math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 11

Speedup: 1.0×

• 50.4% 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: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.058:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \sqrt[3]{{im}^{6} \cdot 0.125}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.058)
   (* (cos re) (+ (* 0.5 (pow im 2.0)) 1.0))
   (if (<= im 2.6e+45)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (cbrt (* (pow im 6.0) 0.125))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.058) {
		tmp = cos(re) * ((0.5 * pow(im, 2.0)) + 1.0);
	} else if (im <= 2.6e+45) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * cbrt((pow(im, 6.0) * 0.125));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.058) {
		tmp = Math.cos(re) * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	} else if (im <= 2.6e+45) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * Math.cbrt((Math.pow(im, 6.0) * 0.125));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.058)
		tmp = Float64(cos(re) * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	elseif (im <= 2.6e+45)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * cbrt(Float64((im ^ 6.0) * 0.125)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.058], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+45], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[Power[N[(N[Power[im, 6.0], $MachinePrecision] * 0.125), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0580000000000000029

   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 81.4%

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

      1. associate-*r* 81.4%

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

      2. distribute-rgt1-in 81.4%

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

   5. Simplified 81.4%

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

   if 0.0580000000000000029 < im < 2.60000000000000007e45

   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 2.60000000000000007e45 < 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 57.6%

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

      1. associate-*r* 57.6%

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

      2. distribute-rgt1-in 57.6%

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

   5. Simplified 57.6%

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

   6. Taylor expanded in im around inf 57.6%

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

      1. associate-*r* 57.6%

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

      2. *-commutative 57.6%

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

   8. Simplified 57.6%

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

      1. add-cbrt-cube 77.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.5 \cdot {im}^{2}\right) \cdot \left(0.5 \cdot {im}^{2}\right)\right) \cdot \left(0.5 \cdot {im}^{2}\right)}} \]

      2. pow1/3 77.6%

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

      3. pow3 77.6%

         \[\leadsto {\color{blue}{\left({\left(0.5 \cdot {im}^{2}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. *-commutative 77.6%

         \[\leadsto {\left({\color{blue}{\left({im}^{2} \cdot 0.5\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. unpow-prod-down 77.6%

         \[\leadsto {\color{blue}{\left({\left({im}^{2}\right)}^{3} \cdot {0.5}^{3}\right)}}^{0.3333333333333333} \]

      6. pow-pow 77.6%

         \[\leadsto {\left(\color{blue}{{im}^{\left(2 \cdot 3\right)}} \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      7. metadata-eval 77.6%

         \[\leadsto {\left({im}^{\color{blue}{6}} \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      8. metadata-eval 77.6%

         \[\leadsto {\left({im}^{6} \cdot \color{blue}{0.125}\right)}^{0.3333333333333333} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \cos re \cdot \color{blue}{{\left({im}^{6} \cdot 0.125\right)}^{0.3333333333333333}} \]
   11. Step-by-step derivation

       1. unpow1/3 77.6%

          \[\leadsto \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]

   12. Simplified 100.0%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.058:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \sqrt[3]{{im}^{6} \cdot 0.125}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.054:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.054)
   (cos re)
   (if (<= im 1.35e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.054) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.054d0) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.054) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.054:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.054)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.054)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.054], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0539999999999999994

   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 65.9%

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

   if 0.0539999999999999994 < im < 1.35000000000000003e154

   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 79.4%

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

   if 1.35000000000000003e154 < 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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.054:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 0.058:\\ \;\;\;\;\cos re \cdot \left(t\_0 + 1\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 0.058)
     (* (cos re) (+ t_0 1.0))
     (if (<= im 1.35e+154)
       (* 0.5 (+ (exp (- im)) (exp im)))
       (* (cos re) t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * pow(im, 2.0);
	double tmp;
	if (im <= 0.058) {
		tmp = cos(re) * (t_0 + 1.0);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im ** 2.0d0)
    if (im <= 0.058d0) then
        tmp = cos(re) * (t_0 + 1.0d0)
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.pow(im, 2.0);
	double tmp;
	if (im <= 0.058) {
		tmp = Math.cos(re) * (t_0 + 1.0);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.pow(im, 2.0)
	tmp = 0
	if im <= 0.058:
		tmp = math.cos(re) * (t_0 + 1.0)
	elif im <= 1.35e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * (im ^ 2.0))
	tmp = 0.0
	if (im <= 0.058)
		tmp = Float64(cos(re) * Float64(t_0 + 1.0));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im ^ 2.0);
	tmp = 0.0;
	if (im <= 0.058)
		tmp = cos(re) * (t_0 + 1.0);
	elseif (im <= 1.35e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.058], N[(N[Cos[re], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0580000000000000029

   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 81.4%

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

      1. associate-*r* 81.4%

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

      2. distribute-rgt1-in 81.4%

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

   5. Simplified 81.4%

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

   if 0.0580000000000000029 < im < 1.35000000000000003e154

   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 79.4%

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

   if 1.35000000000000003e154 < 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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.058:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.054:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.054) (cos re) (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.054) {
		tmp = cos(re);
	} 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 <= 0.054d0) then
        tmp = cos(re)
    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 <= 0.054) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.054:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.054)
		tmp = cos(re);
	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 <= 0.054)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.054], N[Cos[re], $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 < 0.0539999999999999994

   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 65.9%

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

   if 0.0539999999999999994 < 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 re around 0 78.7%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.054:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.9 \cdot 10^{+52}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{im}^{6} \cdot 0.125}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.9e+52) (cos re) (cbrt (* (pow im 6.0) 0.125))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.9e+52) {
		tmp = cos(re);
	} else {
		tmp = cbrt((pow(im, 6.0) * 0.125));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.9e+52) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.cbrt((Math.pow(im, 6.0) * 0.125));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.9e+52)
		tmp = cos(re);
	else
		tmp = cbrt(Float64((im ^ 6.0) * 0.125));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.9e+52], N[Cos[re], $MachinePrecision], N[Power[N[(N[Power[im, 6.0], $MachinePrecision] * 0.125), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.9e52

   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 62.0%

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

   if 2.9e52 < 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 58.7%

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

      1. associate-*r* 58.7%

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

      2. distribute-rgt1-in 58.7%

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

   5. Simplified 58.7%

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

   6. Taylor expanded in im around inf 58.7%

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

      1. associate-*r* 58.7%

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

      2. *-commutative 58.7%

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

   8. Simplified 58.7%

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

   9. Taylor expanded in re around 0 45.8%

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

       1. add-cbrt-cube 79.2%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.5 \cdot {im}^{2}\right) \cdot \left(0.5 \cdot {im}^{2}\right)\right) \cdot \left(0.5 \cdot {im}^{2}\right)}} \]

       2. pow1/3 79.2%

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

       3. pow3 79.2%

          \[\leadsto {\color{blue}{\left({\left(0.5 \cdot {im}^{2}\right)}^{3}\right)}}^{0.3333333333333333} \]

       4. *-commutative 79.2%

          \[\leadsto {\left({\color{blue}{\left({im}^{2} \cdot 0.5\right)}}^{3}\right)}^{0.3333333333333333} \]

       5. unpow-prod-down 79.2%

          \[\leadsto {\color{blue}{\left({\left({im}^{2}\right)}^{3} \cdot {0.5}^{3}\right)}}^{0.3333333333333333} \]

       6. pow-pow 79.2%

          \[\leadsto {\left(\color{blue}{{im}^{\left(2 \cdot 3\right)}} \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

       7. metadata-eval 79.2%

          \[\leadsto {\left({im}^{\color{blue}{6}} \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

       8. metadata-eval 79.2%

          \[\leadsto {\left({im}^{6} \cdot \color{blue}{0.125}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 79.2%

       \[\leadsto \color{blue}{{\left({im}^{6} \cdot 0.125\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 79.2%

          \[\leadsto \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]

   13. Simplified 79.2%

       \[\leadsto \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.9 \cdot 10^{+52}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{im}^{6} \cdot 0.125}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.32 \cdot 10^{+67}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.32e+67) (cos re) (+ (* 0.5 (pow im 2.0)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.32e+67) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * pow(im, 2.0)) + 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 <= 1.32d+67) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * (im ** 2.0d0)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.32e+67) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.pow(im, 2.0)) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.32e+67:
		tmp = math.cos(re)
	else:
		tmp = (0.5 * math.pow(im, 2.0)) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.32e+67)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 * (im ^ 2.0)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.32e+67)
		tmp = cos(re);
	else
		tmp = (0.5 * (im ^ 2.0)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.32e+67], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3200000000000001e67

   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 61.2%

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

   if 1.3200000000000001e67 < 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 62.3%

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

      1. associate-*r* 62.3%

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

      2. distribute-rgt1-in 62.3%

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

   5. Simplified 62.3%

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

   6. Taylor expanded in re around 0 48.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.32 \cdot 10^{+67}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.3 \cdot 10^{+65}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.3e+65) (cos re) (* 0.5 (pow im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.3e+65) {
		tmp = cos(re);
	} else {
		tmp = 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 <= 2.3d+65) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.3e+65) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.3e+65:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.3e+65)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.3e+65)
		tmp = cos(re);
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.3e+65], N[Cos[re], $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.3e65

   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 61.2%

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

   if 2.3e65 < 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 62.3%

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

      1. associate-*r* 62.3%

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

      2. distribute-rgt1-in 62.3%

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

   5. Simplified 62.3%

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

   6. Taylor expanded in im around inf 62.3%

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

      1. associate-*r* 62.3%

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

      2. *-commutative 62.3%

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

   8. Simplified 62.3%

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

   9. Taylor expanded in re around 0 48.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.3 \cdot 10^{+65}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \cos re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.cos(re)

Julia

function code(re, im)
	return cos(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Cos[re], $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 51.0%

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

4. Final simplification 51.0%

   \[\leadsto \cos re \]
5. Add Preprocessing

Alternative 10: 2.3% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

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

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 2.4%

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

   1. pow-base-1 2.4%

      \[\leadsto \log \color{blue}{1} \]

   2. metadata-eval 2.4%

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

6. Simplified 2.4%

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

7. Final simplification 2.4%

   \[\leadsto 0 \]
8. Add Preprocessing

Alternative 11: 28.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

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

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

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 24.2%

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

   1. +-inverses 24.2%

      \[\leadsto \frac{\cos re \cdot -2}{\cos re \cdot -2 + \color{blue}{0}} \]

   2. +-rgt-identity 24.2%

      \[\leadsto \frac{\cos re \cdot -2}{\color{blue}{\cos re \cdot -2}} \]

   3. *-inverses 24.2%

      \[\leadsto \color{blue}{1} \]

6. Simplified 24.2%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 24.2%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024082 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))