math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.3s

Alternatives: 15

Speedup: 1.0×

• 48.8% 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: 92.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := 0.08333333333333333 \cdot {im}^{4}\\ \mathbf{if}\;im \leq 0.68:\\ \;\;\;\;t_0 \cdot \left(\left(2 + im \cdot im\right) + t_1\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 + \left(im \cdot im + \left(t_1 + 0.00023919753086419754\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* 0.08333333333333333 (pow im 4.0))))
   (if (<= im 0.68)
     (* t_0 (+ (+ 2.0 (* im im)) t_1))
     (if (<= im 2e+73)
       (* (+ (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re))))
       (* t_0 (+ 2.0 (+ (* im im) (+ t_1 0.00023919753086419754))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = 0.08333333333333333 * pow(im, 4.0);
	double tmp;
	if (im <= 0.68) {
		tmp = t_0 * ((2.0 + (im * im)) + t_1);
	} else if (im <= 2e+73) {
		tmp = (exp(-im) + exp(im)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = t_0 * (2.0 + ((im * im) + (t_1 + 0.00023919753086419754)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = 0.08333333333333333d0 * (im ** 4.0d0)
    if (im <= 0.68d0) then
        tmp = t_0 * ((2.0d0 + (im * im)) + t_1)
    else if (im <= 2d+73) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = t_0 * (2.0d0 + ((im * im) + (t_1 + 0.00023919753086419754d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double t_1 = 0.08333333333333333 * Math.pow(im, 4.0);
	double tmp;
	if (im <= 0.68) {
		tmp = t_0 * ((2.0 + (im * im)) + t_1);
	} else if (im <= 2e+73) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = t_0 * (2.0 + ((im * im) + (t_1 + 0.00023919753086419754)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = 0.08333333333333333 * math.pow(im, 4.0)
	tmp = 0
	if im <= 0.68:
		tmp = t_0 * ((2.0 + (im * im)) + t_1)
	elif im <= 2e+73:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = t_0 * (2.0 + ((im * im) + (t_1 + 0.00023919753086419754)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(0.08333333333333333 * (im ^ 4.0))
	tmp = 0.0
	if (im <= 0.68)
		tmp = Float64(t_0 * Float64(Float64(2.0 + Float64(im * im)) + t_1));
	elseif (im <= 2e+73)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(t_0 * Float64(2.0 + Float64(Float64(im * im) + Float64(t_1 + 0.00023919753086419754))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = 0.08333333333333333 * (im ^ 4.0);
	tmp = 0.0;
	if (im <= 0.68)
		tmp = t_0 * ((2.0 + (im * im)) + t_1);
	elseif (im <= 2e+73)
		tmp = (exp(-im) + exp(im)) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = t_0 * (2.0 + ((im * im) + (t_1 + 0.00023919753086419754)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.68], N[(t$95$0 * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2e+73], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[(t$95$1 + 0.00023919753086419754), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.680000000000000049

   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 93.6%

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

      1. associate-+r+ 93.6%

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

      2. unpow2 93.6%

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

   4. Simplified 93.6%

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

   if 0.680000000000000049 < im < 1.99999999999999997e73

   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 5.3%

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

      1. associate-*r* 5.3%

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

      2. distribute-rgt-out 84.2%

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

      3. unpow2 84.2%

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

   4. Simplified 84.2%

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

   if 1.99999999999999997e73 < 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. Taylor expanded in re around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-+r+ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. unpow2 100.0%

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

      6. fma-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   8. Applied egg-rr 98.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \left(0.08333333333333333 \cdot {im}^{4} + \color{blue}{0.00023919753086419754}\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.68:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \left(0.08333333333333333 \cdot {im}^{4} + 0.00023919753086419754\right)\right)\right)\\ \end{array} \]

Alternative 3: 92.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.68 \lor \neg \left(im \leq 2 \cdot 10^{+73}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.68) (not (<= im 2e+73)))
   (*
    (* 0.5 (cos re))
    (+ (+ 2.0 (* im im)) (* 0.08333333333333333 (pow im 4.0))))
   (* (+ (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.68) || !(im <= 2e+73)) {
		tmp = (0.5 * cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * pow(im, 4.0)));
	} else {
		tmp = (exp(-im) + exp(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 <= 0.68d0) .or. (.not. (im <= 2d+73))) then
        tmp = (0.5d0 * cos(re)) * ((2.0d0 + (im * im)) + (0.08333333333333333d0 * (im ** 4.0d0)))
    else
        tmp = (exp(-im) + exp(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 <= 0.68) || !(im <= 2e+73)) {
		tmp = (0.5 * Math.cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * Math.pow(im, 4.0)));
	} else {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.68) or not (im <= 2e+73):
		tmp = (0.5 * math.cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * math.pow(im, 4.0)))
	else:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.68) || !(im <= 2e+73))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64(0.08333333333333333 * (im ^ 4.0))));
	else
		tmp = Float64(Float64(exp(Float64(-im)) + exp(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 <= 0.68) || ~((im <= 2e+73)))
		tmp = (0.5 * cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * (im ^ 4.0)));
	else
		tmp = (exp(-im) + exp(im)) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.68], N[Not[LessEqual[im, 2e+73]], $MachinePrecision]], 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], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.680000000000000049 or 1.99999999999999997e73 < 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 94.5%

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

      1. associate-+r+ 94.5%

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

      2. unpow2 94.5%

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

   4. Simplified 94.5%

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

   if 0.680000000000000049 < im < 1.99999999999999997e73

   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 5.3%

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

      1. associate-*r* 5.3%

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

      2. distribute-rgt-out 84.2%

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

      3. unpow2 84.2%

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

   4. Simplified 84.2%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.68 \lor \neg \left(im \leq 2 \cdot 10^{+73}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 4: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.66:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.66)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 4.5e+152)
     (* (+ (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re))))
     (* im (* 0.5 (* (cos re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.66) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 4.5e+152) {
		tmp = (exp(-im) + exp(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 <= 0.66d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 4.5d+152) then
        tmp = (exp(-im) + exp(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 <= 0.66) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 4.5e+152) {
		tmp = (Math.exp(-im) + Math.exp(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 <= 0.66:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 4.5e+152:
		tmp = (math.exp(-im) + math.exp(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 <= 0.66)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 4.5e+152)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.66)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 4.5e+152)
		tmp = (exp(-im) + exp(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, 0.66], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+152], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.660000000000000031

   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 87.4%

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

      1. unpow2 87.4%

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

   4. Simplified 87.4%

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

   if 0.660000000000000031 < im < 4.5000000000000001e152

   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 2.6%

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

      1. associate-*r* 2.6%

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

      2. distribute-rgt-out 79.5%

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

      3. unpow2 79.5%

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

   4. Simplified 79.5%

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

   if 4.5000000000000001e152 < 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. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

      6. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.66:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.66:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.66)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 2.1e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* im (* 0.5 (* (cos re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.66) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 2.1e+154) {
		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 <= 0.66d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 2.1d+154) 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 <= 0.66) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 2.1e+154) {
		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 <= 0.66:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 2.1e+154:
		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 <= 0.66)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 2.1e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.66)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 2.1e+154)
		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, 0.66], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.1e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.660000000000000031

   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 87.4%

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

      1. unpow2 87.4%

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

   4. Simplified 87.4%

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

   if 0.660000000000000031 < im < 2.09999999999999994e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 69.2%

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

      1. *-commutative 69.2%

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

   4. Simplified 69.2%

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

   if 2.09999999999999994e154 < 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. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

      6. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.66:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 78.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ \mathbf{if}\;im \leq 3800000 \lor \neg \left(im \leq 2.15 \cdot 10^{+136}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t_0, -0.25 \cdot \left(t_0 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im))))
   (if (or (<= im 3800000.0) (not (<= im 2.15e+136)))
     (* (* 0.5 (cos re)) t_0)
     (fma 0.5 t_0 (* -0.25 (* t_0 (* re re)))))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double tmp;
	if ((im <= 3800000.0) || !(im <= 2.15e+136)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = fma(0.5, t_0, (-0.25 * (t_0 * (re * re))));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	tmp = 0.0
	if ((im <= 3800000.0) || !(im <= 2.15e+136))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = fma(0.5, t_0, Float64(-0.25 * Float64(t_0 * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[im, 3800000.0], N[Not[LessEqual[im, 2.15e+136]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(0.5 * t$95$0 + N[(-0.25 * N[(t$95$0 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 3.8e6 or 2.1499999999999999e136 < 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 87.6%

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

      1. unpow2 87.6%

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

   4. Simplified 87.6%

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

   if 3.8e6 < im < 2.1499999999999999e136

   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 5.2%

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

      1. unpow2 5.2%

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

   4. Simplified 5.2%

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

   5. Taylor expanded in re around 0 36.6%

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

      1. fma-def 36.6%

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

      2. unpow2 36.6%

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

      3. *-commutative 36.6%

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

      4. unpow2 36.6%

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

      5. unpow2 36.6%

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

   7. Simplified 36.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3800000 \lor \neg \left(im \leq 2.15 \cdot 10^{+136}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, 2 + im \cdot im, -0.25 \cdot \left(\left(2 + im \cdot im\right) \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]

Alternative 7: 78.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5800000000 \lor \neg \left(im \leq 2.15 \cdot 10^{+136}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 5800000000.0) (not (<= im 2.15e+136)))
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (* (* im im) (* -0.25 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 5800000000.0) || !(im <= 2.15e+136)) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else {
		tmp = (im * im) * (-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 <= 5800000000.0d0) .or. (.not. (im <= 2.15d+136))) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else
        tmp = (im * im) * ((-0.25d0) * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 5800000000.0) || !(im <= 2.15e+136)) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else {
		tmp = (im * im) * (-0.25 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 5800000000.0) or not (im <= 2.15e+136):
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	else:
		tmp = (im * im) * (-0.25 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 5800000000.0) || !(im <= 2.15e+136))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(Float64(im * im) * Float64(-0.25 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 5800000000.0) || ~((im <= 2.15e+136)))
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	else
		tmp = (im * im) * (-0.25 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 5800000000.0], N[Not[LessEqual[im, 2.15e+136]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.8e9 or 2.1499999999999999e136 < 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 87.6%

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

      1. unpow2 87.6%

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

   4. Simplified 87.6%

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

   if 5.8e9 < im < 2.1499999999999999e136

   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 5.2%

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

      1. unpow2 5.2%

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

   4. Simplified 5.2%

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

   5. Taylor expanded in re around 0 36.6%

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

      1. fma-def 36.6%

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

      2. unpow2 36.6%

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

      3. *-commutative 36.6%

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

      4. unpow2 36.6%

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

      5. unpow2 36.6%

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

   7. Simplified 36.6%

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

   8. Taylor expanded in re around inf 34.7%

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

      1. *-commutative 34.7%

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

      2. +-commutative 34.7%

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

      3. unpow2 34.7%

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

      4. fma-udef 34.7%

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

      5. associate-*r* 34.7%

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

      6. unpow2 34.7%

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

   10. Simplified 34.7%

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

   11. Taylor expanded in im around inf 34.7%

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

       1. unpow2 34.7%

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

       2. associate-*r* 34.7%

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

       3. unpow2 34.7%

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

   13. Simplified 34.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5800000000 \lor \neg \left(im \leq 2.15 \cdot 10^{+136}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 8: 65.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8500000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+136}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8500000.0)
   (cos re)
   (if (<= im 2.15e+136)
     (* (* im im) (* -0.25 (* re re)))
     (* im (* 0.5 (* (cos re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8500000.0) {
		tmp = cos(re);
	} else if (im <= 2.15e+136) {
		tmp = (im * im) * (-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 <= 8500000.0d0) then
        tmp = cos(re)
    else if (im <= 2.15d+136) then
        tmp = (im * im) * ((-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 <= 8500000.0) {
		tmp = Math.cos(re);
	} else if (im <= 2.15e+136) {
		tmp = (im * im) * (-0.25 * (re * re));
	} else {
		tmp = im * (0.5 * (Math.cos(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8500000.0:
		tmp = math.cos(re)
	elif im <= 2.15e+136:
		tmp = (im * im) * (-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 <= 8500000.0)
		tmp = cos(re);
	elseif (im <= 2.15e+136)
		tmp = Float64(Float64(im * im) * Float64(-0.25 * Float64(re * re)));
	else
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8500000.0)
		tmp = cos(re);
	elseif (im <= 2.15e+136)
		tmp = (im * im) * (-0.25 * (re * re));
	else
		tmp = im * (0.5 * (cos(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8500000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.15e+136], N[(N[(im * im), $MachinePrecision] * N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8.5e6

   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 66.1%

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

   if 8.5e6 < im < 2.1499999999999999e136

   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 5.2%

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

      1. unpow2 5.2%

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

   4. Simplified 5.2%

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

   5. Taylor expanded in re around 0 36.6%

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

      1. fma-def 36.6%

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

      2. unpow2 36.6%

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

      3. *-commutative 36.6%

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

      4. unpow2 36.6%

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

      5. unpow2 36.6%

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

   7. Simplified 36.6%

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

   8. Taylor expanded in re around inf 34.7%

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

      1. *-commutative 34.7%

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

      2. +-commutative 34.7%

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

      3. unpow2 34.7%

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

      4. fma-udef 34.7%

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

      5. associate-*r* 34.7%

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

      6. unpow2 34.7%

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

   10. Simplified 34.7%

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

   11. Taylor expanded in im around inf 34.7%

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

       1. unpow2 34.7%

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

       2. associate-*r* 34.7%

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

       3. unpow2 34.7%

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

   13. Simplified 34.7%

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

   if 2.1499999999999999e136 < 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. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

      6. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8500000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+136}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 62.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 500000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+136}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 500000000.0)
   (cos re)
   (if (<= im 2.15e+136)
     (* (* im im) (* -0.25 (* re re)))
     (+ 1.0 (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 500000000.0) {
		tmp = cos(re);
	} else if (im <= 2.15e+136) {
		tmp = (im * im) * (-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 <= 500000000.0d0) then
        tmp = cos(re)
    else if (im <= 2.15d+136) then
        tmp = (im * im) * ((-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 <= 500000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 2.15e+136) {
		tmp = (im * im) * (-0.25 * (re * re));
	} else {
		tmp = 1.0 + (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 500000000.0:
		tmp = math.cos(re)
	elif im <= 2.15e+136:
		tmp = (im * im) * (-0.25 * (re * re))
	else:
		tmp = 1.0 + (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 500000000.0)
		tmp = cos(re);
	elseif (im <= 2.15e+136)
		tmp = Float64(Float64(im * im) * 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 <= 500000000.0)
		tmp = cos(re);
	elseif (im <= 2.15e+136)
		tmp = (im * im) * (-0.25 * (re * re));
	else
		tmp = 1.0 + (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 500000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.15e+136], N[(N[(im * im), $MachinePrecision] * N[(-0.25 * N[(re * re), $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 < 5e8

   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 66.1%

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

   if 5e8 < im < 2.1499999999999999e136

   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 5.2%

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

      1. unpow2 5.2%

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

   4. Simplified 5.2%

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

   5. Taylor expanded in re around 0 36.6%

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

      1. fma-def 36.6%

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

      2. unpow2 36.6%

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

      3. *-commutative 36.6%

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

      4. unpow2 36.6%

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

      5. unpow2 36.6%

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

   7. Simplified 36.6%

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

   8. Taylor expanded in re around inf 34.7%

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

      1. *-commutative 34.7%

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

      2. +-commutative 34.7%

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

      3. unpow2 34.7%

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

      4. fma-udef 34.7%

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

      5. associate-*r* 34.7%

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

      6. unpow2 34.7%

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

   10. Simplified 34.7%

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

   11. Taylor expanded in im around inf 34.7%

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

       1. unpow2 34.7%

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

       2. associate-*r* 34.7%

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

       3. unpow2 34.7%

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

   13. Simplified 34.7%

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

   if 2.1499999999999999e136 < 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 55.2%

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

      1. distribute-rgt-in 55.2%

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

      2. metadata-eval 55.2%

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

      3. unpow2 55.2%

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

   7. Simplified 55.2%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 500000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+136}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 10: 49.3% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 720000:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 720000.0)
   (+ 1.0 (* 0.5 (* im im)))
   (* re (* (* im im) (* re -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 720000.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = re * ((im * im) * (re * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 720000.0d0) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = re * ((im * im) * (re * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 720000.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = re * ((im * im) * (re * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 720000.0:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = re * ((im * im) * (re * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 720000.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(re * Float64(Float64(im * im) * Float64(re * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 720000.0)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = re * ((im * im) * (re * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 720000.0], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.2e5

   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 75.5%

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

      1. unpow2 75.5%

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

   4. Simplified 75.5%

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

   5. Taylor expanded in re around 0 53.0%

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

      1. distribute-rgt-in 53.0%

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

      2. metadata-eval 53.0%

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

      3. unpow2 53.0%

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

   7. Simplified 53.0%

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

   if 7.2e5 < 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 78.8%

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

      1. unpow2 78.8%

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

   4. Simplified 78.8%

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

   5. Taylor expanded in re around 0 12.2%

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

      1. fma-def 12.2%

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

      2. unpow2 12.2%

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

      3. *-commutative 12.2%

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

      4. unpow2 12.2%

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

      5. unpow2 12.2%

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

   7. Simplified 12.2%

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

   8. Taylor expanded in re around inf 31.2%

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

      1. *-commutative 31.2%

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

      2. +-commutative 31.2%

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

      3. unpow2 31.2%

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

      4. fma-udef 31.2%

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

      5. associate-*r* 31.2%

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

      6. unpow2 31.2%

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

   10. Simplified 31.2%

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

   11. Taylor expanded in im around inf 31.1%

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

       1. unpow2 31.1%

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

       2. associate-*r* 31.1%

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

       3. unpow2 31.1%

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

       4. associate-*l* 31.1%

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

       5. *-commutative 31.1%

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

       6. associate-*l* 31.4%

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

       7. *-commutative 31.4%

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

   13. Simplified 31.4%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 720000:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 11: 39.8% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8500000:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+136}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8500000.0)
   1.0
   (if (<= im 1.9e+136) (* (* re re) -0.5) (* im (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8500000.0) {
		tmp = 1.0;
	} else if (im <= 1.9e+136) {
		tmp = (re * re) * -0.5;
	} else {
		tmp = im * (0.5 * 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 <= 8500000.0d0) then
        tmp = 1.0d0
    else if (im <= 1.9d+136) then
        tmp = (re * re) * (-0.5d0)
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8500000.0) {
		tmp = 1.0;
	} else if (im <= 1.9e+136) {
		tmp = (re * re) * -0.5;
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8500000.0:
		tmp = 1.0
	elif im <= 1.9e+136:
		tmp = (re * re) * -0.5
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8500000.0)
		tmp = 1.0;
	elseif (im <= 1.9e+136)
		tmp = Float64(Float64(re * re) * -0.5);
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8500000.0)
		tmp = 1.0;
	elseif (im <= 1.9e+136)
		tmp = (re * re) * -0.5;
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8500000.0], 1.0, If[LessEqual[im, 1.9e+136], N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8.5e6

   1. Initial program 100.0%

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

   3. Applied egg-rr 35.7%

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

      1. +-inverses 35.7%

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

      2. +-rgt-identity 35.7%

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

      3. *-inverses 35.7%

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

   5. Simplified 35.7%

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

   if 8.5e6 < im < 1.90000000000000007e136

   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 5.2%

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

      1. unpow2 5.2%

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

   4. Simplified 5.2%

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

   5. Taylor expanded in re around 0 36.6%

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

      1. fma-def 36.6%

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

      2. unpow2 36.6%

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

      3. *-commutative 36.6%

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

      4. unpow2 36.6%

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

      5. unpow2 36.6%

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

   7. Simplified 36.6%

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

   8. Taylor expanded in re around inf 34.7%

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

      1. *-commutative 34.7%

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

      2. +-commutative 34.7%

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

      3. unpow2 34.7%

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

      4. fma-udef 34.7%

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

      5. associate-*r* 34.7%

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

      6. unpow2 34.7%

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

   10. Simplified 34.7%

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

   11. Taylor expanded in im around 0 24.3%

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

       1. *-commutative 24.3%

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

       2. unpow2 24.3%

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

   13. Simplified 24.3%

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

   if 1.90000000000000007e136 < 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. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

      6. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 55.2%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8500000:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+136}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 12: 48.8% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+160}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.3e+160) (+ 1.0 (* 0.5 (* im im))) (* (* re re) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.3e+160) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (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 <= 3.3d+160) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = (re * re) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.3e+160) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (re * re) * -0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.3e+160:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = (re * re) * -0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.3e+160)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(Float64(re * re) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.3e+160)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = (re * re) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.3e+160], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.2999999999999997e160

   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 76.3%

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

      1. unpow2 76.3%

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

   4. Simplified 76.3%

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

   5. Taylor expanded in re around 0 48.8%

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

      1. distribute-rgt-in 48.8%

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

      2. metadata-eval 48.8%

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

      3. unpow2 48.8%

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

   7. Simplified 48.8%

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

   if 3.2999999999999997e160 < 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 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 12.4%

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

      1. fma-def 12.4%

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

      2. unpow2 12.4%

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

      3. *-commutative 12.4%

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

      4. unpow2 12.4%

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

      5. unpow2 12.4%

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

   7. Simplified 12.4%

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

   8. Taylor expanded in re around inf 35.3%

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

      1. *-commutative 35.3%

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

      2. +-commutative 35.3%

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

      3. unpow2 35.3%

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

      4. fma-udef 35.3%

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

      5. associate-*r* 35.3%

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

      6. unpow2 35.3%

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

   10. Simplified 35.3%

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

   11. Taylor expanded in im around 0 35.3%

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

       1. *-commutative 35.3%

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

       2. unpow2 35.3%

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

   13. Simplified 35.3%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+160}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -0.5\\ \end{array} \]

Alternative 13: 48.8% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+160}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.3e+160) (+ 1.0 (* 0.5 (* im im))) (+ 1.0 (* (* re re) -0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.3e+160) {
		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 <= 3.3d+160) 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 <= 3.3e+160) {
		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 <= 3.3e+160:
		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 <= 3.3e+160)
		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 (re <= 3.3e+160)
		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[LessEqual[re, 3.3e+160], 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 re < 3.2999999999999997e160

   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 76.3%

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

      1. unpow2 76.3%

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

   4. Simplified 76.3%

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

   5. Taylor expanded in re around 0 48.8%

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

      1. distribute-rgt-in 48.8%

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

      2. metadata-eval 48.8%

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

      3. unpow2 48.8%

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

   7. Simplified 48.8%

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

   if 3.2999999999999997e160 < 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 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 12.4%

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

      1. fma-def 12.4%

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

      2. unpow2 12.4%

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

      3. *-commutative 12.4%

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

      4. unpow2 12.4%

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

      5. unpow2 12.4%

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

   7. Simplified 12.4%

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

   8. Taylor expanded in im around 0 35.3%

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

      1. *-commutative 35.3%

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

      2. unpow2 35.3%

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

   10. Simplified 35.3%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+160}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \]

Alternative 14: 38.7% accurate, 43.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0145:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.0145) 1.0 (* im (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0145) {
		tmp = 1.0;
	} else {
		tmp = im * (0.5 * 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 <= 0.0145d0) then
        tmp = 1.0d0
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0145) {
		tmp = 1.0;
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0145:
		tmp = 1.0
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0145)
		tmp = 1.0;
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0145)
		tmp = 1.0;
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0145], 1.0, N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0145000000000000007

   1. Initial program 100.0%

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

   3. Applied egg-rr 36.5%

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

      1. +-inverses 36.5%

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

      2. +-rgt-identity 36.5%

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

      3. *-inverses 36.5%

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

   5. Simplified 36.5%

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

   if 0.0145000000000000007 < 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 45.4%

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

      1. unpow2 45.4%

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

   4. Simplified 45.4%

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

   5. Taylor expanded in im around inf 45.2%

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

      1. associate-*r* 45.2%

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

      2. unpow2 45.2%

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

      3. associate-*r* 45.2%

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

      4. *-commutative 45.2%

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

      5. associate-*l* 45.2%

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

      6. *-commutative 45.2%

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

   7. Simplified 45.2%

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

   8. Taylor expanded in re around 0 25.2%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0145:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 15: 29.7% 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 27.3%

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

   1. +-inverses 27.3%

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

   2. +-rgt-identity 27.3%

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

   3. *-inverses 27.3%

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

5. Simplified 27.3%

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

6. Final simplification 27.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023274 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))