text: /**
   * \file Geodesic.cpp
   * \brief Implementation for GeographicLib::Geodesic class
   *
   * Copyright (c) Charles Karney (2009-2021) <charles@karney.com> and licensed
   * under the MIT/X11 License.  For more information, see
   * https://geographiclib.sourceforge.io/
   *
   * This is a reformulation of the geodesic problem.  The notation is as
   * follows:
   * - at a general point (no suffix or 1 or 2 as suffix)
   *   - phi = latitude
   *   - beta = latitude on auxiliary sphere
   *   - omega = longitude on auxiliary sphere
   *   - lambda = longitude
   *   - alpha = azimuth of great circle
   *   - sigma = arc length along great circle
   *   - s = distance
   *   - tau = scaled distance (= sigma at multiples of pi/2)
   * - at northwards equator crossing
   *   - beta = phi = 0
   *   - omega = lambda = 0
   *   - alpha = alpha0
   *   - sigma = s = 0
   * - a 12 suffix means a difference, e.g., s12 = s2 - s1.
   * - s and c prefixes mean sin and cos
   **********************************************************************/
  
  #include <GeographicLib/Geodesic.hpp>
  #include <GeographicLib/GeodesicLine.hpp>
  
  #if defined(_MSC_VER)
  // Squelch warnings about potentially uninitialized local variables and
  // constant conditional expressions
  #  pragma warning (disable: 4701 4127)
  #endif
  
  namespace GeographicLib {
  
    using namespace std;
  
    Geodesic::Geodesic(real a, real f)
      : maxit2_(maxit1_ + Math::digits() + 10)
        // Underflow guard.  We require
        //   tiny_ * epsilon() > 0
        //   tiny_ + epsilon() == epsilon()
      , tiny_(sqrt(numeric_limits<real>::min()))
      , tol0_(numeric_limits<real>::epsilon())
        // Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse
        // case 52.784459512564 0 -52.784459512563990912 179.634407464943777557
        // which otherwise failed for Visual Studio 10 (Release and Debug)
      , tol1_(200 * tol0_)
      , tol2_(sqrt(tol0_))
      , tolb_(tol0_ * tol2_)      // Check on bisection interval
      , xthresh_(1000 * tol2_)
      , _a(a)
      , _f(f)
      , _f1(1 - _f)
      , _e2(_f * (2 - _f))
      , _ep2(_e2 / Math::sq(_f1)) // e2 / (1 - e2)
      , _n(_f / ( 2 - _f))
      , _b(_a * _f1)
      , _c2((Math::sq(_a) + Math::sq(_b) *
             (_e2 == 0 ? 1 :
              Math::eatanhe(real(1), (_f < 0 ? -1 : 1) * sqrt(abs(_e2))) / _e2))
            / 2) // authalic radius squared
        // The sig12 threshold for "really short".  Using the auxiliary sphere
        // solution with dnm computed at (bet1 + bet2) / 2, the relative error in
        // the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
        // (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000.  For a
        // given f and sig12, the max error occurs for lines near the pole.  If
        // the old rule for computing dnm = (dn1 + dn2)/2 is used, then the error
        // increases by a factor of 2.)  Setting this equal to epsilon gives
        // sig12 = etol2.  Here 0.1 is a safety factor (error decreased by 100)
        // and max(0.001, abs(f)) stops etol2 getting too large in the nearly
        // spherical case.
      , _etol2(real(0.1) * tol2_ /
               sqrt( max(real(0.001), abs(_f)) * min(real(1), 1 - _f/2) / 2 ))
    {
      if (!(isfinite(_a) && _a > 0))
        throw GeographicErr("Equatorial radius is not positive");
      if (!(isfinite(_b) && _b > 0))
        throw GeographicErr("Polar semi-axis is not positive");
      A3coeff();
      C3coeff();
      C4coeff();
    }
  
    const Geodesic& Geodesic::WGS84() {
      static const Geodesic wgs84(Constants::WGS84_a(), Constants::WGS84_f());
      return wgs84;
    }
  
    Math::real Geodesic::SinCosSeries(bool sinp,
                                      real sinx, real cosx,
                                      const real c[], int n) {
      // Evaluate
      // y = sinp ? sum(c[i] * sin( 2*i    * x), i, 1, n) :
      //            sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
      // using Clenshaw summation.  N.B. c[0] is unused for sin series
      // Approx operation count = (n + 5) mult and (2 * n + 2) add
      c += (n + sinp);            // Point to one beyond last element
      real
        ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
        y0 = n & 1 ? *--c : 0, y1 = 0;          // accumulators for sum
      // Now n is even
      n /= 2;
      while (n--) {
        // Unroll loop x 2, so accumulators return to their original role
        y1 = ar * y0 - y1 + *--c;
        y0 = ar * y1 - y0 + *--c;
      }
      return sinp
        ? 2 * sinx * cosx * y0    // sin(2 * x) * y0
        : cosx * (y0 - y1);       // cos(x) * (y0 - y1)
    }
  
    GeodesicLine Geodesic::Line(real lat1, real lon1, real azi1,
                                unsigned caps) const {
      return GeodesicLine(*this, lat1, lon1, azi1, caps);
    }
  
    Math::real Geodesic::GenDirect(real lat1, real lon1, real azi1,
                                   bool arcmode, real s12_a12, unsigned outmask,
                                   real& lat2, real& lon2, real& azi2,
                                   real& s12, real& m12, real& M12, real& M21,
                                   real& S12) const {
      // Automatically supply DISTANCE_IN if necessary
      if (!arcmode) outmask |= DISTANCE_IN;
      return GeodesicLine(*this, lat1, lon1, azi1, outmask)
        .                         // Note the dot!
        GenPosition(arcmode, s12_a12, outmask,
                    lat2, lon2, azi2, s12, m12, M12, M21, S12);
    }
  
    GeodesicLine Geodesic::GenDirectLine(real lat1, real lon1, real azi1,
                                         bool arcmode, real s12_a12,
                                         unsigned caps) const {
      azi1 = Math::AngNormalize(azi1);
      real salp1, calp1;
      // Guard against underflow in salp0.  Also -0 is converted to +0.
      Math::sincosd(Math::AngRound(azi1), salp1, calp1);
      // Automatically supply DISTANCE_IN if necessary
      if (!arcmode) caps |= DISTANCE_IN;
      return GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1,
                          caps, arcmode, s12_a12);
    }
  
    GeodesicLine Geodesic::DirectLine(real lat1, real lon1, real azi1, real s12,
                                      unsigned caps) const {
      return GenDirectLine(lat1, lon1, azi1, false, s12, caps);
    }
  
    GeodesicLine Geodesic::ArcDirectLine(real lat1, real lon1, real azi1,
                                         real a12, unsigned caps) const {
      return GenDirectLine(lat1, lon1, azi1, true, a12, caps);
    }
  
    Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
                                    unsigned outmask, real& s12,
                                    real& salp1, real& calp1,
                                    real& salp2, real& calp2,
                                    real& m12, real& M12, real& M21,
                                    real& S12) const {
      // Compute longitude difference (AngDiff does this carefully).  Result is
      // in [-180, 180] but -180 is only for west-going geodesics.  180 is for
      // east-going and meridional geodesics.
      real lon12s, lon12 = Math::AngDiff(lon1, lon2, lon12s);
      // Make longitude difference positive.
      int lonsign = lon12 >= 0 ? 1 : -1;
      // If very close to being on the same half-meridian, then make it so.
      lon12 = lonsign * Math::AngRound(lon12);
      lon12s = Math::AngRound((180 - lon12) - lonsign * lon12s);
      real
        lam12 = lon12 * Math::degree(),
        slam12, clam12;
      if (lon12 > 90) {
        Math::sincosd(lon12s, slam12, clam12);
        clam12 = -clam12;
      } else
        Math::sincosd(lon12, slam12, clam12);
  
      // If really close to the equator, treat as on equator.
      lat1 = Math::AngRound(Math::LatFix(lat1));
      lat2 = Math::AngRound(Math::LatFix(lat2));
      // Swap points so that point with higher (abs) latitude is point 1.
      // If one latitude is a nan, then it becomes lat1.
      int swapp = abs(lat1) < abs(lat2) ? -1 : 1;
      if (swapp < 0) {
        lonsign *= -1;
        swap(lat1, lat2);
      }
      // Make lat1 <= 0
      int latsign = lat1 < 0 ? 1 : -1;
      lat1 *= latsign;
      lat2 *= latsign;
      // Now we have
      //
      //     0 <= lon12 <= 180
      //     -90 <= lat1 <= 0
      //     lat1 <= lat2 <= -lat1
      //
      // longsign, swapp, latsign register the transformation to bring the
      // coordinates to this canonical form.  In all cases, 1 means no change was
      // made.  We make these transformations so that there are few cases to
      // check, e.g., on verifying quadrants in atan2.  In addition, this
      // enforces some symmetries in the results returned.
  
      real sbet1, cbet1, sbet2, cbet2, s12x, m12x;
  
      Math::sincosd(lat1, sbet1, cbet1); sbet1 *= _f1;
      // Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12
      // will be <= 2*tiny for two points at the same pole.
      Math::norm(sbet1, cbet1); cbet1 = max(tiny_, cbet1);
  
      Math::sincosd(lat2, sbet2, cbet2); sbet2 *= _f1;
      // Ensure cbet2 = +epsilon at poles
      Math::norm(sbet2, cbet2); cbet2 = max(tiny_, cbet2);
  
      // If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
      // |bet1| - |bet2|.  Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
      // a better measure.  This logic is used in assigning calp2 in Lambda12.
      // Sometimes these quantities vanish and in that case we force bet2 = +/-
      // bet1 exactly.  An example where is is necessary is the inverse problem
      // 48.522876735459 0 -48.52287673545898293 179.599720456223079643
      // which failed with Visual Studio 10 (Release and Debug)
  
      if (cbet1 < -sbet1) {
        if (cbet2 == cbet1)
          sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
      } else {
        if (abs(sbet2) == -sbet1)
          cbet2 = cbet1;
      }
  
      real
        dn1 = sqrt(1 + _ep2 * Math::sq(sbet1)),
        dn2 = sqrt(1 + _ep2 * Math::sq(sbet2));
  
      real a12, sig12;
      // index zero element of this array is unused
      real Ca[nC_];
  
      bool meridian = lat1 == -90 || slam12 == 0;
  
      if (meridian) {
  
        // Endpoints are on a single full meridian, so the geodesic might lie on
        // a meridian.
  
        calp1 = clam12; salp1 = slam12; // Head to the target longitude
        calp2 = 1; salp2 = 0;           // At the target we're heading north
  
        real
          // tan(bet) = tan(sig) * cos(alp)
          ssig1 = sbet1, csig1 = calp1 * cbet1,
          ssig2 = sbet2, csig2 = calp2 * cbet2;
  
        // sig12 = sig2 - sig1
        sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
                                   csig1 * csig2 + ssig1 * ssig2);
        {
          real dummy;
          Lengths(_n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
                  outmask | DISTANCE | REDUCEDLENGTH,
                  s12x, m12x, dummy, M12, M21, Ca);
        }
        // Add the check for sig12 since zero length geodesics might yield m12 <
        // 0.  Test case was
        //
        //    echo 20.001 0 20.001 0 | GeodSolve -i
        //
        // In fact, we will have sig12 > pi/2 for meridional geodesic which is
        // not a shortest path.
        // TODO: investigate m12 < 0 result for aarch/ppc (with -f -p 20)
        // 20.001000000000001 0.000000000000000 180.000000000000000
        // 20.001000000000001 0.000000000000000 180.000000000000000
        // 0.0000000002 0.000000000000001 -0.0000000001
        // 0.99999999999999989 0.99999999999999989 0.000
        if (sig12 < 1 || m12x >= 0) {
          // Need at least 2, to handle 90 0 90 180
          if (sig12 < 3 * tiny_ ||
              // Prevent negative s12 or m12 for short lines
              (sig12 < tol0_ && (s12x < 0 || m12x < 0)))
            sig12 = m12x = s12x = 0;
          m12x *= _b;
          s12x *= _b;
          a12 = sig12 / Math::degree();
        } else
          // m12 < 0, i.e., prolate and too close to anti-podal
          meridian = false;
      }
  
      // somg12 > 1 marks that it needs to be calculated
      real omg12 = 0, somg12 = 2, comg12 = 0;
      if (!meridian &&
          sbet1 == 0 &&   // and sbet2 == 0
          (_f <= 0 || lon12s >= _f * 180)) {
  
        // Geodesic runs along equator
        calp1 = calp2 = 0; salp1 = salp2 = 1;
        s12x = _a * lam12;
        sig12 = omg12 = lam12 / _f1;
        m12x = _b * sin(sig12);
        if (outmask & GEODESICSCALE)
          M12 = M21 = cos(sig12);
        a12 = lon12 / _f1;
  
      } else if (!meridian) {
  
        // Now point1 and point2 belong within a hemisphere bounded by a
        // meridian and geodesic is neither meridional or equatorial.
  
        // Figure a starting point for Newton's method
        real dnm;
        sig12 = InverseStart(sbet1, cbet1, dn1, sbet2, cbet2, dn2,
                             lam12, slam12, clam12,
                             salp1, calp1, salp2, calp2, dnm,
                             Ca);
  
        if (sig12 >= 0) {
          // Short lines (InverseStart sets salp2, calp2, dnm)
          s12x = sig12 * _b * dnm;
          m12x = Math::sq(dnm) * _b * sin(sig12 / dnm);
          if (outmask & GEODESICSCALE)
            M12 = M21 = cos(sig12 / dnm);
          a12 = sig12 / Math::degree();
          omg12 = lam12 / (_f1 * dnm);
        } else {
  
          // Newton's method.  This is a straightforward solution of f(alp1) =
          // lambda12(alp1) - lam12 = 0 with one wrinkle.  f(alp) has exactly one
          // root in the interval (0, pi) and its derivative is positive at the
          // root.  Thus f(alp) is positive for alp > alp1 and negative for alp <
          // alp1.  During the course of the iteration, a range (alp1a, alp1b) is
          // maintained which brackets the root and with each evaluation of
          // f(alp) the range is shrunk, if possible.  Newton's method is
          // restarted whenever the derivative of f is negative (because the new
          // value of alp1 is then further from the solution) or if the new
          // estimate of alp1 lies outside (0,pi); in this case, the new starting
          // guess is taken to be (alp1a + alp1b) / 2.
          //
          // initial values to suppress warnings (if loop is executed 0 times)
          real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, domg12 = 0;
          unsigned numit = 0;
          // Bracketing range
          real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
          for (bool tripn = false, tripb = false;
               numit < maxit2_ || GEOGRAPHICLIB_PANIC;
               ++numit) {
            // the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
            // WGS84 and random input: mean = 2.85, sd = 0.60
            real dv;
            real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
                              slam12, clam12,
                              salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
                              eps, domg12, numit < maxit1_, dv, Ca);
            // Reversed test to allow escape with NaNs
            if (tripb || !(abs(v) >= (tripn ? 8 : 1) * tol0_)) break;
            // Update bracketing values
            if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
              { salp1b = salp1; calp1b = calp1; }
            else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
              { salp1a = salp1; calp1a = calp1; }
            if (numit < maxit1_ && dv > 0) {
              real
                dalp1 = -v/dv;
              real
                sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
                nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
              if (nsalp1 > 0 && abs(dalp1) < Math::pi()) {
                calp1 = calp1 * cdalp1 - salp1 * sdalp1;
                salp1 = nsalp1;
                Math::norm(salp1, calp1);
                // In some regimes we don't get quadratic convergence because
                // slope -> 0.  So use convergence conditions based on epsilon
                // instead of sqrt(epsilon).
                tripn = abs(v) <= 16 * tol0_;
                continue;
              }
            }
            // Either dv was not positive or updated value was outside legal
            // range.  Use the midpoint of the bracket as the next estimate.
            // This mechanism is not needed for the WGS84 ellipsoid, but it does
            // catch problems with more eccentric ellipsoids.  Its efficacy is
            // such for the WGS84 test set with the starting guess set to alp1 =
            // 90deg:
            // the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
            // WGS84 and random input: mean = 4.74, sd = 0.99
            salp1 = (salp1a + salp1b)/2;
            calp1 = (calp1a + calp1b)/2;
            Math::norm(salp1, calp1);
            tripn = false;
            tripb = (abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
                     abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
          }
          {
            real dummy;
            // Ensure that the reduced length and geodesic scale are computed in
            // a "canonical" way, with the I2 integral.
            unsigned lengthmask = outmask |
              (outmask & (REDUCEDLENGTH | GEODESICSCALE) ? DISTANCE : NONE);
            Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
                    cbet1, cbet2, lengthmask, s12x, m12x, dummy, M12, M21, Ca);
          }
          m12x *= _b;
          s12x *= _b;
          a12 = sig12 / Math::degree();
          if (outmask & AREA) {
            // omg12 = lam12 - domg12
            real sdomg12 = sin(domg12), cdomg12 = cos(domg12);
            somg12 = slam12 * cdomg12 - clam12 * sdomg12;
            comg12 = clam12 * cdomg12 + slam12 * sdomg12;
          }
        }
      }
  
      if (outmask & DISTANCE)
        s12 = 0 + s12x;           // Convert -0 to 0
  
      if (outmask & REDUCEDLENGTH)
        m12 = 0 + m12x;           // Convert -0 to 0
  
      if (outmask & AREA) {
        real
          // From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
          salp0 = salp1 * cbet1,
          calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
        real alp12;
        if (calp0 != 0 && salp0 != 0) {
          real
            // From Lambda12: tan(bet) = tan(sig) * cos(alp)
            ssig1 = sbet1, csig1 = calp1 * cbet1,
            ssig2 = sbet2, csig2 = calp2 * cbet2,
            k2 = Math::sq(calp0) * _ep2,
            eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
            // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
            A4 = Math::sq(_a) * calp0 * salp0 * _e2;
          Math::norm(ssig1, csig1);
          Math::norm(ssig2, csig2);
          C4f(eps, Ca);
          real
            B41 = SinCosSeries(false, ssig1, csig1, Ca, nC4_),
            B42 = SinCosSeries(false, ssig2, csig2, Ca, nC4_);
          S12 = A4 * (B42 - B41);
        } else
          // Avoid problems with indeterminate sig1, sig2 on equator
          S12 = 0;
  
        if (!meridian && somg12 > 1) {
          somg12 = sin(omg12); comg12 = cos(omg12);
        }
  
        if (!meridian &&
            // omg12 < 3/4 * pi
            comg12 > -real(0.7071) &&     // Long difference not too big
            sbet2 - sbet1 < real(1.75)) { // Lat difference not too big
          // Use tan(Gamma/2) = tan(omg12/2)
          // * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
          // with tan(x/2) = sin(x)/(1+cos(x))
          real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
          alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
                             domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
        } else {
          // alp12 = alp2 - alp1, used in atan2 so no need to normalize
          real
            salp12 = salp2 * calp1 - calp2 * salp1,
            calp12 = calp2 * calp1 + salp2 * salp1;
          // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
          // salp12 = -0 and alp12 = -180.  However this depends on the sign
          // being attached to 0 correctly.  The following ensures the correct
          // behavior.
          if (salp12 == 0 && calp12 < 0) {
            salp12 = tiny_ * calp1;
            calp12 = -1;
          }
          alp12 = atan2(salp12, calp12);
        }
        S12 += _c2 * alp12;
        S12 *= swapp * lonsign * latsign;
        // Convert -0 to 0
        S12 += 0;
      }
  
      // Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
      if (swapp < 0) {
        swap(salp1, salp2);
        swap(calp1, calp2);
        if (outmask & GEODESICSCALE)
          swap(M12, M21);
      }
  
      salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
      salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
  
      // Returned value in [0, 180]
      return a12;
    }
  
    Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
                                    unsigned outmask,
                                    real& s12, real& azi1, real& azi2,
                                    real& m12, real& M12, real& M21,
                                    real& S12) const {
      outmask &= OUT_MASK;
      real salp1, calp1, salp2, calp2,
        a12 =  GenInverse(lat1, lon1, lat2, lon2,
                          outmask, s12, salp1, calp1, salp2, calp2,
                          m12, M12, M21, S12);
      if (outmask & AZIMUTH) {
        azi1 = Math::atan2d(salp1, calp1);
        azi2 = Math::atan2d(salp2, calp2);
      }
      return a12;
    }
  
    GeodesicLine Geodesic::InverseLine(real lat1, real lon1,
                                       real lat2, real lon2,
                                       unsigned caps) const {
      real t, salp1, calp1, salp2, calp2,
        a12 = GenInverse(lat1, lon1, lat2, lon2,
                         // No need to specify AZIMUTH here
                         0u, t, salp1, calp1, salp2, calp2,
                         t, t, t, t),
        azi1 = Math::atan2d(salp1, calp1);
      // Ensure that a12 can be converted to a distance
      if (caps & (OUT_MASK & DISTANCE_IN)) caps |= DISTANCE;
      return
        GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1, caps, true, a12);
    }
  
    void Geodesic::Lengths(real eps, real sig12,
                           real ssig1, real csig1, real dn1,
                           real ssig2, real csig2, real dn2,
                           real cbet1, real cbet2, unsigned outmask,
                           real& s12b, real& m12b, real& m0,
                           real& M12, real& M21,
                           // Scratch area of the right size
                           real Ca[]) const {
      // Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
      // and m0 = coefficient of secular term in expression for reduced length.
  
      outmask &= OUT_MASK;
      // outmask & DISTANCE: set s12b
      // outmask & REDUCEDLENGTH: set m12b & m0
      // outmask & GEODESICSCALE: set M12 & M21
  
      real m0x = 0, J12 = 0, A1 = 0, A2 = 0;
      real Cb[nC2_ + 1];
      if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
        A1 = A1m1f(eps);
        C1f(eps, Ca);
        if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
          A2 = A2m1f(eps);
          C2f(eps, Cb);
          m0x = A1 - A2;
          A2 = 1 + A2;
        }
        A1 = 1 + A1;
      }
      if (outmask & DISTANCE) {
        real B1 = SinCosSeries(true, ssig2, csig2, Ca, nC1_) -
          SinCosSeries(true, ssig1, csig1, Ca, nC1_);
        // Missing a factor of _b
        s12b = A1 * (sig12 + B1);
        if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
          real B2 = SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
            SinCosSeries(true, ssig1, csig1, Cb, nC2_);
          J12 = m0x * sig12 + (A1 * B1 - A2 * B2);
        }
      } else if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
        // Assume here that nC1_ >= nC2_
        for (int l = 1; l <= nC2_; ++l)
          Cb[l] = A1 * Ca[l] - A2 * Cb[l];
        J12 = m0x * sig12 + (SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
                             SinCosSeries(true, ssig1, csig1, Cb, nC2_));
      }
      if (outmask & REDUCEDLENGTH) {
        m0 = m0x;
        // Missing a factor of _b.
        // Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
        // accurate cancellation in the case of coincident points.
        m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
          csig1 * csig2 * J12;
      }
      if (outmask & GEODESICSCALE) {
        real csig12 = csig1 * csig2 + ssig1 * ssig2;
        real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
        M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
        M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
      }
    }
  
    Math::real Geodesic::Astroid(real x, real y) {
      // Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
      // This solution is adapted from Geocentric::Reverse.
      real k;
      real
        p = Math::sq(x),
        q = Math::sq(y),
        r = (p + q - 1) / 6;
      if ( !(q == 0 && r <= 0) ) {
        real
          // Avoid possible division by zero when r = 0 by multiplying equations
          // for s and t by r^3 and r, resp.
          S = p * q / 4,            // S = r^3 * s
          r2 = Math::sq(r),
          r3 = r * r2,
          // The discriminant of the quadratic equation for T3.  This is zero on
          // the evolute curve p^(1/3)+q^(1/3) = 1
          disc = S * (S + 2 * r3);
        real u = r;
        if (disc >= 0) {
          real T3 = S + r3;
          // Pick the sign on the sqrt to maximize abs(T3).  This minimizes loss
          // of precision due to cancellation.  The result is unchanged because
          // of the way the T is used in definition of u.
          T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3
          // N.B. cbrt always returns the real root.  cbrt(-8) = -2.
          real T = cbrt(T3); // T = r * t
          // T can be zero; but then r2 / T -> 0.
          u += T + (T != 0 ? r2 / T : 0);
        } else {
          // T is complex, but the way u is defined the result is real.
          real ang = atan2(sqrt(-disc), -(S + r3));
          // There are three possible cube roots.  We choose the root which
          // avoids cancellation.  Note that disc < 0 implies that r < 0.
          u += 2 * r * cos(ang / 3);
        }
        real
          v = sqrt(Math::sq(u) + q),    // guaranteed positive
          // Avoid loss of accuracy when u < 0.
          uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive
          w = (uv - q) / (2 * v);           // positive?
        // Rearrange expression for k to avoid loss of accuracy due to
        // subtraction.  Division by 0 not possible because uv > 0, w >= 0.
        k = uv / (sqrt(uv + Math::sq(w)) + w);   // guaranteed positive
      } else {               // q == 0 && r <= 0
        // y = 0 with |x| <= 1.  Handle this case directly.
        // for y small, positive root is k = abs(y)/sqrt(1-x^2)
        k = 0;
      }
      return k;
    }
  
    Math::real Geodesic::InverseStart(real sbet1, real cbet1, real dn1,
                                      real sbet2, real cbet2, real dn2,
                                      real lam12, real slam12, real clam12,
                                      real& salp1, real& calp1,
                                      // Only updated if return val >= 0
                                      real& salp2, real& calp2,
                                      // Only updated for short lines
                                      real& dnm,
                                      // Scratch area of the right size
                                      real Ca[]) const {
      // Return a starting point for Newton's method in salp1 and calp1 (function
      // value is -1).  If Newton's method doesn't need to be used, return also
      // salp2 and calp2 and function value is sig12.
      real
        sig12 = -1,               // Return value
        // bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
        sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
        cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
      real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
      bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
        cbet2 * lam12 < real(0.5);
      real somg12, comg12;
      if (shortline) {
        real sbetm2 = Math::sq(sbet1 + sbet2);
        // sin((bet1+bet2)/2)^2
        // =  (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
        sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2);
        dnm = sqrt(1 + _ep2 * sbetm2);
        real omg12 = lam12 / (_f1 * dnm);
        somg12 = sin(omg12); comg12 = cos(omg12);
      } else {
        somg12 = slam12; comg12 = clam12;
      }
  
      salp1 = cbet2 * somg12;
      calp1 = comg12 >= 0 ?
        sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) :
        sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
  
      real
        ssig12 = hypot(salp1, calp1),
        csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
  
      if (shortline && ssig12 < _etol2) {
        // really short lines
        salp2 = cbet1 * somg12;
        calp2 = sbet12 - cbet1 * sbet2 *
          (comg12 >= 0 ? Math::sq(somg12) / (1 + comg12) : 1 - comg12);
        Math::norm(salp2, calp2);
        // Set return value
        sig12 = atan2(ssig12, csig12);
      } else if (abs(_n) > real(0.1) || // Skip astroid calc if too eccentric
                 csig12 >= 0 ||
                 ssig12 >= 6 * abs(_n) * Math::pi() * Math::sq(cbet1)) {
        // Nothing to do, zeroth order spherical approximation is OK
      } else {
        // Scale lam12 and bet2 to x, y coordinate system where antipodal point
        // is at origin and singular point is at y = 0, x = -1.
        real y, lamscale, betscale;
        // Volatile declaration needed to fix inverse case
        // 56.320923501171 0 -56.320923501171 179.664747671772880215
        // which otherwise fails with g++ 4.4.4 x86 -O3
        GEOGRAPHICLIB_VOLATILE real x;
        real lam12x = atan2(-slam12, -clam12); // lam12 - pi
        if (_f >= 0) {            // In fact f == 0 does not get here
          // x = dlong, y = dlat
          {
            real
              k2 = Math::sq(sbet1) * _ep2,
              eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
            lamscale = _f * cbet1 * A3f(eps) * Math::pi();
          }
          betscale = lamscale * cbet1;
  
          x = lam12x / lamscale;
          y = sbet12a / betscale;
        } else {                  // _f < 0
          // x = dlat, y = dlong
          real
            cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
            bet12a = atan2(sbet12a, cbet12a);
          real m12b, m0, dummy;
          // In the case of lon12 = 180, this repeats a calculation made in
          // Inverse.
          Lengths(_n, Math::pi() + bet12a,
                  sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
                  cbet1, cbet2,
                  REDUCEDLENGTH, dummy, m12b, m0, dummy, dummy, Ca);
          x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi());
          betscale = x < -real(0.01) ? sbet12a / x :
            -_f * Math::sq(cbet1) * Math::pi();
          lamscale = betscale / cbet1;
          y = lam12x / lamscale;
        }
  
        if (y > -tol1_ && x > -1 - xthresh_) {
          // strip near cut
          // Need real(x) here to cast away the volatility of x for min/max
          if (_f >= 0) {
            salp1 = min(real(1), -real(x)); calp1 = - sqrt(1 - Math::sq(salp1));
          } else {
            calp1 = max(real(x > -tol1_ ? 0 : -1), real(x));
            salp1 = sqrt(1 - Math::sq(calp1));
          }
        } else {
          // Estimate alp1, by solving the astroid problem.
          //
          // Could estimate alpha1 = theta + pi/2, directly, i.e.,
          //   calp1 = y/k; salp1 = -x/(1+k);  for _f >= 0
          //   calp1 = x/(1+k); salp1 = -y/k;  for _f < 0 (need to check)
          //
          // However, it's better to estimate omg12 from astroid and use
          // spherical formula to compute alp1.  This reduces the mean number of
          // Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
          // (min 0 max 5).  The changes in the number of iterations are as
          // follows:
          //
          // change percent
          //    1       5
          //    0      78
          //   -1      16
          //   -2       0.6
          //   -3       0.04
          //   -4       0.002
          //
          // The histogram of iterations is (m = number of iterations estimating
          // alp1 directly, n = number of iterations estimating via omg12, total
          // number of trials = 148605):
          //
          //  iter    m      n
          //    0   148    186
          //    1 13046  13845
          //    2 93315 102225
          //    3 36189  32341
          //    4  5396      7
          //    5   455      1
          //    6    56      0
          //
          // Because omg12 is near pi, estimate work with omg12a = pi - omg12
          real k = Astroid(x, y);
          real
            omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
          somg12 = sin(omg12a); comg12 = -cos(omg12a);
          // Update spherical estimate of alp1 using omg12 instead of lam12
          salp1 = cbet2 * somg12;
          calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
        }
      }
      // Sanity check on starting guess.  Backwards check allows NaN through.
      if (!(salp1 <= 0))
        Math::norm(salp1, calp1);
      else {
        salp1 = 1; calp1 = 0;
      }
      return sig12;
    }
  
    Math::real Geodesic::Lambda12(real sbet1, real cbet1, real dn1,
                                  real sbet2, real cbet2, real dn2,
                                  real salp1, real calp1,
                                  real slam120, real clam120,
                                  real& salp2, real& calp2,
                                  real& sig12,
                                  real& ssig1, real& csig1,
                                  real& ssig2, real& csig2,
                                  real& eps, real& domg12,
                                  bool diffp, real& dlam12,
                                  // Scratch area of the right size
                                  real Ca[]) const {
  
      if (sbet1 == 0 && calp1 == 0)
        // Break degeneracy of equatorial line.  This case has already been
        // handled.
        calp1 = -tiny_;
  
      real
        // sin(alp1) * cos(bet1) = sin(alp0)
        salp0 = salp1 * cbet1,
        calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
  
      real somg1, comg1, somg2, comg2, somg12, comg12, lam12;
      // tan(bet1) = tan(sig1) * cos(alp1)
      // tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
      ssig1 = sbet1; somg1 = salp0 * sbet1;
      csig1 = comg1 = calp1 * cbet1;
      Math::norm(ssig1, csig1);
      // Math::norm(somg1, comg1); -- don't need to normalize!
  
      // Enforce symmetries in the case abs(bet2) = -bet1.  Need to be careful
      // about this case, since this can yield singularities in the Newton
      // iteration.
      // sin(alp2) * cos(bet2) = sin(alp0)
      salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
      // calp2 = sqrt(1 - sq(salp2))
      //       = sqrt(sq(calp0) - sq(sbet2)) / cbet2
      // and subst for calp0 and rearrange to give (choose positive sqrt
      // to give alp2 in [0, pi/2]).
      calp2 = cbet2 != cbet1 || abs(sbet2) != -sbet1 ?
        sqrt(Math::sq(calp1 * cbet1) +
             (cbet1 < -sbet1 ?
              (cbet2 - cbet1) * (cbet1 + cbet2) :
              (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
        abs(calp1);
      // tan(bet2) = tan(sig2) * cos(alp2)
      // tan(omg2) = sin(alp0) * tan(sig2).
      ssig2 = sbet2; somg2 = salp0 * sbet2;
      csig2 = comg2 = calp2 * cbet2;
      Math::norm(ssig2, csig2);
      // Math::norm(somg2, comg2); -- don't need to normalize!
  
      // sig12 = sig2 - sig1, limit to [0, pi]
      sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
                                 csig1 * csig2 + ssig1 * ssig2);
  
      // omg12 = omg2 - omg1, limit to [0, pi]
      somg12 = max(real(0), comg1 * somg2 - somg1 * comg2);
      comg12 =              comg1 * comg2 + somg1 * somg2;
      // eta = omg12 - lam120
      real eta = atan2(somg12 * clam120 - comg12 * slam120,
                       comg12 * clam120 + somg12 * slam120);
      real B312;
      real k2 = Math::sq(calp0) * _ep2;
      eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
      C3f(eps, Ca);
      B312 = (SinCosSeries(true, ssig2, csig2, Ca, nC3_-1) -
              SinCosSeries(true, ssig1, csig1, Ca, nC3_-1));
      domg12 = -_f * A3f(eps) * salp0 * (sig12 + B312);
      lam12 = eta + domg12;
  
      if (diffp) {
        if (calp2 == 0)
          dlam12 = - 2 * _f1 * dn1 / sbet1;
        else {
          real dummy;
          Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
                  cbet1, cbet2, REDUCEDLENGTH,
                  dummy, dlam12, dummy, dummy, dummy, Ca);
          dlam12 *= _f1 / (calp2 * cbet2);
        }
      }
  
      return lam12;
    }
  
    Math::real Geodesic::A3f(real eps) const {
      // Evaluate A3
      return Math::polyval(nA3_ - 1, _A3x, eps);
    }
  
    void Geodesic::C3f(real eps, real c[]) const {
      // Evaluate C3 coeffs
      // Elements c[1] thru c[nC3_ - 1] are set
      real mult = 1;
      int o = 0;
      for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
        int m = nC3_ - l - 1;          // order of polynomial in eps
        mult *= eps;
        c[l] = mult * Math::polyval(m, _C3x + o, eps);
        o += m + 1;
      }
      // Post condition: o == nC3x_
    }
  
    void Geodesic::C4f(real eps, real c[]) const {
      // Evaluate C4 coeffs
      // Elements c[0] thru c[nC4_ - 1] are set
      real mult = 1;
      int o = 0;
      for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
        int m = nC4_ - l - 1;          // order of polynomial in eps
        c[l] = mult * Math::polyval(m, _C4x + o, eps);
        o += m + 1;
        mult *= eps;
      }
      // Post condition: o == nC4x_
    }
  
    // The static const coefficient arrays in the following functions are
    // generated by Maxima and give the coefficients of the Taylor expansions for
    // the geodesics.  The convention on the order of these coefficients is as
    // follows:
    //
    //   ascending order in the trigonometric expansion,
    //   then powers of eps in descending order,
    //   finally powers of n in descending order.
    //
    // (For some expansions, only a subset of levels occur.)  For each polynomial
    // of order n at the lowest level, the (n+1) coefficients of the polynomial
    // are followed by a divisor which is applied to the whole polynomial.  In
    // this way, the coefficients are expressible with no round off error.  The
    // sizes of the coefficient arrays are:
    //
    //   A1m1f, A2m1f            = floor(N/2) + 2
    //   C1f, C1pf, C2f, A3coeff = (N^2 + 7*N - 2*floor(N/2)) / 4
    //   C3coeff       = (N - 1) * (N^2 + 7*N - 2*floor(N/2)) / 8
    //   C4coeff       = N * (N + 1) * (N + 5) / 6
    //
    // where N = GEOGRAPHICLIB_GEODESIC_ORDER
    //         = nA1 = nA2 = nC1 = nC1p = nA3 = nC4
  
    // The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
    Math::real Geodesic::A1m1f(real eps) {
      // Generated by Maxima on 2015-05-05 18:08:12-04:00
  #if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1
      static const real coeff[] = {
        // (1-eps)*A1-1, polynomial in eps2 of order 1
        1, 0, 4,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2
      static const real coeff[] = {
        // (1-eps)*A1-1, polynomial in eps2 of order 2
        1, 16, 0, 64,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3
      static const real coeff[] = {
        // (1-eps)*A1-1, polynomial in eps2 of order 3
        1, 4, 64, 0, 256,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4
      static const real coeff[] = {
        // (1-eps)*A1-1, polynomial in eps2 of order 4
        25, 64, 256, 4096, 0, 16384,
      };
  #else
  #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
  #endif
      static_assert(sizeof(coeff) / sizeof(real) == nA1_/2 + 2,
                    "Coefficient array size mismatch in A1m1f");
      int m = nA1_/2;
      real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
      return (t + eps) / (1 - eps);
    }
  
    // The coefficients C1[l] in the Fourier expansion of B1
    void Geodesic::C1f(real eps, real c[]) {
      // Generated by Maxima on 2015-05-05 18:08:12-04:00
  #if GEOGRAPHICLIB_GEODESIC_ORDER == 3
      static const real coeff[] = {
        // C1[1]/eps^1, polynomial in eps2 of order 1
        3, -8, 16,
        // C1[2]/eps^2, polynomial in eps2 of order 0
        -1, 16,
        // C1[3]/eps^3, polynomial in eps2 of order 0
        -1, 48,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
      static const real coeff[] = {
        // C1[1]/eps^1, polynomial in eps2 of order 1
        3, -8, 16,
        // C1[2]/eps^2, polynomial in eps2 of order 1
        1, -2, 32,
        // C1[3]/eps^3, polynomial in eps2 of order 0
        -1, 48,
        // C1[4]/eps^4, polynomial in eps2 of order 0
        -5, 512,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
      static const real coeff[] = {
        // C1[1]/eps^1, polynomial in eps2 of order 2
        -1, 6, -16, 32,
        // C1[2]/eps^2, polynomial in eps2 of order 1
        1, -2, 32,
        // C1[3]/eps^3, polynomial in eps2 of order 1
        9, -16, 768,
        // C1[4]/eps^4, polynomial in eps2 of order 0
        -5, 512,
        // C1[5]/eps^5, polynomial in eps2 of order 0
        -7, 1280,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
      static const real coeff[] = {
        // C1[1]/eps^1, polynomial in eps2 of order 2
        -1, 6, -16, 32,
        // C1[2]/eps^2, polynomial in eps2 of order 2
        -9, 64, -128, 2048,
        // C1[3]/eps^3, polynomial in eps2 of order 1
        9, -16, 768,
        // C1[4]/eps^4, polynomial in eps2 of order 1
        3, -5, 512,
        // C1[5]/eps^5, polynomial in eps2 of order 0
        -7, 1280,
        // C1[6]/eps^6, polynomial in eps2 of order 0
        -7, 2048,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
      static const real coeff[] = {
        // C1[1]/eps^1, polynomial in eps2 of order 3
        19, -64, 384, -1024, 2048,
        // C1[2]/eps^2, polynomial in eps2 of order 2
        -9, 64, -128, 2048,
        // C1[3]/eps^3, polynomial in eps2 of order 2
        -9, 72, -128, 6144,
        // C1[4]/eps^4, polynomial in eps2 of order 1
        3, -5, 512,
        // C1[5]/eps^5, polynomial in eps2 of order 1
        35, -56, 10240,
        // C1[6]/eps^6, polynomial in eps2 of order 0
        -7, 2048,
        // C1[7]/eps^7, polynomial in eps2 of order 0
        -33, 14336,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
      static const real coeff[] = {
        // C1[1]/eps^1, polynomial in eps2 of order 3
        19, -64, 384, -1024, 2048,
        // C1[2]/eps^2, polynomial in eps2 of order 3
        7, -18, 128, -256, 4096,
        // C1[3]/eps^3, polynomial in eps2 of order 2
        -9, 72, -128, 6144,
        // C1[4]/eps^4, polynomial in eps2 of order 2
        -11, 96, -160, 16384,
        // C1[5]/eps^5, polynomial in eps2 of order 1
        35, -56, 10240,
        // C1[6]/eps^6, polynomial in eps2 of order 1
        9, -14, 4096,
        // C1[7]/eps^7, polynomial in eps2 of order 0
        -33, 14336,
        // C1[8]/eps^8, polynomial in eps2 of order 0
        -429, 262144,
      };
  #else
  #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
  #endif
      static_assert(sizeof(coeff) / sizeof(real) ==
                    (nC1_*nC1_ + 7*nC1_ - 2*(nC1_/2)) / 4,
                    "Coefficient array size mismatch in C1f");
      real
        eps2 = Math::sq(eps),
        d = eps;
      int o = 0;
      for (int l = 1; l <= nC1_; ++l) { // l is index of C1p[l]
        int m = (nC1_ - l) / 2;         // order of polynomial in eps^2
        c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
        o += m + 2;
        d *= eps;
      }
      // Post condition: o == sizeof(coeff) / sizeof(real)
    }
  
    // The coefficients C1p[l] in the Fourier expansion of B1p
    void Geodesic::C1pf(real eps, real c[]) {
      // Generated by Maxima on 2015-05-05 18:08:12-04:00
  #if GEOGRAPHICLIB_GEODESIC_ORDER == 3
      static const real coeff[] = {
        // C1p[1]/eps^1, polynomial in eps2 of order 1
        -9, 16, 32,
        // C1p[2]/eps^2, polynomial in eps2 of order 0
        5, 16,
        // C1p[3]/eps^3, polynomial in eps2 of order 0
        29, 96,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
      static const real coeff[] = {
        // C1p[1]/eps^1, polynomial in eps2 of order 1
        -9, 16, 32,
        // C1p[2]/eps^2, polynomial in eps2 of order 1
        -37, 30, 96,
        // C1p[3]/eps^3, polynomial in eps2 of order 0
        29, 96,
        // C1p[4]/eps^4, polynomial in eps2 of order 0
        539, 1536,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
      static const real coeff[] = {
        // C1p[1]/eps^1, polynomial in eps2 of order 2
        205, -432, 768, 1536,
        // C1p[2]/eps^2, polynomial in eps2 of order 1
        -37, 30, 96,
        // C1p[3]/eps^3, polynomial in eps2 of order 1
        -225, 116, 384,
        // C1p[4]/eps^4, polynomial in eps2 of order 0
        539, 1536,
        // C1p[5]/eps^5, polynomial in eps2 of order 0
        3467, 7680,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
      static const real coeff[] = {
        // C1p[1]/eps^1, polynomial in eps2 of order 2
        205, -432, 768, 1536,
        // C1p[2]/eps^2, polynomial in eps2 of order 2
        4005, -4736, 3840, 12288,
        // C1p[3]/eps^3, polynomial in eps2 of order 1
        -225, 116, 384,
        // C1p[4]/eps^4, polynomial in eps2 of order 1
        -7173, 2695, 7680,
        // C1p[5]/eps^5, polynomial in eps2 of order 0
        3467, 7680,
        // C1p[6]/eps^6, polynomial in eps2 of order 0
        38081, 61440,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
      static const real coeff[] = {
        // C1p[1]/eps^1, polynomial in eps2 of order 3
        -4879, 9840, -20736, 36864, 73728,
        // C1p[2]/eps^2, polynomial in eps2 of order 2
        4005, -4736, 3840, 12288,
        // C1p[3]/eps^3, polynomial in eps2 of order 2
        8703, -7200, 3712, 12288,
        // C1p[4]/eps^4, polynomial in eps2 of order 1
        -7173, 2695, 7680,
        // C1p[5]/eps^5, polynomial in eps2 of order 1
        -141115, 41604, 92160,
        // C1p[6]/eps^6, polynomial in eps2 of order 0
        38081, 61440,
        // C1p[7]/eps^7, polynomial in eps2 of order 0
        459485, 516096,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
      static const real coeff[] = {
        // C1p[1]/eps^1, polynomial in eps2 of order 3
        -4879, 9840, -20736, 36864, 73728,
        // C1p[2]/eps^2, polynomial in eps2 of order 3
        -86171, 120150, -142080, 115200, 368640,
        // C1p[3]/eps^3, polynomial in eps2 of order 2
        8703, -7200, 3712, 12288,
        // C1p[4]/eps^4, polynomial in eps2 of order 2
        1082857, -688608, 258720, 737280,
        // C1p[5]/eps^5, polynomial in eps2 of order 1
        -141115, 41604, 92160,
        // C1p[6]/eps^6, polynomial in eps2 of order 1
        -2200311, 533134, 860160,
        // C1p[7]/eps^7, polynomial in eps2 of order 0
        459485, 516096,
        // C1p[8]/eps^8, polynomial in eps2 of order 0
        109167851, 82575360,
      };
  #else
  #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
  #endif
      static_assert(sizeof(coeff) / sizeof(real) ==
                    (nC1p_*nC1p_ + 7*nC1p_ - 2*(nC1p_/2)) / 4,
                    "Coefficient array size mismatch in C1pf");
      real
        eps2 = Math::sq(eps),
        d = eps;
      int o = 0;
      for (int l = 1; l <= nC1p_; ++l) { // l is index of C1p[l]
        int m = (nC1p_ - l) / 2;         // order of polynomial in eps^2
        c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
        o += m + 2;
        d *= eps;
      }
      // Post condition: o == sizeof(coeff) / sizeof(real)
    }
  
    // The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
    Math::real Geodesic::A2m1f(real eps) {
      // Generated by Maxima on 2015-05-29 08:09:47-04:00
  #if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1
      static const real coeff[] = {
        // (eps+1)*A2-1, polynomial in eps2 of order 1
        -3, 0, 4,
      };  // count = 3
  #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2
      static const real coeff[] = {
        // (eps+1)*A2-1, polynomial in eps2 of order 2
        -7, -48, 0, 64,
      };  // count = 4
  #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3
      static const real coeff[] = {
        // (eps+1)*A2-1, polynomial in eps2 of order 3
        -11, -28, -192, 0, 256,
      };  // count = 5
  #elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4
      static const real coeff[] = {
        // (eps+1)*A2-1, polynomial in eps2 of order 4
        -375, -704, -1792, -12288, 0, 16384,
      };  // count = 6
  #else
  #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
  #endif
      static_assert(sizeof(coeff) / sizeof(real) == nA2_/2 + 2,
                    "Coefficient array size mismatch in A2m1f");
      int m = nA2_/2;
      real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
      return (t - eps) / (1 + eps);
    }
  
    // The coefficients C2[l] in the Fourier expansion of B2
    void Geodesic::C2f(real eps, real c[]) {
      // Generated by Maxima on 2015-05-05 18:08:12-04:00
  #if GEOGRAPHICLIB_GEODESIC_ORDER == 3
      static const real coeff[] = {
        // C2[1]/eps^1, polynomial in eps2 of order 1
        1, 8, 16,
        // C2[2]/eps^2, polynomial in eps2 of order 0
        3, 16,
        // C2[3]/eps^3, polynomial in eps2 of order 0
        5, 48,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
      static const real coeff[] = {
        // C2[1]/eps^1, polynomial in eps2 of order 1
        1, 8, 16,
        // C2[2]/eps^2, polynomial in eps2 of order 1
        1, 6, 32,
        // C2[3]/eps^3, polynomial in eps2 of order 0
        5, 48,
        // C2[4]/eps^4, polynomial in eps2 of order 0
        35, 512,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
      static const real coeff[] = {
        // C2[1]/eps^1, polynomial in eps2 of order 2
        1, 2, 16, 32,
        // C2[2]/eps^2, polynomial in eps2 of order 1
        1, 6, 32,
        // C2[3]/eps^3, polynomial in eps2 of order 1
        15, 80, 768,
        // C2[4]/eps^4, polynomial in eps2 of order 0
        35, 512,
        // C2[5]/eps^5, polynomial in eps2 of order 0
        63, 1280,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
      static const real coeff[] = {
        // C2[1]/eps^1, polynomial in eps2 of order 2
        1, 2, 16, 32,
        // C2[2]/eps^2, polynomial in eps2 of order 2
        35, 64, 384, 2048,
        // C2[3]/eps^3, polynomial in eps2 of order 1
        15, 80, 768,
        // C2[4]/eps^4, polynomial in eps2 of order 1
        7, 35, 512,
        // C2[5]/eps^5, polynomial in eps2 of order 0
        63, 1280,
        // C2[6]/eps^6, polynomial in eps2 of order 0
        77, 2048,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
      static const real coeff[] = {
        // C2[1]/eps^1, polynomial in eps2 of order 3
        41, 64, 128, 1024, 2048,
        // C2[2]/eps^2, polynomial in eps2 of order 2
        35, 64, 384, 2048,
        // C2[3]/eps^3, polynomial in eps2 of order 2
        69, 120, 640, 6144,
        // C2[4]/eps^4, polynomial in eps2 of order 1
        7, 35, 512,
        // C2[5]/eps^5, polynomial in eps2 of order 1
        105, 504, 10240,
        // C2[6]/eps^6, polynomial in eps2 of order 0
        77, 2048,
        // C2[7]/eps^7, polynomial in eps2 of order 0
        429, 14336,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
      static const real coeff[] = {
        // C2[1]/eps^1, polynomial in eps2 of order 3
        41, 64, 128, 1024, 2048,
        // C2[2]/eps^2, polynomial in eps2 of order 3
        47, 70, 128, 768, 4096,
        // C2[3]/eps^3, polynomial in eps2 of order 2
        69, 120, 640, 6144,
        // C2[4]/eps^4, polynomial in eps2 of order 2
        133, 224, 1120, 16384,
        // C2[5]/eps^5, polynomial in eps2 of order 1
        105, 504, 10240,
        // C2[6]/eps^6, polynomial in eps2 of order 1
        33, 154, 4096,
        // C2[7]/eps^7, polynomial in eps2 of order 0
        429, 14336,
        // C2[8]/eps^8, polynomial in eps2 of order 0
        6435, 262144,
      };
  #else
  #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
  #endif
      static_assert(sizeof(coeff) / sizeof(real) ==
                    (nC2_*nC2_ + 7*nC2_ - 2*(nC2_/2)) / 4,
                    "Coefficient array size mismatch in C2f");
      real
        eps2 = Math::sq(eps),
        d = eps;
      int o = 0;
      for (int l = 1; l <= nC2_; ++l) { // l is index of C2[l]
        int m = (nC2_ - l) / 2;         // order of polynomial in eps^2
        c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
        o += m + 2;
        d *= eps;
      }
      // Post condition: o == sizeof(coeff) / sizeof(real)
    }
  
    // The scale factor A3 = mean value of (d/dsigma)I3
    void Geodesic::A3coeff() {
      // Generated by Maxima on 2015-05-05 18:08:13-04:00
  #if GEOGRAPHICLIB_GEODESIC_ORDER == 3
      static const real coeff[] = {
        // A3, coeff of eps^2, polynomial in n of order 0
        -1, 4,
        // A3, coeff of eps^1, polynomial in n of order 1
        1, -1, 2,
        // A3, coeff of eps^0, polynomial in n of order 0
        1, 1,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
      static const real coeff[] = {
        // A3, coeff of eps^3, polynomial in n of order 0
        -1, 16,
        // A3, coeff of eps^2, polynomial in n of order 1
        -1, -2, 8,
        // A3, coeff of eps^1, polynomial in n of order 1
        1, -1, 2,
        // A3, coeff of eps^0, polynomial in n of order 0
        1, 1,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
      static const real coeff[] = {
        // A3, coeff of eps^4, polynomial in n of order 0
        -3, 64,
        // A3, coeff of eps^3, polynomial in n of order 1
        -3, -1, 16,
        // A3, coeff of eps^2, polynomial in n of order 2
        3, -1, -2, 8,
        // A3, coeff of eps^1, polynomial in n of order 1
        1, -1, 2,
        // A3, coeff of eps^0, polynomial in n of order 0
        1, 1,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
      static const real coeff[] = {
        // A3, coeff of eps^5, polynomial in n of order 0
        -3, 128,
        // A3, coeff of eps^4, polynomial in n of order 1
        -2, -3, 64,
        // A3, coeff of eps^3, polynomial in n of order 2
        -1, -3, -1, 16,
        // A3, coeff of eps^2, polynomial in n of order 2
        3, -1, -2, 8,
        // A3, coeff of eps^1, polynomial in n of order 1
        1, -1, 2,
        // A3, coeff of eps^0, polynomial in n of order 0
        1, 1,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
      static const real coeff[] = {
        // A3, coeff of eps^6, polynomial in n of order 0
        -5, 256,
        // A3, coeff of eps^5, polynomial in n of order 1
        -5, -3, 128,
        // A3, coeff of eps^4, polynomial in n of order 2
        -10, -2, -3, 64,
        // A3, coeff of eps^3, polynomial in n of order 3
        5, -1, -3, -1, 16,
        // A3, coeff of eps^2, polynomial in n of order 2
        3, -1, -2, 8,
        // A3, coeff of eps^1, polynomial in n of order 1
        1, -1, 2,
        // A3, coeff of eps^0, polynomial in n of order 0
        1, 1,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
      static const real coeff[] = {
        // A3, coeff of eps^7, polynomial in n of order 0
        -25, 2048,
        // A3, coeff of eps^6, polynomial in n of order 1
        -15, -20, 1024,
        // A3, coeff of eps^5, polynomial in n of order 2
        -5, -10, -6, 256,
        // A3, coeff of eps^4, polynomial in n of order 3
        -5, -20, -4, -6, 128,
        // A3, coeff of eps^3, polynomial in n of order 3
        5, -1, -3, -1, 16,
        // A3, coeff of eps^2, polynomial in n of order 2
        3, -1, -2, 8,
        // A3, coeff of eps^1, polynomial in n of order 1
        1, -1, 2,
        // A3, coeff of eps^0, polynomial in n of order 0
        1, 1,
      };
  #else
  #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
  #endif
      static_assert(sizeof(coeff) / sizeof(real) ==
                    (nA3_*nA3_ + 7*nA3_ - 2*(nA3_/2)) / 4,
                    "Coefficient array size mismatch in A3f");
      int o = 0, k = 0;
      for (int j = nA3_ - 1; j >= 0; --j) { // coeff of eps^j
        int m = min(nA3_ - j - 1, j);       // order of polynomial in n
        _A3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
        o += m + 2;
      }
      // Post condition: o == sizeof(coeff) / sizeof(real) && k == nA3x_
    }
  
    // The coefficients C3[l] in the Fourier expansion of B3
    void Geodesic::C3coeff() {
      // Generated by Maxima on 2015-05-05 18:08:13-04:00
  #if GEOGRAPHICLIB_GEODESIC_ORDER == 3
      static const real coeff[] = {
      // C3[1], coeff of eps^2, polynomial in n of order 0
      1, 8,
      // C3[1], coeff of eps^1, polynomial in n of order 1
      -1, 1, 4,
      // C3[2], coeff of eps^2, polynomial in n of order 0
      1, 16,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
      static const real coeff[] = {
      // C3[1], coeff of eps^3, polynomial in n of order 0
      3, 64,
      // C3[1], coeff of eps^2, polynomial in n of order 1
      // This is a case where a leading 0 term has been inserted to maintain the
      // pattern in the orders of the polynomials.
      0, 1, 8,
      // C3[1], coeff of eps^1, polynomial in n of order 1
      -1, 1, 4,
      // C3[2], coeff of eps^3, polynomial in n of order 0
      3, 64,
      // C3[2], coeff of eps^2, polynomial in n of order 1
      -3, 2, 32,
      // C3[3], coeff of eps^3, polynomial in n of order 0
      5, 192,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
      static const real coeff[] = {
      // C3[1], coeff of eps^4, polynomial in n of order 0
      5, 128,
      // C3[1], coeff of eps^3, polynomial in n of order 1
      3, 3, 64,
      // C3[1], coeff of eps^2, polynomial in n of order 2
      -1, 0, 1, 8,
      // C3[1], coeff of eps^1, polynomial in n of order 1
      -1, 1, 4,
      // C3[2], coeff of eps^4, polynomial in n of order 0
      3, 128,
      // C3[2], coeff of eps^3, polynomial in n of order 1
      -2, 3, 64,
      // C3[2], coeff of eps^2, polynomial in n of order 2
      1, -3, 2, 32,
      // C3[3], coeff of eps^4, polynomial in n of order 0
      3, 128,
      // C3[3], coeff of eps^3, polynomial in n of order 1
      -9, 5, 192,
      // C3[4], coeff of eps^4, polynomial in n of order 0
      7, 512,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
      static const real coeff[] = {
      // C3[1], coeff of eps^5, polynomial in n of order 0
      3, 128,
      // C3[1], coeff of eps^4, polynomial in n of order 1
      2, 5, 128,
      // C3[1], coeff of eps^3, polynomial in n of order 2
      -1, 3, 3, 64,
      // C3[1], coeff of eps^2, polynomial in n of order 2
      -1, 0, 1, 8,
      // C3[1], coeff of eps^1, polynomial in n of order 1
      -1, 1, 4,
      // C3[2], coeff of eps^5, polynomial in n of order 0
      5, 256,
      // C3[2], coeff of eps^4, polynomial in n of order 1
      1, 3, 128,
      // C3[2], coeff of eps^3, polynomial in n of order 2
      -3, -2, 3, 64,
      // C3[2], coeff of eps^2, polynomial in n of order 2
      1, -3, 2, 32,
      // C3[3], coeff of eps^5, polynomial in n of order 0
      7, 512,
      // C3[3], coeff of eps^4, polynomial in n of order 1
      -10, 9, 384,
      // C3[3], coeff of eps^3, polynomial in n of order 2
      5, -9, 5, 192,
      // C3[4], coeff of eps^5, polynomial in n of order 0
      7, 512,
      // C3[4], coeff of eps^4, polynomial in n of order 1
      -14, 7, 512,
      // C3[5], coeff of eps^5, polynomial in n of order 0
      21, 2560,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
      static const real coeff[] = {
      // C3[1], coeff of eps^6, polynomial in n of order 0
      21, 1024,
      // C3[1], coeff of eps^5, polynomial in n of order 1
      11, 12, 512,
      // C3[1], coeff of eps^4, polynomial in n of order 2
      2, 2, 5, 128,
      // C3[1], coeff of eps^3, polynomial in n of order 3
      -5, -1, 3, 3, 64,
      // C3[1], coeff of eps^2, polynomial in n of order 2
      -1, 0, 1, 8,
      // C3[1], coeff of eps^1, polynomial in n of order 1
      -1, 1, 4,
      // C3[2], coeff of eps^6, polynomial in n of order 0
      27, 2048,
      // C3[2], coeff of eps^5, polynomial in n of order 1
      1, 5, 256,
      // C3[2], coeff of eps^4, polynomial in n of order 2
      -9, 2, 6, 256,
      // C3[2], coeff of eps^3, polynomial in n of order 3
      2, -3, -2, 3, 64,
      // C3[2], coeff of eps^2, polynomial in n of order 2
      1, -3, 2, 32,
      // C3[3], coeff of eps^6, polynomial in n of order 0
      3, 256,
      // C3[3], coeff of eps^5, polynomial in n of order 1
      -4, 21, 1536,
      // C3[3], coeff of eps^4, polynomial in n of order 2
      -6, -10, 9, 384,
      // C3[3], coeff of eps^3, polynomial in n of order 3
      -1, 5, -9, 5, 192,
      // C3[4], coeff of eps^6, polynomial in n of order 0
      9, 1024,
      // C3[4], coeff of eps^5, polynomial in n of order 1
      -10, 7, 512,
      // C3[4], coeff of eps^4, polynomial in n of order 2
      10, -14, 7, 512,
      // C3[5], coeff of eps^6, polynomial in n of order 0
      9, 1024,
      // C3[5], coeff of eps^5, polynomial in n of order 1
      -45, 21, 2560,
      // C3[6], coeff of eps^6, polynomial in n of order 0
      11, 2048,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
      static const real coeff[] = {
      // C3[1], coeff of eps^7, polynomial in n of order 0
      243, 16384,
      // C3[1], coeff of eps^6, polynomial in n of order 1
      10, 21, 1024,
      // C3[1], coeff of eps^5, polynomial in n of order 2
      3, 11, 12, 512,
      // C3[1], coeff of eps^4, polynomial in n of order 3
      -2, 2, 2, 5, 128,
      // C3[1], coeff of eps^3, polynomial in n of order 3
      -5, -1, 3, 3, 64,
      // C3[1], coeff of eps^2, polynomial in n of order 2
      -1, 0, 1, 8,
      // C3[1], coeff of eps^1, polynomial in n of order 1
      -1, 1, 4,
      // C3[2], coeff of eps^7, polynomial in n of order 0
      187, 16384,
      // C3[2], coeff of eps^6, polynomial in n of order 1
      69, 108, 8192,
      // C3[2], coeff of eps^5, polynomial in n of order 2
      -2, 1, 5, 256,
      // C3[2], coeff of eps^4, polynomial in n of order 3
      -6, -9, 2, 6, 256,
      // C3[2], coeff of eps^3, polynomial in n of order 3
      2, -3, -2, 3, 64,
      // C3[2], coeff of eps^2, polynomial in n of order 2
      1, -3, 2, 32,
      // C3[3], coeff of eps^7, polynomial in n of order 0
      139, 16384,
      // C3[3], coeff of eps^6, polynomial in n of order 1
      -1, 12, 1024,
      // C3[3], coeff of eps^5, polynomial in n of order 2
      -77, -8, 42, 3072,
      // C3[3], coeff of eps^4, polynomial in n of order 3
      10, -6, -10, 9, 384,
      // C3[3], coeff of eps^3, polynomial in n of order 3
      -1, 5, -9, 5, 192,
      // C3[4], coeff of eps^7, polynomial in n of order 0
      127, 16384,
      // C3[4], coeff of eps^6, polynomial in n of order 1
      -43, 72, 8192,
      // C3[4], coeff of eps^5, polynomial in n of order 2
      -7, -40, 28, 2048,
      // C3[4], coeff of eps^4, polynomial in n of order 3
      -7, 20, -28, 14, 1024,
      // C3[5], coeff of eps^7, polynomial in n of order 0
      99, 16384,
      // C3[5], coeff of eps^6, polynomial in n of order 1
      -15, 9, 1024,
      // C3[5], coeff of eps^5, polynomial in n of order 2
      75, -90, 42, 5120,
      // C3[6], coeff of eps^7, polynomial in n of order 0
      99, 16384,
      // C3[6], coeff of eps^6, polynomial in n of order 1
      -99, 44, 8192,
      // C3[7], coeff of eps^7, polynomial in n of order 0
      429, 114688,
      };
  #else
  #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
  #endif
      static_assert(sizeof(coeff) / sizeof(real) ==
                    ((nC3_-1)*(nC3_*nC3_ + 7*nC3_ - 2*(nC3_/2)))/8,
                    "Coefficient array size mismatch in C3coeff");
      int o = 0, k = 0;
      for (int l = 1; l < nC3_; ++l) {        // l is index of C3[l]
        for (int j = nC3_ - 1; j >= l; --j) { // coeff of eps^j
          int m = min(nC3_ - j - 1, j);       // order of polynomial in n
          _C3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
          o += m + 2;
        }
      }
      // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC3x_
    }
  
    void Geodesic::C4coeff() {
      // Generated by Maxima on 2015-05-05 18:08:13-04:00
  #if GEOGRAPHICLIB_GEODESIC_ORDER == 3
      static const real coeff[] = {
        // C4[0], coeff of eps^2, polynomial in n of order 0
        -2, 105,
        // C4[0], coeff of eps^1, polynomial in n of order 1
        16, -7, 35,
        // C4[0], coeff of eps^0, polynomial in n of order 2
        8, -28, 70, 105,
        // C4[1], coeff of eps^2, polynomial in n of order 0
        -2, 105,
        // C4[1], coeff of eps^1, polynomial in n of order 1
        -16, 7, 315,
        // C4[2], coeff of eps^2, polynomial in n of order 0
        4, 525,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
      static const real coeff[] = {
        // C4[0], coeff of eps^3, polynomial in n of order 0
        11, 315,
        // C4[0], coeff of eps^2, polynomial in n of order 1
        -32, -6, 315,
        // C4[0], coeff of eps^1, polynomial in n of order 2
        -32, 48, -21, 105,
        // C4[0], coeff of eps^0, polynomial in n of order 3
        4, 24, -84, 210, 315,
        // C4[1], coeff of eps^3, polynomial in n of order 0
        -1, 105,
        // C4[1], coeff of eps^2, polynomial in n of order 1
        64, -18, 945,
        // C4[1], coeff of eps^1, polynomial in n of order 2
        32, -48, 21, 945,
        // C4[2], coeff of eps^3, polynomial in n of order 0
        -8, 1575,
        // C4[2], coeff of eps^2, polynomial in n of order 1
        -32, 12, 1575,
        // C4[3], coeff of eps^3, polynomial in n of order 0
        8, 2205,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
      static const real coeff[] = {
        // C4[0], coeff of eps^4, polynomial in n of order 0
        4, 1155,
        // C4[0], coeff of eps^3, polynomial in n of order 1
        -368, 121, 3465,
        // C4[0], coeff of eps^2, polynomial in n of order 2
        1088, -352, -66, 3465,
        // C4[0], coeff of eps^1, polynomial in n of order 3
        48, -352, 528, -231, 1155,
        // C4[0], coeff of eps^0, polynomial in n of order 4
        16, 44, 264, -924, 2310, 3465,
        // C4[1], coeff of eps^4, polynomial in n of order 0
        4, 1155,
        // C4[1], coeff of eps^3, polynomial in n of order 1
        80, -99, 10395,
        // C4[1], coeff of eps^2, polynomial in n of order 2
        -896, 704, -198, 10395,
        // C4[1], coeff of eps^1, polynomial in n of order 3
        -48, 352, -528, 231, 10395,
        // C4[2], coeff of eps^4, polynomial in n of order 0
        -8, 1925,
        // C4[2], coeff of eps^3, polynomial in n of order 1
        384, -88, 17325,
        // C4[2], coeff of eps^2, polynomial in n of order 2
        320, -352, 132, 17325,
        // C4[3], coeff of eps^4, polynomial in n of order 0
        -16, 8085,
        // C4[3], coeff of eps^3, polynomial in n of order 1
        -256, 88, 24255,
        // C4[4], coeff of eps^4, polynomial in n of order 0
        64, 31185,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
      static const real coeff[] = {
        // C4[0], coeff of eps^5, polynomial in n of order 0
        97, 15015,
        // C4[0], coeff of eps^4, polynomial in n of order 1
        1088, 156, 45045,
        // C4[0], coeff of eps^3, polynomial in n of order 2
        -224, -4784, 1573, 45045,
        // C4[0], coeff of eps^2, polynomial in n of order 3
        -10656, 14144, -4576, -858, 45045,
        // C4[0], coeff of eps^1, polynomial in n of order 4
        64, 624, -4576, 6864, -3003, 15015,
        // C4[0], coeff of eps^0, polynomial in n of order 5
        100, 208, 572, 3432, -12012, 30030, 45045,
        // C4[1], coeff of eps^5, polynomial in n of order 0
        1, 9009,
        // C4[1], coeff of eps^4, polynomial in n of order 1
        -2944, 468, 135135,
        // C4[1], coeff of eps^3, polynomial in n of order 2
        5792, 1040, -1287, 135135,
        // C4[1], coeff of eps^2, polynomial in n of order 3
        5952, -11648, 9152, -2574, 135135,
        // C4[1], coeff of eps^1, polynomial in n of order 4
        -64, -624, 4576, -6864, 3003, 135135,
        // C4[2], coeff of eps^5, polynomial in n of order 0
        8, 10725,
        // C4[2], coeff of eps^4, polynomial in n of order 1
        1856, -936, 225225,
        // C4[2], coeff of eps^3, polynomial in n of order 2
        -8448, 4992, -1144, 225225,
        // C4[2], coeff of eps^2, polynomial in n of order 3
        -1440, 4160, -4576, 1716, 225225,
        // C4[3], coeff of eps^5, polynomial in n of order 0
        -136, 63063,
        // C4[3], coeff of eps^4, polynomial in n of order 1
        1024, -208, 105105,
        // C4[3], coeff of eps^3, polynomial in n of order 2
        3584, -3328, 1144, 315315,
        // C4[4], coeff of eps^5, polynomial in n of order 0
        -128, 135135,
        // C4[4], coeff of eps^4, polynomial in n of order 1
        -2560, 832, 405405,
        // C4[5], coeff of eps^5, polynomial in n of order 0
        128, 99099,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
      static const real coeff[] = {
        // C4[0], coeff of eps^6, polynomial in n of order 0
        10, 9009,
        // C4[0], coeff of eps^5, polynomial in n of order 1
        -464, 291, 45045,
        // C4[0], coeff of eps^4, polynomial in n of order 2
        -4480, 1088, 156, 45045,
        // C4[0], coeff of eps^3, polynomial in n of order 3
        10736, -224, -4784, 1573, 45045,
        // C4[0], coeff of eps^2, polynomial in n of order 4
        1664, -10656, 14144, -4576, -858, 45045,
        // C4[0], coeff of eps^1, polynomial in n of order 5
        16, 64, 624, -4576, 6864, -3003, 15015,
        // C4[0], coeff of eps^0, polynomial in n of order 6
        56, 100, 208, 572, 3432, -12012, 30030, 45045,
        // C4[1], coeff of eps^6, polynomial in n of order 0
        10, 9009,
        // C4[1], coeff of eps^5, polynomial in n of order 1
        112, 15, 135135,
        // C4[1], coeff of eps^4, polynomial in n of order 2
        3840, -2944, 468, 135135,
        // C4[1], coeff of eps^3, polynomial in n of order 3
        -10704, 5792, 1040, -1287, 135135,
        // C4[1], coeff of eps^2, polynomial in n of order 4
        -768, 5952, -11648, 9152, -2574, 135135,
        // C4[1], coeff of eps^1, polynomial in n of order 5
        -16, -64, -624, 4576, -6864, 3003, 135135,
        // C4[2], coeff of eps^6, polynomial in n of order 0
        -4, 25025,
        // C4[2], coeff of eps^5, polynomial in n of order 1
        -1664, 168, 225225,
        // C4[2], coeff of eps^4, polynomial in n of order 2
        1664, 1856, -936, 225225,
        // C4[2], coeff of eps^3, polynomial in n of order 3
        6784, -8448, 4992, -1144, 225225,
        // C4[2], coeff of eps^2, polynomial in n of order 4
        128, -1440, 4160, -4576, 1716, 225225,
        // C4[3], coeff of eps^6, polynomial in n of order 0
        64, 315315,
        // C4[3], coeff of eps^5, polynomial in n of order 1
        1792, -680, 315315,
        // C4[3], coeff of eps^4, polynomial in n of order 2
        -2048, 1024, -208, 105105,
        // C4[3], coeff of eps^3, polynomial in n of order 3
        -1792, 3584, -3328, 1144, 315315,
        // C4[4], coeff of eps^6, polynomial in n of order 0
        -512, 405405,
        // C4[4], coeff of eps^5, polynomial in n of order 1
        2048, -384, 405405,
        // C4[4], coeff of eps^4, polynomial in n of order 2
        3072, -2560, 832, 405405,
        // C4[5], coeff of eps^6, polynomial in n of order 0
        -256, 495495,
        // C4[5], coeff of eps^5, polynomial in n of order 1
        -2048, 640, 495495,
        // C4[6], coeff of eps^6, polynomial in n of order 0
        512, 585585,
      };
  #elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
      static const real coeff[] = {
        // C4[0], coeff of eps^7, polynomial in n of order 0
        193, 85085,
        // C4[0], coeff of eps^6, polynomial in n of order 1
        4192, 850, 765765,
        // C4[0], coeff of eps^5, polynomial in n of order 2
        20960, -7888, 4947, 765765,
        // C4[0], coeff of eps^4, polynomial in n of order 3
        12480, -76160, 18496, 2652, 765765,
        // C4[0], coeff of eps^3, polynomial in n of order 4
        -154048, 182512, -3808, -81328, 26741, 765765,
        // C4[0], coeff of eps^2, polynomial in n of order 5
        3232, 28288, -181152, 240448, -77792, -14586, 765765,
        // C4[0], coeff of eps^1, polynomial in n of order 6
        96, 272, 1088, 10608, -77792, 116688, -51051, 255255,
        // C4[0], coeff of eps^0, polynomial in n of order 7
        588, 952, 1700, 3536, 9724, 58344, -204204, 510510, 765765,
        // C4[1], coeff of eps^7, polynomial in n of order 0
        349, 2297295,
        // C4[1], coeff of eps^6, polynomial in n of order 1
        -1472, 510, 459459,
        // C4[1], coeff of eps^5, polynomial in n of order 2
        -39840, 1904, 255, 2297295,
        // C4[1], coeff of eps^4, polynomial in n of order 3
        52608, 65280, -50048, 7956, 2297295,
        // C4[1], coeff of eps^3, polynomial in n of order 4
        103744, -181968, 98464, 17680, -21879, 2297295,
        // C4[1], coeff of eps^2, polynomial in n of order 5
        -1344, -13056, 101184, -198016, 155584, -43758, 2297295,
        // C4[1], coeff of eps^1, polynomial in n of order 6
        -96, -272, -1088, -10608, 77792, -116688, 51051, 2297295,
        // C4[2], coeff of eps^7, polynomial in n of order 0
        464, 1276275,
        // C4[2], coeff of eps^6, polynomial in n of order 1
        -928, -612, 3828825,
        // C4[2], coeff of eps^5, polynomial in n of order 2
        64256, -28288, 2856, 3828825,
        // C4[2], coeff of eps^4, polynomial in n of order 3
        -126528, 28288, 31552, -15912, 3828825,
        // C4[2], coeff of eps^3, polynomial in n of order 4
        -41472, 115328, -143616, 84864, -19448, 3828825,
        // C4[2], coeff of eps^2, polynomial in n of order 5
        160, 2176, -24480, 70720, -77792, 29172, 3828825,
        // C4[3], coeff of eps^7, polynomial in n of order 0
        -16, 97461,
        // C4[3], coeff of eps^6, polynomial in n of order 1
        -16384, 1088, 5360355,
        // C4[3], coeff of eps^5, polynomial in n of order 2
        -2560, 30464, -11560, 5360355,
        // C4[3], coeff of eps^4, polynomial in n of order 3
        35840, -34816, 17408, -3536, 1786785,
        // C4[3], coeff of eps^3, polynomial in n of order 4
        7168, -30464, 60928, -56576, 19448, 5360355,
        // C4[4], coeff of eps^7, polynomial in n of order 0
        128, 2297295,
        // C4[4], coeff of eps^6, polynomial in n of order 1
        26624, -8704, 6891885,
        // C4[4], coeff of eps^5, polynomial in n of order 2
        -77824, 34816, -6528, 6891885,
        // C4[4], coeff of eps^4, polynomial in n of order 3
        -32256, 52224, -43520, 14144, 6891885,
        // C4[5], coeff of eps^7, polynomial in n of order 0
        -6784, 8423415,
        // C4[5], coeff of eps^6, polynomial in n of order 1
        24576, -4352, 8423415,
        // C4[5], coeff of eps^5, polynomial in n of order 2
        45056, -34816, 10880, 8423415,
        // C4[6], coeff of eps^7, polynomial in n of order 0
        -1024, 3318315,
        // C4[6], coeff of eps^6, polynomial in n of order 1
        -28672, 8704, 9954945,
        // C4[7], coeff of eps^7, polynomial in n of order 0
        1024, 1640925,
      };
  #else
  #error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
  #endif
      static_assert(sizeof(coeff) / sizeof(real) ==
                    (nC4_ * (nC4_ + 1) * (nC4_ + 5)) / 6,
                    "Coefficient array size mismatch in C4coeff");
      int o = 0, k = 0;
      for (int l = 0; l < nC4_; ++l) {        // l is index of C4[l]
        for (int j = nC4_ - 1; j >= l; --j) { // coeff of eps^j
          int m = nC4_ - j - 1;               // order of polynomial in n
          _C4x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
          o += m + 2;
        }
      }
      // Post condition: o == sizeof(coeff) / sizeof(real) && k == nC4x_
    }
  
  } // namespace GeographicLib
  