math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 15

Speedup: 1.0×

• 49.0% 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: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 1.2e+77)
     (* (+ (exp (- im)) (exp im)) (+ 0.5 (* (* re re) -0.25)))
     (* (cos re) (* (pow im 4.0) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.2e+77) {
		tmp = (exp(-im) + exp(im)) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = cos(re) * (pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 1.2d+77) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = cos(re) * ((im ** 4.0d0) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.2e+77) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 1.2e+77:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = math.cos(re) * (math.pow(im, 4.0) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.2e+77)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(cos(re) * Float64((im ^ 4.0) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 1.2e+77)
		tmp = (exp(-im) + exp(im)) * (0.5 + ((re * re) * -0.25));
	else
		tmp = cos(re) * ((im ^ 4.0) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+77], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.5

   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 82.5%

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

   4. Simplified 82.5%

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

   if 2.5 < im < 1.1999999999999999e77

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in re around 0 7.7%

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

      1. associate-*r* 7.7%

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

      2. distribute-rgt-out 76.9%

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

      3. *-commutative 76.9%

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

      4. unpow2 76.9%

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

   6. Simplified 76.9%

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

   if 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%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   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. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 3: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5)
   (*
    (* 0.5 (cos re))
    (+ (+ 2.0 (* im im)) (* 0.08333333333333333 (pow im 4.0))))
   (if (<= im 1.2e+77)
     (* (+ (exp (- im)) (exp im)) (+ 0.5 (* (* re re) -0.25)))
     (* (cos re) (* (pow im 4.0) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = (0.5 * cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * pow(im, 4.0)));
	} else if (im <= 1.2e+77) {
		tmp = (exp(-im) + exp(im)) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = cos(re) * (pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d0) then
        tmp = (0.5d0 * cos(re)) * ((2.0d0 + (im * im)) + (0.08333333333333333d0 * (im ** 4.0d0)))
    else if (im <= 1.2d+77) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = cos(re) * ((im ** 4.0d0) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = (0.5 * Math.cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * Math.pow(im, 4.0)));
	} else if (im <= 1.2e+77) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5:
		tmp = (0.5 * math.cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * math.pow(im, 4.0)))
	elif im <= 1.2e+77:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = math.cos(re) * (math.pow(im, 4.0) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64(0.08333333333333333 * (im ^ 4.0))));
	elseif (im <= 1.2e+77)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(cos(re) * Float64((im ^ 4.0) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5)
		tmp = (0.5 * cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * (im ^ 4.0)));
	elseif (im <= 1.2e+77)
		tmp = (exp(-im) + exp(im)) * (0.5 + ((re * re) * -0.25));
	else
		tmp = cos(re) * ((im ^ 4.0) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+77], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.5

   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 87.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 87.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   if 2.5 < im < 1.1999999999999999e77

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in re around 0 7.7%

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

      1. associate-*r* 7.7%

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

      2. distribute-rgt-out 76.9%

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

      3. *-commutative 76.9%

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

      4. unpow2 76.9%

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

   6. Simplified 76.9%

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

   if 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%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   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. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 0.053:\\ \;\;\;\;t_0 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 0.053)
     (* t_0 (+ 2.0 (* im im)))
     (if (<= im 1.35e+154)
       (* 0.5 (+ (exp (- im)) (exp im)))
       (* t_0 (* im im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 0.053) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = t_0 * (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) :: tmp
    t_0 = 0.5d0 * cos(re)
    if (im <= 0.053d0) then
        tmp = t_0 * (2.0d0 + (im * im))
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = t_0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 0.053) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = t_0 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 0.053:
		tmp = t_0 * (2.0 + (im * im))
	elif im <= 1.35e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = t_0 * (im * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 0.053)
		tmp = Float64(t_0 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(t_0 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 0.053)
		tmp = t_0 * (2.0 + (im * im));
	elseif (im <= 1.35e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = t_0 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.053], N[(t$95$0 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0529999999999999985

   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 82.8%

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

   4. Simplified 82.8%

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

   if 0.0529999999999999985 < im < 1.35000000000000003e154

   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 77.5%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.053:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 5: 87.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.039:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.039)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 1.2e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* (pow im 4.0) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.039) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.2e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.039d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 1.2d+77) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * ((im ** 4.0d0) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.039) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.2e+77) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.039:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 1.2e+77:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * (math.pow(im, 4.0) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.039)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.2e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64((im ^ 4.0) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.039)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 1.2e+77)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * ((im ^ 4.0) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.039], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0389999999999999999

   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 82.8%

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

   4. Simplified 82.8%

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

   if 0.0389999999999999999 < 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 80.3%

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

   if 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%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   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. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.039:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 6: 81.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 250:\\ \;\;\;\;t_0 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 250.0)
     (* t_0 (+ 2.0 (* im im)))
     (if (<= im 1.35e+154)
       (*
        (pow im 4.0)
        (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))
       (* t_0 (* im im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 250.0) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = pow(im, 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	} else {
		tmp = t_0 * (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) :: tmp
    t_0 = 0.5d0 * cos(re)
    if (im <= 250.0d0) then
        tmp = t_0 * (2.0d0 + (im * im))
    else if (im <= 1.35d+154) then
        tmp = (im ** 4.0d0) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    else
        tmp = t_0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 250.0) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = Math.pow(im, 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	} else {
		tmp = t_0 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 250.0:
		tmp = t_0 * (2.0 + (im * im))
	elif im <= 1.35e+154:
		tmp = math.pow(im, 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	else:
		tmp = t_0 * (im * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 250.0)
		tmp = Float64(t_0 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64((im ^ 4.0) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	else
		tmp = Float64(t_0 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 250.0)
		tmp = t_0 * (2.0 + (im * im));
	elseif (im <= 1.35e+154)
		tmp = (im ^ 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	else
		tmp = t_0 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 250.0], N[(t$95$0 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[Power[im, 4.0], $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 250

   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 82.5%

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

   4. Simplified 82.5%

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

   if 250 < im < 1.35000000000000003e154

   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 57.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 57.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   5. Taylor expanded in im around inf 57.5%

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

      1. associate-*r* 57.5%

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

      2. *-commutative 57.5%

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

      3. associate-*l* 57.5%

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

   7. Simplified 57.5%

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

   8. Taylor expanded in re around 0 5.6%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right) + 0.041666666666666664 \cdot {im}^{4}} \]
   9. Step-by-step derivation

      1. +-commutative 5.6%

         \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4} + -0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right)} \]

      2. associate-*r* 5.6%

         \[\leadsto 0.041666666666666664 \cdot {im}^{4} + \color{blue}{\left(-0.020833333333333332 \cdot {re}^{2}\right) \cdot {im}^{4}} \]

      3. distribute-rgt-out 47.0%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot {re}^{2}\right)} \]

      4. *-commutative 47.0%

         \[\leadsto {im}^{4} \cdot \left(0.041666666666666664 + \color{blue}{{re}^{2} \cdot -0.020833333333333332}\right) \]

      5. unpow2 47.0%

         \[\leadsto {im}^{4} \cdot \left(0.041666666666666664 + \color{blue}{\left(re \cdot re\right)} \cdot -0.020833333333333332\right) \]

   10. Simplified 47.0%

       \[\leadsto \color{blue}{{im}^{4} \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 7: 68.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664 + 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 250.0)
   (cos re)
   (if (<= im 1.45e+76)
     (+ 1.0 (* re (* re -0.5)))
     (if (<= im 1.35e+154)
       (+ (* (pow im 4.0) 0.041666666666666664) 1.0)
       (* (* 0.5 (cos re)) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = cos(re);
	} else if (im <= 1.45e+76) {
		tmp = 1.0 + (re * (re * -0.5));
	} else if (im <= 1.35e+154) {
		tmp = (pow(im, 4.0) * 0.041666666666666664) + 1.0;
	} else {
		tmp = (0.5 * cos(re)) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 250.0d0) then
        tmp = cos(re)
    else if (im <= 1.45d+76) then
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    else if (im <= 1.35d+154) then
        tmp = ((im ** 4.0d0) * 0.041666666666666664d0) + 1.0d0
    else
        tmp = (0.5d0 * cos(re)) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.45e+76) {
		tmp = 1.0 + (re * (re * -0.5));
	} else if (im <= 1.35e+154) {
		tmp = (Math.pow(im, 4.0) * 0.041666666666666664) + 1.0;
	} else {
		tmp = (0.5 * Math.cos(re)) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 250.0:
		tmp = math.cos(re)
	elif im <= 1.45e+76:
		tmp = 1.0 + (re * (re * -0.5))
	elif im <= 1.35e+154:
		tmp = (math.pow(im, 4.0) * 0.041666666666666664) + 1.0
	else:
		tmp = (0.5 * math.cos(re)) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 1.45e+76)
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64((im ^ 4.0) * 0.041666666666666664) + 1.0);
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 1.45e+76)
		tmp = 1.0 + (re * (re * -0.5));
	elseif (im <= 1.35e+154)
		tmp = ((im ^ 4.0) * 0.041666666666666664) + 1.0;
	else
		tmp = (0.5 * cos(re)) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 250.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.45e+76], N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 250

   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 70.2%

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

   if 250 < im < 1.4500000000000001e76

   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 3.3%

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

   3. Taylor expanded in re around 0 11.0%

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

      1. expm1-log1p-u 2.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot {re}^{2}\right)\right)} \]

      2. expm1-udef 2.0%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot {re}^{2}\right)} - 1\right)} \]

      3. *-commutative 2.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{{re}^{2} \cdot -0.5}\right)} - 1\right) \]

      4. unpow2 2.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5\right)} - 1\right) \]

      5. associate-*l* 2.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot \left(re \cdot -0.5\right)}\right)} - 1\right) \]

   5. Applied egg-rr 2.0%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(re \cdot -0.5\right)\right)} - 1\right)} \]
   6. Step-by-step derivation

      1. expm1-def 2.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(re \cdot -0.5\right)\right)\right)} \]

      2. expm1-log1p 11.0%

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

   7. Simplified 11.0%

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

   if 1.4500000000000001e76 < im < 1.35000000000000003e154

   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 95.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 95.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   5. Taylor expanded in re around 0 71.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 71.4%

         \[\leadsto \color{blue}{0.5 \cdot 2 + 0.5 \cdot \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)} \]

      2. metadata-eval 71.4%

         \[\leadsto \color{blue}{1} + 0.5 \cdot \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right) \]

      3. unpow2 71.4%

         \[\leadsto 1 + 0.5 \cdot \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right) \]

      4. *-commutative 71.4%

         \[\leadsto 1 + 0.5 \cdot \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right) \]

   7. Simplified 71.4%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)} \]

   8. Taylor expanded in im around inf 71.4%

      \[\leadsto 1 + \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]

   if 1.35000000000000003e154 < 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%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664 + 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 8: 69.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;{im}^{4} \cdot \left(re \cdot \left(re \cdot -0.020833333333333332\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664 + 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 700.0)
   (cos re)
   (if (<= im 8.2e+76)
     (* (pow im 4.0) (* re (* re -0.020833333333333332)))
     (if (<= im 1.35e+154)
       (+ (* (pow im 4.0) 0.041666666666666664) 1.0)
       (* (* 0.5 (cos re)) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		tmp = cos(re);
	} else if (im <= 8.2e+76) {
		tmp = pow(im, 4.0) * (re * (re * -0.020833333333333332));
	} else if (im <= 1.35e+154) {
		tmp = (pow(im, 4.0) * 0.041666666666666664) + 1.0;
	} else {
		tmp = (0.5 * cos(re)) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 700.0d0) then
        tmp = cos(re)
    else if (im <= 8.2d+76) then
        tmp = (im ** 4.0d0) * (re * (re * (-0.020833333333333332d0)))
    else if (im <= 1.35d+154) then
        tmp = ((im ** 4.0d0) * 0.041666666666666664d0) + 1.0d0
    else
        tmp = (0.5d0 * cos(re)) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		tmp = Math.cos(re);
	} else if (im <= 8.2e+76) {
		tmp = Math.pow(im, 4.0) * (re * (re * -0.020833333333333332));
	} else if (im <= 1.35e+154) {
		tmp = (Math.pow(im, 4.0) * 0.041666666666666664) + 1.0;
	} else {
		tmp = (0.5 * Math.cos(re)) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 700.0:
		tmp = math.cos(re)
	elif im <= 8.2e+76:
		tmp = math.pow(im, 4.0) * (re * (re * -0.020833333333333332))
	elif im <= 1.35e+154:
		tmp = (math.pow(im, 4.0) * 0.041666666666666664) + 1.0
	else:
		tmp = (0.5 * math.cos(re)) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 700.0)
		tmp = cos(re);
	elseif (im <= 8.2e+76)
		tmp = Float64((im ^ 4.0) * Float64(re * Float64(re * -0.020833333333333332)));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64((im ^ 4.0) * 0.041666666666666664) + 1.0);
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 700.0)
		tmp = cos(re);
	elseif (im <= 8.2e+76)
		tmp = (im ^ 4.0) * (re * (re * -0.020833333333333332));
	elseif (im <= 1.35e+154)
		tmp = ((im ^ 4.0) * 0.041666666666666664) + 1.0;
	else
		tmp = (0.5 * cos(re)) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 700.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 8.2e+76], N[(N[Power[im, 4.0], $MachinePrecision] * N[(re * N[(re * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 700

   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 69.9%

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

   if 700 < im < 8.1999999999999997e76

   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 4.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 4.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   5. Taylor expanded in im around inf 4.3%

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

      1. associate-*r* 4.3%

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

      2. *-commutative 4.3%

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

      3. associate-*l* 4.3%

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

   7. Simplified 4.3%

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

   8. Taylor expanded in re around 0 12.9%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right) + 0.041666666666666664 \cdot {im}^{4}} \]
   9. Step-by-step derivation

      1. +-commutative 12.9%

         \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4} + -0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right)} \]

      2. associate-*r* 12.9%

         \[\leadsto 0.041666666666666664 \cdot {im}^{4} + \color{blue}{\left(-0.020833333333333332 \cdot {re}^{2}\right) \cdot {im}^{4}} \]

      3. distribute-rgt-out 12.9%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot {re}^{2}\right)} \]

      4. *-commutative 12.9%

         \[\leadsto {im}^{4} \cdot \left(0.041666666666666664 + \color{blue}{{re}^{2} \cdot -0.020833333333333332}\right) \]

      5. unpow2 12.9%

         \[\leadsto {im}^{4} \cdot \left(0.041666666666666664 + \color{blue}{\left(re \cdot re\right)} \cdot -0.020833333333333332\right) \]

   10. Simplified 12.9%

       \[\leadsto \color{blue}{{im}^{4} \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]

   11. Taylor expanded in re around inf 10.9%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right)} \]
   12. Step-by-step derivation

       1. unpow2 10.9%

          \[\leadsto -0.020833333333333332 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {im}^{4}\right) \]

       2. associate-*r* 10.9%

          \[\leadsto \color{blue}{\left(-0.020833333333333332 \cdot \left(re \cdot re\right)\right) \cdot {im}^{4}} \]

       3. *-commutative 10.9%

          \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot -0.020833333333333332\right)} \cdot {im}^{4} \]

       4. associate-*r* 10.9%

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot -0.020833333333333332\right)\right)} \cdot {im}^{4} \]

       5. *-commutative 10.9%

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(re \cdot \left(re \cdot -0.020833333333333332\right)\right)} \]

   13. Simplified 10.9%

       \[\leadsto \color{blue}{{im}^{4} \cdot \left(re \cdot \left(re \cdot -0.020833333333333332\right)\right)} \]

   if 8.1999999999999997e76 < im < 1.35000000000000003e154

   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 95.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 95.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   5. Taylor expanded in re around 0 71.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 71.4%

         \[\leadsto \color{blue}{0.5 \cdot 2 + 0.5 \cdot \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)} \]

      2. metadata-eval 71.4%

         \[\leadsto \color{blue}{1} + 0.5 \cdot \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right) \]

      3. unpow2 71.4%

         \[\leadsto 1 + 0.5 \cdot \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right) \]

      4. *-commutative 71.4%

         \[\leadsto 1 + 0.5 \cdot \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right) \]

   7. Simplified 71.4%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)} \]

   8. Taylor expanded in im around inf 71.4%

      \[\leadsto 1 + \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]

   if 1.35000000000000003e154 < 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%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;{im}^{4} \cdot \left(re \cdot \left(re \cdot -0.020833333333333332\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664 + 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 9: 82.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 700:\\ \;\;\;\;t_0 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+76}:\\ \;\;\;\;{im}^{4} \cdot \left(re \cdot \left(re \cdot -0.020833333333333332\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664 + 1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 700.0)
     (* t_0 (+ 2.0 (* im im)))
     (if (<= im 2.2e+76)
       (* (pow im 4.0) (* re (* re -0.020833333333333332)))
       (if (<= im 1.35e+154)
         (+ (* (pow im 4.0) 0.041666666666666664) 1.0)
         (* t_0 (* im im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 700.0) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 2.2e+76) {
		tmp = pow(im, 4.0) * (re * (re * -0.020833333333333332));
	} else if (im <= 1.35e+154) {
		tmp = (pow(im, 4.0) * 0.041666666666666664) + 1.0;
	} else {
		tmp = t_0 * (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) :: tmp
    t_0 = 0.5d0 * cos(re)
    if (im <= 700.0d0) then
        tmp = t_0 * (2.0d0 + (im * im))
    else if (im <= 2.2d+76) then
        tmp = (im ** 4.0d0) * (re * (re * (-0.020833333333333332d0)))
    else if (im <= 1.35d+154) then
        tmp = ((im ** 4.0d0) * 0.041666666666666664d0) + 1.0d0
    else
        tmp = t_0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 700.0) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 2.2e+76) {
		tmp = Math.pow(im, 4.0) * (re * (re * -0.020833333333333332));
	} else if (im <= 1.35e+154) {
		tmp = (Math.pow(im, 4.0) * 0.041666666666666664) + 1.0;
	} else {
		tmp = t_0 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 700.0:
		tmp = t_0 * (2.0 + (im * im))
	elif im <= 2.2e+76:
		tmp = math.pow(im, 4.0) * (re * (re * -0.020833333333333332))
	elif im <= 1.35e+154:
		tmp = (math.pow(im, 4.0) * 0.041666666666666664) + 1.0
	else:
		tmp = t_0 * (im * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 700.0)
		tmp = Float64(t_0 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 2.2e+76)
		tmp = Float64((im ^ 4.0) * Float64(re * Float64(re * -0.020833333333333332)));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64((im ^ 4.0) * 0.041666666666666664) + 1.0);
	else
		tmp = Float64(t_0 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 700.0)
		tmp = t_0 * (2.0 + (im * im));
	elseif (im <= 2.2e+76)
		tmp = (im ^ 4.0) * (re * (re * -0.020833333333333332));
	elseif (im <= 1.35e+154)
		tmp = ((im ^ 4.0) * 0.041666666666666664) + 1.0;
	else
		tmp = t_0 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 700.0], N[(t$95$0 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.2e+76], N[(N[Power[im, 4.0], $MachinePrecision] * N[(re * N[(re * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + 1.0), $MachinePrecision], N[(t$95$0 * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 700

   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 82.1%

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

   4. Simplified 82.1%

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

   if 700 < im < 2.2e76

   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 4.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 4.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   5. Taylor expanded in im around inf 4.3%

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

      1. associate-*r* 4.3%

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

      2. *-commutative 4.3%

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

      3. associate-*l* 4.3%

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

   7. Simplified 4.3%

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

   8. Taylor expanded in re around 0 12.9%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right) + 0.041666666666666664 \cdot {im}^{4}} \]
   9. Step-by-step derivation

      1. +-commutative 12.9%

         \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4} + -0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right)} \]

      2. associate-*r* 12.9%

         \[\leadsto 0.041666666666666664 \cdot {im}^{4} + \color{blue}{\left(-0.020833333333333332 \cdot {re}^{2}\right) \cdot {im}^{4}} \]

      3. distribute-rgt-out 12.9%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot {re}^{2}\right)} \]

      4. *-commutative 12.9%

         \[\leadsto {im}^{4} \cdot \left(0.041666666666666664 + \color{blue}{{re}^{2} \cdot -0.020833333333333332}\right) \]

      5. unpow2 12.9%

         \[\leadsto {im}^{4} \cdot \left(0.041666666666666664 + \color{blue}{\left(re \cdot re\right)} \cdot -0.020833333333333332\right) \]

   10. Simplified 12.9%

       \[\leadsto \color{blue}{{im}^{4} \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]

   11. Taylor expanded in re around inf 10.9%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right)} \]
   12. Step-by-step derivation

       1. unpow2 10.9%

          \[\leadsto -0.020833333333333332 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {im}^{4}\right) \]

       2. associate-*r* 10.9%

          \[\leadsto \color{blue}{\left(-0.020833333333333332 \cdot \left(re \cdot re\right)\right) \cdot {im}^{4}} \]

       3. *-commutative 10.9%

          \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot -0.020833333333333332\right)} \cdot {im}^{4} \]

       4. associate-*r* 10.9%

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot -0.020833333333333332\right)\right)} \cdot {im}^{4} \]

       5. *-commutative 10.9%

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(re \cdot \left(re \cdot -0.020833333333333332\right)\right)} \]

   13. Simplified 10.9%

       \[\leadsto \color{blue}{{im}^{4} \cdot \left(re \cdot \left(re \cdot -0.020833333333333332\right)\right)} \]

   if 2.2e76 < im < 1.35000000000000003e154

   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 95.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 95.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   5. Taylor expanded in re around 0 71.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 71.4%

         \[\leadsto \color{blue}{0.5 \cdot 2 + 0.5 \cdot \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)} \]

      2. metadata-eval 71.4%

         \[\leadsto \color{blue}{1} + 0.5 \cdot \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right) \]

      3. unpow2 71.4%

         \[\leadsto 1 + 0.5 \cdot \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right) \]

      4. *-commutative 71.4%

         \[\leadsto 1 + 0.5 \cdot \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right) \]

   7. Simplified 71.4%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)} \]

   8. Taylor expanded in im around inf 71.4%

      \[\leadsto 1 + \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]

   if 1.35000000000000003e154 < 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%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+76}:\\ \;\;\;\;{im}^{4} \cdot \left(re \cdot \left(re \cdot -0.020833333333333332\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664 + 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 10: 65.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+74}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664 + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 250.0)
   (cos re)
   (if (<= im 1.15e+74)
     (+ 1.0 (* re (* re -0.5)))
     (+ (* (pow im 4.0) 0.041666666666666664) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = cos(re);
	} else if (im <= 1.15e+74) {
		tmp = 1.0 + (re * (re * -0.5));
	} else {
		tmp = (pow(im, 4.0) * 0.041666666666666664) + 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 <= 250.0d0) then
        tmp = cos(re)
    else if (im <= 1.15d+74) then
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    else
        tmp = ((im ** 4.0d0) * 0.041666666666666664d0) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.15e+74) {
		tmp = 1.0 + (re * (re * -0.5));
	} else {
		tmp = (Math.pow(im, 4.0) * 0.041666666666666664) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 250.0:
		tmp = math.cos(re)
	elif im <= 1.15e+74:
		tmp = 1.0 + (re * (re * -0.5))
	else:
		tmp = (math.pow(im, 4.0) * 0.041666666666666664) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 1.15e+74)
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	else
		tmp = Float64(Float64((im ^ 4.0) * 0.041666666666666664) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 1.15e+74)
		tmp = 1.0 + (re * (re * -0.5));
	else
		tmp = ((im ^ 4.0) * 0.041666666666666664) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 250.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.15e+74], N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 250

   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 70.2%

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

   if 250 < im < 1.1499999999999999e74

   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 3.3%

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

   3. Taylor expanded in re around 0 11.0%

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

      1. expm1-log1p-u 2.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot {re}^{2}\right)\right)} \]

      2. expm1-udef 2.0%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot {re}^{2}\right)} - 1\right)} \]

      3. *-commutative 2.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{{re}^{2} \cdot -0.5}\right)} - 1\right) \]

      4. unpow2 2.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5\right)} - 1\right) \]

      5. associate-*l* 2.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot \left(re \cdot -0.5\right)}\right)} - 1\right) \]

   5. Applied egg-rr 2.0%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(re \cdot -0.5\right)\right)} - 1\right)} \]
   6. Step-by-step derivation

      1. expm1-def 2.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(re \cdot -0.5\right)\right)\right)} \]

      2. expm1-log1p 11.0%

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

   7. Simplified 11.0%

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

   if 1.1499999999999999e74 < 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 97.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 97.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   5. Taylor expanded in re around 0 66.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 66.2%

         \[\leadsto \color{blue}{0.5 \cdot 2 + 0.5 \cdot \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)} \]

      2. metadata-eval 66.2%

         \[\leadsto \color{blue}{1} + 0.5 \cdot \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right) \]

      3. unpow2 66.2%

         \[\leadsto 1 + 0.5 \cdot \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right) \]

      4. *-commutative 66.2%

         \[\leadsto 1 + 0.5 \cdot \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right) \]

   7. Simplified 66.2%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)} \]

   8. Taylor expanded in im around inf 66.2%

      \[\leadsto 1 + \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+74}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664 + 1\\ \end{array} \]

Alternative 11: 65.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+73}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 250.0)
   (cos re)
   (if (<= im 4.3e+73)
     (+ 1.0 (* re (* re -0.5)))
     (* (pow im 4.0) 0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = cos(re);
	} else if (im <= 4.3e+73) {
		tmp = 1.0 + (re * (re * -0.5));
	} else {
		tmp = pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 250.0d0) then
        tmp = cos(re)
    else if (im <= 4.3d+73) then
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    else
        tmp = (im ** 4.0d0) * 0.041666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = Math.cos(re);
	} else if (im <= 4.3e+73) {
		tmp = 1.0 + (re * (re * -0.5));
	} else {
		tmp = Math.pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 250.0:
		tmp = math.cos(re)
	elif im <= 4.3e+73:
		tmp = 1.0 + (re * (re * -0.5))
	else:
		tmp = math.pow(im, 4.0) * 0.041666666666666664
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 4.3e+73)
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	else
		tmp = Float64((im ^ 4.0) * 0.041666666666666664);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 4.3e+73)
		tmp = 1.0 + (re * (re * -0.5));
	else
		tmp = (im ^ 4.0) * 0.041666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 250.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4.3e+73], N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 250

   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 70.2%

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

   if 250 < im < 4.30000000000000013e73

   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 3.3%

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

   3. Taylor expanded in re around 0 11.0%

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

      1. expm1-log1p-u 2.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot {re}^{2}\right)\right)} \]

      2. expm1-udef 2.0%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot {re}^{2}\right)} - 1\right)} \]

      3. *-commutative 2.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{{re}^{2} \cdot -0.5}\right)} - 1\right) \]

      4. unpow2 2.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5\right)} - 1\right) \]

      5. associate-*l* 2.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot \left(re \cdot -0.5\right)}\right)} - 1\right) \]

   5. Applied egg-rr 2.0%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(re \cdot -0.5\right)\right)} - 1\right)} \]
   6. Step-by-step derivation

      1. expm1-def 2.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(re \cdot -0.5\right)\right)\right)} \]

      2. expm1-log1p 11.0%

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

   7. Simplified 11.0%

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

   if 4.30000000000000013e73 < 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 97.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   4. Simplified 97.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   5. Taylor expanded in im around inf 97.9%

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

      1. associate-*r* 97.9%

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

      2. *-commutative 97.9%

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

      3. associate-*l* 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in re around 0 0.4%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right) + 0.041666666666666664 \cdot {im}^{4}} \]
   9. Step-by-step derivation

      1. +-commutative 0.4%

         \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4} + -0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right)} \]

      2. associate-*r* 0.4%

         \[\leadsto 0.041666666666666664 \cdot {im}^{4} + \color{blue}{\left(-0.020833333333333332 \cdot {re}^{2}\right) \cdot {im}^{4}} \]

      3. distribute-rgt-out 76.0%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot {re}^{2}\right)} \]

      4. *-commutative 76.0%

         \[\leadsto {im}^{4} \cdot \left(0.041666666666666664 + \color{blue}{{re}^{2} \cdot -0.020833333333333332}\right) \]

      5. unpow2 76.0%

         \[\leadsto {im}^{4} \cdot \left(0.041666666666666664 + \color{blue}{\left(re \cdot re\right)} \cdot -0.020833333333333332\right) \]

   10. Simplified 76.0%

       \[\leadsto \color{blue}{{im}^{4} \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)} \]

   11. Taylor expanded in re around 0 66.2%

       \[\leadsto {im}^{4} \cdot \color{blue}{0.041666666666666664} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+73}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \]

Alternative 12: 61.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+150}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 250.0)
   (cos re)
   (if (<= im 5.8e+150) (+ 1.0 (* re (* re -0.5))) (+ 1.0 (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 250.0) {
		tmp = cos(re);
	} else if (im <= 5.8e+150) {
		tmp = 1.0 + (re * (re * -0.5));
	} 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 (im <= 250.0d0) then
        tmp = cos(re)
    else if (im <= 5.8d+150) then
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    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 (im <= 250.0) {
		tmp = Math.cos(re);
	} else if (im <= 5.8e+150) {
		tmp = 1.0 + (re * (re * -0.5));
	} else {
		tmp = 1.0 + (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 250.0:
		tmp = math.cos(re)
	elif im <= 5.8e+150:
		tmp = 1.0 + (re * (re * -0.5))
	else:
		tmp = 1.0 + (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 5.8e+150)
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	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 (im <= 250.0)
		tmp = cos(re);
	elseif (im <= 5.8e+150)
		tmp = 1.0 + (re * (re * -0.5));
	else
		tmp = 1.0 + (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 250.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5.8e+150], N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 250

   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 70.2%

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

   if 250 < im < 5.80000000000000022e150

   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 3.2%

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

   3. Taylor expanded in re around 0 9.7%

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

      1. expm1-log1p-u 1.8%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot {re}^{2}\right)\right)} \]

      2. expm1-udef 1.8%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot {re}^{2}\right)} - 1\right)} \]

      3. *-commutative 1.8%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{{re}^{2} \cdot -0.5}\right)} - 1\right) \]

      4. unpow2 1.8%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5\right)} - 1\right) \]

      5. associate-*l* 1.8%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot \left(re \cdot -0.5\right)}\right)} - 1\right) \]

   5. Applied egg-rr 1.8%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(re \cdot -0.5\right)\right)} - 1\right)} \]
   6. Step-by-step derivation

      1. expm1-def 1.8%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(re \cdot -0.5\right)\right)\right)} \]

      2. expm1-log1p 9.7%

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

   7. Simplified 9.7%

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

   if 5.80000000000000022e150 < 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 96.5%

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

   4. Simplified 96.5%

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

   5. Taylor expanded in re around 0 60.5%

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

      1. distribute-lft-in 60.5%

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

      2. metadata-eval 60.5%

         \[\leadsto \color{blue}{1} + 0.5 \cdot {im}^{2} \]

      3. unpow2 60.5%

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

   7. Simplified 60.5%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+150}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 13: 47.1% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.4 \cdot 10^{+157} \lor \neg \left(re \leq 5.3 \cdot 10^{+247}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 3.4e+157) (not (<= re 5.3e+247)))
   (+ 1.0 (* 0.5 (* im im)))
   (+ 1.0 (* re (* re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 3.4e+157) || !(re <= 5.3e+247)) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 1.0 + (re * (re * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= 3.4d+157) .or. (.not. (re <= 5.3d+247))) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 3.4e+157) || !(re <= 5.3e+247)) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 1.0 + (re * (re * -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 3.4e+157) or not (re <= 5.3e+247):
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = 1.0 + (re * (re * -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 3.4e+157) || !(re <= 5.3e+247))
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 3.4e+157) || ~((re <= 5.3e+247)))
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = 1.0 + (re * (re * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 3.4e+157], N[Not[LessEqual[re, 5.3e+247]], $MachinePrecision]], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.39999999999999979e157 or 5.3000000000000002e247 < re

   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 75.7%

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

   4. Simplified 75.7%

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

   5. Taylor expanded in re around 0 52.5%

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

      1. distribute-lft-in 52.5%

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

      2. metadata-eval 52.5%

         \[\leadsto \color{blue}{1} + 0.5 \cdot {im}^{2} \]

      3. unpow2 52.5%

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

   7. Simplified 52.5%

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

   if 3.39999999999999979e157 < re < 5.3000000000000002e247

   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 40.4%

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

   3. Taylor expanded in re around 0 39.3%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot {re}^{2}\right)\right)} \]

      2. expm1-udef 0.0%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot {re}^{2}\right)} - 1\right)} \]

      3. *-commutative 0.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{{re}^{2} \cdot -0.5}\right)} - 1\right) \]

      4. unpow2 0.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5\right)} - 1\right) \]

      5. associate-*l* 0.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot \left(re \cdot -0.5\right)}\right)} - 1\right) \]

   5. Applied egg-rr 0.0%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(re \cdot -0.5\right)\right)} - 1\right)} \]
   6. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(re \cdot -0.5\right)\right)\right)} \]

      2. expm1-log1p 39.3%

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

   7. Simplified 39.3%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.4 \cdot 10^{+157} \lor \neg \left(re \leq 5.3 \cdot 10^{+247}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \end{array} \]

Alternative 14: 46.8% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 + N[(0.5 * N[(im * 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. Taylor expanded in im around 0 75.4%

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

4. Simplified 75.4%

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

5. Taylor expanded in re around 0 50.5%

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

   1. distribute-lft-in 50.5%

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

   2. metadata-eval 50.5%

      \[\leadsto \color{blue}{1} + 0.5 \cdot {im}^{2} \]

   3. unpow2 50.5%

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

7. Simplified 50.5%

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

8. Final simplification 50.5%

   \[\leadsto 1 + 0.5 \cdot \left(im \cdot im\right) \]

Alternative 15: 28.3% 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 56.3%

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

3. Taylor expanded in re around 0 35.7%

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

4. Final simplification 35.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023199 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))