ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation.
pm_math1mexp::get1mexp Interface Reference

Generate and return the expression 1 - exp(x) robustly (without numerical underflow). More...

Detailed Description

Generate and return the expression 1 - exp(x) robustly (without numerical underflow).

Parameters
[in]x: The input scalar, or array of arbitrary rank, shape, and size, of either
  • type complex of kind any supported by the processor (e.g., CK, CK32, CK64, or CK128), or
  • type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128),
representing the x value whose 1 - exp(x) is to be returned.
[in]control: The input scalar object that can be,
  1. the constant selection or equivalently, an object of type selection_type.
    Specifying this option enables the runtime checks for underflow occurrence via branching and dynamic dispatch.
    Enabling this option can aid runtime efficiency when the x ratio is expected to cause underflow.
    This option avoids the computation of a logarithm term, leading to better runtime efficiency.
    Note that logarithm is highly expensive (on the order of ~200 CPU cycles).
    See the relevant benchmark here.
The presence of this argument is merely for compile-time resolution of the procedures of this generic interface.
(optional. If missing, a sequence control flow will be assumed without dynamic dispatch.)
Returns
onemexp : The output scalar (or array) of the same type and kind (and shape) as the input x representing 1 - exp(x) without underflow.


Possible calling interfaces

use pm_math1mexp, only: get1mexp, selection
onemexp = get1mexp(x)
onemexp = get1mexp(x, control = selection)
Generate and return the expression 1 - exp(x) robustly (without numerical underflow).
This module contains procedures and generic interfaces for computing 1 - exp(x) more precisely for ti...
Warning
The condition x < log(huge(x)) must hold.
When the input arguments are of type complex, this condition must hold for the real components of the numbers.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.
The pure procedure(s) documented herein become impure when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1.
By default, these procedures are pure in release build and impure in debug and testing builds.
Remarks
The procedures under discussion are elemental.
See also
getLog1p
get1mexp
getLogAddExp
getLogSubExp
getLogSumExp


Example usage

1program example
2
3 use pm_kind, only: SK, IK, LK, RKH
4 use pm_io, only: display_type
6 use pm_math1mexp, only: get1mexp, selection
7
8 implicit none
9
10 real :: x, result(3)
11 real(RKH) :: ref
12 real(RKH), allocatable :: inaccuracy(:)
13 real, parameter :: EPS = epsilon(0.)
14 type(display_type) :: disp
15
16 disp = display_type(file = "main.out.F90")
17
18 call disp%skip()
19 call disp%show("x = .9")
20 x = .9
21 call disp%show("ref = 1._RKH - exp(real(x, RKH)) ! reference high-precision value for comparison")
22 ref = 1._RKH - exp(real(x, RKH))
23 call disp%show("ref")
24 call disp%show( ref )
25 call disp%show("result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]")
26 result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
27 call disp%show("result")
28 call disp%show( result )
29 call disp%show("inaccuracy = abs(ref - result)")
30 inaccuracy = abs(ref - result)
31 call disp%show("inaccuracy")
32 call disp%show( inaccuracy )
33 call disp%show("getRankDense(inaccuracy)")
34 call disp%show( getRankDense(inaccuracy) )
35 call disp%skip()
36
37 call disp%skip()
38 call disp%show("x = EPS")
39 x = EPS
40 call disp%show("x")
41 call disp%show( x )
42 call disp%show("ref = 1._RKH - exp(real(x, RKH)) ! reference high-precision value for comparison")
43 ref = 1._RKH - exp(real(x, RKH))
44 call disp%show("ref")
45 call disp%show( ref )
46 call disp%show("result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]")
47 result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
48 call disp%show("result")
49 call disp%show( result )
50 call disp%show("inaccuracy = abs(ref - result)")
51 inaccuracy = abs(ref - result)
52 call disp%show("inaccuracy")
53 call disp%show( inaccuracy )
54 call disp%show("getRankDense(inaccuracy)")
55 call disp%show( getRankDense(inaccuracy) )
56 call disp%skip()
57
58 call disp%skip()
59 call disp%show("x = EPS/2")
60 x = EPS/2
61 call disp%show("x")
62 call disp%show( x )
63 call disp%show("ref = 1._RKH - exp(real(x, RKH))) ! reference high-precision value for comparison")
64 ref = 1._RKH - exp(real(x, RKH))
65 call disp%show("ref")
66 call disp%show( ref )
67 call disp%show("result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]")
68 result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
69 call disp%show("result")
70 call disp%show( result )
71 call disp%show("inaccuracy = abs(ref - result)")
72 inaccuracy = abs(ref - result)
73 call disp%show("inaccuracy")
74 call disp%show( inaccuracy )
75 call disp%show("getRankDense(inaccuracy)")
76 call disp%show( getRankDense(inaccuracy) )
77 call disp%skip()
78
79 call disp%skip()
80 call disp%show("x = -EPS/2")
81 x = -EPS/2
82 call disp%show("x")
83 call disp%show( x )
84 call disp%show("ref = 1._RKH - exp(real(x, RKH)) ! reference high-precision value for comparison")
85 ref = 1._RKH - exp(real(x, RKH))
86 call disp%show("ref")
87 call disp%show( ref )
88 call disp%show("result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]")
89 result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
90 call disp%show("result")
91 call disp%show( result )
92 call disp%show("inaccuracy = abs(ref - result)")
93 inaccuracy = abs(ref - result)
94 call disp%show("inaccuracy")
95 call disp%show( inaccuracy )
96 call disp%show("getRankDense(inaccuracy)")
97 call disp%show( getRankDense(inaccuracy) )
98 call disp%skip()
99
100 call disp%skip()
101 call disp%show("x = sqrt(tiny(x))")
102 x = sqrt(tiny(x))
103 call disp%show("x")
104 call disp%show( x )
105 call disp%show("ref = 1._RKH - exp(real(x, RKH)) ! reference high-precision value for comparison")
106 ref = 1._RKH - exp(real(x, RKH))
107 call disp%show("ref")
108 call disp%show( ref )
109 call disp%show("result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]")
110 result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
111 call disp%show("result")
112 call disp%show( result )
113 call disp%show("inaccuracy = abs(ref - result)")
114 inaccuracy = abs(ref - result)
115 call disp%show("inaccuracy")
116 call disp%show( inaccuracy )
117 call disp%show("getRankDense(inaccuracy)")
118 call disp%show( getRankDense(inaccuracy) )
119 call disp%skip()
120
121 call disp%skip()
122 call disp%show("x = -sqrt(tiny(x))")
123 x = -sqrt(tiny(x))
124 call disp%show("x")
125 call disp%show( x )
126 call disp%show("ref = 1._RKH - exp(real(x, RKH)) ! reference high-precision value for comparison")
127 ref = 1._RKH - exp(real(x, RKH))
128 call disp%show("ref")
129 call disp%show( ref )
130 call disp%show("result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]")
131 result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
132 call disp%show("result")
133 call disp%show( result )
134 call disp%show("inaccuracy = abs(ref - result)")
135 inaccuracy = abs(ref - result)
136 call disp%show("inaccuracy")
137 call disp%show( inaccuracy )
138 call disp%show("getRankDense(inaccuracy)")
139 call disp%show( getRankDense(inaccuracy) )
140 call disp%skip()
141
142 call disp%skip()
143 call disp%show("tiny(0._RKH)")
144 call disp%show( tiny(0._RKH) )
145 call disp%show("[(1._RKH + tiny(0._RKH)), get1mexp(tiny(0._RKH)), get1mexp(tiny(0._RKH), control = selection)]")
146 call disp%show( [(1._RKH + tiny(0._RKH)), get1mexp(tiny(0._RKH)), get1mexp(tiny(0._RKH), control = selection)] )
147 call disp%skip()
148
149 call disp%skip()
150 call disp%show("tiny(0._RKH)")
151 call disp%show( tiny(0._RKH) )
152 call disp%show("[log(1._RKH - tiny(0._RKH)), get1mexp(-tiny(0._RKH)), get1mexp(-tiny(0._RKH), control = selection)]")
153 call disp%show( [log(1._RKH - tiny(0._RKH)), get1mexp(-tiny(0._RKH)), get1mexp(-tiny(0._RKH), control = selection)] )
154 call disp%skip()
155
156end program example
Generate and return the Dense rank of the input scalar string or contiguous array of rank 1 in ascend...
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508
This module contains procedures and generic interfaces for obtaining various rankings of elements of ...
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
Definition: pm_kind.F90:541
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
Definition: pm_kind.F90:540
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
Definition: pm_kind.F90:539
integer, parameter RKH
The scalar integer constant of intrinsic default kind, representing the highest-precision real kind t...
Definition: pm_kind.F90:858
Generate and return an object of type display_type.
Definition: pm_io.F90:10282

Example Unix compile command via Intel ifort compiler
1#!/usr/bin/env sh
2rm main.exe
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example Windows Batch compile command via Intel ifort compiler
1del main.exe
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
4main.exe

Example Unix / MinGW compile command via GNU gfortran compiler
1#!/usr/bin/env sh
2rm main.exe
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example output
1
2x = .9
3ref = 1._RKH - exp(real(x, RKH)) ! reference high-precision value for comparison
4ref
5-1.45960305251544247127912782270480705
6result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
7result
8-1.45960307, -1.45960307, -1.45960307
9inaccuracy = abs(ref - result)
10inaccuracy
11+0.186973260834083721772951929513111672E-7, +0.186973260834083721772951929513111672E-7, +0.186973260834083721772951929513111672E-7
12getRankDense(inaccuracy)
13+1, +1, +1
14
15
16x = EPS
17x
18+0.119209290E-6
19ref = 1._RKH - exp(real(x, RKH)) ! reference high-precision value for comparison
20ref
21-0.119209296656208889945326024661011389E-6
22result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
23result
24-0.119209290E-6, -0.119209290E-6, -0.119209290E-6
25inaccuracy = abs(ref - result)
26inaccuracy
27+0.710542763994532602466101138934800795E-14, +0.710542763994532602466101138934800795E-14, +0.710542763994532602466101138934800795E-14
28getRankDense(inaccuracy)
29+1, +1, +1
30
31
32x = EPS/2
33x
34+0.596046448E-7
35ref = 1._RKH - exp(real(x, RKH))) ! reference high-precision value for comparison
36ref
37-0.596046465517474996932904594500639460E-7
38result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
39result
40-0.119209290E-6, -0.596046448E-7, -0.596046448E-7
41inaccuracy = abs(ref - result)
42inaccuracy
43+0.596046429990337503067095405499360540E-7, +0.177635687469329045945006394598704577E-14, +0.177635687469329045945006394598704577E-14
44getRankDense(inaccuracy)
45+2, +1, +1
46
47
48x = -EPS/2
49x
50-0.596046448E-7
51ref = 1._RKH - exp(real(x, RKH)) ! reference high-precision value for comparison
52ref
53+0.596046429990338208927884783440999782E-7
54result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
55result
56+0.596046448E-7, +0.596046448E-7, +0.596046448E-7
57inaccuracy = abs(ref - result)
58inaccuracy
59+0.177635680410721152165590002177307858E-14, +0.177635680410721152165590002177307858E-14, +0.177635680410721152165590002177307858E-14
60getRankDense(inaccuracy)
61+1, +1, +1
62
63
64x = sqrt(tiny(x))
65x
66+0.108420217E-18
67ref = 1._RKH - exp(real(x, RKH)) ! reference high-precision value for comparison
68ref
69-0.108420217248550443400745280086994171E-18
70result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
71result
72+0.00000000, -0.108420217E-18, -0.108420217E-18
73inaccuracy = abs(ref - result)
74inaccuracy
75+0.108420217248550443400745280086994171E-18, +0.00000000000000000000000000000000000, +0.00000000000000000000000000000000000
76getRankDense(inaccuracy)
77+2, +1, +1
78
79
80x = -sqrt(tiny(x))
81x
82-0.108420217E-18
83ref = 1._RKH - exp(real(x, RKH)) ! reference high-precision value for comparison
84ref
85+0.108420217248550443400745280086994171E-18
86result = [1. - exp(x), get1mexp(x), get1mexp(x, control = selection)]
87result
88+0.00000000, +0.108420217E-18, +0.108420217E-18
89inaccuracy = abs(ref - result)
90inaccuracy
91+0.108420217248550443400745280086994171E-18, +0.00000000000000000000000000000000000, +0.00000000000000000000000000000000000
92getRankDense(inaccuracy)
93+2, +1, +1
94
95
96tiny(0._RKH)
97+0.336210314311209350626267781732175260E-4931
98[(1._RKH + tiny(0._RKH)), get1mexp(tiny(0._RKH)), get1mexp(tiny(0._RKH), control = selection)]
99+1.00000000000000000000000000000000000, -0.336210314311209350626267781732175260E-4931, -0.336210314311209350626267781732175260E-4931
100
101
102tiny(0._RKH)
103+0.336210314311209350626267781732175260E-4931
104[log(1._RKH - tiny(0._RKH)), get1mexp(-tiny(0._RKH)), get1mexp(-tiny(0._RKH), control = selection)]
105+0.00000000000000000000000000000000000, +0.336210314311209350626267781732175260E-4931, +0.336210314311209350626267781732175260E-4931
106
107
Benchmarks:


Benchmark :: The effects of control on runtime efficiency

The following program compares the runtime performance of get1mexp algorithm with and without checking for underflows.
1! Test the performance of `get1mexp()` with and without the selection `control` argument.
2program benchmark
3
4 use pm_bench, only: bench_type
6 use pm_kind, only: IK, LK, RKG => RKD, RK, SK
7 use iso_fortran_env, only: error_unit
8
9 implicit none
10
11 integer(IK) :: i
12 integer(IK) :: iarr
13 integer(IK) :: fileUnit
14 integer(IK) , parameter :: NBENCH = 2_IK
15 real(RKG) :: dummySum = 0._RKG
16 real(RKG) , allocatable :: X(:)
17 real(RKG) :: onemexp
18 type(bench_type) :: bench(NBENCH)
19 logical(LK) :: underflowEnabled
20
21 bench(1) = bench_type(name = SK_"get1mexpSelection", exec = get1mexpSelection, overhead = setOverhead)
22 bench(2) = bench_type(name = SK_"get1mexpSequence", exec = get1mexpSequence, overhead = setOverhead)
23
24 write(*,"(*(g0,:,' '))")
25 write(*,"(*(g0,:,' '))") "get1mexp(...) vs. get1mexp(..., control = selection)"
26 write(*,"(*(g0,:,' '))")
27
28 open(newunit = fileUnit, file = "main.out", status = "replace")
29
30 write(fileUnit, "(*(g0,:,','))") "X", (bench(i)%name, i = 1, NBENCH)
31 X = getLinSpace(x1 = -2 * log(huge(onemexp)), x2 = log(epsilon(onemexp)), count = 20_IK)
32 loopOverArraySize: do iarr = 1, size(X)
33
34 write(*,"(*(g0,:,' '))") "Benchmarking with X", X(iarr)
35 do i = 1, NBENCH
36 bench(i)%timing = bench(i)%getTiming(minsec = 0.07_RK)
37 end do
38 write(fileUnit,"(*(g0,:,','))") X(iarr), (bench(i)%timing%mean, i = 1, NBENCH)
39
40 end do loopOverArraySize
41 write(*,"(*(g0,:,' '))") dummySum
42 write(*,"(*(g0,:,' '))")
43
44 close(fileUnit)
45
46contains
47
48 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
49 ! procedure wrappers.
50 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
51
52 subroutine setOverhead()
53 call getDummy()
54 end subroutine
55
56 subroutine getDummy()
57 dummySum = dummySum + onemexp
58 end subroutine
59
60 subroutine get1mexpSelection()
61 use pm_math1mexp, only: get1mexp, selection
62 onemexp = get1mexp(X(iarr), selection)
63 call getDummy()
64 end subroutine
65
66 subroutine get1mexpSequence()
67 use pm_math1mexp, only: get1mexp
68 onemexp = get1mexp(X(iarr))
69 call getDummy()
70 end subroutine
71
72end program benchmark
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Generate and return an object of type timing_type containing the benchmark timing information and sta...
Definition: pm_bench.F90:574
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
This module contains abstract interfaces and types that facilitate benchmarking of different procedur...
Definition: pm_bench.F90:41
integer, parameter RK
The default real kind in the ParaMonte library: real64 in Fortran, c_double in C-Fortran Interoperati...
Definition: pm_kind.F90:543
integer, parameter RKD
The double precision real kind in Fortran mode. On most platforms, this is an 64-bit real kind.
Definition: pm_kind.F90:568
This is the class for creating benchmark and performance-profiling objects.
Definition: pm_bench.F90:386

Example Unix compile command via Intel ifort compiler
1#!/usr/bin/env sh
2rm main.exe
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example Windows Batch compile command via Intel ifort compiler
1del main.exe
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
4main.exe

Example Unix / MinGW compile command via GNU gfortran compiler
1#!/usr/bin/env sh
2rm main.exe
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Postprocessing of the benchmark output
1#!/usr/bin/env python
2
3import matplotlib.pyplot as plt
4import pandas as pd
5import numpy as np
6
7import os
8dirname = os.path.basename(os.getcwd())
9
10fontsize = 14
11
12df = pd.read_csv("main.out", delimiter = ",")
13colnames = list(df.columns.values)
14
15
18
19ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
20ax = plt.subplot()
21
22for colname in colnames[1:]:
23 plt.plot( df[colnames[0]].values
24 , df[colname].values
25 , linewidth = 2
26 )
27
28plt.xticks(fontsize = fontsize)
29plt.yticks(fontsize = fontsize)
30ax.set_xlabel(colnames[0], fontsize = fontsize)
31ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
32ax.set_title(" vs. ".join(colnames[1:])+"\nLower is better.", fontsize = fontsize)
33#ax.set_xscale("log")
34ax.set_yscale("log")
35plt.minorticks_on()
36plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
37ax.tick_params(axis = "y", which = "minor")
38ax.tick_params(axis = "x", which = "minor")
39ax.legend ( colnames[1:]
40 #, loc='center left'
41 #, bbox_to_anchor=(1, 0.5)
42 , fontsize = fontsize
43 )
44
45plt.tight_layout()
46plt.savefig("benchmark." + dirname + ".runtime.png")
47
48
51
52ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
53ax = plt.subplot()
54
55plt.plot( df[colnames[0]].values
56 , np.ones(len(df[colnames[0]].values))
57 , linestyle = "--"
58 #, color = "black"
59 , linewidth = 2
60 )
61for colname in colnames[2:]:
62 plt.plot( df[colnames[0]].values
63 , df[colname].values / df[colnames[1]].values
64 , linewidth = 2
65 )
66
67plt.xticks(fontsize = fontsize)
68plt.yticks(fontsize = fontsize)
69ax.set_xlabel(colnames[0], fontsize = fontsize)
70ax.set_ylabel("Runtime compared to {}".format(colnames[1]), fontsize = fontsize)
71ax.set_title("Runtime Ratio Comparison. Lower means faster.\nLower than 1 means faster than {}().".format(colnames[1]), fontsize = fontsize)
72#ax.set_xscale("log")
73#ax.set_yscale("log")
74plt.minorticks_on()
75plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
76ax.tick_params(axis = "y", which = "minor")
77ax.tick_params(axis = "x", which = "minor")
78ax.legend ( colnames[1:]
79 #, bbox_to_anchor = (1, 0.5)
80 #, loc = "center left"
81 , fontsize = fontsize
82 )
83
84plt.tight_layout()
85plt.savefig("benchmark." + dirname + ".runtime.ratio.png")

Visualization of the benchmark output

Benchmark moral
  1. If the input value x is known to be smaller than -log(huge) then it is generally beneficial to call the interface with control argument set to selection.
Test:
test_pm_math1mexp


Final Remarks


If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

  1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
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Author:
Amir Shahmoradi, Thursday 1:45 AM, August 22, 2019, Dallas, TX

Definition at line 133 of file pm_math1mexp.F90.


The documentation for this interface was generated from the following file: