sicif.h

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/*                          sicif.c
 *
 *  Sine and cosine integrals
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, Ci, Si;
 *
 * sicif( x, &Si, &Ci );
 *
 *
 * DESCRIPTION:
 *
 * Evaluates the integrals
 *
 *                          x
 *                          -
 *                         |  cos t - 1
 *   Ci(x) = eul + ln x +  |  --------- dt,
 *                         |      t
 *                        -
 *                         0
 *             x
 *             -
 *            |  sin t
 *   Si(x) =  |  ----- dt
 *            |    t
 *           -
 *            0
 *
 * where eul = 0.57721566490153286061 is Euler's constant.
 * The integrals are approximated by rational functions.
 * For x > 8 auxiliary functions f(x) and g(x) are employed
 * such that
 *
 * Ci(x) = f(x) sin(x) - g(x) cos(x)
 * Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)
 *
 *
 * ACCURACY:
 *    Test interval = [0,50].
 * Absolute error, except relative when > 1:
 * arithmetic   function   # trials      peak         rms
 *    IEEE        Si        30000       2.1e-7      4.3e-8
 *    IEEE        Ci        30000       3.9e-7      2.2e-8
 */

/*
 Cephes Math Library Release 2.1:  January, 1989
 Copyright 1984, 1987, 1989 by Stephen L. Moshier
 Direct inquiries to 30 Frost Street, Cambridge, MA 02140
 */

#include "vdt/vdtMath.h"

static const float SN[] = {
    -8.39167827910303881427E-11,
    4.62591714427012837309E-8,
    -9.75759303843632795789E-6,
    9.76945438170435310816E-4,
    -4.13470316229406538752E-2,
    1.00000000000000000302E0,
};
static const float SD[] = {
    2.03269266195951942049E-12,
    1.27997891179943299903E-9,
    4.41827842801218905784E-7,
    9.96412122043875552487E-5,
    1.42085239326149893930E-2,
    9.99999999999999996984E-1,
};

static const float CN[] = {
    2.02524002389102268789E-11,
    -1.35249504915790756375E-8,
    3.59325051419993077021E-6,
    -4.74007206873407909465E-4,
    2.89159652607555242092E-2,
    -1.00000000000000000080E0,
};
static const float CD[] = {
    4.07746040061880559506E-12,
    3.06780997581887812692E-9,
    1.23210355685883423679E-6,
    3.17442024775032769882E-4,
    5.10028056236446052392E-2,
    4.00000000000000000080E0,
};

static const float FN4[] = {
    4.23612862892216586994E0,
    5.45937717161812843388E0,
    1.62083287701538329132E0,
    1.67006611831323023771E-1,
    6.81020132472518137426E-3,
    1.08936580650328664411E-4,
    5.48900223421373614008E-7,
};
static const float FD4[] = {
    /*  1.00000000000000000000E0,*/
    8.16496634205391016773E0,
    7.30828822505564552187E0,
    1.86792257950184183883E0,
    1.78792052963149907262E-1,
    7.01710668322789753610E-3,
    1.10034357153915731354E-4,
    5.48900252756255700982E-7,
};

static const float FN8[] = {
    4.55880873470465315206E-1,
    7.13715274100146711374E-1,
    1.60300158222319456320E-1,
    1.16064229408124407915E-2,
    3.49556442447859055605E-4,
    4.86215430826454749482E-6,
    3.20092790091004902806E-8,
    9.41779576128512936592E-11,
    9.70507110881952024631E-14,
};
static const float FD8[] = {
    /*  1.00000000000000000000E0,*/
    9.17463611873684053703E-1,
    1.78685545332074536321E-1,
    1.22253594771971293032E-2,
    3.58696481881851580297E-4,
    4.92435064317881464393E-6,
    3.21956939101046018377E-8,
    9.43720590350276732376E-11,
    9.70507110881952025725E-14,
};

static const float GN4[] = {
    8.71001698973114191777E-2,
    6.11379109952219284151E-1,
    3.97180296392337498885E-1,
    7.48527737628469092119E-2,
    5.38868681462177273157E-3,
    1.61999794598934024525E-4,
    1.97963874140963632189E-6,
    7.82579040744090311069E-9,
};
static const float GD4[] = {
    /*  1.00000000000000000000E0,*/
    1.64402202413355338886E0,
    6.66296701268987968381E-1,
    9.88771761277688796203E-2,
    6.22396345441768420760E-3,
    1.73221081474177119497E-4,
    2.02659182086343991969E-6,
    7.82579218933534490868E-9,
};

static const float GN8[] = {
    6.97359953443276214934E-1,
    3.30410979305632063225E-1,
    3.84878767649974295920E-2,
    1.71718239052347903558E-3,
    3.48941165502279436777E-5,
    3.47131167084116673800E-7,
    1.70404452782044526189E-9,
    3.85945925430276600453E-12,
    3.14040098946363334640E-15,
};
static const float GD8[] = {
    /*  1.00000000000000000000E0,*/
    1.68548898811011640017E0,
    4.87852258695304967486E-1,
    4.67913194259625806320E-2,
    1.90284426674399523638E-3,
    3.68475504442561108162E-5,
    3.57043223443740838771E-7,
    1.72693748966316146736E-9,
    3.87830166023954706752E-12,
    3.14040098946363335242E-15,
};

inline float polevlf(float xx, const float *coef, int N) {
  float ans, x;
  const float *p;
  int i;

  x = xx;
  p = coef;
  ans = *p++;

  i = N;
  do
    ans = ans * x + *p++;
  while (--i);

  return (ans);
}

/*                          p1evl() */
/*                                          N
 * Evaluate polynomial when coefficient of x  is 1.0.
 * Otherwise same as polevl.
 */
inline float p1evlf(float xx, const float *coef, int N) {
  float ans, x;
  const float *p;
  int i;

  x = xx;
  p = coef;
  ans = x + *p++;
  i = N - 1;

  do
    ans = ans * x + *p++;
  while (--i);

  return (ans);
}

inline int sicif(float xx, float &si, float &ci) {
  const float MAXNUMF = 1.7014117331926442990585209174225846272e38;
  const float PIO2F = 1.5707963267948966192;
  // const float MACHEPF = 5.9604644775390625E-8;
  const float EUL = 0.57721566490153286061;

  float x, z, c, s, f, g;
  int sign;

  x = xx;
  if (x < 0.0f) {
    sign = -1;
    x = -x;
  } else
    sign = 0;

  if (x == 0.0f) {
    si = 0.0;
    ci = -MAXNUMF;
    return (0);
  }

  if (x > 1.0e9f) {
    float su, cu;
    vdt::fast_sincosf(x, su, cu);
    si = PIO2F - cu / x;
    ci = su / x;
    return (0);
  }

  if (x > 4.0f)
    goto asympt;

  z = x * x;
  s = x * polevlf(z, SN, 5) / polevlf(z, SD, 5);
  c = z * polevlf(z, CN, 5) / polevlf(z, CD, 5);

  if (sign)
    s = -s;
  si = s;
  ci = EUL + vdt::fast_logf(x) + c; /* real part if x < 0 */
  return (0);

  /* The auxiliary functions are:
    *
    *
    * *si = *si - PIO2;
    * c = cos(x);
    * s = sin(x);
    *
    * t = *ci * s - *si * c;
    * a = *ci * c + *si * s;
    *
    * *si = t;
    * *ci = -a;
    */

asympt:
  vdt::fast_sincosf(x, s, c);
  z = 1.0f / (x * x);
  if (x < 8.0f) {
    f = polevlf(z, FN4, 6) / (x * p1evlf(z, FD4, 7));
    g = z * polevlf(z, GN4, 7) / p1evlf(z, GD4, 7);
  } else {
    f = polevlf(z, FN8, 8) / (x * p1evlf(z, FD8, 8));
    g = z * polevlf(z, GN8, 8) / p1evlf(z, GD8, 9);
  }
  si = PIO2F - f * c - g * s;
  if (sign)
    si = -(si);
  ci = f * s - g * c;

  return (0);
}