PROCEDURE InvariantsFromDerivatives(
    READONLY v: RealDerivatives; 
    VAR p: RealInvariants;
  ) =
  BEGIN
    WITH
      f = v[0],
      fx = v[1],
      fy = v[2],
      fxx = v[3],
      fxy = v[4],
      fyy = v[5],
    
      gx = fx*fx,
      gy = fy*fy,
      gm = 2.0*fx*fy
    DO
      p[0] := f;
      p[1] := gx + gy;
      p[2] := fxx + fyy;
      p[3] := fxx*gy - fxy*gm + fyy*gx;
      p[4] := fxy*fxy - fxx*fyy
    END;
  END InvariantsFromDerivatives;
  
PROCEDURE DerivativePartialsFromInvariantPartials(
    READONLY v: RealDerivatives;
    READONLY dEdp: RealInvariants;
    VAR (*OUT*) dEdv: RealDerivatives;
  ) =
  BEGIN
    WITH
      fx = v[1],
      fy = v[2],
      fxx = v[3],
      fxy = v[4],
      fyy = v[5],
    
      gx = fx*fx,
      gy = fy*fy,
      gm = 2.0*fx*fy,

      dEdgm  = - dEdp[3]*fxy,
      dEdgx  = dEdp[1] + dEdp[3]*fyy,
      dEdgy  = dEdp[1] + dEdp[3]*fxx
    DO
      dEdv[0] := dEdp[0];
      dEdv[1] := 2.0 * (dEdgm * fy + dEdgx * fx);
      dEdv[2] := 2.0 * (dEdgm * fx + dEdgy * fy);
      dEdv[3] := dEdp[2] + dEdp[3]*gy - dEdp[4]*fyy;
      dEdv[4] := - dEdp[3]*gm + 2.0 * dEdp[4]*fxy;
      dEdv[5] := dEdp[2] + dEdp[3]*gx - dEdp[4]*fxx;
    END;
  END DerivativePartialsFromInvariantPartials;