use the logger everywhere

This converts the last remaining warnings to use the logging system.
Also drops `catand` and replace it with the intrinsic `atan`, which
supports complex as well as real numbers.

Note: before 3eab989d the `catand` function was actually incorrent!
The definition of arctan(z) can be obtained starting from the identity

  d/dz arctan(z) = 1/(1 + z²) = ½ [1/(1 + iz) + 1/(1 - iz)],

integrating and using the definition log(z) = ∫₁^z dz/z,

  arctan(z) = -i/2 [log(1 + iz) - log(1 - iz)].

If log is the principal branch, log(z) = log|z| + i arg(z), then

  arctan(z) = -i/2 log(w) = 1/2 arg(w) -i/2 log|w|

where w = (1 + iz)/(1 - iz). Finally, the real part is

  Re arctan(z) = 1/2 atan2(2Re(z), 1 - |z|²).

The term -|z|² is missing from the `catand` definition of GRAY,
but is present in the original Fortran 77 code from [SLATEC]:
it has probably been lost in the translation.

[SLATEC]: https://people.math.sc.edu/Burkardt/f_src/slatec/slatec.f90

src/eccd.f90

@@ -140,7 +140,8 @@ contains
 
   subroutine setcdcoeff_ncl(zeff,rbx,fc,amu,rhop,cst2,eccdpar)
     use magsurf_data, only : ch,tjp,tlm,njpt,nlmt
-    use dierckx, only : profil
+    use dierckx,      only : profil
+    use logger,       only : log_warning
     implicit none
     integer, parameter :: ksp=3
     real(wp_), intent(in) :: zeff,rbx,fc,amu,rhop
@@ -154,7 +155,14 @@ contains
     call SpitzFuncCoeff(amu,Zeff,fc)
     nlm=nlmt
     call profil(0,tjp,njpt,tlm,nlmt,ch,ksp,ksp,rhop,nlm,chlm,ierr)
-    if(ierr>0) print*,'    Hlambda profil =',ierr
+    if (ierr > 0) then
+      block
+        character(256) :: msg
+        write(msg, '(a, g0)') &
+          'when computing H(λ) `profil` returned ier=', ierr
+        call log_warning(msg, mod='eccd', proc='setcdcoeff_ncl')
+      end block
+    end if
     npar=3+2*nlm
     allocate(eccdpar(npar))
     eccdpar(1)=zeff

src/gray_core.f90

@@ -652,7 +652,6 @@ contains
     ! beam tracing initial conditions  igrad=1
     ! !!!!!! check ray tracing initial conditions   igrad=0 !!!!!!
     use const_and_precisions, only : zero,one,pi,half,two,degree,ui=>im
-    use math,                 only : catand
     use gray_params,          only : gray_parameters
     use beamdata,             only : nray,nrayr,nrayth,rwmax
 
@@ -730,7 +729,7 @@ contains
       dw   =  wwcsi  - wweta
       ts   = -(dk*sin(2*phirrad)       - ui*dw*sin(2*phiwrad))
       tc   =  (dk*cos(2*phirrad)       - ui*dw*cos(2*phiwrad))
-      phic =  half*catand(ts/tc)
+      phic =  half*atan(ts/tc)
       ddd  =  dk*cos(2*(phirrad+phic)) - ui*dw*cos(2*(phiwrad+phic))
       sss  =  sk - ui*sw
       qi1  =  half*(sss + ddd)

src/magsurf_data.f90

@@ -469,6 +469,7 @@ contains
   subroutine contours_psi(h,rcn,zcn,rup,zup,rlw,zlw)
     use const_and_precisions, only : wp_,pi
     use gray_params, only : iequil
+    use logger,      only : log_warning
     use dierckx, only : profil,sproota
     use equilibrium, only : rmaxis,zmaxis,aminor,frhotor,tr,nsr,tz,nsz,cceq, &
        kspl,psiant,psinop,points_tgo
@@ -517,7 +518,14 @@ contains
         zc=zlw+(zup-zlw)*(1.0_wp_-cos(th*(ic-1)))/2.0_wp_
         iopt=1
         call profil(iopt,tr,nsr,tz,nsz,cceq,kspl,kspl,zc,nsr,czc,ier)
-        if(ier.gt.0) print*,'    profil =',ier
+        if (ier > 0) then
+          block
+            character(256) :: msg
+            write(msg, '(a, a, g0)') &
+              'when computing ψ(R,z) contour `profil` returned ier=', ier
+            call log_warning(msg, mod='magsurf_data', proc='contours_psi')
+          end block
+        end if
         val=h*psiant+psinop
         call sproota(val,tr,nsr,czc,zeroc,mest,m,ier)
         if (zeroc(1).gt.rwallm) then

src/math.f90

@@ -1,100 +1,20 @@
+! Miscellaneous mathematical functions
 module math
 
-  use const_and_precisions, only : wp_, zero, one
+implicit none
-  implicit none
 
 contains
 
-  function catand(z)
+  pure function fact(k)
-!***begin prologue  catan
-!***purpose  compute the complex arc tangent.
-!***library   slatec (fnlib)
-!***category  c4a
-!***type      complex (catan-c)
-!***keywords  arc tangent, elementary functions, fnlib, trigonometric
-!***author  fullerton, w., (lanl)
-!***description
-!
-! catan(z) calculates the complex trigonometric arc tangent of z.
-! the result is in units of radians, and the real part is in the first
-! or fourth quadrant.
-!
-!***references  (none)
-!***routines called  (none)
-!***revision history  (yymmdd)
-!   770801  date written
-!   890531  changed all specific intrinsics to generic.  (wrb)
-!   890531  revision date from version 3.2
-!   891214  prologue converted to version 4.0 format.  (bab)
-!   900315  calls to xerror changed to calls to xermsg.  (thj)
-!   900326  removed duplicate information from description section.
-!       (wrb)
-!***end prologue  catan
-    use const_and_precisions, only : comp_eps, pi2=>pihalf, czero, cunit
-    implicit none
-    complex(wp_) :: catand
-    complex(wp_), intent(in) :: z
-    complex(wp_) :: z2
-    real(wp_) :: r,x,y,r2,xans,yans,twoi
-    integer :: i
-    logical, save :: first=.true.
-    integer, save :: nterms
-    real(wp_), save :: rmin, rmax, sqeps
-!***first executable statement  catan
-    if (first) then
-! nterms = log(eps)/log(rbnd) where rbnd = 0.1
-      nterms = int(-0.4343_wp_*log(0.5_wp_*comp_eps) + 1.0_wp_)
-      sqeps = sqrt(comp_eps)
-      rmin = sqrt (1.5_wp_*comp_eps)
-      rmax = 2.0_wp_/comp_eps
-    endif
-    first = .false.
-!
-    r = abs(z)
-    if (r<=0.1_wp_) then
-!
-      catand = z
-      if (r<rmin) return
-!
-      catand = czero
-      z2 = z*z
-      do i=1,nterms
-        twoi = 2*(nterms-i) + 1
-        catand = 1.0_wp_/twoi - z2*catand
-      end do
-      catand = z*catand
-!
-    else if (r<=rmax) then
-      x = real(z)
-      y = aimag(z)
-      r2 = r*r
-      if (r2==one.and.x==zero) print*,'catand, z is +i or -i'
-      if (abs(r2-one)<=sqeps) then
-        if (abs(cunit+z*z) < sqeps) &
-         print*,'catand, answer lt half precision, z**2 close to -1'
-!
-      end if
-      xans = 0.5_wp_*atan2(2.0_wp_*x, one-r2)
-      yans = 0.25_wp_*log((r2+2.0_wp_*y+one)/(r2-2.0_wp_*y+one))
-      catand = cmplx(xans, yans, wp_)
-!
-    else
-      catand = cmplx(pi2, zero, wp_)
-      if (real(z)<zero) catand = cmplx(-pi2, zero, wp_)
-    end if
-  end function catand
-
-
-  function fact(k)
     ! Factorial function k!
-    ! Note: the result is real
+    ! Note: the result is a real number
+    use const_and_precisions, only : wp_
 
     implicit none
     integer, intent(in) :: k
     real(wp_)           :: fact
 
     fact = gamma(real(1 + k, kind=wp_))
-
    end function fact
 
 end module math