math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 14

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

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

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

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

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

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

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{im} + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= (+ (exp (- im)) (exp im)) 4.0)
     (* t_0 (fma im im 2.0))
     (* t_0 (+ (exp im) 3.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if ((exp(-im) + exp(im)) <= 4.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = t_0 * (exp(im) + 3.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (Float64(exp(Float64(-im)) + exp(im)) <= 4.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(t_0 * Float64(exp(im) + 3.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision], 4.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 4

   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 99.9%

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

      1. +-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. fma-define 99.9%

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

   5. Simplified 99.9%

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

   if 4 < (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   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 59.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (+ (exp (- im)) (exp im)) 4.0)
   (+ (cos re) (* 0.5 (* im im)))
   (* (* 0.5 (cos re)) (+ (exp im) 3.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(-im) + exp(im)) <= 4.0) {
		tmp = cos(re) + (0.5 * (im * im));
	} else {
		tmp = (0.5 * cos(re)) * (exp(im) + 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(-im) + exp(im)) <= 4.0d0) then
        tmp = cos(re) + (0.5d0 * (im * im))
    else
        tmp = (0.5d0 * cos(re)) * (exp(im) + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(-im) + Math.exp(im)) <= 4.0) {
		tmp = Math.cos(re) + (0.5 * (im * im));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp(im) + 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(-im) + math.exp(im)) <= 4.0:
		tmp = math.cos(re) + (0.5 * (im * im))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp(im) + 3.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(Float64(-im)) + exp(im)) <= 4.0)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(im) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(-im) + exp(im)) <= 4.0)
		tmp = cos(re) + (0.5 * (im * im));
	else
		tmp = (0.5 * cos(re)) * (exp(im) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision], 4.0], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 4

   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 99.9%

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

   4. Taylor expanded in re around 0 98.8%

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

      1. unpow2 98.8%

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

   6. Applied egg-rr 98.8%

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

   if 4 < (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   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 59.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00065:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00065)
   (+ (cos re) (* 0.5 (* im im)))
   (if (<= im 1.05e+103)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (*
      (* 0.5 (cos re))
      (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00065) {
		tmp = cos(re) + (0.5 * (im * im));
	} else if (im <= 1.05e+103) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = (0.5 * cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00065d0) then
        tmp = cos(re) + (0.5d0 * (im * im))
    else if (im <= 1.05d+103) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = (0.5d0 * cos(re)) * (4.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00065) {
		tmp = Math.cos(re) + (0.5 * (im * im));
	} else if (im <= 1.05e+103) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00065:
		tmp = math.cos(re) + (0.5 * (im * im))
	elif im <= 1.05e+103:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = (0.5 * math.cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00065)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(im * im)));
	elseif (im <= 1.05e+103)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(4.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00065)
		tmp = cos(re) + (0.5 * (im * im));
	elseif (im <= 1.05e+103)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00065], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.05e+103], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.4999999999999997e-4

   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 87.7%

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

   4. Taylor expanded in re around 0 82.4%

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

      1. unpow2 82.4%

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

   6. Applied egg-rr 82.4%

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

   if 6.4999999999999997e-4 < im < 1.0500000000000001e103

   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 78.3%

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

   if 1.0500000000000001e103 < im

   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 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00065:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00065)
   (+ (cos re) (* 0.5 (* im im)))
   (if (<= im 1.05e+103)
     (*
      0.5
      (+
       (exp im)
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
     (*
      (* 0.5 (cos re))
      (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00065) {
		tmp = cos(re) + (0.5 * (im * im));
	} else if (im <= 1.05e+103) {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = (0.5 * cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00065d0) then
        tmp = cos(re) + (0.5d0 * (im * im))
    else if (im <= 1.05d+103) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = (0.5d0 * cos(re)) * (4.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00065) {
		tmp = Math.cos(re) + (0.5 * (im * im));
	} else if (im <= 1.05e+103) {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00065:
		tmp = math.cos(re) + (0.5 * (im * im))
	elif im <= 1.05e+103:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = (0.5 * math.cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00065)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(im * im)));
	elseif (im <= 1.05e+103)
		tmp = Float64(0.5 * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(4.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00065)
		tmp = cos(re) + (0.5 * (im * im));
	elseif (im <= 1.05e+103)
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00065], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.05e+103], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.4999999999999997e-4

   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 87.7%

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

   4. Taylor expanded in re around 0 82.4%

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

      1. unpow2 82.4%

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

   6. Applied egg-rr 82.4%

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

   if 6.4999999999999997e-4 < im < 1.0500000000000001e103

   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 78.3%

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

   4. Taylor expanded in im around 0 78.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + -0.16666666666666666 \cdot im\right) - 1\right)\right)} + e^{im}\right) \]

   if 1.0500000000000001e103 < im

   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 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00065:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(2 + im \cdot \left(im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.6)
   (+ (cos re) (* 0.5 (* im im)))
   (if (<= im 2.7e+154)
     (+ (* 0.5 (exp im)) 1.5)
     (* (cos re) (+ 2.0 (* im (* im 0.25)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6) {
		tmp = cos(re) + (0.5 * (im * im));
	} else if (im <= 2.7e+154) {
		tmp = (0.5 * exp(im)) + 1.5;
	} else {
		tmp = cos(re) * (2.0 + (im * (im * 0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.6d0) then
        tmp = cos(re) + (0.5d0 * (im * im))
    else if (im <= 2.7d+154) then
        tmp = (0.5d0 * exp(im)) + 1.5d0
    else
        tmp = cos(re) * (2.0d0 + (im * (im * 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.6) {
		tmp = Math.cos(re) + (0.5 * (im * im));
	} else if (im <= 2.7e+154) {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	} else {
		tmp = Math.cos(re) * (2.0 + (im * (im * 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6:
		tmp = math.cos(re) + (0.5 * (im * im))
	elif im <= 2.7e+154:
		tmp = (0.5 * math.exp(im)) + 1.5
	else:
		tmp = math.cos(re) * (2.0 + (im * (im * 0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(im * im)));
	elseif (im <= 2.7e+154)
		tmp = Float64(Float64(0.5 * exp(im)) + 1.5);
	else
		tmp = Float64(cos(re) * Float64(2.0 + Float64(im * Float64(im * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.6)
		tmp = cos(re) + (0.5 * (im * im));
	elseif (im <= 2.7e+154)
		tmp = (0.5 * exp(im)) + 1.5;
	else
		tmp = cos(re) * (2.0 + (im * (im * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.7e+154], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(2.0 + N[(im * N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.60000000000000009

   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 87.7%

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

   4. Taylor expanded in re around 0 82.4%

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

      1. unpow2 82.4%

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

   6. Applied egg-rr 82.4%

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

   if 2.60000000000000009 < im < 2.70000000000000006e154

   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 100.0%

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

   5. Taylor expanded in re around 0 80.6%

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

      1. +-commutative 80.6%

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

      2. distribute-lft-in 80.6%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 80.6%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 80.6%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   if 2.70000000000000006e154 < im

   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 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

Alternative 7: 78.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+168}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.6)
   (+ (cos re) (* 0.5 (* im im)))
   (if (<= im 5e+168)
     (+ (* 0.5 (exp im)) 1.5)
     (* (+ 1.0 (* (* re re) -0.5)) (+ 2.0 (* im (+ 0.5 (* im 0.25))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6) {
		tmp = cos(re) + (0.5 * (im * im));
	} else if (im <= 5e+168) {
		tmp = (0.5 * exp(im)) + 1.5;
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.6d0) then
        tmp = cos(re) + (0.5d0 * (im * im))
    else if (im <= 5d+168) then
        tmp = (0.5d0 * exp(im)) + 1.5d0
    else
        tmp = (1.0d0 + ((re * re) * (-0.5d0))) * (2.0d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.6) {
		tmp = Math.cos(re) + (0.5 * (im * im));
	} else if (im <= 5e+168) {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6:
		tmp = math.cos(re) + (0.5 * (im * im))
	elif im <= 5e+168:
		tmp = (0.5 * math.exp(im)) + 1.5
	else:
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(im * im)));
	elseif (im <= 5e+168)
		tmp = Float64(Float64(0.5 * exp(im)) + 1.5);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * -0.5)) * Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.6)
		tmp = cos(re) + (0.5 * (im * im));
	elseif (im <= 5e+168)
		tmp = (0.5 * exp(im)) + 1.5;
	else
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+168], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.60000000000000009

   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 87.7%

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

   4. Taylor expanded in re around 0 82.4%

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

      1. unpow2 82.4%

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

   6. Applied egg-rr 82.4%

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

   if 2.60000000000000009 < im < 4.99999999999999967e168

   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 100.0%

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

   5. Taylor expanded in re around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. distribute-lft-in 82.4%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 82.4%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 82.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   if 4.99999999999999967e168 < im

   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 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 73.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)} \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 73.0%

         \[\leadsto \left(1 + \color{blue}{{re}^{2} \cdot -0.5}\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]

   10. Simplified 73.0%

       \[\leadsto \color{blue}{\left(1 + {re}^{2} \cdot -0.5\right)} \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 73.0%

          \[\leadsto \left(1 + \color{blue}{\left(re \cdot re\right)} \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]

   12. Applied egg-rr 73.0%

       \[\leadsto \left(1 + \color{blue}{\left(re \cdot re\right)} \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+168}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.5:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+167}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.5)
   (cos re)
   (if (<= im 5e+167)
     (+ (* 0.5 (exp im)) 1.5)
     (* (+ 1.0 (* (* re re) -0.5)) (+ 2.0 (* im (+ 0.5 (* im 0.25))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.5) {
		tmp = cos(re);
	} else if (im <= 5e+167) {
		tmp = (0.5 * exp(im)) + 1.5;
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.5d0) then
        tmp = cos(re)
    else if (im <= 5d+167) then
        tmp = (0.5d0 * exp(im)) + 1.5d0
    else
        tmp = (1.0d0 + ((re * re) * (-0.5d0))) * (2.0d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.5) {
		tmp = Math.cos(re);
	} else if (im <= 5e+167) {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.5:
		tmp = math.cos(re)
	elif im <= 5e+167:
		tmp = (0.5 * math.exp(im)) + 1.5
	else:
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.5)
		tmp = cos(re);
	elseif (im <= 5e+167)
		tmp = Float64(Float64(0.5 * exp(im)) + 1.5);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * -0.5)) * Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.5)
		tmp = cos(re);
	elseif (im <= 5e+167)
		tmp = (0.5 * exp(im)) + 1.5;
	else
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.5], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5e+167], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.5

   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 72.0%

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

   if 3.5 < im < 4.9999999999999997e167

   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 100.0%

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

   5. Taylor expanded in re around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. distribute-lft-in 82.4%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 82.4%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 82.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   if 4.9999999999999997e167 < im

   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 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 73.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)} \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 73.0%

         \[\leadsto \left(1 + \color{blue}{{re}^{2} \cdot -0.5}\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]

   10. Simplified 73.0%

       \[\leadsto \color{blue}{\left(1 + {re}^{2} \cdot -0.5\right)} \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 73.0%

          \[\leadsto \left(1 + \color{blue}{\left(re \cdot re\right)} \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]

   12. Applied egg-rr 73.0%

       \[\leadsto \left(1 + \color{blue}{\left(re \cdot re\right)} \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.5:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+167}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+168}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.4e+27)
   (cos re)
   (if (<= im 1e+168)
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))
     (* (+ 1.0 (* (* re re) -0.5)) (+ 2.0 (* im (+ 0.5 (* im 0.25))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.4e+27) {
		tmp = cos(re);
	} else if (im <= 1e+168) {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.4d+27) then
        tmp = cos(re)
    else if (im <= 1d+168) then
        tmp = 2.0d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0)))))
    else
        tmp = (1.0d0 + ((re * re) * (-0.5d0))) * (2.0d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.4e+27) {
		tmp = Math.cos(re);
	} else if (im <= 1e+168) {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.4e+27:
		tmp = math.cos(re)
	elif im <= 1e+168:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	else:
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.4e+27)
		tmp = cos(re);
	elseif (im <= 1e+168)
		tmp = Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * -0.5)) * Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.4e+27)
		tmp = cos(re);
	elseif (im <= 1e+168)
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	else
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.4e+27], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1e+168], N[(2.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.4e27

   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 70.5%

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

   if 3.4e27 < im < 9.9999999999999993e167

   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 100.0%

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

   5. Taylor expanded in re around 0 83.3%

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

      1. +-commutative 83.3%

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

      2. distribute-lft-in 83.3%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 83.3%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 83.3%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 39.6%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 39.6%

         \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   10. Simplified 39.6%

       \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]

   if 9.9999999999999993e167 < im

   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 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 73.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)} \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 73.0%

         \[\leadsto \left(1 + \color{blue}{{re}^{2} \cdot -0.5}\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]

   10. Simplified 73.0%

       \[\leadsto \color{blue}{\left(1 + {re}^{2} \cdot -0.5\right)} \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 73.0%

          \[\leadsto \left(1 + \color{blue}{\left(re \cdot re\right)} \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]

   12. Applied egg-rr 73.0%

       \[\leadsto \left(1 + \color{blue}{\left(re \cdot re\right)} \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+168}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.26:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+168}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.26)
   1.0
   (if (<= im 4e+168)
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))
     (* (+ 1.0 (* (* re re) -0.5)) (+ 2.0 (* im (+ 0.5 (* im 0.25))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.26) {
		tmp = 1.0;
	} else if (im <= 4e+168) {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.26d0) then
        tmp = 1.0d0
    else if (im <= 4d+168) then
        tmp = 2.0d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0)))))
    else
        tmp = (1.0d0 + ((re * re) * (-0.5d0))) * (2.0d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.26) {
		tmp = 1.0;
	} else if (im <= 4e+168) {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	} else {
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.26:
		tmp = 1.0
	elif im <= 4e+168:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	else:
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.26)
		tmp = 1.0;
	elseif (im <= 4e+168)
		tmp = Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(re * re) * -0.5)) * Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.26)
		tmp = 1.0;
	elseif (im <= 4e+168)
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	else
		tmp = (1.0 + ((re * re) * -0.5)) * (2.0 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.26], 1.0, If[LessEqual[im, 4e+168], N[(2.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.26000000000000001

   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 72.0%

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

   4. Taylor expanded in re around 0 40.8%

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

   if 1.26000000000000001 < im < 3.9999999999999997e168

   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 100.0%

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

   5. Taylor expanded in re around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. distribute-lft-in 82.4%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 82.4%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 82.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 35.2%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 35.2%

         \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   10. Simplified 35.2%

       \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]

   if 3.9999999999999997e168 < im

   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 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 73.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)} \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 73.0%

         \[\leadsto \left(1 + \color{blue}{{re}^{2} \cdot -0.5}\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]

   10. Simplified 73.0%

       \[\leadsto \color{blue}{\left(1 + {re}^{2} \cdot -0.5\right)} \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 73.0%

          \[\leadsto \left(1 + \color{blue}{\left(re \cdot re\right)} \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]

   12. Applied egg-rr 73.0%

       \[\leadsto \left(1 + \color{blue}{\left(re \cdot re\right)} \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.26:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+168}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.26:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.26)
   1.0
   (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.26) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.26d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.26) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.26:
		tmp = 1.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.26)
		tmp = 1.0;
	else
		tmp = Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.26)
		tmp = 1.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.26], 1.0, N[(2.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.26000000000000001

   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 72.0%

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

   4. Taylor expanded in re around 0 40.8%

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

   if 1.26000000000000001 < im

   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 100.0%

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

   5. Taylor expanded in re around 0 76.1%

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

      1. +-commutative 76.1%

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

      2. distribute-lft-in 76.1%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 76.1%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 76.1%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 53.5%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 53.5%

         \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   10. Simplified 53.5%

       \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 38.4% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.26:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.26) 1.0 (+ 2.0 (* im (+ 0.5 (* im 0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.26) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.26d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 + (im * (0.5d0 + (im * 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.26) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.26:
		tmp = 1.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * 0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.26)
		tmp = 1.0;
	else
		tmp = Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.26)
		tmp = 1.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.26], 1.0, N[(2.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.26000000000000001

   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 72.0%

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

   4. Taylor expanded in re around 0 40.8%

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

   if 1.26000000000000001 < im

   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 100.0%

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

   5. Taylor expanded in re around 0 76.1%

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

      1. +-commutative 76.1%

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

      2. distribute-lft-in 76.1%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 76.1%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 76.1%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 42.7%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.26:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.5% 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

3. Taylor expanded in im around 0 52.9%

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

4. Taylor expanded in re around 0 30.2%

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

Alternative 14: 9.1% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.75)

C

double code(double re, double im) {
	return 0.75;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.75d0
end function

Java

public static double code(double re, double im) {
	return 0.75;
}

Python

def code(re, im):
	return 0.75

Julia

function code(re, im)
	return 0.75
end

MATLAB

function tmp = code(re, im)
	tmp = 0.75;
end

Wolfram

code[re_, im_] := 0.75

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 re around 0 66.0%

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

5. Applied egg-rr 9.6%

   \[\leadsto 0.5 \cdot \color{blue}{1.5} \]
6. Step-by-step derivation

   1. metadata-eval 9.6%

      \[\leadsto \color{blue}{0.75} \]

7. Applied egg-rr 9.6%

   \[\leadsto \color{blue}{0.75} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024149 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))