cpp conversion: revert geodesic.cpp to geodesic.c

1 files changed, 2104 insertions, 0 deletions

diff --git a/src/geodesic.c b/src/geodesic.c
new file mode 100644
index 00000000..5504cb3b
--- /dev/null
+++ b/src/geodesic.c
@@ -0,0 +1,2104 @@
+/**
+ * \file geodesic.c
+ * \brief Implementation of the geodesic routines in C
+ *
+ * For the full documentation see geodesic.h.
+ **********************************************************************/
+
+/** @cond SKIP */
+
+/*
+ * This is a C implementation of the geodesic algorithms described in
+ *
+ *   C. F. F. Karney,
+ *   Algorithms for geodesics,
+ *   J. Geodesy <b>87</b>, 43--55 (2013);
+ *   https://doi.org/10.1007/s00190-012-0578-z
+ *   Addenda: https://geographiclib.sourceforge.io/geod-addenda.html
+ *
+ * See the comments in geodesic.h for documentation.
+ *
+ * Copyright (c) Charles Karney (2012-2018) <charles@karney.com> and licensed
+ * under the MIT/X11 License.  For more information, see
+ * https://geographiclib.sourceforge.io/
+ */
+
+#include "geodesic.h"
+#ifdef PJ_LIB__
+#include "proj_math.h"
+#else
+#include <math.h>
+#endif
+
+#if !defined(HAVE_C99_MATH)
+#define HAVE_C99_MATH 0
+#endif
+
+#if !defined(__cplusplus)
+#define nullptr 0
+#endif
+
+#define GEOGRAPHICLIB_GEODESIC_ORDER 6
+#define nA1   GEOGRAPHICLIB_GEODESIC_ORDER
+#define nC1   GEOGRAPHICLIB_GEODESIC_ORDER
+#define nC1p  GEOGRAPHICLIB_GEODESIC_ORDER
+#define nA2   GEOGRAPHICLIB_GEODESIC_ORDER
+#define nC2   GEOGRAPHICLIB_GEODESIC_ORDER
+#define nA3   GEOGRAPHICLIB_GEODESIC_ORDER
+#define nA3x  nA3
+#define nC3   GEOGRAPHICLIB_GEODESIC_ORDER
+#define nC3x  ((nC3 * (nC3 - 1)) / 2)
+#define nC4   GEOGRAPHICLIB_GEODESIC_ORDER
+#define nC4x  ((nC4 * (nC4 + 1)) / 2)
+#define nC    (GEOGRAPHICLIB_GEODESIC_ORDER + 1)
+
+typedef double real;
+typedef int boolx;
+
+static unsigned init = 0;
+static const int FALSE = 0;
+static const int TRUE = 1;
+static unsigned digits, maxit1, maxit2;
+static real epsilon, realmin, pi, degree, NaN,
+  tiny, tol0, tol1, tol2, tolb, xthresh;
+
+static void Init() {
+  if (!init) {
+#if defined(__DBL_MANT_DIG__)
+    digits = __DBL_MANT_DIG__;
+#else
+    digits = 53;
+#endif
+#if defined(__DBL_EPSILON__)
+    epsilon = __DBL_EPSILON__;
+#else
+    epsilon = pow(0.5, digits - 1);
+#endif
+#if defined(__DBL_MIN__)
+    realmin = __DBL_MIN__;
+#else
+    realmin = pow(0.5, 1022);
+#endif
+#if defined(M_PI)
+    pi = M_PI;
+#else
+    pi = atan2(0.0, -1.0);
+#endif
+    maxit1 = 20;
+    maxit2 = maxit1 + digits + 10;
+    tiny = sqrt(realmin);
+    tol0 = 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);
+    /* Check on bisection interval */
+    tolb = tol0 * tol2;
+    xthresh = 1000 * tol2;
+    degree = pi/180;
+    #if defined(NAN)
+    NaN = NAN;
+    #else
+    {
+      real minus1 = -1;
+      /* cppcheck-suppress wrongmathcall */
+      NaN = sqrt(minus1);
+    }
+    #endif
+    init = 1;
+  }
+}
+
+enum captype {
+  CAP_NONE = 0U,
+  CAP_C1   = 1U<<0,
+  CAP_C1p  = 1U<<1,
+  CAP_C2   = 1U<<2,
+  CAP_C3   = 1U<<3,
+  CAP_C4   = 1U<<4,
+  CAP_ALL  = 0x1FU,
+  OUT_ALL  = 0x7F80U
+};
+
+static real sq(real x) { return x * x; }
+#if HAVE_C99_MATH
+#define atanhx atanh
+#define copysignx copysign
+#define hypotx hypot
+#define cbrtx cbrt
+#else
+static real log1px(real x) {
+  volatile real
+    y = 1 + x,
+    z = y - 1;
+  /* Here's the explanation for this magic: y = 1 + z, exactly, and z
+   * approx x, thus log(y)/z (which is nearly constant near z = 0) returns
+   * a good approximation to the true log(1 + x)/x.  The multiplication x *
+   * (log(y)/z) introduces little additional error. */
+  return z == 0 ? x : x * log(y) / z;
+}
+
+static real atanhx(real x) {
+  real y = fabs(x);             /* Enforce odd parity */
+  y = log1px(2 * y/(1 - y))/2;
+  return x < 0 ? -y : y;
+}
+
+static real copysignx(real x, real y) {
+  return fabs(x) * (y < 0 || (y == 0 && 1/y < 0) ? -1 : 1);
+}
+
+static real hypotx(real x, real y)
+{ return sqrt(x * x + y * y); }
+
+static real cbrtx(real x) {
+  real y = pow(fabs(x), 1/(real)(3)); /* Return the real cube root */
+  return x < 0 ? -y : y;
+}
+#endif
+
+static real sumx(real u, real v, real* t) {
+  volatile real s = u + v;
+  volatile real up = s - v;
+  volatile real vpp = s - up;
+  up -= u;
+  vpp -= v;
+  if (t) *t = -(up + vpp);
+  /* error-free sum:
+   * u + v =       s      + t
+   *       = round(u + v) + t */
+  return s;
+}
+
+static real polyval(int N, const real p[], real x) {
+  real y = N < 0 ? 0 : *p++;
+  while (--N >= 0) y = y * x + *p++;
+  return y;
+}
+
+/* mimic C++ std::min and std::max */
+static real minx(real a, real b)
+{ return (b < a) ? b : a; }
+
+static real maxx(real a, real b)
+{ return (a < b) ? b : a; }
+
+static void swapx(real* x, real* y)
+{ real t = *x; *x = *y; *y = t; }
+
+static void norm2(real* sinx, real* cosx) {
+  real r = hypotx(*sinx, *cosx);
+  *sinx /= r;
+  *cosx /= r;
+}
+
+static real AngNormalize(real x) {
+#if HAVE_C99_MATH
+  x = remainder(x, (real)(360));
+  return x != -180 ? x : 180;
+#else
+  real y = fmod(x, (real)(360));
+#if defined(_MSC_VER) && _MSC_VER < 1900
+  /*
+   * Before version 14 (2015), Visual Studio had problems dealing
+   * with -0.0.  Specifically
+   *   VC 10,11,12 and 32-bit compile: fmod(-0.0, 360.0) -> +0.0
+   * sincosdx has a similar fix.
+   * python 2.7 on Windows 32-bit machines has the same problem.
+   */
+  if (x == 0) y = x;
+#endif
+  return y <= -180 ? y + 360 : (y <= 180 ? y : y - 360);
+#endif
+}
+
+static real LatFix(real x)
+{ return fabs(x) > 90 ? NaN : x; }
+
+static real AngDiff(real x, real y, real* e) {
+  real t, d = AngNormalize(sumx(AngNormalize(-x), AngNormalize(y), &t));
+  /* Here y - x = d + t (mod 360), exactly, where d is in (-180,180] and
+   * abs(t) <= eps (eps = 2^-45 for doubles).  The only case where the
+   * addition of t takes the result outside the range (-180,180] is d = 180
+   * and t > 0.  The case, d = -180 + eps, t = -eps, can't happen, since
+   * sum would have returned the exact result in such a case (i.e., given t
+   * = 0). */
+  return sumx(d == 180 && t > 0 ? -180 : d, t, e);
+}
+
+static real AngRound(real x) {
+  const real z = 1/(real)(16);
+  volatile real y;
+  if (x == 0) return 0;
+  y = fabs(x);
+  /* The compiler mustn't "simplify" z - (z - y) to y */
+  y = y < z ? z - (z - y) : y;
+  return x < 0 ? -y : y;
+}
+
+static void sincosdx(real x, real* sinx, real* cosx) {
+  /* In order to minimize round-off errors, this function exactly reduces
+   * the argument to the range [-45, 45] before converting it to radians. */
+  real r, s, c; int q;
+#if HAVE_C99_MATH && !defined(__GNUC__)
+  /* Disable for gcc because of bug in glibc version < 2.22, see
+   * https://sourceware.org/bugzilla/show_bug.cgi?id=17569 */
+  r = remquo(x, (real)(90), &q);
+#else
+  r = fmod(x, (real)(360));
+  /* check for NaN */
+  q = r == r ? (int)(floor(r / 90 + (real)(0.5))) : 0;
+  r -= 90 * q;
+#endif
+  /* now abs(r) <= 45 */
+  r *= degree;
+  /* Possibly could call the gnu extension sincos */
+  s = sin(r); c = cos(r);
+#if defined(_MSC_VER) && _MSC_VER < 1900
+  /*
+   * Before version 14 (2015), Visual Studio had problems dealing
+   * with -0.0.  Specifically
+   *   VC 10,11,12 and 32-bit compile: fmod(-0.0, 360.0) -> +0.0
+   *   VC 12       and 64-bit compile:  sin(-0.0)        -> +0.0
+   * AngNormalize has a similar fix.
+   * python 2.7 on Windows 32-bit machines has the same problem.
+   */
+  if (x == 0) s = x;
+#endif
+  switch ((unsigned)q & 3U) {
+  case 0U: *sinx =  s; *cosx =  c; break;
+  case 1U: *sinx =  c; *cosx = -s; break;
+  case 2U: *sinx = -s; *cosx = -c; break;
+  default: *sinx = -c; *cosx =  s; break; /* case 3U */
+  }
+  if (x != 0) { *sinx += (real)(0); *cosx += (real)(0); }
+}
+
+static real atan2dx(real y, real x) {
+  /* In order to minimize round-off errors, this function rearranges the
+   * arguments so that result of atan2 is in the range [-pi/4, pi/4] before
+   * converting it to degrees and mapping the result to the correct
+   * quadrant. */
+  int q = 0; real ang;
+  if (fabs(y) > fabs(x)) { swapx(&x, &y); q = 2; }
+  if (x < 0) { x = -x; ++q; }
+  /* here x >= 0 and x >= abs(y), so angle is in [-pi/4, pi/4] */
+  ang = atan2(y, x) / degree;
+  switch (q) {
+    /* Note that atan2d(-0.0, 1.0) will return -0.  However, we expect that
+     * atan2d will not be called with y = -0.  If need be, include
+     *
+     *   case 0: ang = 0 + ang; break;
+     */
+  case 1: ang = (y >= 0 ? 180 : -180) - ang; break;
+  case 2: ang =  90 - ang; break;
+  case 3: ang = -90 + ang; break;
+  }
+  return ang;
+}
+
+static void A3coeff(struct geod_geodesic* g);
+static void C3coeff(struct geod_geodesic* g);
+static void C4coeff(struct geod_geodesic* g);
+static real SinCosSeries(boolx sinp,
+                         real sinx, real cosx,
+                         const real c[], int n);
+static void Lengths(const struct geod_geodesic* g,
+                    real eps, real sig12,
+                    real ssig1, real csig1, real dn1,
+                    real ssig2, real csig2, real dn2,
+                    real cbet1, real cbet2,
+                    real* ps12b, real* pm12b, real* pm0,
+                    real* pM12, real* pM21,
+                    /* Scratch area of the right size */
+                    real Ca[]);
+static real Astroid(real x, real y);
+static real InverseStart(const struct geod_geodesic* g,
+                         real sbet1, real cbet1, real dn1,
+                         real sbet2, real cbet2, real dn2,
+                         real lam12, real slam12, real clam12,
+                         real* psalp1, real* pcalp1,
+                         /* Only updated if return val >= 0 */
+                         real* psalp2, real* pcalp2,
+                         /* Only updated for short lines */
+                         real* pdnm,
+                         /* Scratch area of the right size */
+                         real Ca[]);
+static real Lambda12(const struct geod_geodesic* g,
+                     real sbet1, real cbet1, real dn1,
+                     real sbet2, real cbet2, real dn2,
+                     real salp1, real calp1,
+                     real slam120, real clam120,
+                     real* psalp2, real* pcalp2,
+                     real* psig12,
+                     real* pssig1, real* pcsig1,
+                     real* pssig2, real* pcsig2,
+                     real* peps,
+                     real* pdomg12,
+                     boolx diffp, real* pdlam12,
+                     /* Scratch area of the right size */
+                     real Ca[]);
+static real A3f(const struct geod_geodesic* g, real eps);
+static void C3f(const struct geod_geodesic* g, real eps, real c[]);
+static void C4f(const struct geod_geodesic* g, real eps, real c[]);
+static real A1m1f(real eps);
+static void C1f(real eps, real c[]);
+static void C1pf(real eps, real c[]);
+static real A2m1f(real eps);
+static void C2f(real eps, real c[]);
+static int transit(real lon1, real lon2);
+static int transitdirect(real lon1, real lon2);
+static void accini(real s[]);
+static void acccopy(const real s[], real t[]);
+static void accadd(real s[], real y);
+static real accsum(const real s[], real y);
+static void accneg(real s[]);
+
+void geod_init(struct geod_geodesic* g, real a, real f) {
+  if (!init) Init();
+  g->a = a;
+  g->f = f;
+  g->f1 = 1 - g->f;
+  g->e2 = g->f * (2 - g->f);
+  g->ep2 = g->e2 / sq(g->f1);   /* e2 / (1 - e2) */
+  g->n = g->f / ( 2 - g->f);
+  g->b = g->a * g->f1;
+  g->c2 = (sq(g->a) + sq(g->b) *
+           (g->e2 == 0 ? 1 :
+            (g->e2 > 0 ? atanhx(sqrt(g->e2)) : atan(sqrt(-g->e2))) /
+            sqrt(fabs(g->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. */
+  g->etol2 = 0.1 * tol2 /
+    sqrt( maxx((real)(0.001), fabs(g->f)) * minx((real)(1), 1 - g->f/2) / 2 );
+
+  A3coeff(g);
+  C3coeff(g);
+  C4coeff(g);
+}
+
+static void geod_lineinit_int(struct geod_geodesicline* l,
+                              const struct geod_geodesic* g,
+                              real lat1, real lon1,
+                              real azi1, real salp1, real calp1,
+                              unsigned caps) {
+  real cbet1, sbet1, eps;
+  l->a = g->a;
+  l->f = g->f;
+  l->b = g->b;
+  l->c2 = g->c2;
+  l->f1 = g->f1;
+  /* If caps is 0 assume the standard direct calculation */
+  l->caps = (caps ? caps : GEOD_DISTANCE_IN | GEOD_LONGITUDE) |
+    /* always allow latitude and azimuth and unrolling of longitude */
+    GEOD_LATITUDE | GEOD_AZIMUTH | GEOD_LONG_UNROLL;
+
+  l->lat1 = LatFix(lat1);
+  l->lon1 = lon1;
+  l->azi1 = azi1;
+  l->salp1 = salp1;
+  l->calp1 = calp1;
+
+  sincosdx(AngRound(l->lat1), &sbet1, &cbet1); sbet1 *= l->f1;
+  /* Ensure cbet1 = +epsilon at poles */
+  norm2(&sbet1, &cbet1); cbet1 = maxx(tiny, cbet1);
+  l->dn1 = sqrt(1 + g->ep2 * sq(sbet1));
+
+  /* Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0), */
+  l->salp0 = l->salp1 * cbet1; /* alp0 in [0, pi/2 - |bet1|] */
+  /* Alt: calp0 = hypot(sbet1, calp1 * cbet1).  The following
+   * is slightly better (consider the case salp1 = 0). */
+  l->calp0 = hypotx(l->calp1, l->salp1 * sbet1);
+  /* Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
+   * sig = 0 is nearest northward crossing of equator.
+   * With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
+   * With bet1 =  pi/2, alp1 = -pi, sig1 =  pi/2
+   * With bet1 = -pi/2, alp1 =  0 , sig1 = -pi/2
+   * Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
+   * With alp0 in (0, pi/2], quadrants for sig and omg coincide.
+   * No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
+   * With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi. */
+  l->ssig1 = sbet1; l->somg1 = l->salp0 * sbet1;
+  l->csig1 = l->comg1 = sbet1 != 0 || l->calp1 != 0 ? cbet1 * l->calp1 : 1;
+  norm2(&l->ssig1, &l->csig1); /* sig1 in (-pi, pi] */
+  /* norm2(somg1, comg1); -- don't need to normalize! */
+
+  l->k2 = sq(l->calp0) * g->ep2;
+  eps = l->k2 / (2 * (1 + sqrt(1 + l->k2)) + l->k2);
+
+  if (l->caps & CAP_C1) {
+    real s, c;
+    l->A1m1 = A1m1f(eps);
+    C1f(eps, l->C1a);
+    l->B11 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C1a, nC1);
+    s = sin(l->B11); c = cos(l->B11);
+    /* tau1 = sig1 + B11 */
+    l->stau1 = l->ssig1 * c + l->csig1 * s;
+    l->ctau1 = l->csig1 * c - l->ssig1 * s;
+    /* Not necessary because C1pa reverts C1a
+     *    B11 = -SinCosSeries(TRUE, stau1, ctau1, C1pa, nC1p); */
+  }
+
+  if (l->caps & CAP_C1p)
+    C1pf(eps, l->C1pa);
+
+  if (l->caps & CAP_C2) {
+    l->A2m1 = A2m1f(eps);
+    C2f(eps, l->C2a);
+    l->B21 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C2a, nC2);
+  }
+
+  if (l->caps & CAP_C3) {
+    C3f(g, eps, l->C3a);
+    l->A3c = -l->f * l->salp0 * A3f(g, eps);
+    l->B31 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C3a, nC3-1);
+  }
+
+  if (l->caps & CAP_C4) {
+    C4f(g, eps, l->C4a);
+    /* Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) */
+    l->A4 = sq(l->a) * l->calp0 * l->salp0 * g->e2;
+    l->B41 = SinCosSeries(FALSE, l->ssig1, l->csig1, l->C4a, nC4);
+  }
+
+  l->a13 = l->s13 = NaN;
+}
+
+void geod_lineinit(struct geod_geodesicline* l,
+                   const struct geod_geodesic* g,
+                   real lat1, real lon1, real azi1, unsigned caps) {
+  real salp1, calp1;
+  azi1 = AngNormalize(azi1);
+  /* Guard against underflow in salp0 */
+  sincosdx(AngRound(azi1), &salp1, &calp1);
+  geod_lineinit_int(l, g, lat1, lon1, azi1, salp1, calp1, caps);
+}
+
+void geod_gendirectline(struct geod_geodesicline* l,
+                        const struct geod_geodesic* g,
+                        real lat1, real lon1, real azi1,
+                        unsigned flags, real s12_a12,
+                        unsigned caps) {
+  geod_lineinit(l, g, lat1, lon1, azi1, caps);
+  geod_gensetdistance(l, flags, s12_a12);
+}
+
+void geod_directline(struct geod_geodesicline* l,
+                        const struct geod_geodesic* g,
+                        real lat1, real lon1, real azi1,
+                        real s12, unsigned caps) {
+  geod_gendirectline(l, g, lat1, lon1, azi1, GEOD_NOFLAGS, s12, caps);
+}
+
+real geod_genposition(const struct geod_geodesicline* l,
+                      unsigned flags, real s12_a12,
+                      real* plat2, real* plon2, real* pazi2,
+                      real* ps12, real* pm12,
+                      real* pM12, real* pM21,
+                      real* pS12) {
+  real lat2 = 0, lon2 = 0, azi2 = 0, s12 = 0,
+    m12 = 0, M12 = 0, M21 = 0, S12 = 0;
+  /* Avoid warning about uninitialized B12. */
+  real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
+  real omg12, lam12, lon12;
+  real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2, dn2;
+  unsigned outmask =
+    (plat2 ? GEOD_LATITUDE : GEOD_NONE) |
+    (plon2 ? GEOD_LONGITUDE : GEOD_NONE) |
+    (pazi2 ? GEOD_AZIMUTH : GEOD_NONE) |
+    (ps12 ? GEOD_DISTANCE : GEOD_NONE) |
+    (pm12 ? GEOD_REDUCEDLENGTH : GEOD_NONE) |
+    (pM12 || pM21 ? GEOD_GEODESICSCALE : GEOD_NONE) |
+    (pS12 ? GEOD_AREA : GEOD_NONE);
+
+  outmask &= l->caps & OUT_ALL;
+  if (!( TRUE /*Init()*/ &&
+         (flags & GEOD_ARCMODE || (l->caps & (GEOD_DISTANCE_IN & OUT_ALL))) ))
+    /* Uninitialized or impossible distance calculation requested */
+    return NaN;
+
+  if (flags & GEOD_ARCMODE) {
+    /* Interpret s12_a12 as spherical arc length */
+    sig12 = s12_a12 * degree;
+    sincosdx(s12_a12, &ssig12, &csig12);
+  } else {
+    /* Interpret s12_a12 as distance */
+    real
+      tau12 = s12_a12 / (l->b * (1 + l->A1m1)),
+      s = sin(tau12),
+      c = cos(tau12);
+    /* tau2 = tau1 + tau12 */
+    B12 = - SinCosSeries(TRUE,
+                         l->stau1 * c + l->ctau1 * s,
+                         l->ctau1 * c - l->stau1 * s,
+                         l->C1pa, nC1p);
+    sig12 = tau12 - (B12 - l->B11);
+    ssig12 = sin(sig12); csig12 = cos(sig12);
+    if (fabs(l->f) > 0.01) {
+      /* Reverted distance series is inaccurate for |f| > 1/100, so correct
+       * sig12 with 1 Newton iteration.  The following table shows the
+       * approximate maximum error for a = WGS_a() and various f relative to
+       * GeodesicExact.
+       *     erri = the error in the inverse solution (nm)
+       *     errd = the error in the direct solution (series only) (nm)
+       *     errda = the error in the direct solution (series + 1 Newton) (nm)
+       *
+       *       f     erri  errd errda
+       *     -1/5    12e6 1.2e9  69e6
+       *     -1/10  123e3  12e6 765e3
+       *     -1/20   1110 108e3  7155
+       *     -1/50  18.63 200.9 27.12
+       *     -1/100 18.63 23.78 23.37
+       *     -1/150 18.63 21.05 20.26
+       *      1/150 22.35 24.73 25.83
+       *      1/100 22.35 25.03 25.31
+       *      1/50  29.80 231.9 30.44
+       *      1/20   5376 146e3  10e3
+       *      1/10  829e3  22e6 1.5e6
+       *      1/5   157e6 3.8e9 280e6 */
+      real serr;
+      ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12;
+      csig2 = l->csig1 * csig12 - l->ssig1 * ssig12;
+      B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1);
+      serr = (1 + l->A1m1) * (sig12 + (B12 - l->B11)) - s12_a12 / l->b;
+      sig12 = sig12 - serr / sqrt(1 + l->k2 * sq(ssig2));
+      ssig12 = sin(sig12); csig12 = cos(sig12);
+      /* Update B12 below */
+    }
+  }
+
+  /* sig2 = sig1 + sig12 */
+  ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12;
+  csig2 = l->csig1 * csig12 - l->ssig1 * ssig12;
+  dn2 = sqrt(1 + l->k2 * sq(ssig2));
+  if (outmask & (GEOD_DISTANCE | GEOD_REDUCEDLENGTH | GEOD_GEODESICSCALE)) {
+    if (flags & GEOD_ARCMODE || fabs(l->f) > 0.01)
+      B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1);
+    AB1 = (1 + l->A1m1) * (B12 - l->B11);
+  }
+  /* sin(bet2) = cos(alp0) * sin(sig2) */
+  sbet2 = l->calp0 * ssig2;
+  /* Alt: cbet2 = hypot(csig2, salp0 * ssig2); */
+  cbet2 = hypotx(l->salp0, l->calp0 * csig2);
+  if (cbet2 == 0)
+    /* I.e., salp0 = 0, csig2 = 0.  Break the degeneracy in this case */
+    cbet2 = csig2 = tiny;
+  /* tan(alp0) = cos(sig2)*tan(alp2) */
+  salp2 = l->salp0; calp2 = l->calp0 * csig2; /* No need to normalize */
+
+  if (outmask & GEOD_DISTANCE)
+    s12 = (flags & GEOD_ARCMODE) ?
+      l->b * ((1 + l->A1m1) * sig12 + AB1) :
+      s12_a12;
+
+  if (outmask & GEOD_LONGITUDE) {
+    real E = copysignx(1, l->salp0); /* east or west going? */
+    /* tan(omg2) = sin(alp0) * tan(sig2) */
+    somg2 = l->salp0 * ssig2; comg2 = csig2;  /* No need to normalize */
+    /* omg12 = omg2 - omg1 */
+    omg12 = (flags & GEOD_LONG_UNROLL)
+      ? E * (sig12
+             - (atan2(    ssig2, csig2) - atan2(    l->ssig1, l->csig1))
+             + (atan2(E * somg2, comg2) - atan2(E * l->somg1, l->comg1)))
+      : atan2(somg2 * l->comg1 - comg2 * l->somg1,
+              comg2 * l->comg1 + somg2 * l->somg1);
+    lam12 = omg12 + l->A3c *
+      ( sig12 + (SinCosSeries(TRUE, ssig2, csig2, l->C3a, nC3-1)
+                 - l->B31));
+    lon12 = lam12 / degree;
+    lon2 = (flags & GEOD_LONG_UNROLL) ? l->lon1 + lon12 :
+      AngNormalize(AngNormalize(l->lon1) + AngNormalize(lon12));
+  }
+
+  if (outmask & GEOD_LATITUDE)
+    lat2 = atan2dx(sbet2, l->f1 * cbet2);
+
+  if (outmask & GEOD_AZIMUTH)
+    azi2 = atan2dx(salp2, calp2);
+
+  if (outmask & (GEOD_REDUCEDLENGTH | GEOD_GEODESICSCALE)) {
+    real
+      B22 = SinCosSeries(TRUE, ssig2, csig2, l->C2a, nC2),
+      AB2 = (1 + l->A2m1) * (B22 - l->B21),
+      J12 = (l->A1m1 - l->A2m1) * sig12 + (AB1 - AB2);
+    if (outmask & GEOD_REDUCEDLENGTH)
+      /* Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
+       * accurate cancellation in the case of coincident points. */
+      m12 = l->b * ((dn2 * (l->csig1 * ssig2) - l->dn1 * (l->ssig1 * csig2))
+                    - l->csig1 * csig2 * J12);
+    if (outmask & GEOD_GEODESICSCALE) {
+      real t = l->k2 * (ssig2 - l->ssig1) * (ssig2 + l->ssig1) /
+        (l->dn1 + dn2);
+      M12 = csig12 + (t *  ssig2 -  csig2 * J12) * l->ssig1 / l->dn1;
+      M21 = csig12 - (t * l->ssig1 - l->csig1 * J12) *  ssig2 /  dn2;
+    }
+  }
+
+  if (outmask & GEOD_AREA) {
+    real
+      B42 = SinCosSeries(FALSE, ssig2, csig2, l->C4a, nC4);
+    real salp12, calp12;
+    if (l->calp0 == 0 || l->salp0 == 0) {
+      /* alp12 = alp2 - alp1, used in atan2 so no need to normalize */
+      salp12 = salp2 * l->calp1 - calp2 * l->salp1;
+      calp12 = calp2 * l->calp1 + salp2 * l->salp1;
+    } else {
+      /* tan(alp) = tan(alp0) * sec(sig)
+       * tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
+       * = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
+       * If csig12 > 0, write
+       *   csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
+       * else
+       *   csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
+       * No need to normalize */
+      salp12 = l->calp0 * l->salp0 *
+        (csig12 <= 0 ? l->csig1 * (1 - csig12) + ssig12 * l->ssig1 :
+         ssig12 * (l->csig1 * ssig12 / (1 + csig12) + l->ssig1));
+      calp12 = sq(l->salp0) + sq(l->calp0) * l->csig1 * csig2;
+    }
+    S12 = l->c2 * atan2(salp12, calp12) + l->A4 * (B42 - l->B41);
+  }
+
+  /* In the pattern
+   *
+   *   if ((outmask & GEOD_XX) && pYY)
+   *     *pYY = YY;
+   *
+   * the second check "&& pYY" is redundant.  It's there to make the CLang
+   * static analyzer happy.
+   */
+  if ((outmask & GEOD_LATITUDE) && plat2)
+    *plat2 = lat2;
+  if ((outmask & GEOD_LONGITUDE) && plon2)
+    *plon2 = lon2;
+  if ((outmask & GEOD_AZIMUTH) && pazi2)
+    *pazi2 = azi2;
+  if ((outmask & GEOD_DISTANCE) && ps12)
+    *ps12 = s12;
+  if ((outmask & GEOD_REDUCEDLENGTH) && pm12)
+    *pm12 = m12;
+  if (outmask & GEOD_GEODESICSCALE) {
+    if (pM12) *pM12 = M12;
+    if (pM21) *pM21 = M21;
+  }
+  if ((outmask & GEOD_AREA) && pS12)
+    *pS12 = S12;
+
+  return (flags & GEOD_ARCMODE) ? s12_a12 : sig12 / degree;
+}
+
+void geod_setdistance(struct geod_geodesicline* l, real s13) {
+  l->s13 = s13;
+  l->a13 = geod_genposition(l, GEOD_NOFLAGS, l->s13, nullptr, nullptr, nullptr, nullptr, nullptr, nullptr, nullptr, nullptr);
+}
+
+static void geod_setarc(struct geod_geodesicline* l, real a13) {
+  l->a13 = a13; l->s13 = NaN;
+  geod_genposition(l, GEOD_ARCMODE, l->a13, nullptr, nullptr, nullptr, &l->s13, nullptr, nullptr, nullptr, nullptr);
+}
+
+void geod_gensetdistance(struct geod_geodesicline* l,
+ unsigned flags, real s13_a13) {
+  (flags & GEOD_ARCMODE) ?
+    geod_setarc(l, s13_a13) :
+    geod_setdistance(l, s13_a13);
+}
+
+void geod_position(const struct geod_geodesicline* l, real s12,
+                   real* plat2, real* plon2, real* pazi2) {
+  geod_genposition(l, FALSE, s12, plat2, plon2, pazi2, nullptr, nullptr, nullptr, nullptr, nullptr);
+}
+
+real geod_gendirect(const struct geod_geodesic* g,
+                    real lat1, real lon1, real azi1,
+                    unsigned flags, real s12_a12,
+                    real* plat2, real* plon2, real* pazi2,
+                    real* ps12, real* pm12, real* pM12, real* pM21,
+                    real* pS12) {
+  struct geod_geodesicline l;
+  unsigned outmask =
+    (plat2 ? GEOD_LATITUDE : GEOD_NONE) |
+    (plon2 ? GEOD_LONGITUDE : GEOD_NONE) |
+    (pazi2 ? GEOD_AZIMUTH : GEOD_NONE) |
+    (ps12 ? GEOD_DISTANCE : GEOD_NONE) |
+    (pm12 ? GEOD_REDUCEDLENGTH : GEOD_NONE) |
+    (pM12 || pM21 ? GEOD_GEODESICSCALE : GEOD_NONE) |
+    (pS12 ? GEOD_AREA : GEOD_NONE);
+
+  geod_lineinit(&l, g, lat1, lon1, azi1,
+                /* Automatically supply GEOD_DISTANCE_IN if necessary */
+                outmask |
+                ((flags & GEOD_ARCMODE) ? GEOD_NONE : GEOD_DISTANCE_IN));
+  return geod_genposition(&l, flags, s12_a12,
+                          plat2, plon2, pazi2, ps12, pm12, pM12, pM21, pS12);
+}
+
+void geod_direct(const struct geod_geodesic* g,
+                 real lat1, real lon1, real azi1,
+                 real s12,
+                 real* plat2, real* plon2, real* pazi2) {
+  geod_gendirect(g, lat1, lon1, azi1, GEOD_NOFLAGS, s12, plat2, plon2, pazi2,
+                 nullptr, nullptr, nullptr, nullptr, nullptr);
+}
+
+static real geod_geninverse_int(const struct geod_geodesic* g,
+                                real lat1, real lon1, real lat2, real lon2,
+                                real* ps12,
+                                real* psalp1, real* pcalp1,
+                                real* psalp2, real* pcalp2,
+                                real* pm12, real* pM12, real* pM21,
+                                real* pS12) {
+  real s12 = 0, m12 = 0, M12 = 0, M21 = 0, S12 = 0;
+  real lon12, lon12s;
+  int latsign, lonsign, swapp;
+  real sbet1, cbet1, sbet2, cbet2, s12x = 0, m12x = 0;
+  real dn1, dn2, lam12, slam12, clam12;
+  real a12 = 0, sig12, calp1 = 0, salp1 = 0, calp2 = 0, salp2 = 0;
+  real Ca[nC];
+  boolx meridian;
+  /* somg12 > 1 marks that it needs to be calculated */
+  real omg12 = 0, somg12 = 2, comg12 = 0;
+
+  unsigned outmask =
+    (ps12 ? GEOD_DISTANCE : GEOD_NONE) |
+    (pm12 ? GEOD_REDUCEDLENGTH : GEOD_NONE) |
+    (pM12 || pM21 ? GEOD_GEODESICSCALE : GEOD_NONE) |
+    (pS12 ? GEOD_AREA : GEOD_NONE);
+
+  outmask &= OUT_ALL;
+  /* 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. */
+  lon12 = AngDiff(lon1, lon2, &lon12s);
+  /* Make longitude difference positive. */
+  lonsign = lon12 >= 0 ? 1 : -1;
+  /* If very close to being on the same half-meridian, then make it so. */
+  lon12 = lonsign * AngRound(lon12);
+  lon12s = AngRound((180 - lon12) - lonsign * lon12s);
+  lam12 = lon12 * degree;
+  if (lon12 > 90) {
+    sincosdx(lon12s, &slam12, &clam12);
+    clam12 = -clam12;
+  } else
+    sincosdx(lon12, &slam12, &clam12);
+
+  /* If really close to the equator, treat as on equator. */
+  lat1 = AngRound(LatFix(lat1));
+  lat2 = AngRound(LatFix(lat2));
+  /* Swap points so that point with higher (abs) latitude is point 1
+   * If one latitude is a nan, then it becomes lat1. */
+  swapp = fabs(lat1) < fabs(lat2) ? -1 : 1;
+  if (swapp < 0) {
+    lonsign *= -1;
+    swapx(&lat1, &lat2);
+  }
+  /* Make lat1 <= 0 */
+  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. */
+
+  sincosdx(lat1, &sbet1, &cbet1); sbet1 *= g->f1;
+  /* Ensure cbet1 = +epsilon at poles */
+  norm2(&sbet1, &cbet1); cbet1 = maxx(tiny, cbet1);
+
+  sincosdx(lat2, &sbet2, &cbet2); sbet2 *= g->f1;
+  /* Ensure cbet2 = +epsilon at poles */
+  norm2(&sbet2, &cbet2); cbet2 = maxx(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 (fabs(sbet2) == -sbet1)
+      cbet2 = cbet1;
+  }
+
+  dn1 = sqrt(1 + g->ep2 * sq(sbet1));
+  dn2 = sqrt(1 + g->ep2 * sq(sbet2));
+
+  meridian = lat1 == -90 || slam12 == 0;
+
+  if (meridian) {
+
+    /* Endpoints are on a single full meridian, so the geodesic might lie on
+     * a meridian. */
+
+    real ssig1, csig1, ssig2, csig2;
+    calp1 = clam12; salp1 = slam12; /* Head to the target longitude */
+    calp2 = 1; salp2 = 0;           /* At the target we're heading north */
+
+    /* tan(bet) = tan(sig) * cos(alp) */
+    ssig1 = sbet1; csig1 = calp1 * cbet1;
+    ssig2 = sbet2; csig2 = calp2 * cbet2;
+
+    /* sig12 = sig2 - sig1 */
+    sig12 = atan2(maxx((real)(0), csig1 * ssig2 - ssig1 * csig2),
+                                  csig1 * csig2 + ssig1 * ssig2);
+    Lengths(g, g->n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
+            cbet1, cbet2, &s12x, &m12x, nullptr,
+            (outmask & GEOD_GEODESICSCALE) ? &M12 : nullptr,
+            (outmask & GEOD_GEODESICSCALE) ? &M21 : nullptr,
+            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. */
+    if (sig12 < 1 || m12x >= 0) {
+      /* Need at least 2, to handle 90 0 90 180 */
+      if (sig12 < 3 * tiny)
+        sig12 = m12x = s12x = 0;
+      m12x *= g->b;
+      s12x *= g->b;
+      a12 = sig12 / degree;
+    } else
+      /* m12 < 0, i.e., prolate and too close to anti-podal */
+      meridian = FALSE;
+  }
+
+  if (!meridian &&
+      sbet1 == 0 &&           /* and sbet2 == 0 */
+      /* Mimic the way Lambda12 works with calp1 = 0 */
+      (g->f <= 0 || lon12s >= g->f * 180)) {
+
+    /* Geodesic runs along equator */
+    calp1 = calp2 = 0; salp1 = salp2 = 1;
+    s12x = g->a * lam12;
+    sig12 = omg12 = lam12 / g->f1;
+    m12x = g->b * sin(sig12);
+    if (outmask & GEOD_GEODESICSCALE)
+      M12 = M21 = cos(sig12);
+    a12 = lon12 / g->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 = 0;
+    sig12 = InverseStart(g, 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 * g->b * dnm;
+      m12x = sq(dnm) * g->b * sin(sig12 / dnm);
+      if (outmask & GEOD_GEODESICSCALE)
+        M12 = M21 = cos(sig12 / dnm);
+      a12 = sig12 / degree;
+      omg12 = lam12 / (g->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. */
+      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;
+      boolx tripn = FALSE;
+      boolx tripb = FALSE;
+      for (; numit < maxit2; ++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 = 0,
+          v = Lambda12(g, sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
+                        slam12, clam12,
+                        &salp2, &calp2, &sig12, &ssig1, &csig1, &ssig2, &csig2,
+                        &eps, &domg12, numit < maxit1, &dv, Ca);
+        /* 2 * tol0 is approximately 1 ulp for a number in [0, pi]. */
+        /* Reversed test to allow escape with NaNs */
+        if (tripb || !(fabs(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 && fabs(dalp1) < pi) {
+            calp1 = calp1 * cdalp1 - salp1 * sdalp1;
+            salp1 = nsalp1;
+            norm2(&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 = fabs(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;
+        norm2(&salp1, &calp1);
+        tripn = FALSE;
+        tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb ||
+                 fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb);
+      }
+      Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
+              cbet1, cbet2, &s12x, &m12x, nullptr,
+              (outmask & GEOD_GEODESICSCALE) ? &M12 : nullptr,
+              (outmask & GEOD_GEODESICSCALE) ? &M21 : nullptr, Ca);
+      m12x *= g->b;
+      s12x *= g->b;
+      a12 = sig12 / degree;
+      if (outmask & GEOD_AREA) {
+        /* omg12 = lam12 - domg12 */
+        real sdomg12 = sin(domg12), cdomg12 = cos(domg12);
+        somg12 = slam12 * cdomg12 - clam12 * sdomg12;
+        comg12 = clam12 * cdomg12 + slam12 * sdomg12;
+      }
+    }
+  }
+
+  if (outmask & GEOD_DISTANCE)
+    s12 = 0 + s12x;             /* Convert -0 to 0 */
+
+  if (outmask & GEOD_REDUCEDLENGTH)
+    m12 = 0 + m12x;             /* Convert -0 to 0 */
+
+  if (outmask & GEOD_AREA) {
+    real
+      /* From Lambda12: sin(alp1) * cos(bet1) = sin(alp0) */
+      salp0 = salp1 * cbet1,
+      calp0 = hypotx(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 = sq(calp0) * g->ep2,
+        eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
+        /* Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0). */
+        A4 = sq(g->a) * calp0 * salp0 * g->e2;
+      real B41, B42;
+      norm2(&ssig1, &csig1);
+      norm2(&ssig2, &csig2);
+      C4f(g, eps, Ca);
+      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 += g->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) {
+    swapx(&salp1, &salp2);
+    swapx(&calp1, &calp2);
+    if (outmask & GEOD_GEODESICSCALE)
+      swapx(&M12, &M21);
+  }
+
+  salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
+  salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
+
+  if (psalp1) *psalp1 = salp1;
+  if (pcalp1) *pcalp1 = calp1;
+  if (psalp2) *psalp2 = salp2;
+  if (pcalp2) *pcalp2 = calp2;
+
+  if (outmask & GEOD_DISTANCE)
+    *ps12 = s12;
+  if (outmask & GEOD_REDUCEDLENGTH)
+    *pm12 = m12;
+  if (outmask & GEOD_GEODESICSCALE) {
+    if (pM12) *pM12 = M12;
+    if (pM21) *pM21 = M21;
+  }
+  if (outmask & GEOD_AREA)
+    *pS12 = S12;
+
+  /* Returned value in [0, 180] */
+  return a12;
+}
+
+real geod_geninverse(const struct geod_geodesic* g,
+                     real lat1, real lon1, real lat2, real lon2,
+                     real* ps12, real* pazi1, real* pazi2,
+                     real* pm12, real* pM12, real* pM21, real* pS12) {
+  real salp1, calp1, salp2, calp2,
+    a12 = geod_geninverse_int(g, lat1, lon1, lat2, lon2, ps12,
+                              &salp1, &calp1, &salp2, &calp2,
+                              pm12, pM12, pM21, pS12);
+  if (pazi1) *pazi1 = atan2dx(salp1, calp1);
+  if (pazi2) *pazi2 = atan2dx(salp2, calp2);
+  return a12;
+}
+
+void geod_inverseline(struct geod_geodesicline* l,
+                      const struct geod_geodesic* g,
+                      real lat1, real lon1, real lat2, real lon2,
+                      unsigned caps) {
+  real salp1, calp1,
+    a12 = geod_geninverse_int(g, lat1, lon1, lat2, lon2, nullptr,
+                              &salp1, &calp1, nullptr, nullptr,
+                              nullptr, nullptr, nullptr, nullptr),
+    azi1 = atan2dx(salp1, calp1);
+  caps = caps ? caps : GEOD_DISTANCE_IN | GEOD_LONGITUDE;
+  /* Ensure that a12 can be converted to a distance */
+  if (caps & (OUT_ALL & GEOD_DISTANCE_IN)) caps |= GEOD_DISTANCE;
+  geod_lineinit_int(l, g, lat1, lon1, azi1, salp1, calp1, caps);
+  geod_setarc(l, a12);
+}
+
+void geod_inverse(const struct geod_geodesic* g,
+                  real lat1, real lon1, real lat2, real lon2,
+                  real* ps12, real* pazi1, real* pazi2) {
+  geod_geninverse(g, lat1, lon1, lat2, lon2, ps12, pazi1, pazi2, nullptr, nullptr, nullptr, nullptr);
+}
+
+real SinCosSeries(boolx 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 */
+  real ar, y0, y1;
+  c += (n + sinp);              /* Point to one beyond last element */
+  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) */
+}
+
+void Lengths(const struct geod_geodesic* g,
+             real eps, real sig12,
+             real ssig1, real csig1, real dn1,
+             real ssig2, real csig2, real dn2,
+             real cbet1, real cbet2,
+             real* ps12b, real* pm12b, real* pm0,
+             real* pM12, real* pM21,
+             /* Scratch area of the right size */
+             real Ca[]) {
+  real m0 = 0, J12 = 0, A1 = 0, A2 = 0;
+  real Cb[nC];
+
+  /* Return m12b = (reduced length)/b; also calculate s12b = distance/b,
+   * and m0 = coefficient of secular term in expression for reduced length. */
+  boolx redlp = pm12b || pm0 || pM12 || pM21;
+  if (ps12b || redlp) {
+    A1 = A1m1f(eps);
+    C1f(eps, Ca);
+    if (redlp) {
+      A2 = A2m1f(eps);
+      C2f(eps, Cb);
+      m0 = A1 - A2;
+      A2 = 1 + A2;
+    }
+    A1 = 1 + A1;
+  }
+  if (ps12b) {
+    real B1 = SinCosSeries(TRUE, ssig2, csig2, Ca, nC1) -
+      SinCosSeries(TRUE, ssig1, csig1, Ca, nC1);
+    /* Missing a factor of b */
+    *ps12b = A1 * (sig12 + B1);
+    if (redlp) {
+      real B2 = SinCosSeries(TRUE, ssig2, csig2, Cb, nC2) -
+        SinCosSeries(TRUE, ssig1, csig1, Cb, nC2);
+      J12 = m0 * sig12 + (A1 * B1 - A2 * B2);
+    }
+  } else if (redlp) {
+    /* Assume here that nC1 >= nC2 */
+    int l;
+    for (l = 1; l <= nC2; ++l)
+      Cb[l] = A1 * Ca[l] - A2 * Cb[l];
+    J12 = m0 * sig12 + (SinCosSeries(TRUE, ssig2, csig2, Cb, nC2) -
+                        SinCosSeries(TRUE, ssig1, csig1, Cb, nC2));
+  }
+  if (pm0) *pm0 = m0;
+  if (pm12b)
+    /* Missing a factor of b.
+     * Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
+     * accurate cancellation in the case of coincident points. */
+    *pm12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
+      csig1 * csig2 * J12;
+  if (pM12 || pM21) {
+    real csig12 = csig1 * csig2 + ssig1 * ssig2;
+    real t = g->ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
+    if (pM12)
+      *pM12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
+    if (pM21)
+      *pM21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
+  }
+}
+
+real 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 = sq(x),
+    q = 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 = 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;
+    real v, uv, w;
+    if (disc >= 0) {
+      real T3 = S + r3, T;
+      /* 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. cbrtx always returns the real root.  cbrtx(-8) = -2. */
+      T = cbrtx(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);
+    }
+    v = sqrt(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 + 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;
+}
+
+real InverseStart(const struct geod_geodesic* g,
+                  real sbet1, real cbet1, real dn1,
+                  real sbet2, real cbet2, real dn2,
+                  real lam12, real slam12, real clam12,
+                  real* psalp1, real* pcalp1,
+                  /* Only updated if return val >= 0 */
+                  real* psalp2, real* pcalp2,
+                  /* Only updated for short lines */
+                  real* pdnm,
+                  /* Scratch area of the right size */
+                  real Ca[]) {
+  real salp1 = 0, calp1 = 0, salp2 = 0, calp2 = 0, dnm = 0;
+
+  /* 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;
+  boolx shortline = cbet12 >= 0 && sbet12 < (real)(0.5) &&
+    cbet2 * lam12 < (real)(0.5);
+  real somg12, comg12, ssig12, csig12;
+#if defined(__GNUC__) && __GNUC__ == 4 &&       \
+  (__GNUC_MINOR__ < 6 || defined(__MINGW32__))
+  /* Volatile declaration needed to fix inverse cases
+   * 88.202499451857 0 -88.202499451857 179.981022032992859592
+   * 89.262080389218 0 -89.262080389218 179.992207982775375662
+   * 89.333123580033 0 -89.333123580032997687 179.99295812360148422
+   * which otherwise fail with g++ 4.4.4 x86 -O3 (Linux)
+   * and g++ 4.4.0 (mingw) and g++ 4.6.1 (tdm mingw). */
+  {
+    volatile real xx1 = sbet2 * cbet1;
+    volatile real xx2 = cbet2 * sbet1;
+    sbet12a = xx1 + xx2;
+  }
+#else
+  sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
+#endif
+  if (shortline) {
+    real sbetm2 = sq(sbet1 + sbet2), omg12;
+    /* sin((bet1+bet2)/2)^2
+     * =  (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2) */
+    sbetm2 /= sbetm2 + sq(cbet1 + cbet2);
+    dnm = sqrt(1 + g->ep2 * sbetm2);
+    omg12 = lam12 / (g->f1 * dnm);
+    somg12 = sin(omg12); comg12 = cos(omg12);
+  } else {
+    somg12 = slam12; comg12 = clam12;
+  }
+
+  salp1 = cbet2 * somg12;
+  calp1 = comg12 >= 0 ?
+    sbet12 + cbet2 * sbet1 * sq(somg12) / (1 + comg12) :
+    sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12);
+
+  ssig12 = hypotx(salp1, calp1);
+  csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
+
+  if (shortline && ssig12 < g->etol2) {
+    /* really short lines */
+    salp2 = cbet1 * somg12;
+    calp2 = sbet12 - cbet1 * sbet2 *
+      (comg12 >= 0 ? sq(somg12) / (1 + comg12) : 1 - comg12);
+    norm2(&salp2, &calp2);
+    /* Set return value */
+    sig12 = atan2(ssig12, csig12);
+  } else if (fabs(g->n) > (real)(0.1) || /* No astroid calc if too eccentric */
+             csig12 >= 0 ||
+             ssig12 >= 6 * fabs(g->n) * pi * 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 */
+    volatile real x;
+    real lam12x = atan2(-slam12, -clam12); /* lam12 - pi */
+    if (g->f >= 0) {            /* In fact f == 0 does not get here */
+      /* x = dlong, y = dlat */
+      {
+        real
+          k2 = sq(sbet1) * g->ep2,
+          eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
+        lamscale = g->f * cbet1 * A3f(g, eps) * 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;
+      /* In the case of lon12 = 180, this repeats a calculation made in
+       * Inverse. */
+      Lengths(g, g->n, pi + bet12a,
+              sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
+              cbet1, cbet2, nullptr, &m12b, &m0, nullptr, nullptr, Ca);
+      x = -1 + m12b / (cbet1 * cbet2 * m0 * pi);
+      betscale = x < -(real)(0.01) ? sbet12a / x :
+        -g->f * sq(cbet1) * pi;
+      lamscale = betscale / cbet1;
+      y = lam12x / lamscale;
+    }
+
+    if (y > -tol1 && x > -1 - xthresh) {
+      /* strip near cut */
+      if (g->f >= 0) {
+        salp1 = minx((real)(1), -(real)(x)); calp1 = - sqrt(1 - sq(salp1));
+      } else {
+        calp1 = maxx((real)(x > -tol1 ? 0 : -1), (real)(x));
+        salp1 = sqrt(1 - 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 * ( g->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 * sq(somg12) / (1 - comg12);
+    }
+  }
+  /* Sanity check on starting guess.  Backwards check allows NaN through. */
+  if (!(salp1 <= 0))
+    norm2(&salp1, &calp1);
+  else {
+    salp1 = 1; calp1 = 0;
+  }
+
+  *psalp1 = salp1;
+  *pcalp1 = calp1;
+  if (shortline)
+    *pdnm = dnm;
+  if (sig12 >= 0) {
+    *psalp2 = salp2;
+    *pcalp2 = calp2;
+  }
+  return sig12;
+}
+
+real Lambda12(const struct geod_geodesic* g,
+              real sbet1, real cbet1, real dn1,
+              real sbet2, real cbet2, real dn2,
+              real salp1, real calp1,
+              real slam120, real clam120,
+              real* psalp2, real* pcalp2,
+              real* psig12,
+              real* pssig1, real* pcsig1,
+              real* pssig2, real* pcsig2,
+              real* peps,
+              real* pdomg12,
+              boolx diffp, real* pdlam12,
+              /* Scratch area of the right size */
+              real Ca[]) {
+  real salp2 = 0, calp2 = 0, sig12 = 0,
+    ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0,
+    domg12 = 0, dlam12 = 0;
+  real salp0, calp0;
+  real somg1, comg1, somg2, comg2, somg12, comg12, lam12;
+  real B312, eta, k2;
+
+  if (sbet1 == 0 && calp1 == 0)
+    /* Break degeneracy of equatorial line.  This case has already been
+     * handled. */
+    calp1 = -tiny;
+
+  /* sin(alp1) * cos(bet1) = sin(alp0) */
+  salp0 = salp1 * cbet1;
+  calp0 = hypotx(calp1, salp1 * sbet1); /* calp0 > 0 */
+
+  /* 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;
+  norm2(&ssig1, &csig1);
+  /* norm2(&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 || fabs(sbet2) != -sbet1 ?
+    sqrt(sq(calp1 * cbet1) +
+         (cbet1 < -sbet1 ?
+          (cbet2 - cbet1) * (cbet1 + cbet2) :
+          (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
+    fabs(calp1);
+  /* tan(bet2) = tan(sig2) * cos(alp2)
+   * tan(omg2) = sin(alp0) * tan(sig2). */
+  ssig2 = sbet2; somg2 = salp0 * sbet2;
+  csig2 = comg2 = calp2 * cbet2;
+  norm2(&ssig2, &csig2);
+  /* norm2(&somg2, &comg2); -- don't need to normalize! */
+
+  /* sig12 = sig2 - sig1, limit to [0, pi] */
+  sig12 = atan2(maxx((real)(0), csig1 * ssig2 - ssig1 * csig2),
+                                csig1 * csig2 + ssig1 * ssig2);
+
+  /* omg12 = omg2 - omg1, limit to [0, pi] */
+  somg12 = maxx((real)(0), comg1 * somg2 - somg1 * comg2);
+  comg12 =                 comg1 * comg2 + somg1 * somg2;
+  /* eta = omg12 - lam120 */
+  eta = atan2(somg12 * clam120 - comg12 * slam120,
+              comg12 * clam120 + somg12 * slam120);
+  k2 = sq(calp0) * g->ep2;
+  eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
+  C3f(g, eps, Ca);
+  B312 = (SinCosSeries(TRUE, ssig2, csig2, Ca, nC3-1) -
+          SinCosSeries(TRUE, ssig1, csig1, Ca, nC3-1));
+  domg12 = -g->f * A3f(g, eps) * salp0 * (sig12 + B312);
+  lam12 = eta + domg12;
+
+  if (diffp) {
+    if (calp2 == 0)
+      dlam12 = - 2 * g->f1 * dn1 / sbet1;
+    else {
+      Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
+              cbet1, cbet2, nullptr, &dlam12, nullptr, nullptr, nullptr, Ca);
+      dlam12 *= g->f1 / (calp2 * cbet2);
+    }
+  }
+
+  *psalp2 = salp2;
+  *pcalp2 = calp2;
+  *psig12 = sig12;
+  *pssig1 = ssig1;
+  *pcsig1 = csig1;
+  *pssig2 = ssig2;
+  *pcsig2 = csig2;
+  *peps = eps;
+  *pdomg12 = domg12;
+  if (diffp)
+    *pdlam12 = dlam12;
+
+  return lam12;
+}
+
+real A3f(const struct geod_geodesic* g, real eps) {
+  /* Evaluate A3 */
+  return polyval(nA3 - 1, g->A3x, eps);
+}
+
+void C3f(const struct geod_geodesic* g, real eps, real c[]) {
+  /* Evaluate C3 coeffs
+   * Elements c[1] through c[nC3 - 1] are set */
+  real mult = 1;
+  int o = 0, l;
+  for (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 * polyval(m, g->C3x + o, eps);
+    o += m + 1;
+  }
+}
+
+void C4f(const struct geod_geodesic* g, real eps, real c[]) {
+  /* Evaluate C4 coeffs
+   * Elements c[0] through c[nC4 - 1] are set */
+  real mult = 1;
+  int o = 0, l;
+  for (l = 0; l < nC4; ++l) {   /* l is index of C4[l] */
+    int m = nC4 - l - 1;        /* order of polynomial in eps */
+    c[l] = mult * polyval(m, g->C4x + o, eps);
+    o += m + 1;
+    mult *= eps;
+  }
+}
+
+/* The scale factor A1-1 = mean value of (d/dsigma)I1 - 1 */
+real A1m1f(real eps)  {
+  static const real coeff[] = {
+    /* (1-eps)*A1-1, polynomial in eps2 of order 3 */
+    1, 4, 64, 0, 256,
+  };
+  int m = nA1/2;
+  real t = polyval(m, coeff, sq(eps)) / coeff[m + 1];
+  return (t + eps) / (1 - eps);
+}
+
+/* The coefficients C1[l] in the Fourier expansion of B1 */
+void C1f(real eps, real c[])  {
+  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,
+  };
+  real
+    eps2 = sq(eps),
+    d = eps;
+  int o = 0, l;
+  for (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 * polyval(m, coeff + o, eps2) / coeff[o + m + 1];
+    o += m + 2;
+    d *= eps;
+  }
+}
+
+/* The coefficients C1p[l] in the Fourier expansion of B1p */
+void C1pf(real eps, real c[])  {
+  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,
+  };
+  real
+    eps2 = sq(eps),
+    d = eps;
+  int o = 0, l;
+  for (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 * polyval(m, coeff + o, eps2) / coeff[o + m + 1];
+    o += m + 2;
+    d *= eps;
+  }
+}
+
+/* The scale factor A2-1 = mean value of (d/dsigma)I2 - 1 */
+real A2m1f(real eps)  {
+  static const real coeff[] = {
+    /* (eps+1)*A2-1, polynomial in eps2 of order 3 */
+    -11, -28, -192, 0, 256,
+  };
+  int m = nA2/2;
+  real t = polyval(m, coeff, sq(eps)) / coeff[m + 1];
+  return (t - eps) / (1 + eps);
+}
+
+/* The coefficients C2[l] in the Fourier expansion of B2 */
+void C2f(real eps, real c[])  {
+  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,
+  };
+  real
+    eps2 = sq(eps),
+    d = eps;
+  int o = 0, l;
+  for (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 * polyval(m, coeff + o, eps2) / coeff[o + m + 1];
+    o += m + 2;
+    d *= eps;
+  }
+}
+
+/* The scale factor A3 = mean value of (d/dsigma)I3 */
+void A3coeff(struct geod_geodesic* g) {
+  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,
+  };
+  int o = 0, k = 0, j;
+  for (j = nA3 - 1; j >= 0; --j) {             /* coeff of eps^j */
+    int m = nA3 - j - 1 < j ? nA3 - j - 1 : j; /* order of polynomial in n */
+    g->A3x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1];
+    o += m + 2;
+  }
+}
+
+/* The coefficients C3[l] in the Fourier expansion of B3 */
+void C3coeff(struct geod_geodesic* g) {
+  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,
+  };
+  int o = 0, k = 0, l, j;
+  for (l = 1; l < nC3; ++l) {                    /* l is index of C3[l] */
+    for (j = nC3 - 1; j >= l; --j) {             /* coeff of eps^j */
+      int m = nC3 - j - 1 < j ? nC3 - j - 1 : j; /* order of polynomial in n */
+      g->C3x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1];
+      o += m + 2;
+    }
+  }
+}
+
+/* The coefficients C4[l] in the Fourier expansion of I4 */
+void C4coeff(struct geod_geodesic* g) {
+  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,
+  };
+  int o = 0, k = 0, l, j;
+  for (l = 0; l < nC4; ++l) {        /* l is index of C4[l] */
+    for (j = nC4 - 1; j >= l; --j) { /* coeff of eps^j */
+      int m = nC4 - j - 1;           /* order of polynomial in n */
+      g->C4x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1];
+      o += m + 2;
+    }
+  }
+}
+
+int transit(real lon1, real lon2) {
+  real lon12;
+  /* Return 1 or -1 if crossing prime meridian in east or west direction.
+   * Otherwise return zero. */
+  /* Compute lon12 the same way as Geodesic::Inverse. */
+  lon1 = AngNormalize(lon1);
+  lon2 = AngNormalize(lon2);
+  lon12 = AngDiff(lon1, lon2, nullptr);
+  return lon1 <= 0 && lon2 > 0 && lon12 > 0 ? 1 :
+    (lon2 <= 0 && lon1 > 0 && lon12 < 0 ? -1 : 0);
+}
+
+int transitdirect(real lon1, real lon2) {
+#if HAVE_C99_MATH
+  lon1 = remainder(lon1, (real)(720));
+  lon2 = remainder(lon2, (real)(720));
+  return ( (lon2 <= 0 && lon2 > -360 ? 1 : 0) -
+           (lon1 <= 0 && lon1 > -360 ? 1 : 0) );
+#else
+  lon1 = fmod(lon1, (real)(720));
+  lon2 = fmod(lon2, (real)(720));
+  return ( ((lon2 <= 0 && lon2 > -360) || lon2 > 360 ? 1 : 0) -
+           ((lon1 <= 0 && lon1 > -360) || lon1 > 360 ? 1 : 0) );
+#endif
+}
+
+void accini(real s[]) {
+  /* Initialize an accumulator; this is an array with two elements. */
+  s[0] = s[1] = 0;
+}
+
+void acccopy(const real s[], real t[]) {
+  /* Copy an accumulator; t = s. */
+  t[0] = s[0]; t[1] = s[1];
+}
+
+void accadd(real s[], real y) {
+  /* Add y to an accumulator. */
+  real u, z = sumx(y, s[1], &u);
+  s[0] = sumx(z, s[0], &s[1]);
+  if (s[0] == 0)
+    s[0] = u;
+  else
+    s[1] = s[1] + u;
+}
+
+real accsum(const real s[], real y) {
+  /* Return accumulator + y (but don't add to accumulator). */
+  real t[2];
+  acccopy(s, t);
+  accadd(t, y);
+  return t[0];
+}
+
+void accneg(real s[]) {
+  /* Negate an accumulator. */
+  s[0] = -s[0]; s[1] = -s[1];
+}
+
+void geod_polygon_init(struct geod_polygon* p, boolx polylinep) {
+  p->polyline = (polylinep != 0);
+  geod_polygon_clear(p);
+}
+
+void geod_polygon_clear(struct geod_polygon* p) {
+  p->lat0 = p->lon0 = p->lat = p->lon = NaN;
+  accini(p->P);
+  accini(p->A);
+  p->num = p->crossings = 0;
+}
+
+void geod_polygon_addpoint(const struct geod_geodesic* g,
+                           struct geod_polygon* p,
+                           real lat, real lon) {
+  lon = AngNormalize(lon);
+  if (p->num == 0) {
+    p->lat0 = p->lat = lat;
+    p->lon0 = p->lon = lon;
+  } else {
+    real s12, S12 = 0;       /* Initialize S12 to stop Visual Studio warning */
+    geod_geninverse(g, p->lat, p->lon, lat, lon,
+                    &s12, nullptr, nullptr, nullptr, nullptr, nullptr, p->polyline ? nullptr : &S12);
+    accadd(p->P, s12);
+    if (!p->polyline) {
+      accadd(p->A, S12);
+      p->crossings += transit(p->lon, lon);
+    }
+    p->lat = lat; p->lon = lon;
+  }
+  ++p->num;
+}
+
+void geod_polygon_addedge(const struct geod_geodesic* g,
+                          struct geod_polygon* p,
+                          real azi, real s) {
+  if (p->num) {              /* Do nothing is num is zero */
+    /* Initialize S12 to stop Visual Studio warning.  Initialization of lat and
+     * lon is to make CLang static analyzer happy. */
+    real lat = 0, lon = 0, S12 = 0;
+    geod_gendirect(g, p->lat, p->lon, azi, GEOD_LONG_UNROLL, s,
+                   &lat, &lon, nullptr,
+                   nullptr, nullptr, nullptr, nullptr, p->polyline ? nullptr : &S12);
+    accadd(p->P, s);
+    if (!p->polyline) {
+      accadd(p->A, S12);
+      p->crossings += transitdirect(p->lon, lon);
+    }
+    p->lat = lat; p->lon = lon;
+    ++p->num;
+  }
+}
+
+unsigned geod_polygon_compute(const struct geod_geodesic* g,
+                              const struct geod_polygon* p,
+                              boolx reverse, boolx sign,
+                              real* pA, real* pP) {
+  real s12, S12, t[2], area0;
+  int crossings;
+  if (p->num < 2) {
+    if (pP) *pP = 0;
+    if (!p->polyline && pA) *pA = 0;
+    return p->num;
+  }
+  if (p->polyline) {
+    if (pP) *pP = p->P[0];
+    return p->num;
+  }
+  geod_geninverse(g, p->lat, p->lon, p->lat0, p->lon0,
+                  &s12, nullptr, nullptr, nullptr, nullptr, nullptr, &S12);
+  if (pP) *pP = accsum(p->P, s12);
+  acccopy(p->A, t);
+  accadd(t, S12);
+  crossings = p->crossings + transit(p->lon, p->lon0);
+  area0 = 4 * pi * g->c2;
+  if (crossings & 1)
+    accadd(t, (t[0] < 0 ? 1 : -1) * area0/2);
+  /* area is with the clockwise sense.  If !reverse convert to
+   * counter-clockwise convention. */
+  if (!reverse)
+    accneg(t);
+  /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */
+  if (sign) {
+    if (t[0] > area0/2)
+      accadd(t, -area0);
+    else if (t[0] <= -area0/2)
+      accadd(t, +area0);
+  } else {
+    if (t[0] >= area0)
+      accadd(t, -area0);
+    else if (t[0] < 0)
+      accadd(t, +area0);
+  }
+  if (pA) *pA = 0 + t[0];
+  return p->num;
+}
+
+unsigned geod_polygon_testpoint(const struct geod_geodesic* g,
+                                const struct geod_polygon* p,
+                                real lat, real lon,
+                                boolx reverse, boolx sign,
+                                real* pA, real* pP) {
+  real perimeter, tempsum, area0;
+  int crossings, i;
+  unsigned num = p->num + 1;
+  if (num == 1) {
+    if (pP) *pP = 0;
+    if (!p->polyline && pA) *pA = 0;
+    return num;
+  }
+  perimeter = p->P[0];
+  tempsum = p->polyline ? 0 : p->A[0];
+  crossings = p->crossings;
+  for (i = 0; i < (p->polyline ? 1 : 2); ++i) {
+    real s12, S12 = 0;       /* Initialize S12 to stop Visual Studio warning */
+    geod_geninverse(g,
+                    i == 0 ? p->lat  : lat, i == 0 ? p->lon  : lon,
+                    i != 0 ? p->lat0 : lat, i != 0 ? p->lon0 : lon,
+                    &s12, nullptr, nullptr, nullptr, nullptr, nullptr, p->polyline ? nullptr : &S12);
+    perimeter += s12;
+    if (!p->polyline) {
+      tempsum += S12;
+      crossings += transit(i == 0 ? p->lon  : lon,
+                           i != 0 ? p->lon0 : lon);
+    }
+  }
+
+  if (pP) *pP = perimeter;
+  if (p->polyline)
+    return num;
+
+  area0 = 4 * pi * g->c2;
+  if (crossings & 1)
+    tempsum += (tempsum < 0 ? 1 : -1) * area0/2;
+  /* area is with the clockwise sense.  If !reverse convert to
+   * counter-clockwise convention. */
+  if (!reverse)
+    tempsum *= -1;
+  /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */
+  if (sign) {
+    if (tempsum > area0/2)
+      tempsum -= area0;
+    else if (tempsum <= -area0/2)
+      tempsum += area0;
+  } else {
+    if (tempsum >= area0)
+      tempsum -= area0;
+    else if (tempsum < 0)
+      tempsum += area0;
+  }
+  if (pA) *pA = 0 + tempsum;
+  return num;
+}
+
+unsigned geod_polygon_testedge(const struct geod_geodesic* g,
+                               const struct geod_polygon* p,
+                               real azi, real s,
+                               boolx reverse, boolx sign,
+                               real* pA, real* pP) {
+  real perimeter, tempsum, area0;
+  int crossings;
+  unsigned num = p->num + 1;
+  if (num == 1) {               /* we don't have a starting point! */
+    if (pP) *pP = NaN;
+    if (!p->polyline && pA) *pA = NaN;
+    return 0;
+  }
+  perimeter = p->P[0] + s;
+  if (p->polyline) {
+    if (pP) *pP = perimeter;
+    return num;
+  }
+
+  tempsum = p->A[0];
+  crossings = p->crossings;
+  {
+    /* Initialization of lat, lon, and S12 is to make CLang static analyzer
+       happy. */
+    real lat = 0, lon = 0, s12, S12 = 0;
+    geod_gendirect(g, p->lat, p->lon, azi, GEOD_LONG_UNROLL, s,
+                   &lat, &lon, nullptr,
+                   nullptr, nullptr, nullptr, nullptr, &S12);
+    tempsum += S12;
+    crossings += transitdirect(p->lon, lon);
+    geod_geninverse(g, lat,  lon, p->lat0,  p->lon0,
+                    &s12, nullptr, nullptr, nullptr, nullptr, nullptr, &S12);
+    perimeter += s12;
+    tempsum += S12;
+    crossings += transit(lon, p->lon0);
+  }
+
+  area0 = 4 * pi * g->c2;
+  if (crossings & 1)
+    tempsum += (tempsum < 0 ? 1 : -1) * area0/2;
+  /* area is with the clockwise sense.  If !reverse convert to
+   * counter-clockwise convention. */
+  if (!reverse)
+    tempsum *= -1;
+  /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */
+  if (sign) {
+    if (tempsum > area0/2)
+      tempsum -= area0;
+    else if (tempsum <= -area0/2)
+      tempsum += area0;
+  } else {
+    if (tempsum >= area0)
+      tempsum -= area0;
+    else if (tempsum < 0)
+      tempsum += area0;
+  }
+  if (pP) *pP = perimeter;
+  if (pA) *pA = 0 + tempsum;
+  return num;
+}
+
+void geod_polygonarea(const struct geod_geodesic* g,
+                      real lats[], real lons[], int n,
+                      real* pA, real* pP) {
+  int i;
+  struct geod_polygon p;
+  geod_polygon_init(&p, FALSE);
+  for (i = 0; i < n; ++i)
+    geod_polygon_addpoint(g, &p, lats[i], lons[i]);
+  geod_polygon_compute(g, &p, FALSE, TRUE, pA, pP);
+}
+
+/** @endcond */