math.cos on complex, imaginary part

Percentage Accurate: 64.9% → 99.8%

Time: 8.3s

Alternatives: 13

Speedup: 2.8×

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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]

Initial Program: 64.9% 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]

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -5 \lor \neg \left(t_0 \leq 0.0005\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -5.0) (not (<= t_0 0.0005)))
     (* t_0 (* 0.5 (sin re)))
     (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 0.0005)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-5.0d0)) .or. (.not. (t_0 <= 0.0005d0))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 0.0005)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -5.0) or not (t_0 <= 0.0005):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -5.0) || !(t_0 <= 0.0005))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -5.0) || ~((t_0 <= 0.0005)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5.0], N[Not[LessEqual[t$95$0, 0.0005]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -5 or 5.0000000000000001e-4 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   if -5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 5.0000000000000001e-4

   1. Initial program 31.4%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 99.8%

         \[\leadsto -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \color{blue}{\left(-\sin re \cdot im\right)} \]

      2. unsub-neg 99.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - \sin re \cdot im} \]

      3. *-commutative 99.8%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \sin re \cdot im \]

      4. associate-*l* 99.8%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \sin re \cdot im \]

      5. distribute-lft-out-- 99.8%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \lor \neg \left(e^{-im} - e^{im} \leq 0.0005\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := {im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \mathbf{if}\;im \leq -9 \cdot 10^{+110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.000195:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.95:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* (pow im 3.0) (* (sin re) -0.16666666666666666))))
   (if (<= im -9e+110)
     t_1
     (if (<= im -0.000195)
       t_0
       (if (<= im 1.95) (* im (- (sin re))) (if (<= im 5.8e+102) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	double t_1 = pow(im, 3.0) * (sin(re) * -0.16666666666666666);
	double tmp;
	if (im <= -9e+110) {
		tmp = t_1;
	} else if (im <= -0.000195) {
		tmp = t_0;
	} else if (im <= 1.95) {
		tmp = im * -sin(re);
	} else if (im <= 5.8e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((exp(-im) - exp(im)) * re)
    t_1 = (im ** 3.0d0) * (sin(re) * (-0.16666666666666666d0))
    if (im <= (-9d+110)) then
        tmp = t_1
    else if (im <= (-0.000195d0)) then
        tmp = t_0
    else if (im <= 1.95d0) then
        tmp = im * -sin(re)
    else if (im <= 5.8d+102) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((Math.exp(-im) - Math.exp(im)) * re);
	double t_1 = Math.pow(im, 3.0) * (Math.sin(re) * -0.16666666666666666);
	double tmp;
	if (im <= -9e+110) {
		tmp = t_1;
	} else if (im <= -0.000195) {
		tmp = t_0;
	} else if (im <= 1.95) {
		tmp = im * -Math.sin(re);
	} else if (im <= 5.8e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	t_1 = math.pow(im, 3.0) * (math.sin(re) * -0.16666666666666666)
	tmp = 0
	if im <= -9e+110:
		tmp = t_1
	elif im <= -0.000195:
		tmp = t_0
	elif im <= 1.95:
		tmp = im * -math.sin(re)
	elif im <= 5.8e+102:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64((im ^ 3.0) * Float64(sin(re) * -0.16666666666666666))
	tmp = 0.0
	if (im <= -9e+110)
		tmp = t_1;
	elseif (im <= -0.000195)
		tmp = t_0;
	elseif (im <= 1.95)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 5.8e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	t_1 = (im ^ 3.0) * (sin(re) * -0.16666666666666666);
	tmp = 0.0;
	if (im <= -9e+110)
		tmp = t_1;
	elseif (im <= -0.000195)
		tmp = t_0;
	elseif (im <= 1.95)
		tmp = im * -sin(re);
	elseif (im <= 5.8e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9e+110], t$95$1, If[LessEqual[im, -0.000195], t$95$0, If[LessEqual[im, 1.95], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 5.8e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.0000000000000005e110 or 5.8000000000000005e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \color{blue}{\left(-\sin re \cdot im\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - \sin re \cdot im} \]

      3. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \sin re \cdot im \]

      4. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \sin re \cdot im \]

      5. distribute-lft-out-- 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{3} \cdot \sin re\right)} \cdot -0.16666666666666666 \]

      3. associate-*l* 100.0%

         \[\leadsto \color{blue}{{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)} \]

   if -9.0000000000000005e110 < im < -1.94999999999999996e-4 or 1.94999999999999996 < im < 5.8000000000000005e102

   1. Initial program 99.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 74.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]

   if -1.94999999999999996e-4 < im < 1.94999999999999996

   1. Initial program 31.4%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 98.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 98.8%

         \[\leadsto \color{blue}{-\sin re \cdot im} \]

      2. *-commutative 98.8%

         \[\leadsto -\color{blue}{im \cdot \sin re} \]

      3. distribute-rgt-neg-in 98.8%

         \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

   4. Simplified 98.8%

      \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+110}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -0.000195:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 1.95:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 3: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := {im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \mathbf{if}\;im \leq -9 \cdot 10^{+110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -2.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.95:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* (pow im 3.0) (* (sin re) -0.16666666666666666))))
   (if (<= im -9e+110)
     t_1
     (if (<= im -2.6)
       t_0
       (if (<= im 1.95)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 5.8e+102) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	double t_1 = pow(im, 3.0) * (sin(re) * -0.16666666666666666);
	double tmp;
	if (im <= -9e+110) {
		tmp = t_1;
	} else if (im <= -2.6) {
		tmp = t_0;
	} else if (im <= 1.95) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 5.8e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((exp(-im) - exp(im)) * re)
    t_1 = (im ** 3.0d0) * (sin(re) * (-0.16666666666666666d0))
    if (im <= (-9d+110)) then
        tmp = t_1
    else if (im <= (-2.6d0)) then
        tmp = t_0
    else if (im <= 1.95d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 5.8d+102) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((Math.exp(-im) - Math.exp(im)) * re);
	double t_1 = Math.pow(im, 3.0) * (Math.sin(re) * -0.16666666666666666);
	double tmp;
	if (im <= -9e+110) {
		tmp = t_1;
	} else if (im <= -2.6) {
		tmp = t_0;
	} else if (im <= 1.95) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 5.8e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	t_1 = math.pow(im, 3.0) * (math.sin(re) * -0.16666666666666666)
	tmp = 0
	if im <= -9e+110:
		tmp = t_1
	elif im <= -2.6:
		tmp = t_0
	elif im <= 1.95:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 5.8e+102:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64((im ^ 3.0) * Float64(sin(re) * -0.16666666666666666))
	tmp = 0.0
	if (im <= -9e+110)
		tmp = t_1;
	elseif (im <= -2.6)
		tmp = t_0;
	elseif (im <= 1.95)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 5.8e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	t_1 = (im ^ 3.0) * (sin(re) * -0.16666666666666666);
	tmp = 0.0;
	if (im <= -9e+110)
		tmp = t_1;
	elseif (im <= -2.6)
		tmp = t_0;
	elseif (im <= 1.95)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 5.8e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9e+110], t$95$1, If[LessEqual[im, -2.6], t$95$0, If[LessEqual[im, 1.95], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.8e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.0000000000000005e110 or 5.8000000000000005e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \color{blue}{\left(-\sin re \cdot im\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - \sin re \cdot im} \]

      3. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \sin re \cdot im \]

      4. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \sin re \cdot im \]

      5. distribute-lft-out-- 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{3} \cdot \sin re\right)} \cdot -0.16666666666666666 \]

      3. associate-*l* 100.0%

         \[\leadsto \color{blue}{{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)} \]

   if -9.0000000000000005e110 < im < -2.60000000000000009 or 1.94999999999999996 < im < 5.8000000000000005e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 75.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]

   if -2.60000000000000009 < im < 1.94999999999999996

   1. Initial program 32.4%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 98.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 98.6%

         \[\leadsto -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \color{blue}{\left(-\sin re \cdot im\right)} \]

      2. unsub-neg 98.6%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - \sin re \cdot im} \]

      3. *-commutative 98.6%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \sin re \cdot im \]

      4. associate-*l* 98.6%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \sin re \cdot im \]

      5. distribute-lft-out-- 98.6%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   4. Simplified 98.6%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+110}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -2.6:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 1.95:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 4: 83.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.4 \lor \neg \left(im \leq 2.5\right):\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.4) (not (<= im 2.5)))
   (* (pow im 3.0) (* (sin re) -0.16666666666666666))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.4) || !(im <= 2.5)) {
		tmp = pow(im, 3.0) * (sin(re) * -0.16666666666666666);
	} else {
		tmp = im * -sin(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 <= (-2.4d0)) .or. (.not. (im <= 2.5d0))) then
        tmp = (im ** 3.0d0) * (sin(re) * (-0.16666666666666666d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.4) || !(im <= 2.5)) {
		tmp = Math.pow(im, 3.0) * (Math.sin(re) * -0.16666666666666666);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.4) or not (im <= 2.5):
		tmp = math.pow(im, 3.0) * (math.sin(re) * -0.16666666666666666)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.4) || !(im <= 2.5))
		tmp = Float64((im ^ 3.0) * Float64(sin(re) * -0.16666666666666666));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.4) || ~((im <= 2.5)))
		tmp = (im ^ 3.0) * (sin(re) * -0.16666666666666666);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.4], N[Not[LessEqual[im, 2.5]], $MachinePrecision]], N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.39999999999999991 or 2.5 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 67.4%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 67.4%

         \[\leadsto -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \color{blue}{\left(-\sin re \cdot im\right)} \]

      2. unsub-neg 67.4%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - \sin re \cdot im} \]

      3. *-commutative 67.4%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \sin re \cdot im \]

      4. associate-*l* 67.4%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \sin re \cdot im \]

      5. distribute-lft-out-- 67.4%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   4. Simplified 67.4%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   5. Taylor expanded in im around inf 67.4%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 67.4%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666} \]

      2. *-commutative 67.4%

         \[\leadsto \color{blue}{\left({im}^{3} \cdot \sin re\right)} \cdot -0.16666666666666666 \]

      3. associate-*l* 67.4%

         \[\leadsto \color{blue}{{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)} \]

   7. Simplified 67.4%

      \[\leadsto \color{blue}{{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)} \]

   if -2.39999999999999991 < im < 2.5

   1. Initial program 31.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 98.5%

      \[\leadsto \color{blue}{-1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 98.5%

         \[\leadsto \color{blue}{-\sin re \cdot im} \]

      2. *-commutative 98.5%

         \[\leadsto -\color{blue}{im \cdot \sin re} \]

      3. distribute-rgt-neg-in 98.5%

         \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

   4. Simplified 98.5%

      \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.4 \lor \neg \left(im \leq 2.5\right):\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 5: 77.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2050000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          0.5
          (+
           (* re (* im (* im (* im -0.3333333333333333))))
           (* re (* im -2.0))))))
   (if (<= im -5.6e-5)
     t_0
     (if (<= im 2050000000.0)
       (* im (- (sin re)))
       (if (<= im 1.6e+105)
         (* im (- (* 0.16666666666666666 (pow re 3.0)) re))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
	double tmp;
	if (im <= -5.6e-5) {
		tmp = t_0;
	} else if (im <= 2050000000.0) {
		tmp = im * -sin(re);
	} else if (im <= 1.6e+105) {
		tmp = im * ((0.16666666666666666 * pow(re, 3.0)) - re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((re * (im * (im * (im * (-0.3333333333333333d0))))) + (re * (im * (-2.0d0))))
    if (im <= (-5.6d-5)) then
        tmp = t_0
    else if (im <= 2050000000.0d0) then
        tmp = im * -sin(re)
    else if (im <= 1.6d+105) then
        tmp = im * ((0.16666666666666666d0 * (re ** 3.0d0)) - re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
	double tmp;
	if (im <= -5.6e-5) {
		tmp = t_0;
	} else if (im <= 2050000000.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 1.6e+105) {
		tmp = im * ((0.16666666666666666 * Math.pow(re, 3.0)) - re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)))
	tmp = 0
	if im <= -5.6e-5:
		tmp = t_0
	elif im <= 2050000000.0:
		tmp = im * -math.sin(re)
	elif im <= 1.6e+105:
		tmp = im * ((0.16666666666666666 * math.pow(re, 3.0)) - re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(re * Float64(im * Float64(im * Float64(im * -0.3333333333333333)))) + Float64(re * Float64(im * -2.0))))
	tmp = 0.0
	if (im <= -5.6e-5)
		tmp = t_0;
	elseif (im <= 2050000000.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 1.6e+105)
		tmp = Float64(im * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
	tmp = 0.0;
	if (im <= -5.6e-5)
		tmp = t_0;
	elseif (im <= 2050000000.0)
		tmp = im * -sin(re);
	elseif (im <= 1.6e+105)
		tmp = im * ((0.16666666666666666 * (re ^ 3.0)) - re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(re * N[(im * N[(im * N[(im * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.6e-5], t$95$0, If[LessEqual[im, 2050000000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.6e+105], N[(im * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.59999999999999992e-5 or 1.6e105 < im

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 69.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]

   3. Taylor expanded in im around 0 57.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(re \cdot {im}^{3}\right) + -2 \cdot \left(re \cdot im\right)\right)} \]
   4. Step-by-step derivation

      1. cube-mult 57.5%

         \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \left(re \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) + -2 \cdot \left(re \cdot im\right)\right) \]

      2. associate-*r* 51.3%

         \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(im \cdot im\right)\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

      3. *-commutative 51.3%

         \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(re \cdot im\right)\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

      4. associate-*r* 51.3%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot im\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

      5. distribute-rgt-out 51.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]

   5. Simplified 51.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 51.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) + \left(re \cdot im\right) \cdot -2\right)} \]

      2. associate-*l* 57.5%

         \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right)} + \left(re \cdot im\right) \cdot -2\right) \]

      3. *-commutative 57.5%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot -0.3333333333333333\right)}\right) + \left(re \cdot im\right) \cdot -2\right) \]

      4. associate-*l* 57.5%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)}\right) + \left(re \cdot im\right) \cdot -2\right) \]

      5. associate-*l* 57.5%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + \color{blue}{re \cdot \left(im \cdot -2\right)}\right) \]

   7. Applied egg-rr 57.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)} \]

   if -5.59999999999999992e-5 < im < 2.05e9

   1. Initial program 32.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 98.1%

      \[\leadsto \color{blue}{-1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 98.1%

         \[\leadsto \color{blue}{-\sin re \cdot im} \]

      2. *-commutative 98.1%

         \[\leadsto -\color{blue}{im \cdot \sin re} \]

      3. distribute-rgt-neg-in 98.1%

         \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

   4. Simplified 98.1%

      \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

   if 2.05e9 < im < 1.6e105

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 2.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 2.8%

         \[\leadsto \color{blue}{-\sin re \cdot im} \]

      2. *-commutative 2.8%

         \[\leadsto -\color{blue}{im \cdot \sin re} \]

      3. distribute-rgt-neg-in 2.8%

         \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

   4. Simplified 2.8%

      \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

   5. Taylor expanded in re around 0 10.7%

      \[\leadsto \color{blue}{-1 \cdot \left(re \cdot im\right) + 0.16666666666666666 \cdot \left({re}^{3} \cdot im\right)} \]
   6. Step-by-step derivation

      1. +-commutative 10.7%

         \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({re}^{3} \cdot im\right) + -1 \cdot \left(re \cdot im\right)} \]

      2. mul-1-neg 10.7%

         \[\leadsto 0.16666666666666666 \cdot \left({re}^{3} \cdot im\right) + \color{blue}{\left(-re \cdot im\right)} \]

      3. unsub-neg 10.7%

         \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({re}^{3} \cdot im\right) - re \cdot im} \]

      4. associate-*r* 10.7%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot im} - re \cdot im \]

      5. distribute-rgt-out-- 28.9%

         \[\leadsto \color{blue}{im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)} \]

   7. Simplified 28.9%

      \[\leadsto \color{blue}{im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2050000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)\\ \end{array} \]

Alternative 6: 77.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{if}\;im \leq -0.000105:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+105}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          0.5
          (+
           (* re (* im (* im (* im -0.3333333333333333))))
           (* re (* im -2.0))))))
   (if (<= im -0.000105)
     t_0
     (if (<= im 1.5e+29)
       (* im (- (sin re)))
       (if (<= im 1.6e+105)
         (* 0.16666666666666666 (* im (pow re 3.0)))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
	double tmp;
	if (im <= -0.000105) {
		tmp = t_0;
	} else if (im <= 1.5e+29) {
		tmp = im * -sin(re);
	} else if (im <= 1.6e+105) {
		tmp = 0.16666666666666666 * (im * pow(re, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((re * (im * (im * (im * (-0.3333333333333333d0))))) + (re * (im * (-2.0d0))))
    if (im <= (-0.000105d0)) then
        tmp = t_0
    else if (im <= 1.5d+29) then
        tmp = im * -sin(re)
    else if (im <= 1.6d+105) then
        tmp = 0.16666666666666666d0 * (im * (re ** 3.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
	double tmp;
	if (im <= -0.000105) {
		tmp = t_0;
	} else if (im <= 1.5e+29) {
		tmp = im * -Math.sin(re);
	} else if (im <= 1.6e+105) {
		tmp = 0.16666666666666666 * (im * Math.pow(re, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)))
	tmp = 0
	if im <= -0.000105:
		tmp = t_0
	elif im <= 1.5e+29:
		tmp = im * -math.sin(re)
	elif im <= 1.6e+105:
		tmp = 0.16666666666666666 * (im * math.pow(re, 3.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(re * Float64(im * Float64(im * Float64(im * -0.3333333333333333)))) + Float64(re * Float64(im * -2.0))))
	tmp = 0.0
	if (im <= -0.000105)
		tmp = t_0;
	elseif (im <= 1.5e+29)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 1.6e+105)
		tmp = Float64(0.16666666666666666 * Float64(im * (re ^ 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
	tmp = 0.0;
	if (im <= -0.000105)
		tmp = t_0;
	elseif (im <= 1.5e+29)
		tmp = im * -sin(re);
	elseif (im <= 1.6e+105)
		tmp = 0.16666666666666666 * (im * (re ^ 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(re * N[(im * N[(im * N[(im * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -0.000105], t$95$0, If[LessEqual[im, 1.5e+29], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.6e+105], N[(0.16666666666666666 * N[(im * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.05e-4 or 1.6e105 < im

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 69.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]

   3. Taylor expanded in im around 0 57.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(re \cdot {im}^{3}\right) + -2 \cdot \left(re \cdot im\right)\right)} \]
   4. Step-by-step derivation

      1. cube-mult 57.5%

         \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \left(re \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) + -2 \cdot \left(re \cdot im\right)\right) \]

      2. associate-*r* 51.3%

         \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(im \cdot im\right)\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

      3. *-commutative 51.3%

         \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(re \cdot im\right)\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

      4. associate-*r* 51.3%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot im\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

      5. distribute-rgt-out 51.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]

   5. Simplified 51.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 51.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) + \left(re \cdot im\right) \cdot -2\right)} \]

      2. associate-*l* 57.5%

         \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right)} + \left(re \cdot im\right) \cdot -2\right) \]

      3. *-commutative 57.5%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot -0.3333333333333333\right)}\right) + \left(re \cdot im\right) \cdot -2\right) \]

      4. associate-*l* 57.5%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)}\right) + \left(re \cdot im\right) \cdot -2\right) \]

      5. associate-*l* 57.5%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + \color{blue}{re \cdot \left(im \cdot -2\right)}\right) \]

   7. Applied egg-rr 57.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)} \]

   if -1.05e-4 < im < 1.5e29

   1. Initial program 34.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 94.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 94.4%

         \[\leadsto \color{blue}{-\sin re \cdot im} \]

      2. *-commutative 94.4%

         \[\leadsto -\color{blue}{im \cdot \sin re} \]

      3. distribute-rgt-neg-in 94.4%

         \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

   4. Simplified 94.4%

      \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

   if 1.5e29 < im < 1.6e105

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 15.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 15.5%

         \[\leadsto -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \color{blue}{\left(-\sin re \cdot im\right)} \]

      2. unsub-neg 15.5%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - \sin re \cdot im} \]

      3. *-commutative 15.5%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \sin re \cdot im \]

      4. associate-*l* 15.5%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \sin re \cdot im \]

      5. distribute-lft-out-- 15.5%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   4. Simplified 15.5%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   5. Taylor expanded in re around 0 2.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 2.3%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {re}^{3}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      2. distribute-rgt-out 37.5%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.16666666666666666 \cdot {re}^{3} + re\right)} \]

      3. *-commutative 37.5%

         \[\leadsto \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \cdot \left(-0.16666666666666666 \cdot {re}^{3} + re\right) \]

      4. *-commutative 37.5%

         \[\leadsto \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(\color{blue}{{re}^{3} \cdot -0.16666666666666666} + re\right) \]

   7. Simplified 37.5%

      \[\leadsto \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left({re}^{3} \cdot -0.16666666666666666 + re\right)} \]

   8. Taylor expanded in im around 0 36.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(re + -0.16666666666666666 \cdot {re}^{3}\right) \cdot im\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 36.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\right) \cdot im} \]

      2. *-commutative 36.8%

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\right)} \]

      3. +-commutative 36.8%

         \[\leadsto im \cdot \left(-1 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {re}^{3} + re\right)}\right) \]

      4. distribute-lft-in 36.8%

         \[\leadsto im \cdot \color{blue}{\left(-1 \cdot \left(-0.16666666666666666 \cdot {re}^{3}\right) + -1 \cdot re\right)} \]

      5. neg-mul-1 36.8%

         \[\leadsto im \cdot \left(\color{blue}{\left(--0.16666666666666666 \cdot {re}^{3}\right)} + -1 \cdot re\right) \]

      6. neg-mul-1 36.8%

         \[\leadsto im \cdot \left(\left(--0.16666666666666666 \cdot {re}^{3}\right) + \color{blue}{\left(-re\right)}\right) \]

      7. unsub-neg 36.8%

         \[\leadsto im \cdot \color{blue}{\left(\left(--0.16666666666666666 \cdot {re}^{3}\right) - re\right)} \]

      8. *-commutative 36.8%

         \[\leadsto im \cdot \left(\left(-\color{blue}{{re}^{3} \cdot -0.16666666666666666}\right) - re\right) \]

      9. distribute-rgt-neg-in 36.8%

         \[\leadsto im \cdot \left(\color{blue}{{re}^{3} \cdot \left(--0.16666666666666666\right)} - re\right) \]

      10. metadata-eval 36.8%

          \[\leadsto im \cdot \left({re}^{3} \cdot \color{blue}{0.16666666666666666} - re\right) \]

   10. Simplified 36.8%

       \[\leadsto \color{blue}{im \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)} \]

   11. Taylor expanded in re around inf 36.4%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({re}^{3} \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.000105:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+105}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)\\ \end{array} \]

Alternative 7: 76.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{-5} \lor \neg \left(im \leq 8.2 \cdot 10^{+79}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5.6e-5) (not (<= im 8.2e+79)))
   (*
    0.5
    (+ (* re (* im (* im (* im -0.3333333333333333)))) (* re (* im -2.0))))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -5.6e-5) || !(im <= 8.2e+79)) {
		tmp = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
	} else {
		tmp = im * -sin(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 <= (-5.6d-5)) .or. (.not. (im <= 8.2d+79))) then
        tmp = 0.5d0 * ((re * (im * (im * (im * (-0.3333333333333333d0))))) + (re * (im * (-2.0d0))))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -5.6e-5) || !(im <= 8.2e+79)) {
		tmp = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -5.6e-5) or not (im <= 8.2e+79):
		tmp = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -5.6e-5) || !(im <= 8.2e+79))
		tmp = Float64(0.5 * Float64(Float64(re * Float64(im * Float64(im * Float64(im * -0.3333333333333333)))) + Float64(re * Float64(im * -2.0))));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -5.6e-5) || ~((im <= 8.2e+79)))
		tmp = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -5.6e-5], N[Not[LessEqual[im, 8.2e+79]], $MachinePrecision]], N[(0.5 * N[(N[(re * N[(im * N[(im * N[(im * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -5.59999999999999992e-5 or 8.2e79 < im

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 68.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]

   3. Taylor expanded in im around 0 55.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(re \cdot {im}^{3}\right) + -2 \cdot \left(re \cdot im\right)\right)} \]
   4. Step-by-step derivation

      1. cube-mult 55.4%

         \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \left(re \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) + -2 \cdot \left(re \cdot im\right)\right) \]

      2. associate-*r* 49.6%

         \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(im \cdot im\right)\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

      3. *-commutative 49.6%

         \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(re \cdot im\right)\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

      4. associate-*r* 49.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot im\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

      5. distribute-rgt-out 49.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]

   5. Simplified 49.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 49.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) + \left(re \cdot im\right) \cdot -2\right)} \]

      2. associate-*l* 55.4%

         \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right)} + \left(re \cdot im\right) \cdot -2\right) \]

      3. *-commutative 55.4%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot -0.3333333333333333\right)}\right) + \left(re \cdot im\right) \cdot -2\right) \]

      4. associate-*l* 55.4%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)}\right) + \left(re \cdot im\right) \cdot -2\right) \]

      5. associate-*l* 55.4%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + \color{blue}{re \cdot \left(im \cdot -2\right)}\right) \]

   7. Applied egg-rr 55.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)} \]

   if -5.59999999999999992e-5 < im < 8.2e79

   1. Initial program 39.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 87.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\sin re \cdot im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 87.4%

         \[\leadsto \color{blue}{-\sin re \cdot im} \]

      2. *-commutative 87.4%

         \[\leadsto -\color{blue}{im \cdot \sin re} \]

      3. distribute-rgt-neg-in 87.4%

         \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

   4. Simplified 87.4%

      \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{-5} \lor \neg \left(im \leq 8.2 \cdot 10^{+79}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 8: 53.5% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  0.5
  (+ (* re (* im (* im (* im -0.3333333333333333)))) (* re (* im -2.0)))))

C

double code(double re, double im) {
	return 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * ((re * (im * (im * (im * (-0.3333333333333333d0))))) + (re * (im * (-2.0d0))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
}

Python

def code(re, im):
	return 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)))

Julia

function code(re, im)
	return Float64(0.5 * Float64(Float64(re * Float64(im * Float64(im * Float64(im * -0.3333333333333333)))) + Float64(re * Float64(im * -2.0))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * ((re * (im * (im * (im * -0.3333333333333333)))) + (re * (im * -2.0)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(re * N[(im * N[(im * N[(im * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in re around 0 49.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]

3. Taylor expanded in im around 0 50.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(re \cdot {im}^{3}\right) + -2 \cdot \left(re \cdot im\right)\right)} \]
4. Step-by-step derivation

   1. cube-mult 50.9%

      \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \left(re \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) + -2 \cdot \left(re \cdot im\right)\right) \]

   2. associate-*r* 48.3%

      \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(im \cdot im\right)\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

   3. *-commutative 48.3%

      \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(re \cdot im\right)\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

   4. associate-*r* 48.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot im\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

   5. distribute-rgt-out 48.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]

5. Simplified 48.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in 48.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) + \left(re \cdot im\right) \cdot -2\right)} \]

   2. associate-*l* 50.9%

      \[\leadsto 0.5 \cdot \left(\color{blue}{re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right)} + \left(re \cdot im\right) \cdot -2\right) \]

   3. *-commutative 50.9%

      \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot -0.3333333333333333\right)}\right) + \left(re \cdot im\right) \cdot -2\right) \]

   4. associate-*l* 50.9%

      \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)}\right) + \left(re \cdot im\right) \cdot -2\right) \]

   5. associate-*l* 50.9%

      \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + \color{blue}{re \cdot \left(im \cdot -2\right)}\right) \]

7. Applied egg-rr 50.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right)} \]

8. Final simplification 50.9%

   \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) + re \cdot \left(im \cdot -2\right)\right) \]

Alternative 9: 50.3% accurate, 23.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(im \cdot re\right) \cdot \left(-2 + -0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (* (* im re) (+ -2.0 (* -0.3333333333333333 (* im im))))))

C

double code(double re, double im) {
	return 0.5 * ((im * re) * (-2.0 + (-0.3333333333333333 * (im * im))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * ((im * re) * ((-2.0d0) + ((-0.3333333333333333d0) * (im * im))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * ((im * re) * (-2.0 + (-0.3333333333333333 * (im * im))));
}

Python

def code(re, im):
	return 0.5 * ((im * re) * (-2.0 + (-0.3333333333333333 * (im * im))))

Julia

function code(re, im)
	return Float64(0.5 * Float64(Float64(im * re) * Float64(-2.0 + Float64(-0.3333333333333333 * Float64(im * im)))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * ((im * re) * (-2.0 + (-0.3333333333333333 * (im * im))));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(im * re), $MachinePrecision] * N[(-2.0 + N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in re around 0 49.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]

3. Taylor expanded in im around 0 50.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(re \cdot {im}^{3}\right) + -2 \cdot \left(re \cdot im\right)\right)} \]
4. Step-by-step derivation

   1. cube-mult 50.9%

      \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \left(re \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) + -2 \cdot \left(re \cdot im\right)\right) \]

   2. associate-*r* 48.3%

      \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(im \cdot im\right)\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

   3. *-commutative 48.3%

      \[\leadsto 0.5 \cdot \left(-0.3333333333333333 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(re \cdot im\right)\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

   4. associate-*r* 48.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot im\right)} + -2 \cdot \left(re \cdot im\right)\right) \]

   5. distribute-rgt-out 48.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]

5. Simplified 48.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(re \cdot im\right) \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]

6. Final simplification 48.3%

   \[\leadsto 0.5 \cdot \left(\left(im \cdot re\right) \cdot \left(-2 + -0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right) \]

Alternative 10: 33.2% accurate, 77.0× speedup

Math

\[\begin{array}{l} \\ \left(-im\right) \cdot re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (- im) re))

C

double code(double re, double im) {
	return -im * re;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -im * re
end function

Java

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

Python

def code(re, im):
	return -im * re

Julia

function code(re, im)
	return Float64(Float64(-im) * re)
end

MATLAB

function tmp = code(re, im)
	tmp = -im * re;
end

Wolfram

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

Derivation

1. Initial program 66.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in im around 0 51.1%

   \[\leadsto \color{blue}{-1 \cdot \left(\sin re \cdot im\right)} \]
3. Step-by-step derivation

   1. mul-1-neg 51.1%

      \[\leadsto \color{blue}{-\sin re \cdot im} \]

   2. *-commutative 51.1%

      \[\leadsto -\color{blue}{im \cdot \sin re} \]

   3. distribute-rgt-neg-in 51.1%

      \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

4. Simplified 51.1%

   \[\leadsto \color{blue}{im \cdot \left(-\sin re\right)} \]

5. Taylor expanded in re around 0 33.1%

   \[\leadsto \color{blue}{-1 \cdot \left(re \cdot im\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 33.1%

      \[\leadsto \color{blue}{-re \cdot im} \]

   2. distribute-rgt-neg-in 33.1%

      \[\leadsto \color{blue}{re \cdot \left(-im\right)} \]

7. Simplified 33.1%

   \[\leadsto \color{blue}{re \cdot \left(-im\right)} \]

8. Final simplification 33.1%

   \[\leadsto \left(-im\right) \cdot re \]

Alternative 11: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.5)

C

double code(double re, double im) {
	return -1.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.5d0
end function

Java

public static double code(double re, double im) {
	return -1.5;
}

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

function tmp = code(re, im)
	tmp = -1.5;
end

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 66.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in re around 0 49.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]

4. Applied egg-rr 3.0%

   \[\leadsto 0.5 \cdot \color{blue}{-3} \]

5. Final simplification 3.0%

   \[\leadsto -1.5 \]

Alternative 12: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -4.96145150637606 \cdot 10^{-8} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -4.96145150637606e-8)

C

double code(double re, double im) {
	return -4.96145150637606e-8;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -4.96145150637606d-8
end function

Java

public static double code(double re, double im) {
	return -4.96145150637606e-8;
}

Python

def code(re, im):
	return -4.96145150637606e-8

Julia

function code(re, im)
	return -4.96145150637606e-8
end

MATLAB

function tmp = code(re, im)
	tmp = -4.96145150637606e-8;
end

Wolfram

code[re_, im_] := -4.96145150637606e-8

Derivation

1. Initial program 66.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in re around 0 49.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]

4. Applied egg-rr 3.0%

   \[\leadsto 0.5 \cdot \color{blue}{-9.92290301275212 \cdot 10^{-8}} \]

5. Final simplification 3.0%

   \[\leadsto -4.96145150637606 \cdot 10^{-8} \]

Alternative 13: 14.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 66.2%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in re around 0 49.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]

4. Applied egg-rr 14.7%

   \[\leadsto 0.5 \cdot \color{blue}{0} \]

5. Final simplification 14.7%

   \[\leadsto 0 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023174 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))