/**
 * Cylindrical Bessel functions of integral order.
 *
 * Copyright:
 *     Based on the CEPHES math library, which is Copyright (C) 1994 Stephen L.
 *       Moshier (moshier@world.std.com).
 *     Some parts copyright (c) 2009-2016 dunnhumby Germany GmbH.
 *     All rights reserved.
 *
 * License:
 *     Tango Dual License: 3-Clause BSD License / Academic Free License v3.0.
 *     See LICENSE_TANGO.txt for details.
 *
 * Authors: Stephen L. Moshier (original C code). Conversion to D by Don Clugston
 *
 */

module ocean.math.Bessel;

import ocean.math.Math;
import ocean.math.IEEE;
import ocean.core.Verify;

version (unittest) import ocean.core.Test;

private {   // Rational polynomial approximations to j0, y0, j1, y1.

// sqrt(j0^2(1/x^2) + y0^2(1/x^2)) = z P(z**2)/Q(z**2), z(x) = 1/sqrt(x)
// Peak error =  1.80e-20
static immutable real[] j0modulusn = [ 0x1.154700ea96e79656p-7, 0x1.72244b6e998cd6fp-4,
   0x1.6ebccf42e9c19fd2p-1, 0x1.6bd844e89cbd639ap+1, 0x1.e812b377c75ebc96p+2,
   0x1.46d69ca24ce76686p+3, 0x1.b756f7234cc67146p+2, 0x1.943a7471eaa50ab2p-2
];

static immutable real[] j0modulusd = [ 0x1.5b84007c37011506p-7, 0x1.cfe76758639bdab4p-4,
   0x1.cbfa09bf71bafc7ep-1, 0x1.c8eafb3836f2eeb4p+1, 0x1.339db78060eb706ep+3,
   0x1.a06530916be8bc7ap+3, 0x1.23bfe7f67a54893p+3,  1.0
];


// atan(y0(x)/j0(x)) = x - pi/4 + x P(x**2)/Q(x**2)
// Peak error =  2.80e-21. Relative error spread =  5.5e-1
static immutable real[] j0phasen = [ -0x1.ccbaf3865bb0985ep-22, -0x1.3a6b175e64bdb82ep-14,
   -0x1.06124b5310cdca28p-8, -0x1.3cebb7ab09cf1b14p-4, -0x1.00156ed209b43c6p-1,
   -0x1.78aa9ba4254ca20cp-1
];

static immutable real[] j0phased = [ 0x1.ccbaf3865bb09918p-19, 0x1.3b5b0e12900d58b8p-11,
   0x1.0897373ff9906f7ep-5, 0x1.450a5b8c552ade4ap-1, 0x1.123e263e7f0e96d2p+2,
   0x1.d82ecca5654be7d2p+2, 1.0
];


// j1(x) = (x^2-r0^2)(x^2-r1^2)(x^2-r2^2) x P(x**2)/Q(x**2), 0 <= x <= 9
// Peak error =  2e-21
static immutable real[] j1n = [ -0x1.2f494fa4e623b1bp+58, 0x1.8289f0a5f1e1a784p+52,
  -0x1.9d773ee29a52c6d8p+45, 0x1.e9394ff57a46071cp+37, -0x1.616c7939904a359p+29,
   0x1.424414b9ee5671eap+20, -0x1.6db34a9892d653e6p+10, 0x1.dcd7412d90a0db86p-1,
   -0x1.1444a1643199ee5ep-12
];

static immutable real[] j1d = [ 0x1.5a1e0a45eb67bacep+75, 0x1.35ee485d62f0ccbap+68,
   0x1.11ee7aad4e4bcd8p+60, 0x1.3adde5dead800244p+51, 0x1.041c413dfbab693p+42,
   0x1.4066d12193fcc082p+32, 0x1.24309d0dc2c4d42ep+22, 0x1.7115bea028dd75f2p+11,
   1.0
];

// sqrt(j1^2(1/x^2) + y1^2(1/x^2)) = z P(z**2)/Q(z**2), z(x) = 1/sqrt(x)
// Peak error =  1.35e=20, Relative error spread =  9.9e0
static immutable real [] j1modulusn = [ 0x1.059262020bf7520ap-6, 0x1.012ffc4d1f5cdbc8p-3,
   0x1.03c17ce18cae596p+0, 0x1.6e0414a7114ae3ccp+1, 0x1.cb047410d229cbc4p+2,
   0x1.4385d04bb718faaap+1, 0x1.914074c30c746222p-2, -0x1.42abe77f6b307aa6p+2
];

static immutable real [] j1modulusd = [ 0x1.47d4e6ad98d8246ep-6, 0x1.42562f48058ff904p-3,
   0x1.44985e2af35c6f9cp+0, 0x1.c6f4a03469c4ef6cp+1, 0x1.1829a060e8d604cp+3,
   0x1.44111c892f9cc84p+1, -0x1.d7c36d7f1e5aef6ap-1, -0x1.8eeafb1ac81c4c06p+2,
   1.0
];

// atan(y1(x)/j1(x))  =  x - 3pi/4 + z P(z**2)/Q(z**2), z(x) = 1/x
// Peak error =  4.83e-21. Relative error spread =  1.9e0
static immutable real [] j1phasen = [ 0x1.ca9f612d83aaa818p-20, 0x1.2e82fcfb7d0fee9ep-12,
   0x1.e28858c1e947506p-7, 0x1.12b8f96e5173d20ep-2, 0x1.965e6a013154c0ap+0,
   0x1.0156a25eaa0dd78p+1
];

static immutable real [] j1phased = [ 0x1.31bf961e57c71ae4p-18, 0x1.9464d8f2abf750a6p-11,
   0x1.446a786bac2131fp-5, 0x1.76caa8513919873cp-1, 0x1.2130b56bc1a563e4p+2,
   0x1.b3cc1a865259dfc6p+2, 0x1p+0
];

}

/***
 *  Bessel function of order zero
 *
 * Returns Bessel function of first kind, order zero of the argument.
 */

 /* The domain is divided into the intervals [0, 9] and
 * (9, infinity). In the first interval the rational approximation
 * is (x^2 - r^2) (x^2 - s^2) (x^2 - t^2) P7(x^2) / Q8(x^2),
 * where r, s, t are the first three zeros of the function.
 * In the second interval the expansion is in terms of the
 * modulus M0(x) = sqrt(J0(x)^2 + Y0(x)^2) and phase  P0(x)
 * = atan(Y0(x)/J0(x)).  M0 is approximated by sqrt(1/x)P7(1/x)/Q7(1/x).
 * The approximation to J0 is M0 * cos(x -  pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)).
 */
real cylBessel_j0(real x)
{

// j0(x) = (x^2-JZ1)(x^2-JZ2)(x^2-JZ3)P(x**2)/Q(x**2), 0 <= x <= 9
// Peak error =  8.49e-22. Relative error spread =  2.2e-3
static immutable real[] j0n = [ -0x1.3e8ff72b890d72d8p+59, 0x1.cc86e3755a4c803p+53,
 -0x1.0ea6f5bac6623616p+47, 0x1.532c6d94d36f2874p+39, -0x1.ef25a232f6c00118p+30,
   0x1.aa0690536c11fc2p+21, -0x1.94e67651cc57535p+11,  0x1.4bfe47ac8411eeb2p+0
];

static immutable real[] j0d = [ 0x1.0096dec5f6560158p+73, 0x1.11705db14995fb9cp+66,
   0x1.220a41c3daaa7a58p+58, 0x1.93c6b48d196c1082p+49, 0x1.9814684a10dbfda2p+40,
   0x1.36f20ec527fccda4p+31, 0x1.634596b9247fc34p+21, 0x1.1d3eb73f90657bfcp+11,
   1.0
];
    real xx, y, z, modulus, phase;

    xx = x * x;
    if ( xx < 81.0L ) {
        static immutable real [] JZ = [5.783185962946784521176L,
            30.47126234366208639908L, 7.488700679069518344489e1L];
        y = (xx - JZ[0]) * (xx - JZ[1]) * (xx - JZ[2]);
        y *= rationalPoly( xx, j0n, j0d);
        return y;
    }

    y = fabs(x);
    xx = 1.0/xx;
    phase = rationalPoly( xx, j0phasen, j0phased);

    z = 1.0/y;
    modulus = rationalPoly( z, j0modulusn, j0modulusd);

    y = modulus * cos( y -  PI_4 + z*phase) / sqrt(y);
    return y;
}

/**
 * Bessel function of the second kind, order zero
 * Also known as the cylindrical Neumann function, order zero.
 *
 * Returns Bessel function of the second kind, of order
 * zero, of the argument.
 */
real cylBessel_y0(real x)
{
/* The domain is divided into the intervals [0, 5>, [5,9> and
 * [9, infinity). In the first interval a rational approximation
 * R(x) is employed to compute y0(x)  = R(x) + 2/pi * log(x) * j0(x).
 *
 * In the second interval, the approximation is
 *     (x - p)(x - q)(x - r)(x - s)P7(x)/Q7(x)
 * where p, q, r, s are zeros of y0(x).
 *
 * The third interval uses the same approximations to modulus
 * and phase as j0(x), whence y0(x) = modulus * sin(phase).
 */

// y0(x) = 2/pi * log(x) * j0(x) + P(z**2)/Q(z**2), 0 <= x <= 5
// Peak error =  8.55e-22. Relative error spread =  2.7e-1
static immutable real[] y0n = [ -0x1.068026b402e2bf7ap+54, 0x1.3a2f7be8c4c8a03ep+55,
 -0x1.89928488d6524792p+51, 0x1.3e3ea2846f756432p+46, -0x1.c8be8d9366867c78p+39,
   0x1.43879530964e5fbap+32, -0x1.bee052fef72a5d8p+23, 0x1.e688c8fe417c24d8p+13
];

static immutable real[] y0d = [ 0x1.bc96c5351e564834p+57, 0x1.6821ac3b4c5209a6p+51,
   0x1.27098b571836ce64p+44, 0x1.41870d2a9b90aa76p+36, 0x1.00394fd321f52f48p+28,
   0x1.317ce3b16d65b27p+19, 0x1.0432b36efe4b20aep+10, 1.0
];

// y0(x) = (x-Y0Z1)(x-Y0Z2)(x-Y0Z3)(x-Y0Z4)P(x)/Q(x), 4.5 <= x <= 9
// Peak error =  2.35e-20. Relative error spread =  7.8e-13
static immutable real[] y059n = [ -0x1.0fce17d26a21f218p+19, -0x1.c6fc144765fdfaa8p+16,
   0x1.3e20237c53c7180ep+19, 0x1.7d14055ff6a493c4p+17, 0x1.b8b694729689d1f4p+12,
   -0x1.1e24596784b6c5cp+12, 0x1.35189cb3ece7ab46p+6, 0x1.9428b3f406b4aa08p+4,
   -0x1.791187b68dd4240ep+0, 0x1.8417216d568b325ep-6
];

static immutable real[] y059d = [ 0x1.17af71a3d4167676p+30, 0x1.a36abbb668c79d6cp+31,
 -0x1.4ff64a14ed73c4d6p+29, 0x1.9d427af195244ffep+26, -0x1.4e85bbbc8d2fd914p+23,
  0x1.ac59b523ae0bd16cp+19, -0x1.8ebda33eaac74518p+15, 0x1.16194a051cd55a12p+11,
   -0x1.f2d714ab48d1bd7ep+5, 1.0
];


    real xx, y, z, modulus, phase;

    if ( x < 0.0 ) return -real.max;
    xx = x * x;
    if ( xx < 81.0L ) {
        if ( xx < 20.25L ) {
            y = M_2_PI * log(x) * cylBessel_j0(x);
            y += rationalPoly( xx, y0n, y0d);
        } else {
            static immutable real [] Y0Z = [3.957678419314857868376e0L, 7.086051060301772697624e0L,
                1.022234504349641701900e1L, 1.336109747387276347827e1L];
            y = (x - Y0Z[0])*(x - Y0Z[1])*(x - Y0Z[2])*(x - Y0Z[3]);
            y *= rationalPoly( x, y059n, y059d);
        }
        return y;
    }

    y = fabs(x);
    xx = 1.0/xx;
    phase = rationalPoly( xx, j0phasen, j0phased);

    z = 1.0/y;
    modulus = rationalPoly( z, j0modulusn, j0modulusd);

    y = modulus * sin( y -  PI_4 + z*phase) / sqrt(y);
    return y;
}

/**
 *  Bessel function of order one
 *
 * Returns Bessel function of order one of the argument.
 */
real cylBessel_j1(real x)
{
/* The domain is divided into the intervals [0, 9] and
 * (9, infinity). In the first interval the rational approximation
 * is (x^2 - r^2) (x^2 - s^2) (x^2 - t^2) x P8(x^2) / Q8(x^2),
 * where r, s, t are the first three zeros of the function.
 * In the second interval the expansion is in terms of the
 * modulus M1(x) = sqrt(J1(x)^2 + Y1(x)^2) and phase  P1(x)
 * = atan(Y1(x)/J1(x)).  M1 is approximated by sqrt(1/x)P7(1/x)/Q8(1/x).
 * The approximation to j1 is M1 * cos(x -  3 pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)).
 */

    real xx, y, z, modulus, phase;

    xx = x * x;
    if ( xx < 81.0L ) {
        static immutable real [] JZ = [1.46819706421238932572e1L,
            4.92184563216946036703e1L, 1.03499453895136580332e2L];
        y = (xx - JZ[0]) * (xx - JZ[1]) * (xx - JZ[2]);
        y *= x * poly( xx, j1n) / poly( xx, j1d);
        return y;
    }
    y = fabs(x);
    xx = 1.0/xx;
    phase = rationalPoly( xx, j1phasen, j1phased);
    z = 1.0/y;
    modulus = rationalPoly( z, j1modulusn, j1modulusd);

    static immutable real M_3PI_4 = 3 * PI_4;

    y = modulus * cos( y -  M_3PI_4 + z*phase) / sqrt(y);
    if( x < 0 )
        y = -y;
    return y;
}

/**
 *  Bessel function of the second kind, order zero
 *
 * Returns Bessel function of the second kind, of order
 * zero, of the argument.
 */
real cylBessel_y1(real x)
{
verify(x>=0.0);

// TODO: should it return -infinity for x<0 ?
/* The domain is divided into the intervals [0, 4.5>, [4.5,9> and
 * [9, infinity). In the first interval a rational approximation
 * R(x) is employed to compute y0(x)  = R(x) + 2/pi * log(x) * j0(x).
 *
 * In the second interval, the approximation is
 *     (x - p)(x - q)(x - r)(x - s)P9(x)/Q10(x)
 * where p, q, r, s are zeros of y1(x).
 *
 * The third interval uses the same approximations to modulus
 * and phase as j1(x), whence y1(x) = modulus * sin(phase).
 *
 * ACCURACY:
 *
 *  Absolute error, when y0(x) < 1; else relative error:
 *
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0, 30       36000       2.7e-19     5.3e-20
 *
 */

// y1(x) = 2/pi * (log(x) * j1(x) - 1/x) + R(x^2) z P(z**2)/Q(z**2)
// 0 <= x <= 4.5, z(x) = x
// Peak error =  7.25e-22. Relative error spread =  4.5e-2
static immutable real [] y1n = [ -0x1.32cab2601090742p+54, 0x1.432ceb7a8eaeff16p+52,
   -0x1.bcebec5a2484d3fap+47, 0x1.cc58f3cb54d6ac66p+41, -0x1.b1255e154d0eec0ep+34,
   0x1.7a337df43298a7c8p+26, -0x1.f77a1afdeff0b62cp+16
];

static immutable real [] y1d = [ 0x1.8733bcfd7236e604p+56, 0x1.5af412c672fd18d4p+50,
   0x1.394ba130685755ep+43, 0x1.7b3321523b24afcp+35, 0x1.52946dac22f61d0cp+27,
   0x1.c9040c6053de5318p+18, 0x1.be5156e6771dba34p+9, 1.0
];


// y1(x) = (x-YZ1)(x-YZ2)(x-YZ3)(x-YZ4)R(x) P(z)/Q(z)
// z(x) = x, 4.5 <= x <= 9
// Peak error =  3.27e-22. Relative error spread =  4.5e-2
static immutable real[] y159n = [ 0x1.2fed87b1e60aa736p+18, -0x1.1a2b18cdb2d1ec5ep+20,
   -0x1.b848827f47b47022p+20, -0x1.b2e422305ea19a86p+20,
   -0x1.e3f82ac304534676p+16, 0x1.47a2cb5e852d657ep+14, 0x1.81b2fc6e44d7be8p+12,
   -0x1.cd861d7b090dd22ep+9, 0x1.588897d683cbfbe2p+5, -0x1.5c7feccf76856bcap-1
];

static immutable real[] y159d = [ 0x1.9b64f2a4d5614462p+26, -0x1.17501e0e38db675ap+30,
   0x1.fe88b567c2911c1cp+31, -0x1.86b1781e04e748d4p+29, 0x1.ccd7d4396f2edbcap+26,
   -0x1.694110c682e5cbcap+23, 0x1.c20f7005b88c789ep+19, -0x1.983a5b4275ab7da8p+15,
   0x1.17c60380490fa1fcp+11, -0x1.ee84c254392634d8p+5, 1.0
];

    real xx, y, z, modulus, phase;

    z = 1.0/x;
    xx = x * x;
    if ( xx < 81.0L ) {
        if ( xx < 20.25L ) {
            y = M_2_PI * (log(x) * cylBessel_j1(x) - z);
            y += x * poly( xx, y1n) / poly( xx, y1d);
        } else {
            static immutable real [] Y1Z =
            [   2.19714132603101703515e0L, 5.42968104079413513277e0L,
                8.59600586833116892643e0L, 1.17491548308398812434e1L];
            y = (x - Y1Z[0])*(x - Y1Z[1])*(x - Y1Z[2])*(x - Y1Z[3]);
            y *= rationalPoly( x, y159n, y159d);
        }
        return y;
    }
    xx = 1.0/xx;
    phase = rationalPoly( xx, j1phasen, j1phased);
    modulus = rationalPoly( z, j1modulusn, j1modulusd);

    static immutable real M_3PI_4 = 3 * PI_4;

    z = modulus * sin( x -  M_3PI_4 + z*phase) / sqrt(x);
    return z;
}

/**
 *  Bessel function of integer order
 *
 * Returns Bessel function of order n, where n is a
 * (possibly negative) integer.
 *
 * The ratio of jn(x) to j0(x) is computed by backward
 * recurrence.  First the ratio jn/jn-1 is found by a
 * continued fraction expansion.  Then the recurrence
 * relating successive orders is applied until j0 or j1 is
 * reached.
 *
 * If n = 0 or 1 the routine for j0 or j1 is called
 * directly.
 *
 * BUGS: Not suitable for large n or x.
 *
 */
real cylBessel_jn(int n, real x )
{
    real pkm2, pkm1, pk, xk, r, ans;
    int k, sign;

    if ( n < 0 ) {
        n = -n;
        if ( (n & 1) == 0 )  /* -1**n */
            sign = 1;
        else
            sign = -1;
    } else
        sign = 1;

    if ( x < 0.0L ) {
        if ( n & 1 )
            sign = -sign;
        x = -x;
    }

    if ( n == 0 )
        return sign * cylBessel_j0(x);
    if ( n == 1 )
        return sign * cylBessel_j1(x);
    // BUG: This code from Cephes is fast, but it makes the Wronksian test fail.
    // (accuracy is 8 bits lower).
    // But, the problem might lie in the n = 2 case in cylBessel_yn().
//    if ( n == 2 )
//        return sign * (2.0L * cylBessel_j1(x) / x  -  cylBessel_j0(x));

    if ( x < real.epsilon )
        return 0;

    /* continued fraction */
    k = 53;
    pk = 2 * (n + k);
    ans = pk;
    xk = x * x;

    do {
        pk -= 2.0L;
        ans = pk - (xk/ans);
    } while( --k > 0 );
    ans = x/ans;

    /* backward recurrence */

    pk = 1.0L;
    pkm1 = 1.0L/ans;
    k = n-1;
    r = 2 * k;

    do  {
        pkm2 = (pkm1 * r  -  pk * x) / x;
        pk = pkm1;
        pkm1 = pkm2;
        r -= 2.0L;
    } while( --k > 0 );

    if ( fabs(pk) > fabs(pkm1) )
        ans = cylBessel_j1(x)/pk;
    else
        ans = cylBessel_j0(x)/pkm1;
    return sign * ans;
}

/**
 *  Bessel function of second kind of integer order
 *
 * Returns Bessel function of order n, where n is a
 * (possibly negative) integer.
 *
 * The function is evaluated by forward recurrence on
 * n, starting with values computed by the routines
 * cylBessel_y0() and cylBessel_y1().
 *
 * If n = 0 or 1 the routine for cylBessel_y0 or cylBessel_y1 is called
 * directly.
 */
real cylBessel_yn(int n, real x)
{
    verify(x>0); // TODO: should it return -infinity for x<=0 ?

    real an, r;
    int k, sign;

    if ( n < 0 ) {
        n = -n;
        if ( (n & 1) == 0 ) /* -1**n */
            sign = 1;
        else
            sign = -1;
    } else
        sign = 1;

    if ( n == 0 )
        return sign * cylBessel_y0(x);
    if ( n == 1 )
        return sign * cylBessel_y1(x);

    /* forward recurrence on n */
    real anm2 = cylBessel_y0(x);
    real anm1 = cylBessel_y1(x);
    k = 1;
    r = 2 * k;
    do {
        an = r * anm1 / x  -  anm2;
        anm2 = anm1;
        anm1 = an;
        r += 2.0L;
        ++k;
    } while( k < n );
    return sign * an;
}

private {
// Evaluate Chebyshev series
double evalCheby(double x, in double [] poly)
{
    double b0, b1, b2;

    b0 = poly[$-1];
    b1 = 0.0;
    for (ptrdiff_t i=poly.length-1; i>=0; --i) {
        b2 = b1;
        b1 = b0;
        b0 = x * b1 - b2 + poly[i];
    }
    return 0.5*(b0-b2);
}
}

/**
 *  Modified Bessel function of order zero
 *
 * Returns modified Bessel function of order zero of the
 * argument.
 *
 * The function is defined as i0(x) = j0( ix ).
 *
 * The range is partitioned into the two intervals [0,8] and
 * (8, infinity).  Chebyshev polynomial expansions are employed
 * in each interval.
 */
double cylBessel_i0(double x)
{
    // Chebyshev coefficients for exp(-x) I0(x) in the interval [0,8].
    // lim(x->0){ exp(-x) I0(x) } = 1.
    static immutable double [] A = [    0x1.5a84e9035a22ap-1,  -0x1.37febc057cd8dp-2,
     0x1.5f7ac77ac88c0p-3,  -0x1.84b70342d06eap-4,   0x1.93e8acea8a32dp-5,
    -0x1.84e9ef121b6f0p-6,   0x1.59961f3dde3ddp-7,  -0x1.1b65e201aa849p-8,
     0x1.adc758a12100ep-10, -0x1.2e2fd1f15eb52p-11,  0x1.8b51b74107cabp-13,
    -0x1.e2b2659c41d5ap-15,  0x1.13f58be9a2859p-16, -0x1.2866fcba56427p-18,
     0x1.2bf24978cf4acp-20, -0x1.1ec638f227f8dp-22,  0x1.03b769d4d6435p-24,
    -0x1.beaf68c0b30abp-27,  0x1.6d903a454cb34p-29, -0x1.1d4fe13ae9556p-31,
     0x1.a98becc743c10p-34, -0x1.2fc957a946abcp-36,  0x1.9fe2fe19bd324p-39,
    -0x1.1164c62ee1af0p-41,  0x1.59b464b262627p-44, -0x1.a5022c297fbebp-47,
     0x1.ee6d893f65ebap-50, -0x1.184eb721ebbb4p-52,  0x1.33362977da589p-55,
    -0x1.45cb72134d0efp-58 ];

    // Chebyshev coefficients for exp(-x) sqrt(x) I0(x)
    // in the inverted interval [8,infinity].
    // lim(x->inf){ exp(-x) sqrt(x) I0(x) } = 1/sqrt(2pi).
    static immutable double [] B = [      0x1.9be62aca809cbp-1,   0x1.b998ca2e59049p-9,
       0x1.20fa378999e52p-14,  0x1.8412bc101c586p-19,  0x1.b8007d9cd616ep-23,
       0x1.8569280d6d56dp-26,  0x1.d2c64a9225b87p-29,  0x1.0f9ccc0f46f75p-31,
       0x1.a24feabe8004fp-37, -0x1.1511d08397425p-35, -0x1.d0fd7357e7bf2p-37,
      -0x1.f904303178d66p-40,  0x1.94347fa268cecp-41,  0x1.b1c8c6b83c073p-42,
       0x1.156ff0d5fc545p-46, -0x1.75d99cf68bb32p-45, -0x1.583fe7e65629ap-47,
       0x1.12a919094e6d7p-48,  0x1.fee7da3eafb1fp-50, -0x1.8aee7d908de38p-52,
      -0x1.4600babd21fe4p-52,  0x1.3f3dd076041cdp-55,  0x1.9be1812d98421p-55,
      -0x1.646da66119130p-58, -0x1.0adb754ca8b19p-57 ];

    double y;

    if (x < 0)
        x = -x;
    if (x <= 8.0) {
        y = (x/2.0) - 2.0;
        return exp(x) * evalCheby( y, A);
    }
    return exp(x) * evalCheby( 32.0/x - 2.0, B) / sqrt(x);
}

/**
 *  Modified Bessel function of order one
 *
 * Returns modified Bessel function of order one of the
 * argument.
 *
 * The function is defined as i1(x) = -i j1( ix ).
 *
 * The range is partitioned into the two intervals [0,8] and
 * (8, infinity).  Chebyshev polynomial expansions are employed
 * in each interval.
*/
double cylBessel_i1(double x)
{
    static immutable double [] A = [       0x1.02a63724a7ffap-2,  -0x1.694d10469192ep-3,
        0x1.a46dad536f53cp-4,  -0x1.b1bbc537c9ebcp-5,   0x1.951e3e7bb2349p-6,
       -0x1.5a29f7913a26ap-7,   0x1.1065349d3a1b4p-8,  -0x1.8cc620b3cd4a4p-10,
        0x1.0c95db6c6df7dp-11, -0x1.533cad3d694fep-13,  0x1.911b542c70d0bp-15,
       -0x1.bd5f9b8debbcfp-17,  0x1.d1c4ed511afc5p-19, -0x1.cc0798363992ap-21,
        0x1.ae344b347d108p-23, -0x1.7dd3e24b8c3e8p-25,  0x1.4258e02395010p-27,
       -0x1.0361b28ea67e6p-29,  0x1.8ea34b43fdf6cp-32, -0x1.2510397eb07dep-34,
        0x1.9cee2b21d3154p-37, -0x1.173835fb70366p-39,  0x1.6af784779d955p-42,
       -0x1.c628e1c8f0b3bp-45,  0x1.11d7f0615290cp-47, -0x1.3eaaa7e0d1573p-50,
        0x1.663e3e593bfacp-53, -0x1.857d0c38a0576p-56,  0x1.99f2a0c3c4014p-59
    ];

    // Chebyshev coefficients for exp(-x) sqrt(x) I1(x)
    // in the inverted interval [8,infinity].
    // lim(x->inf){ exp(-x) sqrt(x) I1(x) } = 1/sqrt(2pi).
    static immutable double [] B = [       0x1.8ea18b55b1514p-1,  -0x1.3fda053fcdb4cp-7,
       -0x1.cfd7f804aa9a6p-14, -0x1.048df49ca0373p-18, -0x1.0dbfd2e9e5443p-22,
       -0x1.c415394bb46c1p-26, -0x1.0790b9ad53528p-28, -0x1.334ca5423dd80p-31,
       -0x1.4dcf9d4504c0cp-36,  0x1.1e1a1f1587865p-35,  0x1.f101f653c457bp-37,
        0x1.1e7d3f6439fa3p-39, -0x1.953e1076ab493p-41, -0x1.cbc458e73e255p-42,
       -0x1.7a9482e6d22a0p-46,  0x1.80d3c26b3281ep-45,  0x1.776e1762d31e8p-47,
       -0x1.12db5138afbc7p-48, -0x1.0efcd8bc4d22ap-49,  0x1.7d68e5f04a2d1p-52,
        0x1.55915fceb588ap-52, -0x1.2806c9c773320p-55, -0x1.acea3b2532277p-55,
        0x1.45b8aea87b950p-58,  0x1.1556db352e8e6p-57  ];

    double y, z;

    z = fabs(x);
    if( z <= 8.0 ) {
        y = (z/2.0) - 2.0;
        z = evalCheby( y, A ) * z * exp(z);
    } else {
        z = exp(z) * evalCheby( 32.0/z - 2.0, B ) / sqrt(z);
    }
    if (x < 0.0 )
        z = -z;
    return z;
}

unittest {
  // argument, result1, result2, derivative. Correct result is result1+result2.
static immutable real [4][] j0_test_points = [
    [8.0L, 1.71646118164062500000E-1L, 4.68897349140609086941E-6L, -2.34636346853914624381E-1L],
    [4.54541015625L, -3.09783935546875000000E-1L, 7.07472668157686463367E-6L, 2.42993657373627558460E-1L],
    [2.85711669921875L, -2.07901000976562500000E-1L, 1.15237285263902751582E-5L, -3.90402225324501311651E-1L],
    [2.0L, 2.23876953125000000000E-1L, 1.38260162356680518275E-5L, -5.76724807756873387202E-1L],
    [1.16415321826934814453125e-10L, 9.99984741210937500000E-1L, 1.52587890624999966119E-5L,
        9.99999999999999999997E-1L],
    [-2.0L, 2.23876953125000000000E-1L,
        1.38260162356680518275E-5L, 5.76724807756873387202E-1L]
];

static immutable real [4][] y0_test_points = [
    [ 8.0L, 2.23510742187500000000E-1L, 1.07472000662205273234E-5L, 1.58060461731247494256E-1L],
    [4.54541015625L, -2.08114624023437500000E-1L, 1.45018823856668874574E-5L, -2.88887645307401250876E-1L],
    [2.85711669921875L, 4.20303344726562500000E-1L, 1.32781607563122276008E-5L, -2.82488638474982469213E-1],
    [2.0L, 5.10360717773437500000E-1L, 1.49548763076195966066E-5L, 1.07032431540937546888E-1L],
    [1.16415321826934814453125e-10L, -1.46357574462890625000E1L, 3.54110537011061127637E-6L,
        5.46852220461145271913E9L]
];

static immutable real [4][] j1_test_points = [
  [ 8.0L, 2.34634399414062500000E-1L, 1.94743985212438127665E-6L,1.42321263780814578043E-1],
  [4.54541015625L, -2.42996215820312500000E-1L, 2.55844668494153980076E-6L, -2.56317734136211337012E-1],
  [2.85711669921875L, 3.90396118164062500000E-1L, 6.10716043881165077013E-6L, -3.44531507106757980441E-1L],
  [2.0L, 5.76721191406250000000E-1L, 3.61635062338720244824E-6L,  -6.44716247372010255494E-2L],
  [1.16415321826934814453125e-10L, 5.820677273504770710133016109466552734375e-11L,
   8.881784197001251337312921818461805735896e-16L, 4.99999999999999999997E-1L],
  [-2.0L, -5.76721191406250000000E-1L, -3.61635062338720244824E-6L, -6.44716247372010255494E-2L]
];

static immutable real [4][] y1_test_points = [
    [8.0L, -1.58065795898437500000E-1L,
        5.33416719000574444473E-6L, 2.43279047103972157309E-1L],
    [4.54541015625L, 2.88879394531250000000E-1L,
        8.25077615125087585195E-6L, -2.71656024771791736625E-1L],
    [2.85711669921875L, 2.82485961914062500000E-1,
        2.67656091996921314433E-6L, 3.21444694221532719737E-1],
    [2.0L, -1.07040405273437500000E-1L,
        7.97373249995311162923E-6L, 5.63891888420213893041E-1],
    [1.16415321826934814453125e-10L, -5.46852220500000000000E9L,
        3.88547280871200700671E-1L, 4.69742480525120196168E19L]
];

    foreach(real [4] t; j0_test_points) {
        test(feqrel(cylBessel_j0(t[0]), t[1]+t[2]) >=real.mant_dig-3);
    }

    foreach(real [4] t; y0_test_points) {
        test(feqrel(cylBessel_y0(t[0]), t[1]+t[2]) >=real.mant_dig-4);
    }
    foreach(real [4] t; j1_test_points) {
        test(feqrel(cylBessel_j1(t[0]), t[1]+t[2]) >=real.mant_dig-3);
    }

    foreach(real [4] t; y1_test_points) {
        test(feqrel(cylBessel_y1(t[0]), t[1]+t[2]) >=real.mant_dig-4);
    }

    // Values from MS Excel, of doubtful accuracy.
    test(fabs(-0.060_409_940_421_649 - cylBessel_j0(173.5)) < 0.000_000_000_1);
    test(fabs(-0.044_733_447_576_5866 - cylBessel_y0(313.25)) < 0.000_000_000_1);
    test(fabs(0.00391280088318945 - cylBessel_j1(123.25)) < 0.000_000_000_1);
    test(fabs(-0.0648628570878951 - cylBessel_j1(-91)) < 0.000_000_000_1);
    test(fabs(-0.0759578537652805 - cylBessel_y1(107.75)) < 0.000_000_000_1);

    test(fabs(13.442_456_516_6771-cylBessel_i0(4.2)) < 0.000_001);
    test(fabs(1.6500020842093e+28-cylBessel_i0(-68)) < 0.000_001e+28);
    test(fabs(4.02746515903173e+10-cylBessel_i1(27)) < 0.000_001e+10);
    test(fabs(-2.83613942886386e-02-cylBessel_i1(-0.0567)) < 0.000_000_001e-2);
}

unittest {

    // Wronksian test for Bessel functions
    void testWronksian(int n, real x)
    {
      real Jnp1 = cylBessel_jn(n + 1, x);
      real Jmn = cylBessel_jn(-n, x);
      real Jn = cylBessel_jn(n, x);
      real Jmnp1 = cylBessel_jn(-(n + 1), x);
      /* This should be trivially zero.  */
      test( fabs(Jnp1 * Jmn + Jn * Jmnp1) == 0);
      if (x < 0.0) {
          x = -x;
          Jn = cylBessel_jn(n, x);
          Jnp1 = cylBessel_jn(n + 1, x);
      }
      real Yn = cylBessel_yn(n, x);
      real Ynp1 = cylBessel_yn(n + 1, x);
      /* The Wronksian.  */
      real w1 = Jnp1 * Yn - Jn * Ynp1;
      /* What the Wronksian should be. */
      real w2 = 2.0 / (PI * x);

      real reldif = feqrel(w1, w2);
      test(reldif >= real.mant_dig-6);
    }

  real delta;
  int n, i, j;

  delta = 0.6 / PI;
  for (n = -30; n <= 30; n++) {
    real x = -30.0;
    while (x < 30.0)    {
        testWronksian (n, x);
        x += delta;
    }
    delta += .00123456;
  }
  test(cylBessel_jn(20, 1e-80)==0);

      // NaN propagation
    test(isIdentical(cylBessel_i1(NaN(0xDEF)), NaN(0xDEF)));
    test(isIdentical(cylBessel_i0(NaN(0x846)), NaN(0x846)));

}