I am quite bit out of my league here, but given that *gcc* and *gfortran* are open source, one does not need to speculate about the method used to evaluate the intrinsic Bessel functions. One can simply have a look!

If I am not mistaken, the files below show what happens under the hood for fortran and (out of curiosity) for C++.

bessel_r8.c.txt (4.6 KB)

bessel_function.tcc.txt (23.1 KB)

```
/* Implementation of the BESSEL_JN and BESSEL_YN transformational
function using a recurrence algorithm.
Copyright (C) 2010-2022 Free Software Foundation, Inc.
Contributed by Tobias Burnus <burnus@net-b.de>
This file is part of the GNU Fortran runtime library (libgfortran).
Libgfortran is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public
License as published by the Free Software Foundation; either
version 3 of the License, or (at your option) any later version.
Libgfortran is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
Under Section 7 of GPL version 3, you are granted additional
permissions described in the GCC Runtime Library Exception, version
3.1, as published by the Free Software Foundation.
You should have received a copy of the GNU General Public License and
a copy of the GCC Runtime Library Exception along with this program;
see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
<http://www.gnu.org/licenses/>. */
#include "libgfortran.h"
#define MATHFUNC(funcname) funcname
#if defined (HAVE_GFC_REAL_8)
#if defined (HAVE_JN)
extern void bessel_jn_r8 (gfc_array_r8 * const restrict ret, int n1,
int n2, GFC_REAL_8 x);
export_proto(bessel_jn_r8);
void
bessel_jn_r8 (gfc_array_r8 * const restrict ret, int n1, int n2, GFC_REAL_8 x)
{
int i;
index_type stride;
GFC_REAL_8 last1, last2, x2rev;
stride = GFC_DESCRIPTOR_STRIDE(ret,0);
if (ret->base_addr == NULL)
{
size_t size = n2 < n1 ? 0 : n2-n1+1;
GFC_DIMENSION_SET(ret->dim[0], 0, size-1, 1);
ret->base_addr = xmallocarray (size, sizeof (GFC_REAL_8));
ret->offset = 0;
}
if (unlikely (n2 < n1))
return;
if (unlikely (compile_options.bounds_check)
&& GFC_DESCRIPTOR_EXTENT(ret,0) != (n2-n1+1))
runtime_error("Incorrect extent in return value of BESSEL_JN "
"(%ld vs. %ld)", (long int) n2-n1,
(long int) GFC_DESCRIPTOR_EXTENT(ret,0));
stride = GFC_DESCRIPTOR_STRIDE(ret,0);
if (unlikely (x == 0))
{
ret->base_addr[0] = 1;
for (i = 1; i <= n2-n1; i++)
ret->base_addr[i*stride] = 0;
return;
}
last1 = MATHFUNC(jn) (n2, x);
ret->base_addr[(n2-n1)*stride] = last1;
if (n1 == n2)
return;
last2 = MATHFUNC(jn) (n2 - 1, x);
ret->base_addr[(n2-n1-1)*stride] = last2;
if (n1 + 1 == n2)
return;
x2rev = GFC_REAL_8_LITERAL(2.)/x;
for (i = n2-n1-2; i >= 0; i--)
{
ret->base_addr[i*stride] = x2rev * (i+1+n1) * last2 - last1;
last1 = last2;
last2 = ret->base_addr[i*stride];
}
}
#endif
#if defined (HAVE_YN)
extern void bessel_yn_r8 (gfc_array_r8 * const restrict ret,
int n1, int n2, GFC_REAL_8 x);
export_proto(bessel_yn_r8);
void
bessel_yn_r8 (gfc_array_r8 * const restrict ret, int n1, int n2,
GFC_REAL_8 x)
{
int i;
index_type stride;
GFC_REAL_8 last1, last2, x2rev;
stride = GFC_DESCRIPTOR_STRIDE(ret,0);
if (ret->base_addr == NULL)
{
size_t size = n2 < n1 ? 0 : n2-n1+1;
GFC_DIMENSION_SET(ret->dim[0], 0, size-1, 1);
ret->base_addr = xmallocarray (size, sizeof (GFC_REAL_8));
ret->offset = 0;
}
if (unlikely (n2 < n1))
return;
if (unlikely (compile_options.bounds_check)
&& GFC_DESCRIPTOR_EXTENT(ret,0) != (n2-n1+1))
runtime_error("Incorrect extent in return value of BESSEL_JN "
"(%ld vs. %ld)", (long int) n2-n1,
(long int) GFC_DESCRIPTOR_EXTENT(ret,0));
stride = GFC_DESCRIPTOR_STRIDE(ret,0);
if (unlikely (x == 0))
{
for (i = 0; i <= n2-n1; i++)
#if defined(GFC_REAL_8_INFINITY)
ret->base_addr[i*stride] = -GFC_REAL_8_INFINITY;
#else
ret->base_addr[i*stride] = -GFC_REAL_8_HUGE;
#endif
return;
}
last1 = MATHFUNC(yn) (n1, x);
ret->base_addr[0] = last1;
if (n1 == n2)
return;
last2 = MATHFUNC(yn) (n1 + 1, x);
ret->base_addr[1*stride] = last2;
if (n1 + 1 == n2)
return;
x2rev = GFC_REAL_8_LITERAL(2.)/x;
for (i = 2; i <= n2 - n1; i++)
{
#if defined(GFC_REAL_8_INFINITY)
if (unlikely (last2 == -GFC_REAL_8_INFINITY))
{
ret->base_addr[i*stride] = -GFC_REAL_8_INFINITY;
}
else
#endif
{
ret->base_addr[i*stride] = x2rev * (i-1+n1) * last2 - last1;
last1 = last2;
last2 = ret->base_addr[i*stride];
}
}
}
#endif
#endif
```