FastMathFunctions.H

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/*!@file Util/FastMathFunctions.H Miscellaneous math functions */

// //////////////////////////////////////////////////////////////////// //
// The iLab Neuromorphic Vision C++ Toolkit - Copyright (C) 2000-2003   //
// by the University of Southern California (USC) and the iLab at USC.  //
// See http://iLab.usc.edu for information about this project.          //
// //////////////////////////////////////////////////////////////////// //
// Major portions of the iLab Neuromorphic Vision Toolkit are protected //
// under the U.S. patent ``Computation of Intrinsic Perceptual Saliency //
// in Visual Environments, and Applications'' by Christof Koch and      //
// Laurent Itti, California Institute of Technology, 2001 (patent       //
// pending; application number 09/912,225 filed July 23, 2001; see      //
// http://pair.uspto.gov/cgi-bin/final/home.pl for current status).     //
// //////////////////////////////////////////////////////////////////// //
// This file is part of the iLab Neuromorphic Vision C++ Toolkit.       //
//                                                                      //
// The iLab Neuromorphic Vision C++ Toolkit is free software; you can   //
// redistribute it and/or modify it under the terms of the GNU General  //
// Public License as published by the Free Software Foundation; either  //
// version 2 of the License, or (at your option) any later version.     //
//                                                                      //
// The iLab Neuromorphic Vision C++ Toolkit 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 General Public License for more details.       //
//                                                                      //
// You should have received a copy of the GNU General Public License    //
// along with the iLab Neuromorphic Vision C++ Toolkit; if not, write   //
// to the Free Software Foundation, Inc., 59 Temple Place, Suite 330,   //
// Boston, MA 02111-1307 USA.                                           //
// //////////////////////////////////////////////////////////////////// //
//
// Primary maintainer for this file: T. Nathan Mundhenk <mundhenk@usc.edu>
// $HeadURL: svn://isvn.usc.edu/software/invt/trunk/saliency/src/Util/FastMathFunctions.H $
// $Id: FastMathFunctions.H 12979 2010-03-07 17:10:23Z lior $
//

#ifndef FASTMATHFUNCTIONS_H_DEFINED
#define FASTMATHFUNCTIONS_H_DEFINED
#include "Util/Assert.H"
#include "Util/Promotions.H"
#include "rutz/compat_cmath.h" // for M_PI, M_E if not present in <cmath>

// ######################################################################
// ##### Fast Square Root functions
// ######################################################################
/*! @name Fast Square Root Methods

    These are several methods for computing a quick square root
    approximation. The order is by decreasing speed / increasing accuracy

    Name           : opperation time relative to sqrt()

    fastSqrt_2     : 1/5
    fastSqrt_Bab   : 1/2
    fastSqrt_Q3    : 2/3
    fastSqrt_Bab_2 : 2/3

    Details:

    fastSqrt_2     - This is a simple log base 2 approximation using a union
                     to int and an 4 simple arith operations.

    fastSqrt_Bab   - This is the fastSqrt_2 with a Babylonian step added
                     to give it a quadratic improvement.

    fastSqrt_Q3    - This uses the Quake 3 InvRoot function with a
                     quick estimation and a single Newton step. This will call
                     invSqrt listed below.

    fastSqrt_Bab_2 - This is the fastSqrt_Bab with two Babylonian Steps instead
                     of just one.

*/
//@{

// For Magic Derivation see:
// Chris Lomont
// http://www.lomont.org/Math/Papers/2003/InvSqrt.pdf
// Credited to Greg Walsh.
// 32  Bit float magic number
#define SQRT_MAGIC_F 0x5f3759df

// 64  Bit float magic number
#ifdef LONG_IS_32_BITS
//Fit into 2 longs
#define SQRT_MAGIC_D 0x5fe6ec85e7de30daLL
#else
#define SQRT_MAGIC_D 0x5fe6ec85e7de30da
#endif

// A constant which can make the fastSqrt_2 method faster
#define KIMS_CONST_F 0x4C000
#define KIMS_CONST_D (KIMS_CONST_F << 29)
// ######################################################################
//! Fast and dirty Log Base 2 appoximiation for square root
/*! This method is most useful if the number is a power of 2. In this case,
    the results are accurate. The largest error tends to be with numbers
    half way between two powers of 2. So as an example:

    fastSqrt_2(2) and fastSqrt_2(4) will give exact answers while
    fastSqrt_2(3) will give 1.5 when the answer should be 1.414

    See: http://en.wikipedia.org/wiki/Methods_of_computing_square_roots

    Time savings using this is approx 1/5 the time of a normal square root.
*/
inline float fastSqrt_2(const float x)
{
  union     // Use a union to operate on a float like an integer
  {
    int i;
    float x;
  } u;
  u.x = x;
  /*
  This line is a compressed version if this:

  u.i -= 1<<23;   // Remove last bit so 1.0 gives 1.0
  // u.i is now an approximation to logbase2(x)
  u.i = u.i >> 1; // divide by 2
  u.i += 1<<29;   // add 64 to exponent: (e+127)/2 =(e/2)+63,
                  // that represents (e/2)-64 but we want e/2
  */
  u.i = (1<<29) + (u.i >> 1) - (1<<22) - KIMS_CONST_F;
  return u.x;
}

//! Fast and dirty Log Base 2 appoximiation for square root
inline double fastSqrt_2(const double x)
{
#ifdef INVT_USE_64_BIT

  union      // Use a union to operate on a float like an integer
  {
#ifdef LONG_IS_32_BITS
    long long i;
#else
    long i;
#endif
    double x;
  } u;
  u.x = x;
  /*
  This line is a compressed version if this:

  u.i -= ((long)1)<<52; // Remove last bit so 1.0 gives 1.0
  // u.i is now an approximation to logbase2(x)
  u.i = u.i >> 1;       // divide by 2
  u.i += ((long)1)<<61; // add 512 to exponent: (e+1023)/2 =(e/2)+511,
                        // that represents (e/2)-512 but we want e/2
  */
#ifdef LONG_IS_32_BITS
  u.i = (((long long)1)<<61) + (u.i >> 1) - (((long long)1)<<51) - KIMS_CONST_D;
#else
  u.i = (((long)1)<<61) + (u.i >> 1) - (((long)1)<<51) - KIMS_CONST_D;
#endif

  return u.x;

#else
  return (double)fastSqrt_2((float)x);
#endif
}

// ######################################################################
//! Fast and dirty Log Base 2 appoximiation with a Babylonian Step Clean up
/*! Time savings using this is approx 1/2 the time of a normal square root. */
inline float fastSqrt_Bab(const float x)
{
  union     // Use a union to operate on a float like an integer
  {
    int i;
    float x;
  } u;
  u.x = x;
  u.i = (1<<29) + (u.i >> 1) - (1<<22) - KIMS_CONST_F;  // Log2 Approximation y0
  u.x = 0.5f * (u.x + x/u.x);                           // One Babylonian Step

  return u.x;
}

//! Fast and dirty Log Base 2 appoximiation with a Babylonian Step Clean up
inline double fastSqrt_Bab(const double x)
{
#ifdef INVT_USE_64_BIT
  union      // Use a union to operate on a float like an integer
  {
#ifdef LONG_IS_32_BITS
    long long i;
#else
    long i;
#endif

    double x;
  } u;
  u.x = x;

#ifdef LONG_IS_32_BITS
  u.i = (((long long)1)<<61) + (u.i >> 1) - (((long long)1)<<51) - KIMS_CONST_D;// Log2 Approximation y0
#else
  u.i = (((long)1)<<61) + (u.i >> 1) - (((long)1)<<51) - KIMS_CONST_D;// Log2 Approximation y0
#endif
  u.x = 0.5F * (u.x + x/u.x);                                         // One Babylonian Step
  return u.x;
#else
  return (double)fastSqrt_Bab((float)x);
#endif
}

// ######################################################################
//! Quake 3 Fast Square Root Approximation
/*! The integer-shift approximation produced a relative error of less than 4%,
    and the error dropped further to 0.15% with one iteration of Newton's
    method on the following line. In computer graphics it is a very efficient
    way to normalize a vector.

    Note: This must be a floating point due to the IEEE method for
    approx. of sqrt. See the double method below which uses a different
    magic constant.

    To get the square root, divide 1 by the final x.
    See: http://en.wikipedia.org/wiki/Methods_of_computing_square_roots
*/
inline float invSqrt(const float x)
{
  const float xhalf = 0.5f*x;

  union // get bits for floating value
  {
    float x;
    int i;
  } u;
  u.x = x;
  u.i = SQRT_MAGIC_F - (u.i >> 1);  // gives initial guess y0
  return u.x*(1.5f - xhalf*u.x*u.x);// Newton step, repeating increases accuracy
}

//! Fast Square Root Approximation
/*! Time savings using this is approx 2/3 the time of a normal square root.*/
inline float fastSqrt_Q3(const float x)
{
  // return 1.0f / invSqrt(x);
  return x * invSqrt(x);
}

//! Quake 3 Fast Square Root Approximation
inline double invSqrt(const double x)
{
#ifdef INVT_USE_64_BIT
  const double xhalf = 0.5F*x;

  union // get bits for floating value
  {
    double x;
#ifdef LONG_IS_32_BITS
    long long i;
#else
    long i;
#endif
  } u;
  u.x = x;
  u.i = SQRT_MAGIC_D - (u.i >> 1);  // gives initial guess y0
  return u.x*(1.5F - xhalf*u.x*u.x);// Newton step, repeating increases accuracy

#else
  return (double)invSqrt((float)x);
#endif
}

//! Fast Square Root Approximation
/*! Time savings using this is approx 2/3 the time of a normal square root.*/
inline double fastSqrt_Q3(const double x)
{
  // return 1.0F / invSqrt(x);
  return x * invSqrt(x);
}

// ######################################################################
//! Fast and dirty Log Base 2 appoximiation with two Babylonian Steps
/*! Time savings using this is approx 2/3 the time of a normal square root.

    This one gives fairly good estimates with a 50% speed up:

      | sqrt(x)       | Result     | Actual Answer using libm

    0 | 1.000000      | 1.000000   | 1.000000
    1 | 2.000000      | 1.414216   | 1.414214
    2 | 8.000000      | 2.828431   | 2.828427
    3 | 100.000000    | 10.000000  | 10.000000
    4 | 3.141593      | 1.772454   | 1.772454
    5 | 100000.000000 | 316.227783 | 316.227753
    6 | 0.333333      | 0.577350   | 0.577350
*/
inline float fastSqrt_Bab_2(const float x)
{
  union     // Use a union to operate on a float like an integer
  {
    int i;
    float x;
  } u;
  u.x = x;
  u.i = (1<<29) + (u.i >> 1) - (1<<22) - KIMS_CONST_F; // Log2 Approximation y0

  // Two Babylonian Steps (simplified from:)
  // u.x = 0.5f * (u.x + x/u.x);
  // u.x = 0.5f * (u.x + x/u.x);
  u.x =       u.x + x/u.x;              // First Babylonian Step
  u.x = 0.25f*u.x + x/u.x;              // Second Babylonian Step

  return u.x;
}

//! Fast and dirty Log Base 2 appoximiation with two Babylonian Steps
inline double fastSqrt_Bab_2(const double x)
{
#ifdef INVT_USE_64_BIT
  union       // Use a union to operate on a float like an integer
  {
#ifdef LONG_IS_32_BITS
    long long i;
#else
    long i;
#endif
    double x;
  } u;
  u.x = x;
#ifdef LONG_IS_32_BITS
  u.i = (((long long)1)<<61) + (u.i >> 1) - (((long long)1)<<51) - KIMS_CONST_D; // Log2 Approximation y0
#else
  u.i = (((long)1)<<61) + (u.i >> 1) - (((long)1)<<51) - KIMS_CONST_D; // Log2 Approximation y0
#endif

  // Two Babylonian Steps (simplified from:)
  // u.x = 0.5F * (u.x + x/u.x);
  // u.x = 0.5F * (u.x + x/u.x);
  u.x =       u.x + x/u.x;                 // First Babylonian Step
  u.x = 0.25F*u.x + x/u.x;                 // Second Babylonian Step
  return u.x;

#else
  return (double)fastSqrt_Bab_2((float)x);
#endif
}

// ######################################################################
//! This just creates a default function call

// If we are using the fast math methods then call them, otherwise redirect
// to the standard methods. Election to use fast math is set in configure
// using the switch --enable-fastmath-func

/* Speed up test:
test-fastMath::main: >>> Time Int SQRT BASELINE - 431742 mcs
test-fastMath::main: >>> Time Int SQRT - 102694 mcs
test-fastMath::main: >>> Time Float SQRT BASELINE - 175870 mcs
test-fastMath::main: >>> Time Float SQRT - 122220 mcs
*/

#ifdef INVT_USE_FASTMATH

//! This just creates a default function call
inline char fastSqrt(const char x)
{
  return static_cast<char>(fastSqrt_2((float)x));
}

//! This just creates a default function call
inline int fastSqrt(const int x)
{
  return static_cast<int>(fastSqrt_Bab((float)x));
}

//! This just creates a default function call
inline float fastSqrt(const float x)
{
  return fastSqrt_Bab_2(x);
}

//! This just creates a default function call
inline double fastSqrt(const double x)
{
  return fastSqrt_Bab_2(x);
}

#else

//! This just creates a default function call
inline char fastSqrt(const char x)
{
  return static_cast<char>(sqrt(x));
}

//! This just creates a default function call
inline int fastSqrt(const int x)
{
  return static_cast<int>(sqrt(x));
}

//! This just creates a default function call
inline float fastSqrt(const float x)
{
  return static_cast<float>(sqrt(x));
}

//! This just creates a default function call
inline double fastSqrt(const double x)
{
  return static_cast<double>(sqrt(x));
}

#define ACOS_TABLE_SIZE  513
inline float fastacos(int x)
{
//Got this from opencv
static const float acos_table[ACOS_TABLE_SIZE] = {
    3.14159265f, 3.05317551f, 3.01651113f, 2.98834964f, 2.96458497f, 2.94362719f,
    2.92466119f, 2.90720289f, 2.89093699f, 2.87564455f, 2.86116621f, 2.84738169f,
    2.83419760f, 2.82153967f, 2.80934770f, 2.79757211f, 2.78617145f, 2.77511069f,
    2.76435988f, 2.75389319f, 2.74368816f, 2.73372510f, 2.72398665f, 2.71445741f,
    2.70512362f, 2.69597298f, 2.68699438f, 2.67817778f, 2.66951407f, 2.66099493f,
    2.65261279f, 2.64436066f, 2.63623214f, 2.62822133f, 2.62032277f, 2.61253138f,
    2.60484248f, 2.59725167f, 2.58975488f, 2.58234828f, 2.57502832f, 2.56779164f,
    2.56063509f, 2.55355572f, 2.54655073f, 2.53961750f, 2.53275354f, 2.52595650f,
    2.51922417f, 2.51255441f, 2.50594525f, 2.49939476f, 2.49290115f, 2.48646269f,
    2.48007773f, 2.47374472f, 2.46746215f, 2.46122860f, 2.45504269f, 2.44890314f,
    2.44280867f, 2.43675809f, 2.43075025f, 2.42478404f, 2.41885841f, 2.41297232f,
    2.40712480f, 2.40131491f, 2.39554173f, 2.38980439f, 2.38410204f, 2.37843388f,
    2.37279910f, 2.36719697f, 2.36162673f, 2.35608768f, 2.35057914f, 2.34510044f,
    2.33965094f, 2.33423003f, 2.32883709f, 2.32347155f, 2.31813284f, 2.31282041f,
    2.30753373f, 2.30227228f, 2.29703556f, 2.29182309f, 2.28663439f, 2.28146900f,
    2.27632647f, 2.27120637f, 2.26610827f, 2.26103177f, 2.25597646f, 2.25094195f,
    2.24592786f, 2.24093382f, 2.23595946f, 2.23100444f, 2.22606842f, 2.22115104f,
    2.21625199f, 2.21137096f, 2.20650761f, 2.20166166f, 2.19683280f, 2.19202074f,
    2.18722520f, 2.18244590f, 2.17768257f, 2.17293493f, 2.16820274f, 2.16348574f,
    2.15878367f, 2.15409630f, 2.14942338f, 2.14476468f, 2.14011997f, 2.13548903f,
    2.13087163f, 2.12626757f, 2.12167662f, 2.11709859f, 2.11253326f, 2.10798044f,
    2.10343994f, 2.09891156f, 2.09439510f, 2.08989040f, 2.08539725f, 2.08091550f,
    2.07644495f, 2.07198545f, 2.06753681f, 2.06309887f, 2.05867147f, 2.05425445f,
    2.04984765f, 2.04545092f, 2.04106409f, 2.03668703f, 2.03231957f, 2.02796159f,
    2.02361292f, 2.01927344f, 2.01494300f, 2.01062146f, 2.00630870f, 2.00200457f,
    1.99770895f, 1.99342171f, 1.98914271f, 1.98487185f, 1.98060898f, 1.97635399f,
    1.97210676f, 1.96786718f, 1.96363511f, 1.95941046f, 1.95519310f, 1.95098292f,
    1.94677982f, 1.94258368f, 1.93839439f, 1.93421185f, 1.93003595f, 1.92586659f,
    1.92170367f, 1.91754708f, 1.91339673f, 1.90925250f, 1.90511432f, 1.90098208f,
    1.89685568f, 1.89273503f, 1.88862003f, 1.88451060f, 1.88040664f, 1.87630806f,
    1.87221477f, 1.86812668f, 1.86404371f, 1.85996577f, 1.85589277f, 1.85182462f,
    1.84776125f, 1.84370256f, 1.83964848f, 1.83559892f, 1.83155381f, 1.82751305f,
    1.82347658f, 1.81944431f, 1.81541617f, 1.81139207f, 1.80737194f, 1.80335570f,
    1.79934328f, 1.79533460f, 1.79132959f, 1.78732817f, 1.78333027f, 1.77933581f,
    1.77534473f, 1.77135695f, 1.76737240f, 1.76339101f, 1.75941271f, 1.75543743f,
    1.75146510f, 1.74749565f, 1.74352900f, 1.73956511f, 1.73560389f, 1.73164527f,
    1.72768920f, 1.72373560f, 1.71978441f, 1.71583556f, 1.71188899f, 1.70794462f,
    1.70400241f, 1.70006228f, 1.69612416f, 1.69218799f, 1.68825372f, 1.68432127f,
    1.68039058f, 1.67646160f, 1.67253424f, 1.66860847f, 1.66468420f, 1.66076139f,
    1.65683996f, 1.65291986f, 1.64900102f, 1.64508338f, 1.64116689f, 1.63725148f,
    1.63333709f, 1.62942366f, 1.62551112f, 1.62159943f, 1.61768851f, 1.61377831f,
    1.60986877f, 1.60595982f, 1.60205142f, 1.59814349f, 1.59423597f, 1.59032882f,
    1.58642196f, 1.58251535f, 1.57860891f, 1.57470259f, 1.57079633f, 1.56689007f,
    1.56298375f, 1.55907731f, 1.55517069f, 1.55126383f, 1.54735668f, 1.54344917f,
    1.53954124f, 1.53563283f, 1.53172389f, 1.52781434f, 1.52390414f, 1.51999323f,
    1.51608153f, 1.51216900f, 1.50825556f, 1.50434117f, 1.50042576f, 1.49650927f,
    1.49259163f, 1.48867280f, 1.48475270f, 1.48083127f, 1.47690845f, 1.47298419f,
    1.46905841f, 1.46513106f, 1.46120207f, 1.45727138f, 1.45333893f, 1.44940466f,
    1.44546850f, 1.44153038f, 1.43759024f, 1.43364803f, 1.42970367f, 1.42575709f,
    1.42180825f, 1.41785705f, 1.41390346f, 1.40994738f, 1.40598877f, 1.40202755f,
    1.39806365f, 1.39409701f, 1.39012756f, 1.38615522f, 1.38217994f, 1.37820164f,
    1.37422025f, 1.37023570f, 1.36624792f, 1.36225684f, 1.35826239f, 1.35426449f,
    1.35026307f, 1.34625805f, 1.34224937f, 1.33823695f, 1.33422072f, 1.33020059f,
    1.32617649f, 1.32214834f, 1.31811607f, 1.31407960f, 1.31003885f, 1.30599373f,
    1.30194417f, 1.29789009f, 1.29383141f, 1.28976803f, 1.28569989f, 1.28162688f,
    1.27754894f, 1.27346597f, 1.26937788f, 1.26528459f, 1.26118602f, 1.25708205f,
    1.25297262f, 1.24885763f, 1.24473698f, 1.24061058f, 1.23647833f, 1.23234015f,
    1.22819593f, 1.22404557f, 1.21988898f, 1.21572606f, 1.21155670f, 1.20738080f,
    1.20319826f, 1.19900898f, 1.19481283f, 1.19060973f, 1.18639955f, 1.18218219f,
    1.17795754f, 1.17372548f, 1.16948589f, 1.16523866f, 1.16098368f, 1.15672081f,
    1.15244994f, 1.14817095f, 1.14388370f, 1.13958808f, 1.13528396f, 1.13097119f,
    1.12664966f, 1.12231921f, 1.11797973f, 1.11363107f, 1.10927308f, 1.10490563f,
    1.10052856f, 1.09614174f, 1.09174500f, 1.08733820f, 1.08292118f, 1.07849378f,
    1.07405585f, 1.06960721f, 1.06514770f, 1.06067715f, 1.05619540f, 1.05170226f,
    1.04719755f, 1.04268110f, 1.03815271f, 1.03361221f, 1.02905939f, 1.02449407f,
    1.01991603f, 1.01532509f, 1.01072102f, 1.00610363f, 1.00147268f, 0.99682798f,
    0.99216928f, 0.98749636f, 0.98280898f, 0.97810691f, 0.97338991f, 0.96865772f,
    0.96391009f, 0.95914675f, 0.95436745f, 0.94957191f, 0.94475985f, 0.93993099f,
    0.93508504f, 0.93022170f, 0.92534066f, 0.92044161f, 0.91552424f, 0.91058821f,
    0.90563319f, 0.90065884f, 0.89566479f, 0.89065070f, 0.88561619f, 0.88056088f,
    0.87548438f, 0.87038629f, 0.86526619f, 0.86012366f, 0.85495827f, 0.84976956f,
    0.84455709f, 0.83932037f, 0.83405893f, 0.82877225f, 0.82345981f, 0.81812110f,
    0.81275556f, 0.80736262f, 0.80194171f, 0.79649221f, 0.79101352f, 0.78550497f,
    0.77996593f, 0.77439569f, 0.76879355f, 0.76315878f, 0.75749061f, 0.75178826f,
    0.74605092f, 0.74027775f, 0.73446785f, 0.72862033f, 0.72273425f, 0.71680861f,
    0.71084240f, 0.70483456f, 0.69878398f, 0.69268952f, 0.68654996f, 0.68036406f,
    0.67413051f, 0.66784794f, 0.66151492f, 0.65512997f, 0.64869151f, 0.64219789f,
    0.63564741f, 0.62903824f, 0.62236849f, 0.61563615f, 0.60883911f, 0.60197515f,
    0.59504192f, 0.58803694f, 0.58095756f, 0.57380101f, 0.56656433f, 0.55924437f,
    0.55183778f, 0.54434099f, 0.53675018f, 0.52906127f, 0.52126988f, 0.51337132f,
    0.50536051f, 0.49723200f, 0.48897987f, 0.48059772f, 0.47207859f, 0.46341487f,
    0.45459827f, 0.44561967f, 0.43646903f, 0.42713525f, 0.41760600f, 0.40786755f,
    0.39790449f, 0.38769946f, 0.37723277f, 0.36648196f, 0.35542120f, 0.34402054f,
    0.33224495f, 0.32005298f, 0.30739505f, 0.29421096f, 0.28042645f, 0.26594810f,
    0.25065566f, 0.23438976f, 0.21693146f, 0.19796546f, 0.17700769f, 0.15324301f,
    0.12508152f, 0.08841715f, 0.00000000f
};

if (x >= 0 && x<ACOS_TABLE_SIZE)
  return acos_table[x];
else
  return 0;

}

//@}
#endif
// ######################################################################
// ##### Fast Log Function
// ######################################################################
/*! @Fast Log Function
*/
//@{
static const double TBL_LOG_P[] = {
/* ONE   */  1.0,
/* TWO52 */  4503599627370496.0,
/* LN2HI */  6.93147180369123816490e-01,        /* 3fe62e42, fee00000 */
/* LN2LO */  1.90821492927058770002e-10,        /* 3dea39ef, 35793c76 */
/* A1    */ -6.88821452420390473170286327331268694251775741577e-0002,
/* A2    */  1.97493380704769294631262255279580131173133850098e+0000,
/* A3    */  2.24963218866067560242072431719861924648284912109e+0000,
/* A4    */ -9.02975906958474405783476868236903101205825805664e-0001,
/* A5    */ -1.47391630715542865104339398385491222143173217773e+0000,
/* A6    */  1.86846544648220058704168877738993614912033081055e+0000,
/* A7    */  1.82277370459347465292410106485476717352867126465e+0000,
/* A8    */  1.25295479915214102994980294170090928673744201660e+0000,
/* A9    */  1.96709676945198275177517643896862864494323730469e+0000,
/* A10   */ -4.00127989749189894030934055990655906498432159424e-0001,
/* A11   */  3.01675528558798333733648178167641162872314453125e+0000,
/* A12   */ -9.52325445049240770778453679668018594384193420410e-0001,
/* B1    */ -1.25041641589283658575482149899471551179885864258e-0001,
/* B2    */  1.87161713283355151891381127914642725337613123482e+0000,
/* B3    */ -1.89082956295731507978530316904652863740921020508e+0000,
/* B4    */ -2.50562891673640253387134180229622870683670043945e+0000,
/* B5    */  1.64822828085258366037635369139024987816810607910e+0000,
/* B6    */ -1.24409107065868340669112512841820716857910156250e+0000,
/* B7    */  1.70534231658220414296067701798165217041969299316e+0000,
/* B8    */  1.99196833784655646937267192697618156671524047852e+0000,
};

#define        ONE   TBL_LOG_P[0]
#define        TWO52 TBL_LOG_P[1]
#define        LN2HI TBL_LOG_P[2]
#define        LN2LO TBL_LOG_P[3]
#define        A1    TBL_LOG_P[4]
#define        A2    TBL_LOG_P[5]
#define        A3    TBL_LOG_P[6]
#define        A4    TBL_LOG_P[7]
#define        A5    TBL_LOG_P[8]
#define        A6    TBL_LOG_P[9]
#define        A7    TBL_LOG_P[10]
#define        A8    TBL_LOG_P[11]
#define        A9    TBL_LOG_P[12]
#define        A10   TBL_LOG_P[13]
#define        A11   TBL_LOG_P[14]
#define        A12   TBL_LOG_P[15]
#define        B1    TBL_LOG_P[16]
#define        B2    TBL_LOG_P[17]
#define        B3    TBL_LOG_P[18]
#define        B4    TBL_LOG_P[19]
#define        B5    TBL_LOG_P[20]
#define        B6    TBL_LOG_P[21]
#define        B7    TBL_LOG_P[22]
#define        B8    TBL_LOG_P[23]

static const double _TBL_log[] = {
9.47265623608246343e-02, 1.05567010464380857e+01, -2.35676082856530300e+00,
9.66796869131412717e-02, 1.03434344062203838e+01, -2.33635196153499791e+00,
9.86328118117651004e-02, 1.01386139321308306e+01, -2.31635129573594156e+00,
1.00585936733578435e-01, 9.94174764856737347e+00, -2.29674282498938709e+00,
1.02539062499949152e-01, 9.75238095238578850e+00, -2.27751145544242561e+00,
1.04492186859904843e-01, 9.57009351656812157e+00, -2.25864297726331742e+00,
1.06445312294918631e-01, 9.39449543094380957e+00, -2.24012392529694537e+00,
1.08398437050250693e-01, 9.22522526350104144e+00, -2.22194160843615762e+00,
1.10351562442130582e-01, 9.06194690740703912e+00, -2.20408398741152212e+00,
1.12304686894746625e-01, 8.90434787407592943e+00, -2.18653968262558962e+00,
1.14257811990227776e-01, 8.75213679118525256e+00, -2.16929787526329321e+00,
1.16210936696872255e-01, 8.60504207627572093e+00, -2.15234831939887172e+00,
1.18164061975360682e-01, 8.46280995492959498e+00, -2.13568126444263484e+00,
1.20117187499996322e-01, 8.32520325203277523e+00, -2.11928745022706622e+00,
1.22070312499895098e-01, 8.19200000000703987e+00, -2.10315806829801133e+00,
1.24023436774175100e-01, 8.06299217317146599e+00, -2.08728472499318229e+00,
1.26953123746900931e-01, 7.87692315467275872e+00, -2.06393736501443570e+00,
1.30859374098123454e-01, 7.64179109744297769e+00, -2.03363201254049386e+00,
1.34765623780674720e-01, 7.42028992220936967e+00, -2.00421812948999545e+00,
1.38671874242985771e-01, 7.21126764500034501e+00, -1.97564475345722457e+00,
1.42578124148616536e-01, 7.01369867201821506e+00, -1.94786518986246371e+00,
1.46484374166731490e-01, 6.82666670549979404e+00, -1.92083651719164372e+00,
1.50390624434435488e-01, 6.64935067435644189e+00, -1.89451920694646070e+00,
1.54296874339723084e-01, 6.48101268596180624e+00, -1.86887677685174936e+00,
1.58203124999987427e-01, 6.32098765432149001e+00, -1.84387547036714849e+00,
1.62109374999815342e-01, 6.16867469880220742e+00, -1.81948401724404896e+00,
1.66015624243955634e-01, 6.02352943919619310e+00, -1.79567337310324682e+00,
1.69921874302298687e-01, 5.88505749542848644e+00, -1.77241651049093640e+00,
1.73828124315277527e-01, 5.75280901142480605e+00, -1.74968825924644555e+00,
1.77734374286237506e-01, 5.62637364896854919e+00, -1.72746512253855222e+00,
1.81640624146994889e-01, 5.50537636993989743e+00, -1.70572513658236602e+00,
1.85546874316304788e-01, 5.38947370406942916e+00, -1.68444773712372431e+00,
1.89453124405085355e-01, 5.27835053203882509e+00, -1.66361364967629299e+00,
1.93359374570531595e-01, 5.17171718320401652e+00, -1.64320477712600699e+00,
1.97265624263334577e-01, 5.06930694962380368e+00, -1.62320411193263148e+00,
2.01171874086291030e-01, 4.97087380898513764e+00, -1.60359564135180399e+00,
2.05078123979995308e-01, 4.87619050044336610e+00, -1.58436427985572159e+00,
2.08984373896073439e-01, 4.78504675424820736e+00, -1.56549579585994181e+00,
2.12890623963011144e-01, 4.69724772930228163e+00, -1.54697674768135762e+00,
2.16796874723889421e-01, 4.61261261848719517e+00, -1.52879442500076479e+00,
2.20703124198150608e-01, 4.53097346778917753e+00, -1.51093680996032553e+00,
2.24609374375627030e-01, 4.45217392541970725e+00, -1.49339249945607477e+00,
2.28515625000094036e-01, 4.37606837606657528e+00, -1.47615069024134016e+00,
2.32421873924349737e-01, 4.30252102831546246e+00, -1.45920113655598627e+00,
2.36328123935216378e-01, 4.23140497774241098e+00, -1.44253408394829741e+00,
2.40234375000066919e-01, 4.16260162601510064e+00, -1.42614026966681173e+00,
2.44140623863132178e-01, 4.09600001907347711e+00, -1.41001089239381727e+00,
2.48046874999894917e-01, 4.03149606299383390e+00, -1.39413753858134015e+00,
2.53906248590769879e-01, 3.93846156032078243e+00, -1.37079018013412401e+00,
2.61718748558906533e-01, 3.82089554342693294e+00, -1.34048483059486401e+00,
2.69531249159214337e-01, 3.71014493910979404e+00, -1.31107094300173976e+00,
2.77343749428383191e-01, 3.60563381024826013e+00, -1.28249756949928795e+00,
2.85156249289339359e-01, 3.50684932380819214e+00, -1.25471800582335113e+00,
2.92968749999700462e-01, 3.41333333333682321e+00, -1.22768933094427446e+00,
3.00781248554318814e-01, 3.32467534065511261e+00, -1.20137202743229921e+00,
3.08593748521894806e-01, 3.24050634463533127e+00, -1.17572959680235023e+00,
3.16406249999639899e-01, 3.16049382716409077e+00, -1.15072828980826181e+00,
3.24218749999785061e-01, 3.08433734939963511e+00, -1.12633683668362750e+00,
3.32031248841858584e-01, 3.01176471638753718e+00, -1.10252619147729547e+00,
3.39843749265406558e-01, 2.94252874199264314e+00, -1.07926932798654107e+00,
3.47656249999834799e-01, 2.87640449438338930e+00, -1.05654107474789782e+00,
3.55468749999899247e-01, 2.81318681318761055e+00, -1.03431793796299587e+00,
3.63281249999864997e-01, 2.75268817204403371e+00, -1.01257795132667816e+00,
3.71093749064121570e-01, 2.69473684890124421e+00, -9.91300555400967731e-01,
3.78906249999751032e-01, 2.63917525773369288e+00, -9.70466465976836723e-01,
3.86718748879039009e-01, 2.58585859335407608e+00, -9.50057597243619156e-01,
3.94531249999987899e-01, 2.53465346534661240e+00, -9.30056927638333697e-01,
4.02343749999485523e-01, 2.48543689320706163e+00, -9.10448456251205407e-01,
4.10156249578856991e-01, 2.43809524059864202e+00, -8.91217095348825872e-01,
4.17968749447214571e-01, 2.39252336765021800e+00, -8.72348611340208357e-01,
4.25781248601723117e-01, 2.34862386092395203e+00, -8.53829565534445223e-01,
4.33593749393073047e-01, 2.30630630953458038e+00, -8.35647244566987801e-01,
4.41406248572254134e-01, 2.26548673299152270e+00, -8.17789629001761220e-01,
4.49218749348472501e-01, 2.22608695975035964e+00, -8.00245317566669279e-01,
4.57031249277175089e-01, 2.18803419149470768e+00, -7.83003511263371976e-01,
4.64843748529596368e-01, 2.15126051100659366e+00, -7.66053954531254355e-01,
4.72656248830947701e-01, 2.11570248457175136e+00, -7.49386901356188240e-01,
4.80468748609962581e-01, 2.08130081902951236e+00, -7.32993092000230995e-01,
4.88281249241778237e-01, 2.04800000318021258e+00, -7.16863708730099525e-01,
4.96093748931098810e-01, 2.01574803583926521e+00, -7.00990360175606675e-01,
5.07812497779701388e-01, 1.96923077784079825e+00, -6.77642998396260410e-01,
5.23437498033319737e-01, 1.91044776837204044e+00, -6.47337648285891021e-01,
5.39062498006593560e-01, 1.85507247062801328e+00, -6.17923763020271188e-01,
5.54687498964024250e-01, 1.80281690477552603e+00, -5.89350388745976339e-01,
5.70312499806522322e-01, 1.75342465812909332e+00, -5.61570823110474571e-01,
5.85937497921867001e-01, 1.70666667271966777e+00, -5.34542153929987052e-01,
6.01562498226483444e-01, 1.66233766723853860e+00, -5.08224845014116688e-01,
6.17187498682654212e-01, 1.62025316801528496e+00, -4.82582413587029357e-01,
6.32812500000264566e-01, 1.58024691357958624e+00, -4.57581109246760320e-01,
6.48437499353274216e-01, 1.54216867623689291e+00, -4.33189657120379490e-01,
6.64062498728508976e-01, 1.50588235582451335e+00, -4.09379009344016609e-01,
6.79687498865382267e-01, 1.47126437027210688e+00, -3.86122146934356092e-01,
6.95312498728747119e-01, 1.43820224982050338e+00, -3.63393896015796081e-01,
7.10937499999943157e-01, 1.40659340659351906e+00, -3.41170757402847080e-01,
7.26562499999845568e-01, 1.37634408602179792e+00, -3.19430770766573779e-01,
7.42187500000120126e-01, 1.34736842105241350e+00, -2.98153372318914478e-01,
7.57812499999581890e-01, 1.31958762886670744e+00, -2.77319285416786077e-01,
7.73437498602746576e-01, 1.29292929526503420e+00, -2.56910415591577124e-01,
7.89062500000142664e-01, 1.26732673267303819e+00, -2.36909747078176913e-01,
8.04687500000259015e-01, 1.24271844660154174e+00, -2.17301275689659512e-01,
8.20312499999677036e-01, 1.21904761904809900e+00, -1.98069913762487504e-01,
8.35937499999997113e-01, 1.19626168224299478e+00, -1.79201429457714445e-01,
8.51562499999758749e-01, 1.17431192660583728e+00, -1.60682381690756770e-01,
8.67187500000204725e-01, 1.15315315315288092e+00, -1.42500062607046951e-01,
8.82812500000407896e-01, 1.13274336283133503e+00, -1.24642445206814556e-01,
8.98437499999816813e-01, 1.11304347826109651e+00, -1.07098135556570995e-01,
9.14062499999708455e-01, 1.09401709401744296e+00, -8.98563291221800009e-02,
9.29687500000063949e-01, 1.07563025210076635e+00, -7.29067708080189947e-02,
9.45312499999844014e-01, 1.05785123966959604e+00, -5.62397183230410880e-02,
9.60937500000120459e-01, 1.04065040650393459e+00, -3.98459085470743157e-02,
9.76562499999976685e-01, 1.02400000000002445e+00, -2.37165266173399170e-02,
9.92187500000169420e-01, 1.00787401574785940e+00, -7.84317746085513856e-03,
1.01562500000004907e+00, 9.84615384615337041e-01,  1.55041865360135717e-02,
1.04687500000009237e+00, 9.55223880596930641e-01,  4.58095360313824362e-02,
1.07812500000002154e+00, 9.27536231884039442e-01,  7.52234212376075018e-02,
1.10937499999982481e+00, 9.01408450704367703e-01,  1.03796793681485644e-01,
1.14062500000007416e+00, 8.76712328767066285e-01,  1.31576357788784293e-01,
1.17187500000009659e+00, 8.53333333333263000e-01,  1.58605030176721007e-01,
1.20312499999950173e+00, 8.31168831169175393e-01,  1.84922338493597849e-01,
1.23437500000022027e+00, 8.10126582278336449e-01,  2.10564769107528083e-01,
1.26562500000064615e+00, 7.90123456789720069e-01,  2.35566071313277448e-01,
1.29687500000144706e+00, 7.71084337348537208e-01,  2.59957524438041876e-01,
1.32812499999945932e+00, 7.52941176470894757e-01,  2.83768173130237500e-01,
1.35937500055846350e+00, 7.35632183605830825e-01,  3.07025035705735583e-01,
1.39062499999999467e+00, 7.19101123595508374e-01,  3.29753286372464149e-01,
1.42187500000017564e+00, 7.03296703296616421e-01,  3.51976423157301710e-01,
1.45312500161088876e+00, 6.88172042247866766e-01,  3.73716410902152685e-01,
1.48437500134602307e+00, 6.73684209915422660e-01,  3.94993809147663466e-01,
1.51562499999932343e+00, 6.59793814433284220e-01,  4.15827895143264570e-01,
1.54687500000028200e+00, 6.46464646464528614e-01,  4.36236766775100371e-01,
1.57812500000061906e+00, 6.33663366336385092e-01,  4.56237433481979870e-01,
1.60937500243255216e+00, 6.21359222361793417e-01,  4.75845906381452632e-01,
1.64062500000026312e+00, 6.09523809523711768e-01,  4.95077266798011895e-01,
1.67187500000027911e+00, 5.98130841121395473e-01,  5.13945751102401260e-01,
1.70312500224662178e+00, 5.87155962528224662e-01,  5.32464800188589216e-01,
1.73437500283893620e+00, 5.76576575632799071e-01,  5.50647119589526390e-01,
1.76562500399259092e+00, 5.66371680135198341e-01,  5.68504737613959144e-01,
1.79687500443862880e+00, 5.56521737755718449e-01,  5.86049047473771623e-01,
1.82812500114411280e+00, 5.47008546666207462e-01,  6.03290852063923744e-01,
1.85937500250667465e+00, 5.37815125325376786e-01,  6.20240411099985067e-01,
1.89062500504214515e+00, 5.28925618424108568e-01,  6.36907464903988974e-01,
1.92187500371610143e+00, 5.20325202245941476e-01,  6.53301273946326866e-01,
1.95312500494870611e+00, 5.11999998702726389e-01,  6.69430656476366792e-01,
1.98437500351688123e+00, 5.03937006980894941e-01,  6.85304004871206018e-01,
2.03125000000003997e+00, 4.92307692307682621e-01,  7.08651367095930240e-01,
2.09375000579615866e+00, 4.77611938976327366e-01,  7.38956719359554093e-01,
2.15625000000061062e+00, 4.63768115941897652e-01,  7.68370601797816022e-01,
2.21875000323311955e+00, 4.50704224695355204e-01,  7.96943975698769513e-01,
2.28125000853738547e+00, 4.38356162743050726e-01,  8.24723542091080120e-01,
2.34374999999916556e+00, 4.26666666666818573e-01,  8.51752210736227866e-01,
2.40625000438447856e+00, 4.15584414827170512e-01,  8.78069520876078258e-01,
2.46875000884389584e+00, 4.05063289688167072e-01,  9.03711953249632494e-01,
2.53124999999940403e+00, 3.95061728395154743e-01,  9.28713251872476775e-01,
2.59375000434366632e+00, 3.85542168029044230e-01,  9.53104706671537905e-01,
2.65625000734081196e+00, 3.76470587194880080e-01,  9.76915356454189698e-01,
2.71875000787161980e+00, 3.67816090889081959e-01,  1.00017221875016560e+00,
2.78125001557333462e+00, 3.59550559784484969e-01,  1.02290047253181449e+00,
2.84375001147093220e+00, 3.51648350229895601e-01,  1.04512360775085789e+00,
2.90625000771072894e+00, 3.44086020592463127e-01,  1.06686359300668343e+00,
2.96875001371853831e+00, 3.36842103706616824e-01,  1.08814099342179560e+00,
3.03125000512624965e+00, 3.29896906658595002e-01,  1.10897507739479018e+00,
3.09375001373132807e+00, 3.23232321797685962e-01,  1.12938395177327244e+00,
3.15625001204422961e+00, 3.16831681959289180e-01,  1.14938461785752644e+00,
3.21875000888250318e+00, 3.10679610793130057e-01,  1.16899308818952186e+00,
3.28125000000102052e+00, 3.04761904761809976e-01,  1.18822444735810784e+00,
3.34375001587649123e+00, 2.99065419140752298e-01,  1.20709293641028914e+00,
3.40625000791328070e+00, 2.93577980969346064e-01,  1.22561198175258212e+00,
3.46875000615970519e+00, 2.88288287776354346e-01,  1.24379430028837845e+00,
3.53125000516822674e+00, 2.83185840293502689e-01,  1.26165191737618265e+00,
3.59375001425228779e+00, 2.78260868461675415e-01,  1.27919622952937750e+00,
3.65625001719730669e+00, 2.73504272217836075e-01,  1.29643803670156643e+00,
3.71875000856489324e+00, 2.68907562405871714e-01,  1.31338759261496740e+00,
3.78125001788371806e+00, 2.64462808666557803e-01,  1.33005464752659286e+00,
3.84375001532508964e+00, 2.60162600588744020e-01,  1.34644845655970613e+00,
3.90625000429340918e+00, 2.55999999718627136e-01,  1.36257783560168733e+00,
3.96875001912740766e+00, 2.51968502722644594e-01,  1.37845118847836900e+00,
4.06250002536431332e+00, 2.46153844616978895e-01,  1.40179855389937913e+00,
4.18750001743208244e+00, 2.38805969155131859e-01,  1.43210390131407017e+00,
4.31250002253733200e+00, 2.31884056759177282e-01,  1.46151778758352613e+00,
4.43750000671406397e+00, 2.25352112335092170e-01,  1.49009115631456268e+00,
4.56250002627485340e+00, 2.19178080929562313e-01,  1.51787072466748185e+00,
4.68750001185115028e+00, 2.13333332793974317e-01,  1.54489939382477459e+00,
4.81250001682742301e+00, 2.07792207065640028e-01,  1.57121670311050998e+00,
4.93750000000042366e+00, 2.02531645569602875e-01,  1.59685913022732606e+00,
5.06249999999927613e+00, 1.97530864197559108e-01,  1.62186043243251454e+00,
5.18750002327641901e+00, 1.92771083472381588e-01,  1.64625189004383721e+00,
5.31250002381002329e+00, 1.88235293273997795e-01,  1.67006253873242194e+00,
5.43750000000577405e+00, 1.83908045976816203e-01,  1.69331939641586438e+00,
5.56250002193114934e+00, 1.79775280190080267e-01,  1.71604765143503712e+00,
5.68749999999938005e+00, 1.75824175824194989e-01,  1.73827078427695980e+00,
5.81250002749782002e+00, 1.72043009938785768e-01,  1.76001077564428243e+00,
5.93749999999874767e+00, 1.68421052631614471e-01,  1.78128816936054868e+00,
6.06250001966917473e+00, 1.64948453073088669e-01,  1.80212225950800153e+00,
6.18750003004243609e+00, 1.61616160831459688e-01,  1.82253113275015188e+00,
6.31250002448351388e+00, 1.58415840969730465e-01,  1.84253179848005466e+00,
6.43750001359968849e+00, 1.55339805497076044e-01,  1.86214026810242750e+00,
6.56250003345742350e+00, 1.52380951604072529e-01,  1.88137163301601618e+00,
6.68750002403557531e+00, 1.49532709742937614e-01,  1.90024011581622965e+00,
6.81250003423489581e+00, 1.46788990088028509e-01,  1.91875916501466826e+00,
6.93750003062940923e+00, 1.44144143507740546e-01,  1.93694148348760287e+00,
7.06250002747386052e+00, 1.41592919803171097e-01,  1.95479910036266347e+00,
7.18750003617887856e+00, 1.39130434082284093e-01,  1.97234341115705192e+00,
7.31250000000050537e+00, 1.36752136752127301e-01,  1.98958521255804399e+00,
7.43750002212249761e+00, 1.34453781112678528e-01,  2.00653477384620160e+00,
7.56250003604752941e+00, 1.32231404328381430e-01,  2.02320182812357530e+00,
7.68750005007207449e+00, 1.30081299965731312e-01,  2.03959563964607682e+00,
7.81249996125652668e+00, 1.28000000634773070e-01,  2.05572501010335529e+00,
7.93750005224239974e+00, 1.25984251139310915e-01,  2.07159837080052966e+00,
8.12500004244456164e+00, 1.23076922433975874e-01,  2.09494573343974722e+00,
8.37500006149772425e+00, 1.19402984197849338e-01,  2.12525108505414195e+00,
8.62500006593247370e+00, 1.15942028099206410e-01,  2.15466497056176820e+00,
8.87500007743793873e+00, 1.12676055354884341e-01,  2.18323834408688100e+00,
9.12500001754142609e+00, 1.09589040885222130e-01,  2.21101790139090326e+00,
9.37500007707016181e+00, 1.06666665789779500e-01,  2.23804658007729174e+00,
9.62500004426353151e+00, 1.03896103418305616e-01,  2.26436388477265638e+00,
9.87500006518495788e+00, 1.01265822116353585e-01,  2.29000631738819393e+00,
1.01250000000026539e+01, 9.87654320987395445e-02,  2.31500761299286495e+00,
1.03750000409819823e+01, 9.63855417879450060e-02,  2.33939907006683256e+00,
1.06250000362555337e+01, 9.41176467376672460e-02,  2.36320971822276604e+00,
1.08750000879032314e+01, 9.19540222452362582e-02,  2.38646658505780351e+00,
1.11250000697274576e+01, 8.98876398860551373e-02,  2.40919483431994053e+00,
1.13750000462194141e+01, 8.79120875548795450e-02,  2.43141796890025930e+00,
1.16250000714972366e+01, 8.60215048472860316e-02,  2.45315795762371991e+00,
1.18750000788855150e+01, 8.42105257563797310e-02,  2.47443535656369562e+00,
1.21250000895724916e+01, 8.24742261948517991e-02,  2.49526944421096886e+00,
1.23750000985058719e+01, 8.08080801648427965e-02,  2.51567831641482442e+00,
1.26250000894226950e+01, 7.92079202310506381e-02,  2.53567898224440924e+00,
1.28750000768594433e+01, 7.76699024489580225e-02,  2.55528745251946532e+00,
1.31250000578007420e+01, 7.61904758549435401e-02,  2.57451881288155349e+00,
1.33750000809310077e+01, 7.47663546877819496e-02,  2.59338729883298669e+00,
1.36250000915049636e+01, 7.33944949199294983e-02,  2.61190634726526838e+00,
1.38750000830616607e+01, 7.20720716406179490e-02,  2.63008866561892418e+00,
1.41249999999960103e+01, 7.07964601770111474e-02,  2.64794627703222218e+00,
1.43750000290097564e+01, 6.95652172509168693e-02,  2.66549058870148414e+00,
1.46250000868097665e+01, 6.83760679702078294e-02,  2.68273239905363070e+00,
1.48750000966053975e+01, 6.72268903196987927e-02,  2.69968195792617394e+00,
1.51250001097012756e+01, 6.61157019998031836e-02,  2.71634901116988203e+00,
1.53750000510427132e+01, 6.50406501905787804e-02,  2.73274281701243282e+00,
1.56250001080665442e+01, 6.39999995573594382e-02,  2.74887220253872400e+00,
1.58750000434989929e+01, 6.29921258116476201e-02,  2.76474554751884938e+00,
1.62500000641781739e+01, 6.15384612954199342e-02,  2.78809291272517257e+00,
1.67500001015987401e+01, 5.97014921751882754e-02,  2.81839826433667184e+00,
1.72500001048300184e+01, 5.79710141404578272e-02,  2.84781214955447126e+00,
1.77500001262529885e+01, 5.63380277682904579e-02,  2.87638552303426920e+00,
1.82500001543340602e+01, 5.47945200845665337e-02,  2.90416508848516131e+00,
1.87500001096404212e+01, 5.33333330214672482e-02,  2.93119375826390893e+00,
1.92500001680268191e+01, 5.19480514946147609e-02,  2.95751106946245912e+00,
1.97500000329124035e+01, 5.06329113080278073e-02,  2.98315349301358168e+00,
2.02500001270002485e+01, 4.93827157396732261e-02,  3.00815479982416534e+00,
2.07500001519906796e+01, 4.81927707313324349e-02,  3.03254625400155930e+00,
2.12500001425219267e+01, 4.70588232137922752e-02,  3.05635690207734001e+00,
2.17500000758314478e+01, 4.59770113339538697e-02,  3.07961376102119644e+00,
2.22500001767207358e+01, 4.49438198677525880e-02,  3.10234201655475417e+00,
2.27500001365873317e+01, 4.39560436921389575e-02,  3.12456515140079816e+00,
2.32500001697599998e+01, 4.30107523741288036e-02,  3.14630513933487066e+00,
2.37500001766865303e+01, 4.21052628446554611e-02,  3.16758253792008304e+00,
};

#define        HIWORD                1
#define        LOWORD                0

//! Fast Log - Stripped version of libm's Log(x)
/*! This is the libm log stripped down with slightly less precision and
    no handling for inf, 0 etc.

    For an idea of how this works see:

    http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_07_dinechin.pdf

    This is about 15% faster than the standard log()
*/
#ifdef INVT_USE_FASTMATH
inline double fastLog(const double x)
{
  union
  {
    int    i[2];
    double x;
  } u;

  u.x = x;

  /* get exponent part, store in dn */
  const int hx = u.i[HIWORD];     // Get the top half of the double
  const int ix = hx & 0x7fffffff; // condition ix
  double dn    = (double)((ix >> 20) - 0x3ff); // Shift to get E

  /* Now we take advantage of the float point identity:

     log(x) = E × log(2) + log(y)

     Where E is the exponent part of the float and y is from the mentisa.

     y is then set to be on the interval from 11/16 to 23/16 using
     a bit shift to divide it by 2.

     log(y) needs to be scaled and approximated on a table. Then we use the
     Horner scheme polynomial to approximated the final log
  */

  const double dn1 = dn * LN2HI;                       // Low Order E * log(2)
  int i            = (ix & 0x000fffff) | 0x3ff00000;   // scale x to [1,2]
  u.i[HIWORD]      = i;                                // copy back E
  i                = (i - 0x3fb80000) >> 15;           // mantisa table addr
  const double *tb = (double *)_TBL_log + (i + i + i); // Get table const's
  const double s   = (u.x - tb[0]) * tb[1];            // initial estimate
  dn               = dn * LN2LO + tb[2];               // High order E * log(2)

  // 8th order Polynomial in Horner form
  return (dn1 + (dn + ((B1 * s) * (B2 + s * (B3 + s))) *
                 (((B4 + s * B5) + (s * s) * (B6 + s)) *
                  (B7 + s * (B8 + s)))));

};
#else
inline double fastLog(const double x)
{
  return log(x);
}
#endif
#undef        ONE
#undef        TWO52
#undef        LN2HI
#undef        LN2LO
#undef        A1
#undef        A2
#undef        A3
#undef        A4
#undef        A5
#undef        A6
#undef        A7
#undef        A8
#undef        A9
#undef        A10
#undef        A11
#undef        A12
#undef        B1
#undef        B2
#undef        B3
#undef        B4
#undef        B5
#undef        B6
#undef        B7
#undef        B8
#undef        HIWORD
#undef        LOWORD

#endif