math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 10

Speedup: 1.0×

• 49.3% 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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.112:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \sqrt[3]{{im}^{6} \cdot 0.125}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.112)
   (* (cos re) (+ (* 0.5 (pow im 2.0)) 1.0))
   (if (<= im 2.4e+51)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (cbrt (* (pow im 6.0) 0.125))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.112) {
		tmp = cos(re) * ((0.5 * pow(im, 2.0)) + 1.0);
	} else if (im <= 2.4e+51) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * cbrt((pow(im, 6.0) * 0.125));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.112) {
		tmp = Math.cos(re) * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	} else if (im <= 2.4e+51) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * Math.cbrt((Math.pow(im, 6.0) * 0.125));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.112)
		tmp = Float64(cos(re) * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	elseif (im <= 2.4e+51)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * cbrt(Float64((im ^ 6.0) * 0.125)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.112], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.4e+51], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[Power[N[(N[Power[im, 6.0], $MachinePrecision] * 0.125), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.112000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.0%

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

      1. associate-*r* 87.0%

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

      2. distribute-rgt1-in 87.0%

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

   5. Simplified 87.0%

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

   if 0.112000000000000002 < im < 2.3999999999999999e51

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 71.0%

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

   if 2.3999999999999999e51 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 64.9%

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

      1. associate-*r* 64.9%

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

      2. distribute-rgt1-in 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in im around inf 64.9%

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

      1. associate-*r* 64.9%

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

      2. *-commutative 64.9%

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

   8. Simplified 64.9%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow3 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow-prod-down 100.0%

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

      6. pow-pow 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow1/3 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.112:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \sqrt[3]{{im}^{6} \cdot 0.125}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.12:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 1.86 \cdot 10^{+154}:\\ \;\;\;\;{im}^{2} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.12)
   (* (cos re) (+ (* 0.5 (pow im 2.0)) 1.0))
   (if (<= im 2.5e+133)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (if (<= im 1.86e+154)
       (* (pow im 2.0) (+ 0.5 (* -0.25 (pow re 2.0))))
       (* (cos re) (* im (* 0.5 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.12) {
		tmp = cos(re) * ((0.5 * pow(im, 2.0)) + 1.0);
	} else if (im <= 2.5e+133) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else if (im <= 1.86e+154) {
		tmp = pow(im, 2.0) * (0.5 + (-0.25 * pow(re, 2.0)));
	} else {
		tmp = cos(re) * (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 <= 0.12d0) then
        tmp = cos(re) * ((0.5d0 * (im ** 2.0d0)) + 1.0d0)
    else if (im <= 2.5d+133) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else if (im <= 1.86d+154) then
        tmp = (im ** 2.0d0) * (0.5d0 + ((-0.25d0) * (re ** 2.0d0)))
    else
        tmp = cos(re) * (im * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.12) {
		tmp = Math.cos(re) * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	} else if (im <= 2.5e+133) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else if (im <= 1.86e+154) {
		tmp = Math.pow(im, 2.0) * (0.5 + (-0.25 * Math.pow(re, 2.0)));
	} else {
		tmp = Math.cos(re) * (im * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.12:
		tmp = math.cos(re) * ((0.5 * math.pow(im, 2.0)) + 1.0)
	elif im <= 2.5e+133:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	elif im <= 1.86e+154:
		tmp = math.pow(im, 2.0) * (0.5 + (-0.25 * math.pow(re, 2.0)))
	else:
		tmp = math.cos(re) * (im * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.12)
		tmp = Float64(cos(re) * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	elseif (im <= 2.5e+133)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	elseif (im <= 1.86e+154)
		tmp = Float64((im ^ 2.0) * Float64(0.5 + Float64(-0.25 * (re ^ 2.0))));
	else
		tmp = Float64(cos(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.12)
		tmp = cos(re) * ((0.5 * (im ^ 2.0)) + 1.0);
	elseif (im <= 2.5e+133)
		tmp = 0.5 * (exp(-im) + exp(im));
	elseif (im <= 1.86e+154)
		tmp = (im ^ 2.0) * (0.5 + (-0.25 * (re ^ 2.0)));
	else
		tmp = cos(re) * (im * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.12], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+133], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.86e+154], N[(N[Power[im, 2.0], $MachinePrecision] * N[(0.5 + N[(-0.25 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.12

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.0%

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

      1. associate-*r* 87.0%

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

      2. distribute-rgt1-in 87.0%

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

   5. Simplified 87.0%

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

   if 0.12 < im < 2.4999999999999998e133

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 76.6%

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

   if 2.4999999999999998e133 < im < 1.86000000000000014e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 8.8%

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

      1. associate-*r* 8.8%

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

      2. distribute-rgt1-in 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in im around inf 8.8%

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

      1. associate-*r* 8.8%

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

      2. *-commutative 8.8%

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

   8. Simplified 8.8%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow3 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow-prod-down 100.0%

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

      6. pow-pow 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow1/3 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in re around 0 82.2%

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

       1. *-commutative 82.2%

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

       2. associate-*r* 82.2%

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

       3. distribute-rgt-out 82.2%

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

       4. +-commutative 82.2%

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

   15. Simplified 82.2%

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

   if 1.86000000000000014e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \cos re\right)} \]
   7. 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%

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

   8. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow3 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow-prod-down 100.0%

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

      6. pow-pow 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow1/3 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]
   13. Step-by-step derivation

       1. *-commutative 100.0%

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

       2. cbrt-prod 100.0%

          \[\leadsto \cos re \cdot \color{blue}{\left(\sqrt[3]{0.125} \cdot \sqrt[3]{{im}^{6}}\right)} \]

       3. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\color{blue}{0.25 \cdot 0.5}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       4. metadata-eval 100.0%

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

       5. rem-square-sqrt 100.0%

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

       6. pow2 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left(\color{blue}{{\left(\sqrt{0.5}\right)}^{2}} \cdot 0.5\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       7. rem-square-sqrt 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       8. pow2 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{0.5}\right)}^{2}}\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       9. rem-square-sqrt 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       10. pow2 100.0%

           \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt{0.5}\right)}^{2}}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       11. add-cbrt-cube 100.0%

           \[\leadsto \cos re \cdot \left(\color{blue}{{\left(\sqrt{0.5}\right)}^{2}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       12. pow2 100.0%

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

       13. rem-square-sqrt 100.0%

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

       14. sqr-pow 100.0%

           \[\leadsto \cos re \cdot \left(0.5 \cdot \sqrt[3]{\color{blue}{{im}^{\left(\frac{6}{2}\right)} \cdot {im}^{\left(\frac{6}{2}\right)}}}\right) \]

       15. metadata-eval 100.0%

           \[\leadsto \cos re \cdot \left(0.5 \cdot \sqrt[3]{{im}^{\color{blue}{3}} \cdot {im}^{\left(\frac{6}{2}\right)}}\right) \]

       16. metadata-eval 100.0%

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

       17. pow-prod-down 100.0%

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

       18. unpow2 100.0%

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

       19. pow3 100.0%

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

       20. add-cbrt-cube 100.0%

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

       21. unpow2 100.0%

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

       22. associate-*r* 100.0%

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

   14. Applied egg-rr 100.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.12:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 1.86 \cdot 10^{+154}:\\ \;\;\;\;{im}^{2} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.18:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.86 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.18)
   (* (cos re) (+ (* 0.5 (pow im 2.0)) 1.0))
   (if (<= im 1.86e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.18) {
		tmp = cos(re) * ((0.5 * pow(im, 2.0)) + 1.0);
	} else if (im <= 1.86e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (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 <= 0.18d0) then
        tmp = cos(re) * ((0.5d0 * (im ** 2.0d0)) + 1.0d0)
    else if (im <= 1.86d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * (im * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.18) {
		tmp = Math.cos(re) * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	} else if (im <= 1.86e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * (im * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.18:
		tmp = math.cos(re) * ((0.5 * math.pow(im, 2.0)) + 1.0)
	elif im <= 1.86e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * (im * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.18)
		tmp = Float64(cos(re) * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	elseif (im <= 1.86e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.18)
		tmp = cos(re) * ((0.5 * (im ^ 2.0)) + 1.0);
	elseif (im <= 1.86e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * (im * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.18], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.86e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.17999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.0%

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

      1. associate-*r* 87.0%

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

      2. distribute-rgt1-in 87.0%

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

   5. Simplified 87.0%

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

   if 0.17999999999999999 < im < 1.86000000000000014e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 68.3%

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

   if 1.86000000000000014e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \cos re\right)} \]
   7. 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%

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

   8. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow3 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow-prod-down 100.0%

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

      6. pow-pow 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow1/3 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]
   13. Step-by-step derivation

       1. *-commutative 100.0%

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

       2. cbrt-prod 100.0%

          \[\leadsto \cos re \cdot \color{blue}{\left(\sqrt[3]{0.125} \cdot \sqrt[3]{{im}^{6}}\right)} \]

       3. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\color{blue}{0.25 \cdot 0.5}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       4. metadata-eval 100.0%

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

       5. rem-square-sqrt 100.0%

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

       6. pow2 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left(\color{blue}{{\left(\sqrt{0.5}\right)}^{2}} \cdot 0.5\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       7. rem-square-sqrt 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       8. pow2 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{0.5}\right)}^{2}}\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       9. rem-square-sqrt 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       10. pow2 100.0%

           \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt{0.5}\right)}^{2}}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       11. add-cbrt-cube 100.0%

           \[\leadsto \cos re \cdot \left(\color{blue}{{\left(\sqrt{0.5}\right)}^{2}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       12. pow2 100.0%

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

       13. rem-square-sqrt 100.0%

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

       14. sqr-pow 100.0%

           \[\leadsto \cos re \cdot \left(0.5 \cdot \sqrt[3]{\color{blue}{{im}^{\left(\frac{6}{2}\right)} \cdot {im}^{\left(\frac{6}{2}\right)}}}\right) \]

       15. metadata-eval 100.0%

           \[\leadsto \cos re \cdot \left(0.5 \cdot \sqrt[3]{{im}^{\color{blue}{3}} \cdot {im}^{\left(\frac{6}{2}\right)}}\right) \]

       16. metadata-eval 100.0%

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

       17. pow-prod-down 100.0%

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

       18. unpow2 100.0%

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

       19. pow3 100.0%

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

       20. add-cbrt-cube 100.0%

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

       21. unpow2 100.0%

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

       22. associate-*r* 100.0%

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

   14. Applied egg-rr 100.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.18:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.86 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.112:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.112)
   (cos re)
   (if (<= im 2e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.112) {
		tmp = cos(re);
	} else if (im <= 2e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (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 <= 0.112d0) then
        tmp = cos(re)
    else if (im <= 2d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * (im * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.112) {
		tmp = Math.cos(re);
	} else if (im <= 2e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * (im * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.112:
		tmp = math.cos(re)
	elif im <= 2e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * (im * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.112)
		tmp = cos(re);
	elseif (im <= 2e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.112)
		tmp = cos(re);
	elseif (im <= 2e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * (im * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.112], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.112000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 73.6%

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

   if 0.112000000000000002 < im < 2.00000000000000007e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 68.3%

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

   if 2.00000000000000007e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \cos re\right)} \]
   7. 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%

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

   8. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow3 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow-prod-down 100.0%

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

      6. pow-pow 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow1/3 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]
   13. Step-by-step derivation

       1. *-commutative 100.0%

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

       2. cbrt-prod 100.0%

          \[\leadsto \cos re \cdot \color{blue}{\left(\sqrt[3]{0.125} \cdot \sqrt[3]{{im}^{6}}\right)} \]

       3. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\color{blue}{0.25 \cdot 0.5}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       4. metadata-eval 100.0%

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

       5. rem-square-sqrt 100.0%

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

       6. pow2 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left(\color{blue}{{\left(\sqrt{0.5}\right)}^{2}} \cdot 0.5\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       7. rem-square-sqrt 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       8. pow2 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{0.5}\right)}^{2}}\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       9. rem-square-sqrt 100.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       10. pow2 100.0%

           \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt{0.5}\right)}^{2}}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       11. add-cbrt-cube 100.0%

           \[\leadsto \cos re \cdot \left(\color{blue}{{\left(\sqrt{0.5}\right)}^{2}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       12. pow2 100.0%

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

       13. rem-square-sqrt 100.0%

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

       14. sqr-pow 100.0%

           \[\leadsto \cos re \cdot \left(0.5 \cdot \sqrt[3]{\color{blue}{{im}^{\left(\frac{6}{2}\right)} \cdot {im}^{\left(\frac{6}{2}\right)}}}\right) \]

       15. metadata-eval 100.0%

           \[\leadsto \cos re \cdot \left(0.5 \cdot \sqrt[3]{{im}^{\color{blue}{3}} \cdot {im}^{\left(\frac{6}{2}\right)}}\right) \]

       16. metadata-eval 100.0%

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

       17. pow-prod-down 100.0%

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

       18. unpow2 100.0%

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

       19. pow3 100.0%

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

       20. add-cbrt-cube 100.0%

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

       21. unpow2 100.0%

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

       22. associate-*r* 100.0%

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

   14. Applied egg-rr 100.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.112:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.4) (cos re) (* (cos re) (* im (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = cos(re);
	} else {
		tmp = cos(re) * (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 <= 1.4d0) then
        tmp = cos(re)
    else
        tmp = cos(re) * (im * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.cos(re) * (im * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.4:
		tmp = math.cos(re)
	else:
		tmp = math.cos(re) * (im * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.4)
		tmp = cos(re);
	else
		tmp = Float64(cos(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.4)
		tmp = cos(re);
	else
		tmp = cos(re) * (im * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.4], N[Cos[re], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 73.6%

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

   if 1.3999999999999999 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 53.4%

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

      1. associate-*r* 53.4%

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

      2. distribute-rgt1-in 53.4%

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

   5. Simplified 53.4%

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

   6. Taylor expanded in im around inf 53.4%

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

      1. associate-*r* 53.4%

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

      2. *-commutative 53.4%

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

   8. Simplified 53.4%

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

      1. add-cbrt-cube 82.0%

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

      2. pow1/3 82.0%

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

      3. pow3 82.0%

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

      4. *-commutative 82.0%

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

      5. unpow-prod-down 82.0%

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

      6. pow-pow 82.0%

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

      7. metadata-eval 82.0%

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

      8. metadata-eval 82.0%

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

   10. Applied egg-rr 82.0%

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

       1. unpow1/3 82.0%

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

   12. Simplified 82.0%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt[3]{{im}^{6} \cdot 0.125}} \]
   13. Step-by-step derivation

       1. *-commutative 82.0%

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

       2. cbrt-prod 82.0%

          \[\leadsto \cos re \cdot \color{blue}{\left(\sqrt[3]{0.125} \cdot \sqrt[3]{{im}^{6}}\right)} \]

       3. metadata-eval 82.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\color{blue}{0.25 \cdot 0.5}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       4. metadata-eval 82.0%

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

       5. rem-square-sqrt 82.0%

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

       6. pow2 82.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left(\color{blue}{{\left(\sqrt{0.5}\right)}^{2}} \cdot 0.5\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       7. rem-square-sqrt 82.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       8. pow2 82.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{0.5}\right)}^{2}}\right) \cdot 0.5} \cdot \sqrt[3]{{im}^{6}}\right) \]

       9. rem-square-sqrt 82.0%

          \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       10. pow2 82.0%

           \[\leadsto \cos re \cdot \left(\sqrt[3]{\left({\left(\sqrt{0.5}\right)}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt{0.5}\right)}^{2}}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       11. add-cbrt-cube 82.0%

           \[\leadsto \cos re \cdot \left(\color{blue}{{\left(\sqrt{0.5}\right)}^{2}} \cdot \sqrt[3]{{im}^{6}}\right) \]

       12. pow2 82.0%

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

       13. rem-square-sqrt 82.0%

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

       14. sqr-pow 82.0%

           \[\leadsto \cos re \cdot \left(0.5 \cdot \sqrt[3]{\color{blue}{{im}^{\left(\frac{6}{2}\right)} \cdot {im}^{\left(\frac{6}{2}\right)}}}\right) \]

       15. metadata-eval 82.0%

           \[\leadsto \cos re \cdot \left(0.5 \cdot \sqrt[3]{{im}^{\color{blue}{3}} \cdot {im}^{\left(\frac{6}{2}\right)}}\right) \]

       16. metadata-eval 82.0%

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

       17. pow-prod-down 82.0%

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

       18. unpow2 82.0%

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

       19. pow3 82.0%

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

       20. add-cbrt-cube 53.4%

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

       21. unpow2 53.4%

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

       22. associate-*r* 53.4%

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

   14. Applied egg-rr 53.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.06 \cdot 10^{+37}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.06e+37) (cos re) (* 0.5 (pow im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.06e+37) {
		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 <= 2.06d+37) 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 <= 2.06e+37) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.06e+37:
		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 <= 2.06e+37)
		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 <= 2.06e+37)
		tmp = cos(re);
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.06e+37], N[Cos[re], $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.05999999999999988e37

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 70.1%

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

   if 2.05999999999999988e37 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 61.7%

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

      1. associate-*r* 61.7%

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

      2. distribute-rgt1-in 61.7%

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

   5. Simplified 61.7%

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

   6. Taylor expanded in im around inf 61.7%

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

      1. associate-*r* 61.7%

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

      2. *-commutative 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in re around 0 49.1%

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

       1. *-commutative 49.1%

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

   11. Simplified 49.1%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.06 \cdot 10^{+37}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.2% 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. Add Preprocessing

3. Taylor expanded in im around 0 54.6%

   \[\leadsto \color{blue}{\cos re} \]
4. Add Preprocessing

Alternative 9: 29.1% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 30.3%

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

   1. +-inverses 30.3%

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

   2. +-rgt-identity 30.3%

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

   3. *-inverses 30.3%

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

6. Simplified 30.3%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Alternative 10: 2.4% 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%

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

4. Applied egg-rr 2.4%

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

   1. pow-base-1 2.4%

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

   2. metadata-eval 2.4%

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

6. Simplified 2.4%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))