  VN5(-1.4213390571557850962e-6f);
    const SimdFloat  VN4(0.000041603292906656984871f);
    const SimdFloat  VN3(-0.00013134036773265025626f);
    const SimdFloat  VN2(0.038657983986041781264f);
    const SimdFloat  VN1(0.11285044772717598220f);
    const SimdFloat  VN0(1.1283802385263030286f);

    const SimdFloat  VD3(0.0066752224023576045451f);
    const SimdFloat  VD2(0.078647795836373922256f);
    const SimdFloat  VD1(0.43336185284710920150f);
    const SimdFloat  VD0(1.0f);

    SimdFloat        z4;
    SimdFloat        polyVN0, polyVN1, polyVD0, polyVD1;

    z4             = z2 * z2;

    polyVD1        = fma(VD3, z4, VD1);
    polyVD0        = fma(VD2, z4, VD0);
    polyVD0        = fma(polyVD1, z2, polyVD0);

    polyVD0        = inv(polyVD0);

    polyVN0        = fma(VN6, z4, VN4);
    polyVN1        = fma(VN5, z4, VN3);
    polyVN0        = fma(polyVN0, z4, VN2);
    polyVN1        = fma(polyVN1, z4, VN1);
    polyVN0        = fma(polyVN0, z4, VN0);
    polyVN0        = fma(polyVN1, z2, polyVN0);

    return polyVN0 * polyVD0;
}
#endif

/*! \} */

#if GMX_SIMD_HAVE_DOUBLE


/*! \name Double precision SIMD math functions
 *
 *  \note In most cases you should use the real-precision functions instead.
 *  \{
 */

/****************************************
 * DOUBLE PRECISION SIMD MATH FUNCTIONS *
 ****************************************/

#if !GMX_SIMD_HAVE_NATIVE_COPYSIGN_DOUBLE
/*! \brief Composes floating point value with the magnitude of x and the sign of y.
 *
 * \param x Values to set sign for
 * \param y Values used to set sign
 * \return  Magnitude of x, sign of y
 */
static inline SimdDouble gmx_simdcall
copysign(SimdDouble x, SimdDouble y)
{
#if GMX_SIMD_HAVE_LOGICAL
    return abs(x) | (SimdDouble(GMX_DOUBLE_NEGZERO) & y);
#else
    return blend(abs(x), -abs(x), (y < setZero()));
#endif
}
#endif

#if !GMX_SIMD_HAVE_NATIVE_RSQRT_ITER_DOUBLE
/*! \brief Perform one Newton-Raphson iteration to improve 1/sqrt(x) for SIMD double.
 *
 * This is a low-level routine that should only be used by SIMD math routine
 * that evaluates the inverse square root.
 *
 *  \param lu Approximation of 1/sqrt(x), typically obtained from lookup.
 *  \param x  The reference (starting) value x for which we want 1/sqrt(x).
 *  \return   An improved approximation with roughly twice as many bits of accuracy.
 */
static inline SimdDouble gmx_simdcall
rsqrtIter(SimdDouble lu, SimdDouble x)
{
    SimdDouble tmp1 = x*lu;
    SimdDouble tmp2 = SimdDouble(-0.5)*lu;
    tmp1 = fma(tmp1, lu, SimdDouble(-3.0));
    return tmp1*tmp2;
}
#endif

/*! \brief Calculate 1/sqrt(x) for SIMD double.
 *
 *  \param x Argument that must be >0. This routine does not check arguments.
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid.
 */
static inline SimdDouble gmx_simdcall
invsqrt(SimdDouble x)
{
    SimdDouble lu = rsqrt(x);
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*8 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
    return lu;
}

/*! \brief Calculate 1/sqrt(x) for two SIMD doubles.
 *
 * \param x0  First set of arguments, x0 must be positive - no argument checking.
 * \param x1  Second set of arguments, x1 must be positive - no argument checking.
 * \param[out] out0  Result 1/sqrt(x0)
 * \param[out] out1  Result 1/sqrt(x1)
 *
 *  In particular for double precision we can sometimes calculate square root
 *  pairs slightly faster by using single precision until the very last step.
 */
static inline void gmx_simdcall
invsqrtPair(SimdDouble x0,    SimdDouble x1,
            SimdDouble *out0, SimdDouble *out1)
{
#if GMX_SIMD_HAVE_FLOAT && (GMX_SIMD_FLOAT_WIDTH == 2*GMX_SIMD_DOUBLE_WIDTH) && (GMX_SIMD_RSQRT_BITS < 22)
    SimdFloat  xf  = cvtDD2F(x0, x1);
    SimdFloat  luf = rsqrt(xf);
    SimdDouble lu0, lu1;
    // Intermediate target is single - mantissa+1 bits
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    luf = rsqrtIter(luf, xf);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    luf = rsqrtIter(luf, xf);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    luf = rsqrtIter(luf, xf);
#endif
    cvtF2DD(luf, &lu0, &lu1);
    // Last iteration(s) performed in double - if we had 22 bits, this gets us to 44 (~1e-15)
#if (GMX_SIMD_ACCURACY_BITS_SINGLE < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu0 = rsqrtIter(lu0, x0);
    lu1 = rsqrtIter(lu1, x1);
#endif
#if (GMX_SIMD_ACCURACY_BITS_SINGLE*2 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu0 = rsqrtIter(lu0, x0);
    lu1 = rsqrtIter(lu1, x1);
#endif
    *out0 = lu0;
    *out1 = lu1;
#else
    *out0 = invsqrt(x0);
    *out1 = invsqrt(x1);
#endif
}

#if !GMX_SIMD_HAVE_NATIVE_RCP_ITER_DOUBLE
/*! \brief Perform one Newton-Raphson iteration to improve 1/x for SIMD double.
 *
 * This is a low-level routine that should only be used by SIMD math routine
 * that evaluates the reciprocal.
 *
 *  \param lu Approximation of 1/x, typically obtained from lookup.
 *  \param x  The reference (starting) value x for which we want 1/x.
 *  \return   An improved approximation with roughly twice as many bits of accuracy.
 */
static inline SimdDouble gmx_simdcall
rcpIter(SimdDouble lu, SimdDouble x)
{
    return lu*fnma(lu, x, SimdDouble(2.0));
}
#endif

/*! \brief Calculate 1/x for SIMD double.
 *
 *  \param x Argument that must be nonzero. This routine does not check arguments.
 *  \return 1/x. Result is undefined if your argument was invalid.
 */
static inline SimdDouble gmx_simdcall
inv(SimdDouble x)
{
    SimdDouble lu = rcp(x);
#if (GMX_SIMD_RCP_BITS < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*2 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*4 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*8 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rcpIter(lu, x);
#endif
    return lu;
}

/*! \brief Division for SIMD doubles
 *
 * \param nom    Nominator
 * \param denom  Denominator
 *
 * \return nom/denom
 *
 * \note This function does not use any masking to avoid problems with
 *       zero values in the denominator.
 */
static inline SimdDouble gmx_simdcall
operator/(SimdDouble nom, SimdDouble denom)
{
    return nom*inv(denom);
}


/*! \brief Calculate 1/sqrt(x) for masked entries of SIMD double.
 *
 *  This routine only evaluates 1/sqrt(x) for elements for which mask is true.
 *  Illegal values in the masked-out elements will not lead to
 *  floating-point exceptions.
 *
 *  \param x Argument that must be >0 for masked-in entries
 *  \param m Mask
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid or
 *          entry was not masked, and 0.0 for masked-out entries.
 */
static inline SimdDouble
maskzInvsqrt(SimdDouble x, SimdDBool m)
{
    SimdDouble lu = maskzRsqrt(x, m);
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*8 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
    return lu;
}

/*! \brief Calculate 1/x for SIMD double, masked version.
 *
 *  \param x Argument that must be nonzero for non-masked entries.
 *  \param m Mask
 *  \return 1/x for elements where m is true, or 0.0 for masked-out entries.
 */
static inline SimdDouble gmx_simdcall
maskzInv(SimdDouble x, SimdDBool m)
{
    SimdDouble lu = maskzRcp(x, m);
#if (GMX_SIMD_RCP_BITS < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*2 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*4 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*8 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rcpIter(lu, x);
#endif
    return lu;
}


/*! \brief Calculate sqrt(x) correctly for SIMD doubles, including argument 0.0.
 *
 *  \param x Argument that must be >=0.
 *  \return sqrt(x). If x=0, the result will correctly be set to 0.
 *          if x>0 && x<float_min, the result will incorrectly be set to 0.
 *          The result is undefined if the input value is negative.
 */
static inline SimdDouble gmx_simdcall
sqrt(SimdDouble x)
{
    // As we might use a float version of rsqrt, we mask out small values
    return x * maskzInvsqrt(x, SimdDouble(GMX_FLOAT_MIN) <= x);
}

#if !GMX_SIMD_HAVE_NATIVE_LOG_DOUBLE
/*! \brief SIMD double log(x). This is the natural logarithm.
 *
 * \param x Argument, should be >0.
 * \result The natural logarithm of x. Undefined if argument is invalid.
 */
static inline SimdDouble gmx_simdcall
log(SimdDouble x)
{
    const SimdDouble  one(1.0);
    const SimdDouble  two(2.0);
    const SimdDouble  invsqrt2(1.0/std::sqrt(2.0));
    const SimdDouble  corr(0.693147180559945286226764);
    const SimdDouble  CL15(0.148197055177935105296783);
    const SimdDouble  CL13(0.153108178020442575739679);
    const SimdDouble  CL11(0.181837339521549679055568);
    const SimdDouble  CL9(0.22222194152736701733275);
    const SimdDouble  CL7(0.285714288030134544449368);
    const SimdDouble  CL5(0.399999999989941956712869);
    const SimdDouble  CL3(0.666666666666685503450651);
    const SimdDouble  CL1(2.0);
    SimdDouble        fExp, x2, p;
    SimdDBool         m;
    SimdDInt32        iExp;

    x     = frexp(x, &iExp);
    fExp  = cvtI2R(iExp);

    m     = x < invsqrt2;
    // Adjust to non-IEEE format for x<1/sqrt(2): exponent -= 1, mantissa *= 2.0
    fExp  = fExp - selectByMask(one, m);
    x     = x * blend(one, two, m);

    x     = (x-one) * inv( x+one );
    x2    = x * x;

    p     = fma(CL15, x2, CL13);
    p     = fma(p, x2, CL11);
    p     = fma(p, x2, CL9);
    p     = fma(p, x2, CL7);
    p     = fma(p, x2, CL5);
    p     = fma(p, x2, CL3);
    p     = fma(p, x2, CL1);
    p     = fma(p, x, corr * fExp);

    return p;
}
#endif

#if !GMX_SIMD_HAVE_NATIVE_EXP2_DOUBLE
/*! \brief SIMD double 2^x.
 *
 * \param x Argument.
 * \result 2^x. Undefined if input argument caused overflow.
 */
static inline SimdDouble gmx_simdcall
exp2(SimdDouble x)
{
    const SimdDouble  arglimit(1022.0);
    const SimdDouble  CE11(4.435280790452730022081181e-10);
    const SimdDouble  CE10(7.074105630863314448024247e-09);
    const SimdDouble  CE9(1.017819803432096698472621e-07);
    const SimdDouble  CE8(1.321543308956718799557863e-06);
    const SimdDouble  CE7(0.00001525273348995851746990884);
    const SimdDouble  CE6(0.0001540353046251466849082632);
    const SimdDouble  CE5(0.001333355814678995257307880);
    const SimdDouble  CE4(0.009618129107588335039176502);
    const SimdDouble  CE3(0.05550410866481992147457793);
    const SimdDouble  CE2(0.2402265069591015620470894);
    const SimdDouble  CE1(0.6931471805599453304615075);
    const SimdDouble  one(1.0);

    SimdDouble        intpart;
    SimdDouble        fexppart;
    SimdDouble        p;
    SimdDBool         m;

    fexppart  = ldexp(one, cvtR2I(x));
    intpart   = round(x);
    m         = abs(x) <= arglimit;
    fexppart  = selectByMask(fexppart, m);
    x         = x - intpart;

    p         = fma(CE11, x, CE10);
    p         = fma(p, x, CE9);
    p         = fma(p, x, CE8);
    p         = fma(p, x, CE7);
    p         = fma(p, x, CE6);
    p         = fma(p, x, CE5);
    p         = fma(p, x, CE4);
    p         = fma(p, x, CE3);
    p         = fma(p, x, CE2);
    p         = fma(p, x, CE1);
    p         = fma(p, x, one);
    x         = p * fexppart;
    return x;
}
#endif

#if !GMX_SIMD_HAVE_NATIVE_EXP_DOUBLE
/*! \brief SIMD double exp(x).
 *
 * In addition to scaling the argument for 2^x this routine correctly does
 * extended precision arithmetics to improve accuracy.
 *
 * \param x Argument.
 * \result exp(x). Undefined if input argument caused overflow,
 * which can happen if abs(x) \> 7e13.
 */
static inline SimdDouble gmx_simdcall
exp(SimdDouble x)
{
    const SimdDouble  argscale(1.44269504088896340735992468100);
    const SimdDouble  arglimit(1022.0);
    const SimdDouble  invargscale0(-0.69314718055966295651160180568695068359375);
    const SimdDouble  invargscale1(-2.8235290563031577122588448175013436025525412068e-13);
    const SimdDouble  CE12(2.078375306791423699350304e-09);
    const SimdDouble  CE11(2.518173854179933105218635e-08);
    const SimdDouble  CE10(2.755842049600488770111608e-07);
    const SimdDouble  CE9(2.755691815216689746619849e-06);
    const SimdDouble  CE8(2.480158383706245033920920e-05);
    const SimdDouble  CE7(0.0001984127043518048611841321);
    const SimdDouble  CE6(0.001388888889360258341755930);
    const SimdDouble  CE5(0.008333333332907368102819109);
    const SimdDouble  CE4(0.04166666666663836745814631);
    const SimdDouble  CE3(0.1666666666666796929434570);
    const SimdDouble  CE2(0.5);
    const SimdDouble  one(1.0);
    SimdDouble        fexppart;
    SimdDouble        intpart;
    SimdDouble        y, p;
    SimdDBool         m;

    y         = x * argscale;
    fexppart  = ldexp(one, cvtR2I(y));
    intpart   = round(y);
    m         = (abs(y) <= arglimit);
    fexppart  = selectByMask(fexppart, m);

    // Extended precision arithmetics
    x         = fma(invargscale0, intpart, x);
    x         = fma(invargscale1, intpart, x);

    p         = fma(CE12, x, CE11);
    p         = fma(p, x, CE10);
    p         = fma(p, x, CE9);
    p         = fma(p, x, CE8);
    p         = fma(p, x, CE7);
    p         = fma(p, x, CE6);
    p         = fma(p, x, CE5);
    p         = fma(p, x, CE4);
    p         = fma(p, x, CE3);
    p         = fma(p, x, CE2);
    p         = fma(p, x * x, x);
    x         = fma(p, fexppart, fexppart);

    return x;
}
#endif

/*! \brief SIMD double erf(x).
 *
 * \param x The value to calculate erf(x) for.
 * \result erf(x)
 *
 * This routine achieves very close to full precision, but we do not care about
 * the last bit or the subnormal result range.
 */
static inline SimdDouble gmx_simdcall
erf(SimdDouble x)
{
    // Coefficients for minimax approximation of erf(x)=x*(CAoffset + P(x^2)/Q(x^2)) in range [-0.75,0.75]
    const SimdDouble CAP4(-0.431780540597889301512e-4);
    const SimdDouble CAP3(-0.00578562306260059236059);
    const SimdDouble CAP2(-0.028593586920219752446);
    const SimdDouble CAP1(-0.315924962948621698209);
    const SimdDouble CAP0(0.14952975608477029151);

    const SimdDouble CAQ5(-0.374089300177174709737e-5);
    const SimdDouble CAQ4(0.00015126584532155383535);
    const SimdDouble CAQ3(0.00536692680669480725423);
    const SimdDouble CAQ2(0.0668686825594046122636);
    const SimdDouble CAQ1(0.402604990869284362773);
    // CAQ0 == 1.0
    const SimdDouble CAoffset(0.9788494110107421875);

    // Coefficients for minimax approximation of erfc(x)=exp(-x^2)*x*(P(x-1)/Q(x-1)) in range [1.0,4.5]
    const SimdDouble CBP6(2.49650423685462752497647637088e-10);
    const SimdDouble CBP5(0.00119770193298159629350136085658);
    const SimdDouble CBP4(0.0164944422378370965881008942733);
    const SimdDouble CBP3(0.0984581468691775932063932439252);
    const SimdDouble CBP2(0.317364595806937763843589437418);
    const SimdDouble CBP1(0.554167062641455850932670067075);
    const SimdDouble CBP0(0.427583576155807163756925301060);
    const SimdDouble CBQ7(0.00212288829699830145976198384930);
    const SimdDouble CBQ6(0.0334810979522685300554606393425);
    const SimdDouble CBQ5(0.2361713785181450957579508850717);
    const SimdDouble CBQ4(0.955364736493055670530981883072);
    const SimdDouble CBQ3(2.36815675631420037315349279199);
    const SimdDouble CBQ2(3.55261649184083035537184223542);
    const SimdDouble CBQ1(2.93501136050160872574376997993);
    // CBQ0 == 1.0

    // Coefficients for minimax approximation of erfc(x)=exp(-x^2)/x*(P(1/x)/Q(1/x)) in range [4.5,inf]
    const SimdDouble CCP6(-2.8175401114513378771);
    const SimdDouble CCP5(-3.22729451764143718517);
    const SimdDouble CCP4(-2.5518551727311523996);
    const SimdDouble CCP3(-0.687717681153649930619);
    const SimdDouble CCP2(-0.212652252872804219852);
    const SimdDouble CCP1(0.0175389834052493308818);
    const SimdDouble CCP0(0.00628057170626964891937);

    const SimdDouble CCQ6(5.48409182238641741584);
    const SimdDouble CCQ5(13.5064170191802889145);
    const SimdDouble CCQ4(22.9367376522880577224);
    const SimdDouble CCQ3(15.930646027911794143);
    const SimdDouble CCQ2(11.0567237927800161565);
    const SimdDouble CCQ1(2.79257750980575282228);
    // CCQ0 == 1.0
    const SimdDouble CCoffset(0.5579090118408203125);

    const SimdDouble one(1.0);
    const SimdDouble two(2.0);

    SimdDouble       xabs, x2, x4, t, t2, w, w2;
    SimdDouble       PolyAP0, PolyAP1, PolyAQ0, PolyAQ1;
    SimdDouble       PolyBP0, PolyBP1, PolyBQ0, PolyBQ1;
    SimdDouble       PolyCP0, PolyCP1, PolyCQ0, PolyCQ1;
    SimdDouble       res_erf, res_erfcB, res_erfcC, res_erfc, res;
    SimdDouble       expmx2;
    SimdDBool        mask, mask_erf, notmask_erf;

    // Calculate erf()
    xabs        = abs(x);
    mask_erf    = (xabs < one);
    notmask_erf = (one <= xabs);
    x2          = x * x;
    x4          = x2 * x2;

    PolyAP0  = fma(CAP4, x4, CAP2);
    PolyAP1  = fma(CAP3, x4, CAP1);
    PolyAP0  = fma(PolyAP0, x4, CAP0);
    PolyAP0  = fma(PolyAP1, x2, PolyAP0);

    PolyAQ1  = fma(CAQ5, x4, CAQ3);
    PolyAQ0  = fma(CAQ4, x4, CAQ2);
    PolyAQ1  = fma(PolyAQ1, x4, CAQ1);
    PolyAQ0  = fma(PolyAQ0, x4, one);
    PolyAQ0  = fma(PolyAQ1, x2, PolyAQ0);

    res_erf  = PolyAP0 * maskzInv(PolyAQ0, mask_erf);
    res_erf  = CAoffset + res_erf;
    res_erf  = x * res_erf;

    // Calculate erfc() in range [1,4.5]
    t       = xabs - one;
    t2      = t * t;

    PolyBP0  = fma(CBP6, t2, CBP4);
    PolyBP1  = fma(CBP5, t2, CBP3);
    PolyBP0  = fma(PolyBP0, t2, CBP2);
    PolyBP1  = fma(PolyBP1, t2, CBP1);
    PolyBP0  = fma(PolyBP0, t2, CBP0);
    PolyBP0  = fma(PolyBP1, t, PolyBP0);

    PolyBQ1 = fma(CBQ7, t2, CBQ5);
    PolyBQ0 = fma(CBQ6, t2, CBQ4);
    PolyBQ1 = fma(PolyBQ1, t2, CBQ3);
    PolyBQ0 = fma(PolyBQ0, t2, CBQ2);
    PolyBQ1 = fma(PolyBQ1, t2, CBQ1);
    PolyBQ0 = fma(PolyBQ0, t2, one);
    PolyBQ0 = fma(PolyBQ1, t, PolyBQ0);

    // The denominator polynomial can be zero outside the range
    res_erfcB = PolyBP0 * maskzInv(PolyBQ0, notmask_erf);

    res_erfcB = res_erfcB * xabs;

    // Calculate erfc() in range [4.5,inf]
    w       = maskzInv(xabs, notmask_erf);
    w2      = w * w;

    PolyCP0  = fma(CCP6, w2, CCP4);
    PolyCP1  = fma(CCP5, w2, CCP3);
    PolyCP0  = fma(PolyCP0, w2, CCP2);
    PolyCP1  = fma(PolyCP1, w2, CCP1);
    PolyCP0  = fma(PolyCP0, w2, CCP0);
    PolyCP0  = fma(PolyCP1, w, PolyCP0);

    PolyCQ0  = fma(CCQ6, w2, CCQ4);
    PolyCQ1  = fma(CCQ5, w2, CCQ3);
    PolyCQ0  = fma(PolyCQ0, w2, CCQ2);
    PolyCQ1  = fma(PolyCQ1, w2, CCQ1);
    PolyCQ0  = fma(PolyCQ0, w2, one);
    PolyCQ0  = fma(PolyCQ1, w, PolyCQ0);

    expmx2   = exp( -x2 );

    // The denominator polynomial can be zero outside the range
    res_erfcC = PolyCP0 * maskzInv(PolyCQ0, notmask_erf);
    res_erfcC = res_erfcC + CCoffset;
    res_erfcC = res_erfcC * w;

    mask     = (SimdDouble(4.5) < xabs);
    res_erfc = blend(res_erfcB, res_erfcC, mask);

    res_erfc = res_erfc * expmx2;

    // erfc(x<0) = 2-erfc(|x|)
    mask     = (x < setZero());
    res_erfc = blend(res_erfc, two - res_erfc, mask);

    // Select erf() or erfc()
    res  = blend(one - res_erfc, res_erf, mask_erf);

    return res;
}

/*! \brief SIMD double erfc(x).
 *
 * \param x The value to calculate erfc(x) for.
 * \result erfc(x)
 *
 * This routine achieves full precision (bar the last bit) over most of the
 * input range, but for large arguments where the result is getting close
 * to the minimum representable numbers we accept slightly larger errors
 * (think results that are in the ballpark of 10^-200 for double)
 * since that is not relevant for MD.
 */
static inline SimdDouble gmx_simdcall
erfc(SimdDouble x)
{
    // Coefficients for minimax approximation of erf(x)=x*(CAoffset + P(x^2)/Q(x^2)) in range [-0.75,0.75]
    const SimdDouble CAP4(-0.431780540597889301512e-4);
    const SimdDouble CAP3(-0.00578562306260059236059);
    const SimdDouble CAP2(-0.028593586920219752446);
    const SimdDouble CAP1(-0.315924962948621698209);
    const SimdDouble CAP0(0.14952975608477029151);

    const SimdDouble CAQ5(-0.374089300177174709737e-5);
    const SimdDouble CAQ4(0.00015126584532155383535);
    const SimdDouble CAQ3(0.00536692680669480725423);
    const SimdDouble CAQ2(0.0668686825594046122636);
    const SimdDouble CAQ1(0.402604990869284362773);
    // CAQ0 == 1.0
    const SimdDouble CAoffset(0.9788494110107421875);

    // Coefficients for minimax approximation of erfc(x)=exp(-x^2)*x*(P(x-1)/Q(x-1)) in range [1.0,4.5]
    const SimdDouble CBP6(2.49650423685462752497647637088e-10);
    const SimdDouble CBP5(0.00119770193298159629350136085658);
    const SimdDouble CBP4(0.0164944422378370965881008942733);
    const SimdDouble CBP3(0.0984581468691775932063932439252);
    const SimdDouble CBP2(0.317364595806937763843589437418);
    const SimdDouble CBP1(0.554167062641455850932670067075);
    const SimdDouble CBP0(0.427583576155807163756925301060);
    const SimdDouble CBQ7(0.00212288829699830145976198384930);
    const SimdDouble CBQ6(0.0334810979522685300554606393425);
    const SimdDouble CBQ5(0.2361713785181450957579508850717);
    const SimdDouble CBQ4(0.955364736493055670530981883072);
    const SimdDouble CBQ3(2.36815675631420037315349279199);
    const SimdDouble CBQ2(3.55261649184083035537184223542);
    const SimdDouble CBQ1(2.93501136050160872574376997993);
    // CBQ0 == 1.0

    // Coefficients for minimax approximation of erfc(x)=exp(-x^2)/x*(P(1/x)/Q(1/x)) in range [4.5,inf]
    const SimdDouble CCP6(-2.8175401114513378771);
    const SimdDouble CCP5(-3.22729451764143718517);
    const SimdDouble CCP4(-2.5518551727311523996);
    const SimdDouble CCP3(-0.687717681153649930619);
    const SimdDouble CCP2(-0.212652252872804219852);
    const SimdDouble CCP1(0.0175389834052493308818);
    const SimdDouble CCP0(0.00628057170626964891937);

    const SimdDouble CCQ6(5.48409182238641741584);
    const SimdDouble CCQ5(13.5064170191802889145);
    const SimdDouble CCQ4(22.9367376522880577224);
    const SimdDouble CCQ3(15.930646027911794143);
    const SimdDouble CCQ2(11.0567237927800161565);
    const SimdDouble CCQ1(2.79257750980575282228);
    // CCQ0 == 1.0
    const SimdDouble CCoffset(0.5579090118408203125);

    const SimdDouble one(1.0);
    const SimdDouble two(2.0);

    SimdDouble       xabs, x2, x4, t, t2, w, w2;
    SimdDouble       PolyAP0, PolyAP1, PolyAQ0, PolyAQ1;
    SimdDouble       PolyBP0, PolyBP1, PolyBQ0, PolyBQ1;
    SimdDouble       PolyCP0, PolyCP1, PolyCQ0, PolyCQ1;
    SimdDouble       res_erf, res_erfcB, res_erfcC, res_erfc, res;
    SimdDouble       expmx2;
    SimdDBool        mask, mask_erf, notmask_erf;

    // Calculate erf()
    xabs        = abs(x);
    mask_erf    = (xabs < one);
    notmask_erf = (one <= xabs);
    x2          = x * x;
    x4          = x2 * x2;

    PolyAP0  = fma(CAP4, x4, CAP2);
    PolyAP1  = fma(CAP3, x4, CAP1);
    PolyAP0  = fma(PolyAP0, x4, CAP0);
    PolyAP0  = fma(PolyAP1, x2, PolyAP0);
    PolyAQ1  = fma(CAQ5, x4, CAQ3);
    PolyAQ0  = fma(CAQ4, x4, CAQ2);
    PolyAQ1  = fma(PolyAQ1, x4, CAQ1);
    PolyAQ0  = fma(PolyAQ0, x4, one);
    PolyAQ0  = fma(PolyAQ1, x2, PolyAQ0);

    res_erf  = PolyAP0 * maskzInv(PolyAQ0, mask_erf);
    res_erf  = CAoffset + res_erf;
    res_erf  = x * res_erf;

    // Calculate erfc() in range [1,4.5]
    t       = xabs - one;
    t2      = t * t;

    PolyBP0  = fma(CBP6, t2, CBP4);
    PolyBP1  = fma(CBP5, t2, CBP3);
    PolyBP0  = fma(PolyBP0, t2, CBP2);
    PolyBP1  = fma(PolyBP1, t2, CBP1);
    PolyBP0  = fma(PolyBP0, t2, CBP0);
    PolyBP0  = fma(PolyBP1, t, PolyBP0);

    PolyBQ1 = fma(CBQ7, t2, CBQ5);
    PolyBQ0 = fma(CBQ6, t2, CBQ4);
    PolyBQ1 = fma(PolyBQ1, t2, CBQ3);
    PolyBQ0 = fma(PolyBQ0, t2, CBQ2);
    PolyBQ1 = fma(PolyBQ1, t2, CBQ1);
    PolyBQ0 = fma(PolyBQ0, t2, one);
    PolyBQ0 = fma(PolyBQ1, t, PolyBQ0);

    // The denominator polynomial can be zero outside the range
    res_erfcB = PolyBP0 * maskzInv(PolyBQ0, notmask_erf);

    res_erfcB = res_erfcB * xabs;

    // Calculate erfc() in range [4.5,inf]
    w       = maskzInv(xabs, xabs != setZero());
    w2      = w * w;

    PolyCP0  = fma(CCP6, w2, CCP4);
    PolyCP1  = fma(CCP5, w2, CCP3);
    PolyCP0  = fma(PolyCP0, w2, CCP2);
    PolyCP1  = fma(PolyCP1, w2, CCP1);
    PolyCP0  = fma(PolyCP0, w2, CCP0);
    PolyCP0  = fma(PolyCP1, w, PolyCP0);

    PolyCQ0  = fma(CCQ6, w2, CCQ4);
    PolyCQ1  = fma(CCQ5, w2, CCQ3);
    PolyCQ0  = fma(PolyCQ0, w2, CCQ2);
    PolyCQ1  = fma(PolyCQ1, w2, CCQ1);
    PolyCQ0  = fma(PolyCQ0, w2, one);
    PolyCQ0  = fma(PolyCQ1, w, PolyCQ0);

    expmx2   = exp( -x2 );

    // The denominator polynomial can be zero outside the range
    res_erfcC = PolyCP0 * maskzInv(PolyCQ0, notmask_erf);
    res_erfcC = res_erfcC + CCoffset;
    res_erfcC = res_erfcC * w;

    mask     = (SimdDouble(4.5) < xabs);
    res_erfc = blend(res_erfcB, res_erfcC, mask);

    res_erfc = res_erfc * expmx2;

    // erfc(x<0) = 2-erfc(|x|)
    mask     = (x < setZero());
    res_erfc = blend(res_erfc, two - res_erfc, mask);

    // Select erf() or erfc()
    res  = blend(res_erfc, one - res_erf, mask_erf);

    return res;
}

/*! \brief SIMD double sin \& cos.
 *
 * \param x The argument to evaluate sin/cos for
 * \param[out] sinval Sin(x)
 * \param[out] cosval Cos(x)
 *
 * This version achieves close to machine precision, but for very large
 * magnitudes of the argument we inherently begin to lose accuracy due to the
 * argument reduction, despite using extended precision arithmetics internally.
 */
static inline void gmx_simdcall
sincos(SimdDouble x, SimdDouble *sinval, SimdDouble *cosval)
{
    // Constants to subtract Pi/4*x from y while minimizing precision loss
    const SimdDouble  argred0(-2*0.78539816290140151978);
    const SimdDouble  argred1(-2*4.9604678871439933374e-10);
    const SimdDouble  argred2(-2*1.1258708853173288931e-18);
    const SimdDouble  argred3(-2*1.7607799325916000908e-27);
    const SimdDouble  two_over_pi(2.0/M_PI);
    const SimdDouble  const_sin5( 1.58938307283228937328511e-10);
    const SimdDouble  const_sin4(-2.50506943502539773349318e-08);
    const SimdDouble  const_sin3( 2.75573131776846360512547e-06);
    const SimdDouble  const_sin2(-0.000198412698278911770864914);
    const SimdDouble  const_sin1( 0.0083333333333191845961746);
    const SimdDouble  const_sin0(-0.166666666666666130709393);

    const SimdDouble  const_cos7(-1.13615350239097429531523e-11);
    const SimdDouble  const_cos6( 2.08757471207040055479366e-09);
    const SimdDouble  const_cos5(-2.75573144028847567498567e-07);
    const SimdDouble  const_cos4( 2.48015872890001867311915e-05);
    const SimdDouble  const_cos3(-0.00138888888888714019282329);
    const SimdDouble  const_cos2( 0.0416666666666665519592062);
    const SimdDouble  half(0.5);
    const SimdDouble  one(1.0);
    SimdDouble        ssign, csign;
    SimdDouble        x2, y, z, psin, pcos, sss, ccc;
    SimdDBool         mask;
#if GMX_SIMD_HAVE_DINT32_ARITHMETICS && GMX_SIMD_HAVE_LOGICAL
    const SimdDInt32  ione(1);
    const SimdDInt32  itwo(2);
    SimdDInt32        iy;

    z       = x * two_over_pi;
    iy      = cvtR2I(z);
    y       = round(z);

    mask    = cvtIB2B((iy & ione) == setZero());
    ssign   = selectByMask(SimdDouble(GMX_DOUBLE_NEGZERO), cvtIB2B((iy & itwo) == itwo));
    csign   = selectByMask(SimdDouble(GMX_DOUBLE_NEGZERO), cvtIB2B(((iy + ione) & itwo) == itwo));
#else
    const SimdDouble  quarter(0.25);
    const SimdDouble  minusquarter(-0.25);
    SimdDouble        q;
    SimdDBool         m1, m2, m3;

    /* The most obvious way to find the arguments quadrant in the unit circle
     * to calculate the sign is to use integer arithmetic, but that is not
     * present in all SIMD implementations. As an alternative, we have devised a
     * pure floating-point algorithm that uses truncation for argument reduction
     * so that we get a new value 0<=q<1 over the unit circle, and then
     * do floating-point comparisons with fractions. This is likely to be
     * slightly slower (~10%) due to the longer latencies of floating-point, so
     * we only use it when integer SIMD arithmetic is not present.
     */
    ssign   = x;
    x       = abs(x);
    // It is critical that half-way cases are rounded down
    z       = fma(x, two_over_pi, half);
    y       = trunc(z);
    q       = z * quarter;
    q       = q - trunc(q);
    /* z now starts at 0.0 for x=-pi/4 (although neg. values cannot occur), and
     * then increased by 1.0 as x increases by 2*Pi, when it resets to 0.0.
     * This removes the 2*Pi periodicity without using any integer arithmetic.
     * First check if y had the value 2 or 3, set csign if true.
     */
    q       = q - half;
    /* If we have logical operations we can work directly on the signbit, which
     * saves instructions. Otherwise we need to represent signs as +1.0/-1.0.
     * Thus, if you are altering defines to debug alternative code paths, the
     * two GMX_SIMD_HAVE_LOGICAL sections in this routine must either both be
     * active or inactive - you will get errors if only one is used.
     */
#    if GMX_SIMD_HAVE_LOGICAL
    ssign   = ssign & SimdDouble(GMX_DOUBLE_NEGZERO);
    csign   = andNot(q, SimdDouble(GMX_DOUBLE_NEGZERO));
    ssign   = ssign ^ csign;
#    else
    ssign   = copysign(SimdDouble(1.0), ssign);
    csign   = copysign(SimdDouble(1.0), q);
    csign   = -csign;
    ssign   = ssign * csign;    // swap ssign if csign was set.
#    endif
    // Check if y had value 1 or 3 (remember we subtracted 0.5 from q)
    m1      = (q < minusquarter);
    m2      = (setZero() <= q);
    m3      = (q < quarter);
    m2      = m2 && m3;
    mask    = m1 || m2;
    // where mask is FALSE, swap sign.
    csign   = csign * blend(SimdDouble(-1.0), one, mask);
#endif
    x       = fma(y, argred0, x);
    x       = fma(y, argred1, x);
    x       = fma(y, argred2, x);
    x       = fma(y, argred3, x);
    x2      = x * x;

    psin    = fma(const_sin5, x2, const_sin4);
    psin    = fma(psin, x2, const_sin3);
    psin    = fma(psin, x2, const_sin2);
    psin    = fma(psin, x2, const_sin1);
    psin    = fma(psin, x2, const_sin0);
    psin    = fma(psin, x2 * x, x);

    pcos    = fma(const_cos7, x2, const_cos6);
    pcos    = fma(pcos, x2, const_cos5);
    pcos    = fma(pcos, x2, const_cos4);
    pcos    = fma(pcos, x2, const_cos3);
    pcos    = fma(pcos, x2, const_cos2);
    pcos    = fms(pcos, x2, half);
    pcos    = fma(pcos, x2, one);

    sss     = blend(pcos, psin, mask);
    ccc     = blend(psin, pcos, mask);
    // See comment for GMX_SIMD_HAVE_LOGICAL section above.
#if GMX_SIMD_HAVE_LOGICAL
    *sinval = sss ^ ssign;
    *cosval = ccc ^ csign;
#else
    *sinval = sss * ssign;
    *cosval = ccc * csign;
#endif
}

/*! \brief SIMD double sin(x).
 *
 * \param x The argument to evaluate sin for
 * \result Sin(x)
 *
 * \attention Do NOT call both sin & cos if you need both results, since each of them
 * will then call \ref sincos and waste a factor 2 in performance.
 */
static inline SimdDouble gmx_simdcall
sin(SimdDouble x)
{
    SimdDouble s, c;
    sincos(x, &s, &c);
    return s;
}

/*! \brief SIMD double cos(x).
 *
 * \param x The argument to evaluate cos for
 * \result Cos(x)
 *
 * \attention Do NOT call both sin & cos if you need both results, since each of them
 * will then call \ref sincos and waste a factor 2 in performance.
 */
static inline SimdDouble gmx_simdcall
cos(SimdDouble x)
{
    SimdDouble s, c;
    sincos(x, &s, &c);
    return c;
}

/*! \brief SIMD double tan(x).
 *
 * \param x The argument to evaluate tan for
 * \result Tan(x)
 */
static inline SimdDouble gmx_simdcall
tan(SimdDouble x)
{
    const SimdDouble  argred0(-2*0.78539816290140151978);
    const SimdDouble  argred1(-2*4.9604678871439933374e-10);
    const SimdDouble  argred2(-2*1.1258708853173288931e-18);
    const SimdDouble  argred3(-2*1.7607799325916000908e-27);
    const SimdDouble  two_over_pi(2.0/M_PI);
    const SimdDouble  CT15(1.01419718511083373224408e-05);
    const SimdDouble  CT14(-2.59519791585924697698614e-05);
    const SimdDouble  CT13(5.23388081915899855325186e-05);
    const SimdDouble  CT12(-3.05033014433946488225616e-05);
    const SimdDouble  CT11(7.14707504084242744267497e-05);
    const SimdDouble  CT10(8.09674518280159187045078e-05);
    const SimdDouble  CT9(0.000244884931879331847054404);
    const SimdDouble  CT8(0.000588505168743587154904506);
    const SimdDouble  CT7(0.00145612788922812427978848);
    const SimdDouble  CT6(0.00359208743836906619142924);
    const SimdDouble  CT5(0.00886323944362401618113356);
    const SimdDouble  CT4(0.0218694882853846389592078);
    const SimdDouble  CT3(0.0539682539781298417636002);
    const SimdDouble  CT2(0.133333333333125941821962);
    const SimdDouble  CT1(0.333333333333334980164153);

    SimdDouble        x2, p, y, z;
    SimdDBool         m;

#if GMX_SIMD_HAVE_DINT32_ARITHMETICS && GMX_SIMD_HAVE_LOGICAL
    SimdDInt32  iy;
    SimdDInt32  ione(1);

    z       = x * two_over_pi;
    iy      = cvtR2I(z);
    y       = round(z);
    m       = cvtIB2B((iy & ione) == ione);

    x       = fma(y, argred0, x);
    x       = fma(y, argred1, x);
    x       = fma(y, argred2, x);
    x       = fma(y, argred3, x);
    x       = selectByMask(SimdDouble(GMX_DOUBLE_NEGZERO), m) ^ x;
#else
    const SimdDouble  quarter(0.25);
    const SimdDouble  half(0.5);
    const SimdDouble  threequarter(0.75);
    SimdDouble        w, q;
    SimdDBool         m1, m2, m3;

    w       = abs(x);
    z       = fma(w, two_over_pi, half);
    y       = trunc(z);
    q       = z * quarter;
    q       = q - trunc(q);
    m1      = (quarter <= q);
    m2      = (q < half);
    m3      = (threequarter <= q);
    m1      = m1 && m2;
    m       = m1 || m3;
    w       = fma(y, argred0, w);
    w       = fma(y, argred1, w);
    w       = fma(y, argred2, w);
    w       = fma(y, argred3, w);

    w       = blend(w, -w, m);
    x       = w * copysign( SimdDouble(1.0), x );
#endif
    x2      = x * x;
    p       = fma(CT15, x2, CT14);
    p       = fma(p, x2, CT13);
    p       = fma(p, x2, CT12);
    p       = fma(p, x2, CT11);
    p       = fma(p, x2, CT10);
    p       = fma(p, x2, CT9);
    p       = fma(p, x2, CT8);
    p       = fma(p, x2, CT7);
    p       = fma(p, x2, CT6);
    p       = fma(p, x2, CT5);
    p       = fma(p, x2, CT4);
    p       = fma(p, x2, CT3);
    p       = fma(p, x2, CT2);
    p       = fma(p, x2, CT1);
    p       = fma(x2, p * x, x);

    p       = blend( p, maskzInv(p, m), m);
    return p;
}

/*! \brief SIMD double asin(x).
 *
 * \param x The argument to evaluate asin for
 * \result Asin(x)
 */
static inline SimdDouble gmx_simdcall
asin(SimdDouble x)
{
    // Same algorithm as cephes library
    const SimdDouble limit1(0.625);
    const SimdDouble limit2(1e-8);
    const SimdDouble one(1.0);
    const SimdDouble quarterpi(M_PI/4.0);
    const SimdDouble morebits(6.123233995736765886130e-17);

    const SimdDouble P5(4.253011369004428248960e-3);
    const SimdDouble P4(-6.019598008014123785661e-1);
    const SimdDouble P3(5.444622390564711410273e0);
    const SimdDouble P2(-1.626247967210700244449e1);
    const SimdDouble P1(1.956261983317594739197e1);
    const SimdDouble P0(-8.198089802484824371615e0);

    const SimdDouble Q4(-1.474091372988853791896e1);
    const SimdDouble Q3(7.049610280856842141659e1);
    const SimdDouble Q2(-1.471791292232726029859e2);
    const SimdDouble Q1(1.395105614657485689735e2);
    const SimdDouble Q0(-4.918853881490881290097e1);

    const SimdDouble R4(2.967721961301243206100e-3);
    const SimdDouble R3(-5.634242780008963776856e-1);
    const SimdDouble R2(6.968710824104713396794e0);
    const SimdDouble R1(-2.556901049652824852289e1);
    const SimdDouble R0(2.853665548261061424989e1);

    const SimdDouble S3(-2.194779531642920639778e1);
    const SimdDouble S2(1.470656354026814941758e2);
    const SimdDouble S1(-3.838770957603691357202e2);
    const SimdDouble S0(3.424398657913078477438e2);

    SimdDouble       xabs;
    SimdDouble       zz, ww, z, q, w, zz2, ww2;
    SimdDouble       PA, PB;
    SimdDouble       QA, QB;
    SimdDouble       RA, RB;
    SimdDouble       SA, SB;
    SimdDouble       nom, denom;
    SimdDBool        mask, mask2;

    xabs  = abs(x);

    mask  = (limit1 < xabs);

    zz    = one - xabs;
    ww    = xabs * xabs;
    zz2   = zz * zz;
    ww2   = ww * ww;

    // R
    RA    = fma(R4, zz2, R2);
    RB    = fma(R3, zz2, R1);
    RA    = fma(RA, zz2, R0);
    RA    = fma(RB, zz, RA);

    // S, SA = zz2
    SB    = fma(S3, zz2, S1);
    SA    = zz2 +  S2;
    SA    = fma(SA, zz2, S0);
    SA    = fma(SB, zz, SA);

    // P
    PA    = fma(P5, ww2, P3);
    PB    = fma(P4, ww2, P2);
    PA    = fma(PA, ww2, P1);
    PB    = fma(PB, ww2, P0);
    PA    = fma(PA, ww, PB);

    // Q, QA = ww2
    QB    = fma(Q4, ww2, Q2);
    QA    = ww2 + Q3;
    QA    = fma(QA, ww2, Q1);
    QB    = fma(QB, ww2, Q0);
    QA    = fma(QA, ww, QB);

    RA    = RA * zz;
    PA    = PA * ww;

    nom   = blend( PA, RA, mask );
    denom = blend( QA, SA, mask );

    mask2 = (limit2 < xabs);
    q     = nom * maskzInv(denom, mask2);

    zz    = zz + zz;
    zz    = sqrt(zz);
    z     = quarterpi - zz;
    zz    = fms(zz, q, morebits);
    z     = z - zz;
    z     = z + quarterpi;

    w     = xabs * q;
    w     = w + xabs;

    z     = blend( w, z, mask );

    z     = blend( xabs, z, mask2 );

    z = copysign(z, x);

    return z;
}

/*! \brief SIMD double acos(x).
 *
 * \param x The argument to evaluate acos for
 * \result Acos(x)
 */
static inline SimdDouble gmx_simdcall
acos(SimdDouble x)
{
    const SimdDouble one(1.0);
    const SimdDouble half(0.5);
    const SimdDouble quarterpi0(7.85398163397448309616e-1);
    const SimdDouble quarterpi1(6.123233995736765886130e-17);

    SimdDBool        mask1;
    SimdDouble       z, z1, z2;

    mask1 = (half < x);
    z1    = half * (one - x);
    z1    = sqrt(z1);
    z     = blend( x, z1, mask1 );

    z     = asin(z);

    z1    = z + z;

    z2    = quarterpi0 - z;
    z2    = z2 + quarterpi1;
    z2    = z2 + quarterpi0;

    z     = blend(z2, z1, mask1);

    return z;
}

/*! \brief SIMD double asin(x).
 *
 * \param x The argument to evaluate atan for
 * \result Atan(x), same argument/value range as standard math library.
 */
static inline SimdDouble gmx_simdcall
atan(SimdDouble x)
{
    // Same algorithm as cephes library
    const SimdDouble limit1(0.66);
    const SimdDouble limit2(2.41421356237309504880);
    const SimdDouble quarterpi(M_PI/4.0);
    const SimdDouble halfpi(M_PI/2.0);
    const SimdDouble mone(-1.0);
    const SimdDouble morebits1(0.5*6.123233995736765886130E-17);
    const SimdDouble morebits2(6.123233995736765886130E-17);

    const SimdDouble P4(-8.750608600031904122785E-1);
    const SimdDouble P3(-1.615753718733365076637E1);
    const SimdDouble P2(-7.500855792314704667340E1);
    const SimdDouble P1(-1.228866684490136173410E2);
    const SimdDouble P0(-6.485021904942025371773E1);

    const SimdDouble Q4(2.485846490142306297962E1);
    const SimdDouble Q3(1.650270098316988542046E2);
    const SimdDouble Q2(4.328810604912902668951E2);
    const SimdDouble Q1(4.853903996359136964868E2);
    const SimdDouble Q0(1.945506571482613964425E2);

    SimdDouble       y, xabs, t1, t2;
    SimdDouble       z, z2;
    SimdDouble       P_A, P_B, Q_A, Q_B;
    SimdDBool        mask1, mask2;

    xabs   = abs(x);

    mask1  = (limit1 < xabs);
    mask2  = (limit2 < xabs);

    t1     = (xabs + mone) * maskzInv(xabs - mone, mask1);
    t2     = mone * maskzInv(xabs, mask2);

    y      = selectByMask(quarterpi, mask1);
    y      = blend(y, halfpi, mask2);
    xabs   = blend(xabs, t1, mask1);
    xabs   = blend(xabs, t2, mask2);

    z      = xabs * xabs;
    z2     = z * z;

    P_A    = fma(P4, z2, P2);
    P_B    = fma(P3, z2, P1);
    P_A    = fma(P_A, z2, P0);
    P_A    = fma(P_B, z, P_A);

    // Q_A = z2
    Q_B    = fma(Q4, z2, Q2);
    Q_A    = z2 + Q3;
    Q_A    = fma(Q_A, z2, Q1);
    Q_B    = fma(Q_B, z2, Q0);
    Q_A    = fma(Q_A, z, Q_B);

    z      = z * P_A;
    z      = z * inv(Q_A);
    z      = fma(z, xabs, xabs);

    t1     = selectByMask(morebits1, mask1);
    t1     = blend(t1, morebits2, mask2);

    z      = z + t1;
    y      = y + z;

    y      = copysign(y, x);

    return y;
}

/*! \brief SIMD double atan2(y,x).
 *
 * \param y Y component of vector, any quartile
 * \param x X component of vector, any quartile
 * \result Atan(y,x), same argument/value range as standard math library.
 *
 * \note This routine should provide correct results for all finite
 * non-zero or positive-zero arguments. However, negative zero arguments will
 * be treated as positive zero, which means the return value will deviate from
 * the standard math library atan2(y,x) for those cases. That should not be
 * of any concern in Gromacs, and in particular it will not affect calculations
 * of angles from vectors.
 */
static inline SimdDouble gmx_simdcall
atan2(SimdDouble y, SimdDouble x)
{
    const SimdDouble pi(M_PI);
    const SimdDouble halfpi(M_PI/2.0);
    SimdDouble       xinv, p, aoffset;
    SimdDBool        mask_xnz, mask_ynz, mask_xlt0, mask_ylt0;

    mask_xnz  = x != setZero();
    mask_ynz  = y != setZero();
    mask_xlt0 = (x < setZero());
    mask_ylt0 = (y < setZero());

    aoffset   = selectByNotMask(halfpi, mask_xnz);
    aoffset   = selectByMask(aoffset, mask_ynz);

    aoffset   = blend(aoffset, pi, mask_xlt0);
    aoffset   = blend(aoffset, -aoffset, mask_ylt0);

    xinv      = maskzInv(x, mask_xnz);
    p         = y * xinv;
    p         = atan(p);
    p         = p + aoffset;

    return p;
}


/*! \brief Calculate the force correction due to PME analytically in SIMD double.
 *
 * \param z2 This should be the value \f$(r \beta)^2\f$, where r is your
 *           interaction distance and beta the ewald splitting parameters.
 * \result Correction factor to coulomb force.
 *
 * This routine is meant to enable analytical evaluation of the
 * direct-space PME electrostatic force to avoid tables. For details, see the
 * single precision function.
 */
static inline SimdDouble gmx_simdcall
pmeForceCorrection(SimdDouble z2)
{
    const SimdDouble  FN10(-8.0072854618360083154e-14);
    const SimdDouble  FN9(1.1859116242260148027e-11);
    const SimdDouble  FN8(-8.1490406329798423616e-10);
    const SimdDouble  FN7(3.4404793543907847655e-8);
    const SimdDouble  FN6(-9.9471420832602741006e-7);
    const SimdDouble  FN5(0.000020740315999115847456);
    const SimdDouble  FN4(-0.00031991745139313364005);
    const SimdDouble  FN3(0.0035074449373659008203);
    const SimdDouble  FN2(-0.031750380176100813405);
    const SimdDouble  FN1(0.13884101728898463426);
    const SimdDouble  FN0(-0.75225277815249618847);

    const SimdDouble  FD5(0.000016009278224355026701);
    const SimdDouble  FD4(0.00051055686934806966046);
    const SimdDouble  FD3(0.0081803507497974289008);
    const SimdDouble  FD2(0.077181146026670287235);
    const SimdDouble  FD1(0.41543303143712535988);
    const SimdDouble  FD0(1.0);

    SimdDouble        z4;
    SimdDouble        polyFN0, polyFN1, polyFD0, polyFD1;

    z4             = z2 * z2;

    polyFD1        = fma(FD5, z4, FD3);
    polyFD1        = fma(polyFD1, z4, FD1);
    polyFD1        = polyFD1 * z2;
    polyFD0        = fma(FD4, z4, FD2);
    polyFD0        = fma(polyFD0, z4, FD0);
    polyFD0        = polyFD0 + polyFD1;

    polyFD0        = inv(polyFD0);

    polyFN0        = fma(FN10, z4, FN8);
    polyFN0        = fma(polyFN0, z4, FN6);
    polyFN0        = fma(polyFN0, z4, FN4);
    polyFN0        = fma(polyFN0, z4, FN2);
    polyFN0        = fma(polyFN0, z4, FN0);
    polyFN1        = fma(FN9, z4, FN7);
    polyFN1        = fma(polyFN1, z4, FN5);
    polyFN1        = fma(polyFN1, z4, FN3);
    polyFN1        = fma(polyFN1, z4, FN1);
    polyFN0        = fma(polyFN1, z2, polyFN0);


    return polyFN0 * polyFD0;
}



/*! \brief Calculate the potential correction due to PME analytically in SIMD double.
 *
 * \param z2 This should be the value \f$(r \beta)^2\f$, where r is your
 *           interaction distance and beta the ewald splitting parameters.
 * \result Correction factor to coulomb force.
 *
 * This routine is meant to enable analytical evaluation of the
 * direct-space PME electrostatic potential to avoid tables. For details, see the
 * single precision function.
 */
static inline SimdDouble gmx_simdcall
pmePotentialCorrection(SimdDouble z2)
{
    const SimdDouble  VN9(-9.3723776169321855475e-13);
    const SimdDouble  VN8(1.2280156762674215741e-10);
    const SimdDouble  VN7(-7.3562157912251309487e-9);
    const SimdDouble  VN6(2.6215886208032517509e-7);
    const SimdDouble  VN5(-4.9532491651265819499e-6);
    const SimdDouble  VN4(0.00025907400778966060389);
    const SimdDouble  VN3(0.0010585044856156469792);
    const SimdDouble  VN2(0.045247661136833092885);
    const SimdDouble  VN1(0.11643931522926034421);
    const SimdDouble  VN0(1.1283791671726767970);

    const SimdDouble  VD5(0.000021784709867336150342);
    const SimdDouble  VD4(0.00064293662010911388448);
    const SimdDouble  VD3(0.0096311444822588683504);
    const SimdDouble  VD2(0.085608012351550627051);
    const SimdDouble  VD1(0.43652499166614811084);
    const SimdDouble  VD0(1.0);

    SimdDouble        z4;
    SimdDouble        polyVN0, polyVN1, polyVD0, polyVD1;

    z4             = z2 * z2;

    polyVD1        = fma(VD5, z4, VD3);
    polyVD0        = fma(VD4, z4, VD2);
    polyVD1        = fma(polyVD1, z4, VD1);
    polyVD0        = fma(polyVD0, z4, VD0);
    polyVD0        = fma(polyVD1, z2, polyVD0);

    polyVD0        = inv(polyVD0);

    polyVN1        = fma(VN9, z4, VN7);
    polyVN0        = fma(VN8, z4, VN6);
    polyVN1        = fma(polyVN1, z4, VN5);
    polyVN0        = fma(polyVN0, z4, VN4);
    polyVN1        = fma(polyVN1, z4, VN3);
    polyVN0        = fma(polyVN0, z4, VN2);
    polyVN1        = fma(polyVN1, z4, VN1);
    polyVN0        = fma(polyVN0, z4, VN0);
    polyVN0        = fma(polyVN1, z2, polyVN0);

    return polyVN0 * polyVD0;
}

/*! \} */


/*! \name SIMD math functions for double prec. data, single prec. accuracy
 *
 *  \note In some cases we do not need full double accuracy of individual
 *        SIMD math functions, although the data is stored in double precision
 *        SIMD registers. This might be the case for special algorithms, or
 *        if the architecture does not support single precision.
 *        Since the full double precision evaluation of math functions
 *        typically require much more expensive polynomial approximations
 *        these functions implement the algorithms used in the single precision
 *        SIMD math functions, but they operate on double precision
 *        SIMD variables.
 *
 *  \{
 */

/*********************************************************************
 * SIMD MATH FUNCTIONS WITH DOUBLE PREC. DATA, SINGLE PREC. ACCURACY *
 *********************************************************************/

/*! \brief Calculate 1/sqrt(x) for SIMD double, but in single accuracy.
 *
 *  \param x Argument that must be >0. This routine does not check arguments.
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid.
 */
static inline SimdDouble gmx_simdcall
invsqrtSingleAccuracy(SimdDouble x)
{
    SimdDouble lu = rsqrt(x);
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
    return lu;
}

/*! \brief 1/sqrt(x) for masked-in entries of SIMD double, but in single accuracy.
 *
 *  This routine only evaluates 1/sqrt(x) for elements for which mask is true.
 *  Illegal values in the masked-out elements will not lead to
 *  floating-point exceptions.
 *
 *  \param x Argument that must be >0 for masked-in entries
 *  \param m Mask
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid or
 *          entry was not masked, and 0.0 for masked-out entries.
 */
static inline SimdDouble
maskzInvsqrtSingleAccuracy(SimdDouble x, SimdDBool m)
{
    SimdDouble lu = maskzRsqrt(x, m);
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
    return lu;
}

/*! \brief Calculate 1/sqrt(x) for two SIMD doubles, but single accuracy.
 *
 * \param x0  First set of arguments, x0 must be positive - no argument checking.
 * \param x1  Second set of arguments, x1 must be positive - no argument checking.
 * \param[out] out0  Result 1/sqrt(x0)
 * \param[out] out1  Result 1/sqrt(x1)
 *
 *  In particular for double precision we can sometimes calculate square root
 *  pairs slightly faster by using single precision until the very last step.
 */
static inline void gmx_simdcall
invsqrtPairSingleAccuracy(SimdDouble x0,    SimdDouble x1,
                          SimdDouble *out0, SimdDouble *out1)
{
#if GMX_SIMD_HAVE_FLOAT && (GMX_SIMD_FLOAT_WIDTH == 2*GMX_SIMD_DOUBLE_WIDTH) && (GMX_SIMD_RSQRT_BITS < 22)
    SimdFloat  xf  = cvtDD2F(x0, x1);
    SimdFloat  luf = rsqrt(xf);
    SimdDouble lu0, lu1;
    // Intermediate target is single - mantissa+1 bits
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    luf = rsqrtIter(luf, xf);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    luf = rsqrtIter(luf, xf);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    luf = rsqrtIter(luf, xf);
#endif
    cvtF2DD(luf, &lu0, &lu1);
    // We now have single-precision accuracy values in lu0/lu1
    *out0 = lu0;
    *out1 = lu1;
#else
    *out0 = invsqrtSingleAccuracy(x0);
    *out1 = invsqrtSingleAccuracy(x1);
#endif
}

/*! \brief Calculate 1/x for SIMD double, but in single accuracy.
 *
 *  \param x Argument that must be nonzero. This routine does not check arguments.
 *  \return 1/x. Result is undefined if your argument was invalid.
 */
static inline SimdDouble gmx_simdcall
invSingleAccuracy(SimdDouble x)
{
    SimdDouble lu = rcp(x);
#if (GMX_SIMD_RCP_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
    return lu;
}

/*! \brief 1/x for masked entries of SIMD double, single accuracy.
 *
 *  \param x Argument that must be nonzero for non-masked entries.
 *  \param m Mask
 *  \return 1/x for elements where m is true, or 0.0 for masked-out entries.
 */
static inline SimdDouble gmx_simdcall
maskzInvSingleAccuracy(SimdDouble x, SimdDBool m)
{
    SimdDouble lu = maskzRcp(x, m);
#if (GMX_SIMD_RCP_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
#if (GMX_SIMD_RCP_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rcpIter(lu, x);
#endif
    return lu;
}


/*! \brief Calculate sqrt(x) (correct for 0.0) for SIMD double, single accuracy.
 *
 *  \param x Argument that must be >=0.
 *  \return sqrt(x). If x<float_min, the result will correctly be set to 0.
 *          The result is undefined if the input value is negative.
 */
static inline SimdDouble gmx_simdcall
sqrtSingleAccuracy(SimdDouble x)
{
    return x * maskzInvsqrtSingleAccuracy(x, SimdDouble(GMX_FLOAT_MIN) <= x);
}

/*! \brief SIMD log(x). Double precision SIMD data, single accuracy.
 *
 * \param x Argument, should be >0.
 * \result The natural logarithm of x. Undefined if argument is invalid.
 */
static inline SimdDouble gmx_simdcall
logSingleAccuracy(SimdDouble x)
{
    const SimdDouble  one(1.0);
    const SimdDouble  two(2.0);
    const SimdDouble  sqrt2(std::sqrt(2.0));
    const SimdDouble  corr(0.693147180559945286226764);
    const SimdDouble  CL9(0.2371599674224853515625);
    const SimdDouble  CL7(0.285279005765914916992188);
    const SimdDouble  CL5(0.400005519390106201171875);
    const SimdDouble  CL3(0.666666567325592041015625);
    const SimdDouble  CL1(2.0);
    SimdDouble        fexp, x2, p;
    SimdDInt32        iexp;
    SimdDBool         mask;

    x     = frexp(x, &iexp);
    fexp  = cvtI2R(iexp);

    mask  = (x < sqrt2);
    // Adjust to non-IEEE format for x<sqrt(2): exponent -= 1, mantissa *= 2.0
    fexp  = fexp - selectByMask(one, mask);
    x     = x * blend(one, two, mask);

    x     = (x - one) * invSingleAccuracy( x + one );
    x2    = x * x;

    p     = fma(CL9, x2, CL7);
    p     = fma(p, x2, CL5);
    p     = fma(p, x2, CL3);
    p     = fma(p, x2, CL1);
    p     = fma(p, x, corr * fexp);

    return p;
}

/*! \brief SIMD 2^x. Double precision SIMD data, single accuracy.
 *
 * \param x Argument.
 * \result 2^x. Undefined if input argument caused overflow.
 */
static inline SimdDouble gmx_simdcall
exp2SingleAccuracy(SimdDouble x)
{
    // Lower bound: Disallow numbers that would lead to an IEEE fp exponent reaching +-127.
    const SimdDouble  arglimit = SimdDouble(126.0);
    const SimdDouble  CC6(0.0001534581200287996416911311);
    const SimdDouble  CC5(0.001339993121934088894618990);
    const SimdDouble  CC4(0.009618488957115180159497841);
    const SimdDouble  CC3(0.05550328776964726865751735);
    const SimdDouble  CC2(0.2402264689063408646490722);
    const SimdDouble  CC1(0.6931472057372680777553816);
    const SimdDouble  one(1.0);

    SimdDouble        intpart;
    SimdDouble        p;
    SimdDBool         valuemask;
    SimdDInt32        ix;

    ix        = cvtR2I(x);
    intpart   = round(x);
    valuemask = (abs(x) <= arglimit);
    x         = x - intpart;

    p         = fma(CC6, x, CC5);
    p         = fma(p, x, CC4);
    p         = fma(p, x, CC3);
    p         = fma(p, x, CC2);
    p         = fma(p, x, CC1);
    p         = fma(p, x, one);
    x         = ldexp(p, ix);
    x         = selectByMask(x, valuemask);

    return x;
}

/*! \brief SIMD exp(x). Double precision SIMD data, single accuracy.
 *
 * \param x Argument.
 * \result exp(x). Undefined if input argument caused overflow.
 */
static inline SimdDouble gmx_simdcall
expSingleAccuracy(SimdDouble x)
{
    const SimdDouble  argscale(1.44269504088896341);
    // Lower bound: Disallow numbers that would lead to an IEEE fp exponent reaching +-127
    const SimdDouble  arglimit(126.0);
    const SimdDouble  invargscale(-0.69314718055994528623);
    const SimdDouble  CC4(0.00136324646882712841033936);
    const SimdDouble  CC3(0.00836596917361021041870117);
    const SimdDouble  CC2(0.0416710823774337768554688);
    const SimdDouble  CC1(0.166665524244308471679688);
    const SimdDouble  CC0(0.499999850988388061523438);
    const SimdDouble  one(1.0);
    SimdDouble        intpart;
    SimdDouble        y, p;
    SimdDBool         valuemask;
    SimdDInt32        iy;

    y         = x * argscale;
    iy        = cvtR2I(y);
    intpart   = round(y);        // use same rounding algorithm here
    valuemask = (abs(y) <= arglimit);

    // Extended precision arithmetics not needed since
    // we have double precision and only need single accuracy.
    x         = fma(invargscale, intpart, x);

    p         = fma(CC4, x, CC3);
    p         = fma(p, x, CC2);
    p         = fma(p, x, CC1);
    p         = fma(p, x, CC0);
    p         = fma(x*x, p, x);
    p         = p + one;
    x         = ldexp(p, iy);
    x         = selectByMask(x, valuemask);
    return x;
}

/*! \brief SIMD erf(x). Double precision SIMD data, single accuracy.
 *
 * \param x The value to calculate erf(x) for.
 * \result erf(x)
 *
 * This routine achieves very close to single precision, but we do not care about
 * the last bit or the subnormal result range.
 */
static inline SimdDouble gmx_simdcall
erfSingleAccuracy(SimdDouble x)
{
    // Coefficients for minimax approximation of erf(x)=x*P(x^2) in range [-1,1]
    const SimdDouble  CA6(7.853861353153693e-5);
    const SimdDouble  CA5(-8.010193625184903e-4);
    const SimdDouble  CA4(5.188327685732524e-3);
    const SimdDouble  CA3(-2.685381193529856e-2);
    const SimdDouble  CA2(1.128358514861418e-1);
    const SimdDouble  CA1(-3.761262582423300e-1);
    const SimdDouble  CA0(1.128379165726710);
    // Coefficients for minimax approximation of erfc(x)=Exp(-x^2)*P((1/(x-1))^2) in range [0.67,2]
    const SimdDouble  CB9(-0.0018629930017603923);
    const SimdDouble  CB8(0.003909821287598495);
    const SimdDouble  CB7(-0.0052094582210355615);
    const SimdDouble  CB6(0.005685614362160572);
    const SimdDouble  CB5(-0.0025367682853477272);
    const SimdDouble  CB4(-0.010199799682318782);
    const SimdDouble  CB3(0.04369575504816542);
    const SimdDouble  CB2(-0.11884063474674492);
    const SimdDouble  CB1(0.2732120154030589);
    const SimdDouble  CB0(0.42758357702025784);
    // Coefficients for minimax approximation of erfc(x)=Exp(-x^2)*(1/x)*P((1/x)^2) in range [2,9.19]
    const SimdDouble  CC10(-0.0445555913112064);
    const SimdDouble  CC9(0.21376355144663348);
    const SimdDouble  CC8(-0.3473187200259257);
    const SimdDouble  CC7(0.016690861551248114);
    const SimdDouble  CC6(0.7560973182491192);
    const SimdDouble  CC5(-1.2137903600145787);
    const SimdDouble  CC4(0.8411872321232948);
    const SimdDouble  CC3(-0.08670413896296343);
    const SimdDouble  CC2(-0.27124782687240334);
    const SimdDouble  CC1(-0.0007502488047806069);
    const SimdDouble  CC0(0.5642114853803148);
    const SimdDouble  one(1.0);
    const SimdDouble  two(2.0);

    SimdDouble        x2, x4, y;
    SimdDouble        t, t2, w, w2;
    SimdDouble        pA0, pA1, pB0, pB1, pC0, pC1;
    SimdDouble        expmx2;
    SimdDouble        res_erf, res_erfc, res;
    SimdDBool         mask, msk_erf;

    // Calculate erf()
    x2   = x * x;
    x4   = x2 * x2;

    pA0  = fma(CA6, x4, CA4);
    pA1  = fma(CA5, x4, CA3);
    pA0  = fma(pA0, x4, CA2);
    pA1  = fma(pA1, x4, CA1);
    pA0  = pA0 * x4;
    pA0  = fma(pA1, x2, pA0);
    // Constant term must come last for precision reasons
    pA0  = pA0 + CA0;

    res_erf = x * pA0;

    // Calculate erfc
    y       = abs(x);
    msk_erf = (SimdDouble(0.75) <= y);
    t       = maskzInvSingleAccuracy(y, msk_erf);
    w       = t - one;
    t2      = t * t;
    w2      = w * w;

    expmx2  = expSingleAccuracy( -y*y );

    pB1  = fma(CB9, w2, CB7);
    pB0  = fma(CB8, w2, CB6);
    pB1  = fma(pB1, w2, CB5);
    pB0  = fma(pB0, w2, CB4);
    pB1  = fma(pB1, w2, CB3);
    pB0  = fma(pB0, w2, CB2);
    pB1  = fma(pB1, w2, CB1);
    pB0  = fma(pB0, w2, CB0);
    pB0  = fma(pB1, w, pB0);

    pC0  = fma(CC10, t2, CC8);
    pC1  = fma(CC9, t2, CC7);
    pC0  = fma(pC0, t2, CC6);
    pC1  = fma(pC1, t2, CC5);
    pC0  = fma(pC0, t2, CC4);
    pC1  = fma(pC1, t2, CC3);
    pC0  = fma(pC0, t2, CC2);
    pC1  = fma(pC1, t2, CC1);

    pC0  = fma(pC0, t2, CC0);
    pC0  = fma(pC1, t, pC0);
    pC0  = pC0 * t;

    // Select pB0 or pC0 for erfc()
    mask     = (two < y);
    res_erfc = blend(pB0, pC0, mask);
    res_erfc = res_erfc * expmx2;

    // erfc(x<0) = 2-erfc(|x|)
    mask     = (x < setZero());
    res_erfc = blend(res_erfc, two - res_erfc, mask);

    // Select erf() or erfc()
    mask = (y < SimdDouble(0.75));
    res  = blend(one - res_erfc, res_erf, mask);

    return res;
}

/*! \brief SIMD erfc(x). Double precision SIMD data, single accuracy.
 *
 * \param x The value to calculate erfc(x) for.
 * \result erfc(x)
 *
 * This routine achieves singleprecision (bar the last bit) over most of the
 * input range, but for large arguments where the result is getting close
 * to the minimum representable numbers we accept slightly larger errors
 * (think results that are in the ballpark of 10^-30) since that is not
 * relevant for MD.
 */
static inline SimdDouble gmx_simdcall
erfcSingleAccuracy(SimdDouble x)
{
    // Coefficients for minimax approximation of erf(x)=x*P(x^2) in range [-1,1]
    const SimdDouble  CA6(7.853861353153693e-5);
    const SimdDouble  CA5(-8.010193625184903e-4);
    const SimdDouble  CA4(5.188327685732524e-3);
    const SimdDouble  CA3(-2.685381193529856e-2);
    const SimdDouble  CA2(1.128358514861418e-1);
    const SimdDouble  CA1(-3.761262582423300e-1);
    const SimdDouble  CA0(1.128379165726710);
    // Coefficients for minimax approximation of erfc(x)=Exp(-x^2)*P((1/(x-1))^2) in range [0.67,2]
    const SimdDouble  CB9(-0.0018629930017603923);
    const SimdDouble  CB8(0.003909821287598495);
    const SimdDouble  CB7(-0.0052094582210355615);
    const SimdDouble  CB6(0.005685614362160572);
    const SimdDouble  CB5(-0.0025367682853477272);
    const SimdDouble  CB4(-0.010199799682318782);
    const SimdDouble  CB3(0.04369575504816542);
    const SimdDouble  CB2(-0.11884063474674492);
    const SimdDouble  CB1(0.2732120154030589);
    const SimdDouble  CB0(0.42758357702025784);
    // Coefficients for minimax approximation of erfc(x)=Exp(-x^2)*(1/x)*P((1/x)^2) in range [2,9.19]
    const SimdDouble  CC10(-0.0445555913112064);
    const SimdDouble  CC9(0.21376355144663348);
    const SimdDouble  CC8(-0.3473187200259257);
    const SimdDouble  CC7(0.016690861551248114);
    const SimdDouble  CC6(0.7560973182491192);
    const SimdDouble  CC5(-1.2137903600145787);
    const SimdDouble  CC4(0.8411872321232948);
    const SimdDouble  CC3(-0.08670413896296343);
    const SimdDouble  CC2(-0.27124782687240334);
    const SimdDouble  CC1(-0.0007502488047806069);
    const SimdDouble  CC0(0.5642114853803148);
    const SimdDouble  one(1.0);
    const SimdDouble  two(2.0);

    SimdDouble        x2, x4, y;
    SimdDouble        t, t2, w, w2;
    SimdDouble        pA0, pA1, pB0, pB1, pC0, pC1;
    SimdDouble        expmx2;
    SimdDouble        res_erf, res_erfc, res;
    SimdDBool         mask, msk_erf;

    // Calculate erf()
    x2     = x * x;
    x4     = x2 * x2;

    pA0  = fma(CA6, x4, CA4);
    pA1  = fma(CA5, x4, CA3);
    pA0  = fma(pA0, x4, CA2);
    pA1  = fma(pA1, x4, CA1);
    pA1  = pA1 * x2;
    pA0  = fma(pA0, x4, pA1);
    // Constant term must come last for precision reasons
    pA0  = pA0 + CA0;

    res_erf = x * pA0;

    // Calculate erfc
    y       = abs(x);
    msk_erf = (SimdDouble(0.75) <= y);
    t       = maskzInvSingleAccuracy(y, msk_erf);
    w       = t - one;
    t2      = t * t;
    w2      = w * w;

    expmx2  = expSingleAccuracy( -y*y );

    pB1  = fma(CB9, w2, CB7);
    pB0  = fma(CB8, w2, CB6);
    pB1  = fma(pB1, w2, CB5);
    pB0  = fma(pB0, w2, CB4);
    pB1  = fma(pB1, w2, CB3);
    pB0  = fma(pB0, w2, CB2);
    pB1  = fma(pB1, w2, CB1);
    pB0  = fma(pB0, w2, CB0);
    pB0  = fma(pB1, w, pB0);

    pC0  = fma(CC10, t2, CC8);
    pC1  = fma(CC9, t2, CC7);
    pC0  = fma(pC0, t2, CC6);
    pC1  = fma(pC1, t2, CC5);
    pC0  = fma(pC0, t2, CC4);
    pC1  = fma(pC1, t2, CC3);
    pC0  = fma(pC0, t2, CC2);
    pC1  = fma(pC1, t2, CC1);

    pC0  = fma(pC0, t2, CC0);
    pC0  = fma(pC1, t, pC0);
    pC0  = pC0 * t;

    // Select pB0 or pC0 for erfc()
    mask     = (two < y);
    res_erfc = blend(pB0, pC0, mask);
    res_erfc = res_erfc * expmx2;

    // erfc(x<0) = 2-erfc(|x|)
    mask     = (x < setZero());
    res_erfc = blend(res_erfc, two - res_erfc, mask);

    // Select erf() or erfc()
    mask = (y < SimdDouble(0.75));
    res  = blend(res_erfc, one - res_erf, mask);

    return res;
}

/*! \brief SIMD sin \& cos. Double precision SIMD data, single accuracy.
 *
 * \param x The argument to evaluate sin/cos for
 * \param[out] sinval Sin(x)
 * \param[out] cosval Cos(x)
 */
static inline void gmx_simdcall
sinCosSingleAccuracy(SimdDouble x, SimdDouble *sinval, SimdDouble *cosval)
{
    // Constants to subtract Pi/4*x from y while minimizing precision loss
    const SimdDouble  argred0(2*0.78539816290140151978);
    const SimdDouble  argred1(2*4.9604678871439933374e-10);
    const SimdDouble  argred2(2*1.1258708853173288931e-18);
    const SimdDouble  two_over_pi(2.0/M_PI);
    const SimdDouble  const_sin2(-1.9515295891e-4);
    const SimdDouble  const_sin1( 8.3321608736e-3);
    const SimdDouble  const_sin0(-1.6666654611e-1);
    const SimdDouble  const_cos2( 2.443315711809948e-5);
    const SimdDouble  const_cos1(-1.388731625493765e-3);
    const SimdDouble  const_cos0( 4.166664568298827e-2);

    const SimdDouble  half(0.5);
    const SimdDouble  one(1.0);
    SimdDouble        ssign, csign;
    SimdDouble        x2, y, z, psin, pcos, sss, ccc;
    SimdDBool         mask;

#if GMX_SIMD_HAVE_DINT32_ARITHMETICS && GMX_SIMD_HAVE_LOGICAL
    const SimdDInt32  ione(1);
    const SimdDInt32  itwo(2);
    SimdDInt32        iy;

    z       = x * two_over_pi;
    iy      = cvtR2I(z);
    y       = round(z);

    mask    = cvtIB2B((iy & ione) == setZero());
    ssign   = selectByMask(SimdDouble(GMX_DOUBLE_NEGZERO), cvtIB2B((iy & itwo) == itwo));
    csign   = selectByMask(SimdDouble(GMX_DOUBLE_NEGZERO), cvtIB2B(((iy + ione) & itwo) == itwo));
#else
    const SimdDouble  quarter(0.25);
    const SimdDouble  minusquarter(-0.25);
    SimdDouble        q;
    SimdDBool         m1, m2, m3;

    /* The most obvious way to find the arguments quadrant in the unit circle
     * to calculate the sign is to use integer arithmetic, but that is not
     * present in all SIMD implementations. As an alternative, we have devised a
     * pure floating-point algorithm that uses truncation for argument reduction
     * so that we get a new value 0<=q<1 over the unit circle, and then
     * do floating-point comparisons with fractions. This is likely to be
     * slightly slower (~10%) due to the longer latencies of floating-point, so
     * we only use it when integer SIMD arithmetic is not present.
     */
    ssign   = x;
    x       = abs(x);
    // It is critical that half-way cases are rounded down
    z       = fma(x, two_over_pi, half);
    y       = trunc(z);
    q       = z * quarter;
    q       = q - trunc(q);
    /* z now starts at 0.0 for x=-pi/4 (although neg. values cannot occur), and
     * then increased by 1.0 as x increases by 2*Pi, when it resets to 0.0.
     * This removes the 2*Pi periodicity without using any integer arithmetic.
     * First check if y had the value 2 or 3, set csign if true.
     */
    q       = q - half;
    /* If we have logical operations we can work directly on the signbit, which
     * saves instructions. Otherwise we need to represent signs as +1.0/-1.0.
     * Thus, if you are altering defines to debug alternative code paths, the
     * two GMX_SIMD_HAVE_LOGICAL sections in this routine must either both be
     * active or inactive - you will get errors if only one is used.
     */
#    if GMX_SIMD_HAVE_LOGICAL
    ssign   = ssign & SimdDouble(GMX_DOUBLE_NEGZERO);
    csign   = andNot(q, SimdDouble(GMX_DOUBLE_NEGZERO));
    ssign   = ssign ^ csign;
#    else
    ssign   = copysign(SimdDouble(1.0), ssign);
    csign   = copysign(SimdDouble(1.0), q);
    csign   = -csign;
    ssign   = ssign * csign;    // swap ssign if csign was set.
#    endif
    // Check if y had value 1 or 3 (remember we subtracted 0.5 from q)
    m1      = (q < minusquarter);
    m2      = (setZero() <= q);
    m3      = (q < quarter);
    m2      = m2 && m3;
    mask    = m1 || m2;
    // where mask is FALSE, swap sign.
    csign   = csign * blend(SimdDouble(-1.0), one, mask);
#endif
    x       = fnma(y, argred0, x);
    x       = fnma(y, argred1, x);
    x       = fnma(y, argred2, x);
    x2      = x * x;

    psin    = fma(const_sin2, x2, const_sin1);
    psin    = fma(psin, x2, const_sin0);
    psin    = fma(psin, x * x2, x);
    pcos    = fma(const_cos2, x2, const_cos1);
    pcos    = fma(pcos, x2, const_cos0);
    pcos    = fms(pcos, x2, half);
    pcos    = fma(pcos, x2, one);

    sss     = blend(pcos, psin, mask);
    ccc     = blend(psin, pcos, mask);
    // See comment for GMX_SIMD_HAVE_LOGICAL section above.
#if GMX_SIMD_HAVE_LOGICAL
    *sinval = sss ^ ssign;
    *cosval = ccc ^ csign;
#else
    *sinval = sss * ssign;
    *cosval = ccc * csign;
#endif
}

/*! \brief SIMD sin(x). Double precision SIMD data, single accuracy.
 *
 * \param x The argument to evaluate sin for
 * \result Sin(x)
 *
 * \attention Do NOT call both sin & cos if you need both results, since each of them
 * will then call \ref sincos and waste a factor 2 in performance.
 */
static inline SimdDouble gmx_simdcall
sinSingleAccuracy(SimdDouble x)
{
    SimdDouble s, c;
    sinCosSingleAccuracy(x, &s, &c);
    return s;
}

/*! \brief SIMD cos(x). Double precision SIMD data, single accuracy.
 *
 * \param x The argument to evaluate cos for
 * \result Cos(x)
 *
 * \attention Do NOT call both sin & cos if you need both results, since each of them
 * will then call \ref sincos and waste a factor 2 in performance.
 */
static inline SimdDouble gmx_simdcall
cosSingleAccuracy(SimdDouble x)
{
    SimdDouble s, c;
    sinCosSingleAccuracy(x, &s, &c);
    return c;
}

/*! \brief SIMD tan(x). Double precision SIMD data, single accuracy.
 *
 * \param x The argument to evaluate tan for
 * \result Tan(x)
 */
static inline SimdDouble gmx_simdcall
tanSingleAccuracy(SimdDouble x)
{
    const SimdDouble  argred0(2*0.78539816290140151978);
    const SimdDouble  argred1(2*4.9604678871439933374e-10);
    const SimdDouble  argred2(2*1.1258708853173288931e-18);
    const SimdDouble  two_over_pi(2.0/M_PI);
    const SimdDouble  CT6(0.009498288995810566122993911);
    const SimdDouble  CT5(0.002895755790837379295226923);
    const SimdDouble  CT4(0.02460087336161924491836265);
    const SimdDouble  CT3(0.05334912882656359828045988);
    const SimdDouble  CT2(0.1333989091464957704418495);
    const SimdDouble  CT1(0.3333307599244198227797507);

    SimdDouble        x2, p, y, z;
    SimdDBool         mask;

#if GMX_SIMD_HAVE_DINT32_ARITHMETICS && GMX_SIMD_HAVE_LOGICAL
    SimdDInt32  iy;
    SimdDInt32  ione(1);

    z       = x * two_over_pi;
    iy      = cvtR2I(z);
    y       = round(z);
    mask    = cvtIB2B((iy & ione) == ione);

    x       = fnma(y, argred0, x);
    x       = fnma(y, argred1, x);
    x       = fnma(y, argred2, x);
    x       = selectByMask(SimdDouble(GMX_DOUBLE_NEGZERO), mask) ^ x;
#else
    const SimdDouble  quarter(0.25);
    const SimdDouble  half(0.5);
    const SimdDouble  threequarter(0.75);
    SimdDouble        w, q;
    SimdDBool         m1, m2, m3;

    w       = abs(x);
    z       = fma(w, two_over_pi, half);
    y       = trunc(z);
    q       = z * quarter;
    q       = q - trunc(q);
    m1      = (quarter <= q);
    m2      = (q < half);
    m3      = (threequarter <= q);
    m1      = m1 && m2;
    mask    = m1 || m3;
    w       = fnma(y, argred0, w);
    w       = fnma(y, argred1, w);
    w       = fnma(y, argred2, w);

    w       = blend(w, -w, mask);
    x       = w * copysign( SimdDouble(1.0), x );
#endif
    x2      = x * x;
    p       = fma(CT6, x2, CT5);
    p       = fma(p, x2, CT4);
    p       = fma(p, x2, CT3);
    p       = fma(p, x2, CT2);
    p       = fma(p, x2, CT1);
    p       = fma(x2, p * x, x);

    p       = blend( p, maskzInvSingleAccuracy(p, mask), mask);
    return p;
}

/*! \brief SIMD asin(x). Double precision SIMD data, single accuracy.
 *
 * \param x The argument to evaluate asin for
 * \result Asin(x)
 */
static inline SimdDouble gmx_simdcall
asinSingleAccuracy(SimdDouble x)
{
    const SimdDouble limitlow(1e-4);
    const SimdDouble half(0.5);
    const SimdDouble one(1.0);
    const SimdDouble halfpi(M_PI/2.0);
    const SimdDouble CC5(4.2163199048E-2);
    const SimdDouble CC4(2.4181311049E-2);
    const SimdDouble CC3(4.5470025998E-2);
    const SimdDouble CC2(7.4953002686E-2);
    const SimdDouble CC1(1.6666752422E-1);
    SimdDouble       xabs;
    SimdDouble       z, z1, z2, q, q1, q2;
    SimdDouble       pA, pB;
    SimdDBool        mask, mask2;

    xabs  = abs(x);
    mask  = (half < xabs);
    z1    = half * (one - xabs);
    mask2 = (xabs < one);
    q1    = z1 * maskzInvsqrtSingleAccuracy(z1, mask2);
    q2    = xabs;
    z2    = q2 * q2;
    z     = blend(z2, z1, mask);
    q     = blend(q2, q1, mask);

    z2    = z * z;
    pA    = fma(CC5, z2, CC3);
    pB    = fma(CC4, z2, CC2);
    pA    = fma(pA, z2, CC1);
    pA    = pA * z;
    z     = fma(pB, z2, pA);
    z     = fma(z, q, q);
    q2    = halfpi - z;
    q2    = q2 - z;
    z     = blend(z, q2, mask);

    mask  = (limitlow < xabs);
    z     = blend( xabs, z, mask );
    z     = copysign(z, x);

    return z;
}

/*! \brief SIMD acos(x). Double precision SIMD data, single accuracy.
 *
 * \param x The argument to evaluate acos for
 * \result Acos(x)
 */
static inline SimdDouble gmx_simdcall
acosSingleAccuracy(SimdDouble x)
{
    const SimdDouble one(1.0);
    const SimdDouble half(0.5);
    const SimdDouble pi(M_PI);
    const SimdDouble halfpi(M_PI/2.0);
    SimdDouble       xabs;
    SimdDouble       z, z1, z2, z3;
    SimdDBool        mask1, mask2, mask3;

    xabs  = abs(x);
    mask1 = (half < xabs);
    mask2 = (setZero() < x);

    z     = half * (one - xabs);
    mask3 = (xabs < one);
    z     = z * maskzInvsqrtSingleAccuracy(z, mask3);
    z     = blend(x, z, mask1);
    z     = asinSingleAccuracy(z);

    z2    = z + z;
    z1    = pi - z2;
    z3    = halfpi - z;
    z     = blend(z1, z2, mask2);
    z     = blend(z3, z, mask1);

    return z;
}

/*! \brief SIMD asin(x). Double precision SIMD data, single accuracy.
 *
 * \param x The argument to evaluate atan for
 * \result Atan(x), same argument/value range as standard math library.
 */
static inline SimdDouble gmx_simdcall
atanSingleAccuracy(SimdDouble x)
{
    const SimdDouble halfpi(M_PI/2);
    const SimdDouble CA17(0.002823638962581753730774);
    const SimdDouble CA15(-0.01595690287649631500244);
    const SimdDouble CA13(0.04250498861074447631836);
    const SimdDouble CA11(-0.07489009201526641845703);
    const SimdDouble CA9(0.1063479334115982055664);
    const SimdDouble CA7(-0.1420273631811141967773);
    const SimdDouble CA5(0.1999269574880599975585);
    const SimdDouble CA3(-0.3333310186862945556640);
    SimdDouble       x2, x3, x4, pA, pB;
    SimdDBool        mask, mask2;

    mask  = (x < setZero());
    x     = abs(x);
    mask2 = (SimdDouble(1.0) < x);
    x     = blend(x, maskzInvSingleAccuracy(x, mask2), mask2);

    x2    = x * x;
    x3    = x2 * x;
    x4    = x2 * x2;
    pA    = fma(CA17, x4, CA13);
    pB    = fma(CA15, x4, CA11);
    pA    = fma(pA, x4, CA9);
    pB    = fma(pB, x4, CA7);
    pA    = fma(pA, x4, CA5);
    pB    = fma(pB, x4, CA3);
    pA    = fma(pA, x2, pB);
    pA    = fma(pA, x3, x);

    pA    = blend(pA, halfpi - pA, mask2);
    pA    = blend(pA, -pA, mask);

    return pA;
}

/*! \brief SIMD atan2(y,x). Double precision SIMD data, single accuracy.
 *
 * \param y Y component of vector, any quartile
 * \param x X component of vector, any quartile
 * \result Atan(y,x), same argument/value range as standard math library.
 *
 * \note This routine should provide correct results for all finite
 * non-zero or positive-zero arguments. However, negative zero arguments will
 * be treated as positive zero, which means the return value will deviate from
 * the standard math library atan2(y,x) for those cases. That should not be
 * of any concern in Gromacs, and in particular it will not affect calculations
 * of angles from vectors.
 */
static inline SimdDouble gmx_simdcall
atan2SingleAccuracy(SimdDouble y, SimdDouble x)
{
    const SimdDouble pi(M_PI);
    const SimdDouble halfpi(M_PI/2.0);
    SimdDouble       xinv, p, aoffset;
    SimdDBool        mask_xnz, mask_ynz, mask_xlt0, mask_ylt0;

    mask_xnz  = x != setZero();
    mask_ynz  = y != setZero();
    mask_xlt0 = (x < setZero());
    mask_ylt0 = (y < setZero());

    aoffset   = selectByNotMask(halfpi, mask_xnz);
    aoffset   = selectByMask(aoffset, mask_ynz);

    aoffset   = blend(aoffset, pi, mask_xlt0);
    aoffset   = blend(aoffset, -aoffset, mask_ylt0);

    xinv      = maskzInvSingleAccuracy(x, mask_xnz);
    p         = y * xinv;
    p         = atanSingleAccuracy(p);
    p         = p + aoffset;

    return p;
}

/*! \brief Analytical PME force correction, double SIMD data, single accuracy.
 *
 * \param z2 \f$(r \beta)^2\f$ - see below for details.
 * \result Correction factor to coulomb force - see below for details.
 *
 * This routine is meant to enable analytical evaluation of the
 * direct-space PME electrostatic force to avoid tables.
 *
 * The direct-space potential should be \f$ \mbox{erfc}(\beta r)/r\f$, but there
 * are some problems evaluating that:
 *
 * First, the error function is difficult (read: expensive) to
 * approxmiate accurately for intermediate to large arguments, and
 * this happens already in ranges of \f$(\beta r)\f$ that occur in simulations.
 * Second, we now try to avoid calculating potentials in Gromacs but
 * use forces directly.
 *
 * We can simply things slight by noting that the PME part is really
 * a correction to the normal Coulomb force since \f$\mbox{erfc}(z)=1-\mbox{erf}(z)\f$, i.e.
 * \f[
 * V = \frac{1}{r} - \frac{\mbox{erf}(\beta r)}{r}
 * \f]
 * The first term we already have from the inverse square root, so
 * that we can leave out of this routine.
 *
 * For pme tolerances of 1e-3 to 1e-8 and cutoffs of 0.5nm to 1.8nm,
 * the argument \f$beta r\f$ will be in the range 0.15 to ~4. Use your
 * favorite plotting program to realize how well-behaved \f$\frac{\mbox{erf}(z)}{z}\f$ is
 * in this range!
 *
 * We approximate \f$f(z)=\mbox{erf}(z)/z\f$ with a rational minimax polynomial.
 * However, it turns out it is more efficient to approximate \f$f(z)/z\f$ and
 * then only use even powers. This is another minor optimization, since
 * we actually \a want \f$f(z)/z\f$, because it is going to be multiplied by
 * the vector between the two atoms to get the vectorial force. The
 * fastest flops are the ones we can avoid calculating!
 *
 * So, here's how it should be used:
 *
 * 1. Calculate \f$r^2\f$.
 * 2. Multiply by \f$\beta^2\f$, so you get \f$z^2=(\beta r)^2\f$.
 * 3. Evaluate this routine with \f$z^2\f$ as the argument.
 * 4. The return value is the expression:
 *
 * \f[
 *    \frac{2 \exp{-z^2}}{\sqrt{\pi} z^2}-\frac{\mbox{erf}(z)}{z^3}
 * \f]
 *
 * 5. Multiply the entire expression by \f$\beta^3\f$. This will get you
 *
 *  \f[
 *    \frac{2 \beta^3 \exp(-z^2)}{\sqrt{\pi} z^2} - \frac{\beta^3 \mbox{erf}(z)}{z^3}
 *  \f]
 *
 *    or, switching back to \f$r\f$ (since \f$z=r \beta\f$):
 *
 *  \f[
 *    \frac{2 \beta \exp(-r^2 \beta^2)}{\sqrt{\pi} r^2} - \frac{\mbox{erf}(r \beta)}{r^3}
 *  \f]
 *
 *    With a bit of math exercise you should be able to confirm that
 *    this is exactly
 *
 *  \f[
 *   \frac{\frac{d}{dr}\left( \frac{\mbox{erf}(\beta r)}{r} \right)}{r}
 *  \f]
 *
 * 6. Add the result to \f$r^{-3}\f$, multiply by the product of the charges,
 *    and you have your force (divided by \f$r\f$). A final multiplication
 *    with the vector connecting the two particles and you have your
 *    vectorial force to add to the particles.
 *
 * This approximation achieves an accuracy slightly lower than 1e-6; when
 * added to \f$1/r\f$ the error will be insignificant.
 *
 */
static SimdDouble gmx_simdcall
pmeForceCorrectionSingleAccuracy(SimdDouble z2)
{
    const SimdDouble  FN6(-1.7357322914161492954e-8);
    const SimdDouble  FN5(1.4703624142580877519e-6);
    const SimdDouble  FN4(-0.000053401640219807709149);
    const SimdDouble  FN3(0.0010054721316683106153);
    const SimdDouble  FN2(-0.019278317264888380590);
    const SimdDouble  FN1(0.069670166153766424023);
    const SimdDouble  FN0(-0.75225204789749321333);

    const SimdDouble  FD4(0.0011193462567257629232);
    const SimdDouble  FD3(0.014866955030185295499);
    const SimdDouble  FD2(0.11583842382862377919);
    const SimdDouble  FD1(0.50736591960530292870);
    const SimdDouble  FD0(1.0);

    SimdDouble        z4;
    SimdDouble        polyFN0, polyFN1, polyFD0, polyFD1;

    z4             = z2 * z2;

    polyFD0        = fma(FD4, z4, FD2);
    polyFD1        = fma(FD3, z4, FD1);
    polyFD0        = fma(polyFD0, z4, FD0);
    polyFD0        = fma(polyFD1, z2, polyFD0);

    polyFD0        = invSingleAccuracy(polyFD0);

    polyFN0        = fma(FN6, z4, FN4);
    polyFN1        = fma(FN5, z4, FN3);
    polyFN0        = fma(polyFN0, z4, FN2);
    polyFN1        = fma(polyFN1, z4, FN1);
    polyFN0        = fma(polyFN0, z4, FN0);
    polyFN0        = fma(polyFN1, z2, polyFN0);

    return polyFN0 * polyFD0;
}



/*! \brief Analytical PME potential correction, double SIMD data, single accuracy.
 *
 * \param z2 \f$(r \beta)^2\f$ - see below for details.
 * \result Correction factor to coulomb potential - see below for details.
 *
 * This routine calculates \f$\mbox{erf}(z)/z\f$, although you should provide \f$z^2\f$
 * as the input argument.
 *
 * Here's how it should be used:
 *
 * 1. Calculate \f$r^2\f$.
 * 2. Multiply by \f$\beta^2\f$, so you get \f$z^2=\beta^2*r^2\f$.
 * 3. Evaluate this routine with z^2 as the argument.
 * 4. The return value is the expression:
 *
 *  \f[
 *   \frac{\mbox{erf}(z)}{z}
 *  \f]
 *
 * 5. Multiply the entire expression by beta and switching back to \f$r\f$ (since \f$z=r \beta\f$):
 *
 *  \f[
 *    \frac{\mbox{erf}(r \beta)}{r}
 *  \f]
 *
 * 6. Subtract the result from \f$1/r\f$, multiply by the product of the charges,
 *    and you have your potential.
 *
 * This approximation achieves an accuracy slightly lower than 1e-6; when
 * added to \f$1/r\f$ the error will be insignificant.
 */
static SimdDouble gmx_simdcall
pmePotentialCorrectionSingleAccuracy(SimdDouble z2)
{
    const SimdDouble  VN6(1.9296833005951166339e-8);
    const SimdDouble  VN5(-1.4213390571557850962e-6);
    const SimdDouble  VN4(0.000041603292906656984871);
    const SimdDouble  VN3(-0.00013134036773265025626);
    const SimdDouble  VN2(0.038657983986041781264);
    const SimdDouble  VN1(0.11285044772717598220);
    const SimdDouble  VN0(1.1283802385263030286);

    const SimdDouble  VD3(0.0066752224023576045451);
    const SimdDouble  VD2(0.078647795836373922256);
    const SimdDouble  VD1(0.43336185284710920150);
    const SimdDouble  VD0(1.0);

    SimdDouble        z4;
    SimdDouble        polyVN0, polyVN1, polyVD0, polyVD1;

    z4             = z2 * z2;

    polyVD1        = fma(VD3, z4, VD1);
    polyVD0        = fma(VD2, z4, VD0);
    polyVD0        = fma(polyVD1, z2, polyVD0);

    polyVD0        = invSingleAccuracy(polyVD0);

    polyVN0        = fma(VN6, z4, VN4);
    polyVN1        = fma(VN5, z4, VN3);
    polyVN0        = fma(polyVN0, z4, VN2);
    polyVN1        = fma(polyVN1, z4, VN1);
    polyVN0        = fma(polyVN0, z4, VN0);
    polyVN0        = fma(polyVN1, z2, polyVN0);

    return polyVN0 * polyVD0;
}

#endif


/*! \name SIMD4 math functions
 *
 * \note Only a subset of the math functions are implemented for SIMD4.
 *  \{
 */


#if GMX_SIMD4_HAVE_FLOAT

/*************************************************************************
 * SINGLE PRECISION SIMD4 MATH FUNCTIONS - JUST A SMALL SUBSET SUPPORTED *
 *************************************************************************/

/*! \brief Perform one Newton-Raphson iteration to improve 1/sqrt(x) for SIMD4 float.
 *
 * This is a low-level routine that should only be used by SIMD math routine
 * that evaluates the inverse square root.
 *
 *  \param lu Approximation of 1/sqrt(x), typically obtained from lookup.
 *  \param x  The reference (starting) value x for which we want 1/sqrt(x).
 *  \return   An improved approximation with roughly twice as many bits of accuracy.
 */
static inline Simd4Float gmx_simdcall
rsqrtIter(Simd4Float lu, Simd4Float x)
{
    Simd4Float tmp1 = x*lu;
    Simd4Float tmp2 = Simd4Float(-0.5f)*lu;
    tmp1 = fma(tmp1, lu, Simd4Float(-3.0f));
    return tmp1*tmp2;
}

/*! \brief Calculate 1/sqrt(x) for SIMD4 float.
 *
 *  \param x Argument that must be >0. This routine does not check arguments.
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid.
 */
static inline Simd4Float gmx_simdcall
invsqrt(Simd4Float x)
{
    Simd4Float lu = rsqrt(x);
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
    return lu;
}


#endif // GMX_SIMD4_HAVE_FLOAT



#if GMX_SIMD4_HAVE_DOUBLE
/*************************************************************************
 * DOUBLE PRECISION SIMD4 MATH FUNCTIONS - JUST A SMALL SUBSET SUPPORTED *
 *************************************************************************/

/*! \brief Perform one Newton-Raphson iteration to improve 1/sqrt(x) for SIMD4 double.
 *
 * This is a low-level routine that should only be used by SIMD math routine
 * that evaluates the inverse square root.
 *
 *  \param lu Approximation of 1/sqrt(x), typically obtained from lookup.
 *  \param x  The reference (starting) value x for which we want 1/sqrt(x).
 *  \return   An improved approximation with roughly twice as many bits of accuracy.
 */
static inline Simd4Double gmx_simdcall
rsqrtIter(Simd4Double lu, Simd4Double x)
{
    Simd4Double tmp1 = x*lu;
    Simd4Double tmp2 = Simd4Double(-0.5f)*lu;
    tmp1             = fma(tmp1, lu, Simd4Double(-3.0f));
    return tmp1*tmp2;
}

/*! \brief Calculate 1/sqrt(x) for SIMD4 double.
 *
 *  \param x Argument that must be >0. This routine does not check arguments.
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid.
 */
static inline Simd4Double gmx_simdcall
invsqrt(Simd4Double x)
{
    Simd4Double lu = rsqrt(x);
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*8 < GMX_SIMD_ACCURACY_BITS_DOUBLE)
    lu = rsqrtIter(lu, x);
#endif
    return lu;
}


/**********************************************************************
 * SIMD4 MATH FUNCTIONS WITH DOUBLE PREC. DATA, SINGLE PREC. ACCURACY *
 **********************************************************************/

/*! \brief Calculate 1/sqrt(x) for SIMD4 double, but in single accuracy.
 *
 *  \param x Argument that must be >0. This routine does not check arguments.
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid.
 */
static inline Simd4Double gmx_simdcall
invsqrtSingleAccuracy(Simd4Double x)
{
    Simd4Double lu = rsqrt(x);
#if (GMX_SIMD_RSQRT_BITS < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*2 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
#if (GMX_SIMD_RSQRT_BITS*4 < GMX_SIMD_ACCURACY_BITS_SINGLE)
    lu = rsqrtIter(lu, x);
#endif
    return lu;
}



#endif // GMX_SIMD4_HAVE_DOUBLE

/*! \} */

#if GMX_SIMD_HAVE_FLOAT
/*! \brief Calculate 1/sqrt(x) for SIMD float, only targeting single accuracy.
 *
 *  \param x Argument that must be >0. This routine does not check arguments.
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid.
 */
static inline SimdFloat gmx_simdcall
invsqrtSingleAccuracy(SimdFloat x)
{
    return invsqrt(x);
}

/*! \brief Calculate 1/sqrt(x) for masked SIMD floats, only targeting single accuracy.
 *
 *  This routine only evaluates 1/sqrt(x) for elements for which mask is true.
 *  Illegal values in the masked-out elements will not lead to
 *  floating-point exceptions.
 *
 *  \param x Argument that must be >0 for masked-in entries
 *  \param m Mask
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid or
 *          entry was not masked, and 0.0 for masked-out entries.
 */
static inline SimdFloat
maskzInvsqrtSingleAccuracy(SimdFloat x, SimdFBool m)
{
    return maskzInvsqrt(x, m);
}

/*! \brief Calculate 1/sqrt(x) for two SIMD floats, only targeting single accuracy.
 *
 * \param x0  First set of arguments, x0 must be positive - no argument checking.
 * \param x1  Second set of arguments, x1 must be positive - no argument checking.
 * \param[out] out0  Result 1/sqrt(x0)
 * \param[out] out1  Result 1/sqrt(x1)
 *
 *  In particular for double precision we can sometimes calculate square root
 *  pairs slightly faster by using single precision until the very last step.
 */
static inline void gmx_simdcall
invsqrtPairSingleAccuracy(SimdFloat x0,    SimdFloat x1,
                          SimdFloat *out0, SimdFloat *out1)
{
    return invsqrtPair(x0, x1, out0, out1);
}

/*! \brief Calculate 1/x for SIMD float, only targeting single accuracy.
 *
 *  \param x Argument that must be nonzero. This routine does not check arguments.
 *  \return 1/x. Result is undefined if your argument was invalid.
 */
static inline SimdFloat gmx_simdcall
invSingleAccuracy(SimdFloat x)
{
    return inv(x);
}


/*! \brief Calculate 1/x for masked SIMD floats, only targeting single accuracy.
 *
 *  \param x Argument that must be nonzero for non-masked entries.
 *  \param m Mask
 *  \return 1/x for elements where m is true, or 0.0 for masked-out entries.
 */
static inline SimdFloat
maskzInvSingleAccuracy(SimdFloat x, SimdFBool m)
{
    return maskzInv(x, m);
}

/*! \brief Calculate sqrt(x) for SIMD float, only targeting single accuracy.
 *
 *  \param x Argument that must be >=0.
 *  \return sqrt(x). If x=0, the result will correctly be set to 0.
 *          The result is undefined if the input value is negative.
 */
static inline SimdFloat gmx_simdcall
sqrtSingleAccuracy(SimdFloat x)
{
    return sqrt(x);
}

/*! \brief SIMD float log(x), only targeting single accuracy. This is the natural logarithm.
 *
 * \param x Argument, should be >0.
 * \result The natural logarithm of x. Undefined if argument is invalid.
 */
static inline SimdFloat gmx_simdcall
logSingleAccuracy(SimdFloat x)
{
    return log(x);
}

/*! \brief SIMD float 2^x, only targeting single accuracy.
 *
 * \param x Argument.
 * \result 2^x. Undefined if input argument caused overflow.
 */
static inline SimdFloat gmx_simdcall
exp2SingleAccuracy(SimdFloat x)
{
    return exp2(x);
}

/*! \brief SIMD float e^x, only targeting single accuracy.
 *
 * \param x Argument.
 * \result exp(x). Undefined if input argument caused overflow.
 */
static inline SimdFloat gmx_simdcall
expSingleAccuracy(SimdFloat x)
{
    return exp(x);
}

/*! \brief SIMD float erf(x), only targeting single accuracy.
 *
 * \param x The value to calculate erf(x) for.
 * \result erf(x)
 *
 * This routine achieves very close to single precision, but we do not care about
 * the last bit or the subnormal result range.
 */
static inline SimdFloat gmx_simdcall
erfSingleAccuracy(SimdFloat x)
{
    return erf(x);
}

/*! \brief SIMD float erfc(x), only targeting single accuracy.
 *
 * \param x The value to calculate erfc(x) for.
 * \result erfc(x)
 *
 * This routine achieves singleprecision (bar the last bit) over most of the
 * input range, but for large arguments where the result is getting close
 * to the minimum representable numbers we accept slightly larger errors
 * (think results that are in the ballpark of 10^-30) since that is not
 * relevant for MD.
 */
static inline SimdFloat gmx_simdcall
erfcSingleAccuracy(SimdFloat x)
{
    return erfc(x);
}

/*! \brief SIMD float sin \& cos, only targeting single accuracy.
 *
 * \param x The argument to evaluate sin/cos for
 * \param[out] sinval Sin(x)
 * \param[out] cosval Cos(x)
 */
static inline void gmx_simdcall
sinCosSingleAccuracy(SimdFloat x, SimdFloat *sinval, SimdFloat *cosval)
{
    sincos(x, sinval, cosval);
}

/*! \brief SIMD float sin(x), only targeting single accuracy.
 *
 * \param x The argument to evaluate sin for
 * \result Sin(x)
 *
 * \attention Do NOT call both sin & cos if you need both results, since each of them
 * will then call \ref sincos and waste a factor 2 in performance.
 */
static inline SimdFloat gmx_simdcall
sinSingleAccuracy(SimdFloat x)
{
    return sin(x);
}

/*! \brief SIMD float cos(x), only targeting single accuracy.
 *
 * \param x The argument to evaluate cos for
 * \result Cos(x)
 *
 * \attention Do NOT call both sin & cos if you need both results, since each of them
 * will then call \ref sincos and waste a factor 2 in performance.
 */
static inline SimdFloat gmx_simdcall
cosSingleAccuracy(SimdFloat x)
{
    return cos(x);
}

/*! \brief SIMD float tan(x), only targeting single accuracy.
 *
 * \param x The argument to evaluate tan for
 * \result Tan(x)
 */
static inline SimdFloat gmx_simdcall
tanSingleAccuracy(SimdFloat x)
{
    return tan(x);
}

/*! \brief SIMD float asin(x), only targeting single accuracy.
 *
 * \param x The argument to evaluate asin for
 * \result Asin(x)
 */
static inline SimdFloat gmx_simdcall
asinSingleAccuracy(SimdFloat x)
{
    return asin(x);
}

/*! \brief SIMD float acos(x), only targeting single accuracy.
 *
 * \param x The argument to evaluate acos for
 * \result Acos(x)
 */
static inline SimdFloat gmx_simdcall
acosSingleAccuracy(SimdFloat x)
{
    return acos(x);
}

/*! \brief SIMD float atan(x), only targeting single accuracy.
 *
 * \param x The argument to evaluate atan for
 * \result Atan(x), same argument/value range as standard math library.
 */
static inline SimdFloat gmx_simdcall
atanSingleAccuracy(SimdFloat x)
{
    return atan(x);
}

/*! \brief SIMD float atan2(y,x), only targeting single accuracy.
 *
 * \param y Y component of vector, any quartile
 * \param x X component of vector, any quartile
 * \result Atan(y,x), same argument/value range as standard math library.
 *
 * \note This routine should provide correct results for all finite
 * non-zero or positive-zero arguments. However, negative zero arguments will
 * be treated as positive zero, which means the return value will deviate from
 * the standard math library atan2(y,x) for those cases. That should not be
 * of any concern in Gromacs, and in particular it will not affect calculations
 * of angles from vectors.
 */
static inline SimdFloat gmx_simdcall
atan2SingleAccuracy(SimdFloat y, SimdFloat x)
{
    return atan2(y, x);
}

/*! \brief SIMD Analytic PME force correction, only targeting single accuracy.
 *
 * \param z2 \f$(r \beta)^2\f$ - see default single precision version for details.
 * \result Correction factor to coulomb force.
 */
static inline SimdFloat gmx_simdcall
pmeForceCorrectionSingleAccuracy(SimdFloat z2)
{
    return pmeForceCorrection(z2);
}

/*! \brief SIMD Analytic PME potential correction, only targeting single accuracy.
 *
 * \param z2 \f$(r \beta)^2\f$ - see default single precision version for details.
 * \result Correction factor to coulomb force.
 */
static inline SimdFloat gmx_simdcall
pmePotentialCorrectionSingleAccuracy(SimdFloat z2)
{
    return pmePotentialCorrection(z2);
}
#endif // GMX_SIMD_HAVE_FLOAT

#if GMX_SIMD4_HAVE_FLOAT
/*! \brief Calculate 1/sqrt(x) for SIMD4 float, only targeting single accuracy.
 *
 *  \param x Argument that must be >0. This routine does not check arguments.
 *  \return 1/sqrt(x). Result is undefined if your argument was invalid.
 */
static inline Simd4Float gmx_simdcall
invsqrtSingleAccuracy(Simd4Float x)
{
    return invsqrt(x);
}
#endif // GMX_SIMD4_HAVE_FLOAT

/*! \}   end of addtogroup module_simd */
/*! \endcond  end of condition libabl */

#endif // GMX_SIMD

}      // namespace gmx

#endif // GMX_SIMD_SIMD_MATH_H