math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 11

Speedup: 1.0×

• 49.7% 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: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0054:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+37}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0054)
   (* (* 0.5 (cos re)) (fma im im 2.0))
   (if (<= im 7.5e+37)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (sqrt (* (pow im 8.0) 0.001736111111111111))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0054) {
		tmp = (0.5 * cos(re)) * fma(im, im, 2.0);
	} else if (im <= 7.5e+37) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * sqrt((pow(im, 8.0) * 0.001736111111111111));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0054)
		tmp = Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0));
	elseif (im <= 7.5e+37)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * sqrt(Float64((im ^ 8.0) * 0.001736111111111111)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0054], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.5e+37], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[Sqrt[N[(N[Power[im, 8.0], $MachinePrecision] * 0.001736111111111111), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0054000000000000003

   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 86.4%

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

      1. +-commutative 86.4%

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

      2. unpow2 86.4%

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

      3. fma-define 86.4%

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

   5. Simplified 86.4%

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

   if 0.0054000000000000003 < im < 7.5000000000000003e37

   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 90.0%

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

   if 7.5000000000000003e37 < 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 91.6%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 91.6%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 91.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 91.6%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 91.6%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 91.6%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   6. Taylor expanded in im around inf 91.6%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 91.6%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \cos re} \]

   8. Simplified 91.6%

      \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \cos re} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 91.6%

         \[\leadsto \color{blue}{\left(\sqrt{0.041666666666666664 \cdot {im}^{4}} \cdot \sqrt{0.041666666666666664 \cdot {im}^{4}}\right)} \cdot \cos re \]

      2. sqrt-unprod 98.3%

         \[\leadsto \color{blue}{\sqrt{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)}} \cdot \cos re \]

      3. pow1/2 98.3%

         \[\leadsto \color{blue}{{\left(\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\right)}^{0.5}} \cdot \cos re \]

      4. *-commutative 98.3%

         \[\leadsto {\left(\color{blue}{\left({im}^{4} \cdot 0.041666666666666664\right)} \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\right)}^{0.5} \cdot \cos re \]

      5. *-commutative 98.3%

         \[\leadsto {\left(\left({im}^{4} \cdot 0.041666666666666664\right) \cdot \color{blue}{\left({im}^{4} \cdot 0.041666666666666664\right)}\right)}^{0.5} \cdot \cos re \]

      6. swap-sqr 98.3%

         \[\leadsto {\color{blue}{\left(\left({im}^{4} \cdot {im}^{4}\right) \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)\right)}}^{0.5} \cdot \cos re \]

      7. pow-prod-up 98.3%

         \[\leadsto {\left(\color{blue}{{im}^{\left(4 + 4\right)}} \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)\right)}^{0.5} \cdot \cos re \]

      8. metadata-eval 98.3%

         \[\leadsto {\left({im}^{\color{blue}{8}} \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)\right)}^{0.5} \cdot \cos re \]

      9. metadata-eval 98.3%

         \[\leadsto {\left({im}^{8} \cdot \color{blue}{0.001736111111111111}\right)}^{0.5} \cdot \cos re \]

   10. Applied egg-rr 98.3%

       \[\leadsto \color{blue}{{\left({im}^{8} \cdot 0.001736111111111111\right)}^{0.5}} \cdot \cos re \]
   11. Step-by-step derivation

       1. unpow1/2 98.3%

          \[\leadsto \color{blue}{\sqrt{{im}^{8} \cdot 0.001736111111111111}} \cdot \cos re \]

   12. Simplified 98.3%

       \[\leadsto \color{blue}{\sqrt{{im}^{8} \cdot 0.001736111111111111}} \cdot \cos re \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0054:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+37}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 14200000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+61}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 14200000.0)
   (cos re)
   (if (<= im 8.2e+61)
     (+ 0.25 (pow re -2.0))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 14200000.0) {
		tmp = cos(re);
	} else if (im <= 8.2e+61) {
		tmp = 0.25 + pow(re, -2.0);
	} else {
		tmp = 0.041666666666666664 * (cos(re) * 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 <= 14200000.0d0) then
        tmp = cos(re)
    else if (im <= 8.2d+61) then
        tmp = 0.25d0 + (re ** (-2.0d0))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 14200000.0) {
		tmp = Math.cos(re);
	} else if (im <= 8.2e+61) {
		tmp = 0.25 + Math.pow(re, -2.0);
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 14200000.0:
		tmp = math.cos(re)
	elif im <= 8.2e+61:
		tmp = 0.25 + math.pow(re, -2.0)
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 14200000.0)
		tmp = cos(re);
	elseif (im <= 8.2e+61)
		tmp = Float64(0.25 + (re ^ -2.0));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 14200000.0)
		tmp = cos(re);
	elseif (im <= 8.2e+61)
		tmp = 0.25 + (re ^ -2.0);
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 14200000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 8.2e+61], N[(0.25 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.42e7

   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 69.0%

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

   if 1.42e7 < im < 8.19999999999999944e61

   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 2.7%

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

   5. Taylor expanded in re around 0 11.9%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 11.9%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 11.9%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   9. Applied egg-rr 47.0%

      \[\leadsto 0.25 + \color{blue}{{re}^{-2}} \]

   if 8.19999999999999944e61 < 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 96.5%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 96.5%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 96.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 96.5%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 96.5%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 96.5%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   6. Taylor expanded in im around inf 96.5%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 14200000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+61}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.6e-13)
   (cos re)
   (if (<= im 2.9e+71)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.6e-13) {
		tmp = cos(re);
	} else if (im <= 2.9e+71) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * 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 <= 3.6d-13) then
        tmp = cos(re)
    else if (im <= 2.9d+71) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.6e-13) {
		tmp = Math.cos(re);
	} else if (im <= 2.9e+71) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.6e-13:
		tmp = math.cos(re)
	elif im <= 2.9e+71:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.6e-13)
		tmp = cos(re);
	elseif (im <= 2.9e+71)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.6e-13)
		tmp = cos(re);
	elseif (im <= 2.9e+71)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.6e-13], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.9e+71], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.5999999999999998e-13

   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 69.2%

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

   if 3.5999999999999998e-13 < im < 2.90000000000000007e71

   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 82.4%

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

   if 2.90000000000000007e71 < 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%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0054:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0054)
   (* (* 0.5 (cos re)) (fma im im 2.0))
   (if (<= im 2.9e+71)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0054) {
		tmp = (0.5 * cos(re)) * fma(im, im, 2.0);
	} else if (im <= 2.9e+71) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0054)
		tmp = Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0));
	elseif (im <= 2.9e+71)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0054], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.9e+71], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0054000000000000003

   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 86.4%

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

      1. +-commutative 86.4%

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

      2. unpow2 86.4%

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

      3. fma-define 86.4%

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

   5. Simplified 86.4%

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

   if 0.0054000000000000003 < im < 2.90000000000000007e71

   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 80.0%

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

   if 2.90000000000000007e71 < 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%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re, \cos re\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re, \cos re\right) \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \cos re\right) \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \cos re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \cos re\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \cos re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \cos re\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0054:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 14200000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 14200000.0)
   (cos re)
   (if (<= im 1.35e+154) (+ 0.25 (pow re -2.0)) (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 14200000.0) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.25 + pow(re, -2.0);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 14200000.0)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(0.25 + (re ^ -2.0));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 14200000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.25 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.42e7

   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 69.0%

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

   if 1.42e7 < 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

   4. Applied egg-rr 2.4%

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

   5. Taylor expanded in re around 0 12.5%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 12.5%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 12.5%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   9. Applied egg-rr 29.2%

      \[\leadsto 0.25 + \color{blue}{{re}^{-2}} \]

   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%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 85.7%

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

      1. +-commutative 85.7%

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

      2. unpow2 85.7%

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

      3. fma-undefine 85.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   8. Simplified 85.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 14200000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 14200000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 14200000.0) (cos re) (+ 0.25 (pow re -2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 14200000.0) {
		tmp = cos(re);
	} else {
		tmp = 0.25 + pow(re, -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 <= 14200000.0d0) then
        tmp = cos(re)
    else
        tmp = 0.25d0 + (re ** (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 14200000.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.25 + Math.pow(re, -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 14200000.0:
		tmp = math.cos(re)
	else:
		tmp = 0.25 + math.pow(re, -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 14200000.0)
		tmp = cos(re);
	else
		tmp = Float64(0.25 + (re ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 14200000.0)
		tmp = cos(re);
	else
		tmp = 0.25 + (re ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 14200000.0], N[Cos[re], $MachinePrecision], N[(0.25 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.42e7

   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 69.0%

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

   if 1.42e7 < im

   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 2.6%

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

   5. Taylor expanded in re around 0 15.0%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 15.0%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

   7. Simplified 15.0%

      \[\leadsto \color{blue}{0.25 + {re}^{2} \cdot 0.25} \]

   9. Applied egg-rr 30.1%

      \[\leadsto 0.25 + \color{blue}{{re}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 14200000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + {re}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.9% 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 52.5%

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

4. Final simplification 52.5%

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

Alternative 9: 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 10: 8.0% 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 8.0%

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

5. Taylor expanded in re around 0 8.1%

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

6. Final simplification 8.1%

   \[\leadsto 0.25 \]
7. 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 27.6%

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

   1. +-inverses 27.6%

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

   2. +-rgt-identity 27.6%

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

   3. *-inverses 27.6%

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

6. Simplified 27.6%

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

7. Final simplification 27.6%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))