math.sin on complex, imaginary part

Percentage Accurate: 54.5% → 99.3%

Time: 10.0s

Alternatives: 14

Speedup: 2.8×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-14)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos 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 <= -((double) INFINITY)) || !(t_0 <= 5e-14)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e-14)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(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 <= -math.inf) or not (t_0 <= 5e-14):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(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 <= Float64(-Inf)) || !(t_0 <= 5e-14))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(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 <= -Inf) || ~((t_0 <= 5e-14)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(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, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-14]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[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 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 5.0000000000000002e-14 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 5.0000000000000002e-14

   1. Initial program 7.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.25 \cdot 10^{+173} \lor \neg \left(im \leq -0.05 \lor \neg \left(im \leq 0.122\right) \land im \leq 5.5 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.25e+173)
         (not (or (<= im -0.05) (and (not (<= im 0.122)) (<= im 5.5e+102)))))
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* 0.5 (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.25e+173) || !((im <= -0.05) || (!(im <= 0.122) && (im <= 5.5e+102)))) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (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 ((im <= (-1.25d+173)) .or. (.not. (im <= (-0.05d0)) .or. (.not. (im <= 0.122d0)) .and. (im <= 5.5d+102))) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = 0.5d0 * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.25e+173) || !((im <= -0.05) || (!(im <= 0.122) && (im <= 5.5e+102)))) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.25e+173) or not ((im <= -0.05) or (not (im <= 0.122) and (im <= 5.5e+102))):
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.25e+173) || !((im <= -0.05) || (!(im <= 0.122) && (im <= 5.5e+102))))
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.25e+173) || ~(((im <= -0.05) || (~((im <= 0.122)) && (im <= 5.5e+102)))))
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = 0.5 * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.25e+173], N[Not[Or[LessEqual[im, -0.05], And[N[Not[LessEqual[im, 0.122]], $MachinePrecision], LessEqual[im, 5.5e+102]]]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.25000000000000009e173 or -0.050000000000000003 < im < 0.122 or 5.49999999999999981e102 < im

   1. Initial program 37.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

   if -1.25000000000000009e173 < im < -0.050000000000000003 or 0.122 < im < 5.49999999999999981e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 81.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.25 \cdot 10^{+173} \lor \neg \left(im \leq -0.05 \lor \neg \left(im \leq 0.122\right) \land im \leq 5.5 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 3: 95.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -5.7 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -220:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 0.1 \lor \neg \left(im \leq 5.5 \cdot 10^{+102}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im)))
        (t_1 (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -5.7e+102)
     t_1
     (if (<= im -220.0)
       (* t_0 (+ 0.5 (* re (* re -0.25))))
       (if (or (<= im 0.1) (not (<= im 5.5e+102))) t_1 (* 0.5 t_0))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -5.7e+102) {
		tmp = t_1;
	} else if (im <= -220.0) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else if ((im <= 0.1) || !(im <= 5.5e+102)) {
		tmp = t_1;
	} else {
		tmp = 0.5 * 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) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-5.7d+102)) then
        tmp = t_1
    else if (im <= (-220.0d0)) then
        tmp = t_0 * (0.5d0 + (re * (re * (-0.25d0))))
    else if ((im <= 0.1d0) .or. (.not. (im <= 5.5d+102))) then
        tmp = t_1
    else
        tmp = 0.5d0 * t_0
    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 t_1 = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -5.7e+102) {
		tmp = t_1;
	} else if (im <= -220.0) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else if ((im <= 0.1) || !(im <= 5.5e+102)) {
		tmp = t_1;
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -5.7e+102:
		tmp = t_1
	elif im <= -220.0:
		tmp = t_0 * (0.5 + (re * (re * -0.25)))
	elif (im <= 0.1) or not (im <= 5.5e+102):
		tmp = t_1
	else:
		tmp = 0.5 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -5.7e+102)
		tmp = t_1;
	elseif (im <= -220.0)
		tmp = Float64(t_0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	elseif ((im <= 0.1) || !(im <= 5.5e+102))
		tmp = t_1;
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -5.7e+102)
		tmp = t_1;
	elseif (im <= -220.0)
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	elseif ((im <= 0.1) || ~((im <= 5.5e+102)))
		tmp = t_1;
	else
		tmp = 0.5 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.7e+102], t$95$1, If[LessEqual[im, -220.0], N[(t$95$0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 0.1], N[Not[LessEqual[im, 5.5e+102]], $MachinePrecision]], t$95$1, N[(0.5 * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.6999999999999999e102 or -220 < im < 0.10000000000000001 or 5.49999999999999981e102 < im

   1. Initial program 42.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 42.5%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in im around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*l* 99.4%

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

      5. distribute-lft-out-- 99.4%

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

   6. Simplified 99.4%

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

   if -5.6999999999999999e102 < im < -220

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 77.3%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 77.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 77.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 77.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 77.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 77.3%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   if 0.10000000000000001 < im < 5.49999999999999981e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 84.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.7 \cdot 10^{+102}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -220:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 0.1 \lor \neg \left(im \leq 5.5 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4: 87.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.3 \cdot 10^{+210}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -0.023 \lor \neg \left(im \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.3e+210)
   (* (+ (* -0.5 (* re re)) 1.0) (- (* (pow im 3.0) -0.16666666666666666) im))
   (if (or (<= im -0.023) (not (<= im 9e-5)))
     (* 0.5 (- (exp (- im)) (exp im)))
     (* im (- (cos re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.3e+210) {
		tmp = ((-0.5 * (re * re)) + 1.0) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if ((im <= -0.023) || !(im <= 9e-5)) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = im * -cos(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 <= (-3.3d+210)) then
        tmp = (((-0.5d0) * (re * re)) + 1.0d0) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if ((im <= (-0.023d0)) .or. (.not. (im <= 9d-5))) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3.3e+210) {
		tmp = ((-0.5 * (re * re)) + 1.0) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if ((im <= -0.023) || !(im <= 9e-5)) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3.3e+210:
		tmp = ((-0.5 * (re * re)) + 1.0) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif (im <= -0.023) or not (im <= 9e-5):
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3.3e+210)
		tmp = Float64(Float64(Float64(-0.5 * Float64(re * re)) + 1.0) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif ((im <= -0.023) || !(im <= 9e-5))
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3.3e+210)
		tmp = ((-0.5 * (re * re)) + 1.0) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif ((im <= -0.023) || ~((im <= 9e-5)))
		tmp = 0.5 * (exp(-im) - exp(im));
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.3e+210], N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -0.023], N[Not[LessEqual[im, 9e-5]], $MachinePrecision]], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.29999999999999995e210

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. associate--l+ 0.0%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      3. distribute-lft1-in 92.9%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      4. unpow2 92.9%

         \[\leadsto \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

   9. Simplified 92.9%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

   if -3.29999999999999995e210 < im < -0.023 or 9.00000000000000057e-5 < im

   1. Initial program 99.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 78.2%

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

   if -0.023 < im < 9.00000000000000057e-5

   1. Initial program 6.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in im around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-lft-neg-in 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.3 \cdot 10^{+210}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -0.023 \lor \neg \left(im \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 5: 77.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := \left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left(t_0 - im\right)\\ \mathbf{if}\;im \leq -5 \cdot 10^{+214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -2.15 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{+17} \lor \neg \left(im \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1 (* (+ (* -0.5 (* re re)) 1.0) (- t_0 im))))
   (if (<= im -5e+214)
     t_1
     (if (<= im -2.15e+96)
       t_0
       (if (or (<= im -6.4e+17) (not (<= im 9e-5)))
         t_1
         (* im (- (cos re))))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = ((-0.5 * (re * re)) + 1.0) * (t_0 - im);
	double tmp;
	if (im <= -5e+214) {
		tmp = t_1;
	} else if (im <= -2.15e+96) {
		tmp = t_0;
	} else if ((im <= -6.4e+17) || !(im <= 9e-5)) {
		tmp = t_1;
	} else {
		tmp = im * -cos(re);
	}
	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 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = (((-0.5d0) * (re * re)) + 1.0d0) * (t_0 - im)
    if (im <= (-5d+214)) then
        tmp = t_1
    else if (im <= (-2.15d+96)) then
        tmp = t_0
    else if ((im <= (-6.4d+17)) .or. (.not. (im <= 9d-5))) then
        tmp = t_1
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double t_1 = ((-0.5 * (re * re)) + 1.0) * (t_0 - im);
	double tmp;
	if (im <= -5e+214) {
		tmp = t_1;
	} else if (im <= -2.15e+96) {
		tmp = t_0;
	} else if ((im <= -6.4e+17) || !(im <= 9e-5)) {
		tmp = t_1;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = ((-0.5 * (re * re)) + 1.0) * (t_0 - im)
	tmp = 0
	if im <= -5e+214:
		tmp = t_1
	elif im <= -2.15e+96:
		tmp = t_0
	elif (im <= -6.4e+17) or not (im <= 9e-5):
		tmp = t_1
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(Float64(Float64(-0.5 * Float64(re * re)) + 1.0) * Float64(t_0 - im))
	tmp = 0.0
	if (im <= -5e+214)
		tmp = t_1;
	elseif (im <= -2.15e+96)
		tmp = t_0;
	elseif ((im <= -6.4e+17) || !(im <= 9e-5))
		tmp = t_1;
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = ((-0.5 * (re * re)) + 1.0) * (t_0 - im);
	tmp = 0.0;
	if (im <= -5e+214)
		tmp = t_1;
	elseif (im <= -2.15e+96)
		tmp = t_0;
	elseif ((im <= -6.4e+17) || ~((im <= 9e-5)))
		tmp = t_1;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$0 - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5e+214], t$95$1, If[LessEqual[im, -2.15e+96], t$95$0, If[Or[LessEqual[im, -6.4e+17], N[Not[LessEqual[im, 9e-5]], $MachinePrecision]], t$95$1, N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.99999999999999953e214 or -2.15000000000000001e96 < im < -6.4e17 or 9.00000000000000057e-5 < im

   1. Initial program 99.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in im around 0 65.0%

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

      1. mul-1-neg 65.0%

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

      2. unsub-neg 65.0%

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

      3. *-commutative 65.0%

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

      4. associate-*l* 65.0%

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

      5. distribute-lft-out-- 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in re around 0 11.3%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. associate--l+ 11.3%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      2. associate-*r* 11.3%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      3. distribute-lft1-in 63.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      4. unpow2 63.0%

         \[\leadsto \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

   9. Simplified 63.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

   if -4.99999999999999953e214 < im < -2.15000000000000001e96

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 88.1%

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

      1. mul-1-neg 88.1%

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

      2. unsub-neg 88.1%

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

      3. *-commutative 88.1%

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

      4. associate-*l* 88.1%

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

      5. distribute-lft-out-- 88.1%

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

   6. Simplified 88.1%

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

   7. Taylor expanded in re around 0 81.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   8. Taylor expanded in im around inf 81.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]

   if -6.4e17 < im < 9.00000000000000057e-5

   1. Initial program 9.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 9.9%

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

   3. Simplified 9.9%

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

   4. Taylor expanded in im around 0 96.0%

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

      1. mul-1-neg 96.0%

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

      2. *-commutative 96.0%

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

      3. distribute-lft-neg-in 96.0%

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

   6. Simplified 96.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+214}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -2.15 \cdot 10^{+96}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{+17} \lor \neg \left(im \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 6: 54.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := \left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{if}\;im \leq -2.4 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{+15}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1 (- (* (* re re) (* im 0.5)) im)))
   (if (<= im -2.4e+94)
     t_0
     (if (<= im -6.4e+17)
       t_1
       (if (<= im 7.6e+15) (- im) (if (<= im 1.7e+106) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = ((re * re) * (im * 0.5)) - im;
	double tmp;
	if (im <= -2.4e+94) {
		tmp = t_0;
	} else if (im <= -6.4e+17) {
		tmp = t_1;
	} else if (im <= 7.6e+15) {
		tmp = -im;
	} else if (im <= 1.7e+106) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = ((re * re) * (im * 0.5d0)) - im
    if (im <= (-2.4d+94)) then
        tmp = t_0
    else if (im <= (-6.4d+17)) then
        tmp = t_1
    else if (im <= 7.6d+15) then
        tmp = -im
    else if (im <= 1.7d+106) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double t_1 = ((re * re) * (im * 0.5)) - im;
	double tmp;
	if (im <= -2.4e+94) {
		tmp = t_0;
	} else if (im <= -6.4e+17) {
		tmp = t_1;
	} else if (im <= 7.6e+15) {
		tmp = -im;
	} else if (im <= 1.7e+106) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = ((re * re) * (im * 0.5)) - im
	tmp = 0
	if im <= -2.4e+94:
		tmp = t_0
	elif im <= -6.4e+17:
		tmp = t_1
	elif im <= 7.6e+15:
		tmp = -im
	elif im <= 1.7e+106:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im)
	tmp = 0.0
	if (im <= -2.4e+94)
		tmp = t_0;
	elseif (im <= -6.4e+17)
		tmp = t_1;
	elseif (im <= 7.6e+15)
		tmp = Float64(-im);
	elseif (im <= 1.7e+106)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = ((re * re) * (im * 0.5)) - im;
	tmp = 0.0;
	if (im <= -2.4e+94)
		tmp = t_0;
	elseif (im <= -6.4e+17)
		tmp = t_1;
	elseif (im <= 7.6e+15)
		tmp = -im;
	elseif (im <= 1.7e+106)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -2.4e+94], t$95$0, If[LessEqual[im, -6.4e+17], t$95$1, If[LessEqual[im, 7.6e+15], (-im), If[LessEqual[im, 1.7e+106], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.39999999999999983e94 or 1.69999999999999997e106 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 95.8%

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

      1. mul-1-neg 95.8%

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

      2. unsub-neg 95.8%

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

      3. *-commutative 95.8%

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

      4. associate-*l* 95.8%

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

      5. distribute-lft-out-- 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in re around 0 73.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   8. Taylor expanded in im around inf 73.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]

   if -2.39999999999999983e94 < im < -6.4e17 or 7.6e15 < im < 1.69999999999999997e106

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.7%

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

      1. mul-1-neg 3.7%

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

      2. *-commutative 3.7%

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

      3. distribute-lft-neg-in 3.7%

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

   6. Simplified 3.7%

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

   7. Taylor expanded in re around 0 29.0%

      \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 29.0%

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

      2. +-commutative 29.0%

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

      3. unsub-neg 29.0%

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

      4. *-commutative 29.0%

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

      5. associate-*l* 29.0%

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

      6. unpow2 29.0%

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

   9. Simplified 29.0%

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

   if -6.4e17 < im < 7.6e15

   1. Initial program 13.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 13.0%

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

   3. Simplified 13.0%

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

   4. Taylor expanded in im around 0 93.7%

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

      1. mul-1-neg 93.7%

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

      2. unsub-neg 93.7%

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

      3. *-commutative 93.7%

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

      4. associate-*l* 93.7%

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

      5. distribute-lft-out-- 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in re around 0 46.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   8. Taylor expanded in im around 0 46.0%

      \[\leadsto \color{blue}{-1 \cdot im} \]
   9. Step-by-step derivation

      1. neg-mul-1 46.0%

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

   10. Simplified 46.0%

       \[\leadsto \color{blue}{-im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.4 \cdot 10^{+94}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{+15}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+106}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 7: 75.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -1.9 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -1.9e+96)
     t_0
     (if (<= im -6.4e+17)
       (- (* (* re re) (* im 0.5)) im)
       (if (<= im 4.7e-5) (* im (- (cos re))) (- t_0 im))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -1.9e+96) {
		tmp = t_0;
	} else if (im <= -6.4e+17) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 4.7e-5) {
		tmp = im * -cos(re);
	} else {
		tmp = t_0 - 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 = (im ** 3.0d0) * (-0.16666666666666666d0)
    if (im <= (-1.9d+96)) then
        tmp = t_0
    else if (im <= (-6.4d+17)) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else if (im <= 4.7d-5) then
        tmp = im * -cos(re)
    else
        tmp = t_0 - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -1.9e+96) {
		tmp = t_0;
	} else if (im <= -6.4e+17) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 4.7e-5) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -1.9e+96:
		tmp = t_0
	elif im <= -6.4e+17:
		tmp = ((re * re) * (im * 0.5)) - im
	elif im <= 4.7e-5:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0 - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -1.9e+96)
		tmp = t_0;
	elseif (im <= -6.4e+17)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	elseif (im <= 4.7e-5)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -1.9e+96)
		tmp = t_0;
	elseif (im <= -6.4e+17)
		tmp = ((re * re) * (im * 0.5)) - im;
	elseif (im <= 4.7e-5)
		tmp = im * -cos(re);
	else
		tmp = t_0 - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -1.9e+96], t$95$0, If[LessEqual[im, -6.4e+17], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 4.7e-5], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(t$95$0 - im), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.9000000000000001e96

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 91.9%

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

      1. mul-1-neg 91.9%

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

      2. unsub-neg 91.9%

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

      3. *-commutative 91.9%

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

      4. associate-*l* 91.9%

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

      5. distribute-lft-out-- 91.9%

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

   6. Simplified 91.9%

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

   7. Taylor expanded in re around 0 71.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   8. Taylor expanded in im around inf 71.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]

   if -1.9000000000000001e96 < im < -6.4e17

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.6%

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

      1. mul-1-neg 3.6%

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

      2. *-commutative 3.6%

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

      3. distribute-lft-neg-in 3.6%

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

   6. Simplified 3.6%

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

   7. Taylor expanded in re around 0 37.3%

      \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 37.3%

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

      2. +-commutative 37.3%

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

      3. unsub-neg 37.3%

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

      4. *-commutative 37.3%

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

      5. associate-*l* 37.3%

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

      6. unpow2 37.3%

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

   9. Simplified 37.3%

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

   if -6.4e17 < im < 4.69999999999999972e-5

   1. Initial program 9.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 9.9%

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

   3. Simplified 9.9%

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

   4. Taylor expanded in im around 0 96.0%

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

      1. mul-1-neg 96.0%

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

      2. *-commutative 96.0%

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

      3. distribute-lft-neg-in 96.0%

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

   6. Simplified 96.0%

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

   if 4.69999999999999972e-5 < im

   1. Initial program 99.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in im around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. unsub-neg 70.6%

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

      3. *-commutative 70.6%

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

      4. associate-*l* 70.6%

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

      5. distribute-lft-out-- 70.6%

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

   6. Simplified 70.6%

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

   7. Taylor expanded in re around 0 52.4%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.9 \cdot 10^{+96}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 8: 75.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+61}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -2.1e+96)
     t_0
     (if (<= im -6.4e+17)
       (- (* (* re re) (* im 0.5)) im)
       (if (<= im 1.2e+61) (* im (- (cos re))) t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -2.1e+96) {
		tmp = t_0;
	} else if (im <= -6.4e+17) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 1.2e+61) {
		tmp = im * -cos(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 = (im ** 3.0d0) * (-0.16666666666666666d0)
    if (im <= (-2.1d+96)) then
        tmp = t_0
    else if (im <= (-6.4d+17)) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else if (im <= 1.2d+61) then
        tmp = im * -cos(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -2.1e+96) {
		tmp = t_0;
	} else if (im <= -6.4e+17) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 1.2e+61) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -2.1e+96:
		tmp = t_0
	elif im <= -6.4e+17:
		tmp = ((re * re) * (im * 0.5)) - im
	elif im <= 1.2e+61:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -2.1e+96)
		tmp = t_0;
	elseif (im <= -6.4e+17)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	elseif (im <= 1.2e+61)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -2.1e+96)
		tmp = t_0;
	elseif (im <= -6.4e+17)
		tmp = ((re * re) * (im * 0.5)) - im;
	elseif (im <= 1.2e+61)
		tmp = im * -cos(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -2.1e+96], t$95$0, If[LessEqual[im, -6.4e+17], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 1.2e+61], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.1000000000000001e96 or 1.1999999999999999e61 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 87.4%

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

      1. mul-1-neg 87.4%

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

      2. unsub-neg 87.4%

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

      3. *-commutative 87.4%

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

      4. associate-*l* 87.4%

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

      5. distribute-lft-out-- 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in re around 0 66.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   8. Taylor expanded in im around inf 66.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]

   if -2.1000000000000001e96 < im < -6.4e17

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.6%

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

      1. mul-1-neg 3.6%

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

      2. *-commutative 3.6%

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

      3. distribute-lft-neg-in 3.6%

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

   6. Simplified 3.6%

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

   7. Taylor expanded in re around 0 37.3%

      \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 37.3%

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

      2. +-commutative 37.3%

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

      3. unsub-neg 37.3%

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

      4. *-commutative 37.3%

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

      5. associate-*l* 37.3%

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

      6. unpow2 37.3%

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

   9. Simplified 37.3%

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

   if -6.4e17 < im < 1.1999999999999999e61

   1. Initial program 16.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 16.5%

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

   3. Simplified 16.5%

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

   4. Taylor expanded in im around 0 89.6%

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

      1. mul-1-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. distribute-lft-neg-in 89.6%

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

   6. Simplified 89.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+96}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{+17}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+61}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 9: 36.7% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot 0.75\\ \mathbf{if}\;re \leq 4.2 \cdot 10^{+75}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{+126}:\\ \;\;\;\;\frac{2.25 - t_0 \cdot t_0}{-1.5 - t_0}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) 0.75)))
   (if (<= re 4.2e+75)
     (- (* (* re re) (* im 0.5)) im)
     (if (<= re 1.75e+126)
       (/ (- 2.25 (* t_0 t_0)) (- -1.5 t_0))
       (- (* re (* re (* im 0.5))) im)))))

C

double code(double re, double im) {
	double t_0 = (re * re) * 0.75;
	double tmp;
	if (re <= 4.2e+75) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (re <= 1.75e+126) {
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	} else {
		tmp = (re * (re * (im * 0.5))) - 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 = (re * re) * 0.75d0
    if (re <= 4.2d+75) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else if (re <= 1.75d+126) then
        tmp = (2.25d0 - (t_0 * t_0)) / ((-1.5d0) - t_0)
    else
        tmp = (re * (re * (im * 0.5d0))) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) * 0.75;
	double tmp;
	if (re <= 4.2e+75) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (re <= 1.75e+126) {
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	} else {
		tmp = (re * (re * (im * 0.5))) - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) * 0.75
	tmp = 0
	if re <= 4.2e+75:
		tmp = ((re * re) * (im * 0.5)) - im
	elif re <= 1.75e+126:
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0)
	else:
		tmp = (re * (re * (im * 0.5))) - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * 0.75)
	tmp = 0.0
	if (re <= 4.2e+75)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	elseif (re <= 1.75e+126)
		tmp = Float64(Float64(2.25 - Float64(t_0 * t_0)) / Float64(-1.5 - t_0));
	else
		tmp = Float64(Float64(re * Float64(re * Float64(im * 0.5))) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) * 0.75;
	tmp = 0.0;
	if (re <= 4.2e+75)
		tmp = ((re * re) * (im * 0.5)) - im;
	elseif (re <= 1.75e+126)
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	else
		tmp = (re * (re * (im * 0.5))) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]}, If[LessEqual[re, 4.2e+75], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[re, 1.75e+126], N[(N[(2.25 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.5 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 4.19999999999999997e75

   1. Initial program 51.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in im around 0 54.4%

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

      1. mul-1-neg 54.4%

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

      2. *-commutative 54.4%

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

      3. distribute-lft-neg-in 54.4%

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

   6. Simplified 54.4%

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

   7. Taylor expanded in re around 0 37.0%

      \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 37.0%

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

      2. +-commutative 37.0%

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

      3. unsub-neg 37.0%

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

      4. *-commutative 37.0%

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

      5. associate-*l* 37.0%

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

      6. unpow2 37.0%

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

   9. Simplified 37.0%

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

   if 4.19999999999999997e75 < re < 1.7500000000000001e126

   1. Initial program 64.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in re around 0 1.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 1.8%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 1.8%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 26.8%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 26.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 26.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 26.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 26.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 26.8%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 3.7%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in 3.7%

         \[\leadsto \color{blue}{-3 \cdot 0.5 + -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      2. flip-+ 38.3%

         \[\leadsto \color{blue}{\frac{\left(-3 \cdot 0.5\right) \cdot \left(-3 \cdot 0.5\right) - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)}} \]

      3. metadata-eval 38.3%

         \[\leadsto \frac{\color{blue}{-1.5} \cdot \left(-3 \cdot 0.5\right) - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      4. metadata-eval 38.3%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{-1.5} - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      5. metadata-eval 38.3%

         \[\leadsto \frac{\color{blue}{2.25} - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      6. *-commutative 38.3%

         \[\leadsto \frac{2.25 - \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)} \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      7. *-commutative 38.3%

         \[\leadsto \frac{2.25 - \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right) \cdot \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      8. associate-*r* 38.3%

         \[\leadsto \frac{2.25 - \left(\color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3\right) \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      9. associate-*l* 38.3%

         \[\leadsto \frac{2.25 - \color{blue}{\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)\right)} \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      10. metadata-eval 38.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \color{blue}{0.75}\right) \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      11. associate-*r* 38.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      12. associate-*l* 38.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)\right)}}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      13. metadata-eval 38.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{0.75}\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      14. metadata-eval 38.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{\color{blue}{-1.5} - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      15. *-commutative 38.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3}} \]

      16. associate-*r* 38.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3} \]

      17. associate-*l* 38.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)}} \]

      18. metadata-eval 38.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot \color{blue}{0.75}} \]

   10. Applied egg-rr 38.3%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}} \]

   if 1.7500000000000001e126 < re

   1. Initial program 50.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in im around 0 54.8%

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

      1. mul-1-neg 54.8%

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

      2. *-commutative 54.8%

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

      3. distribute-lft-neg-in 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in re around 0 21.6%

      \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 21.6%

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

      2. +-commutative 21.6%

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

      3. unsub-neg 21.6%

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

      4. *-commutative 21.6%

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

      5. associate-*l* 21.6%

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

      6. unpow2 21.6%

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

   9. Simplified 21.6%

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

       1. sub-neg 21.6%

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

       2. associate-*l* 21.7%

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

   11. Applied egg-rr 21.7%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) + \left(-im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.2 \cdot 10^{+75}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{+126}:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \end{array} \]

Alternative 10: 36.6% accurate, 34.3× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (- (* (* re re) (* im 0.5)) im))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 51.7%

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

3. Simplified 51.7%

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

4. Taylor expanded in im around 0 54.0%

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

   1. mul-1-neg 54.0%

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

   2. *-commutative 54.0%

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

   3. distribute-lft-neg-in 54.0%

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

6. Simplified 54.0%

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

7. Taylor expanded in re around 0 34.3%

   \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
8. Step-by-step derivation

   1. neg-mul-1 34.3%

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

   2. +-commutative 34.3%

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

   3. unsub-neg 34.3%

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

   4. *-commutative 34.3%

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

   5. associate-*l* 34.3%

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

   6. unpow2 34.3%

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

9. Simplified 34.3%

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

10. Final simplification 34.3%

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

Alternative 11: 36.6% accurate, 34.3× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (- (* re (* re (* im 0.5))) im))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 51.7%

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

3. Simplified 51.7%

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

4. Taylor expanded in im around 0 54.0%

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

   1. mul-1-neg 54.0%

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

   2. *-commutative 54.0%

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

   3. distribute-lft-neg-in 54.0%

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

6. Simplified 54.0%

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

7. Taylor expanded in re around 0 34.3%

   \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
8. Step-by-step derivation

   1. neg-mul-1 34.3%

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

   2. +-commutative 34.3%

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

   3. unsub-neg 34.3%

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

   4. *-commutative 34.3%

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

   5. associate-*l* 34.3%

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

   6. unpow2 34.3%

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

9. Simplified 34.3%

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

    1. sub-neg 34.3%

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

    2. associate-*l* 34.3%

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

11. Applied egg-rr 34.3%

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

12. Final simplification 34.3%

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

Alternative 12: 31.2% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.65 \cdot 10^{+182}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.65e+182) (- im) (* (* re re) 0.75)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.65e+182) {
		tmp = -im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.65d+182) then
        tmp = -im
    else
        tmp = (re * re) * 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.65e+182) {
		tmp = -im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.65e+182:
		tmp = -im
	else:
		tmp = (re * re) * 0.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.65e+182)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(re * re) * 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.65e+182)
		tmp = -im;
	else
		tmp = (re * re) * 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.65e+182], (-im), N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.65e182

   1. Initial program 51.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in im around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. unsub-neg 85.1%

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

      3. *-commutative 85.1%

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

      4. associate-*l* 85.1%

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

      5. distribute-lft-out-- 85.1%

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

   6. Simplified 85.1%

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

   7. Taylor expanded in re around 0 53.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   8. Taylor expanded in im around 0 29.8%

      \[\leadsto \color{blue}{-1 \cdot im} \]
   9. Step-by-step derivation

      1. neg-mul-1 29.8%

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

   10. Simplified 29.8%

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

   if 1.65e182 < re

   1. Initial program 54.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 22.2%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 22.2%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 22.2%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 22.2%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 22.2%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 22.2%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in 30.6%

         \[\leadsto \color{blue}{-3 \cdot 0.5 + -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      2. flip-+ 0.0%

         \[\leadsto \color{blue}{\frac{\left(-3 \cdot 0.5\right) \cdot \left(-3 \cdot 0.5\right) - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)}} \]

      3. metadata-eval 0.0%

         \[\leadsto \frac{\color{blue}{-1.5} \cdot \left(-3 \cdot 0.5\right) - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      4. metadata-eval 0.0%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{-1.5} - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      5. metadata-eval 0.0%

         \[\leadsto \frac{\color{blue}{2.25} - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      6. *-commutative 0.0%

         \[\leadsto \frac{2.25 - \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)} \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      7. *-commutative 0.0%

         \[\leadsto \frac{2.25 - \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right) \cdot \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      8. associate-*r* 0.0%

         \[\leadsto \frac{2.25 - \left(\color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3\right) \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      9. associate-*l* 0.0%

         \[\leadsto \frac{2.25 - \color{blue}{\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)\right)} \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      10. metadata-eval 0.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \color{blue}{0.75}\right) \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      11. associate-*r* 0.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      12. associate-*l* 0.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)\right)}}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      13. metadata-eval 0.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{0.75}\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      14. metadata-eval 0.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{\color{blue}{-1.5} - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      15. *-commutative 0.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3}} \]

      16. associate-*r* 0.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3} \]

      17. associate-*l* 0.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)}} \]

      18. metadata-eval 0.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot \color{blue}{0.75}} \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}} \]

   11. Taylor expanded in re around inf 30.6%

       \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 30.6%

          \[\leadsto 0.75 \cdot \color{blue}{\left(re \cdot re\right)} \]

   13. Simplified 30.6%

       \[\leadsto \color{blue}{0.75 \cdot \left(re \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.65 \cdot 10^{+182}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \]

Alternative 13: 29.4% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 51.7%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 51.7%

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

3. Simplified 51.7%

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

4. Taylor expanded in im around 0 84.4%

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

   1. mul-1-neg 84.4%

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

   2. unsub-neg 84.4%

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

   3. *-commutative 84.4%

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

   4. associate-*l* 84.4%

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

   5. distribute-lft-out-- 84.4%

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

6. Simplified 84.4%

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

7. Taylor expanded in re around 0 50.2%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

8. Taylor expanded in im around 0 27.3%

   \[\leadsto \color{blue}{-1 \cdot im} \]
9. Step-by-step derivation

   1. neg-mul-1 27.3%

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

10. Simplified 27.3%

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

11. Final simplification 27.3%

    \[\leadsto -im \]

Alternative 14: 2.9% accurate, 309.0× 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 51.7%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 51.7%

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

3. Simplified 51.7%

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

4. Taylor expanded in re around 0 2.4%

   \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
5. Step-by-step derivation

   1. *-commutative 2.4%

      \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

   2. associate-*r* 2.4%

      \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

   3. distribute-rgt-out 38.7%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

   4. +-commutative 38.7%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

   5. *-commutative 38.7%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

   6. unpow2 38.7%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

   7. associate-*l* 38.7%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

6. Simplified 38.7%

   \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

8. Applied egg-rr 8.2%

   \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

9. Taylor expanded in re around 0 2.9%

   \[\leadsto \color{blue}{-1.5} \]

10. Final simplification 2.9%

    \[\leadsto -1.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - 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 = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - 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.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(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 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[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[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023201 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))