math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 10

Speedup: 1.0×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
4. Add Preprocessing

Alternative 2: 84.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.024 \lor \neg \left(im \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.024) (not (<= im 1.4e+154)))
   (* (cos re) (+ (* 0.5 (* im im)) 1.0))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.024) || !(im <= 1.4e+154)) {
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	} 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 <= 0.024d0) .or. (.not. (im <= 1.4d+154))) then
        tmp = cos(re) * ((0.5d0 * (im * im)) + 1.0d0)
    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 <= 0.024) || !(im <= 1.4e+154)) {
		tmp = Math.cos(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.024) or not (im <= 1.4e+154):
		tmp = math.cos(re) * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.024) || !(im <= 1.4e+154))
		tmp = Float64(cos(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	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 <= 0.024) || ~((im <= 1.4e+154)))
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.024], N[Not[LessEqual[im, 1.4e+154]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $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 < 0.024 or 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 85.8%

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

      1. associate-*r* 85.8%

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

      2. distribute-rgt1-in 85.8%

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

      3. *-commutative 85.8%

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

   5. Applied egg-rr 85.8%

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

      1. unpow2 25.3%

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

   7. Applied egg-rr 85.8%

      \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5 + 1\right) \cdot \cos re \]

   if 0.024 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 83.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.024 \lor \neg \left(im \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.027:\\ \;\;\;\;\cos re + 0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.027)
   (+ (cos re) (* 0.5 (* (cos re) (* im im))))
   (if (<= im 1.4e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (+ (* 0.5 (* im im)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.027) {
		tmp = cos(re) + (0.5 * (cos(re) * (im * im)));
	} else if (im <= 1.4e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.027d0) then
        tmp = cos(re) + (0.5d0 * (cos(re) * (im * im)))
    else if (im <= 1.4d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * ((0.5d0 * (im * im)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.027) {
		tmp = Math.cos(re) + (0.5 * (Math.cos(re) * (im * im)));
	} else if (im <= 1.4e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.027:
		tmp = math.cos(re) + (0.5 * (math.cos(re) * (im * im)))
	elif im <= 1.4e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * ((0.5 * (im * im)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.027)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(cos(re) * Float64(im * im))));
	elseif (im <= 1.4e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.027)
		tmp = cos(re) + (0.5 * (cos(re) * (im * im)));
	elseif (im <= 1.4e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.027], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0269999999999999997

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 83.0%

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

      1. unpow2 15.5%

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

   5. Applied egg-rr 83.0%

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

   if 0.0269999999999999997 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 83.3%

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

   if 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt1-in 100.0%

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

      3. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow2 75.7%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5 + 1\right) \cdot \cos re \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.027:\\ \;\;\;\;\cos re + 0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8.4 \cdot 10^{+74} \lor \neg \left(im \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{{im}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 8.4e+74) (not (<= im 1.4e+154)))
   (* (cos re) (+ (* 0.5 (* im im)) 1.0))
   (* 0.5 (sqrt (pow im 4.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 8.4e+74) || !(im <= 1.4e+154)) {
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.5 * sqrt(pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 8.4d+74) .or. (.not. (im <= 1.4d+154))) then
        tmp = cos(re) * ((0.5d0 * (im * im)) + 1.0d0)
    else
        tmp = 0.5d0 * sqrt((im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 8.4e+74) || !(im <= 1.4e+154)) {
		tmp = Math.cos(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = 0.5 * Math.sqrt(Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 8.4e+74) or not (im <= 1.4e+154):
		tmp = math.cos(re) * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = 0.5 * math.sqrt(math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 8.4e+74) || !(im <= 1.4e+154))
		tmp = Float64(cos(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	else
		tmp = Float64(0.5 * sqrt((im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 8.4e+74) || ~((im <= 1.4e+154)))
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	else
		tmp = 0.5 * sqrt((im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 8.4e+74], N[Not[LessEqual[im, 1.4e+154]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[Power[im, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.3999999999999995e74 or 1.4e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 82.0%

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

      1. associate-*r* 82.0%

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

      2. distribute-rgt1-in 82.0%

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

      3. *-commutative 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. unpow2 24.3%

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

   7. Applied egg-rr 82.0%

      \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5 + 1\right) \cdot \cos re \]

   if 8.3999999999999995e74 < im < 1.4e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 6.0%

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

   4. Taylor expanded in im around inf 6.0%

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

      1. *-commutative 6.0%

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

      2. associate-*r* 6.0%

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

      3. *-commutative 6.0%

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

   6. Simplified 6.0%

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

   7. Taylor expanded in re around 0 5.4%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 5.4%

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

      2. sqrt-unprod 79.4%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt{{im}^{2} \cdot {im}^{2}}} \]

      3. pow-prod-up 79.4%

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

      4. metadata-eval 79.4%

         \[\leadsto 0.5 \cdot \sqrt{{im}^{\color{blue}{4}}} \]

   9. Applied egg-rr 79.4%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{{im}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.4 \cdot 10^{+74} \lor \neg \left(im \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{{im}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+22} \lor \neg \left(im \leq 6.6 \cdot 10^{+145}\right):\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + {re}^{2} \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 7.5e+22) (not (<= im 6.6e+145)))
   (* (cos re) (+ (* 0.5 (* im im)) 1.0))
   (* (* im im) (+ 0.5 (* (pow re 2.0) -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 7.5e+22) || !(im <= 6.6e+145)) {
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = (im * im) * (0.5 + (pow(re, 2.0) * -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 ((im <= 7.5d+22) .or. (.not. (im <= 6.6d+145))) then
        tmp = cos(re) * ((0.5d0 * (im * im)) + 1.0d0)
    else
        tmp = (im * im) * (0.5d0 + ((re ** 2.0d0) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 7.5e+22) || !(im <= 6.6e+145)) {
		tmp = Math.cos(re) * ((0.5 * (im * im)) + 1.0);
	} else {
		tmp = (im * im) * (0.5 + (Math.pow(re, 2.0) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 7.5e+22) or not (im <= 6.6e+145):
		tmp = math.cos(re) * ((0.5 * (im * im)) + 1.0)
	else:
		tmp = (im * im) * (0.5 + (math.pow(re, 2.0) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 7.5e+22) || !(im <= 6.6e+145))
		tmp = Float64(cos(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64((re ^ 2.0) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 7.5e+22) || ~((im <= 6.6e+145)))
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	else
		tmp = (im * im) * (0.5 + ((re ^ 2.0) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 7.5e+22], N[Not[LessEqual[im, 6.6e+145]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[Power[re, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.5000000000000002e22 or 6.60000000000000054e145 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 84.0%

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

      1. associate-*r* 84.0%

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

      2. distribute-rgt1-in 84.0%

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

      3. *-commutative 84.0%

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

   5. Applied egg-rr 84.0%

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

      1. unpow2 24.9%

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

   7. Applied egg-rr 84.0%

      \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5 + 1\right) \cdot \cos re \]

   if 7.5000000000000002e22 < im < 6.60000000000000054e145

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 5.3%

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

   4. Taylor expanded in im around inf 5.3%

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

      1. *-commutative 5.3%

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

      2. associate-*r* 5.3%

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

      3. *-commutative 5.3%

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

   6. Simplified 5.3%

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

   7. Taylor expanded in re around 0 23.3%

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

      1. *-commutative 23.3%

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

      2. associate-*l* 23.3%

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

      3. *-commutative 23.3%

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

      4. distribute-lft-out 23.3%

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

   9. Simplified 23.3%

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

       1. unpow2 4.4%

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

   11. Applied egg-rr 23.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+22} \lor \neg \left(im \leq 6.6 \cdot 10^{+145}\right):\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + {re}^{2} \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0255:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0255) (cos re) (+ 1.0 (* 0.5 (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0255) {
		tmp = cos(re);
	} else {
		tmp = 1.0 + (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0255d0) then
        tmp = cos(re)
    else
        tmp = 1.0d0 + (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0255) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.0 + (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0255:
		tmp = math.cos(re)
	else:
		tmp = 1.0 + (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0255)
		tmp = cos(re);
	else
		tmp = Float64(1.0 + Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0255)
		tmp = cos(re);
	else
		tmp = 1.0 + (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0255], N[Cos[re], $MachinePrecision], N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0254999999999999984

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 66.5%

      \[\leadsto \color{blue}{\cos re} \]

   if 0.0254999999999999984 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 57.5%

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

   4. Taylor expanded in re around 0 43.8%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0255:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (cos re) (+ (* 0.5 (* im im)) 1.0)))

C

double code(double re, double im) {
	return cos(re) * ((0.5 * (im * im)) + 1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re) * ((0.5d0 * (im * im)) + 1.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.cos(re) * ((0.5 * (im * im)) + 1.0);
}

Python

def code(re, im):
	return math.cos(re) * ((0.5 * (im * im)) + 1.0)

Julia

function code(re, im)
	return Float64(cos(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
end

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0 76.3%

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

   1. associate-*r* 76.3%

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

   2. distribute-rgt1-in 76.3%

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

   3. *-commutative 76.3%

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

5. Applied egg-rr 76.3%

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

   1. unpow2 22.9%

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

7. Applied egg-rr 76.3%

   \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5 + 1\right) \cdot \cos re \]

8. Final simplification 76.3%

   \[\leadsto \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \]
9. Add Preprocessing

Alternative 8: 60.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+31}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.6e+31) (cos re) (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.6e+31) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (im * 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 <= 3.6d+31) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.6e+31) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.6e+31:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.6e+31)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.6e+31)
		tmp = cos(re);
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.6e+31], N[Cos[re], $MachinePrecision], N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.59999999999999996e31

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 64.5%

      \[\leadsto \color{blue}{\cos re} \]

   if 3.59999999999999996e31 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 62.9%

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

   4. Taylor expanded in im around inf 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*r* 62.9%

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

      3. *-commutative 62.9%

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

   6. Simplified 62.9%

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

   7. Taylor expanded in re around 0 47.8%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 47.8%

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

   9. Applied egg-rr 47.8%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+31}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 17.3% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.71:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.71) 0.25 (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.71) {
		tmp = 0.25;
	} else {
		tmp = 0.5 * (im * 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 <= 0.71d0) then
        tmp = 0.25d0
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.71) {
		tmp = 0.25;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.71:
		tmp = 0.25
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.71)
		tmp = 0.25;
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.71)
		tmp = 0.25;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.71], 0.25, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.70999999999999996

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 10.1%

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

   5. Taylor expanded in re around 0 10.2%

      \[\leadsto \color{blue}{0.25} \]

   if 0.70999999999999996 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 57.5%

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

   4. Taylor expanded in im around inf 57.5%

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

      1. *-commutative 57.5%

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

      2. associate-*r* 57.5%

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

      3. *-commutative 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in re around 0 43.8%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 43.8%

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

   9. Applied egg-rr 43.8%

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

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.71:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 8.1% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

4. Applied egg-rr 8.1%

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

5. Taylor expanded in re around 0 8.2%

   \[\leadsto \color{blue}{0.25} \]

6. Final simplification 8.2%

   \[\leadsto 0.25 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))