math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 15

Speedup: 1.0×

• 49.4% 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

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

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 99.2%

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

   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 56.4%

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

4. Final simplification 77.3%

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

Alternative 3: 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 99.2%

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

      1. +-commutative 99.2%

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

      2. unpow2 99.2%

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

      3. fma-define 99.2%

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

   5. Simplified 99.2%

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

   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 56.4%

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

4. Final simplification 77.3%

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

Alternative 4: 75.7% 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 98.3%

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

   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 56.4%

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

4. Final simplification 76.9%

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

Alternative 5: 73.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0006:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0006)
   (cos re)
   (if (<= im 1.02e+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 <= 0.0006) {
		tmp = cos(re);
	} else if (im <= 1.02e+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 <= 0.0006d0) then
        tmp = cos(re)
    else if (im <= 1.02d+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 <= 0.0006) {
		tmp = Math.cos(re);
	} else if (im <= 1.02e+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 <= 0.0006:
		tmp = math.cos(re)
	elif im <= 1.02e+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 <= 0.0006)
		tmp = cos(re);
	elseif (im <= 1.02e+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 <= 0.0006)
		tmp = cos(re);
	elseif (im <= 1.02e+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, 0.0006], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.02e+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 < 5.99999999999999947e-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 67.9%

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

   if 5.99999999999999947e-4 < im < 1.01999999999999991e103

   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 80.8%

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

   if 1.01999999999999991e103 < 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 6: 73.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00176:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00176)
   (cos re)
   (if (<= im 1.02e+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 <= 0.00176) {
		tmp = cos(re);
	} else if (im <= 1.02e+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 <= 0.00176d0) then
        tmp = cos(re)
    else if (im <= 1.02d+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 <= 0.00176) {
		tmp = Math.cos(re);
	} else if (im <= 1.02e+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 <= 0.00176:
		tmp = math.cos(re)
	elif im <= 1.02e+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 <= 0.00176)
		tmp = cos(re);
	elseif (im <= 1.02e+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 <= 0.00176)
		tmp = cos(re);
	elseif (im <= 1.02e+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, 0.00176], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.02e+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 < 0.00176000000000000006

   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 67.9%

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

   if 0.00176000000000000006 < im < 1.01999999999999991e103

   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 80.8%

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

   4. Taylor expanded in im around 0 79.7%

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

   if 1.01999999999999991e103 < 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 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00176:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.7)
   (cos re)
   (if (<= im 1.9e+154)
     (+ (* 0.5 (exp im)) 1.5)
     (* (* 0.5 (cos re)) (+ 4.0 (* im (* 0.5 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.7) {
		tmp = cos(re);
	} else if (im <= 1.9e+154) {
		tmp = (0.5 * exp(im)) + 1.5;
	} 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 <= 2.7d0) then
        tmp = cos(re)
    else if (im <= 1.9d+154) then
        tmp = (0.5d0 * exp(im)) + 1.5d0
    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 <= 2.7) {
		tmp = Math.cos(re);
	} else if (im <= 1.9e+154) {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im * (0.5 * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.7:
		tmp = math.cos(re)
	elif im <= 1.9e+154:
		tmp = (0.5 * math.exp(im)) + 1.5
	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 <= 2.7)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = Float64(Float64(0.5 * exp(im)) + 1.5);
	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 <= 2.7)
		tmp = cos(re);
	elseif (im <= 1.9e+154)
		tmp = (0.5 * exp(im)) + 1.5;
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im * (0.5 * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.7], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 1.5), $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 < 2.7000000000000002

   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 67.8%

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

   if 2.7000000000000002 < 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 73.5%

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

      1. +-commutative 73.5%

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

      2. distribute-lft-in 73.5%

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

      3. metadata-eval 73.5%

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

   7. Simplified 73.5%

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

   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 8: 69.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.3:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.3) (cos re) (+ (* 0.5 (exp im)) 1.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.3) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * exp(im)) + 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.3d0) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * exp(im)) + 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.3) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.3:
		tmp = math.cos(re)
	else:
		tmp = (0.5 * math.exp(im)) + 1.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.3)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 * exp(im)) + 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.3)
		tmp = cos(re);
	else
		tmp = (0.5 * exp(im)) + 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.3], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.2999999999999998

   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 67.8%

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

   if 2.2999999999999998 < 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 79.2%

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

      1. +-commutative 79.2%

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

      2. distribute-lft-in 79.2%

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

      3. metadata-eval 79.2%

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

   7. Simplified 79.2%

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

Alternative 9: 64.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot 0.25\right)\\ \mathbf{if}\;im \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;2 + \frac{\left(0.5 \cdot im\right) \cdot \left(0.5 \cdot im\right) - t\_0 \cdot t\_0}{0.5 \cdot im - t\_0}\\ \mathbf{else}:\\ \;\;\;\;2 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im 0.25))))
   (if (<= im 1.8e+26)
     (cos re)
     (if (<= im 2.7e+154)
       (+ 2.0 (/ (- (* (* 0.5 im) (* 0.5 im)) (* t_0 t_0)) (- (* 0.5 im) t_0)))
       (+ 2.0 t_0)))))

C

double code(double re, double im) {
	double t_0 = im * (im * 0.25);
	double tmp;
	if (im <= 1.8e+26) {
		tmp = cos(re);
	} else if (im <= 2.7e+154) {
		tmp = 2.0 + ((((0.5 * im) * (0.5 * im)) - (t_0 * t_0)) / ((0.5 * im) - t_0));
	} else {
		tmp = 2.0 + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im * 0.25d0)
    if (im <= 1.8d+26) then
        tmp = cos(re)
    else if (im <= 2.7d+154) then
        tmp = 2.0d0 + ((((0.5d0 * im) * (0.5d0 * im)) - (t_0 * t_0)) / ((0.5d0 * im) - t_0))
    else
        tmp = 2.0d0 + t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * 0.25);
	double tmp;
	if (im <= 1.8e+26) {
		tmp = Math.cos(re);
	} else if (im <= 2.7e+154) {
		tmp = 2.0 + ((((0.5 * im) * (0.5 * im)) - (t_0 * t_0)) / ((0.5 * im) - t_0));
	} else {
		tmp = 2.0 + t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * 0.25)
	tmp = 0
	if im <= 1.8e+26:
		tmp = math.cos(re)
	elif im <= 2.7e+154:
		tmp = 2.0 + ((((0.5 * im) * (0.5 * im)) - (t_0 * t_0)) / ((0.5 * im) - t_0))
	else:
		tmp = 2.0 + t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * 0.25))
	tmp = 0.0
	if (im <= 1.8e+26)
		tmp = cos(re);
	elseif (im <= 2.7e+154)
		tmp = Float64(2.0 + Float64(Float64(Float64(Float64(0.5 * im) * Float64(0.5 * im)) - Float64(t_0 * t_0)) / Float64(Float64(0.5 * im) - t_0)));
	else
		tmp = Float64(2.0 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * 0.25);
	tmp = 0.0;
	if (im <= 1.8e+26)
		tmp = cos(re);
	elseif (im <= 2.7e+154)
		tmp = 2.0 + ((((0.5 * im) * (0.5 * im)) - (t_0 * t_0)) / ((0.5 * im) - t_0));
	else
		tmp = 2.0 + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.8e+26], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.7e+154], N[(2.0 + N[(N[(N[(N[(0.5 * im), $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * im), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.80000000000000012e26

   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 65.8%

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

   if 1.80000000000000012e26 < 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 71.4%

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

      1. +-commutative 71.4%

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

      2. distribute-lft-in 71.4%

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

      3. metadata-eval 71.4%

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

   7. Simplified 71.4%

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

   8. Taylor expanded in im around 0 3.7%

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

      1. distribute-lft-in 3.7%

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

      2. flip-+ 44.1%

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

      3. *-commutative 44.1%

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

      4. *-commutative 44.1%

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

      5. *-commutative 44.1%

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

   10. Applied egg-rr 44.1%

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

   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 re around 0 84.2%

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

      1. +-commutative 84.2%

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

      2. distribute-lft-in 84.2%

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

      3. metadata-eval 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in im around 0 84.2%

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

   9. Taylor expanded in im around inf 84.2%

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

       1. *-commutative 84.2%

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

   11. Simplified 84.2%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;2 + \frac{\left(0.5 \cdot im\right) \cdot \left(0.5 \cdot im\right) - \left(im \cdot \left(im \cdot 0.25\right)\right) \cdot \left(im \cdot \left(im \cdot 0.25\right)\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(im \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot 0.25\right)\\ \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;2 + \frac{\left(0.5 \cdot im\right) \cdot \left(0.5 \cdot im\right) - t\_0 \cdot t\_0}{0.5 \cdot im - t\_0}\\ \mathbf{else}:\\ \;\;\;\;2 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im 0.25))))
   (if (<= im 1.3)
     1.0
     (if (<= im 2.7e+154)
       (+ 2.0 (/ (- (* (* 0.5 im) (* 0.5 im)) (* t_0 t_0)) (- (* 0.5 im) t_0)))
       (+ 2.0 t_0)))))

C

double code(double re, double im) {
	double t_0 = im * (im * 0.25);
	double tmp;
	if (im <= 1.3) {
		tmp = 1.0;
	} else if (im <= 2.7e+154) {
		tmp = 2.0 + ((((0.5 * im) * (0.5 * im)) - (t_0 * t_0)) / ((0.5 * im) - t_0));
	} else {
		tmp = 2.0 + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im * 0.25d0)
    if (im <= 1.3d0) then
        tmp = 1.0d0
    else if (im <= 2.7d+154) then
        tmp = 2.0d0 + ((((0.5d0 * im) * (0.5d0 * im)) - (t_0 * t_0)) / ((0.5d0 * im) - t_0))
    else
        tmp = 2.0d0 + t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * 0.25);
	double tmp;
	if (im <= 1.3) {
		tmp = 1.0;
	} else if (im <= 2.7e+154) {
		tmp = 2.0 + ((((0.5 * im) * (0.5 * im)) - (t_0 * t_0)) / ((0.5 * im) - t_0));
	} else {
		tmp = 2.0 + t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * 0.25)
	tmp = 0
	if im <= 1.3:
		tmp = 1.0
	elif im <= 2.7e+154:
		tmp = 2.0 + ((((0.5 * im) * (0.5 * im)) - (t_0 * t_0)) / ((0.5 * im) - t_0))
	else:
		tmp = 2.0 + t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * 0.25))
	tmp = 0.0
	if (im <= 1.3)
		tmp = 1.0;
	elseif (im <= 2.7e+154)
		tmp = Float64(2.0 + Float64(Float64(Float64(Float64(0.5 * im) * Float64(0.5 * im)) - Float64(t_0 * t_0)) / Float64(Float64(0.5 * im) - t_0)));
	else
		tmp = Float64(2.0 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * 0.25);
	tmp = 0.0;
	if (im <= 1.3)
		tmp = 1.0;
	elseif (im <= 2.7e+154)
		tmp = 2.0 + ((((0.5 * im) * (0.5 * im)) - (t_0 * t_0)) / ((0.5 * im) - t_0));
	else
		tmp = 2.0 + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.3], 1.0, If[LessEqual[im, 2.7e+154], N[(2.0 + N[(N[(N[(N[(0.5 * im), $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * im), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.30000000000000004

   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 67.8%

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

   4. Taylor expanded in re around 0 38.7%

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

   if 1.30000000000000004 < 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 73.5%

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

      1. +-commutative 73.5%

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

      2. distribute-lft-in 73.5%

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

      3. metadata-eval 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in im around 0 3.7%

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

      1. distribute-lft-in 3.7%

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

      2. flip-+ 37.0%

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

      3. *-commutative 37.0%

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

      4. *-commutative 37.0%

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

      5. *-commutative 37.0%

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

   10. Applied egg-rr 37.0%

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

   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 re around 0 84.2%

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

      1. +-commutative 84.2%

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

      2. distribute-lft-in 84.2%

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

      3. metadata-eval 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in im around 0 84.2%

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

   9. Taylor expanded in im around inf 84.2%

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

       1. *-commutative 84.2%

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

   11. Simplified 84.2%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;2 + \frac{\left(0.5 \cdot im\right) \cdot \left(0.5 \cdot im\right) - \left(im \cdot \left(im \cdot 0.25\right)\right) \cdot \left(im \cdot \left(im \cdot 0.25\right)\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(im \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.3% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.9 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+88}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.9e+16)
   1.0
   (if (<= im 4.2e+88)
     (+ 1.0 (* (* re re) -0.5))
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.9e+16) {
		tmp = 1.0;
	} else if (im <= 4.2e+88) {
		tmp = 1.0 + ((re * re) * -0.5);
	} 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 <= 5.9d+16) then
        tmp = 1.0d0
    else if (im <= 4.2d+88) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    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 <= 5.9e+16) {
		tmp = 1.0;
	} else if (im <= 4.2e+88) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.9e+16:
		tmp = 1.0
	elif im <= 4.2e+88:
		tmp = 1.0 + ((re * re) * -0.5)
	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 <= 5.9e+16)
		tmp = 1.0;
	elseif (im <= 4.2e+88)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	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 <= 5.9e+16)
		tmp = 1.0;
	elseif (im <= 4.2e+88)
		tmp = 1.0 + ((re * re) * -0.5);
	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, 5.9e+16], 1.0, If[LessEqual[im, 4.2e+88], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $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 3 regimes

2. if im < 5.9e16

   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 66.4%

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

   4. Taylor expanded in re around 0 38.0%

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

   if 5.9e16 < im < 4.2e88

   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 3.1%

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

   4. Taylor expanded in re around 0 25.3%

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

      1. *-commutative 25.3%

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

   6. Simplified 25.3%

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

      1. unpow2 25.3%

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

   8. Applied egg-rr 25.3%

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

   if 4.2e88 < 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 80.4%

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

      1. +-commutative 80.4%

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

      2. distribute-lft-in 80.4%

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

      3. metadata-eval 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in im around 0 73.2%

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

      1. *-commutative 73.2%

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

   10. Simplified 73.2%

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

Alternative 12: 39.0% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.9 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+153}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.9e+16)
   1.0
   (if (<= im 1.8e+153)
     (+ 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 <= 5.9e+16) {
		tmp = 1.0;
	} else if (im <= 1.8e+153) {
		tmp = 1.0 + ((re * re) * -0.5);
	} 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 <= 5.9d+16) then
        tmp = 1.0d0
    else if (im <= 1.8d+153) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    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 <= 5.9e+16) {
		tmp = 1.0;
	} else if (im <= 1.8e+153) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.9e+16:
		tmp = 1.0
	elif im <= 1.8e+153:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = 2.0 + (im * (0.5 + (im * 0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.9e+16)
		tmp = 1.0;
	elseif (im <= 1.8e+153)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	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 <= 5.9e+16)
		tmp = 1.0;
	elseif (im <= 1.8e+153)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.9e+16], 1.0, If[LessEqual[im, 1.8e+153], 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]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.9e16

   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 66.4%

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

   4. Taylor expanded in re around 0 38.0%

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

   if 5.9e16 < im < 1.8e153

   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 3.1%

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

   4. Taylor expanded in re around 0 21.7%

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

      1. *-commutative 21.7%

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

   6. Simplified 21.7%

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

      1. unpow2 21.7%

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

   8. Applied egg-rr 21.7%

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

   if 1.8e153 < 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 84.2%

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

      1. +-commutative 84.2%

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

      2. distribute-lft-in 84.2%

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

      3. metadata-eval 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in im around 0 84.2%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.9 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+153}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.0% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.9 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+152}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.9e+16)
   1.0
   (if (<= im 3.3e+152)
     (+ 1.0 (* (* re re) -0.5))
     (+ 2.0 (* im (* im 0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.9e+16) {
		tmp = 1.0;
	} else if (im <= 3.3e+152) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = 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 <= 5.9d+16) then
        tmp = 1.0d0
    else if (im <= 3.3d+152) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    else
        tmp = 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 <= 5.9e+16) {
		tmp = 1.0;
	} else if (im <= 3.3e+152) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = 2.0 + (im * (im * 0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.9e+16:
		tmp = 1.0
	elif im <= 3.3e+152:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = 2.0 + (im * (im * 0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.9e+16)
		tmp = 1.0;
	elseif (im <= 3.3e+152)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	else
		tmp = 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 <= 5.9e+16)
		tmp = 1.0;
	elseif (im <= 3.3e+152)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = 2.0 + (im * (im * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.9e+16], 1.0, If[LessEqual[im, 3.3e+152], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(im * N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.9e16

   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 66.4%

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

   4. Taylor expanded in re around 0 38.0%

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

   if 5.9e16 < im < 3.3000000000000001e152

   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 3.1%

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

   4. Taylor expanded in re around 0 21.7%

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

      1. *-commutative 21.7%

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

   6. Simplified 21.7%

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

      1. unpow2 21.7%

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

   8. Applied egg-rr 21.7%

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

   if 3.3000000000000001e152 < 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 84.2%

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

      1. +-commutative 84.2%

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

      2. distribute-lft-in 84.2%

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

      3. metadata-eval 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in im around 0 84.2%

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

   9. Taylor expanded in im around inf 84.2%

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

       1. *-commutative 84.2%

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

   11. Simplified 84.2%

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

Alternative 14: 31.3% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.26 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.26e+17) 1.0 (+ 1.0 (* (* re re) -0.5))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.26e+17) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.26d+17) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.26e+17) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.26e+17:
		tmp = 1.0
	else:
		tmp = 1.0 + ((re * re) * -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.26e+17)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.26e+17)
		tmp = 1.0;
	else
		tmp = 1.0 + ((re * re) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.26e+17], 1.0, N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.26e17

   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 66.4%

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

   4. Taylor expanded in re around 0 38.0%

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

   if 1.26e17 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 3.1%

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

   4. Taylor expanded in re around 0 13.9%

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

      1. *-commutative 13.9%

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

   6. Simplified 13.9%

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

      1. unpow2 13.9%

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

   8. Applied egg-rr 13.9%

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

Alternative 15: 28.6% 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 49.6%

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

4. Taylor expanded in re around 0 28.6%

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

Reproduce

herbie shell --seed 2024188 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))