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+Add cuberoot function
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  Added the new functions cuberoot and intrinsic_functions_unit_tests to the
MOM_intrinsic_functions module, and call intrinsic_functions_unit_tests from
unit_tests to confirm that this new function works as intended.  Separately,
cuberoot was tested by replacing expressions like A**(1./3.) with cuberoot(A) in
MOM_energetic_PBL and verifying that the answers only change at roundoff, but
that it can give bitwise identical results when the argument is scaled by an
integer power of 8 and then unscaled by the corresponding integer power of 2,
but that change will occur in a subsequent commit as it can change answers
depending on an ANSWER_DATE flag.  With this commit, cuberoot is not yet being
used so all answers are bitwise identical, although there are new publicly
visible routines.
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Hallberg-NOAA committed Jan 5, 2024
1 parent 5137442 commit 56432ee
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13 changes: 8 additions & 5 deletions src/core/MOM_unit_tests.F90
Original file line number Diff line number Diff line change
Expand Up @@ -4,15 +4,16 @@ module MOM_unit_tests
! This file is part of MOM6. See LICENSE.md for the license.

use MOM_error_handler, only : MOM_error, FATAL, is_root_pe

use MOM_string_functions, only : string_functions_unit_tests
use MOM_remapping, only : remapping_unit_tests
use MOM_hor_bnd_diffusion, only : near_boundary_unit_tests
use MOM_intrinsic_functions, only : intrinsic_functions_unit_tests
use MOM_mixed_layer_restrat, only : mixedlayer_restrat_unit_tests
use MOM_neutral_diffusion, only : neutral_diffusion_unit_tests
use MOM_random, only : random_unit_tests
use MOM_hor_bnd_diffusion, only : near_boundary_unit_tests
use MOM_remapping, only : remapping_unit_tests
use MOM_string_functions, only : string_functions_unit_tests
use MOM_CFC_cap, only : CFC_cap_unit_tests
use MOM_EOS, only : EOS_unit_tests
use MOM_mixed_layer_restrat, only : mixedlayer_restrat_unit_tests

implicit none ; private

public unit_tests
Expand All @@ -36,6 +37,8 @@ subroutine unit_tests(verbosity)
"MOM_unit_tests: EOS_unit_tests FAILED")
if (remapping_unit_tests(verbose)) call MOM_error(FATAL, &
"MOM_unit_tests: remapping_unit_tests FAILED")
if (intrinsic_functions_unit_tests(verbose)) call MOM_error(FATAL, &
"MOM_unit_tests: intrinsic_functions_unit_tests FAILED")
if (neutral_diffusion_unit_tests(verbose)) call MOM_error(FATAL, &
"MOM_unit_tests: neutralDiffusionUnitTests FAILED")
if (random_unit_tests(verbose)) call MOM_error(FATAL, &
Expand Down
138 changes: 137 additions & 1 deletion src/framework/MOM_intrinsic_functions.F90
Original file line number Diff line number Diff line change
Expand Up @@ -4,9 +4,12 @@ module MOM_intrinsic_functions

! This file is part of MOM6. See LICENSE.md for the license.

use iso_fortran_env, only : stdout=>output_unit, stderr=>error_unit

implicit none ; private

public :: invcosh
public :: invcosh, cuberoot
public :: intrinsic_functions_unit_tests

contains

Expand All @@ -25,4 +28,137 @@ function invcosh(x)

end function invcosh

!> Returns the cube root of a real argument at roundoff accuracy, in a form that works properly with
!! rescaling of the argument by integer powers of 8. If the argument is a NaN, a NaN is returned.
elemental function cuberoot(x) result(root)
real, intent(in) :: x !< The argument of cuberoot in arbitrary units cubed [A3]
real :: root !< The real cube root of x in arbitrary units [A]

real :: asx ! The absolute value of x rescaled by an integer power of 8 to put it into
! the range from 0.125 < asx <= 1.0, in ambiguous units cubed [B3]
real :: root_asx ! The cube root of asx [B]
real :: num ! The numerator of an expression for the evolving estimate of the cube root of asx
! in arbitrary units that can grow or shrink with each iteration [B C]
real :: den ! The denominator of an expression for the evolving estimate of the cube root of asx
! in arbitrary units that can grow or shrink with each iteration [C]
real :: num_prev ! The numerator of an expression for the previous iteration of the evolving estimate
! of the cube root of asx in arbitrary units that can grow or shrink with each iteration [B D]
real :: den_prev ! The denominator of an expression for the previous iteration of the evolving estimate of
! the cube root of asx in arbitrary units that can grow or shrink with each iteration [D]
integer :: ex_3 ! One third of the exponent part of x, used to rescale x to get a.
integer :: itt

if ((x >= 0.0) .eqv. (x <= 0.0)) then
! Return 0 for an input of 0, or NaN for a NaN input.
root = x
else
ex_3 = exponent(x) / 3
asx = scale(abs(x), -3*ex_3)
if (asx > 1.0) then
asx = 0.125 * asx ; ex_3 = ex_3 + 1
elseif (asx <= 0.125) then
! This is retained in case ((3.0 / 3.0) == 1.0) is not exactly true, which might lead
! to convergence to slightly different values for root for asx = 0.125 or asx = 1.0.
asx = 8.0 * asx ; ex_3 = ex_3 - 1
endif

if (asx == 1.0) then
root_asx = 1.0
else
! This first estimate is one iteration of Newton's method with a starting guess of 1. It is
! always an over-estimate of the true solution, but it is a good approximation for asx near 1.
num = (2.0 + asx)
den = 3.0
! Iteratively determine Root = asx**1/3 using Newton's method, noting that in this case Newton's
! method converges monotonically from above and needs no bounding. For the range of asx from
! 0.125 to 1.0 with the first guess used above, 6 iterations suffice to converge to roundoff.
do itt=1,9
! Newton's method iterates estimates as Root = Root - (Root**3 - asx) / (3.0 * Root**2), or
! equivalently as Root = (2.0*Root**2 + asx) / (3.0 * Root**2).
! Keeping the estimates in a fractional form Root = num / den allows this calculation with
! fewer (or no) real divisions during the iterations before doing a single real division
! at the end, and it is therefore more computationally efficient.

! Because successive estimates of the numerator and denominator tend to be the cube of their
! predecessors, the numerator and denominator need to be rescaled by division when they get
! too large or small to avoid overflow or underflow in the convergence test below. Were
! this being done for 32 bit reals, the values to compare with would be about 1.0e9 and 1.0e-9.
if ((den > 1.0e75) .or. (den < 1.0e-75)) then
num = num / den ; den = 1.0
endif

num_prev = num ; den_prev = den
num = (2.0 * num_prev**3 + asx * den_prev**3)
den = 3.0 * (den_prev * num_prev**2)

! If successive estimates of root are identical, this is an exact or converged solution.
if ((num * den_prev == num_prev * den) .or. (itt == 9)) then
root_asx = num / den
exit
endif

! If the estimates are increasing, this also indicates convergence, but for a more subtle
! reason. Because Newton's method converges monotonically from above (at least for infinite
! precision math), the only reason why this estimate could increase is if the iterations
! have converged to a roundoff-level limit cycle around an irrational or otherwise
! unrepresentable solution, with values only changing in the last bit or two. If so, we
! should stop iterating and accept the one of the current or previous solutions, both of
! which will be within numerical roundoff of the true solution.
if (num * den_prev > num_prev * den) then
root_asx = num / den
! Pick the more accurate of the last two iterations.
! Given that both of the two previous iterations are within roundoff of the true
! solution, this next step might be overkill.
if ( abs(den_prev**3*root_asx**3 - den_prev**3*asx) > abs(num_prev**3 - den_prev**3*asx) ) then
! The previous iteration was slightly more accurate, so use that for root_asx.
root_asx = num_prev / den_prev
endif
exit
endif
enddo
endif

root = sign(scale(root_asx, ex_3), x)
endif

end function cuberoot

!> Returns true if any unit test of intrinsic_functions fails, or false if they all pass.
logical function intrinsic_functions_unit_tests(verbose)
logical, intent(in) :: verbose !< If true, write results to stdout

! Local variables
real :: testval ! A test value for self-consistency testing [nondim]
logical :: fail, v

fail = .false.
v = verbose
write(stdout,*) '==== MOM_intrinsic_functions: intrinsic_functions_unit_tests ==='

fail = fail .or. Test_cuberoot(v, 1.2345678901234e9)
fail = fail .or. Test_cuberoot(v, -9.8765432109876e-21)
fail = fail .or. Test_cuberoot(v, 64.0)
fail = fail .or. Test_cuberoot(v, -0.5000000000001)
fail = fail .or. Test_cuberoot(v, 0.0)

end function intrinsic_functions_unit_tests

!> True if the cube of cuberoot(val) does not closely match val. False otherwise.
logical function Test_cuberoot(verbose, val)
logical, intent(in) :: verbose !< If true, write results to stdout
real, intent(in) :: val !< The real value to test, in arbitrary units [A]
! Local variables
real :: diff ! The difference between val and the cube root of its cube.

diff = val - cuberoot(val**3)
Test_cuberoot = (abs(diff) > 2.0e-15*abs(val))

if (Test_cuberoot) then
write(stdout, '("For val = ",ES22.15,", (val - cuberoot(val**3))) = ",ES9.2," <-- FAIL")') val, diff
elseif (verbose) then
write(stdout, '("For val = ",ES22.15,", (val - cuberoot(val**3))) = ",ES9.2)') val, diff

endif
end function Test_cuberoot

end module MOM_intrinsic_functions

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