math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 10

Speedup: 1.0×

• 48.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 2: 85.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 0.0068:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 0.0068)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.32e+154)
       (* (* 0.5 re) (+ (exp (- im)) (exp im)))
       (* t_0 (pow im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 0.0068) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.32e+154) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = t_0 * pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 0.0068)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.32e+154)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(t_0 * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.0068], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.32e+154], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 3: 66.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 245:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.015625}{{re}^{6}}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 245.0)
   (sin re)
   (if (<= im 1.32e+154)
     (cbrt (/ 0.015625 (pow re 6.0)))
     (* (* 0.5 (sin re)) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 245.0) {
		tmp = sin(re);
	} else if (im <= 1.32e+154) {
		tmp = cbrt((0.015625 / pow(re, 6.0)));
	} else {
		tmp = (0.5 * sin(re)) * pow(im, 2.0);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 245.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.32e+154) {
		tmp = Math.cbrt((0.015625 / Math.pow(re, 6.0)));
	} else {
		tmp = (0.5 * Math.sin(re)) * Math.pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 245.0)
		tmp = sin(re);
	elseif (im <= 1.32e+154)
		tmp = cbrt(Float64(0.015625 / (re ^ 6.0)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 245.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.32e+154], N[Power[N[(0.015625 / N[Power[re, 6.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 4: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 245:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.015625}{{re}^{6}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 245.0)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.32e+154)
       (cbrt (/ 0.015625 (pow re 6.0)))
       (* t_0 (pow im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 245.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.32e+154) {
		tmp = cbrt((0.015625 / pow(re, 6.0)));
	} else {
		tmp = t_0 * pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 245.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.32e+154)
		tmp = cbrt(Float64(0.015625 / (re ^ 6.0)));
	else
		tmp = Float64(t_0 * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 245.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.32e+154], N[Power[N[(0.015625 / N[Power[re, 6.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(t$95$0 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 5: 62.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 245:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+141}:\\ \;\;\;\;\sqrt[3]{\frac{0.015625}{{re}^{6}}}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 245.0)
   (sin re)
   (if (<= im 1.7e+141)
     (cbrt (/ 0.015625 (pow re 6.0)))
     (* re (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 245.0) {
		tmp = sin(re);
	} else if (im <= 1.7e+141) {
		tmp = cbrt((0.015625 / pow(re, 6.0)));
	} else {
		tmp = re * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 245.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.7e+141) {
		tmp = Math.cbrt((0.015625 / Math.pow(re, 6.0)));
	} else {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 245.0)
		tmp = sin(re);
	elseif (im <= 1.7e+141)
		tmp = cbrt(Float64(0.015625 / (re ^ 6.0)));
	else
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 245.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.7e+141], N[Power[N[(0.015625 / N[Power[re, 6.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 6: 61.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.8 \cdot 10^{+22}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.8e+22)
   (sin re)
   (if (<= im 1.95e+141) (/ (/ 0.25 re) re) (* re (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.8e+22) {
		tmp = sin(re);
	} else if (im <= 1.95e+141) {
		tmp = (0.25 / re) / re;
	} else {
		tmp = re * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.8d+22) then
        tmp = sin(re)
    else if (im <= 1.95d+141) then
        tmp = (0.25d0 / re) / re
    else
        tmp = re * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.8e+22) {
		tmp = Math.sin(re);
	} else if (im <= 1.95e+141) {
		tmp = (0.25 / re) / re;
	} else {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.8e+22:
		tmp = math.sin(re)
	elif im <= 1.95e+141:
		tmp = (0.25 / re) / re
	else:
		tmp = re * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.8e+22)
		tmp = sin(re);
	elseif (im <= 1.95e+141)
		tmp = Float64(Float64(0.25 / re) / re);
	else
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.8e+22)
		tmp = sin(re);
	elseif (im <= 1.95e+141)
		tmp = (0.25 / re) / re;
	else
		tmp = re * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.8e+22], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.95e+141], N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 7: 54.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 245:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 245.0) (sin re) (+ 0.08333333333333333 (* (/ 0.5 re) (/ 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 245.0) {
		tmp = sin(re);
	} else {
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 245.0d0) then
        tmp = sin(re)
    else
        tmp = 0.08333333333333333d0 + ((0.5d0 / re) * (0.5d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 245.0) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 245.0:
		tmp = math.sin(re)
	else:
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 245.0)
		tmp = sin(re);
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.5 / re) * Float64(0.5 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 245.0)
		tmp = sin(re);
	else
		tmp = 0.08333333333333333 + ((0.5 / re) * (0.5 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 245.0], N[Sin[re], $MachinePrecision], N[(0.08333333333333333 + N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 8: 29.7% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.55 \cdot 10^{+21}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.55e+21) re (/ (/ 0.25 re) re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.55e+21) {
		tmp = re;
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.55d+21) then
        tmp = re
    else
        tmp = (0.25d0 / re) / re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.55e+21) {
		tmp = re;
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.55e+21:
		tmp = re
	else:
		tmp = (0.25 / re) / re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.55e+21)
		tmp = re;
	else
		tmp = Float64(Float64(0.25 / re) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.55e+21)
		tmp = re;
	else
		tmp = (0.25 / re) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.55e+21], re, N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 9: 27.6% accurate, 100.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.7:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 2.7) re 0.08333333333333333))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.7) {
		tmp = re;
	} else {
		tmp = 0.08333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.7d0) then
        tmp = re
    else
        tmp = 0.08333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.7) {
		tmp = re;
	} else {
		tmp = 0.08333333333333333;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.7:
		tmp = re
	else:
		tmp = 0.08333333333333333
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.7)
		tmp = re;
	else
		tmp = 0.08333333333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.7)
		tmp = re;
	else
		tmp = 0.08333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.7], re, 0.08333333333333333]

Derivation

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1. Add Preprocessing

Alternative 10: 4.3% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.08333333333333333)

C

double code(double re, double im) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.08333333333333333d0
end function

Java

public static double code(double re, double im) {
	return 0.08333333333333333;
}

Python

def code(re, im):
	return 0.08333333333333333

Julia

function code(re, im)
	return 0.08333333333333333
end

MATLAB

function tmp = code(re, im)
	tmp = 0.08333333333333333;
end

Wolfram

code[re_, im_] := 0.08333333333333333

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2023350 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))