math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.6s

Alternatives: 19

Speedup: 1.0×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0)))
   (exp re)
   (* (cos im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.0) || ~((exp(re) <= 2.0)))
		tmp = exp(re);
	else
		tmp = cos(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 2 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 86.4%

      \[\leadsto \color{blue}{e^{re}} \]

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.6%

      \[\leadsto \color{blue}{\cos im + re \cdot \cos im} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0))) (exp re) (cos im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = cos(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = cos(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.cos(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.0) || ~((exp(re) <= 2.0)))
		tmp = exp(re);
	else
		tmp = cos(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Cos[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 2 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 86.4%

      \[\leadsto \color{blue}{e^{re}} \]

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 98.6%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.024:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+90}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.024)
   (exp re)
   (if (<= re 350.0)
     (*
      (cos im)
      (+ (+ re 1.0) (* (+ 0.5 (* re 0.16666666666666666)) (* re re))))
     (if (<= re 2.1e+90)
       (* (exp re) (+ 1.0 (* im (* im -0.5))))
       (* (cos im) (* 0.16666666666666666 (* re (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.024) {
		tmp = exp(re);
	} else if (re <= 350.0) {
		tmp = cos(im) * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)));
	} else if (re <= 2.1e+90) {
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = cos(im) * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.024d0)) then
        tmp = exp(re)
    else if (re <= 350.0d0) then
        tmp = cos(im) * ((re + 1.0d0) + ((0.5d0 + (re * 0.16666666666666666d0)) * (re * re)))
    else if (re <= 2.1d+90) then
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    else
        tmp = cos(im) * (0.16666666666666666d0 * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.024) {
		tmp = Math.exp(re);
	} else if (re <= 350.0) {
		tmp = Math.cos(im) * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)));
	} else if (re <= 2.1e+90) {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = Math.cos(im) * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.024:
		tmp = math.exp(re)
	elif re <= 350.0:
		tmp = math.cos(im) * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)))
	elif re <= 2.1e+90:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	else:
		tmp = math.cos(im) * (0.16666666666666666 * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.024)
		tmp = exp(re);
	elseif (re <= 350.0)
		tmp = Float64(cos(im) * Float64(Float64(re + 1.0) + Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * Float64(re * re))));
	elseif (re <= 2.1e+90)
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = Float64(cos(im) * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.024)
		tmp = exp(re);
	elseif (re <= 350.0)
		tmp = cos(im) * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)));
	elseif (re <= 2.1e+90)
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	else
		tmp = cos(im) * (0.16666666666666666 * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.024], N[Exp[re], $MachinePrecision], If[LessEqual[re, 350.0], N[(N[Cos[im], $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+90], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.024

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.024 < re < 350

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 98.4%

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

      1. distribute-rgt-in 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-+r+ 98.4%

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

      4. distribute-rgt1-in 98.4%

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

      5. *-commutative 98.4%

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

      6. *-commutative 98.4%

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

      7. associate-*l* 98.4%

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

      8. +-commutative 98.4%

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

      9. associate-*r* 98.4%

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

      10. distribute-rgt-out 98.4%

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

      11. associate-*l* 98.4%

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

      12. distribute-lft-out 98.4%

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

   5. Simplified 98.4%

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

   if 350 < re < 2.09999999999999981e90

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 0.0%

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

      1. *-lft-identity 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 82.6%

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

      4. *-commutative 82.6%

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

      5. unpow2 82.6%

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

      6. associate-*l* 82.6%

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

   5. Simplified 82.6%

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

   if 2.09999999999999981e90 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 98.0%

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

      1. distribute-rgt-in 98.0%

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

      2. *-commutative 98.0%

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

      3. associate-+r+ 98.0%

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

      4. distribute-rgt1-in 98.0%

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

      5. *-commutative 98.0%

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

      6. *-commutative 98.0%

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

      7. associate-*l* 98.0%

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

      8. +-commutative 98.0%

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

      9. associate-*r* 98.0%

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

      10. distribute-rgt-out 98.0%

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

      11. associate-*l* 98.0%

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

      12. distribute-lft-out 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in re around inf 98.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 98.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} \]

      2. cube-mult 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.024:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+90}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0205:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+90}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0205)
   (exp re)
   (if (<= re 350.0)
     (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (if (<= re 2.1e+90)
       (* (exp re) (+ 1.0 (* im (* im -0.5))))
       (* (cos im) (* 0.16666666666666666 (* re (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0205) {
		tmp = exp(re);
	} else if (re <= 350.0) {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 2.1e+90) {
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = cos(im) * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0205d0)) then
        tmp = exp(re)
    else if (re <= 350.0d0) then
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else if (re <= 2.1d+90) then
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    else
        tmp = cos(im) * (0.16666666666666666d0 * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0205) {
		tmp = Math.exp(re);
	} else if (re <= 350.0) {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 2.1e+90) {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = Math.cos(im) * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0205:
		tmp = math.exp(re)
	elif re <= 350.0:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	elif re <= 2.1e+90:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	else:
		tmp = math.cos(im) * (0.16666666666666666 * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0205)
		tmp = exp(re);
	elseif (re <= 350.0)
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	elseif (re <= 2.1e+90)
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = Float64(cos(im) * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0205)
		tmp = exp(re);
	elseif (re <= 350.0)
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	elseif (re <= 2.1e+90)
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	else
		tmp = cos(im) * (0.16666666666666666 * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0205], N[Exp[re], $MachinePrecision], If[LessEqual[re, 350.0], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+90], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.0205000000000000009

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.0205000000000000009 < re < 350

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 98.3%

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

      1. *-rgt-identity 98.3%

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

      2. distribute-rgt-in 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*l* 98.3%

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

      5. *-commutative 98.3%

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

      6. associate-*l* 98.3%

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

      7. distribute-lft-out 98.3%

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

      8. distribute-lft-out 98.3%

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

      9. *-commutative 98.3%

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

      10. distribute-rgt1-in 98.3%

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

      11. +-commutative 98.3%

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

      12. *-commutative 98.3%

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

   5. Simplified 98.3%

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

   if 350 < re < 2.09999999999999981e90

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 0.0%

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

      1. *-lft-identity 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 82.6%

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

      4. *-commutative 82.6%

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

      5. unpow2 82.6%

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

      6. associate-*l* 82.6%

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

   5. Simplified 82.6%

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

   if 2.09999999999999981e90 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 98.0%

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

      1. distribute-rgt-in 98.0%

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

      2. *-commutative 98.0%

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

      3. associate-+r+ 98.0%

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

      4. distribute-rgt1-in 98.0%

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

      5. *-commutative 98.0%

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

      6. *-commutative 98.0%

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

      7. associate-*l* 98.0%

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

      8. +-commutative 98.0%

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

      9. associate-*r* 98.0%

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

      10. distribute-rgt-out 98.0%

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

      11. associate-*l* 98.0%

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

      12. distribute-lft-out 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in re around inf 98.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 98.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} \]

      2. cube-mult 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0205:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+90}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0003:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+90}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0003)
   (exp re)
   (if (<= re 350.0)
     (* (cos im) (+ re 1.0))
     (if (<= re 2.1e+90)
       (* (exp re) (+ 1.0 (* im (* im -0.5))))
       (* (cos im) (* 0.16666666666666666 (* re (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0003) {
		tmp = exp(re);
	} else if (re <= 350.0) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 2.1e+90) {
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = cos(im) * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0003d0)) then
        tmp = exp(re)
    else if (re <= 350.0d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 2.1d+90) then
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    else
        tmp = cos(im) * (0.16666666666666666d0 * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0003) {
		tmp = Math.exp(re);
	} else if (re <= 350.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 2.1e+90) {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = Math.cos(im) * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0003:
		tmp = math.exp(re)
	elif re <= 350.0:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 2.1e+90:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	else:
		tmp = math.cos(im) * (0.16666666666666666 * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0003)
		tmp = exp(re);
	elseif (re <= 350.0)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 2.1e+90)
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = Float64(cos(im) * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0003)
		tmp = exp(re);
	elseif (re <= 350.0)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 2.1e+90)
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	else
		tmp = cos(im) * (0.16666666666666666 * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0003], N[Exp[re], $MachinePrecision], If[LessEqual[re, 350.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+90], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.99999999999999974e-4

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{e^{re}} \]

   if -2.99999999999999974e-4 < re < 350

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 98.1%

      \[\leadsto \color{blue}{\cos im + re \cdot \cos im} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in 98.1%

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

      2. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   if 350 < re < 2.09999999999999981e90

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 0.0%

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

      1. *-lft-identity 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 82.6%

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

      4. *-commutative 82.6%

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

      5. unpow2 82.6%

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

      6. associate-*l* 82.6%

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

   5. Simplified 82.6%

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

   if 2.09999999999999981e90 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 98.0%

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

      1. distribute-rgt-in 98.0%

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

      2. *-commutative 98.0%

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

      3. associate-+r+ 98.0%

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

      4. distribute-rgt1-in 98.0%

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

      5. *-commutative 98.0%

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

      6. *-commutative 98.0%

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

      7. associate-*l* 98.0%

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

      8. +-commutative 98.0%

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

      9. associate-*r* 98.0%

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

      10. distribute-rgt-out 98.0%

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

      11. associate-*l* 98.0%

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

      12. distribute-lft-out 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in re around inf 98.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 98.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} \]

      2. cube-mult 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0003:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+90}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00068:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0036:\\ \;\;\;\;\frac{-\cos im}{re + -1}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00068)
   (exp re)
   (if (<= re 0.0036)
     (/ (- (cos im)) (+ re -1.0))
     (if (<= re 1.25e+150) (exp re) (* (cos im) (+ 1.0 (* re (* re 0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00068) {
		tmp = exp(re);
	} else if (re <= 0.0036) {
		tmp = -cos(im) / (re + -1.0);
	} else if (re <= 1.25e+150) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (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 (re <= (-0.00068d0)) then
        tmp = exp(re)
    else if (re <= 0.0036d0) then
        tmp = -cos(im) / (re + (-1.0d0))
    else if (re <= 1.25d+150) then
        tmp = exp(re)
    else
        tmp = cos(im) * (1.0d0 + (re * (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00068) {
		tmp = Math.exp(re);
	} else if (re <= 0.0036) {
		tmp = -Math.cos(im) / (re + -1.0);
	} else if (re <= 1.25e+150) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re * (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00068:
		tmp = math.exp(re)
	elif re <= 0.0036:
		tmp = -math.cos(im) / (re + -1.0)
	elif re <= 1.25e+150:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re * (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00068)
		tmp = exp(re);
	elseif (re <= 0.0036)
		tmp = Float64(Float64(-cos(im)) / Float64(re + -1.0));
	elseif (re <= 1.25e+150)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00068)
		tmp = exp(re);
	elseif (re <= 0.0036)
		tmp = -cos(im) / (re + -1.0);
	elseif (re <= 1.25e+150)
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re * (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00068], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0036], N[((-N[Cos[im], $MachinePrecision]) / N[(re + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.25e+150], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.8e-4 or 0.0035999999999999999 < re < 1.25000000000000002e150

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 93.4%

      \[\leadsto \color{blue}{e^{re}} \]

   if -6.8e-4 < re < 0.0035999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.6%

      \[\leadsto \color{blue}{\cos im + re \cdot \cos im} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. flip-+ 99.6%

         \[\leadsto \color{blue}{\frac{re \cdot re - 1 \cdot 1}{re - 1}} \cdot \cos im \]

      3. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\left(re \cdot re - 1 \cdot 1\right) \cdot \cos im}{re - 1}} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{\left(re \cdot re - \color{blue}{1}\right) \cdot \cos im}{re - 1} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\left(re \cdot re - 1\right) \cdot \cos im}{re - 1}} \]

   8. Taylor expanded in re around 0 99.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \cos im}}{re - 1} \]
   9. Step-by-step derivation

      1. mul-1-neg 99.6%

         \[\leadsto \frac{\color{blue}{-\cos im}}{re - 1} \]

      2. neg-sub0 99.6%

         \[\leadsto \frac{\color{blue}{0 - \cos im}}{re - 1} \]

   10. Simplified 99.6%

       \[\leadsto \frac{\color{blue}{0 - \cos im}}{re - 1} \]

   if 1.25000000000000002e150 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 97.5%

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

      1. *-rgt-identity 97.5%

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

      2. distribute-rgt-in 97.5%

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

      3. *-commutative 97.5%

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

      4. associate-*l* 97.5%

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

      5. *-commutative 97.5%

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

      6. associate-*l* 97.5%

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

      7. distribute-lft-out 97.5%

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

      8. distribute-lft-out 97.5%

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

      9. *-commutative 97.5%

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

      10. distribute-rgt1-in 97.5%

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

      11. +-commutative 97.5%

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

      12. *-commutative 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in re around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. associate-*r* 97.5%

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

   8. Simplified 97.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00068:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0036:\\ \;\;\;\;\frac{-\cos im}{re + -1}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0014:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0165:\\ \;\;\;\;\frac{-\cos im}{re + -1}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0014)
   (exp re)
   (if (<= re 0.0165)
     (/ (- (cos im)) (+ re -1.0))
     (if (<= re 1.25e+150) (exp re) (* (cos im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0014) {
		tmp = exp(re);
	} else if (re <= 0.0165) {
		tmp = -cos(im) / (re + -1.0);
	} else if (re <= 1.25e+150) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (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 (re <= (-0.0014d0)) then
        tmp = exp(re)
    else if (re <= 0.0165d0) then
        tmp = -cos(im) / (re + (-1.0d0))
    else if (re <= 1.25d+150) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0014) {
		tmp = Math.exp(re);
	} else if (re <= 0.0165) {
		tmp = -Math.cos(im) / (re + -1.0);
	} else if (re <= 1.25e+150) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0014:
		tmp = math.exp(re)
	elif re <= 0.0165:
		tmp = -math.cos(im) / (re + -1.0)
	elif re <= 1.25e+150:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0014)
		tmp = exp(re);
	elseif (re <= 0.0165)
		tmp = Float64(Float64(-cos(im)) / Float64(re + -1.0));
	elseif (re <= 1.25e+150)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0014)
		tmp = exp(re);
	elseif (re <= 0.0165)
		tmp = -cos(im) / (re + -1.0);
	elseif (re <= 1.25e+150)
		tmp = exp(re);
	else
		tmp = cos(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0014], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0165], N[((-N[Cos[im], $MachinePrecision]) / N[(re + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.25e+150], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00139999999999999999 or 0.016500000000000001 < re < 1.25000000000000002e150

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 93.4%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.00139999999999999999 < re < 0.016500000000000001

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.6%

      \[\leadsto \color{blue}{\cos im + re \cdot \cos im} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. flip-+ 99.6%

         \[\leadsto \color{blue}{\frac{re \cdot re - 1 \cdot 1}{re - 1}} \cdot \cos im \]

      3. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{\left(re \cdot re - 1 \cdot 1\right) \cdot \cos im}{re - 1}} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{\left(re \cdot re - \color{blue}{1}\right) \cdot \cos im}{re - 1} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\left(re \cdot re - 1\right) \cdot \cos im}{re - 1}} \]

   8. Taylor expanded in re around 0 99.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \cos im}}{re - 1} \]
   9. Step-by-step derivation

      1. mul-1-neg 99.6%

         \[\leadsto \frac{\color{blue}{-\cos im}}{re - 1} \]

      2. neg-sub0 99.6%

         \[\leadsto \frac{\color{blue}{0 - \cos im}}{re - 1} \]

   10. Simplified 99.6%

       \[\leadsto \frac{\color{blue}{0 - \cos im}}{re - 1} \]

   if 1.25000000000000002e150 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 97.5%

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

      1. *-rgt-identity 97.5%

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

      2. distribute-rgt-in 97.5%

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

      3. *-commutative 97.5%

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

      4. associate-*l* 97.5%

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

      5. *-commutative 97.5%

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

      6. associate-*l* 97.5%

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

      7. distribute-lft-out 97.5%

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

      8. distribute-lft-out 97.5%

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

      9. *-commutative 97.5%

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

      10. distribute-rgt1-in 97.5%

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

      11. +-commutative 97.5%

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

      12. *-commutative 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in re around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*l* 97.5%

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

      4. unpow2 97.5%

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

      5. associate-*r* 97.5%

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

   8. Simplified 97.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0014:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0165:\\ \;\;\;\;\frac{-\cos im}{re + -1}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0044:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.00225:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0044)
   (exp re)
   (if (<= re 0.00225)
     (* (cos im) (+ re 1.0))
     (if (<= re 1.25e+150) (exp re) (* (cos im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0044) {
		tmp = exp(re);
	} else if (re <= 0.00225) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 1.25e+150) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (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 (re <= (-0.0044d0)) then
        tmp = exp(re)
    else if (re <= 0.00225d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 1.25d+150) then
        tmp = exp(re)
    else
        tmp = cos(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0044) {
		tmp = Math.exp(re);
	} else if (re <= 0.00225) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 1.25e+150) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0044:
		tmp = math.exp(re)
	elif re <= 0.00225:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 1.25e+150:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0044)
		tmp = exp(re);
	elseif (re <= 0.00225)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 1.25e+150)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0044)
		tmp = exp(re);
	elseif (re <= 0.00225)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 1.25e+150)
		tmp = exp(re);
	else
		tmp = cos(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0044], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.00225], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.25e+150], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00440000000000000027 or 0.00224999999999999983 < re < 1.25000000000000002e150

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 93.4%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.00440000000000000027 < re < 0.00224999999999999983

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.6%

      \[\leadsto \color{blue}{\cos im + re \cdot \cos im} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   if 1.25000000000000002e150 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 97.5%

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

      1. *-rgt-identity 97.5%

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

      2. distribute-rgt-in 97.5%

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

      3. *-commutative 97.5%

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

      4. associate-*l* 97.5%

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

      5. *-commutative 97.5%

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

      6. associate-*l* 97.5%

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

      7. distribute-lft-out 97.5%

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

      8. distribute-lft-out 97.5%

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

      9. *-commutative 97.5%

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

      10. distribute-rgt1-in 97.5%

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

      11. +-commutative 97.5%

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

      12. *-commutative 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in re around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*l* 97.5%

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

      4. unpow2 97.5%

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

      5. associate-*r* 97.5%

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

   8. Simplified 97.5%

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

Alternative 10: 67.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -2100000:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot \left(\frac{0.5}{-re} - 0.25\right)\right)\\ \mathbf{elif}\;re \leq 680:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+77}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{t\_0 + 0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot t\_0\right)\right)}{re \cdot re + \left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot 0.25\right) - re \cdot 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re re))))
   (if (<= re -2100000.0)
     (* (* im im) (* (* re re) (- (/ 0.5 (- re)) 0.25)))
     (if (<= re 680.0)
       (cos im)
       (if (<= re 1.65e+77)
         (+
          (* im (* (* re re) (+ (* -0.5 (/ im re)) (* im -0.25))))
          (+
           1.0
           (/
            (+ t_0 (* 0.125 (* (* re re) (* re t_0))))
            (+ (* re re) (* (* re re) (- (* re (* re 0.25)) (* re 0.5)))))))
         (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double t_0 = re * (re * re);
	double tmp;
	if (re <= -2100000.0) {
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25));
	} else if (re <= 680.0) {
		tmp = cos(im);
	} else if (re <= 1.65e+77) {
		tmp = (im * ((re * re) * ((-0.5 * (im / re)) + (im * -0.25)))) + (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5))))));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	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 = re * (re * re)
    if (re <= (-2100000.0d0)) then
        tmp = (im * im) * ((re * re) * ((0.5d0 / -re) - 0.25d0))
    else if (re <= 680.0d0) then
        tmp = cos(im)
    else if (re <= 1.65d+77) then
        tmp = (im * ((re * re) * (((-0.5d0) * (im / re)) + (im * (-0.25d0))))) + (1.0d0 + ((t_0 + (0.125d0 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25d0)) - (re * 0.5d0))))))
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * re);
	double tmp;
	if (re <= -2100000.0) {
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25));
	} else if (re <= 680.0) {
		tmp = Math.cos(im);
	} else if (re <= 1.65e+77) {
		tmp = (im * ((re * re) * ((-0.5 * (im / re)) + (im * -0.25)))) + (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5))))));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * re)
	tmp = 0
	if re <= -2100000.0:
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25))
	elif re <= 680.0:
		tmp = math.cos(im)
	elif re <= 1.65e+77:
		tmp = (im * ((re * re) * ((-0.5 * (im / re)) + (im * -0.25)))) + (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5))))))
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * re))
	tmp = 0.0
	if (re <= -2100000.0)
		tmp = Float64(Float64(im * im) * Float64(Float64(re * re) * Float64(Float64(0.5 / Float64(-re)) - 0.25)));
	elseif (re <= 680.0)
		tmp = cos(im);
	elseif (re <= 1.65e+77)
		tmp = Float64(Float64(im * Float64(Float64(re * re) * Float64(Float64(-0.5 * Float64(im / re)) + Float64(im * -0.25)))) + Float64(1.0 + Float64(Float64(t_0 + Float64(0.125 * Float64(Float64(re * re) * Float64(re * t_0)))) / Float64(Float64(re * re) + Float64(Float64(re * re) * Float64(Float64(re * Float64(re * 0.25)) - Float64(re * 0.5)))))));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * re);
	tmp = 0.0;
	if (re <= -2100000.0)
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25));
	elseif (re <= 680.0)
		tmp = cos(im);
	elseif (re <= 1.65e+77)
		tmp = (im * ((re * re) * ((-0.5 * (im / re)) + (im * -0.25)))) + (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5))))));
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2100000.0], N[(N[(im * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(0.5 / (-re)), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 680.0], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1.65e+77], N[(N[(im * N[(N[(re * re), $MachinePrecision] * N[(N[(-0.5 * N[(im / re), $MachinePrecision]), $MachinePrecision] + N[(im * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(t$95$0 + N[(0.125 * N[(N[(re * re), $MachinePrecision] * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(re * re), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(N[(re * N[(re * 0.25), $MachinePrecision]), $MachinePrecision] - N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.1e6

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 1.9%

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

      1. *-rgt-identity 1.9%

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

      2. distribute-rgt-in 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*l* 1.9%

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

      5. *-commutative 1.9%

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

      6. associate-*l* 1.9%

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

      7. distribute-lft-out 1.9%

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

      8. distribute-lft-out 1.9%

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

      9. *-commutative 1.9%

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

      10. distribute-rgt1-in 1.9%

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

      11. +-commutative 1.9%

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

      12. *-commutative 1.9%

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

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0 0.9%

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

      1. +-commutative 0.9%

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

      2. distribute-rgt-in 0.9%

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

      3. *-commutative 0.9%

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

      4. distribute-rgt-in 0.9%

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

      5. associate-+l+ 0.9%

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

   8. Simplified 0.9%

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

   9. Taylor expanded in re around inf 0.9%

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

       1. unpow2 0.9%

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

       2. *-commutative 0.9%

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

   11. Simplified 0.9%

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

   12. Taylor expanded in im around inf 10.2%

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

       1. mul-1-neg 10.2%

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

       2. distribute-rgt-neg-in 10.2%

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

       3. unpow2 10.2%

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

       4. *-commutative 10.2%

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

       5. distribute-rgt-neg-in 10.2%

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

       6. +-commutative 10.2%

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

       7. associate-*r/ 10.2%

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

       8. metadata-eval 10.2%

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

       9. mul-1-neg 10.2%

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

       10. unpow2 10.2%

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

   14. Simplified 10.2%

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

   if -2.1e6 < re < 680

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 96.4%

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

   if 680 < re < 1.6499999999999999e77

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 3.9%

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

      1. *-rgt-identity 3.9%

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

      2. distribute-rgt-in 3.9%

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

      3. *-commutative 3.9%

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

      4. associate-*l* 3.9%

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

      5. *-commutative 3.9%

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

      6. associate-*l* 3.9%

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

      7. distribute-lft-out 3.9%

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

      8. distribute-lft-out 3.9%

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

      9. *-commutative 3.9%

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

      10. distribute-rgt1-in 3.9%

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

      11. +-commutative 3.9%

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

      12. *-commutative 3.9%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in im around 0 16.9%

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

      1. +-commutative 16.9%

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

      2. distribute-rgt-in 16.9%

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

      3. *-commutative 16.9%

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

      4. distribute-rgt-in 16.9%

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

      5. associate-+l+ 16.9%

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

   8. Simplified 16.9%

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

   9. Taylor expanded in re around inf 16.9%

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

       1. unpow2 16.9%

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

       2. *-commutative 16.9%

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

   11. Simplified 16.9%

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

       1. distribute-rgt-in 16.9%

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

       2. *-un-lft-identity 16.9%

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

       3. flip3-+ 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \color{blue}{\frac{{re}^{3} + {\left(\left(re \cdot 0.5\right) \cdot re\right)}^{3}}{re \cdot re + \left(\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}}\right) \]

       4. cube-mult 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{re \cdot \left(re \cdot re\right)} + {\left(\left(re \cdot 0.5\right) \cdot re\right)}^{3}}{re \cdot re + \left(\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

       5. *-commutative 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)}}^{3}}{re \cdot re + \left(\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

       6. *-commutative 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

       7. *-commutative 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

       8. *-commutative 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)}\right)}\right) \]

   13. Applied egg-rr 39.7%

       \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \color{blue}{\frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}}\right) \]
   14. Step-by-step derivation

       1. +-commutative 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{{\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3} + re \cdot \left(re \cdot re\right)}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       2. associate-*r* 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{{\color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)}}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       3. unpow2 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{{\left(\color{blue}{{re}^{2}} \cdot 0.5\right)}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       4. *-commutative 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{{\color{blue}{\left(0.5 \cdot {re}^{2}\right)}}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       5. cube-prod 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{{0.5}^{3} \cdot {\left({re}^{2}\right)}^{3}} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       6. metadata-eval 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{0.125} \cdot {\left({re}^{2}\right)}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       7. cube-unmult 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right)} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       8. unpow2 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       9. unpow2 39.7%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {re}^{2}\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       10. associate-*l* 39.7%

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot {re}^{2}\right)\right)}\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       11. unpow2 39.7%

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

   15. Simplified 39.7%

       \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \color{blue}{\frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot 0.25\right) - re \cdot 0.5\right)}}\right) \]

   if 1.6499999999999999e77 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 94.1%

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

      1. distribute-rgt-in 94.1%

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

      2. *-commutative 94.1%

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

      3. associate-+r+ 94.1%

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

      4. distribute-rgt1-in 94.1%

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

      5. *-commutative 94.1%

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

      6. *-commutative 94.1%

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

      7. associate-*l* 94.1%

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

      8. +-commutative 94.1%

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

      9. associate-*r* 94.1%

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

      10. distribute-rgt-out 94.1%

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

      11. associate-*l* 94.1%

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

      12. distribute-lft-out 94.1%

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

   5. Simplified 94.1%

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

   6. Taylor expanded in im around 0 68.6%

      \[\leadsto \color{blue}{1 + \left(re + {re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. *-rgt-identity 68.6%

         \[\leadsto 1 + \left(\color{blue}{re \cdot 1} + {re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

      2. unpow2 68.6%

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

      3. +-commutative 68.6%

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

      4. *-commutative 68.6%

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

      5. associate-*l* 68.6%

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

      6. *-commutative 68.6%

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

      7. +-commutative 68.6%

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

      8. distribute-lft-in 68.6%

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

   8. Simplified 68.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2100000:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot \left(\frac{0.5}{-re} - 0.25\right)\right)\\ \mathbf{elif}\;re \leq 680:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+77}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + 0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)}{re \cdot re + \left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot 0.25\right) - re \cdot 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -350:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot \left(\frac{0.5}{-re} - 0.25\right)\right)\\ \mathbf{elif}\;re \leq 320 \lor \neg \left(re \leq 1.65 \cdot 10^{+77}\right):\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{t\_0 + 0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot t\_0\right)\right)}{re \cdot re + \left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot 0.25\right) - re \cdot 0.5\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re re))))
   (if (<= re -350.0)
     (* (* im im) (* (* re re) (- (/ 0.5 (- re)) 0.25)))
     (if (or (<= re 320.0) (not (<= re 1.65e+77)))
       (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
       (+
        (* im (* (* re re) (+ (* -0.5 (/ im re)) (* im -0.25))))
        (+
         1.0
         (/
          (+ t_0 (* 0.125 (* (* re re) (* re t_0))))
          (+ (* re re) (* (* re re) (- (* re (* re 0.25)) (* re 0.5)))))))))))

C

double code(double re, double im) {
	double t_0 = re * (re * re);
	double tmp;
	if (re <= -350.0) {
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25));
	} else if ((re <= 320.0) || !(re <= 1.65e+77)) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (im * ((re * re) * ((-0.5 * (im / re)) + (im * -0.25)))) + (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5))))));
	}
	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 = re * (re * re)
    if (re <= (-350.0d0)) then
        tmp = (im * im) * ((re * re) * ((0.5d0 / -re) - 0.25d0))
    else if ((re <= 320.0d0) .or. (.not. (re <= 1.65d+77))) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    else
        tmp = (im * ((re * re) * (((-0.5d0) * (im / re)) + (im * (-0.25d0))))) + (1.0d0 + ((t_0 + (0.125d0 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25d0)) - (re * 0.5d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * re);
	double tmp;
	if (re <= -350.0) {
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25));
	} else if ((re <= 320.0) || !(re <= 1.65e+77)) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (im * ((re * re) * ((-0.5 * (im / re)) + (im * -0.25)))) + (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * re)
	tmp = 0
	if re <= -350.0:
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25))
	elif (re <= 320.0) or not (re <= 1.65e+77):
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	else:
		tmp = (im * ((re * re) * ((-0.5 * (im / re)) + (im * -0.25)))) + (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * re))
	tmp = 0.0
	if (re <= -350.0)
		tmp = Float64(Float64(im * im) * Float64(Float64(re * re) * Float64(Float64(0.5 / Float64(-re)) - 0.25)));
	elseif ((re <= 320.0) || !(re <= 1.65e+77))
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	else
		tmp = Float64(Float64(im * Float64(Float64(re * re) * Float64(Float64(-0.5 * Float64(im / re)) + Float64(im * -0.25)))) + Float64(1.0 + Float64(Float64(t_0 + Float64(0.125 * Float64(Float64(re * re) * Float64(re * t_0)))) / Float64(Float64(re * re) + Float64(Float64(re * re) * Float64(Float64(re * Float64(re * 0.25)) - Float64(re * 0.5)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * re);
	tmp = 0.0;
	if (re <= -350.0)
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25));
	elseif ((re <= 320.0) || ~((re <= 1.65e+77)))
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	else
		tmp = (im * ((re * re) * ((-0.5 * (im / re)) + (im * -0.25)))) + (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -350.0], N[(N[(im * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(0.5 / (-re)), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 320.0], N[Not[LessEqual[re, 1.65e+77]], $MachinePrecision]], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(N[(re * re), $MachinePrecision] * N[(N[(-0.5 * N[(im / re), $MachinePrecision]), $MachinePrecision] + N[(im * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(t$95$0 + N[(0.125 * N[(N[(re * re), $MachinePrecision] * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(re * re), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(N[(re * N[(re * 0.25), $MachinePrecision]), $MachinePrecision] - N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -350

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 1.9%

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

      1. *-rgt-identity 1.9%

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

      2. distribute-rgt-in 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*l* 1.9%

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

      5. *-commutative 1.9%

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

      6. associate-*l* 1.9%

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

      7. distribute-lft-out 1.9%

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

      8. distribute-lft-out 1.9%

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

      9. *-commutative 1.9%

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

      10. distribute-rgt1-in 1.9%

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

      11. +-commutative 1.9%

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

      12. *-commutative 1.9%

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

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0 0.9%

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

      1. +-commutative 0.9%

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

      2. distribute-rgt-in 0.9%

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

      3. *-commutative 0.9%

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

      4. distribute-rgt-in 0.9%

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

      5. associate-+l+ 0.9%

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

   8. Simplified 0.9%

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

   9. Taylor expanded in re around inf 0.9%

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

       1. unpow2 0.9%

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

       2. *-commutative 0.9%

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

   11. Simplified 0.9%

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

   12. Taylor expanded in im around inf 10.1%

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

       1. mul-1-neg 10.1%

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

       2. distribute-rgt-neg-in 10.1%

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

       3. unpow2 10.1%

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

       4. *-commutative 10.1%

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

       5. distribute-rgt-neg-in 10.1%

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

       6. +-commutative 10.1%

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

       7. associate-*r/ 10.1%

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

       8. metadata-eval 10.1%

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

       9. mul-1-neg 10.1%

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

       10. unpow2 10.1%

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

   14. Simplified 10.1%

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

   if -350 < re < 320 or 1.6499999999999999e77 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 97.8%

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

      1. distribute-rgt-in 97.8%

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

      2. *-commutative 97.8%

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

      3. associate-+r+ 97.8%

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

      4. distribute-rgt1-in 97.8%

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

      5. *-commutative 97.8%

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

      6. *-commutative 97.8%

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

      7. associate-*l* 97.8%

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

      8. +-commutative 97.8%

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

      9. associate-*r* 97.8%

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

      10. distribute-rgt-out 97.8%

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

      11. associate-*l* 97.8%

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

      12. distribute-lft-out 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in im around 0 56.4%

      \[\leadsto \color{blue}{1 + \left(re + {re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. *-rgt-identity 56.4%

         \[\leadsto 1 + \left(\color{blue}{re \cdot 1} + {re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

      2. unpow2 56.4%

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

      3. +-commutative 56.4%

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

      4. *-commutative 56.4%

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

      5. associate-*l* 56.4%

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

      6. *-commutative 56.4%

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

      7. +-commutative 56.4%

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

      8. distribute-lft-in 56.4%

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

   8. Simplified 56.4%

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

   if 320 < re < 1.6499999999999999e77

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 4.0%

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

      1. *-rgt-identity 4.0%

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

      2. distribute-rgt-in 4.0%

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

      3. *-commutative 4.0%

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

      4. associate-*l* 4.0%

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

      5. *-commutative 4.0%

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

      6. associate-*l* 4.0%

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

      7. distribute-lft-out 4.0%

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

      8. distribute-lft-out 4.0%

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

      9. *-commutative 4.0%

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

      10. distribute-rgt1-in 4.0%

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

      11. +-commutative 4.0%

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

      12. *-commutative 4.0%

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

   5. Simplified 4.0%

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

   6. Taylor expanded in im around 0 16.3%

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

      1. +-commutative 16.3%

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

      2. distribute-rgt-in 16.3%

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

      3. *-commutative 16.3%

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

      4. distribute-rgt-in 16.3%

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

      5. associate-+l+ 16.3%

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

   8. Simplified 16.3%

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

   9. Taylor expanded in re around inf 16.3%

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

       1. unpow2 16.3%

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

       2. *-commutative 16.3%

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

   11. Simplified 16.3%

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

       1. distribute-rgt-in 16.3%

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

       2. *-un-lft-identity 16.3%

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

       3. flip3-+ 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \color{blue}{\frac{{re}^{3} + {\left(\left(re \cdot 0.5\right) \cdot re\right)}^{3}}{re \cdot re + \left(\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}}\right) \]

       4. cube-mult 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{re \cdot \left(re \cdot re\right)} + {\left(\left(re \cdot 0.5\right) \cdot re\right)}^{3}}{re \cdot re + \left(\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

       5. *-commutative 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)}}^{3}}{re \cdot re + \left(\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

       6. *-commutative 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

       7. *-commutative 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

       8. *-commutative 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)}\right)}\right) \]

   13. Applied egg-rr 38.1%

       \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \color{blue}{\frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}}\right) \]
   14. Step-by-step derivation

       1. +-commutative 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{{\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3} + re \cdot \left(re \cdot re\right)}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       2. associate-*r* 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{{\color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)}}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       3. unpow2 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{{\left(\color{blue}{{re}^{2}} \cdot 0.5\right)}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       4. *-commutative 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{{\color{blue}{\left(0.5 \cdot {re}^{2}\right)}}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       5. cube-prod 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{{0.5}^{3} \cdot {\left({re}^{2}\right)}^{3}} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       6. metadata-eval 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{0.125} \cdot {\left({re}^{2}\right)}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       7. cube-unmult 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right)} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       8. unpow2 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       9. unpow2 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {re}^{2}\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       10. associate-*l* 38.1%

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot {re}^{2}\right)\right)}\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       11. unpow2 38.1%

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

   15. Simplified 38.1%

       \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \color{blue}{\frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot 0.25\right) - re \cdot 0.5\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -350:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot \left(\frac{0.5}{-re} - 0.25\right)\right)\\ \mathbf{elif}\;re \leq 320 \lor \neg \left(re \leq 1.65 \cdot 10^{+77}\right):\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + 0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)}{re \cdot re + \left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot 0.25\right) - re \cdot 0.5\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -380:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot \left(\frac{0.5}{-re} - 0.25\right)\right)\\ \mathbf{elif}\;re \leq 310 \lor \neg \left(re \leq 1.65 \cdot 10^{+77}\right):\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{t\_0 + 0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot t\_0\right)\right)}{re \cdot re + \left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot 0.25\right) - re \cdot 0.5\right)}\right) + im \cdot \left(im \cdot \left(-0.5 \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re re))))
   (if (<= re -380.0)
     (* (* im im) (* (* re re) (- (/ 0.5 (- re)) 0.25)))
     (if (or (<= re 310.0) (not (<= re 1.65e+77)))
       (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
       (+
        (+
         1.0
         (/
          (+ t_0 (* 0.125 (* (* re re) (* re t_0))))
          (+ (* re re) (* (* re re) (- (* re (* re 0.25)) (* re 0.5))))))
        (* im (* im (* -0.5 (+ 1.0 (* re (+ 1.0 (* re 0.5))))))))))))

C

double code(double re, double im) {
	double t_0 = re * (re * re);
	double tmp;
	if (re <= -380.0) {
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25));
	} else if ((re <= 310.0) || !(re <= 1.65e+77)) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5)))))) + (im * (im * (-0.5 * (1.0 + (re * (1.0 + (re * 0.5)))))));
	}
	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 = re * (re * re)
    if (re <= (-380.0d0)) then
        tmp = (im * im) * ((re * re) * ((0.5d0 / -re) - 0.25d0))
    else if ((re <= 310.0d0) .or. (.not. (re <= 1.65d+77))) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    else
        tmp = (1.0d0 + ((t_0 + (0.125d0 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25d0)) - (re * 0.5d0)))))) + (im * (im * ((-0.5d0) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * re);
	double tmp;
	if (re <= -380.0) {
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25));
	} else if ((re <= 310.0) || !(re <= 1.65e+77)) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5)))))) + (im * (im * (-0.5 * (1.0 + (re * (1.0 + (re * 0.5)))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * re)
	tmp = 0
	if re <= -380.0:
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25))
	elif (re <= 310.0) or not (re <= 1.65e+77):
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	else:
		tmp = (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5)))))) + (im * (im * (-0.5 * (1.0 + (re * (1.0 + (re * 0.5)))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * re))
	tmp = 0.0
	if (re <= -380.0)
		tmp = Float64(Float64(im * im) * Float64(Float64(re * re) * Float64(Float64(0.5 / Float64(-re)) - 0.25)));
	elseif ((re <= 310.0) || !(re <= 1.65e+77))
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(t_0 + Float64(0.125 * Float64(Float64(re * re) * Float64(re * t_0)))) / Float64(Float64(re * re) + Float64(Float64(re * re) * Float64(Float64(re * Float64(re * 0.25)) - Float64(re * 0.5)))))) + Float64(im * Float64(im * Float64(-0.5 * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * re);
	tmp = 0.0;
	if (re <= -380.0)
		tmp = (im * im) * ((re * re) * ((0.5 / -re) - 0.25));
	elseif ((re <= 310.0) || ~((re <= 1.65e+77)))
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	else
		tmp = (1.0 + ((t_0 + (0.125 * ((re * re) * (re * t_0)))) / ((re * re) + ((re * re) * ((re * (re * 0.25)) - (re * 0.5)))))) + (im * (im * (-0.5 * (1.0 + (re * (1.0 + (re * 0.5)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -380.0], N[(N[(im * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(0.5 / (-re)), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 310.0], N[Not[LessEqual[re, 1.65e+77]], $MachinePrecision]], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(t$95$0 + N[(0.125 * N[(N[(re * re), $MachinePrecision] * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(re * re), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * N[(N[(re * N[(re * 0.25), $MachinePrecision]), $MachinePrecision] - N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im * N[(im * N[(-0.5 * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -380

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 1.9%

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

      1. *-rgt-identity 1.9%

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

      2. distribute-rgt-in 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*l* 1.9%

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

      5. *-commutative 1.9%

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

      6. associate-*l* 1.9%

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

      7. distribute-lft-out 1.9%

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

      8. distribute-lft-out 1.9%

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

      9. *-commutative 1.9%

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

      10. distribute-rgt1-in 1.9%

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

      11. +-commutative 1.9%

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

      12. *-commutative 1.9%

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

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0 0.9%

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

      1. +-commutative 0.9%

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

      2. distribute-rgt-in 0.9%

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

      3. *-commutative 0.9%

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

      4. distribute-rgt-in 0.9%

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

      5. associate-+l+ 0.9%

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

   8. Simplified 0.9%

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

   9. Taylor expanded in re around inf 0.9%

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

       1. unpow2 0.9%

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

       2. *-commutative 0.9%

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

   11. Simplified 0.9%

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

   12. Taylor expanded in im around inf 10.1%

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

       1. mul-1-neg 10.1%

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

       2. distribute-rgt-neg-in 10.1%

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

       3. unpow2 10.1%

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

       4. *-commutative 10.1%

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

       5. distribute-rgt-neg-in 10.1%

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

       6. +-commutative 10.1%

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

       7. associate-*r/ 10.1%

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

       8. metadata-eval 10.1%

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

       9. mul-1-neg 10.1%

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

       10. unpow2 10.1%

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

   14. Simplified 10.1%

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

   if -380 < re < 310 or 1.6499999999999999e77 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 97.8%

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

      1. distribute-rgt-in 97.8%

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

      2. *-commutative 97.8%

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

      3. associate-+r+ 97.8%

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

      4. distribute-rgt1-in 97.8%

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

      5. *-commutative 97.8%

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

      6. *-commutative 97.8%

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

      7. associate-*l* 97.8%

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

      8. +-commutative 97.8%

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

      9. associate-*r* 97.8%

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

      10. distribute-rgt-out 97.8%

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

      11. associate-*l* 97.8%

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

      12. distribute-lft-out 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in im around 0 56.4%

      \[\leadsto \color{blue}{1 + \left(re + {re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. *-rgt-identity 56.4%

         \[\leadsto 1 + \left(\color{blue}{re \cdot 1} + {re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

      2. unpow2 56.4%

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

      3. +-commutative 56.4%

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

      4. *-commutative 56.4%

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

      5. associate-*l* 56.4%

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

      6. *-commutative 56.4%

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

      7. +-commutative 56.4%

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

      8. distribute-lft-in 56.4%

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

   8. Simplified 56.4%

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

   if 310 < re < 1.6499999999999999e77

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 4.0%

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

      1. *-rgt-identity 4.0%

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

      2. distribute-rgt-in 4.0%

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

      3. *-commutative 4.0%

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

      4. associate-*l* 4.0%

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

      5. *-commutative 4.0%

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

      6. associate-*l* 4.0%

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

      7. distribute-lft-out 4.0%

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

      8. distribute-lft-out 4.0%

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

      9. *-commutative 4.0%

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

      10. distribute-rgt1-in 4.0%

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

      11. +-commutative 4.0%

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

      12. *-commutative 4.0%

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

   5. Simplified 4.0%

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

   6. Taylor expanded in im around 0 16.3%

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

      1. +-commutative 16.3%

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

      2. distribute-rgt-in 16.3%

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

      3. *-commutative 16.3%

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

      4. distribute-rgt-in 16.3%

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

      5. associate-+l+ 16.3%

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

   8. Simplified 16.3%

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

      1. distribute-rgt-in 16.3%

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

      2. *-un-lft-identity 16.3%

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

      3. flip3-+ 38.1%

         \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \color{blue}{\frac{{re}^{3} + {\left(\left(re \cdot 0.5\right) \cdot re\right)}^{3}}{re \cdot re + \left(\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}}\right) \]

      4. cube-mult 38.1%

         \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{re \cdot \left(re \cdot re\right)} + {\left(\left(re \cdot 0.5\right) \cdot re\right)}^{3}}{re \cdot re + \left(\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

      5. *-commutative 38.1%

         \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)}}^{3}}{re \cdot re + \left(\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

      6. *-commutative 38.1%

         \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \cdot \left(\left(re \cdot 0.5\right) \cdot re\right) - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

      7. *-commutative 38.1%

         \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} - re \cdot \left(\left(re \cdot 0.5\right) \cdot re\right)\right)}\right) \]

      8. *-commutative 38.1%

         \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)}\right)}\right) \]

   10. Applied egg-rr 38.1%

       \[\leadsto im \cdot \left(im \cdot \left(-0.5 \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\right)\right) + \left(1 + \color{blue}{\frac{re \cdot \left(re \cdot re\right) + {\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. +-commutative 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{{\left(re \cdot \left(re \cdot 0.5\right)\right)}^{3} + re \cdot \left(re \cdot re\right)}}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       2. associate-*r* 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{{\color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)}}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       3. unpow2 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{{\left(\color{blue}{{re}^{2}} \cdot 0.5\right)}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       4. *-commutative 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{{\color{blue}{\left(0.5 \cdot {re}^{2}\right)}}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       5. cube-prod 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{{0.5}^{3} \cdot {\left({re}^{2}\right)}^{3}} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       6. metadata-eval 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{\color{blue}{0.125} \cdot {\left({re}^{2}\right)}^{3} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       7. cube-unmult 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right)} + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       8. unpow2 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       9. unpow2 38.1%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {re}^{2}\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       10. associate-*l* 38.1%

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot {re}^{2}\right)\right)}\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

       11. unpow2 38.1%

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(-0.5 \cdot \frac{im}{re} + im \cdot -0.25\right)\right) + \left(1 + \frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(\left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right) - re \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\right)}\right) \]

   12. Simplified 38.1%

       \[\leadsto im \cdot \left(im \cdot \left(-0.5 \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\right)\right) + \left(1 + \color{blue}{\frac{0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right) + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot 0.25\right) - re \cdot 0.5\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -380:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot \left(\frac{0.5}{-re} - 0.25\right)\right)\\ \mathbf{elif}\;re \leq 310 \lor \neg \left(re \leq 1.65 \cdot 10^{+77}\right):\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{re \cdot \left(re \cdot re\right) + 0.125 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)}{re \cdot re + \left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot 0.25\right) - re \cdot 0.5\right)}\right) + im \cdot \left(im \cdot \left(-0.5 \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 305:\\ \;\;\;\;1\\ \mathbf{elif}\;re \leq 1.18 \cdot 10^{+157}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 305.0)
   1.0
   (if (<= re 1.18e+157)
     (* re (+ 1.0 (* -0.5 (* im im))))
     (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 305.0) {
		tmp = 1.0;
	} else if (re <= 1.18e+157) {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	} else {
		tmp = 1.0 + (re * (1.0 + (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 (re <= 305.0d0) then
        tmp = 1.0d0
    else if (re <= 1.18d+157) then
        tmp = re * (1.0d0 + ((-0.5d0) * (im * im)))
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 305.0) {
		tmp = 1.0;
	} else if (re <= 1.18e+157) {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 305.0:
		tmp = 1.0
	elif re <= 1.18e+157:
		tmp = re * (1.0 + (-0.5 * (im * im)))
	else:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 305.0)
		tmp = 1.0;
	elseif (re <= 1.18e+157)
		tmp = Float64(re * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 305.0)
		tmp = 1.0;
	elseif (re <= 1.18e+157)
		tmp = re * (1.0 + (-0.5 * (im * im)));
	else
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 305.0], 1.0, If[LessEqual[re, 1.18e+157], N[(re * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 305

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 63.7%

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

      1. *-rgt-identity 63.7%

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

      2. distribute-rgt-in 63.7%

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

      3. *-commutative 63.7%

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

      4. associate-*l* 63.7%

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

      5. *-commutative 63.7%

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

      6. associate-*l* 63.7%

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

      7. distribute-lft-out 63.7%

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

      8. distribute-lft-out 63.7%

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

      9. *-commutative 63.7%

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

      10. distribute-rgt1-in 63.7%

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

      11. +-commutative 63.7%

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

      12. *-commutative 63.7%

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

   5. Simplified 63.7%

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

   6. Taylor expanded in im around 0 30.5%

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

      1. +-commutative 30.5%

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

      2. distribute-rgt-in 30.5%

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

      3. *-commutative 30.5%

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

      4. distribute-rgt-in 30.5%

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

      5. associate-+l+ 30.5%

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

   8. Simplified 30.5%

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

   9. Taylor expanded in re around inf 30.5%

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

       1. unpow2 30.5%

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

       2. *-commutative 30.5%

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

   11. Simplified 30.5%

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

   12. Taylor expanded in re around 0 34.0%

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

   if 305 < re < 1.1799999999999999e157

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 3.8%

      \[\leadsto \color{blue}{\cos im + re \cdot \cos im} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in 3.8%

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

      2. *-commutative 3.8%

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

   5. Simplified 3.8%

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

   6. Taylor expanded in re around inf 3.8%

      \[\leadsto \color{blue}{re \cdot \cos im} \]
   7. Step-by-step derivation

      1. *-commutative 3.8%

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

   8. Simplified 3.8%

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

   9. Taylor expanded in im around 0 18.5%

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

       1. *-lft-identity 18.5%

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

       2. associate-*r* 18.5%

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

       3. distribute-rgt-in 18.5%

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

       4. unpow2 18.5%

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

   11. Simplified 18.5%

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

   if 1.1799999999999999e157 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-rgt1-in 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 71.9%

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

      1. distribute-rgt-in 71.9%

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

      2. *-commutative 71.9%

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

      3. distribute-rgt-in 71.9%

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

   8. Simplified 71.9%

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

Alternative 14: 40.2% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 260:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 260.0) 1.0 (* (* re re) (+ 0.5 (* (* im im) -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 260.0) {
		tmp = 1.0;
	} else {
		tmp = (re * re) * (0.5 + ((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 (re <= 260.0d0) then
        tmp = 1.0d0
    else
        tmp = (re * re) * (0.5d0 + ((im * im) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 260.0) {
		tmp = 1.0;
	} else {
		tmp = (re * re) * (0.5 + ((im * im) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 260.0:
		tmp = 1.0
	else:
		tmp = (re * re) * (0.5 + ((im * im) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 260.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(re * re) * Float64(0.5 + Float64(Float64(im * im) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 260.0)
		tmp = 1.0;
	else
		tmp = (re * re) * (0.5 + ((im * im) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 260.0], 1.0, N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 260

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 63.7%

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

      1. *-rgt-identity 63.7%

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

      2. distribute-rgt-in 63.7%

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

      3. *-commutative 63.7%

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

      4. associate-*l* 63.7%

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

      5. *-commutative 63.7%

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

      6. associate-*l* 63.7%

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

      7. distribute-lft-out 63.7%

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

      8. distribute-lft-out 63.7%

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

      9. *-commutative 63.7%

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

      10. distribute-rgt1-in 63.7%

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

      11. +-commutative 63.7%

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

      12. *-commutative 63.7%

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

   5. Simplified 63.7%

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

   6. Taylor expanded in im around 0 30.5%

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

      1. +-commutative 30.5%

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

      2. distribute-rgt-in 30.5%

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

      3. *-commutative 30.5%

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

      4. distribute-rgt-in 30.5%

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

      5. associate-+l+ 30.5%

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

   8. Simplified 30.5%

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

   9. Taylor expanded in re around inf 30.5%

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

       1. unpow2 30.5%

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

       2. *-commutative 30.5%

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

   11. Simplified 30.5%

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

   12. Taylor expanded in re around 0 34.0%

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

   if 260 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 51.8%

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

      1. *-rgt-identity 51.8%

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

      2. distribute-rgt-in 51.8%

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

      3. *-commutative 51.8%

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

      4. associate-*l* 51.8%

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

      5. *-commutative 51.8%

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

      6. associate-*l* 51.8%

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

      7. distribute-lft-out 51.8%

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

      8. distribute-lft-out 51.8%

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

      9. *-commutative 51.8%

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

      10. distribute-rgt1-in 51.8%

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

      11. +-commutative 51.8%

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

      12. *-commutative 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in im around 0 7.3%

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

      1. +-commutative 7.3%

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

      2. distribute-rgt-in 7.3%

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

      3. *-commutative 7.3%

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

      4. distribute-rgt-in 7.3%

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

      5. associate-+l+ 7.3%

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

   8. Simplified 7.3%

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

   9. Taylor expanded in re around inf 43.5%

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

       1. *-commutative 43.5%

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

       2. *-commutative 43.5%

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

       3. unpow2 43.5%

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

       4. unpow2 43.5%

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

   11. Simplified 43.5%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 260:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 31.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 31.5) 1.0 (* re (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 31.5) {
		tmp = 1.0;
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 31.5d0) then
        tmp = 1.0d0
    else
        tmp = re * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 31.5) {
		tmp = 1.0;
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 31.5:
		tmp = 1.0
	else:
		tmp = re * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 31.5)
		tmp = 1.0;
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 31.5)
		tmp = 1.0;
	else
		tmp = re * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 31.5], 1.0, N[(re * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 31.5

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 63.7%

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

      1. *-rgt-identity 63.7%

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

      2. distribute-rgt-in 63.7%

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

      3. *-commutative 63.7%

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

      4. associate-*l* 63.7%

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

      5. *-commutative 63.7%

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

      6. associate-*l* 63.7%

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

      7. distribute-lft-out 63.7%

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

      8. distribute-lft-out 63.7%

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

      9. *-commutative 63.7%

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

      10. distribute-rgt1-in 63.7%

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

      11. +-commutative 63.7%

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

      12. *-commutative 63.7%

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

   5. Simplified 63.7%

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

   6. Taylor expanded in im around 0 30.5%

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

      1. +-commutative 30.5%

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

      2. distribute-rgt-in 30.5%

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

      3. *-commutative 30.5%

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

      4. distribute-rgt-in 30.5%

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

      5. associate-+l+ 30.5%

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

   8. Simplified 30.5%

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

   9. Taylor expanded in re around inf 30.5%

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

       1. unpow2 30.5%

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

       2. *-commutative 30.5%

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

   11. Simplified 30.5%

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

   12. Taylor expanded in re around 0 34.0%

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

   if 31.5 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 5.4%

      \[\leadsto \color{blue}{\cos im + re \cdot \cos im} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in 5.4%

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

      2. *-commutative 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in re around inf 5.4%

      \[\leadsto \color{blue}{re \cdot \cos im} \]
   7. Step-by-step derivation

      1. *-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in im around 0 24.5%

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

       1. *-lft-identity 24.5%

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

       2. associate-*r* 24.5%

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

       3. distribute-rgt-in 24.5%

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

       4. unpow2 24.5%

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

   11. Simplified 24.5%

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

Alternative 16: 41.5% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ 1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	return 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
end function

Java

public static double code(double re, double im) {
	return 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
}

Python

def code(re, im):
	return 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))

Julia

function code(re, im)
	return Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 64.3%

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

   1. distribute-rgt-in 64.3%

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

   2. *-commutative 64.3%

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

   3. associate-+r+ 64.3%

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

   4. distribute-rgt1-in 64.3%

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

   5. *-commutative 64.3%

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

   6. *-commutative 64.3%

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

   7. associate-*l* 64.3%

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

   8. +-commutative 64.3%

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

   9. associate-*r* 64.3%

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

   10. distribute-rgt-out 64.3%

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

   11. associate-*l* 64.3%

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

   12. distribute-lft-out 64.3%

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

5. Simplified 64.3%

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

6. Taylor expanded in im around 0 37.4%

   \[\leadsto \color{blue}{1 + \left(re + {re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right)} \]
7. Step-by-step derivation

   1. *-rgt-identity 37.4%

      \[\leadsto 1 + \left(\color{blue}{re \cdot 1} + {re}^{2} \cdot \left(0.5 + 0.16666666666666666 \cdot re\right)\right) \]

   2. unpow2 37.4%

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

   3. +-commutative 37.4%

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

   4. *-commutative 37.4%

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

   5. associate-*l* 37.4%

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

   6. *-commutative 37.4%

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

   7. +-commutative 37.4%

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

   8. distribute-lft-in 37.4%

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

8. Simplified 37.4%

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

9. Final simplification 37.4%

   \[\leadsto 1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \]
10. Add Preprocessing

Alternative 17: 31.7% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 75:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 75.0) 1.0 (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 75.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 75.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((-0.5d0) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 75.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 75.0:
		tmp = 1.0
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 75.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 75.0)
		tmp = 1.0;
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 75.0], 1.0, N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 75

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 63.7%

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

      1. *-rgt-identity 63.7%

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

      2. distribute-rgt-in 63.7%

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

      3. *-commutative 63.7%

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

      4. associate-*l* 63.7%

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

      5. *-commutative 63.7%

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

      6. associate-*l* 63.7%

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

      7. distribute-lft-out 63.7%

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

      8. distribute-lft-out 63.7%

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

      9. *-commutative 63.7%

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

      10. distribute-rgt1-in 63.7%

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

      11. +-commutative 63.7%

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

      12. *-commutative 63.7%

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

   5. Simplified 63.7%

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

   6. Taylor expanded in im around 0 30.5%

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

      1. +-commutative 30.5%

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

      2. distribute-rgt-in 30.5%

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

      3. *-commutative 30.5%

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

      4. distribute-rgt-in 30.5%

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

      5. associate-+l+ 30.5%

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

   8. Simplified 30.5%

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

   9. Taylor expanded in re around inf 30.5%

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

       1. unpow2 30.5%

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

       2. *-commutative 30.5%

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

   11. Simplified 30.5%

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

   12. Taylor expanded in re around 0 34.0%

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

   if 75 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 51.8%

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

      1. *-rgt-identity 51.8%

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

      2. distribute-rgt-in 51.8%

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

      3. *-commutative 51.8%

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

      4. associate-*l* 51.8%

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

      5. *-commutative 51.8%

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

      6. associate-*l* 51.8%

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

      7. distribute-lft-out 51.8%

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

      8. distribute-lft-out 51.8%

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

      9. *-commutative 51.8%

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

      10. distribute-rgt1-in 51.8%

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

      11. +-commutative 51.8%

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

      12. *-commutative 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in im around 0 7.3%

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

      1. +-commutative 7.3%

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

      2. distribute-rgt-in 7.3%

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

      3. *-commutative 7.3%

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

      4. distribute-rgt-in 7.3%

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

      5. associate-+l+ 7.3%

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

   8. Simplified 7.3%

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

   9. Taylor expanded in re around 0 18.0%

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

       1. unpow2 18.0%

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

   11. Simplified 18.0%

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

Alternative 18: 29.0% accurate, 67.7× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (+ re 1.0))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re + 1.0

Julia

function code(re, im)
	return Float64(re + 1.0)
end

MATLAB

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

Wolfram

code[re_, im_] := N[(re + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 48.0%

   \[\leadsto \color{blue}{\cos im + re \cdot \cos im} \]
4. Step-by-step derivation

   1. distribute-rgt1-in 48.0%

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

   2. *-commutative 48.0%

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

5. Simplified 48.0%

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

6. Taylor expanded in im around 0 25.7%

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

   1. +-commutative 25.7%

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

8. Simplified 25.7%

   \[\leadsto \color{blue}{re + 1} \]
9. Add Preprocessing

Alternative 19: 28.5% accurate, 203.0× 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%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 60.5%

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

   1. *-rgt-identity 60.5%

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

   2. distribute-rgt-in 60.5%

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

   3. *-commutative 60.5%

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

   4. associate-*l* 60.5%

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

   5. *-commutative 60.5%

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

   6. associate-*l* 60.5%

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

   7. distribute-lft-out 60.5%

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

   8. distribute-lft-out 60.5%

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

   9. *-commutative 60.5%

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

   10. distribute-rgt1-in 60.5%

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

   11. +-commutative 60.5%

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

   12. *-commutative 60.5%

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

5. Simplified 60.5%

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

6. Taylor expanded in im around 0 24.2%

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

   1. +-commutative 24.2%

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

   2. distribute-rgt-in 24.2%

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

   3. *-commutative 24.2%

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

   4. distribute-rgt-in 24.2%

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

   5. associate-+l+ 24.2%

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

8. Simplified 24.2%

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

9. Taylor expanded in re around inf 24.3%

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

    1. unpow2 24.3%

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

    2. *-commutative 24.3%

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

11. Simplified 24.3%

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

12. Taylor expanded in re around 0 25.5%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))