math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.5s

Alternatives: 19

Speedup: 1.0×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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. Final simplification 100.0%

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

Alternative 2: 95.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := 0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{if}\;im \leq -6.5 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.00096:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-29}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ (exp (- im)) (exp im))))
        (t_1 (* 0.041666666666666664 (* (cos re) (pow im 4.0)))))
   (if (<= im -6.5e+97)
     t_1
     (if (<= im -0.00096)
       t_0
       (if (<= im 1.45e-29) (cos re) (if (<= im 1.2e+77) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) + exp(im));
	double t_1 = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	double tmp;
	if (im <= -6.5e+97) {
		tmp = t_1;
	} else if (im <= -0.00096) {
		tmp = t_0;
	} else if (im <= 1.45e-29) {
		tmp = cos(re);
	} else if (im <= 1.2e+77) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) + exp(im))
    t_1 = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    if (im <= (-6.5d+97)) then
        tmp = t_1
    else if (im <= (-0.00096d0)) then
        tmp = t_0
    else if (im <= 1.45d-29) then
        tmp = cos(re)
    else if (im <= 1.2d+77) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) + Math.exp(im));
	double t_1 = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	double tmp;
	if (im <= -6.5e+97) {
		tmp = t_1;
	} else if (im <= -0.00096) {
		tmp = t_0;
	} else if (im <= 1.45e-29) {
		tmp = Math.cos(re);
	} else if (im <= 1.2e+77) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) + math.exp(im))
	t_1 = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	tmp = 0
	if im <= -6.5e+97:
		tmp = t_1
	elif im <= -0.00096:
		tmp = t_0
	elif im <= 1.45e-29:
		tmp = math.cos(re)
	elif im <= 1.2e+77:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)))
	t_1 = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)))
	tmp = 0.0
	if (im <= -6.5e+97)
		tmp = t_1;
	elseif (im <= -0.00096)
		tmp = t_0;
	elseif (im <= 1.45e-29)
		tmp = cos(re);
	elseif (im <= 1.2e+77)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) + exp(im));
	t_1 = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	tmp = 0.0;
	if (im <= -6.5e+97)
		tmp = t_1;
	elseif (im <= -0.00096)
		tmp = t_0;
	elseif (im <= 1.45e-29)
		tmp = cos(re);
	elseif (im <= 1.2e+77)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.5e+97], t$95$1, If[LessEqual[im, -0.00096], t$95$0, If[LessEqual[im, 1.45e-29], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.2e+77], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.4999999999999999e97 or 1.1999999999999999e77 < im

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -6.4999999999999999e97 < im < -9.60000000000000024e-4 or 1.45000000000000012e-29 < im < 1.1999999999999999e77

   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 89.2%

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

   if -9.60000000000000024e-4 < im < 1.45000000000000012e-29

   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 99.6%

      \[\leadsto \color{blue}{\cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+97}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -0.00096:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-29}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 3: 95.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := 0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{if}\;im \leq -6.5 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.065:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-29}:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ (exp (- im)) (exp im))))
        (t_1 (* 0.041666666666666664 (* (cos re) (pow im 4.0)))))
   (if (<= im -6.5e+97)
     t_1
     (if (<= im -0.065)
       t_0
       (if (<= im 1.45e-29)
         (* (cos re) (fma im (* 0.5 im) 1.0))
         (if (<= im 1.15e+77) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) + exp(im));
	double t_1 = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	double tmp;
	if (im <= -6.5e+97) {
		tmp = t_1;
	} else if (im <= -0.065) {
		tmp = t_0;
	} else if (im <= 1.45e-29) {
		tmp = cos(re) * fma(im, (0.5 * im), 1.0);
	} else if (im <= 1.15e+77) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)))
	t_1 = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)))
	tmp = 0.0
	if (im <= -6.5e+97)
		tmp = t_1;
	elseif (im <= -0.065)
		tmp = t_0;
	elseif (im <= 1.45e-29)
		tmp = Float64(cos(re) * fma(im, Float64(0.5 * im), 1.0));
	elseif (im <= 1.15e+77)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.5e+97], t$95$1, If[LessEqual[im, -0.065], t$95$0, If[LessEqual[im, 1.45e-29], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.4999999999999999e97 or 1.14999999999999997e77 < im

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -6.4999999999999999e97 < im < -0.065000000000000002 or 1.45000000000000012e-29 < 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 89.2%

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

   if -0.065000000000000002 < im < 1.45000000000000012e-29

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. unpow2 99.9%

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

      3. fma-udef 99.9%

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

      4. *-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in re around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. distribute-lft1-in 99.9%

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

      4. *-commutative 99.9%

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

      5. unpow2 99.9%

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

      6. associate-*l* 99.9%

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

      7. fma-def 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+97}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -0.065:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-29}:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 4: 87.7% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.2) (not (<= im 2.2)))
   (* 0.041666666666666664 (* (cos re) (pow im 4.0)))
   (cos re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.2) || !(im <= 2.2)) {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	} else {
		tmp = cos(re);
	}
	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)) .or. (.not. (im <= 2.2d0))) then
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    else
        tmp = cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.2) || !(im <= 2.2)) {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	} else {
		tmp = Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.2) or not (im <= 2.2):
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	else:
		tmp = math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.2) || !(im <= 2.2))
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	else
		tmp = cos(re);
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.2], N[Not[LessEqual[im, 2.2]], $MachinePrecision]], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cos[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.2000000000000002 or 2.2000000000000002 < im

   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 81.5%

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

   4. Simplified 81.5%

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

   5. Taylor expanded in im around inf 81.5%

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

      1. *-commutative 81.5%

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

   7. Simplified 81.5%

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

   if -2.2000000000000002 < im < 2.2000000000000002

   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 99.6%

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

4. Final simplification 89.6%

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

Alternative 5: 84.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ t_1 := 0.041666666666666664 \cdot {im}^{4}\\ \mathbf{if}\;im \leq -1.4 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6.2 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3800000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* im (* 0.5 im))))
        (t_1 (* 0.041666666666666664 (pow im 4.0))))
   (if (<= im -1.4e+152)
     t_0
     (if (<= im -6.2e+19)
       t_1
       (if (<= im 3800000000.0) (cos re) (if (<= im 1.9e+154) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (im * (0.5 * im));
	double t_1 = 0.041666666666666664 * pow(im, 4.0);
	double tmp;
	if (im <= -1.4e+152) {
		tmp = t_0;
	} else if (im <= -6.2e+19) {
		tmp = t_1;
	} else if (im <= 3800000000.0) {
		tmp = cos(re);
	} else if (im <= 1.9e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(re) * (im * (0.5d0 * im))
    t_1 = 0.041666666666666664d0 * (im ** 4.0d0)
    if (im <= (-1.4d+152)) then
        tmp = t_0
    else if (im <= (-6.2d+19)) then
        tmp = t_1
    else if (im <= 3800000000.0d0) then
        tmp = cos(re)
    else if (im <= 1.9d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (im * (0.5 * im));
	double t_1 = 0.041666666666666664 * Math.pow(im, 4.0);
	double tmp;
	if (im <= -1.4e+152) {
		tmp = t_0;
	} else if (im <= -6.2e+19) {
		tmp = t_1;
	} else if (im <= 3800000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.9e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (im * (0.5 * im))
	t_1 = 0.041666666666666664 * math.pow(im, 4.0)
	tmp = 0
	if im <= -1.4e+152:
		tmp = t_0
	elif im <= -6.2e+19:
		tmp = t_1
	elif im <= 3800000000.0:
		tmp = math.cos(re)
	elif im <= 1.9e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(im * Float64(0.5 * im)))
	t_1 = Float64(0.041666666666666664 * (im ^ 4.0))
	tmp = 0.0
	if (im <= -1.4e+152)
		tmp = t_0;
	elseif (im <= -6.2e+19)
		tmp = t_1;
	elseif (im <= 3800000000.0)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (im * (0.5 * im));
	t_1 = 0.041666666666666664 * (im ^ 4.0);
	tmp = 0.0;
	if (im <= -1.4e+152)
		tmp = t_0;
	elseif (im <= -6.2e+19)
		tmp = t_1;
	elseif (im <= 3800000000.0)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.4e+152], t$95$0, If[LessEqual[im, -6.2e+19], t$95$1, If[LessEqual[im, 3800000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.9e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.4000000000000001e152 or 1.8999999999999999e154 < im

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 98.9%

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

      1. associate-*r* 98.9%

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

      2. unpow2 98.9%

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

      3. fma-udef 98.9%

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

      4. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in im around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l* 98.9%

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

      3. unpow2 98.9%

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

      4. associate-*l* 98.9%

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

   10. Simplified 98.9%

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

   if -1.4000000000000001e152 < im < -6.2e19 or 3.8e9 < im < 1.8999999999999999e154

   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 61.4%

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

   4. Simplified 61.4%

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

   5. Taylor expanded in im around inf 61.4%

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

      1. *-commutative 61.4%

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

   7. Simplified 61.4%

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

   8. Taylor expanded in re around 0 53.2%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]

   if -6.2e19 < im < 3.8e9

   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 97.9%

      \[\leadsto \color{blue}{\cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.4 \cdot 10^{+152}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -6.2 \cdot 10^{+19}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{elif}\;im \leq 3800000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 78.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 \cdot {im}^{4}\\ t_1 := \left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{if}\;im \leq -1 \cdot 10^{+216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -4.8 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 32000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 10^{+219}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.041666666666666664 (pow im 4.0)))
        (t_1 (* (* im im) (+ 0.5 (* (* re re) -0.25)))))
   (if (<= im -1e+216)
     t_1
     (if (<= im -4.8e+18)
       t_0
       (if (<= im 32000000000.0)
         (cos re)
         (if (<= im 1e+188) t_0 (if (<= im 1e+219) t_1 (* 0.5 (* im im)))))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 * pow(im, 4.0);
	double t_1 = (im * im) * (0.5 + ((re * re) * -0.25));
	double tmp;
	if (im <= -1e+216) {
		tmp = t_1;
	} else if (im <= -4.8e+18) {
		tmp = t_0;
	} else if (im <= 32000000000.0) {
		tmp = cos(re);
	} else if (im <= 1e+188) {
		tmp = t_0;
	} else if (im <= 1e+219) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.041666666666666664d0 * (im ** 4.0d0)
    t_1 = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    if (im <= (-1d+216)) then
        tmp = t_1
    else if (im <= (-4.8d+18)) then
        tmp = t_0
    else if (im <= 32000000000.0d0) then
        tmp = cos(re)
    else if (im <= 1d+188) then
        tmp = t_0
    else if (im <= 1d+219) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.041666666666666664 * Math.pow(im, 4.0);
	double t_1 = (im * im) * (0.5 + ((re * re) * -0.25));
	double tmp;
	if (im <= -1e+216) {
		tmp = t_1;
	} else if (im <= -4.8e+18) {
		tmp = t_0;
	} else if (im <= 32000000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 1e+188) {
		tmp = t_0;
	} else if (im <= 1e+219) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 * math.pow(im, 4.0)
	t_1 = (im * im) * (0.5 + ((re * re) * -0.25))
	tmp = 0
	if im <= -1e+216:
		tmp = t_1
	elif im <= -4.8e+18:
		tmp = t_0
	elif im <= 32000000000.0:
		tmp = math.cos(re)
	elif im <= 1e+188:
		tmp = t_0
	elif im <= 1e+219:
		tmp = t_1
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 * (im ^ 4.0))
	t_1 = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)))
	tmp = 0.0
	if (im <= -1e+216)
		tmp = t_1;
	elseif (im <= -4.8e+18)
		tmp = t_0;
	elseif (im <= 32000000000.0)
		tmp = cos(re);
	elseif (im <= 1e+188)
		tmp = t_0;
	elseif (im <= 1e+219)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 * (im ^ 4.0);
	t_1 = (im * im) * (0.5 + ((re * re) * -0.25));
	tmp = 0.0;
	if (im <= -1e+216)
		tmp = t_1;
	elseif (im <= -4.8e+18)
		tmp = t_0;
	elseif (im <= 32000000000.0)
		tmp = cos(re);
	elseif (im <= 1e+188)
		tmp = t_0;
	elseif (im <= 1e+219)
		tmp = t_1;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1e+216], t$95$1, If[LessEqual[im, -4.8e+18], t$95$0, If[LessEqual[im, 32000000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1e+188], t$95$0, If[LessEqual[im, 1e+219], t$95$1, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1e216 or 1e188 < im < 9.99999999999999965e218

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

      3. fma-udef 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*l* 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in re around 0 0.0%

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

       1. associate-*r* 0.0%

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

       2. distribute-rgt-out 93.1%

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

       3. unpow2 93.1%

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

       4. *-commutative 93.1%

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

       5. unpow2 93.1%

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

   13. Simplified 93.1%

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

   if -1e216 < im < -4.8e18 or 3.2e10 < im < 1e188

   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 73.3%

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

   4. Simplified 73.3%

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

   5. Taylor expanded in im around inf 73.3%

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

      1. *-commutative 73.3%

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

   7. Simplified 73.3%

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

   8. Taylor expanded in re around 0 61.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]

   if -4.8e18 < im < 3.2e10

   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 97.9%

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

   if 9.99999999999999965e218 < im

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

      3. fma-udef 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*l* 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in re around 0 95.0%

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

       1. unpow2 95.0%

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

   13. Simplified 95.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot im\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+216}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq -4.8 \cdot 10^{+18}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{elif}\;im \leq 32000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+188}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{elif}\;im \leq 10^{+219}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 7: 71.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := 0.25 + \left(re \cdot re\right) \cdot 0.25\\ t_2 := \left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{if}\;im \leq -1.95 \cdot 10^{+167}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -1.45 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 580:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+188}:\\ \;\;\;\;1 + t_0\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+219}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im)))
        (t_1 (+ 0.25 (* (* re re) 0.25)))
        (t_2 (* (* im im) (+ 0.5 (* (* re re) -0.25)))))
   (if (<= im -1.95e+167)
     t_2
     (if (<= im -1.45e+20)
       t_1
       (if (<= im 580.0)
         (cos re)
         (if (<= im 6.6e+134)
           t_1
           (if (<= im 5e+188) (+ 1.0 t_0) (if (<= im 1.8e+219) t_2 t_0))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = 0.25 + ((re * re) * 0.25);
	double t_2 = (im * im) * (0.5 + ((re * re) * -0.25));
	double tmp;
	if (im <= -1.95e+167) {
		tmp = t_2;
	} else if (im <= -1.45e+20) {
		tmp = t_1;
	} else if (im <= 580.0) {
		tmp = cos(re);
	} else if (im <= 6.6e+134) {
		tmp = t_1;
	} else if (im <= 5e+188) {
		tmp = 1.0 + t_0;
	} else if (im <= 1.8e+219) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (im * im)
    t_1 = 0.25d0 + ((re * re) * 0.25d0)
    t_2 = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    if (im <= (-1.95d+167)) then
        tmp = t_2
    else if (im <= (-1.45d+20)) then
        tmp = t_1
    else if (im <= 580.0d0) then
        tmp = cos(re)
    else if (im <= 6.6d+134) then
        tmp = t_1
    else if (im <= 5d+188) then
        tmp = 1.0d0 + t_0
    else if (im <= 1.8d+219) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = 0.25 + ((re * re) * 0.25);
	double t_2 = (im * im) * (0.5 + ((re * re) * -0.25));
	double tmp;
	if (im <= -1.95e+167) {
		tmp = t_2;
	} else if (im <= -1.45e+20) {
		tmp = t_1;
	} else if (im <= 580.0) {
		tmp = Math.cos(re);
	} else if (im <= 6.6e+134) {
		tmp = t_1;
	} else if (im <= 5e+188) {
		tmp = 1.0 + t_0;
	} else if (im <= 1.8e+219) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im * im)
	t_1 = 0.25 + ((re * re) * 0.25)
	t_2 = (im * im) * (0.5 + ((re * re) * -0.25))
	tmp = 0
	if im <= -1.95e+167:
		tmp = t_2
	elif im <= -1.45e+20:
		tmp = t_1
	elif im <= 580.0:
		tmp = math.cos(re)
	elif im <= 6.6e+134:
		tmp = t_1
	elif im <= 5e+188:
		tmp = 1.0 + t_0
	elif im <= 1.8e+219:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(0.25 + Float64(Float64(re * re) * 0.25))
	t_2 = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)))
	tmp = 0.0
	if (im <= -1.95e+167)
		tmp = t_2;
	elseif (im <= -1.45e+20)
		tmp = t_1;
	elseif (im <= 580.0)
		tmp = cos(re);
	elseif (im <= 6.6e+134)
		tmp = t_1;
	elseif (im <= 5e+188)
		tmp = Float64(1.0 + t_0);
	elseif (im <= 1.8e+219)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im * im);
	t_1 = 0.25 + ((re * re) * 0.25);
	t_2 = (im * im) * (0.5 + ((re * re) * -0.25));
	tmp = 0.0;
	if (im <= -1.95e+167)
		tmp = t_2;
	elseif (im <= -1.45e+20)
		tmp = t_1;
	elseif (im <= 580.0)
		tmp = cos(re);
	elseif (im <= 6.6e+134)
		tmp = t_1;
	elseif (im <= 5e+188)
		tmp = 1.0 + t_0;
	elseif (im <= 1.8e+219)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 + N[(N[(re * re), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.95e+167], t$95$2, If[LessEqual[im, -1.45e+20], t$95$1, If[LessEqual[im, 580.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 6.6e+134], t$95$1, If[LessEqual[im, 5e+188], N[(1.0 + t$95$0), $MachinePrecision], If[LessEqual[im, 1.8e+219], t$95$2, t$95$0]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -1.9499999999999999e167 or 5.0000000000000001e188 < im < 1.80000000000000003e219

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

      3. fma-udef 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*l* 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in re around 0 0.0%

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

       1. associate-*r* 0.0%

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

       2. distribute-rgt-out 86.8%

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

       3. unpow2 86.8%

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

       4. *-commutative 86.8%

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

       5. unpow2 86.8%

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

   13. Simplified 86.8%

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

   if -1.9499999999999999e167 < im < -1.45e20 or 580 < im < 6.6e134

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.8%

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

   4. Taylor expanded in re around 0 25.9%

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

      1. *-commutative 25.9%

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

      2. unpow2 25.9%

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

   6. Simplified 25.9%

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

   if -1.45e20 < im < 580

   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 97.9%

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

   if 6.6e134 < im < 5.0000000000000001e188

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 75.0%

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

      1. associate-*r* 75.0%

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

      2. unpow2 75.0%

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

      3. fma-udef 75.0%

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

      4. *-commutative 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in re around 0 63.5%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 63.5%

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

   10. Simplified 63.5%

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

   if 1.80000000000000003e219 < im

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

      3. fma-udef 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*l* 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in re around 0 95.0%

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

       1. unpow2 95.0%

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

   13. Simplified 95.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot im\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.95 \cdot 10^{+167}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq -1.45 \cdot 10^{+20}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{elif}\;im \leq 580:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+134}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+188}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+219}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 8: 49.6% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := 0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{if}\;im \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 310:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im))) (t_1 (+ 0.25 (* (* re re) 0.25))))
   (if (<= im -9.5e+153)
     t_0
     (if (<= im -4.1e+18)
       t_1
       (if (<= im 310.0) 1.0 (if (<= im 6.5e+135) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = 0.25 + ((re * re) * 0.25);
	double tmp;
	if (im <= -9.5e+153) {
		tmp = t_0;
	} else if (im <= -4.1e+18) {
		tmp = t_1;
	} else if (im <= 310.0) {
		tmp = 1.0;
	} else if (im <= 6.5e+135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (im * im)
    t_1 = 0.25d0 + ((re * re) * 0.25d0)
    if (im <= (-9.5d+153)) then
        tmp = t_0
    else if (im <= (-4.1d+18)) then
        tmp = t_1
    else if (im <= 310.0d0) then
        tmp = 1.0d0
    else if (im <= 6.5d+135) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = 0.25 + ((re * re) * 0.25);
	double tmp;
	if (im <= -9.5e+153) {
		tmp = t_0;
	} else if (im <= -4.1e+18) {
		tmp = t_1;
	} else if (im <= 310.0) {
		tmp = 1.0;
	} else if (im <= 6.5e+135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im * im)
	t_1 = 0.25 + ((re * re) * 0.25)
	tmp = 0
	if im <= -9.5e+153:
		tmp = t_0
	elif im <= -4.1e+18:
		tmp = t_1
	elif im <= 310.0:
		tmp = 1.0
	elif im <= 6.5e+135:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(0.25 + Float64(Float64(re * re) * 0.25))
	tmp = 0.0
	if (im <= -9.5e+153)
		tmp = t_0;
	elseif (im <= -4.1e+18)
		tmp = t_1;
	elseif (im <= 310.0)
		tmp = 1.0;
	elseif (im <= 6.5e+135)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im * im);
	t_1 = 0.25 + ((re * re) * 0.25);
	tmp = 0.0;
	if (im <= -9.5e+153)
		tmp = t_0;
	elseif (im <= -4.1e+18)
		tmp = t_1;
	elseif (im <= 310.0)
		tmp = 1.0;
	elseif (im <= 6.5e+135)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 + N[(N[(re * re), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9.5e+153], t$95$0, If[LessEqual[im, -4.1e+18], t$95$1, If[LessEqual[im, 310.0], 1.0, If[LessEqual[im, 6.5e+135], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.4999999999999995e153 or 6.5000000000000003e135 < im

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 94.4%

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

      1. associate-*r* 94.4%

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

      2. unpow2 94.4%

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

      3. fma-udef 94.4%

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

      4. *-commutative 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in im around inf 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l* 94.4%

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

      3. unpow2 94.4%

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

      4. associate-*l* 94.4%

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

   10. Simplified 94.4%

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

   11. Taylor expanded in re around 0 69.7%

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

       1. unpow2 69.7%

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

   13. Simplified 69.7%

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

   if -9.4999999999999995e153 < im < -4.1e18 or 310 < im < 6.5000000000000003e135

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.8%

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

   4. Taylor expanded in re around 0 24.4%

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

      1. *-commutative 24.4%

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

      2. unpow2 24.4%

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

   6. Simplified 24.4%

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

   if -4.1e18 < im < 310

   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 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in im around inf 3.4%

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

      1. *-commutative 3.4%

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

   7. Simplified 3.4%

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

   9. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{elif}\;im \leq 310:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+135}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 9: 49.5% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := re \cdot re - re\\ \mathbf{if}\;im \leq -1.9 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im))) (t_1 (- (* re re) re)))
   (if (<= im -1.9e+153)
     t_0
     (if (<= im -1.4e+19)
       t_1
       (if (<= im 360.0) 1.0 (if (<= im 1.05e+136) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = (re * re) - re;
	double tmp;
	if (im <= -1.9e+153) {
		tmp = t_0;
	} else if (im <= -1.4e+19) {
		tmp = t_1;
	} else if (im <= 360.0) {
		tmp = 1.0;
	} else if (im <= 1.05e+136) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (im * im)
    t_1 = (re * re) - re
    if (im <= (-1.9d+153)) then
        tmp = t_0
    else if (im <= (-1.4d+19)) then
        tmp = t_1
    else if (im <= 360.0d0) then
        tmp = 1.0d0
    else if (im <= 1.05d+136) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = (re * re) - re;
	double tmp;
	if (im <= -1.9e+153) {
		tmp = t_0;
	} else if (im <= -1.4e+19) {
		tmp = t_1;
	} else if (im <= 360.0) {
		tmp = 1.0;
	} else if (im <= 1.05e+136) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im * im)
	t_1 = (re * re) - re
	tmp = 0
	if im <= -1.9e+153:
		tmp = t_0
	elif im <= -1.4e+19:
		tmp = t_1
	elif im <= 360.0:
		tmp = 1.0
	elif im <= 1.05e+136:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(Float64(re * re) - re)
	tmp = 0.0
	if (im <= -1.9e+153)
		tmp = t_0;
	elseif (im <= -1.4e+19)
		tmp = t_1;
	elseif (im <= 360.0)
		tmp = 1.0;
	elseif (im <= 1.05e+136)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im * im);
	t_1 = (re * re) - re;
	tmp = 0.0;
	if (im <= -1.9e+153)
		tmp = t_0;
	elseif (im <= -1.4e+19)
		tmp = t_1;
	elseif (im <= 360.0)
		tmp = 1.0;
	elseif (im <= 1.05e+136)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(re * re), $MachinePrecision] - re), $MachinePrecision]}, If[LessEqual[im, -1.9e+153], t$95$0, If[LessEqual[im, -1.4e+19], t$95$1, If[LessEqual[im, 360.0], 1.0, If[LessEqual[im, 1.05e+136], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.89999999999999983e153 or 1.05e136 < im

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 94.4%

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

      1. associate-*r* 94.4%

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

      2. unpow2 94.4%

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

      3. fma-udef 94.4%

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

      4. *-commutative 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in im around inf 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l* 94.4%

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

      3. unpow2 94.4%

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

      4. associate-*l* 94.4%

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

   10. Simplified 94.4%

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

   11. Taylor expanded in re around 0 69.7%

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

       1. unpow2 69.7%

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

   13. Simplified 69.7%

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

   if -1.89999999999999983e153 < im < -1.4e19 or 360 < im < 1.05e136

   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 0.0%

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

   4. Simplified 62.7%

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

   6. Applied egg-rr 23.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, -re\right)} \]
   7. Step-by-step derivation

      1. fma-neg 23.7%

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

   8. Simplified 23.7%

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

   if -1.4e19 < im < 360

   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 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in im around inf 3.4%

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

      1. *-commutative 3.4%

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

   7. Simplified 3.4%

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

   9. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.9 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;re \cdot re - re\\ \mathbf{elif}\;im \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+136}:\\ \;\;\;\;re \cdot re - re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 10: 49.0% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{+145} \lor \neg \left(re \leq 3.3 \cdot 10^{+174}\right):\\ \;\;\;\;re \cdot re - re\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -6e+145) (not (<= re 3.3e+174)))
   (- (* re re) re)
   (+ 1.0 (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -6e+145) || !(re <= 3.3e+174)) {
		tmp = (re * re) - re;
	} else {
		tmp = 1.0 + (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-6d+145)) .or. (.not. (re <= 3.3d+174))) then
        tmp = (re * re) - re
    else
        tmp = 1.0d0 + (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -6e+145) || !(re <= 3.3e+174)) {
		tmp = (re * re) - re;
	} else {
		tmp = 1.0 + (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -6e+145) or not (re <= 3.3e+174):
		tmp = (re * re) - re
	else:
		tmp = 1.0 + (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -6e+145) || !(re <= 3.3e+174))
		tmp = Float64(Float64(re * re) - re);
	else
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -6e+145) || ~((re <= 3.3e+174)))
		tmp = (re * re) - re;
	else
		tmp = 1.0 + (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -6e+145], N[Not[LessEqual[re, 3.3e+174]], $MachinePrecision]], N[(N[(re * re), $MachinePrecision] - re), $MachinePrecision], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.0000000000000005e145 or 3.3000000000000001e174 < re

   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 0.6%

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

   4. Simplified 24.8%

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

   6. Applied egg-rr 38.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, -re\right)} \]
   7. Step-by-step derivation

      1. fma-neg 38.1%

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

   8. Simplified 38.1%

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

   if -6.0000000000000005e145 < re < 3.3000000000000001e174

   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 90.8%

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

   4. Simplified 90.8%

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

   5. Taylor expanded in im around 0 76.1%

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

      1. associate-*r* 76.1%

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

      2. unpow2 76.1%

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

      3. fma-udef 76.1%

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

      4. *-commutative 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in re around 0 55.2%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 55.2%

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

   10. Simplified 55.2%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{+145} \lor \neg \left(re \leq 3.3 \cdot 10^{+174}\right):\\ \;\;\;\;re \cdot re - re\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 11: 47.4% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.00041 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.00041) (not (<= im 1.4))) (* 0.5 (* im im)) 1.0))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.00041) || !(im <= 1.4)) {
		tmp = 0.5 * (im * im);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-0.00041d0)) .or. (.not. (im <= 1.4d0))) then
        tmp = 0.5d0 * (im * im)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.00041) || !(im <= 1.4)) {
		tmp = 0.5 * (im * im);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.00041) or not (im <= 1.4):
		tmp = 0.5 * (im * im)
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.00041) || !(im <= 1.4))
		tmp = Float64(0.5 * Float64(im * im));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.00041) || ~((im <= 1.4)))
		tmp = 0.5 * (im * im);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.00041], N[Not[LessEqual[im, 1.4]], $MachinePrecision]], N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if im < -4.0999999999999999e-4 or 1.3999999999999999 < im

   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 81.7%

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

   4. Simplified 81.7%

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

   5. Taylor expanded in im around 0 56.4%

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

      1. associate-*r* 56.4%

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

      2. unpow2 56.4%

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

      3. fma-udef 56.4%

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

      4. *-commutative 56.4%

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

   7. Simplified 56.4%

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

   8. Taylor expanded in im around inf 55.8%

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

      1. *-commutative 55.8%

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

      2. associate-*l* 55.8%

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

      3. unpow2 55.8%

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

      4. associate-*l* 55.8%

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

   10. Simplified 55.8%

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

   11. Taylor expanded in re around 0 41.4%

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

       1. unpow2 41.4%

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

   13. Simplified 41.4%

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

   if -4.0999999999999999e-4 < im < 1.3999999999999999

   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 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 3.3%

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

      1. *-commutative 3.3%

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

   7. Simplified 3.3%

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

   9. Applied egg-rr 54.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00041 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 3.9% 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 im around 0 89.8%

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

4. Simplified 89.8%

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

5. Taylor expanded in im around inf 46.7%

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

   1. *-commutative 46.7%

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

7. Simplified 46.7%

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

9. Applied egg-rr 3.3%

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

10. Final simplification 3.3%

    \[\leadsto -1 \]

Alternative 13: 7.4% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.041666666666666664 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.041666666666666664)

C

double code(double re, double im) {
	return 0.041666666666666664;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.041666666666666664d0
end function

Java

public static double code(double re, double im) {
	return 0.041666666666666664;
}

Python

def code(re, im):
	return 0.041666666666666664

Julia

function code(re, im)
	return 0.041666666666666664
end

MATLAB

function tmp = code(re, im)
	tmp = 0.041666666666666664;
end

Wolfram

code[re_, im_] := 0.041666666666666664

Derivation

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 89.8%

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

4. Simplified 89.8%

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

5. Taylor expanded in im around inf 46.7%

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

   1. *-commutative 46.7%

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

7. Simplified 46.7%

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

9. Applied egg-rr 7.1%

   \[\leadsto \color{blue}{0.041666666666666664} \]

10. Final simplification 7.1%

    \[\leadsto 0.041666666666666664 \]

Alternative 14: 7.5% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0625)

C

double code(double re, double im) {
	return 0.0625;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0625d0
end function

Java

public static double code(double re, double im) {
	return 0.0625;
}

Python

def code(re, im):
	return 0.0625

Julia

function code(re, im)
	return 0.0625
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0625;
end

Wolfram

code[re_, im_] := 0.0625

Derivation

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 89.8%

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

4. Simplified 89.8%

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

5. Taylor expanded in im around inf 46.7%

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

   1. *-commutative 46.7%

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

7. Simplified 46.7%

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

9. Applied egg-rr 7.3%

   \[\leadsto \color{blue}{0.0625} \]

10. Final simplification 7.3%

    \[\leadsto 0.0625 \]

Alternative 15: 7.8% 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 im around 0 89.8%

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

4. Simplified 89.8%

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

5. Taylor expanded in im around inf 46.7%

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

   1. *-commutative 46.7%

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

7. Simplified 46.7%

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

9. Applied egg-rr 7.5%

   \[\leadsto \color{blue}{0.125} \]

10. Final simplification 7.5%

    \[\leadsto 0.125 \]

Alternative 16: 8.2% 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%

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

3. Applied egg-rr 7.7%

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

4. Taylor expanded in re around 0 7.8%

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

5. Final simplification 7.8%

   \[\leadsto 0.25 \]

Alternative 17: 8.8% 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 im around 0 89.8%

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

4. Simplified 89.8%

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

5. Taylor expanded in im around inf 46.7%

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

   1. *-commutative 46.7%

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

7. Simplified 46.7%

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

9. Applied egg-rr 8.3%

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

10. Final simplification 8.3%

    \[\leadsto 0.5 \]

Alternative 18: 9.2% 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 im around 0 89.8%

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

4. Simplified 89.8%

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

5. Taylor expanded in im around inf 46.7%

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

   1. *-commutative 46.7%

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

7. Simplified 46.7%

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

9. Applied egg-rr 8.7%

   \[\leadsto \color{blue}{0.75} \]

10. Final simplification 8.7%

    \[\leadsto 0.75 \]

Alternative 19: 29.1% 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 im around 0 89.8%

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

4. Simplified 89.8%

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

5. Taylor expanded in im around inf 46.7%

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

   1. *-commutative 46.7%

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

7. Simplified 46.7%

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

9. Applied egg-rr 25.6%

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

10. Final simplification 25.6%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023196 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))