math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 13

Speedup: 1.0×

• 49.9% 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: 72.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00136:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00136)
   (cos re)
   (if (<= im 1.35e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00136) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00136d0) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00136) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00136:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00136)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00136)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00136], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00136

   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 65.8%

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

   if 0.00136 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 80.0%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00136:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]

Alternative 3: 85.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 0.0031:\\ \;\;\;\;\cos re \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 0.0031)
     (* (cos re) (+ t_0 1.0))
     (if (<= im 1.35e+154)
       (* 0.5 (+ (exp (- im)) (exp im)))
       (* (cos re) t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * pow(im, 2.0);
	double tmp;
	if (im <= 0.0031) {
		tmp = cos(re) * (t_0 + 1.0);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im ** 2.0d0)
    if (im <= 0.0031d0) then
        tmp = cos(re) * (t_0 + 1.0d0)
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.pow(im, 2.0);
	double tmp;
	if (im <= 0.0031) {
		tmp = Math.cos(re) * (t_0 + 1.0);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.pow(im, 2.0)
	tmp = 0
	if im <= 0.0031:
		tmp = math.cos(re) * (t_0 + 1.0)
	elif im <= 1.35e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * (im ^ 2.0))
	tmp = 0.0
	if (im <= 0.0031)
		tmp = Float64(cos(re) * Float64(t_0 + 1.0));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im ^ 2.0);
	tmp = 0.0;
	if (im <= 0.0031)
		tmp = cos(re) * (t_0 + 1.0);
	elseif (im <= 1.35e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.0031], N[(N[Cos[re], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00309999999999999989

   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 85.1%

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

      1. associate-*r* 85.1%

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

      2. distribute-rgt1-in 85.1%

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

   4. Simplified 85.1%

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

   if 0.00309999999999999989 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 80.0%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0031:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]

Alternative 4: 69.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.1e-5) (cos re) (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.1e-5) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (exp(-im) + exp(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 <= 3.1d-5) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.1e-5) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.1e-5:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.1e-5)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.1e-5)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.1e-5], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.10000000000000014e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 65.8%

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

   if 3.10000000000000014e-5 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 85.9%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 5: 66.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{im}^{6} \cdot 0.125}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.2e+27) (cos re) (cbrt (* (pow im 6.0) 0.125))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.2e+27) {
		tmp = cos(re);
	} else {
		tmp = cbrt((pow(im, 6.0) * 0.125));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.2e+27) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.cbrt((Math.pow(im, 6.0) * 0.125));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.2e+27)
		tmp = cos(re);
	else
		tmp = cbrt(Float64((im ^ 6.0) * 0.125));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.2e+27], N[Cos[re], $MachinePrecision], N[Power[N[(N[Power[im, 6.0], $MachinePrecision] * 0.125), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.19999999999999966e27

   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 64.2%

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

   if 7.19999999999999966e27 < 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 59.8%

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

      1. associate-*r* 59.8%

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

      2. distribute-rgt1-in 59.8%

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

   4. Simplified 59.8%

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

   5. Taylor expanded in im around inf 59.8%

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

      1. associate-*r* 59.8%

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

      2. *-commutative 59.8%

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

   7. Simplified 59.8%

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

   8. Taylor expanded in re around 0 54.3%

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

      1. add-cbrt-cube 78.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.5 \cdot {im}^{2}\right) \cdot \left(0.5 \cdot {im}^{2}\right)\right) \cdot \left(0.5 \cdot {im}^{2}\right)}} \]

      2. unpow3 78.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(0.5 \cdot {im}^{2}\right)}^{3}}} \]

      3. pow1/3 78.3%

         \[\leadsto \color{blue}{{\left({\left(0.5 \cdot {im}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \]

      4. *-commutative 78.3%

         \[\leadsto {\left({\color{blue}{\left({im}^{2} \cdot 0.5\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. cube-prod 78.3%

         \[\leadsto {\color{blue}{\left({\left({im}^{2}\right)}^{3} \cdot {0.5}^{3}\right)}}^{0.3333333333333333} \]

      6. pow3 78.3%

         \[\leadsto {\left(\color{blue}{\left(\left({im}^{2} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      7. unpow2 78.3%

         \[\leadsto {\left(\left(\left({im}^{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      8. unswap-sqr 78.3%

         \[\leadsto {\left(\color{blue}{\left(\left({im}^{2} \cdot im\right) \cdot \left({im}^{2} \cdot im\right)\right)} \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      9. pow-plus 78.3%

         \[\leadsto {\left(\left(\color{blue}{{im}^{\left(2 + 1\right)}} \cdot \left({im}^{2} \cdot im\right)\right) \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      10. metadata-eval 78.3%

          \[\leadsto {\left(\left({im}^{\color{blue}{3}} \cdot \left({im}^{2} \cdot im\right)\right) \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      11. pow-plus 78.3%

          \[\leadsto {\left(\left({im}^{3} \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      12. metadata-eval 78.3%

          \[\leadsto {\left(\left({im}^{3} \cdot {im}^{\color{blue}{3}}\right) \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      13. pow-sqr 78.3%

          \[\leadsto {\left(\color{blue}{{im}^{\left(2 \cdot 3\right)}} \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      14. metadata-eval 78.3%

          \[\leadsto {\left({im}^{\color{blue}{6}} \cdot {0.5}^{3}\right)}^{0.3333333333333333} \]

      15. metadata-eval 78.3%

          \[\leadsto {\left({im}^{6} \cdot \color{blue}{0.125}\right)}^{0.3333333333333333} \]

   10. Applied egg-rr 78.3%

       \[\leadsto \color{blue}{{\left({im}^{6} \cdot 0.125\right)}^{0.3333333333333333}} \]
   11. Step-by-step derivation

       1. unpow1/3 78.3%

          \[\leadsto \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]

   12. Simplified 78.3%

       \[\leadsto \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{im}^{6} \cdot 0.125}\\ \end{array} \]

Alternative 6: 61.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+22}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+129}:\\ \;\;\;\;0.25 + {re}^{2} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.6e+22)
   (cos re)
   (if (<= im 3.5e+129) (+ 0.25 (* (pow re 2.0) 0.25)) (* 0.5 (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.6e+22) {
		tmp = cos(re);
	} else if (im <= 3.5e+129) {
		tmp = 0.25 + (pow(re, 2.0) * 0.25);
	} else {
		tmp = 0.5 * pow(im, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.6d+22) then
        tmp = cos(re)
    else if (im <= 3.5d+129) then
        tmp = 0.25d0 + ((re ** 2.0d0) * 0.25d0)
    else
        tmp = 0.5d0 * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.6e+22) {
		tmp = Math.cos(re);
	} else if (im <= 3.5e+129) {
		tmp = 0.25 + (Math.pow(re, 2.0) * 0.25);
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.6e+22:
		tmp = math.cos(re)
	elif im <= 3.5e+129:
		tmp = 0.25 + (math.pow(re, 2.0) * 0.25)
	else:
		tmp = 0.5 * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.6e+22)
		tmp = cos(re);
	elseif (im <= 3.5e+129)
		tmp = Float64(0.25 + Float64((re ^ 2.0) * 0.25));
	else
		tmp = Float64(0.5 * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.6e+22)
		tmp = cos(re);
	elseif (im <= 3.5e+129)
		tmp = 0.25 + ((re ^ 2.0) * 0.25);
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.6e+22], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.5e+129], N[(0.25 + N[(N[Power[re, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.6e22

   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 64.5%

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

   if 3.6e22 < im < 3.4999999999999998e129

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.6%

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

   4. Taylor expanded in re around 0 19.5%

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

      1. *-commutative 19.5%

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

   6. Simplified 19.5%

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

   if 3.4999999999999998e129 < 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 92.5%

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

      1. associate-*r* 92.5%

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

      2. distribute-rgt1-in 92.5%

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

   4. Simplified 92.5%

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

   5. Taylor expanded in im around inf 92.5%

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

      1. associate-*r* 92.5%

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

      2. *-commutative 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in re around 0 84.4%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+22}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+129}:\\ \;\;\;\;0.25 + {re}^{2} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]

Alternative 7: 61.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+22}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+129}:\\ \;\;\;\;0.25 + {re}^{2} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.6e+22)
   (cos re)
   (if (<= im 2.9e+129)
     (+ 0.25 (* (pow re 2.0) 0.25))
     (+ (* 0.5 (pow im 2.0)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.6e+22) {
		tmp = cos(re);
	} else if (im <= 2.9e+129) {
		tmp = 0.25 + (pow(re, 2.0) * 0.25);
	} else {
		tmp = (0.5 * pow(im, 2.0)) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.6d+22) then
        tmp = cos(re)
    else if (im <= 2.9d+129) then
        tmp = 0.25d0 + ((re ** 2.0d0) * 0.25d0)
    else
        tmp = (0.5d0 * (im ** 2.0d0)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.6e+22) {
		tmp = Math.cos(re);
	} else if (im <= 2.9e+129) {
		tmp = 0.25 + (Math.pow(re, 2.0) * 0.25);
	} else {
		tmp = (0.5 * Math.pow(im, 2.0)) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.6e+22:
		tmp = math.cos(re)
	elif im <= 2.9e+129:
		tmp = 0.25 + (math.pow(re, 2.0) * 0.25)
	else:
		tmp = (0.5 * math.pow(im, 2.0)) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.6e+22)
		tmp = cos(re);
	elseif (im <= 2.9e+129)
		tmp = Float64(0.25 + Float64((re ^ 2.0) * 0.25));
	else
		tmp = Float64(Float64(0.5 * (im ^ 2.0)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.6e+22)
		tmp = cos(re);
	elseif (im <= 2.9e+129)
		tmp = 0.25 + ((re ^ 2.0) * 0.25);
	else
		tmp = (0.5 * (im ^ 2.0)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.6e+22], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.9e+129], N[(0.25 + N[(N[Power[re, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.6e22

   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 64.5%

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

   if 3.6e22 < im < 2.90000000000000003e129

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.6%

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

   4. Taylor expanded in re around 0 19.5%

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

      1. *-commutative 19.5%

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

   6. Simplified 19.5%

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

   if 2.90000000000000003e129 < 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 92.5%

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

      1. associate-*r* 92.5%

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

      2. distribute-rgt1-in 92.5%

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

   4. Simplified 92.5%

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

   5. Taylor expanded in re around 0 84.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+22}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+129}:\\ \;\;\;\;0.25 + {re}^{2} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2} + 1\\ \end{array} \]

Alternative 8: 61.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0031:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+144}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0031)
   (cos re)
   (if (<= im 8e+144)
     (+ 1.0 (* (pow re 2.0) -0.5))
     (+ (* 0.5 (pow im 2.0)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0031) {
		tmp = cos(re);
	} else if (im <= 8e+144) {
		tmp = 1.0 + (pow(re, 2.0) * -0.5);
	} else {
		tmp = (0.5 * pow(im, 2.0)) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0031d0) then
        tmp = cos(re)
    else if (im <= 8d+144) then
        tmp = 1.0d0 + ((re ** 2.0d0) * (-0.5d0))
    else
        tmp = (0.5d0 * (im ** 2.0d0)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0031) {
		tmp = Math.cos(re);
	} else if (im <= 8e+144) {
		tmp = 1.0 + (Math.pow(re, 2.0) * -0.5);
	} else {
		tmp = (0.5 * Math.pow(im, 2.0)) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0031:
		tmp = math.cos(re)
	elif im <= 8e+144:
		tmp = 1.0 + (math.pow(re, 2.0) * -0.5)
	else:
		tmp = (0.5 * math.pow(im, 2.0)) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0031)
		tmp = cos(re);
	elseif (im <= 8e+144)
		tmp = Float64(1.0 + Float64((re ^ 2.0) * -0.5));
	else
		tmp = Float64(Float64(0.5 * (im ^ 2.0)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0031)
		tmp = cos(re);
	elseif (im <= 8e+144)
		tmp = 1.0 + ((re ^ 2.0) * -0.5);
	else
		tmp = (0.5 * (im ^ 2.0)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0031], N[Cos[re], $MachinePrecision], If[LessEqual[im, 8e+144], N[(1.0 + N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00309999999999999989

   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 65.8%

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

   if 0.00309999999999999989 < im < 8.00000000000000019e144

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.8%

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

      1. associate-*r* 4.8%

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

      2. distribute-rgt1-in 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in re around 0 16.1%

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

   6. Taylor expanded in im around 0 15.2%

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

      1. *-commutative 15.2%

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

   8. Simplified 15.2%

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

   if 8.00000000000000019e144 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 91.2%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0031:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+144}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2} + 1\\ \end{array} \]

Alternative 9: 60.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7e+26) (cos re) (* 0.5 (pow im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7e+26) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * pow(im, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7d+26) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7e+26) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7e+26:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7e+26)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7e+26)
		tmp = cos(re);
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7e+26], N[Cos[re], $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.9999999999999998e26

   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 64.2%

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

   if 6.9999999999999998e26 < 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 59.8%

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

      1. associate-*r* 59.8%

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

      2. distribute-rgt1-in 59.8%

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

   4. Simplified 59.8%

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

   5. Taylor expanded in im around inf 59.8%

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

      1. associate-*r* 59.8%

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

      2. *-commutative 59.8%

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

   7. Simplified 59.8%

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

   8. Taylor expanded in re around 0 54.3%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]

Alternative 10: 50.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \cos re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (cos re))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return Math.cos(re);
}

Python

def code(re, im):
	return math.cos(re)

Julia

function code(re, im)
	return cos(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Cos[re], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 50.1%

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

3. Final simplification 50.1%

   \[\leadsto \cos re \]

Alternative 11: 2.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

3. Applied egg-rr 2.3%

   \[\leadsto \color{blue}{\log \left({1}^{\cos re}\right)} \]
4. Step-by-step derivation

   1. pow-base-1 2.3%

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

   2. metadata-eval 2.3%

      \[\leadsto \color{blue}{0} \]

5. Simplified 2.3%

   \[\leadsto \color{blue}{0} \]

6. Final simplification 2.3%

   \[\leadsto 0 \]

Alternative 12: 8.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

3. Applied egg-rr 7.8%

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

4. Taylor expanded in re around 0 8.0%

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

5. Final simplification 8.0%

   \[\leadsto 0.25 \]

Alternative 13: 28.4% 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 30.1%

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

   1. +-inverses 30.1%

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

   2. +-rgt-identity 30.1%

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

   3. *-inverses 30.1%

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

5. Simplified 30.1%

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

6. Final simplification 30.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023305 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))