Exponential and related functions

Exponential function \(\exp(x) = e^x\)

ctx.exp(z)

where ctx is ctx_pm (see Python contexts for details), ctx53, ctxcpp, ctxflint (see .NET contexts for details).

Returns \(\exp(x)\), the exponential function of \(x\). See also Wikipedia [1342], MathWorld [919], NIST [515], Ehrhardt [309] (4.2.34), Flint [804], Flint [794], Mpmath [573].

For complex numbers, the exponential function satisfies \(\exp(x + iy) = e^x (\cos y + i \sin y)\).

\(\quad\) \(\quad\)

3D wpf plot: real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex sine function \(z = \sin(x + iy)\), with \(-6 \le x \le 6\) (blue axis), \(-6 \le y \le 6\) (red axis), \(-10 \le z \le 10\) (green axis). Function values are loglog-transformed.

An example with real input:

>>> from xlcalcnet import *
>>> x = -5.1; dps = 90
>>> for ctx in ctxlistreal: print(ctx.fmtname + ': ' + ctx.fmt(ctx.exp(x)))
    fpm:  0.00609674656551564
    mpm:  0.00609674656551563610713456478542490178906004208066809730839371569595697636773934832576013656
    dpm:  0.00609674656551563610713456478542490178906004208066809730839371569595697636773934832576013656
    ipm: [0.0060967465655156361071345647854249017890600420806680973083937156959569763677393483257601365613434,
          0.0060967465655156361071345647854249017890600420806680973083937156959569763677393483257601365690139]
    gpm:  0.0060967465655156361071345647854249017890600420806680973083937156959569763677393483257601365623
    apm: [0.00609674656551563610713456478542490178906004208066809730839371569595697636773934832576013657 +/- 8.90e-93]
  dreal:  0.00609674656551564
  sreal:  0.006096747
  dreal:  0.00609674656551564
  ereal:  0.0060967465655156361078
  qreal:  0.0060967465655156361071345647854249
  oreal:  0.0060967465655156361071345647854249017890600420806680973083937156959569763
  mreal:  0.00609674656551563610713456478542490178906004208066809730839371569595697636773934832576013657
 sflint:  0.006096747
 dflint:  0.00609674656551564
 eflint:  0.0060967465655156361078
 qflint:  0.0060967465655156361071345647854249
 oflint:  0.0060967465655156361071345647854249017890600420806680973083937156959569764
 mflint:  0.00609674656551563610713456478542490178906004208066809730839371569595697636773934832576013657
 aflint: [0.0060967465655156361071345647854249017890600420806680973083937156959569763677393483257601366 +/- 4.16e-92]

An example with complex input, using C#-style formatting for complex numbers for the result

>>> from xlcalcnet import *
>>> x = -5.1+2j; gui.setdps(50)
>>> for ctx in [fpm, mpm, cmath53, qcplx]: print(ctx.fmtname + ': ' + ctx.fmt(ctx.exp(x)))
    fpm:  (-0.00253714179646899, 0.00554375596403168)
    mpm:  (-0.0025371417964689871432868572987332349386956711932343, 0.0055437559640316803155849317223325676631201581872807)
cmath53:  (-0.00253714179646899, 0.00554375596403168)
  qcplx:  (-0.00253714179646898714328685729873324, 0.00554375596403168031558493172233257)

The above and additional examples can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder for Python (real input), Python (complex input), and C# (real or complex input).

Auxiliary function \(\mathrm{expj}(x) = e^{ix}\)

ctx.expj(z)

where ctx is math53, mathc53, ctxcpp, ctxflint.

Returns \(e^{iz} = \cos(z) + i \sin(z)\). See also Wikipedia [1492], MathWorld [1094], Mpmath [752].

An example with real input:

>>> from xlcalcnet import *
>>> x = -5.1; dps = 50
>>> for ctx in gui.ctxlist_real: print(ctx.fmtname + ': ' + ctx.cplxctx.fmt(ctx.expj(x)))
    fpm:  (0.37797774271298, 0.925814682327732)
    mpm:  (0.37797774271298056332057555292898167089864157613427, 0.92581468232773229694614624754486331250940301635561)
    dpm:  (0.37797774271298056332057555292898167089864157613428, 0.9258146823277322969461462475448633125094030163556)
    ipm: ([0.3779777427129805633205755529289816708986415761342730157, 0.3779777427129805633205755529289816708986415761342837068],
          [0.9258146823277322969461462475448633125094030163556011089, 0.9258146823277322969461462475448633125094030163556064544])
    gpm:  (0.3779777427129805633205755529289816708986415761342737, 0.9258146823277322969461462475448633125094030163556051)
    apm: ([0.3779777427129805633205755529289816708986415761343 +/- 3.78e-50], [0.9258146823277322969461462475448633125094030163556 +/- 1.06e-50])
  dreal:  (0.37797774271298, 0.925814682327732)
  sreal:  (0.3779777, 0.9258147)
  dreal:  (0.37797774271298, 0.925814682327732)
  ereal:  (0.37797774271298056325, 0.92581468232773229699)
  qreal:  (0.377977742712980563320575552928981, 0.925814682327732296946146247544863)
  oreal:  (0.37797774271298056332057555292898167089864157613427757054176691301095088, 0.92581468232773229694614624754486331250940301635560394879705455193228186)
  mreal:  (0.37797774271298056332057555292898167089864157613429, 0.92581468232773229694614624754486331250940301635559)
 sflint:  (0.3779777, 0.9258147)
 dflint:  (0.37797774271298, 0.925814682327732)
 eflint:  (0.37797774271298056325, 0.92581468232773229699)
 qflint:  (0.377977742712980563320575552928981, 0.925814682327732296946146247544863)
 oflint:  (0.37797774271298056332057555292898167089864157613427757054176691301095088, 0.92581468232773229694614624754486331250940301635560394879705455193228185)
 mflint:  (0.37797774271298056332057555292898167089864157613429, 0.92581468232773229694614624754486331250940301635559)
 aflint: ([0.3779777427129805633205755529289816708986415761343 +/- 2.92e-50], [0.9258146823277322969461462475448633125094030163556 +/- 2.44e-50])

An example with complex input, using C#-style formatting for complex numbers for the result

>>> from xlcalcnet import *
>>> x = -5.1+2j; gui.setdps(50)
>>> for ctx in [fpm, mpm, cmath53, qcplx]: print(ctx.fmtname + ': ' + ctx.fmt(ctx.expj(x)))
    fpm:  (0.0511537248671967, 0.125295392257438)
    mpm:  (0.0511537248671967434898253949927467610442454662229, 0.1252953922574382533359994793180146319527429028572)
cmath53:  (0.0511537248671967, 0.125295392257438)
  qcplx:  (0.0511537248671967434898253949927467, 0.125295392257438253335999479318014)

The above and additional examples can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder for Python (real input), Python (complex input), and C# (real or complex input).

Auxiliary function \(\mathrm{expjpi}(x) = e^{i \pi x} = (-1)^x\)

ctx.expjpi(z)

where ctx is math53, mathc53, ctxcpp or ctxflint.

Returns \(e^{i \pi z} = \cos(\pi z) + i \sin(\pi z)\). See also Wikipedia [1492], MathWorld [1094], Flint [794], Mpmath [753].

Evaluation is accurate near zeros (see also cospi() and sinpi()):

An example with real input:

>>> from xlcalcnet import *
>>> x = -5.1; dps = 50
>>> for ctx in gui.ctxlist_real: print(ctx.fmtname + ': ' + ctx.cplxctx.fmt(ctx.expjpi(x)))
    fpm:  (-0.951056516295154, 0.309016994374946)
    mpm:  (-0.95105651629515357211643933337938214340569863412575, 0.30901699437494742410229341718281905886015458990287)
    dpm:  (-0.95105651629515357211643933337938214340569863412575, 0.30901699437494742410229341718281905886015458990288)
    ipm: ([-0.951056516295153572116439333379382143405698634125765038, -0.951056516295153572116439333379382143405698634125736974], [0.3090169943749474241022934171828190588601545899028378188, 0.3090169943749474241022934171828190588601545899029200063])
    gpm:  (-0.9510565162951535721164393333793821434056986341257517, 0.3090169943749474241022934171828190588601545899028786)
    apm: ([-0.9510565162951535721164393333793821434056986341258 +/- 5.79e-50], [0.3090169943749474241022934171828190588601545899029 +/- 6.59e-50])
  dreal:  (-0.951056516295154, 0.309016994374946)
  sreal:  (-0.9510566, 0.3090167)
  dreal:  (-0.951056516295154, 0.309016994374946)
  ereal:  (-0.95105651629515357222, 0.30901699437494742386)
  qreal:  (-0.951056516295153572116439333379382, 0.309016994374947424102293417182818)
  oreal:  (-0.95105651629515357211643933337938214340569863412575022244730564443015319, 0.30901699437494742410229341718281905886015458990288143106772431135263019)
  mreal:  (-0.95105651629515357211643933337938214340569863412573, 0.30901699437494742410229341718281905886015458990293)
 sflint:  (-0.9510566, 0.3090167)
 dflint:  (-0.951056516295154, 0.309016994374946)
 eflint:  (-0.95105651629515357222, 0.30901699437494742383)
 qflint:  (-0.951056516295153572116439333379382, 0.309016994374947424102293417182818)
 oflint:  (-0.95105651629515357211643933337938214340569863412575022244730564443015318, 0.30901699437494742410229341718281905886015458990288143106772431135263019)
 mflint:  (-0.95105651629515357211643933337938214340569863412573, 0.30901699437494742410229341718281905886015458990292)
 aflint: ([-0.9510565162951535721164393333793821434056986341257 +/- 6.44e-50], [0.3090169943749474241022934171828190588601545899029 +/- 8.93e-50])

An example with complex input, using C#-style formatting for complex numbers for the result

>>> from xlcalcnet import *
>>> x = -5.1+2j; gui.setdps(50)
>>> for ctx in [fpm, mpm, cmath53, qcplx]: print(ctx.fmtname + ': ' + ctx.fmt(ctx.expjpi(x)))
    fpm:  (-0.00177604357879891, 0.000577071540119742)
    mpm:  (-0.0017760435787989049642054631852541036755960880447526, 0.00057707154011974403113330884851121775451896357682577)
cmath53:  (-0.00177604357725158, 0.000577071540703855)
  qcplx:  (-0.00177604357879890496420546318529556, 0.000577071540119744031133308848517094)

The above and additional examples can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder for Python (real input), Python (complex input), and C# (real or complex input).

Exponential function with base \(10\), \(\mathrm{exp10}(x) = 10^z\)

ctx.exp10(x)

where ctx is math53, mathc53, ctxcpp or ctxflint.

Returns \(\mathrm{exp10}(x) = 10^z = \exp(x \cdot \log(10))\), the base-10 exponential function of \(z\). See also Wikipedia [1348], MathWorld [919], NIST [515], Ehrhardt [309] (4.2.36), Mpmath [579].

An example with real input:

>>> from xlcalcnet import *
>>> x = -5.1; dps = 90
>>>     for ctx in gui.ctxlist_real: print(ctx.fmtname + ': ' + ctx.fmt(ctx.exp10(x))
    fpm:  7.94328234724282E-06
    mpm:  0.00000794328234724281502065918282836387932588960631755484332092323929316955697191487537497095251
    dpm:  0.00000794328234724281502065918282836387932588960631755484332092323929316955697191487537497095252
    ipm: [0.0000079432823472428150206591828283638793258896063175548433209232392931695569719148753749709524789891,
          0.0000079432823472428150206591828283638793258896063175548433209232392931695569719148753749709525445325]
    gpm:  7.9432823472428150206591828283638793258896063175548433209232392931695569719148753749709525146e-06
    apm: [7.943282347242815020659182828363879325889606317554843320923239293169556971914875374970953e-6 +/- 5.13e-94]
  dreal:  7.94328234724282E-06
  sreal:  7.943281E-06
  dreal:  7.94328234724281E-06
  ereal:  7.9432823472428150204e-06
  qreal:  7.94328234724281502065918282836389e-06
  oreal:  7.9432823472428150206591828283638793258896063175548433209232392931695569e-06
  mreal:  7.94328234724281502065918282836387932588960631755484332092323929316955697191487537497095253E-06
 sflint:  7.943284E-06
 dflint:  7.94328234724282E-06
 eflint:  7.9432823472428150221e-06
 qflint:  7.94328234724281502065918282836388e-06
 oflint:  7.9432823472428150206591828283638793258896063175548433209232392931695572e-06
 mflint:  7.94328234724281502065918282836387932588960631755484332092323929316955697191487537497095253E-06
 aflint: [7.943282347242815020659182828363879325889606317554843320923239293169556971914875374970953e-6 +/- 5.60e-94]

An example with complex input, using C#-style formatting for complex numbers for the result

>>> from xlcalcnet import *
>>> x = -5.1+2j; gui.setdps(50)
>>> for ctx in [fpm, mpm, cmath53, qcplx]: print(ctx.fmtname + ': ' + ctx.fmt(ctx.exp10(x)))
    fpm:  (-8.50038314869339E-07, -7.89766859973711E-06)
    mpm:  (-0.0000008500383148693386092398053603383376816809556042249, -0.0000078976685997371034342479584101601100277547297776704)
cmath53:  (-8.50038314869336E-07, -7.89766859973711E-06)
  qcplx:  (-8.50038314869338609239805360338339e-07, -7.89766859973710343424795841016012e-06)

The above and additional examples can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder for Python (real input), Python (complex input), and C# (real or complex input).

Exponential function with base \(2\), \(\mathrm{exp2}(x) = 2^x\)

ctx.exp2(x)

where ctx is math53, mathc53, ctxcpp or ctxflint.

Returns \(\mathrm{exp2}(x) = 2^x = \exp(x \cdot \log(2))\), the base-2 exponential function of \(x\). See also Wikipedia [1349], MathWorld [919], NIST [515], Ehrhardt [309] (4.2.35).

An example with real input:

>>> from xlcalcnet import *
>>> x = -5.1; dps = 90
>>> for ctx in gui.ctxlist_real: print(ctx.fmtname + ': ' + ctx.fmt(ctx.exp2(x)))
    fpm:  0.0291572809855252
    mpm:  0.0291572809855252317494169770671856927196009363859841887156405432964186568847051108626158585
    dpm:  0.0291572809855252317494169770671856927196009363859841887156405432964186568847051108626158585
    ipm: [0.0291572809855252317494169770671856927196009363859841887156405432964186568847051108626158585305,
          0.029157280985525231749416977067185692719600936385984188715640543296418656884705110862615858591864]
    gpm:  0.029157280985525231749416977067185692719600936385984188715640543296418656884705110862615858538
    apm: [0.0291572809855252317494169770671856927196009363859841887156405432964186568847051108626158586 +/- 8.98e-92]
  dreal:  0.0291572809855252
  sreal:  0.02915728
  dreal:  0.0291572809855252
  ereal:  0.029157280985525231751
  qreal:  0.0291572809855252317494169770671857
  oreal:  0.029157280985525231749416977067185692719600936385984188715640543296418657
  mreal:  0.0291572809855252317494169770671856927196009363859841887156405432964186568847051108626158586
 sflint:  0.02915728
 dflint:  0.0291572809855252
 eflint:  0.029157280985525231751
 qflint:  0.0291572809855252317494169770671857
 oflint:  0.029157280985525231749416977067185692719600936385984188715640543296418657
 mflint:  0.0291572809855252317494169770671856927196009363859841887156405432964186568847051108626158586
 aflint: [0.029157280985525231749416977067185692719600936385984188715640543296418656884705110862615859 +/- 5.38e-91]

An example with complex input, using C#-style formatting for complex numbers for the result

>>> from xlcalcnet import *
>>> x = -5.1+2j; gui.setdps(50)
>>> for ctx in [fpm, mpm, cmath53, qcplx]: print(ctx.fmtname + ': ' + ctx.fmt(ctx.exp2(x)))
    fpm:  (0.00534910656134485, 0.0286624160437366)
    mpm:  (0.0053491065613448526660310040295797895499084770752861, 0.028662416043736589994269252987760995841700161519681)
cmath53:  (0.00534910656134486, 0.0286624160437366)
  qcplx:  (0.00534910656134485266603100402957979, 0.028662416043736589994269252987761)

The above and additional examples can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder for Python (real input), Python (complex input), and C# (real or complex input).

Auxiliary function \(\mathrm{expm1}(x) = e^x-1\)

ctx.expm1(x)

where ctx is math53, mathc53, ctxcpp or ctxflint.

Returns \(\mathrm{expm1}(x) = \exp(x)-1 = e^x-1\), computed accurately also for small \(x\). See also Wikipedia [1341], MathWorld [919], NIST [515], BoostMath [81], Ehrhardt [309] (4.2.37), Flint [804], Flint [794], Mpmath [574].

An example with real input:

>>> from xlcalcnet import *
>>> x = -5.1; dps = 90
>>> for ctx in gui.ctxlist_real: print(ctx.fmtname + ': ' + ctx.fmt(ctx.expm1(x)))
    fpm:  -0.993903253434484
    mpm:  -0.993903253434484363892865435214575098210939957919331902691606284304043023632260651674239863
    dpm:  -0.993903253434484363892865435214575098210939957919331902691606284304043023632260651674239863
    ipm: [-0.99390325343448436389286543521457509821093995791933190269160628430404302363226065167423986352495,
          -0.99390325343448436389286543521457509821093995791933190269160628430404302363226065167423986340222]
    gpm:  -0.9939032534344843638928654352145750982109399579193319026916062843040430236322606516742398634
    apm: [-0.993903253434484363892865435214575098210939957919331902691606284304043023632260651674239863 +/- 5.31e-91]
  dreal:  -0.993903253434484
  sreal:  -0.9939033
  dreal:  -0.993903253434484
  ereal:  -0.99390325343448436389
  qreal:  -0.993903253434484363892865435214575
  oreal:  -0.99390325343448436389286543521457509821093995791933190269160628430404302
  mreal:  -0.993903253434484363892865435214575098210939957919331902691606284304043023632260651674239864
 sflint:  -0.9939033
 dflint:  -0.993903253434484
 eflint:  -0.99390325343448436389
 qflint:  -0.993903253434484363892865435214575
 oflint:  -0.99390325343448436389286543521457509821093995791933190269160628430404302
 mflint:  -0.993903253434484363892865435214575098210939957919331902691606284304043023632260651674239864
 aflint: [-0.993903253434484363892865435214575098210939957919331902691606284304043023632260651674239863 +/- 6.53e-91]

An example with complex input, using C#-style formatting for complex numbers for the result

>>> from xlcalcnet import *
>>> x = -5.1+2j; gui.setdps(50)
>>> for ctx in [fpm, mpm, cmath53, qcplx]: print(ctx.fmtname + ': ' + ctx.fmt(ctx.expm1(x)))
    fpm:  (-1.00253714179647, 0.00554375596403168)
    mpm:  (-1.0025371417964689871432868572987332349386956711932, 0.0055437559640316803155849317223325676631201581872807)
cmath53:  (-1.00253714179647, 0.00554375596403168)
  qcplx:  (-1.00253714179646898714328685729873, 0.00554375596403168031558493172233257)

The above and additional examples can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder for Python (real input), Python (complex input), and C# (real or complex input).

Auxiliary function \(\mathrm{exp10m1}(x) = 10^x - 1\)

ctx.exp10m1(x)

where ctx is math53, mathc53, ctxcpp or ctxflint.

Returns \(10^x - 1 = \mathrm{expm1}(x \cdot \log(10))\). See also expm1().

An example with real input:

>>> from xlcalcnet import *
>>> x = -5.1; dps = 90
>>> for ctx in gui.ctxlist_real: print(ctx.fmtname + ': ' + ctx.fmt(ctx.exp10m1(x)))
    fpm:  -0.999992056717653
    mpm:  -0.999992056717652757184979340817171636120674110393682445156679076760706830443028085124625029
    dpm:  -0.999992056717652757184979340817171636120674110393682445156679076760706830443028085124625029
    ipm: [-0.99999205671765275718497934081717163612067411039368244515667907676070683044302808512462502915426,
          -0.99999205671765275718497934081717163612067411039368244515667907676070683044302808512462502903153]
    gpm:  -0.99999205671765275718497934081717163612067411039368244515667907676070683044302808512462502903
    apm: [-0.999992056717652757184979340817171636120674110393682445156679076760706830443028085124625029 +/- 1.55e-91]
  dreal:  -0.999992056717653
  sreal:  -0.9999921
  dreal:  -0.999992056717653
  ereal:  -0.99999205671765275719
  qreal:  -0.999992056717652757184979340817172
  oreal:  -0.99999205671765275718497934081717163612067411039368244515667907676070683
  mreal:  -0.999992056717652757184979340817171636120674110393682445156679076760706830443028085124625029
 sflint:  -0.9999921
 dflint:  -0.999992056717653
 eflint:  -0.99999205671765275719
 qflint:  -0.999992056717652757184979340817172
 oflint:  -0.99999205671765275718497934081717163612067411039368244515667907676070683
 mflint:  -0.999992056717652757184979340817171636120674110393682445156679076760706830443028085124625029
 aflint: [-0.999992056717652757184979340817171636120674110393682445156679076760706830443028085124625029 +/- 7.06e-91]

An example with complex input, using C#-style formatting for complex numbers for the result

>>> from xlcalcnet import *
>>> x = -5.1+2j; gui.setdps(50)
>>> for ctx in [fpm, mpm, cmath53, qcplx]: print(ctx.fmtname + ': ' + ctx.fmt(ctx.exp10m1(x)))
    fpm:  (-1.00000085003831, -7.89766859973711E-06)
    mpm:  (-1.0000008500383148693386092398053603383376816809556, -0.0000078976685997371034342479584101601100277547297776704)
cmath53:  (-1.00000085003831, -7.8976685997371E-06)
  qcplx:  (-1.00000085003831486933860923980536, -7.89766859973710343424795841016012e-06)

The above and additional examples can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder for Python (real input), Python (complex input), and C# (real or complex input).

Auxiliary function \(\mathrm{exp2m1}(x) = 2^x - 1\)

ctx.exp2m1(x)

where ctx is math53, mathc53, ctxcpp or ctxflint.

Returns \(2^x - 1 = \mathrm{expm1}(x \cdot \log(2))\). See also expm1().

An example with real input:

>>> from xlcalcnet import *
>>> x = -5.1; dps = 90
>>> for ctx in gui.ctxlist_real: print(ctx.fmtname + ': ' + ctx.fmt(ctx.exp2m1(x)))
    fpm:  -0.970842719014475
    mpm:  -0.970842719014474768250583022932814307280399063614015811284359456703581343115294889137384141
    dpm:  -0.970842719014474768250583022932814307280399063614015811284359456703581343115294889137384141
    ipm: [-0.9708427190144747682505830229328143072803990636140158112843594567035813431152948891373841415347,
          -0.97084271901447476825058302293281430728039906361401581128435945670358134311529488913738414128924]
    gpm:  -0.97084271901447476825058302293281430728039906361401581128435945670358134311529488913738414141
    apm: [-0.970842719014474768250583022932814307280399063614015811284359456703581343115294889137384141 +/- 5.79e-91]
  dreal:  -0.970842719014475
  sreal:  -0.9708427
  dreal:  -0.970842719014475
  ereal:  -0.97084271901447476827
  qreal:  -0.970842719014474768250583022932814
  oreal:  -0.97084271901447476825058302293281430728039906361401581128435945670358134
  mreal:  -0.970842719014474768250583022932814307280399063614015811284359456703581343115294889137384142
 sflint:  -0.9708427
 dflint:  -0.970842719014475
 eflint:  -0.97084271901447476827
 qflint:  -0.970842719014474768250583022932814
 oflint:  -0.97084271901447476825058302293281430728039906361401581128435945670358134
 mflint:  -0.970842719014474768250583022932814307280399063614015811284359456703581343115294889137384142
 aflint: [-0.970842719014474768250583022932814307280399063614015811284359456703581343115294889137384141 +/- 7.68e-91]

An example with complex input, using C#-style formatting for complex numbers for the result

>>> from xlcalcnet import *
>>> x = -5.1+2j; gui.setdps(50)
>>> for ctx in [fpm, mpm, cmath53, qcplx]: print(ctx.fmtname + ': ' + ctx.fmt(ctx.exp2m1(x)))
    fpm:  (-0.994650893438655, 0.0286624160437366)
    mpm:  (-0.99465089343865514733396899597042021045009152292471, 0.028662416043736589994269252987760995841700161519681)
cmath53:  (-0.994650893438655, 0.0286624160437366)
  qcplx:  (-0.99465089343865514733396899597042, 0.028662416043736589994269252987761)

The above and additional examples can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder for Python (real input), Python (complex input), and C# (real or complex input).