math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 15

Speedup: 1.0×

• 49.6% 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: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;t\_0 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{im} + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= (+ (exp (- im)) (exp im)) 4.0)
     (*
      t_0
      (+
       (exp im)
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
     (* 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 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    if ((exp(-im) + exp(im)) <= 4.0d0) then
        tmp = t_0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = t_0 * (exp(im) + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if ((Math.exp(-im) + Math.exp(im)) <= 4.0) {
		tmp = t_0 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (Math.exp(im) + 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if (math.exp(-im) + math.exp(im)) <= 4.0:
		tmp = t_0 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = t_0 * (math.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 * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	else
		tmp = Float64(t_0 * Float64(exp(im) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if ((exp(-im) + exp(im)) <= 4.0)
		tmp = t_0 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = t_0 * (exp(im) + 3.0);
	end
	tmp_2 = 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[(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[(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 99.6%

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

   if 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 48.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% 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 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. fma-define 99.6%

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

   5. Simplified 99.6%

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

   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 48.2%

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

4. Final simplification 76.7%

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

Alternative 4: 74.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (+ (exp (- im)) (exp im)) 4.0)
   (cos re)
   (* (* 0.5 (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);
	} 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)
    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);
	} 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)
	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 = cos(re);
	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);
	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[Cos[re], $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 98.7%

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

   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 48.2%

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

4. Final simplification 76.2%

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

Alternative 5: 71.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00295:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00295)
   (cos re)
   (if (<= im 1.9e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (* 0.5 (cos re)) (+ 4.0 (* im (* 0.5 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00295) {
		tmp = cos(re);
	} else if (im <= 1.9e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = (0.5 * cos(re)) * (4.0 + (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 <= 0.00295d0) then
        tmp = cos(re)
    else if (im <= 1.9d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = (0.5d0 * cos(re)) * (4.0d0 + (im * (0.5d0 * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00295) {
		tmp = Math.cos(re);
	} else if (im <= 1.9e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im * (0.5 * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00295:
		tmp = math.cos(re)
	elif im <= 1.9e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = (0.5 * math.cos(re)) * (4.0 + (im * (0.5 * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00295)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(4.0 + Float64(im * Float64(0.5 * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00295)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im * (0.5 * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00295], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.9e+154], 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[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00294999999999999993

   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 0.00294999999999999993 < im < 1.8999999999999999e154

   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 96.9%

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

   if 1.8999999999999999e154 < 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%

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

   6. Taylor expanded in im around inf 100.0%

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

Alternative 6: 71.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.2:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.2)
   (cos re)
   (if (<= im 1.9e+154)
     (+ 1.5 (* 0.5 (exp im)))
     (* (* 0.5 (cos re)) (+ 4.0 (* im (* 0.5 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.2) {
		tmp = cos(re);
	} else if (im <= 1.9e+154) {
		tmp = 1.5 + (0.5 * exp(im));
	} else {
		tmp = (0.5 * cos(re)) * (4.0 + (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 <= 2.2d0) then
        tmp = cos(re)
    else if (im <= 1.9d+154) then
        tmp = 1.5d0 + (0.5d0 * exp(im))
    else
        tmp = (0.5d0 * cos(re)) * (4.0d0 + (im * (0.5d0 * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.2) {
		tmp = Math.cos(re);
	} else if (im <= 1.9e+154) {
		tmp = 1.5 + (0.5 * Math.exp(im));
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im * (0.5 * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.2:
		tmp = math.cos(re)
	elif im <= 1.9e+154:
		tmp = 1.5 + (0.5 * math.exp(im))
	else:
		tmp = (0.5 * math.cos(re)) * (4.0 + (im * (0.5 * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.2)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = Float64(1.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(4.0 + Float64(im * Float64(0.5 * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.2)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = 1.5 + (0.5 * exp(im));
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im * (0.5 * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.2], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(1.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.2000000000000002

   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 69.9%

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

   if 2.2000000000000002 < im < 1.8999999999999999e154

   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 96.8%

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

      1. distribute-lft-in 96.8%

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

      2. metadata-eval 96.8%

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

   7. Simplified 96.8%

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

   if 1.8999999999999999e154 < 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%

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

   6. Taylor expanded in im around inf 100.0%

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

Alternative 7: 68.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.2) (cos re) (+ 1.5 (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.2) {
		tmp = cos(re);
	} else {
		tmp = 1.5 + (0.5 * exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.2d0) then
        tmp = cos(re)
    else
        tmp = 1.5d0 + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.2) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.2:
		tmp = math.cos(re)
	else:
		tmp = 1.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.2)
		tmp = cos(re);
	else
		tmp = Float64(1.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.2)
		tmp = cos(re);
	else
		tmp = 1.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.2], N[Cos[re], $MachinePrecision], N[(1.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.2000000000000002

   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 69.9%

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

   if 2.2000000000000002 < 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 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. metadata-eval 90.6%

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

   7. Simplified 90.6%

      \[\leadsto \color{blue}{1.5 + 0.5 \cdot e^{im}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 950000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 950000.0)
   (cos re)
   (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 950000.0) {
		tmp = cos(re);
	} 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 <= 950000.0d0) then
        tmp = cos(re)
    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 <= 950000.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 950000.0:
		tmp = math.cos(re)
	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 <= 950000.0)
		tmp = cos(re);
	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 <= 950000.0)
		tmp = cos(re);
	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, 950000.0], N[Cos[re], $MachinePrecision], 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 < 9.5e5

   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 69.9%

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

   if 9.5e5 < 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 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. metadata-eval 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in im around 0 60.6%

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

      1. *-commutative 60.6%

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

   10. Simplified 60.6%

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

Alternative 9: 40.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.95:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.95)
   (*
    0.5
    (+
     (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
     (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))
   (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.95) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} 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 <= 1.95d0) then
        tmp = 0.5d0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    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 <= 1.95) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.95:
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	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 <= 1.95)
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	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 <= 1.95)
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	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, 1.95], N[(0.5 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < 1.94999999999999996

   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 63.3%

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

   4. Taylor expanded in im around 0 52.0%

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

   5. Taylor expanded in im around 0 37.7%

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

      1. *-commutative 37.7%

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

   7. Simplified 37.7%

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

   if 1.94999999999999996 < 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 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. metadata-eval 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in im around 0 60.6%

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

      1. *-commutative 60.6%

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

   10. Simplified 60.6%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.95:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.3% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85)
   (*
    0.5
    (+
     (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
     (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
   (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} 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 <= 1.85d0) then
        tmp = 0.5d0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    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 <= 1.85) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85:
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))))
	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 <= 1.85)
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	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 <= 1.85)
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	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, 1.85], N[(0.5 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < 1.8500000000000001

   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 63.3%

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

   4. Taylor expanded in im around 0 52.0%

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

   5. Taylor expanded in im around 0 52.0%

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

   if 1.8500000000000001 < 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 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. metadata-eval 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in im around 0 60.6%

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

      1. *-commutative 60.6%

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

   10. Simplified 60.6%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(im + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75)
   (*
    0.5
    (+
     (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
     (+ im 1.0)))
   (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (im + 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 <= 1.75d0) then
        tmp = 0.5d0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (im + 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 <= 1.75) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (im + 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 <= 1.75:
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (im + 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 <= 1.75)
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + Float64(im + 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 <= 1.75)
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (im + 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, 1.75], N[(0.5 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < 1.75

   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 63.3%

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

   4. Taylor expanded in im around 0 52.0%

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

   5. Taylor expanded in im around 0 51.9%

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

   if 1.75 < 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 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. metadata-eval 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in im around 0 60.6%

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

      1. *-commutative 60.6%

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

   10. Simplified 60.6%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(im + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.4% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.25)
   1.0
   (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.25) {
		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 <= 1.25d0) 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 <= 1.25) {
		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 <= 1.25:
		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 <= 1.25)
		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 <= 1.25)
		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, 1.25], 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 < 1.25

   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 69.9%

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

   4. Taylor expanded in re around 0 38.2%

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

   if 1.25 < 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 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. metadata-eval 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in im around 0 60.6%

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

      1. *-commutative 60.6%

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

   10. Simplified 60.6%

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

Alternative 13: 37.4% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.2) 1.0 (+ 2.0 (* im (+ 0.5 (* im 0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.2) {
		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 <= 1.2d0) 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 <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.2:
		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 <= 1.2)
		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 <= 1.2)
		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, 1.2], 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 < 1.19999999999999996

   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 69.9%

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

   4. Taylor expanded in re around 0 38.2%

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

   if 1.19999999999999996 < 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 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. metadata-eval 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in im around 0 37.5%

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

      1. *-commutative 37.5%

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

   10. Simplified 37.5%

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

Alternative 14: 28.8% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 56.1%

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

4. Taylor expanded in re around 0 30.9%

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

Alternative 15: 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 68.9%

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

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 2024180 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))