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, 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)

pm_math1mexp::get1mexp
Generate and return the expression 1 - exp(x) robustly (without numerical underflow).
Definition: pm_math1mexp.F90:133

pm_math1mexp
This module contains procedures and generic interfaces for computing 1 - exp(x) more precisely for ti...
Definition: pm_math1mexp.F90:42

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

5 use pm_arrayRank, only: getRankDense

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

pm_arrayRank::getRankDense
Generate and return the Dense rank of the input scalar string or contiguous array of rank 1 in ascend...
Definition: pm_arrayRank.F90:626

pm_io::show
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726

pm_io::skip
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508

pm_arrayRank
This module contains procedures and generic interfaces for obtaining various rankings of elements of ...
Definition: pm_arrayRank.F90:137

pm_io
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252

pm_io::disp
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393

pm_kind
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268

pm_kind::LK
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
Definition: pm_kind.F90:541

pm_kind::IK
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

pm_kind::SK
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

pm_kind::RKH
integer, parameter RKH
The scalar integer constant of intrinsic default kind, representing the highest-precision real kind t...
Definition: pm_kind.F90:858

pm_io::display_type
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.

! Test the performance of `get1mexp()` with and without the selection `control` argument.
program benchmark
 
    use pm_bench, only: bench_type
    use pm_arraySpace, only: getLinSpace
    use pm_kind, only: IK, LK, RKG => RKD, RK, SK
    use iso_fortran_env, only: error_unit
 
    implicit none
 
    integer(IK)                         :: i
    integer(IK)                         :: iarr
    integer(IK)                         :: fileUnit
    integer(IK) , parameter             :: NBENCH = 2_IK
    real(RKG)                           :: dummySum = 0._RKG
    real(RKG)   , allocatable           :: X(:)
    real(RKG)                           :: onemexp
    type(bench_type)                    :: bench(NBENCH)
    logical(LK)                         :: underflowEnabled
 
    bench(1) = bench_type(name = SK_"get1mexpSelection", exec = get1mexpSelection, overhead = setOverhead)
    bench(2) = bench_type(name = SK_"get1mexpSequence", exec = get1mexpSequence, overhead = setOverhead)
 
    write(*,"(*(g0,:,' '))")
    write(*,"(*(g0,:,' '))") "get1mexp(...) vs. get1mexp(..., control = selection)"
    write(*,"(*(g0,:,' '))")
 
    open(newunit = fileUnit, file = "main.out", status = "replace")
 
        write(fileUnit, "(*(g0,:,','))") "X", (bench(i)%name, i = 1, NBENCH)
        X = getLinSpace(x1 = -2 * log(huge(onemexp)), x2 = log(epsilon(onemexp)), count = 20_IK)
        loopOverArraySize: do iarr = 1, size(X)
 
            write(*,"(*(g0,:,' '))") "Benchmarking with X", X(iarr)
            do i = 1, NBENCH
                bench(i)%timing = bench(i)%getTiming(minsec = 0.07_RK)
            end do
            write(fileUnit,"(*(g0,:,','))") X(iarr), (bench(i)%timing%mean, i = 1, NBENCH)
 
        end do loopOverArraySize
        write(*,"(*(g0,:,' '))") dummySum
        write(*,"(*(g0,:,' '))")
 
    close(fileUnit)
 
contains
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! procedure wrappers.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    subroutine setOverhead()
        call getDummy()
    end subroutine
 
    subroutine getDummy()
        dummySum = dummySum + onemexp
    end subroutine
 
    subroutine get1mexpSelection()
        use pm_math1mexp, only: get1mexp, selection
        onemexp = get1mexp(X(iarr), selection)
        call getDummy()
    end subroutine
 
    subroutine get1mexpSequence()
        use pm_math1mexp, only: get1mexp
        onemexp = get1mexp(X(iarr))
        call getDummy()
    end subroutine
 
end program benchmark

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Postprocessing of the benchmark output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
 
import os
dirname = os.path.basename(os.getcwd()) 
 
fontsize = 14
 
df = pd.read_csv("main.out", delimiter = ",")
colnames = list(df.columns.values)
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
for colname in colnames[1:]:
    plt.plot( df[colnames[0]].values
            , df[colname].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
ax.set_title(" vs. ".join(colnames[1:])+"\nLower is better.", fontsize = fontsize)
#ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( colnames[1:]
           #, loc='center left'
           #, bbox_to_anchor=(1, 0.5)
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".runtime.png")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
plt.plot( df[colnames[0]].values
        , np.ones(len(df[colnames[0]].values))
        , linestyle = "--"
       #, color = "black"
        , linewidth = 2
        )
for colname in colnames[2:]:
    plt.plot( df[colnames[0]].values
            , df[colname].values / df[colnames[1]].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Runtime compared to {}".format(colnames[1]), fontsize = fontsize)
ax.set_title("Runtime Ratio Comparison. Lower means faster.\nLower than 1 means faster than {}().".format(colnames[1]), fontsize = fontsize)
#ax.set_xscale("log")
#ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( colnames[1:]
           #, bbox_to_anchor = (1, 0.5)
           #, loc = "center left"
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".runtime.ratio.png")

Visualization of the benchmark output ⛓

Benchmark moral ⛓

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.

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.
If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Copyright: Computational Data Science Lab

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:

src/fortran/main/pm_math1mexp.F90