math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 14

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (cos re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

double code(double re, double im) {
	return cos(re) * fma(0.5, exp(im), (0.5 / exp(im)));
}

Julia

function code(re, im)
	return Float64(cos(re) * fma(0.5, exp(im), Float64(0.5 / exp(im))))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $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. Final simplification 100.0%

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

Alternative 3: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.3)
   (* (* (cos re) 0.5) (fma im im 2.0))
   (* (cos re) (fma 0.5 (exp im) 0.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = (cos(re) * 0.5) * fma(im, im, 2.0);
	} else {
		tmp = cos(re) * fma(0.5, exp(im), 0.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.3)
		tmp = Float64(Float64(cos(re) * 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(cos(re) * fma(0.5, exp(im), 0.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.3], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.0%

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

      1. +-commutative 78.0%

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

      2. unpow2 78.0%

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

      3. fma-def 78.0%

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

   4. Simplified 78.0%

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

   if 1.30000000000000004 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, 0\right)\\ \end{array} \]

Alternative 4: 73.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4000.0)
   (cos re)
   (if (<= im 1.15e+77)
     (* 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 <= 4000.0) {
		tmp = cos(re);
	} else if (im <= 1.15e+77) {
		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 <= 4000.0d0) then
        tmp = cos(re)
    else if (im <= 1.15d+77) 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 <= 4000.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.15e+77) {
		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 <= 4000.0:
		tmp = math.cos(re)
	elif im <= 1.15e+77:
		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 <= 4000.0)
		tmp = cos(re);
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-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 <= 4000.0)
		tmp = cos(re);
	elseif (im <= 1.15e+77)
		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, 4000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.15e+77], 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 < 4e3

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.6%

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

   if 4e3 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 75.0%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + \left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \cos re} \]

      2. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + 0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)} + \cos re \]

      3. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + 0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)\right) + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right)} \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.001388888888888889 \cdot {im}^{6}\right) \cdot \cos re} + 0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)\right) + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \left(\left(0.001388888888888889 \cdot {im}^{6}\right) \cdot \cos re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \cos re}\right) + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      6. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(0.001388888888888889 \cdot {im}^{6} + 0.041666666666666664 \cdot {im}^{4}\right)} + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      7. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right)} + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      8. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \cos re} + \cos re\right) \]

      9. distribute-lft1-in 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \cos re} \]

      10. fma-def 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \cos re \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re} \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \cos re} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]

   10. Taylor expanded in re around inf 100.0%

       \[\leadsto \cos re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right) + \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

   11. 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 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 5: 85.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4000:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4000.0)
   (* (* (cos re) 0.5) (fma im im 2.0))
   (if (<= im 1.15e+77)
     (* 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 <= 4000.0) {
		tmp = (cos(re) * 0.5) * fma(im, im, 2.0);
	} else if (im <= 1.15e+77) {
		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 <= 4000.0)
		tmp = Float64(Float64(cos(re) * 0.5) * fma(im, im, 2.0));
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4000.0], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], 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 < 4e3

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 76.8%

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

      1. +-commutative 76.8%

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

      2. unpow2 76.8%

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

      3. fma-def 76.8%

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

   4. Simplified 76.8%

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

   if 4e3 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 75.0%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + \left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \cos re} \]

      2. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + 0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)} + \cos re \]

      3. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + 0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)\right) + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right)} \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.001388888888888889 \cdot {im}^{6}\right) \cdot \cos re} + 0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)\right) + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \left(\left(0.001388888888888889 \cdot {im}^{6}\right) \cdot \cos re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \cos re}\right) + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      6. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(0.001388888888888889 \cdot {im}^{6} + 0.041666666666666664 \cdot {im}^{4}\right)} + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      7. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right)} + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      8. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \cos re} + \cos re\right) \]

      9. distribute-lft1-in 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \cos re} \]

      10. fma-def 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \cos re \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re} \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \cos re} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]

   10. Taylor expanded in re around inf 100.0%

       \[\leadsto \cos re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right) + \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

   11. 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 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4000:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 6: 68.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.2) (cos re) (* 0.041666666666666664 (* (cos re) (pow im 4.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.2) {
		tmp = cos(re);
	} 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 <= 2.2d0) then
        tmp = cos(re)
    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 <= 2.2) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.2:
		tmp = math.cos(re)
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.2)
		tmp = cos(re);
	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 <= 2.2)
		tmp = cos(re);
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.2], N[Cos[re], $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.2000000000000002

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 63.5%

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

   if 2.2000000000000002 < im

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 77.3%

      \[\leadsto \color{blue}{\cos re + \left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 77.3%

         \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \cos re} \]

      2. associate-+r+ 77.3%

         \[\leadsto \color{blue}{\left(\left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + 0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)\right) + 0.5 \cdot \left({im}^{2} \cdot \cos re\right)\right)} + \cos re \]

      3. associate-+l+ 77.3%

         \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right) + 0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)\right) + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right)} \]

      4. associate-*r* 77.3%

         \[\leadsto \left(\color{blue}{\left(0.001388888888888889 \cdot {im}^{6}\right) \cdot \cos re} + 0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)\right) + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      5. associate-*r* 77.3%

         \[\leadsto \left(\left(0.001388888888888889 \cdot {im}^{6}\right) \cdot \cos re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \cos re}\right) + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      6. distribute-rgt-out 77.3%

         \[\leadsto \color{blue}{\cos re \cdot \left(0.001388888888888889 \cdot {im}^{6} + 0.041666666666666664 \cdot {im}^{4}\right)} + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      7. fma-def 77.3%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right)} + \left(0.5 \cdot \left({im}^{2} \cdot \cos re\right) + \cos re\right) \]

      8. associate-*r* 77.3%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \cos re} + \cos re\right) \]

      9. distribute-lft1-in 77.3%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \cos re} \]

      10. fma-def 77.3%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \cos re \]

   6. Simplified 77.3%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.001388888888888889, {im}^{6}, 0.041666666666666664 \cdot {im}^{4}\right) + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re} \]

   7. Taylor expanded in im around 0 74.5%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]
   8. Step-by-step derivation

      1. associate-*r* 74.5%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \cos re} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]

      2. *-commutative 74.5%

         \[\leadsto \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]

   9. Simplified 74.5%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} + \mathsf{fma}\left(0.5, {im}^{2}, 1\right) \cdot \cos re \]

   10. Taylor expanded in re around inf 74.5%

       \[\leadsto \cos re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right) + \color{blue}{\cos re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

   11. Taylor expanded in im around inf 74.5%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 7: 59.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{+41}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.8e+41) (cos re) (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.8e+41) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.8e+41)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.8e+41], N[Cos[re], $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.8000000000000001e41

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 59.0%

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

   if 3.8000000000000001e41 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 69.6%

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

   3. Taylor expanded in im around 0 44.5%

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

      1. +-commutative 44.5%

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

      2. unpow2 44.5%

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

      3. fma-def 44.5%

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

   5. Simplified 44.5%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{+41}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]

Alternative 8: 50.5% 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%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 46.8%

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

5. Final simplification 46.8%

   \[\leadsto \cos re \]

Alternative 9: 3.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -0.5

Julia

function code(re, im)
	return -0.5
end

MATLAB

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

Wolfram

code[re_, im_] := -0.5

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 66.1%

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

4. Applied egg-rr 3.6%

   \[\leadsto 0.5 \cdot \color{blue}{-1} \]

5. Final simplification 3.6%

   \[\leadsto -0.5 \]

Alternative 10: 7.6% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.125)

C

double code(double re, double im) {
	return 0.125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.125d0
end function

Java

public static double code(double re, double im) {
	return 0.125;
}

Python

def code(re, im):
	return 0.125

Julia

function code(re, im)
	return 0.125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.125;
end

Wolfram

code[re_, im_] := 0.125

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 66.1%

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

4. Applied egg-rr 7.2%

   \[\leadsto 0.5 \cdot \color{blue}{0.25} \]

5. Final simplification 7.2%

   \[\leadsto 0.125 \]

Alternative 11: 7.9% 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. Taylor expanded in re around 0 66.1%

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

4. Applied egg-rr 7.5%

   \[\leadsto 0.5 \cdot \color{blue}{0.5} \]

5. Final simplification 7.5%

   \[\leadsto 0.25 \]

Alternative 12: 8.5% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.5

Julia

function code(re, im)
	return 0.5
end

MATLAB

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

Wolfram

code[re_, im_] := 0.5

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 66.1%

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

4. Applied egg-rr 8.0%

   \[\leadsto 0.5 \cdot \color{blue}{1} \]

5. Final simplification 8.0%

   \[\leadsto 0.5 \]

Alternative 13: 9.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.75)

C

double code(double re, double im) {
	return 0.75;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.75d0
end function

Java

public static double code(double re, double im) {
	return 0.75;
}

Python

def code(re, im):
	return 0.75

Julia

function code(re, im)
	return 0.75
end

MATLAB

function tmp = code(re, im)
	tmp = 0.75;
end

Wolfram

code[re_, im_] := 0.75

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 66.1%

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

4. Applied egg-rr 8.4%

   \[\leadsto 0.5 \cdot \color{blue}{1.5} \]

5. Final simplification 8.4%

   \[\leadsto 0.75 \]

Alternative 14: 28.1% 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. Taylor expanded in re around 0 66.1%

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

3. Taylor expanded in im around 0 28.2%

   \[\leadsto 0.5 \cdot \color{blue}{2} \]

4. Final simplification 28.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))