math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 15

Speedup: 1.0×

• 49.1% 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: 93.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.95:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{6} \cdot \left(\cos re \cdot 0.001388888888888889\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.95)
   (*
    (* 0.5 (cos re))
    (+ (+ 2.0 (* im im)) (* 0.08333333333333333 (pow im 4.0))))
   (if (<= im 4e+48)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (pow im 6.0) (* (cos re) 0.001388888888888889)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.95) {
		tmp = (0.5 * cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * pow(im, 4.0)));
	} else if (im <= 4e+48) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = pow(im, 6.0) * (cos(re) * 0.001388888888888889);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.95d0) then
        tmp = (0.5d0 * cos(re)) * ((2.0d0 + (im * im)) + (0.08333333333333333d0 * (im ** 4.0d0)))
    else if (im <= 4d+48) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = (im ** 6.0d0) * (cos(re) * 0.001388888888888889d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.95) {
		tmp = (0.5 * Math.cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * Math.pow(im, 4.0)));
	} else if (im <= 4e+48) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.pow(im, 6.0) * (Math.cos(re) * 0.001388888888888889);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.95:
		tmp = (0.5 * math.cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * math.pow(im, 4.0)))
	elif im <= 4e+48:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.pow(im, 6.0) * (math.cos(re) * 0.001388888888888889)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.95)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64(0.08333333333333333 * (im ^ 4.0))));
	elseif (im <= 4e+48)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64((im ^ 6.0) * Float64(cos(re) * 0.001388888888888889));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.95)
		tmp = (0.5 * cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * (im ^ 4.0)));
	elseif (im <= 4e+48)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = (im ^ 6.0) * (cos(re) * 0.001388888888888889);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.95], 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, 4e+48], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 6.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.94999999999999996

   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 93.0%

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

      1. associate-+r+ 93.0%

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

      2. unpow2 93.0%

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

   4. Simplified 93.0%

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

   if 0.94999999999999996 < im < 4.00000000000000018e48

   1. Initial program 99.8%

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

   2. Taylor expanded in re around 0 59.9%

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

      1. *-commutative 59.9%

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

   4. Simplified 59.9%

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

   if 4.00000000000000018e48 < 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%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + \left(0.08333333333333333 \cdot {im}^{4} + 0.002777777777777778 \cdot {im}^{6}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + {im}^{2}\right) + \left(0.08333333333333333 \cdot {im}^{4} + 0.002777777777777778 \cdot {im}^{6}\right)\right)} \]

      2. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + \color{blue}{im \cdot im}\right) + \left(0.08333333333333333 \cdot {im}^{4} + 0.002777777777777778 \cdot {im}^{6}\right)\right) \]

      3. fma-def 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + \color{blue}{\mathsf{fma}\left(0.08333333333333333, {im}^{4}, 0.002777777777777778 \cdot {im}^{6}\right)}\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + \mathsf{fma}\left(0.08333333333333333, {im}^{4}, 0.002777777777777778 \cdot {im}^{6}\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{6}\right) \cdot 0.001388888888888889} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{6} \cdot \cos re\right)} \cdot 0.001388888888888889 \]

      3. associate-*l* 100.0%

         \[\leadsto \color{blue}{{im}^{6} \cdot \left(\cos re \cdot 0.001388888888888889\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{6} \cdot \left(\cos re \cdot 0.001388888888888889\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.95:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{6} \cdot \left(\cos re \cdot 0.001388888888888889\right)\\ \end{array} \]

Alternative 3: 72.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{-10}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.6e-10)
   (cos re)
   (if (<= im 2.05e+152)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* im (* (* 0.5 (cos re)) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.6e-10) {
		tmp = cos(re);
	} else if (im <= 2.05e+152) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = im * ((0.5 * cos(re)) * 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 <= 5.6d-10) then
        tmp = cos(re)
    else if (im <= 2.05d+152) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = im * ((0.5d0 * cos(re)) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.6e-10) {
		tmp = Math.cos(re);
	} else if (im <= 2.05e+152) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = im * ((0.5 * Math.cos(re)) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.6e-10:
		tmp = math.cos(re)
	elif im <= 2.05e+152:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = im * ((0.5 * math.cos(re)) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.6e-10)
		tmp = cos(re);
	elseif (im <= 2.05e+152)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(im * Float64(Float64(0.5 * cos(re)) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.6e-10)
		tmp = cos(re);
	elseif (im <= 2.05e+152)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = im * ((0.5 * cos(re)) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.6e-10], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.05e+152], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.60000000000000031e-10

   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 61.6%

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

   if 5.60000000000000031e-10 < im < 2.0499999999999999e152

   1. Initial program 99.8%

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

   2. Taylor expanded in re around 0 73.9%

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

      1. *-commutative 73.9%

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

   4. Simplified 73.9%

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

   if 2.0499999999999999e152 < 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 97.0%

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

      1. unpow2 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in im around inf 97.0%

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

      1. associate-*r* 97.0%

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

      2. *-commutative 97.0%

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

      3. unpow2 97.0%

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

      4. associate-*l* 97.0%

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

   7. Simplified 97.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{-10}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \]

Alternative 4: 74.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{-10}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{6} \cdot \left(\cos re \cdot 0.001388888888888889\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.6e-10)
   (cos re)
   (if (<= im 4e+48)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (pow im 6.0) (* (cos re) 0.001388888888888889)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.6e-10) {
		tmp = cos(re);
	} else if (im <= 4e+48) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = pow(im, 6.0) * (cos(re) * 0.001388888888888889);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 5.6d-10) then
        tmp = cos(re)
    else if (im <= 4d+48) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = (im ** 6.0d0) * (cos(re) * 0.001388888888888889d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.6e-10) {
		tmp = Math.cos(re);
	} else if (im <= 4e+48) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.pow(im, 6.0) * (Math.cos(re) * 0.001388888888888889);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.6e-10:
		tmp = math.cos(re)
	elif im <= 4e+48:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.pow(im, 6.0) * (math.cos(re) * 0.001388888888888889)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.6e-10)
		tmp = cos(re);
	elseif (im <= 4e+48)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64((im ^ 6.0) * Float64(cos(re) * 0.001388888888888889));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.6e-10)
		tmp = cos(re);
	elseif (im <= 4e+48)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = (im ^ 6.0) * (cos(re) * 0.001388888888888889);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.6e-10], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4e+48], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 6.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.60000000000000031e-10

   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 61.6%

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

   if 5.60000000000000031e-10 < im < 4.00000000000000018e48

   1. Initial program 99.6%

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

   2. Taylor expanded in re around 0 57.9%

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

      1. *-commutative 57.9%

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

   4. Simplified 57.9%

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

   if 4.00000000000000018e48 < 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%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + \left(0.08333333333333333 \cdot {im}^{4} + 0.002777777777777778 \cdot {im}^{6}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + {im}^{2}\right) + \left(0.08333333333333333 \cdot {im}^{4} + 0.002777777777777778 \cdot {im}^{6}\right)\right)} \]

      2. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + \color{blue}{im \cdot im}\right) + \left(0.08333333333333333 \cdot {im}^{4} + 0.002777777777777778 \cdot {im}^{6}\right)\right) \]

      3. fma-def 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + \color{blue}{\mathsf{fma}\left(0.08333333333333333, {im}^{4}, 0.002777777777777778 \cdot {im}^{6}\right)}\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + \mathsf{fma}\left(0.08333333333333333, {im}^{4}, 0.002777777777777778 \cdot {im}^{6}\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{6}\right) \cdot 0.001388888888888889} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{6} \cdot \cos re\right)} \cdot 0.001388888888888889 \]

      3. associate-*l* 100.0%

         \[\leadsto \color{blue}{{im}^{6} \cdot \left(\cos re \cdot 0.001388888888888889\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{6} \cdot \left(\cos re \cdot 0.001388888888888889\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{-10}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{6} \cdot \left(\cos re \cdot 0.001388888888888889\right)\\ \end{array} \]

Alternative 5: 78.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := 2 + im \cdot im\\ \mathbf{if}\;im \leq 1020:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t_1, -0.25 \cdot \left(t_1 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(t_0 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (+ 2.0 (* im im))))
   (if (<= im 1020.0)
     (* t_0 t_1)
     (if (<= im 1.35e+154)
       (fma 0.5 t_1 (* -0.25 (* t_1 (* re re))))
       (* im (* t_0 im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = 2.0 + (im * im);
	double tmp;
	if (im <= 1020.0) {
		tmp = t_0 * t_1;
	} else if (im <= 1.35e+154) {
		tmp = fma(0.5, t_1, (-0.25 * (t_1 * (re * re))));
	} else {
		tmp = im * (t_0 * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(2.0 + Float64(im * im))
	tmp = 0.0
	if (im <= 1020.0)
		tmp = Float64(t_0 * t_1);
	elseif (im <= 1.35e+154)
		tmp = fma(0.5, t_1, Float64(-0.25 * Float64(t_1 * Float64(re * re))));
	else
		tmp = Float64(im * Float64(t_0 * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1020.0], N[(t$95$0 * t$95$1), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * t$95$1 + N[(-0.25 * N[(t$95$1 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(t$95$0 * im), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1020

   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 81.0%

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

      1. unpow2 81.0%

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

   4. Simplified 81.0%

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

   if 1020 < 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 6.2%

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

      1. unpow2 6.2%

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

   4. Simplified 6.2%

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

   5. Taylor expanded in re around 0 25.0%

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

      1. fma-def 25.0%

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

      2. unpow2 25.0%

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

      3. *-commutative 25.0%

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

      4. unpow2 25.0%

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

      5. unpow2 25.0%

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

   7. Simplified 25.0%

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

   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%

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

      1. unpow2 100.0%

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

   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%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1020:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(0.5, 2 + im \cdot im, -0.25 \cdot \left(\left(2 + im \cdot im\right) \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \]

Alternative 6: 66.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1020:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1020.0)
   (cos re)
   (if (<= im 1.95e+154)
     (* im (* im (fma -0.25 (* re re) 0.5)))
     (* im (* (* 0.5 (cos re)) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1020.0) {
		tmp = cos(re);
	} else if (im <= 1.95e+154) {
		tmp = im * (im * fma(-0.25, (re * re), 0.5));
	} else {
		tmp = im * ((0.5 * cos(re)) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1020.0)
		tmp = cos(re);
	elseif (im <= 1.95e+154)
		tmp = Float64(im * Float64(im * fma(-0.25, Float64(re * re), 0.5)));
	else
		tmp = Float64(im * Float64(Float64(0.5 * cos(re)) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1020.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.95e+154], N[(im * N[(im * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1020

   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 61.1%

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

   if 1020 < im < 1.9500000000000001e154

   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 6.2%

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

      1. unpow2 6.2%

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

   4. Simplified 6.2%

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

   5. Taylor expanded in re around 0 25.0%

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

      1. fma-def 25.0%

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

      2. unpow2 25.0%

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

      3. *-commutative 25.0%

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

      4. unpow2 25.0%

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

      5. unpow2 25.0%

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

   7. Simplified 25.0%

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

   8. Taylor expanded in im around inf 25.0%

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

      1. unpow2 25.0%

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

      2. *-commutative 25.0%

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

      3. unpow2 25.0%

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

      4. associate-*l* 25.0%

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

      5. +-commutative 25.0%

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

      6. fma-def 25.0%

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

   10. Simplified 25.0%

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

   if 1.9500000000000001e154 < 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%

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

      1. unpow2 100.0%

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

   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%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1020:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \]

Alternative 7: 78.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 1100:\\ \;\;\;\;t_0 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(t_0 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 1100.0)
     (* t_0 (+ 2.0 (* im im)))
     (if (<= im 2.7e+154)
       (* im (* im (fma -0.25 (* re re) 0.5)))
       (* im (* t_0 im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 1100.0) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 2.7e+154) {
		tmp = im * (im * fma(-0.25, (re * re), 0.5));
	} else {
		tmp = im * (t_0 * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 1100.0)
		tmp = Float64(t_0 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 2.7e+154)
		tmp = Float64(im * Float64(im * fma(-0.25, Float64(re * re), 0.5)));
	else
		tmp = Float64(im * Float64(t_0 * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1100.0], N[(t$95$0 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.7e+154], N[(im * N[(im * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(t$95$0 * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1100

   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 81.0%

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

      1. unpow2 81.0%

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

   4. Simplified 81.0%

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

   if 1100 < im < 2.70000000000000006e154

   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 6.2%

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

      1. unpow2 6.2%

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

   4. Simplified 6.2%

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

   5. Taylor expanded in re around 0 25.0%

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

      1. fma-def 25.0%

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

      2. unpow2 25.0%

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

      3. *-commutative 25.0%

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

      4. unpow2 25.0%

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

      5. unpow2 25.0%

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

   7. Simplified 25.0%

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

   8. Taylor expanded in im around inf 25.0%

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

      1. unpow2 25.0%

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

      2. *-commutative 25.0%

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

      3. unpow2 25.0%

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

      4. associate-*l* 25.0%

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

      5. +-commutative 25.0%

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

      6. fma-def 25.0%

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

   10. Simplified 25.0%

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

   if 2.70000000000000006e154 < 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%

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

      1. unpow2 100.0%

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

   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%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1100:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \]

Alternative 8: 66.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2800:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.56 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2800.0)
   (cos re)
   (if (<= im 2.56e+154)
     (* (* im im) (+ 0.5 (* -0.25 (* re re))))
     (* im (* (* 0.5 (cos re)) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2800.0) {
		tmp = cos(re);
	} else if (im <= 2.56e+154) {
		tmp = (im * im) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = im * ((0.5 * cos(re)) * 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 <= 2800.0d0) then
        tmp = cos(re)
    else if (im <= 2.56d+154) then
        tmp = (im * im) * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = im * ((0.5d0 * cos(re)) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2800.0) {
		tmp = Math.cos(re);
	} else if (im <= 2.56e+154) {
		tmp = (im * im) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = im * ((0.5 * Math.cos(re)) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2800.0:
		tmp = math.cos(re)
	elif im <= 2.56e+154:
		tmp = (im * im) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = im * ((0.5 * math.cos(re)) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2800.0)
		tmp = cos(re);
	elseif (im <= 2.56e+154)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(im * Float64(Float64(0.5 * cos(re)) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2800.0)
		tmp = cos(re);
	elseif (im <= 2.56e+154)
		tmp = (im * im) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = im * ((0.5 * cos(re)) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2800.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.56e+154], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2800

   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 61.1%

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

   if 2800 < im < 2.5600000000000001e154

   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 6.2%

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

      1. unpow2 6.2%

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

   4. Simplified 6.2%

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

   5. Taylor expanded in re around 0 25.0%

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

      1. fma-def 25.0%

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

      2. unpow2 25.0%

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

      3. *-commutative 25.0%

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

      4. unpow2 25.0%

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

      5. unpow2 25.0%

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

   7. Simplified 25.0%

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

   8. Taylor expanded in im around inf 25.0%

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

      1. unpow2 25.0%

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

      2. *-commutative 25.0%

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

      3. unpow2 25.0%

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

   10. Simplified 25.0%

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

   if 2.5600000000000001e154 < 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%

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

      1. unpow2 100.0%

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

   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%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2800:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.56 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \cos re\right) \cdot im\right)\\ \end{array} \]

Alternative 9: 63.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1020:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+154} \lor \neg \left(im \leq 5 \cdot 10^{+240}\right) \land im \leq 1.16 \cdot 10^{+275}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1020.0)
   (cos re)
   (if (or (<= im 2e+154) (and (not (<= im 5e+240)) (<= im 1.16e+275)))
     (* (* im im) (+ 0.5 (* -0.25 (* re re))))
     (+ 1.0 (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1020.0) {
		tmp = cos(re);
	} else if ((im <= 2e+154) || (!(im <= 5e+240) && (im <= 1.16e+275))) {
		tmp = (im * im) * (0.5 + (-0.25 * (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 (im <= 1020.0d0) then
        tmp = cos(re)
    else if ((im <= 2d+154) .or. (.not. (im <= 5d+240)) .and. (im <= 1.16d+275)) then
        tmp = (im * im) * (0.5d0 + ((-0.25d0) * (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 (im <= 1020.0) {
		tmp = Math.cos(re);
	} else if ((im <= 2e+154) || (!(im <= 5e+240) && (im <= 1.16e+275))) {
		tmp = (im * im) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = 1.0 + (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1020.0:
		tmp = math.cos(re)
	elif (im <= 2e+154) or (not (im <= 5e+240) and (im <= 1.16e+275)):
		tmp = (im * im) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = 1.0 + (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1020.0)
		tmp = cos(re);
	elseif ((im <= 2e+154) || (!(im <= 5e+240) && (im <= 1.16e+275)))
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(-0.25 * Float64(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 (im <= 1020.0)
		tmp = cos(re);
	elseif ((im <= 2e+154) || (~((im <= 5e+240)) && (im <= 1.16e+275)))
		tmp = (im * im) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = 1.0 + (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1020.0], N[Cos[re], $MachinePrecision], If[Or[LessEqual[im, 2e+154], And[N[Not[LessEqual[im, 5e+240]], $MachinePrecision], LessEqual[im, 1.16e+275]]], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $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 < 1020

   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 61.1%

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

   if 1020 < im < 2.00000000000000007e154 or 5.0000000000000003e240 < im < 1.16e275

   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 32.6%

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

      1. unpow2 32.6%

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

   4. Simplified 32.6%

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

   5. Taylor expanded in re around 0 18.0%

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

      1. fma-def 18.0%

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

      2. unpow2 18.0%

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

      3. *-commutative 18.0%

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

      4. unpow2 18.0%

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

      5. unpow2 18.0%

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

   7. Simplified 18.0%

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

   8. Taylor expanded in im around inf 43.0%

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

      1. unpow2 43.0%

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

      2. *-commutative 43.0%

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

      3. unpow2 43.0%

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

   10. Simplified 43.0%

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

   if 2.00000000000000007e154 < im < 5.0000000000000003e240 or 1.16e275 < 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%

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

      1. unpow2 100.0%

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

   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 re around 0 84.2%

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

      1. distribute-rgt-in 84.2%

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

      2. metadata-eval 84.2%

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

      3. unpow2 84.2%

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

   7. Simplified 84.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1020:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+154} \lor \neg \left(im \leq 5 \cdot 10^{+240}\right) \land im \leq 1.16 \cdot 10^{+275}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 10: 50.1% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1100 \lor \neg \left(im \leq 1.5 \cdot 10^{+154} \lor \neg \left(im \leq 3.2 \cdot 10^{+241}\right) \land im \leq 10^{+275}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 1100.0)
         (not
          (or (<= im 1.5e+154) (and (not (<= im 3.2e+241)) (<= im 1e+275)))))
   (+ 1.0 (* 0.5 (* im im)))
   (* (* im im) (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 1100.0) || !((im <= 1.5e+154) || (!(im <= 3.2e+241) && (im <= 1e+275)))) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (im * im) * (0.5 + (-0.25 * (re * 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 <= 1100.0d0) .or. (.not. (im <= 1.5d+154) .or. (.not. (im <= 3.2d+241)) .and. (im <= 1d+275))) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = (im * im) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 1100.0) || !((im <= 1.5e+154) || (!(im <= 3.2e+241) && (im <= 1e+275)))) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (im * im) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 1100.0) or not ((im <= 1.5e+154) or (not (im <= 3.2e+241) and (im <= 1e+275))):
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = (im * im) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 1100.0) || !((im <= 1.5e+154) || (!(im <= 3.2e+241) && (im <= 1e+275))))
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 1100.0) || ~(((im <= 1.5e+154) || (~((im <= 3.2e+241)) && (im <= 1e+275)))))
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = (im * im) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 1100.0], N[Not[Or[LessEqual[im, 1.5e+154], And[N[Not[LessEqual[im, 3.2e+241]], $MachinePrecision], LessEqual[im, 1e+275]]]], $MachinePrecision]], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1100 or 1.50000000000000013e154 < im < 3.20000000000000004e241 or 9.9999999999999996e274 < 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 82.6%

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

      1. unpow2 82.6%

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

   4. Simplified 82.6%

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

   5. Taylor expanded in re around 0 49.1%

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

      1. distribute-rgt-in 49.1%

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

      2. metadata-eval 49.1%

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

      3. unpow2 49.1%

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

   7. Simplified 49.1%

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

   if 1100 < im < 1.50000000000000013e154 or 3.20000000000000004e241 < im < 9.9999999999999996e274

   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 32.6%

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

      1. unpow2 32.6%

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

   4. Simplified 32.6%

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

   5. Taylor expanded in re around 0 18.0%

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

      1. fma-def 18.0%

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

      2. unpow2 18.0%

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

      3. *-commutative 18.0%

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

      4. unpow2 18.0%

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

      5. unpow2 18.0%

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

   7. Simplified 18.0%

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

   8. Taylor expanded in im around inf 43.0%

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

      1. unpow2 43.0%

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

      2. *-commutative 43.0%

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

      3. unpow2 43.0%

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

   10. Simplified 43.0%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1100 \lor \neg \left(im \leq 1.5 \cdot 10^{+154} \lor \neg \left(im \leq 3.2 \cdot 10^{+241}\right) \land im \leq 10^{+275}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 11: 49.3% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1020 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot 10077696\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 1020.0) (not (<= im 1.35e+154)))
   (+ 1.0 (* 0.5 (* im im)))
   (* (+ 0.5 (* -0.25 (* re re))) 10077696.0)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 1020.0) || !(im <= 1.35e+154)) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * 10077696.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 1020.0d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = (0.5d0 + ((-0.25d0) * (re * re))) * 10077696.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 1020.0) || !(im <= 1.35e+154)) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * 10077696.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 1020.0) or not (im <= 1.35e+154):
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = (0.5 + (-0.25 * (re * re))) * 10077696.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 1020.0) || !(im <= 1.35e+154))
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(Float64(0.5 + Float64(-0.25 * Float64(re * re))) * 10077696.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 1020.0) || ~((im <= 1.35e+154)))
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = (0.5 + (-0.25 * (re * re))) * 10077696.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 1020.0], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 10077696.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1020 or 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 83.3%

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

      1. unpow2 83.3%

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

   4. Simplified 83.3%

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

   5. Taylor expanded in re around 0 50.3%

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

      1. distribute-rgt-in 50.3%

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

      2. metadata-eval 50.3%

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

      3. unpow2 50.3%

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

   7. Simplified 50.3%

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

   if 1020 < 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 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt-out 69.6%

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

      3. unpow2 69.6%

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

   4. Simplified 69.6%

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

   6. Applied egg-rr 23.3%

      \[\leadsto \color{blue}{10077696} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1020 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot 10077696\\ \end{array} \]

Alternative 12: 49.3% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2800 \lor \neg \left(im \leq 1.3 \cdot 10^{+154}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 2800.0) (not (<= im 1.3e+154)))
   (+ 1.0 (* 0.5 (* im im)))
   (+ 1.0 (* (* re re) -0.5))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 2800.0) || !(im <= 1.3e+154)) {
		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 ((im <= 2800.0d0) .or. (.not. (im <= 1.3d+154))) 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 ((im <= 2800.0) || !(im <= 1.3e+154)) {
		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 (im <= 2800.0) or not (im <= 1.3e+154):
		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 ((im <= 2800.0) || !(im <= 1.3e+154))
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 2800.0) || ~((im <= 1.3e+154)))
		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[im, 2800.0], N[Not[LessEqual[im, 1.3e+154]], $MachinePrecision]], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2800 or 1.29999999999999994e154 < 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 83.3%

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

      1. unpow2 83.3%

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

   4. Simplified 83.3%

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

   5. Taylor expanded in re around 0 50.3%

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

      1. distribute-rgt-in 50.3%

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

      2. metadata-eval 50.3%

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

      3. unpow2 50.3%

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

   7. Simplified 50.3%

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

   if 2800 < im < 1.29999999999999994e154

   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 6.2%

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

      1. unpow2 6.2%

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

   4. Simplified 6.2%

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

   5. Taylor expanded in re around 0 25.0%

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

      1. fma-def 25.0%

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

      2. unpow2 25.0%

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

      3. *-commutative 25.0%

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

      4. unpow2 25.0%

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

      5. unpow2 25.0%

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

   7. Simplified 25.0%

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

   8. Taylor expanded in im around 0 23.3%

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

      1. +-commutative 23.3%

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

      2. unpow2 23.3%

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

   10. Simplified 23.3%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2800 \lor \neg \left(im \leq 1.3 \cdot 10^{+154}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \]

Alternative 13: 47.8% 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 76.4%

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

   1. unpow2 76.4%

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

4. Simplified 76.4%

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

5. Taylor expanded in re around 0 46.2%

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

   1. distribute-rgt-in 46.2%

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

   2. metadata-eval 46.2%

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

   3. unpow2 46.2%

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

7. Simplified 46.2%

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

8. Final simplification 46.2%

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

Alternative 14: 3.8% 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 76.4%

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

   1. unpow2 76.4%

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

4. Simplified 76.4%

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

6. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{-2 + \cos re} \]
7. Step-by-step derivation

   1. +-commutative 3.4%

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

8. Simplified 3.4%

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

9. Taylor expanded in re around 0 3.9%

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

10. Final simplification 3.9%

    \[\leadsto -1 \]

Alternative 15: 29.2% 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%

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

3. Applied egg-rr 24.9%

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

   1. +-inverses 24.9%

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

   2. +-rgt-identity 24.9%

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

   3. *-inverses 24.9%

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

5. Simplified 24.9%

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

6. Final simplification 24.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023208 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))