math.sin on complex, imaginary part

Percentage Accurate: 53.9% → 99.5%

Time: 9.0s

Alternatives: 15

Speedup: 2.9×

• 49.2% 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: 53.9% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e-6)))
     (* (* 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 <= 1e-6)) {
		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 <= 1e-6)) {
		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 <= 1e-6):
		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 <= 1e-6))
		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 <= 1e-6)))
		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, 1e-6]], $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 9.99999999999999955e-7 < (-.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)) < 9.99999999999999955e-7

   1. Initial program 8.1%

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

      1. sub0-neg 8.1%

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

   3. Simplified 8.1%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{+91} \lor \neg \left(im \leq -0.3 \lor \neg \left(im \leq 14000\right) \land im \leq 1.08 \cdot 10^{+98}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -8e+91)
         (not (or (<= im -0.3) (and (not (<= im 14000.0)) (<= im 1.08e+98)))))
   (* (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 <= -8e+91) || !((im <= -0.3) || (!(im <= 14000.0) && (im <= 1.08e+98)))) {
		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 <= (-8d+91)) .or. (.not. (im <= (-0.3d0)) .or. (.not. (im <= 14000.0d0)) .and. (im <= 1.08d+98))) 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 <= -8e+91) || !((im <= -0.3) || (!(im <= 14000.0) && (im <= 1.08e+98)))) {
		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 <= -8e+91) or not ((im <= -0.3) or (not (im <= 14000.0) and (im <= 1.08e+98))):
		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 <= -8e+91) || !((im <= -0.3) || (!(im <= 14000.0) && (im <= 1.08e+98))))
		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 <= -8e+91) || ~(((im <= -0.3) || (~((im <= 14000.0)) && (im <= 1.08e+98)))))
		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, -8e+91], N[Not[Or[LessEqual[im, -0.3], And[N[Not[LessEqual[im, 14000.0]], $MachinePrecision], LessEqual[im, 1.08e+98]]]], $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 < -8.00000000000000064e91 or -0.299999999999999989 < im < 14000 or 1.07999999999999997e98 < im

   1. Initial program 44.7%

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

      1. sub0-neg 44.7%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in im around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-*l* 98.2%

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

      5. distribute-lft-out-- 98.2%

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

   6. Simplified 98.2%

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

   if -8.00000000000000064e91 < im < -0.299999999999999989 or 14000 < im < 1.07999999999999997e98

   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 88.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{+91} \lor \neg \left(im \leq -0.3 \lor \neg \left(im \leq 14000\right) \land im \leq 1.08 \cdot 10^{+98}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 3: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{if}\;im \leq -6.1 \cdot 10^{+168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 14000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0))))
   (if (<= im -6.1e+168)
     t_1
     (if (<= im -0.0005)
       t_0
       (if (<= im 14000.0)
         (* (cos re) (- im))
         (if (<= im 1.15e+140) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -6.1e+168) {
		tmp = t_1;
	} else if (im <= -0.0005) {
		tmp = t_0;
	} else if (im <= 14000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1.15e+140) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (cos(re) * (9.0d0 - (im * im))) / (im + (-3.0d0))
    if (im <= (-6.1d+168)) then
        tmp = t_1
    else if (im <= (-0.0005d0)) then
        tmp = t_0
    else if (im <= 14000.0d0) then
        tmp = cos(re) * -im
    else if (im <= 1.15d+140) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = (Math.cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -6.1e+168) {
		tmp = t_1;
	} else if (im <= -0.0005) {
		tmp = t_0;
	} else if (im <= 14000.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.15e+140) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = (math.cos(re) * (9.0 - (im * im))) / (im + -3.0)
	tmp = 0
	if im <= -6.1e+168:
		tmp = t_1
	elif im <= -0.0005:
		tmp = t_0
	elif im <= 14000.0:
		tmp = math.cos(re) * -im
	elif im <= 1.15e+140:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	tmp = 0.0
	if (im <= -6.1e+168)
		tmp = t_1;
	elseif (im <= -0.0005)
		tmp = t_0;
	elseif (im <= 14000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.15e+140)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	tmp = 0.0;
	if (im <= -6.1e+168)
		tmp = t_1;
	elseif (im <= -0.0005)
		tmp = t_0;
	elseif (im <= 14000.0)
		tmp = cos(re) * -im;
	elseif (im <= 1.15e+140)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.1e+168], t$95$1, If[LessEqual[im, -0.0005], t$95$0, If[LessEqual[im, 14000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.15e+140], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.1000000000000002e168 or 1.14999999999999995e140 < 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 0 100.0%

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

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

   8. Applied egg-rr 7.4%

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

      1. *-commutative 7.4%

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

      2. flip-- 98.5%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 98.5%

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

      4. metadata-eval 98.5%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 98.5%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 98.5%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -6.1000000000000002e168 < im < -5.0000000000000001e-4 or 14000 < im < 1.14999999999999995e140

   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 83.1%

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

   if -5.0000000000000001e-4 < im < 14000

   1. Initial program 8.8%

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

      1. sub0-neg 8.8%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in im around 0 99.0%

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

      1. mul-1-neg 99.0%

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

      2. *-commutative 99.0%

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

      3. distribute-lft-neg-in 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.1 \cdot 10^{+168}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -0.0005:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 14000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+140}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 4: 84.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ t_1 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0005:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (- (* (pow im 3.0) -0.16666666666666666) im)
          (+ (* -0.5 (* re re)) 1.0)))
        (t_1 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0))))
   (if (<= im -2.1e+153)
     t_1
     (if (<= im -3.1e-7)
       t_0
       (if (<= im 0.0005)
         (* (cos re) (- im))
         (if (<= im 1.32e+154) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = ((pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	double t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -2.1e+153) {
		tmp = t_1;
	} else if (im <= -3.1e-7) {
		tmp = t_0;
	} else if (im <= 0.0005) {
		tmp = cos(re) * -im;
	} else if (im <= 1.32e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((im ** 3.0d0) * (-0.16666666666666666d0)) - im) * (((-0.5d0) * (re * re)) + 1.0d0)
    t_1 = (cos(re) * (9.0d0 - (im * im))) / (im + (-3.0d0))
    if (im <= (-2.1d+153)) then
        tmp = t_1
    else if (im <= (-3.1d-7)) then
        tmp = t_0
    else if (im <= 0.0005d0) then
        tmp = cos(re) * -im
    else if (im <= 1.32d+154) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = ((Math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	double t_1 = (Math.cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -2.1e+153) {
		tmp = t_1;
	} else if (im <= -3.1e-7) {
		tmp = t_0;
	} else if (im <= 0.0005) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.32e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0)
	t_1 = (math.cos(re) * (9.0 - (im * im))) / (im + -3.0)
	tmp = 0
	if im <= -2.1e+153:
		tmp = t_1
	elif im <= -3.1e-7:
		tmp = t_0
	elif im <= 0.0005:
		tmp = math.cos(re) * -im
	elif im <= 1.32e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0))
	t_1 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	tmp = 0.0
	if (im <= -2.1e+153)
		tmp = t_1;
	elseif (im <= -3.1e-7)
		tmp = t_0;
	elseif (im <= 0.0005)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (((im ^ 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	tmp = 0.0;
	if (im <= -2.1e+153)
		tmp = t_1;
	elseif (im <= -3.1e-7)
		tmp = t_0;
	elseif (im <= 0.0005)
		tmp = cos(re) * -im;
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.1e+153], t$95$1, If[LessEqual[im, -3.1e-7], t$95$0, If[LessEqual[im, 0.0005], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.32e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.10000000000000017e153 or 1.31999999999999998e154 < 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 0 100.0%

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

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

   8. Applied egg-rr 7.2%

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

      1. *-commutative 7.2%

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

      2. flip-- 98.6%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 98.6%

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

      4. metadata-eval 98.6%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 98.6%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 98.6%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -2.10000000000000017e153 < im < -3.1e-7 or 5.0000000000000001e-4 < im < 1.31999999999999998e154

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 34.0%

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

      1. mul-1-neg 34.0%

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

      2. unsub-neg 34.0%

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

      3. *-commutative 34.0%

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

      4. associate-*l* 34.0%

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

      5. distribute-lft-out-- 34.0%

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

   6. Simplified 34.0%

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

   7. Taylor expanded in re around 0 14.4%

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

      1. associate--l+ 14.4%

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

      2. associate-*r* 14.4%

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

      3. distribute-lft1-in 35.8%

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

      4. unpow2 35.8%

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

   9. Simplified 35.8%

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

   if -3.1e-7 < im < 5.0000000000000001e-4

   1. Initial program 7.6%

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

      1. sub0-neg 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{elif}\;im \leq 0.0005:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 5: 83.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot \left(-0.16666666666666666 \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\right)\\ t_1 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -5.8 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 720:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* (pow im 3.0) (* -0.16666666666666666 (+ (* -0.5 (* re re)) 1.0))))
        (t_1 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0))))
   (if (<= im -2.1e+153)
     t_1
     (if (<= im -5.8e+34)
       t_0
       (if (<= im 720.0)
         (* (cos re) (- im))
         (if (<= im 1.32e+154) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * (-0.16666666666666666 * ((-0.5 * (re * re)) + 1.0));
	double t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -2.1e+153) {
		tmp = t_1;
	} else if (im <= -5.8e+34) {
		tmp = t_0;
	} else if (im <= 720.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1.32e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * ((-0.16666666666666666d0) * (((-0.5d0) * (re * re)) + 1.0d0))
    t_1 = (cos(re) * (9.0d0 - (im * im))) / (im + (-3.0d0))
    if (im <= (-2.1d+153)) then
        tmp = t_1
    else if (im <= (-5.8d+34)) then
        tmp = t_0
    else if (im <= 720.0d0) then
        tmp = cos(re) * -im
    else if (im <= 1.32d+154) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * (-0.16666666666666666 * ((-0.5 * (re * re)) + 1.0));
	double t_1 = (Math.cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -2.1e+153) {
		tmp = t_1;
	} else if (im <= -5.8e+34) {
		tmp = t_0;
	} else if (im <= 720.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.32e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * (-0.16666666666666666 * ((-0.5 * (re * re)) + 1.0))
	t_1 = (math.cos(re) * (9.0 - (im * im))) / (im + -3.0)
	tmp = 0
	if im <= -2.1e+153:
		tmp = t_1
	elif im <= -5.8e+34:
		tmp = t_0
	elif im <= 720.0:
		tmp = math.cos(re) * -im
	elif im <= 1.32e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * Float64(-0.16666666666666666 * Float64(Float64(-0.5 * Float64(re * re)) + 1.0)))
	t_1 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	tmp = 0.0
	if (im <= -2.1e+153)
		tmp = t_1;
	elseif (im <= -5.8e+34)
		tmp = t_0;
	elseif (im <= 720.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * (-0.16666666666666666 * ((-0.5 * (re * re)) + 1.0));
	t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	tmp = 0.0;
	if (im <= -2.1e+153)
		tmp = t_1;
	elseif (im <= -5.8e+34)
		tmp = t_0;
	elseif (im <= 720.0)
		tmp = cos(re) * -im;
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * N[(-0.16666666666666666 * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.1e+153], t$95$1, If[LessEqual[im, -5.8e+34], t$95$0, If[LessEqual[im, 720.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.32e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.10000000000000017e153 or 1.31999999999999998e154 < 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 0 100.0%

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

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

   8. Applied egg-rr 7.2%

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

      1. *-commutative 7.2%

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

      2. flip-- 98.6%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 98.6%

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

      4. metadata-eval 98.6%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 98.6%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 98.6%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -2.10000000000000017e153 < im < -5.8000000000000003e34 or 720 < im < 1.31999999999999998e154

   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 35.7%

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

      1. mul-1-neg 35.7%

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

      2. unsub-neg 35.7%

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

      3. *-commutative 35.7%

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

      4. associate-*l* 35.7%

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

      5. distribute-lft-out-- 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in re around 0 13.9%

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

      1. associate--l+ 13.9%

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

      2. associate-*r* 13.9%

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

      3. distribute-lft1-in 37.9%

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

      4. unpow2 37.9%

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

   9. Simplified 37.9%

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

   10. Taylor expanded in im around inf 37.9%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\left(1 + -0.5 \cdot {re}^{2}\right) \cdot {im}^{3}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 37.9%

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

       2. unpow2 37.9%

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

   12. Simplified 37.9%

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

   if -5.8000000000000003e34 < im < 720

   1. Initial program 11.4%

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

      1. sub0-neg 11.4%

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

   3. Simplified 11.4%

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

   4. Taylor expanded in im around 0 96.2%

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

      1. mul-1-neg 96.2%

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

      2. *-commutative 96.2%

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

      3. distribute-lft-neg-in 96.2%

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

   6. Simplified 96.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -5.8 \cdot 10^{+34}:\\ \;\;\;\;{im}^{3} \cdot \left(-0.16666666666666666 \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\right)\\ \mathbf{elif}\;im \leq 720:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;{im}^{3} \cdot \left(-0.16666666666666666 \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 6: 78.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{if}\;im \leq -7 \cdot 10^{+168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 1.8:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0))))
   (if (<= im -7e+168)
     t_0
     (if (<= im -3.1e-7)
       (- (* (pow im 3.0) -0.16666666666666666) im)
       (if (<= im 1.8) (* (cos re) (- im)) t_0)))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -7e+168) {
		tmp = t_0;
	} else if (im <= -3.1e-7) {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	} else if (im <= 1.8) {
		tmp = cos(re) * -im;
	} 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 = (cos(re) * (9.0d0 - (im * im))) / (im + (-3.0d0))
    if (im <= (-7d+168)) then
        tmp = t_0
    else if (im <= (-3.1d-7)) then
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    else if (im <= 1.8d0) then
        tmp = cos(re) * -im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -7e+168) {
		tmp = t_0;
	} else if (im <= -3.1e-7) {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	} else if (im <= 1.8) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.cos(re) * (9.0 - (im * im))) / (im + -3.0)
	tmp = 0
	if im <= -7e+168:
		tmp = t_0
	elif im <= -3.1e-7:
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	elif im <= 1.8:
		tmp = math.cos(re) * -im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	tmp = 0.0
	if (im <= -7e+168)
		tmp = t_0;
	elseif (im <= -3.1e-7)
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.8)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	tmp = 0.0;
	if (im <= -7e+168)
		tmp = t_0;
	elseif (im <= -3.1e-7)
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	elseif (im <= 1.8)
		tmp = cos(re) * -im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7e+168], t$95$0, If[LessEqual[im, -3.1e-7], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 1.8], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.0000000000000004e168 or 1.80000000000000004 < 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 78.2%

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

      1. mul-1-neg 78.2%

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

      2. unsub-neg 78.2%

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

      3. *-commutative 78.2%

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

      4. associate-*l* 78.2%

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

      5. distribute-lft-out-- 78.2%

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

   6. Simplified 78.2%

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

   8. Applied egg-rr 6.4%

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

      1. *-commutative 6.4%

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

      2. flip-- 72.3%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 72.3%

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

      4. metadata-eval 72.3%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 72.3%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 72.3%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -7.0000000000000004e168 < im < -3.1e-7

   1. Initial program 99.3%

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

      1. sub0-neg 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in im around 0 51.3%

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

      1. mul-1-neg 51.3%

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

      2. unsub-neg 51.3%

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

      3. *-commutative 51.3%

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

      4. associate-*l* 51.3%

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

      5. distribute-lft-out-- 51.3%

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

   6. Simplified 51.3%

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

   7. Taylor expanded in re around 0 42.6%

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

   if -3.1e-7 < im < 1.80000000000000004

   1. Initial program 7.6%

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

      1. sub0-neg 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7 \cdot 10^{+168}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 1.8:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 7: 75.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.1 \cdot 10^{-7} \lor \neg \left(im \leq 6.2 \cdot 10^{+14}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.1e-7) (not (<= im 6.2e+14)))
   (- (* (pow im 3.0) -0.16666666666666666) im)
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.1e-7) || !(im <= 6.2e+14)) {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = cos(re) * -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 <= (-3.1d-7)) .or. (.not. (im <= 6.2d+14))) then
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.1e-7) || !(im <= 6.2e+14)) {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.1e-7) or not (im <= 6.2e+14):
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.1e-7) || !(im <= 6.2e+14))
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.1e-7) || ~((im <= 6.2e+14)))
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.1e-7], N[Not[LessEqual[im, 6.2e+14]], $MachinePrecision]], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.1e-7 or 6.2e14 < 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 70.7%

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

      1. mul-1-neg 70.7%

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

      2. unsub-neg 70.7%

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

      3. *-commutative 70.7%

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

      4. associate-*l* 70.7%

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

      5. distribute-lft-out-- 70.7%

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

   6. Simplified 70.7%

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

   7. Taylor expanded in re around 0 55.7%

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

   if -3.1e-7 < im < 6.2e14

   1. Initial program 8.3%

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

      1. sub0-neg 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in im around 0 99.0%

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

      1. mul-1-neg 99.0%

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

      2. *-commutative 99.0%

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

      3. distribute-lft-neg-in 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.1 \cdot 10^{-7} \lor \neg \left(im \leq 6.2 \cdot 10^{+14}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 8: 53.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.6 \lor \neg \left(im \leq 10^{-7}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.6) (not (<= im 1e-7)))
   (* (pow im 3.0) -0.16666666666666666)
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.6) || !(im <= 1e-7)) {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = -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 <= (-2.6d0)) .or. (.not. (im <= 1d-7))) then
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.6) || !(im <= 1e-7)) {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.6) or not (im <= 1e-7):
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.6) || !(im <= 1e-7))
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.6) || ~((im <= 1e-7)))
		tmp = (im ^ 3.0) * -0.16666666666666666;
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.6], N[Not[LessEqual[im, 1e-7]], $MachinePrecision]], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -2.60000000000000009 or 9.9999999999999995e-8 < 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 70.2%

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

      1. mul-1-neg 70.2%

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

      2. unsub-neg 70.2%

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

      3. *-commutative 70.2%

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

      4. associate-*l* 70.2%

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

      5. distribute-lft-out-- 70.2%

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

   6. Simplified 70.2%

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

   7. Taylor expanded in re around 0 54.5%

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

   8. Taylor expanded in im around inf 54.5%

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

   if -2.60000000000000009 < im < 9.9999999999999995e-8

   1. Initial program 7.6%

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

      1. sub0-neg 7.6%

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

   3. Simplified 7.6%

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

   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%

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

   7. Taylor expanded in re around 0 59.8%

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

   8. Taylor expanded in im around 0 59.7%

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

      1. neg-mul-1 59.7%

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

   10. Simplified 59.7%

       \[\leadsto \color{blue}{-im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.6 \lor \neg \left(im \leq 10^{-7}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 9: 75.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.7 \cdot 10^{+46} \lor \neg \left(im \leq 1.55 \cdot 10^{+14}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.7e+46) (not (<= im 1.55e+14)))
   (* (pow im 3.0) -0.16666666666666666)
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.7e+46) || !(im <= 1.55e+14)) {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = cos(re) * -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 <= (-3.7d+46)) .or. (.not. (im <= 1.55d+14))) then
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.7e+46) || !(im <= 1.55e+14)) {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.7e+46) or not (im <= 1.55e+14):
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.7e+46) || !(im <= 1.55e+14))
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.7e+46) || ~((im <= 1.55e+14)))
		tmp = (im ^ 3.0) * -0.16666666666666666;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.7e+46], N[Not[LessEqual[im, 1.55e+14]], $MachinePrecision]], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.6999999999999999e46 or 1.55e14 < 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 75.8%

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

      1. mul-1-neg 75.8%

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

      2. unsub-neg 75.8%

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

      3. *-commutative 75.8%

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

      4. associate-*l* 75.8%

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

      5. distribute-lft-out-- 75.8%

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

   6. Simplified 75.8%

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

   7. Taylor expanded in re around 0 59.5%

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

   8. Taylor expanded in im around inf 59.5%

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

   if -3.6999999999999999e46 < im < 1.55e14

   1. Initial program 14.5%

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

      1. sub0-neg 14.5%

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

   3. Simplified 14.5%

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

   4. Taylor expanded in im around 0 92.9%

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

      1. mul-1-neg 92.9%

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

      2. *-commutative 92.9%

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

      3. distribute-lft-neg-in 92.9%

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

   6. Simplified 92.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.7 \cdot 10^{+46} \lor \neg \left(im \leq 1.55 \cdot 10^{+14}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 10: 36.2% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot 0.75\\ \mathbf{if}\;re \leq 1.05 \cdot 10^{+92}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;re \leq 4 \cdot 10^{+145}:\\ \;\;\;\;\frac{2.25 - t_0 \cdot t_0}{-1.5 - t_0}\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) 0.75)))
   (if (<= re 1.05e+92)
     (- (* (* re re) (* im 0.5)) im)
     (if (<= re 4e+145)
       (/ (- 2.25 (* t_0 t_0)) (- -1.5 t_0))
       (* 27.0 (+ 0.5 (* re (* re -0.25))))))))

C

double code(double re, double im) {
	double t_0 = (re * re) * 0.75;
	double tmp;
	if (re <= 1.05e+92) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (re <= 4e+145) {
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	} else {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	}
	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 <= 1.05d+92) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else if (re <= 4d+145) then
        tmp = (2.25d0 - (t_0 * t_0)) / ((-1.5d0) - t_0)
    else
        tmp = 27.0d0 * (0.5d0 + (re * (re * (-0.25d0))))
    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 <= 1.05e+92) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (re <= 4e+145) {
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	} else {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) * 0.75
	tmp = 0
	if re <= 1.05e+92:
		tmp = ((re * re) * (im * 0.5)) - im
	elif re <= 4e+145:
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0)
	else:
		tmp = 27.0 * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * 0.75)
	tmp = 0.0
	if (re <= 1.05e+92)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	elseif (re <= 4e+145)
		tmp = Float64(Float64(2.25 - Float64(t_0 * t_0)) / Float64(-1.5 - t_0));
	else
		tmp = Float64(27.0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) * 0.75;
	tmp = 0.0;
	if (re <= 1.05e+92)
		tmp = ((re * re) * (im * 0.5)) - im;
	elseif (re <= 4e+145)
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	else
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]}, If[LessEqual[re, 1.05e+92], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[re, 4e+145], N[(N[(2.25 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.5 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(27.0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.04999999999999993e92

   1. Initial program 51.2%

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

      1. sub0-neg 51.2%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in im around 0 55.5%

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

      1. mul-1-neg 55.5%

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

      2. *-commutative 55.5%

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

      3. distribute-lft-neg-in 55.5%

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

   6. Simplified 55.5%

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

   7. Taylor expanded in re around 0 40.5%

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

      1. neg-mul-1 40.5%

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

      2. +-commutative 40.5%

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

      3. unsub-neg 40.5%

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

      4. *-commutative 40.5%

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

      5. associate-*l* 40.5%

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

      6. unpow2 40.5%

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

   9. Simplified 40.5%

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

   if 1.04999999999999993e92 < re < 4e145

   1. Initial program 73.5%

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

      1. sub0-neg 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in re around 0 1.1%

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

      1. *-commutative 1.1%

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

      2. associate-*r* 1.1%

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

      3. distribute-rgt-out 34.5%

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

      4. +-commutative 34.5%

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

      5. *-commutative 34.5%

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

      6. unpow2 34.5%

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

      7. associate-*l* 34.5%

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

   6. Simplified 34.5%

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

   8. Applied egg-rr 3.5%

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

      1. distribute-lft-in 3.5%

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

      2. flip-+ 42.5%

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

      3. metadata-eval 42.5%

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

      4. metadata-eval 42.5%

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

      5. metadata-eval 42.5%

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

      6. *-commutative 42.5%

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

      7. *-commutative 42.5%

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

      8. associate-*r* 42.5%

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

      9. associate-*l* 42.5%

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

      10. metadata-eval 42.5%

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

      11. associate-*r* 42.5%

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

      12. associate-*l* 42.5%

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

      13. metadata-eval 42.5%

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

      14. metadata-eval 42.5%

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

      15. *-commutative 42.5%

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

      16. associate-*r* 42.5%

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

      17. associate-*l* 42.5%

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

      18. metadata-eval 42.5%

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

   10. Applied egg-rr 42.5%

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

   if 4e145 < re

   1. Initial program 51.0%

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

      1. sub0-neg 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in re around 0 0.2%

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

      1. *-commutative 0.2%

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

      2. associate-*r* 0.2%

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

      3. distribute-rgt-out 17.4%

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

      4. +-commutative 17.4%

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

      5. *-commutative 17.4%

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

      6. unpow2 17.4%

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

      7. associate-*l* 17.4%

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

   6. Simplified 17.4%

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

   8. Applied egg-rr 29.2%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{+92}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;re \leq 4 \cdot 10^{+145}:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 11: 32.2% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.7 \cdot 10^{+144}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.7e+144) (- im) (* 27.0 (+ 0.5 (* re (* re -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.7e+144) {
		tmp = -im;
	} else {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.7d+144) then
        tmp = -im
    else
        tmp = 27.0d0 * (0.5d0 + (re * (re * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.7e+144) {
		tmp = -im;
	} else {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.7e+144:
		tmp = -im
	else:
		tmp = 27.0 * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.7e+144)
		tmp = Float64(-im);
	else
		tmp = Float64(27.0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.7e+144)
		tmp = -im;
	else
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.7e+144], (-im), N[(27.0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.7e144

   1. Initial program 52.4%

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

      1. sub0-neg 52.4%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in im around 0 85.3%

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

      1. mul-1-neg 85.3%

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

      2. unsub-neg 85.3%

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

      3. *-commutative 85.3%

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

      4. associate-*l* 85.3%

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

      5. distribute-lft-out-- 85.3%

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

   6. Simplified 85.3%

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

   7. Taylor expanded in re around 0 60.5%

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

   8. Taylor expanded in im around 0 36.1%

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

      1. neg-mul-1 36.1%

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

   10. Simplified 36.1%

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

   if 1.7e144 < re

   1. Initial program 51.0%

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

      1. sub0-neg 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in re around 0 0.2%

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

      1. *-commutative 0.2%

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

      2. associate-*r* 0.2%

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

      3. distribute-rgt-out 17.4%

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

      4. +-commutative 17.4%

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

      5. *-commutative 17.4%

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

      6. unpow2 17.4%

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

      7. associate-*l* 17.4%

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

   6. Simplified 17.4%

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

   8. Applied egg-rr 29.2%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.7 \cdot 10^{+144}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 12: 35.9% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{+197}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.05e+197)
   (- (* (* re re) (* im 0.5)) im)
   (* 27.0 (+ 0.5 (* re (* re -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.05e+197) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.05d+197) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else
        tmp = 27.0d0 * (0.5d0 + (re * (re * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.05e+197) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.05e+197:
		tmp = ((re * re) * (im * 0.5)) - im
	else:
		tmp = 27.0 * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.05e+197)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	else
		tmp = Float64(27.0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.05e+197)
		tmp = ((re * re) * (im * 0.5)) - im;
	else
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.05e+197], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(27.0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.05000000000000003e197

   1. Initial program 52.2%

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

      1. sub0-neg 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in im around 0 54.6%

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

      1. mul-1-neg 54.6%

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

      2. *-commutative 54.6%

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

      3. distribute-lft-neg-in 54.6%

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

   6. Simplified 54.6%

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

   7. Taylor expanded in re around 0 39.2%

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

      1. neg-mul-1 39.2%

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

      2. +-commutative 39.2%

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

      3. unsub-neg 39.2%

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

      4. *-commutative 39.2%

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

      5. associate-*l* 39.2%

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

      6. unpow2 39.2%

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

   9. Simplified 39.2%

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

   if 1.05000000000000003e197 < re

   1. Initial program 52.4%

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

      1. sub0-neg 52.4%

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

   3. Simplified 52.4%

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

   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 11.1%

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

      4. +-commutative 11.1%

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

      5. *-commutative 11.1%

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

      6. unpow2 11.1%

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

      7. associate-*l* 11.1%

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

   6. Simplified 11.1%

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

   8. Applied egg-rr 39.7%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{+197}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 13: 32.0% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.2 \cdot 10^{+168}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.2e+168) (- im) (* (* re re) 0.75)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.2e+168) {
		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 <= 2.2d+168) 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 <= 2.2e+168) {
		tmp = -im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.2e+168:
		tmp = -im
	else:
		tmp = (re * re) * 0.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.2e+168)
		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 <= 2.2e+168)
		tmp = -im;
	else
		tmp = (re * re) * 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.2e+168], (-im), N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.2000000000000002e168

   1. Initial program 52.4%

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

      1. sub0-neg 52.4%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in im around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unsub-neg 85.5%

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

      3. *-commutative 85.5%

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

      4. associate-*l* 85.5%

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

      5. distribute-lft-out-- 85.5%

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

   6. Simplified 85.5%

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

   7. Taylor expanded in re around 0 60.4%

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

   8. Taylor expanded in im around 0 35.6%

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

      1. neg-mul-1 35.6%

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

   10. Simplified 35.6%

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

   if 2.2000000000000002e168 < re

   1. Initial program 50.8%

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

      1. sub0-neg 50.8%

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

   3. Simplified 50.8%

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

   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 20.0%

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

      4. +-commutative 20.0%

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

      5. *-commutative 20.0%

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

      6. unpow2 20.0%

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

      7. associate-*l* 20.0%

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

   6. Simplified 20.0%

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

   8. Applied egg-rr 16.9%

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

   9. Taylor expanded in re around inf 16.9%

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

       1. unpow2 16.9%

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

   11. Simplified 16.9%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.2 \cdot 10^{+168}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \]

Alternative 14: 30.0% 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 52.2%

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

   1. sub0-neg 52.2%

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

3. Simplified 52.2%

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

4. Taylor expanded in im around 0 85.4%

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

   1. mul-1-neg 85.4%

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

   2. unsub-neg 85.4%

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

   3. *-commutative 85.4%

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

   4. associate-*l* 85.4%

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

   5. distribute-lft-out-- 85.4%

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

6. Simplified 85.4%

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

7. Taylor expanded in re around 0 57.2%

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

8. Taylor expanded in im around 0 33.0%

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

   1. neg-mul-1 33.0%

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

10. Simplified 33.0%

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

11. Final simplification 33.0%

    \[\leadsto -im \]

Alternative 15: 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 52.2%

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

   1. sub0-neg 52.2%

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

3. Simplified 52.2%

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

4. Taylor expanded in re around 0 2.8%

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

   1. *-commutative 2.8%

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

   2. associate-*r* 2.8%

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

   3. distribute-rgt-out 36.7%

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

   4. +-commutative 36.7%

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

   5. *-commutative 36.7%

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

   6. unpow2 36.7%

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

   7. associate-*l* 36.7%

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

6. Simplified 36.7%

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

8. Applied egg-rr 6.4%

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

9. Taylor expanded in re around 0 2.7%

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

10. Final simplification 2.7%

    \[\leadsto -1.5 \]

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 2023188 
(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))))