math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 21.3s

Alternatives: 26

Speedup: 1.0×

• 25.8% 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: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;re \leq -0.038:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.047:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (cos im)
          (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))
   (if (<= re -0.038)
     (exp re)
     (if (<= re 0.047) t_0 (if (<= re 1.02e+103) (exp re) t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.038) {
		tmp = exp(re);
	} else if (re <= 0.047) {
		tmp = t_0;
	} else if (re <= 1.02e+103) {
		tmp = exp(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    if (re <= (-0.038d0)) then
        tmp = exp(re)
    else if (re <= 0.047d0) then
        tmp = t_0
    else if (re <= 1.02d+103) then
        tmp = exp(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.038) {
		tmp = Math.exp(re);
	} else if (re <= 0.047) {
		tmp = t_0;
	} else if (re <= 1.02e+103) {
		tmp = Math.exp(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	tmp = 0
	if re <= -0.038:
		tmp = math.exp(re)
	elif re <= 0.047:
		tmp = t_0
	elif re <= 1.02e+103:
		tmp = math.exp(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
	tmp = 0.0
	if (re <= -0.038)
		tmp = exp(re);
	elseif (re <= 0.047)
		tmp = t_0;
	elseif (re <= 1.02e+103)
		tmp = exp(re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	tmp = 0.0;
	if (re <= -0.038)
		tmp = exp(re);
	elseif (re <= 0.047)
		tmp = t_0;
	elseif (re <= 1.02e+103)
		tmp = exp(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.038], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.047], t$95$0, If[LessEqual[re, 1.02e+103], N[Exp[re], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0379999999999999991 or 0.047 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 96.1%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 96.1%

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

   if -0.0379999999999999991 < re < 0.047 or 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.038:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.047:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -0.015:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.027:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))
   (if (<= re -0.015)
     (exp re)
     (if (<= re 0.027) t_0 (if (<= re 1.85e+154) (exp re) t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.015) {
		tmp = exp(re);
	} else if (re <= 0.027) {
		tmp = t_0;
	} else if (re <= 1.85e+154) {
		tmp = exp(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    if (re <= (-0.015d0)) then
        tmp = exp(re)
    else if (re <= 0.027d0) then
        tmp = t_0
    else if (re <= 1.85d+154) then
        tmp = exp(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.015) {
		tmp = Math.exp(re);
	} else if (re <= 0.027) {
		tmp = t_0;
	} else if (re <= 1.85e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	tmp = 0
	if re <= -0.015:
		tmp = math.exp(re)
	elif re <= 0.027:
		tmp = t_0
	elif re <= 1.85e+154:
		tmp = math.exp(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))))
	tmp = 0.0
	if (re <= -0.015)
		tmp = exp(re);
	elseif (re <= 0.027)
		tmp = t_0;
	elseif (re <= 1.85e+154)
		tmp = exp(re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	tmp = 0.0;
	if (re <= -0.015)
		tmp = exp(re);
	elseif (re <= 0.027)
		tmp = t_0;
	elseif (re <= 1.85e+154)
		tmp = exp(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$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, -0.015], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.027], t$95$0, If[LessEqual[re, 1.85e+154], N[Exp[re], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.014999999999999999 or 0.0269999999999999997 < re < 1.84999999999999997e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 93.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 93.9%

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

   if -0.014999999999999999 < re < 0.0269999999999999997 or 1.84999999999999997e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. *-lowering-*.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 99.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.015:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.027:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00031:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.024:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+47}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00031)
   (exp re)
   (if (<= re 0.024)
     (* (cos im) (+ re 1.0))
     (if (<= re 3e+47) (exp re) (* (exp re) (+ 1.0 (* -0.5 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00031) {
		tmp = exp(re);
	} else if (re <= 0.024) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 3e+47) {
		tmp = exp(re);
	} else {
		tmp = exp(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 <= (-0.00031d0)) then
        tmp = exp(re)
    else if (re <= 0.024d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 3d+47) then
        tmp = exp(re)
    else
        tmp = exp(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 <= -0.00031) {
		tmp = Math.exp(re);
	} else if (re <= 0.024) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 3e+47) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00031:
		tmp = math.exp(re)
	elif re <= 0.024:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 3e+47:
		tmp = math.exp(re)
	else:
		tmp = math.exp(re) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00031)
		tmp = exp(re);
	elseif (re <= 0.024)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 3e+47)
		tmp = exp(re);
	else
		tmp = Float64(exp(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 <= -0.00031)
		tmp = exp(re);
	elseif (re <= 0.024)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 3e+47)
		tmp = exp(re);
	else
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00031], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.024], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3e+47], N[Exp[re], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.1e-4 or 0.024 < re < 3.0000000000000001e47

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 98.5%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 98.5%

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

   if -3.1e-4 < re < 0.024

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 99.4%

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

   if 3.0000000000000001e47 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00031:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.024:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+47}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00034:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.024:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+121}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00034)
   (exp re)
   (if (<= re 0.024)
     (* (cos im) (+ re 1.0))
     (if (<= re 6e+121)
       (exp re)
       (*
        (+ 1.0 (* -0.5 (* im im)))
        (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00034) {
		tmp = exp(re);
	} else if (re <= 0.024) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 6e+121) {
		tmp = exp(re);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / 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 <= (-0.00034d0)) then
        tmp = exp(re)
    else if (re <= 0.024d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 6d+121) then
        tmp = exp(re)
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00034) {
		tmp = Math.exp(re);
	} else if (re <= 0.024) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 6e+121) {
		tmp = Math.exp(re);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00034:
		tmp = math.exp(re)
	elif re <= 0.024:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 6e+121:
		tmp = math.exp(re)
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00034)
		tmp = exp(re);
	elseif (re <= 0.024)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 6e+121)
		tmp = exp(re);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00034)
		tmp = exp(re);
	elseif (re <= 0.024)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 6e+121)
		tmp = exp(re);
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00034], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.024], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6e+121], N[Exp[re], $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.4e-4 or 0.024 < re < 6.0000000000000005e121

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 95.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 95.9%

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

   if -3.4e-4 < re < 0.024

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 99.4%

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

   if 6.0000000000000005e121 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 83.8%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 83.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 83.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00034:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.024:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+121}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.4 \cdot 10^{-8}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.024:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 10^{+122}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.4e-8)
   (exp re)
   (if (<= re 0.024)
     (cos im)
     (if (<= re 1e+122)
       (exp re)
       (*
        (+ 1.0 (* -0.5 (* im im)))
        (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.4e-8) {
		tmp = exp(re);
	} else if (re <= 0.024) {
		tmp = cos(im);
	} else if (re <= 1e+122) {
		tmp = exp(re);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / 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 <= (-5.4d-8)) then
        tmp = exp(re)
    else if (re <= 0.024d0) then
        tmp = cos(im)
    else if (re <= 1d+122) then
        tmp = exp(re)
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.4e-8) {
		tmp = Math.exp(re);
	} else if (re <= 0.024) {
		tmp = Math.cos(im);
	} else if (re <= 1e+122) {
		tmp = Math.exp(re);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.4e-8:
		tmp = math.exp(re)
	elif re <= 0.024:
		tmp = math.cos(im)
	elif re <= 1e+122:
		tmp = math.exp(re)
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.4e-8)
		tmp = exp(re);
	elseif (re <= 0.024)
		tmp = cos(im);
	elseif (re <= 1e+122)
		tmp = exp(re);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.4e-8)
		tmp = exp(re);
	elseif (re <= 0.024)
		tmp = cos(im);
	elseif (re <= 1e+122)
		tmp = exp(re);
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.4e-8], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.024], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1e+122], N[Exp[re], $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.40000000000000005e-8 or 0.024 < re < 1.00000000000000001e122

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 95.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 95.9%

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

   if -5.40000000000000005e-8 < re < 0.024

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 99.1%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 99.1%

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

   if 1.00000000000000001e122 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 83.8%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 83.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 83.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.4 \cdot 10^{-8}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.024:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 10^{+122}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re + \left(1 - re\right)\\ t_1 := re \cdot \left(re \cdot re\right)\\ t_2 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\\ t_3 := re \cdot \left(-1 - t\_2\right)\\ \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{t\_0}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 0.062:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\left(1 + t\_1 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot t\_1\right)\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)}{1 + t\_1 \cdot \left(t\_1 + -1\right)}}{t\_0}\\ \mathbf{elif}\;re \leq 4.45 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(1 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25\right) \cdot \left(1 + re \cdot \left(\left(1 + t\_2\right) \cdot t\_3\right)\right)}{\left(1 - im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 + t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* re re) (- 1.0 re)))
        (t_1 (* re (* re re)))
        (t_2 (* re (+ 0.5 (* re 0.16666666666666666))))
        (t_3 (* re (- -1.0 t_2))))
   (if (<= re -1.34e+154)
     (/ (+ 1.0 (* (* im im) (+ -0.5 (* (* im im) 0.041666666666666664)))) t_0)
     (if (<= re -600000.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (if (<= re 0.062)
         (cos im)
         (if (<= re 2.35e+51)
           (/
            (/
             (*
              (+ 1.0 (* t_1 (* (* re re) (* re t_1))))
              (+
               1.0
               (* (* im im) (+ -0.5 (* im (* im 0.041666666666666664))))))
             (+ 1.0 (* t_1 (+ t_1 -1.0))))
            t_0)
           (if (<= re 4.45e+102)
             (/
              (*
               (- 1.0 (* (* im (* im (* im im))) 0.25))
               (+ 1.0 (* re (* (+ 1.0 t_2) t_3))))
              (* (- 1.0 (* im (* im -0.5))) (+ 1.0 t_3)))
             (*
              (+ 1.0 (* -0.5 (* im im)))
              (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re))))))))))))

C

double code(double re, double im) {
	double t_0 = (re * re) + (1.0 - re);
	double t_1 = re * (re * re);
	double t_2 = re * (0.5 + (re * 0.16666666666666666));
	double t_3 = re * (-1.0 - t_2);
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0;
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 0.062) {
		tmp = cos(im);
	} else if (re <= 2.35e+51) {
		tmp = (((1.0 + (t_1 * ((re * re) * (re * t_1)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_1 * (t_1 + -1.0)))) / t_0;
	} else if (re <= 4.45e+102) {
		tmp = ((1.0 - ((im * (im * (im * im))) * 0.25)) * (1.0 + (re * ((1.0 + t_2) * t_3)))) / ((1.0 - (im * (im * -0.5))) * (1.0 + t_3));
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (re * re) + (1.0d0 - re)
    t_1 = re * (re * re)
    t_2 = re * (0.5d0 + (re * 0.16666666666666666d0))
    t_3 = re * ((-1.0d0) - t_2)
    if (re <= (-1.34d+154)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + ((im * im) * 0.041666666666666664d0)))) / t_0
    else if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 0.062d0) then
        tmp = cos(im)
    else if (re <= 2.35d+51) then
        tmp = (((1.0d0 + (t_1 * ((re * re) * (re * t_1)))) * (1.0d0 + ((im * im) * ((-0.5d0) + (im * (im * 0.041666666666666664d0)))))) / (1.0d0 + (t_1 * (t_1 + (-1.0d0))))) / t_0
    else if (re <= 4.45d+102) then
        tmp = ((1.0d0 - ((im * (im * (im * im))) * 0.25d0)) * (1.0d0 + (re * ((1.0d0 + t_2) * t_3)))) / ((1.0d0 - (im * (im * (-0.5d0)))) * (1.0d0 + t_3))
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) + (1.0 - re);
	double t_1 = re * (re * re);
	double t_2 = re * (0.5 + (re * 0.16666666666666666));
	double t_3 = re * (-1.0 - t_2);
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0;
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 0.062) {
		tmp = Math.cos(im);
	} else if (re <= 2.35e+51) {
		tmp = (((1.0 + (t_1 * ((re * re) * (re * t_1)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_1 * (t_1 + -1.0)))) / t_0;
	} else if (re <= 4.45e+102) {
		tmp = ((1.0 - ((im * (im * (im * im))) * 0.25)) * (1.0 + (re * ((1.0 + t_2) * t_3)))) / ((1.0 - (im * (im * -0.5))) * (1.0 + t_3));
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) + (1.0 - re)
	t_1 = re * (re * re)
	t_2 = re * (0.5 + (re * 0.16666666666666666))
	t_3 = re * (-1.0 - t_2)
	tmp = 0
	if re <= -1.34e+154:
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0
	elif re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 0.062:
		tmp = math.cos(im)
	elif re <= 2.35e+51:
		tmp = (((1.0 + (t_1 * ((re * re) * (re * t_1)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_1 * (t_1 + -1.0)))) / t_0
	elif re <= 4.45e+102:
		tmp = ((1.0 - ((im * (im * (im * im))) * 0.25)) * (1.0 + (re * ((1.0 + t_2) * t_3)))) / ((1.0 - (im * (im * -0.5))) * (1.0 + t_3))
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) + Float64(1.0 - re))
	t_1 = Float64(re * Float64(re * re))
	t_2 = Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))
	t_3 = Float64(re * Float64(-1.0 - t_2))
	tmp = 0.0
	if (re <= -1.34e+154)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(Float64(im * im) * 0.041666666666666664)))) / t_0);
	elseif (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 0.062)
		tmp = cos(im);
	elseif (re <= 2.35e+51)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(t_1 * Float64(Float64(re * re) * Float64(re * t_1)))) * Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(im * Float64(im * 0.041666666666666664)))))) / Float64(1.0 + Float64(t_1 * Float64(t_1 + -1.0)))) / t_0);
	elseif (re <= 4.45e+102)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(im * Float64(im * Float64(im * im))) * 0.25)) * Float64(1.0 + Float64(re * Float64(Float64(1.0 + t_2) * t_3)))) / Float64(Float64(1.0 - Float64(im * Float64(im * -0.5))) * Float64(1.0 + t_3)));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) + (1.0 - re);
	t_1 = re * (re * re);
	t_2 = re * (0.5 + (re * 0.16666666666666666));
	t_3 = re * (-1.0 - t_2);
	tmp = 0.0;
	if (re <= -1.34e+154)
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0;
	elseif (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 0.062)
		tmp = cos(im);
	elseif (re <= 2.35e+51)
		tmp = (((1.0 + (t_1 * ((re * re) * (re * t_1)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_1 * (t_1 + -1.0)))) / t_0;
	elseif (re <= 4.45e+102)
		tmp = ((1.0 - ((im * (im * (im * im))) * 0.25)) * (1.0 + (re * ((1.0 + t_2) * t_3)))) / ((1.0 - (im * (im * -0.5))) * (1.0 + t_3));
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] + N[(1.0 - re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(re * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.34e+154], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.062], N[Cos[im], $MachinePrecision], If[LessEqual[re, 2.35e+51], N[(N[(N[(N[(1.0 + N[(t$95$1 * N[(N[(re * re), $MachinePrecision] * N[(re * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[re, 4.45e+102], N[(N[(N[(1.0 - N[(N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(N[(1.0 + t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if re < -1.34000000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 1.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 1.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 1.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, re\right)}, \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{re}\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       12. *-lowering-*.f64 52.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

   13. Simplified 52.4%

       \[\leadsto \frac{\color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}}{re \cdot re + \left(1 - re\right)} \]

   if -1.34000000000000001e154 < re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 61.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 61.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 0.062

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 96.9%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 96.9%

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

   if 0.062 < re < 2.3500000000000001e51

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 6.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 6.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 26.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 26.5%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 31.4%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   12. Applied egg-rr 51.8%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)}{1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(-1 + re \cdot \left(re \cdot re\right)\right)}}}{re \cdot re + \left(1 - re\right)} \]

   if 2.3500000000000001e51 < re < 4.4499999999999999e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 85.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 6.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 6.4%

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

   10. Applied egg-rr 85.7%

       \[\leadsto \color{blue}{\frac{\left(1 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25\right) \cdot \left(1 - re \cdot \left(\left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)\right)}{\left(1 - im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 - re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)}} \]

   if 4.4499999999999999e102 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 82.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 82.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 82.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 82.5%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 82.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 82.5%

       \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 0.062:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)}{1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(re \cdot \left(re \cdot re\right) + -1\right)}}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq 4.45 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(1 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25\right) \cdot \left(1 + re \cdot \left(\left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \cdot \left(re \cdot \left(-1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)\right)}{\left(1 - im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 + re \cdot \left(-1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\\ t_1 := re \cdot \left(-1 - t\_0\right)\\ t_2 := re \cdot re + \left(1 - re\right)\\ t_3 := re \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{t\_2}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\left(1 + t\_3 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot t\_3\right)\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)}{1 + t\_3 \cdot \left(t\_3 + -1\right)}}{t\_2}\\ \mathbf{elif}\;re \leq 3.85 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(1 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25\right) \cdot \left(1 + re \cdot \left(\left(1 + t\_0\right) \cdot t\_1\right)\right)}{\left(1 - im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* re 0.16666666666666666))))
        (t_1 (* re (- -1.0 t_0)))
        (t_2 (+ (* re re) (- 1.0 re)))
        (t_3 (* re (* re re))))
   (if (<= re -1.34e+154)
     (/ (+ 1.0 (* (* im im) (+ -0.5 (* (* im im) 0.041666666666666664)))) t_2)
     (if (<= re -600000.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (if (<= re 2.35e+51)
         (/
          (/
           (*
            (+ 1.0 (* t_3 (* (* re re) (* re t_3))))
            (+ 1.0 (* (* im im) (+ -0.5 (* im (* im 0.041666666666666664))))))
           (+ 1.0 (* t_3 (+ t_3 -1.0))))
          t_2)
         (if (<= re 3.85e+102)
           (/
            (*
             (- 1.0 (* (* im (* im (* im im))) 0.25))
             (+ 1.0 (* re (* (+ 1.0 t_0) t_1))))
            (* (- 1.0 (* im (* im -0.5))) (+ 1.0 t_1)))
           (*
            (+ 1.0 (* -0.5 (* im im)))
            (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re)))))))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = re * (-1.0 - t_0);
	double t_2 = (re * re) + (1.0 - re);
	double t_3 = re * (re * re);
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_2;
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 2.35e+51) {
		tmp = (((1.0 + (t_3 * ((re * re) * (re * t_3)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_3 * (t_3 + -1.0)))) / t_2;
	} else if (re <= 3.85e+102) {
		tmp = ((1.0 - ((im * (im * (im * im))) * 0.25)) * (1.0 + (re * ((1.0 + t_0) * t_1)))) / ((1.0 - (im * (im * -0.5))) * (1.0 + t_1));
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = re * (0.5d0 + (re * 0.16666666666666666d0))
    t_1 = re * ((-1.0d0) - t_0)
    t_2 = (re * re) + (1.0d0 - re)
    t_3 = re * (re * re)
    if (re <= (-1.34d+154)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + ((im * im) * 0.041666666666666664d0)))) / t_2
    else if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 2.35d+51) then
        tmp = (((1.0d0 + (t_3 * ((re * re) * (re * t_3)))) * (1.0d0 + ((im * im) * ((-0.5d0) + (im * (im * 0.041666666666666664d0)))))) / (1.0d0 + (t_3 * (t_3 + (-1.0d0))))) / t_2
    else if (re <= 3.85d+102) then
        tmp = ((1.0d0 - ((im * (im * (im * im))) * 0.25d0)) * (1.0d0 + (re * ((1.0d0 + t_0) * t_1)))) / ((1.0d0 - (im * (im * (-0.5d0)))) * (1.0d0 + t_1))
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = re * (-1.0 - t_0);
	double t_2 = (re * re) + (1.0 - re);
	double t_3 = re * (re * re);
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_2;
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 2.35e+51) {
		tmp = (((1.0 + (t_3 * ((re * re) * (re * t_3)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_3 * (t_3 + -1.0)))) / t_2;
	} else if (re <= 3.85e+102) {
		tmp = ((1.0 - ((im * (im * (im * im))) * 0.25)) * (1.0 + (re * ((1.0 + t_0) * t_1)))) / ((1.0 - (im * (im * -0.5))) * (1.0 + t_1));
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (re * 0.16666666666666666))
	t_1 = re * (-1.0 - t_0)
	t_2 = (re * re) + (1.0 - re)
	t_3 = re * (re * re)
	tmp = 0
	if re <= -1.34e+154:
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_2
	elif re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 2.35e+51:
		tmp = (((1.0 + (t_3 * ((re * re) * (re * t_3)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_3 * (t_3 + -1.0)))) / t_2
	elif re <= 3.85e+102:
		tmp = ((1.0 - ((im * (im * (im * im))) * 0.25)) * (1.0 + (re * ((1.0 + t_0) * t_1)))) / ((1.0 - (im * (im * -0.5))) * (1.0 + t_1))
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))
	t_1 = Float64(re * Float64(-1.0 - t_0))
	t_2 = Float64(Float64(re * re) + Float64(1.0 - re))
	t_3 = Float64(re * Float64(re * re))
	tmp = 0.0
	if (re <= -1.34e+154)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(Float64(im * im) * 0.041666666666666664)))) / t_2);
	elseif (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 2.35e+51)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(t_3 * Float64(Float64(re * re) * Float64(re * t_3)))) * Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(im * Float64(im * 0.041666666666666664)))))) / Float64(1.0 + Float64(t_3 * Float64(t_3 + -1.0)))) / t_2);
	elseif (re <= 3.85e+102)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(im * Float64(im * Float64(im * im))) * 0.25)) * Float64(1.0 + Float64(re * Float64(Float64(1.0 + t_0) * t_1)))) / Float64(Float64(1.0 - Float64(im * Float64(im * -0.5))) * Float64(1.0 + t_1)));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (re * 0.16666666666666666));
	t_1 = re * (-1.0 - t_0);
	t_2 = (re * re) + (1.0 - re);
	t_3 = re * (re * re);
	tmp = 0.0;
	if (re <= -1.34e+154)
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_2;
	elseif (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 2.35e+51)
		tmp = (((1.0 + (t_3 * ((re * re) * (re * t_3)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_3 * (t_3 + -1.0)))) / t_2;
	elseif (re <= 3.85e+102)
		tmp = ((1.0 - ((im * (im * (im * im))) * 0.25)) * (1.0 + (re * ((1.0 + t_0) * t_1)))) / ((1.0 - (im * (im * -0.5))) * (1.0 + t_1));
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(re * re), $MachinePrecision] + N[(1.0 - re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.34e+154], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.35e+51], N[(N[(N[(N[(1.0 + N[(t$95$3 * N[(N[(re * re), $MachinePrecision] * N[(re * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$3 * N[(t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[re, 3.85e+102], N[(N[(N[(1.0 - N[(N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -1.34000000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 1.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 1.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 1.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, re\right)}, \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{re}\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       12. *-lowering-*.f64 52.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

   13. Simplified 52.4%

       \[\leadsto \frac{\color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}}{re \cdot re + \left(1 - re\right)} \]

   if -1.34000000000000001e154 < re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 61.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 61.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 2.3500000000000001e51

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 84.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 84.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 43.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 43.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 44.4%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   12. Applied egg-rr 47.3%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)}{1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(-1 + re \cdot \left(re \cdot re\right)\right)}}}{re \cdot re + \left(1 - re\right)} \]

   if 2.3500000000000001e51 < re < 3.85000000000000007e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 85.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 6.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 6.4%

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

   10. Applied egg-rr 85.7%

       \[\leadsto \color{blue}{\frac{\left(1 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25\right) \cdot \left(1 - re \cdot \left(\left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)\right)}{\left(1 - im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 - re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)}} \]

   if 3.85000000000000007e102 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 82.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 82.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 82.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 82.5%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 82.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 82.5%

       \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)}{1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(re \cdot \left(re \cdot re\right) + -1\right)}}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq 3.85 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(1 - \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot 0.25\right) \cdot \left(1 + re \cdot \left(\left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \cdot \left(re \cdot \left(-1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)\right)}{\left(1 - im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 + re \cdot \left(-1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re + \left(1 - re\right)\\ t_1 := re \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{t\_0}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\left(1 + t\_1 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot t\_1\right)\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)}{1 + t\_1 \cdot \left(t\_1 + -1\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* re re) (- 1.0 re))) (t_1 (* re (* re re))))
   (if (<= re -1.34e+154)
     (/ (+ 1.0 (* (* im im) (+ -0.5 (* (* im im) 0.041666666666666664)))) t_0)
     (if (<= re -600000.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (if (<= re 1.15e+51)
         (/
          (/
           (*
            (+ 1.0 (* t_1 (* (* re re) (* re t_1))))
            (+ 1.0 (* (* im im) (+ -0.5 (* im (* im 0.041666666666666664))))))
           (+ 1.0 (* t_1 (+ t_1 -1.0))))
          t_0)
         (*
          (+ 1.0 (* -0.5 (* im im)))
          (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re))))))))))

C

double code(double re, double im) {
	double t_0 = (re * re) + (1.0 - re);
	double t_1 = re * (re * re);
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0;
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.15e+51) {
		tmp = (((1.0 + (t_1 * ((re * re) * (re * t_1)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_1 * (t_1 + -1.0)))) / t_0;
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (re * re) + (1.0d0 - re)
    t_1 = re * (re * re)
    if (re <= (-1.34d+154)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + ((im * im) * 0.041666666666666664d0)))) / t_0
    else if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 1.15d+51) then
        tmp = (((1.0d0 + (t_1 * ((re * re) * (re * t_1)))) * (1.0d0 + ((im * im) * ((-0.5d0) + (im * (im * 0.041666666666666664d0)))))) / (1.0d0 + (t_1 * (t_1 + (-1.0d0))))) / t_0
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) + (1.0 - re);
	double t_1 = re * (re * re);
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0;
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.15e+51) {
		tmp = (((1.0 + (t_1 * ((re * re) * (re * t_1)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_1 * (t_1 + -1.0)))) / t_0;
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) + (1.0 - re)
	t_1 = re * (re * re)
	tmp = 0
	if re <= -1.34e+154:
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0
	elif re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 1.15e+51:
		tmp = (((1.0 + (t_1 * ((re * re) * (re * t_1)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_1 * (t_1 + -1.0)))) / t_0
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) + Float64(1.0 - re))
	t_1 = Float64(re * Float64(re * re))
	tmp = 0.0
	if (re <= -1.34e+154)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(Float64(im * im) * 0.041666666666666664)))) / t_0);
	elseif (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 1.15e+51)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(t_1 * Float64(Float64(re * re) * Float64(re * t_1)))) * Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(im * Float64(im * 0.041666666666666664)))))) / Float64(1.0 + Float64(t_1 * Float64(t_1 + -1.0)))) / t_0);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) + (1.0 - re);
	t_1 = re * (re * re);
	tmp = 0.0;
	if (re <= -1.34e+154)
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0;
	elseif (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 1.15e+51)
		tmp = (((1.0 + (t_1 * ((re * re) * (re * t_1)))) * (1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664)))))) / (1.0 + (t_1 * (t_1 + -1.0)))) / t_0;
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] + N[(1.0 - re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.34e+154], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.15e+51], N[(N[(N[(N[(1.0 + N[(t$95$1 * N[(N[(re * re), $MachinePrecision] * N[(re * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.34000000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 1.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 1.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 1.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, re\right)}, \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{re}\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       12. *-lowering-*.f64 52.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

   13. Simplified 52.4%

       \[\leadsto \frac{\color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}}{re \cdot re + \left(1 - re\right)} \]

   if -1.34000000000000001e154 < re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 61.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 61.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 1.15000000000000003e51

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 85.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 85.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 44.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 44.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 44.7%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   12. Applied egg-rr 47.6%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)}{1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(-1 + re \cdot \left(re \cdot re\right)\right)}}}{re \cdot re + \left(1 - re\right)} \]

   if 1.15000000000000003e51 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 71.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 71.8%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 71.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 71.8%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)}{1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(re \cdot \left(re \cdot re\right) + -1\right)}}{re \cdot re + \left(1 - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + t\_0 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot t\_0\right)\right)}{\left(1 + t\_0 \cdot \left(t\_0 + -1\right)\right) \cdot \left(\left(1 + re \cdot re\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re re))))
   (if (<= re -1.34e+154)
     (/
      (+ 1.0 (* (* im im) (+ -0.5 (* (* im im) 0.041666666666666664))))
      (+ (* re re) (- 1.0 re)))
     (if (<= re -600000.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (if (<= re 3.5e+38)
         (/
          (+ 1.0 (* t_0 (* (* re re) (* re t_0))))
          (* (+ 1.0 (* t_0 (+ t_0 -1.0))) (- (+ 1.0 (* re re)) re)))
         (*
          (+ 1.0 (* -0.5 (* im im)))
          (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re))))))))))

C

double code(double re, double im) {
	double t_0 = re * (re * re);
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re));
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 3.5e+38) {
		tmp = (1.0 + (t_0 * ((re * re) * (re * t_0)))) / ((1.0 + (t_0 * (t_0 + -1.0))) * ((1.0 + (re * re)) - re));
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	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 <= (-1.34d+154)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + ((im * im) * 0.041666666666666664d0)))) / ((re * re) + (1.0d0 - re))
    else if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 3.5d+38) then
        tmp = (1.0d0 + (t_0 * ((re * re) * (re * t_0)))) / ((1.0d0 + (t_0 * (t_0 + (-1.0d0)))) * ((1.0d0 + (re * re)) - re))
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    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 <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re));
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 3.5e+38) {
		tmp = (1.0 + (t_0 * ((re * re) * (re * t_0)))) / ((1.0 + (t_0 * (t_0 + -1.0))) * ((1.0 + (re * re)) - re));
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * re)
	tmp = 0
	if re <= -1.34e+154:
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re))
	elif re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 3.5e+38:
		tmp = (1.0 + (t_0 * ((re * re) * (re * t_0)))) / ((1.0 + (t_0 * (t_0 + -1.0))) * ((1.0 + (re * re)) - re))
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * re))
	tmp = 0.0
	if (re <= -1.34e+154)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(Float64(im * im) * 0.041666666666666664)))) / Float64(Float64(re * re) + Float64(1.0 - re)));
	elseif (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 3.5e+38)
		tmp = Float64(Float64(1.0 + Float64(t_0 * Float64(Float64(re * re) * Float64(re * t_0)))) / Float64(Float64(1.0 + Float64(t_0 * Float64(t_0 + -1.0))) * Float64(Float64(1.0 + Float64(re * re)) - re)));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * re);
	tmp = 0.0;
	if (re <= -1.34e+154)
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re));
	elseif (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 3.5e+38)
		tmp = (1.0 + (t_0 * ((re * re) * (re * t_0)))) / ((1.0 + (t_0 * (t_0 + -1.0))) * ((1.0 + (re * re)) - re));
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.34e+154], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(re * re), $MachinePrecision] + N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.5e+38], N[(N[(1.0 + N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(t$95$0 * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(re * re), $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.34000000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 1.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 1.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 1.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, re\right)}, \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{re}\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       12. *-lowering-*.f64 52.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

   13. Simplified 52.4%

       \[\leadsto \frac{\color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}}{re \cdot re + \left(1 - re\right)} \]

   if -1.34000000000000001e154 < re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 61.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 61.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 3.50000000000000002e38

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 88.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 88.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 42.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 42.5%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 45.3%

          \[\leadsto \mathsf{+.f64}\left(re, \color{blue}{1}\right) \]

   11. Simplified 45.3%

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

   13. Applied egg-rr 47.6%

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

   if 3.50000000000000002e38 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 79.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 79.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 65.2%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 65.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 65.2%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)}{\left(1 + \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(re \cdot \left(re \cdot re\right) + -1\right)\right) \cdot \left(\left(1 + re \cdot re\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\\ t_1 := re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot t\_0}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{+47}:\\ \;\;\;\;t\_1 + \left(1 + \left(1 + t\_1\right) \cdot \left(im \cdot \left(im \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ -0.5 (* (* im im) 0.041666666666666664)))
        (t_1 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
   (if (<= re -1.34e+154)
     (/ (+ 1.0 (* (* im im) t_0)) (+ (* re re) (- 1.0 re)))
     (if (<= re -600000.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (if (<= re 3.3e+47)
         (+ t_1 (+ 1.0 (* (+ 1.0 t_1) (* im (* im t_0)))))
         (*
          (+ 1.0 (* -0.5 (* im im)))
          (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re))))))))))

C

double code(double re, double im) {
	double t_0 = -0.5 + ((im * im) * 0.041666666666666664);
	double t_1 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * t_0)) / ((re * re) + (1.0 - re));
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 3.3e+47) {
		tmp = t_1 + (1.0 + ((1.0 + t_1) * (im * (im * t_0))));
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) + ((im * im) * 0.041666666666666664d0)
    t_1 = re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))
    if (re <= (-1.34d+154)) then
        tmp = (1.0d0 + ((im * im) * t_0)) / ((re * re) + (1.0d0 - re))
    else if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 3.3d+47) then
        tmp = t_1 + (1.0d0 + ((1.0d0 + t_1) * (im * (im * t_0))))
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 + ((im * im) * 0.041666666666666664);
	double t_1 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * t_0)) / ((re * re) + (1.0 - re));
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 3.3e+47) {
		tmp = t_1 + (1.0 + ((1.0 + t_1) * (im * (im * t_0))));
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 + ((im * im) * 0.041666666666666664)
	t_1 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))
	tmp = 0
	if re <= -1.34e+154:
		tmp = (1.0 + ((im * im) * t_0)) / ((re * re) + (1.0 - re))
	elif re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 3.3e+47:
		tmp = t_1 + (1.0 + ((1.0 + t_1) * (im * (im * t_0))))
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 + Float64(Float64(im * im) * 0.041666666666666664))
	t_1 = Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))
	tmp = 0.0
	if (re <= -1.34e+154)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * t_0)) / Float64(Float64(re * re) + Float64(1.0 - re)));
	elseif (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 3.3e+47)
		tmp = Float64(t_1 + Float64(1.0 + Float64(Float64(1.0 + t_1) * Float64(im * Float64(im * t_0)))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 + ((im * im) * 0.041666666666666664);
	t_1 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	tmp = 0.0;
	if (re <= -1.34e+154)
		tmp = (1.0 + ((im * im) * t_0)) / ((re * re) + (1.0 - re));
	elseif (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 3.3e+47)
		tmp = t_1 + (1.0 + ((1.0 + t_1) * (im * (im * t_0))));
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.34e+154], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(re * re), $MachinePrecision] + N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.3e+47], N[(t$95$1 + N[(1.0 + N[(N[(1.0 + t$95$1), $MachinePrecision] * N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.34000000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 1.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 1.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 1.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, re\right)}, \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{re}\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       12. *-lowering-*.f64 52.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

   13. Simplified 52.4%

       \[\leadsto \frac{\color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}}{re \cdot re + \left(1 - re\right)} \]

   if -1.34000000000000001e154 < re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 61.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 61.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 3.2999999999999999e47

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 86.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 86.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{24} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{24} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right) + \color{blue}{{im}^{2}} \cdot \left(\frac{-1}{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{24} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{24} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{24} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)}\right) \]

   8. Simplified 45.6%

      \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) + \left(1 + \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\right)} \]

   if 3.2999999999999999e47 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 70.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 70.4%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 70.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 70.4%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{+47}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) + \left(1 + \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re + \left(1 - re\right)\\ t_1 := 1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\\ \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{t\_1}{t\_0}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{+17}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{t\_1 \cdot \left(re \cdot \left(re \cdot re\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* re re) (- 1.0 re)))
        (t_1
         (+ 1.0 (* (* im im) (+ -0.5 (* (* im im) 0.041666666666666664))))))
   (if (<= re -1.34e+154)
     (/ t_1 t_0)
     (if (<= re -600000.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (if (<= re 1.06e+17)
         (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
         (if (<= re 7.5e+121)
           (/ (* t_1 (* re (* re re))) t_0)
           (*
            (+ 1.0 (* -0.5 (* im im)))
            (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re)))))))))))

C

double code(double re, double im) {
	double t_0 = (re * re) + (1.0 - re);
	double t_1 = 1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)));
	double tmp;
	if (re <= -1.34e+154) {
		tmp = t_1 / t_0;
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.06e+17) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else if (re <= 7.5e+121) {
		tmp = (t_1 * (re * (re * re))) / t_0;
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (re * re) + (1.0d0 - re)
    t_1 = 1.0d0 + ((im * im) * ((-0.5d0) + ((im * im) * 0.041666666666666664d0)))
    if (re <= (-1.34d+154)) then
        tmp = t_1 / t_0
    else if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 1.06d+17) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    else if (re <= 7.5d+121) then
        tmp = (t_1 * (re * (re * re))) / t_0
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) + (1.0 - re);
	double t_1 = 1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)));
	double tmp;
	if (re <= -1.34e+154) {
		tmp = t_1 / t_0;
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.06e+17) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else if (re <= 7.5e+121) {
		tmp = (t_1 * (re * (re * re))) / t_0;
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) + (1.0 - re)
	t_1 = 1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))
	tmp = 0
	if re <= -1.34e+154:
		tmp = t_1 / t_0
	elif re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 1.06e+17:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	elif re <= 7.5e+121:
		tmp = (t_1 * (re * (re * re))) / t_0
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) + Float64(1.0 - re))
	t_1 = Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(Float64(im * im) * 0.041666666666666664))))
	tmp = 0.0
	if (re <= -1.34e+154)
		tmp = Float64(t_1 / t_0);
	elseif (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 1.06e+17)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	elseif (re <= 7.5e+121)
		tmp = Float64(Float64(t_1 * Float64(re * Float64(re * re))) / t_0);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) + (1.0 - re);
	t_1 = 1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)));
	tmp = 0.0;
	if (re <= -1.34e+154)
		tmp = t_1 / t_0;
	elseif (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 1.06e+17)
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	elseif (re <= 7.5e+121)
		tmp = (t_1 * (re * (re * re))) / t_0;
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] + N[(1.0 - re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.34e+154], N[(t$95$1 / t$95$0), $MachinePrecision], If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.06e+17], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.5e+121], N[(N[(t$95$1 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -1.34000000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 1.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 1.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 1.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, re\right)}, \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{re}\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       12. *-lowering-*.f64 52.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

   13. Simplified 52.4%

       \[\leadsto \frac{\color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}}{re \cdot re + \left(1 - re\right)} \]

   if -1.34000000000000001e154 < re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 61.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 61.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 1.06e17

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 53.4%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 53.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 48.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 48.5%

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

   if 1.06e17 < re < 7.49999999999999965e121

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 3.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 3.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 27.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 27.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({re}^{3}\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, re\right)}, \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(re \cdot \left(re \cdot re\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{re}, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(re \cdot {re}^{2}\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left({re}^{2}\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{re}, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot re\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{re}\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       18. *-lowering-*.f64 39.2%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

   13. Simplified 39.2%

       \[\leadsto \frac{\color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)}}{re \cdot re + \left(1 - re\right)} \]

   if 7.49999999999999965e121 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 83.8%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 83.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 83.8%

       \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{+17}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \cdot \left(re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.08 \cdot 10^{+26}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 1.36 \cdot 10^{+48}:\\ \;\;\;\;\frac{\left(1 + re \cdot \left(re \cdot re\right)\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)}{\left(1 + re \cdot re\right) - re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.35e+154)
   (/
    (+ 1.0 (* (* im im) (+ -0.5 (* (* im im) 0.041666666666666664))))
    (+ (* re re) (- 1.0 re)))
   (if (<= re -600000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 1.08e+26)
       (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
       (if (<= re 1.36e+48)
         (/
          (*
           (+ 1.0 (* re (* re re)))
           (* 0.041666666666666664 (* im (* im (* im im)))))
          (- (+ 1.0 (* re re)) re))
         (*
          (+ 1.0 (* -0.5 (* im im)))
          (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.35e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re));
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.08e+26) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else if (re <= 1.36e+48) {
		tmp = ((1.0 + (re * (re * re))) * (0.041666666666666664 * (im * (im * (im * im))))) / ((1.0 + (re * re)) - re);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / 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 <= (-1.35d+154)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + ((im * im) * 0.041666666666666664d0)))) / ((re * re) + (1.0d0 - re))
    else if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 1.08d+26) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    else if (re <= 1.36d+48) then
        tmp = ((1.0d0 + (re * (re * re))) * (0.041666666666666664d0 * (im * (im * (im * im))))) / ((1.0d0 + (re * re)) - re)
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.35e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re));
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.08e+26) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else if (re <= 1.36e+48) {
		tmp = ((1.0 + (re * (re * re))) * (0.041666666666666664 * (im * (im * (im * im))))) / ((1.0 + (re * re)) - re);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.35e+154:
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re))
	elif re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 1.08e+26:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	elif re <= 1.36e+48:
		tmp = ((1.0 + (re * (re * re))) * (0.041666666666666664 * (im * (im * (im * im))))) / ((1.0 + (re * re)) - re)
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.35e+154)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(Float64(im * im) * 0.041666666666666664)))) / Float64(Float64(re * re) + Float64(1.0 - re)));
	elseif (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 1.08e+26)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	elseif (re <= 1.36e+48)
		tmp = Float64(Float64(Float64(1.0 + Float64(re * Float64(re * re))) * Float64(0.041666666666666664 * Float64(im * Float64(im * Float64(im * im))))) / Float64(Float64(1.0 + Float64(re * re)) - re));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.35e+154)
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re));
	elseif (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 1.08e+26)
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	elseif (re <= 1.36e+48)
		tmp = ((1.0 + (re * (re * re))) * (0.041666666666666664 * (im * (im * (im * im))))) / ((1.0 + (re * re)) - re);
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.35e+154], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(re * re), $MachinePrecision] + N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.08e+26], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.36e+48], N[(N[(N[(1.0 + N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(re * re), $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 1.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 1.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 1.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, re\right)}, \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{re}\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       12. *-lowering-*.f64 52.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

   13. Simplified 52.4%

       \[\leadsto \frac{\color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}}{re \cdot re + \left(1 - re\right)} \]

   if -1.35000000000000003e154 < re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 61.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 61.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 1.08e26

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 54.5%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 54.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 47.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 47.4%

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

   if 1.08e26 < re < 1.3599999999999999e48

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 3.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 3.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 46.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 46.4%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]
   12. Step-by-step derivation

       1. flip3-+ N/A

          \[\leadsto \frac{{re}^{3} + {1}^{3}}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)} \cdot \left(\color{blue}{\frac{1}{24}} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

       2. cube-unmult N/A

          \[\leadsto \frac{re \cdot \left(re \cdot re\right) + {1}^{3}}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)} \cdot \left(\frac{1}{24} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \frac{re \cdot \left(re \cdot re\right) + 1}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)} \cdot \left(\frac{1}{24} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \frac{1 + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)} \cdot \left(\frac{1}{24} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{1 + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)} \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\frac{1}{24}}\right) \]

       6. associate-*r* N/A

          \[\leadsto \frac{1 + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)}\right) \]

       7. associate-*r* N/A

          \[\leadsto \frac{1 + re \cdot \left(re \cdot re\right)}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right) \]

       8. associate-*l/ N/A

          \[\leadsto \frac{\left(1 + re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

       9. metadata-eval N/A

          \[\leadsto \frac{\left(1 + re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)}{re \cdot re + \left(1 - \color{blue}{re} \cdot 1\right)} \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{\left(1 + re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)}{re \cdot re + \left(1 - re\right)} \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right)\right)\right)\right), \color{blue}{\left(re \cdot re + \left(1 - re\right)\right)}\right) \]

   13. Applied egg-rr 56.5%

       \[\leadsto \color{blue}{\frac{\left(1 + re \cdot \left(re \cdot re\right)\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)}{\left(1 + re \cdot re\right) - re}} \]

   if 1.3599999999999999e48 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 70.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 70.4%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 70.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 70.4%

       \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.08 \cdot 10^{+26}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 1.36 \cdot 10^{+48}:\\ \;\;\;\;\frac{\left(1 + re \cdot \left(re \cdot re\right)\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)}{\left(1 + re \cdot re\right) - re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re + \left(1 - re\right)\\ \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{t\_0}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* re re) (- 1.0 re))))
   (if (<= re -1.34e+154)
     (/ (+ 1.0 (* (* im im) (+ -0.5 (* (* im im) 0.041666666666666664)))) t_0)
     (if (<= re -600000.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (if (<= re 1.1e+122)
         (/
          (*
           (+ 1.0 (* (* im im) (+ -0.5 (* im (* im 0.041666666666666664)))))
           (+ 1.0 (* re (* re re))))
          t_0)
         (*
          (+ 1.0 (* -0.5 (* im im)))
          (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re))))))))))

C

double code(double re, double im) {
	double t_0 = (re * re) + (1.0 - re);
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0;
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.1e+122) {
		tmp = ((1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664))))) * (1.0 + (re * (re * re)))) / t_0;
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	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) + (1.0d0 - re)
    if (re <= (-1.34d+154)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + ((im * im) * 0.041666666666666664d0)))) / t_0
    else if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 1.1d+122) then
        tmp = ((1.0d0 + ((im * im) * ((-0.5d0) + (im * (im * 0.041666666666666664d0))))) * (1.0d0 + (re * (re * re)))) / t_0
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) + (1.0 - re);
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0;
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.1e+122) {
		tmp = ((1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664))))) * (1.0 + (re * (re * re)))) / t_0;
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) + (1.0 - re)
	tmp = 0
	if re <= -1.34e+154:
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0
	elif re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 1.1e+122:
		tmp = ((1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664))))) * (1.0 + (re * (re * re)))) / t_0
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) + Float64(1.0 - re))
	tmp = 0.0
	if (re <= -1.34e+154)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(Float64(im * im) * 0.041666666666666664)))) / t_0);
	elseif (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 1.1e+122)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(im * Float64(im * 0.041666666666666664))))) * Float64(1.0 + Float64(re * Float64(re * re)))) / t_0);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) + (1.0 - re);
	tmp = 0.0;
	if (re <= -1.34e+154)
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / t_0;
	elseif (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 1.1e+122)
		tmp = ((1.0 + ((im * im) * (-0.5 + (im * (im * 0.041666666666666664))))) * (1.0 + (re * (re * re)))) / t_0;
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] + N[(1.0 - re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.34e+154], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.1e+122], N[(N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.34000000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 1.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 1.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 1.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, re\right)}, \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{re}\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       12. *-lowering-*.f64 52.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

   13. Simplified 52.4%

       \[\leadsto \frac{\color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}}{re \cdot re + \left(1 - re\right)} \]

   if -1.34000000000000001e154 < re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 61.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 61.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 1.1e122

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 78.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 78.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 42.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 42.2%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 44.2%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   if 1.1e122 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 83.8%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 83.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 83.8%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.34e+154)
   (/
    (+ 1.0 (* (* im im) (+ -0.5 (* (* im im) 0.041666666666666664))))
    (+ (* re re) (- 1.0 re)))
   (if (<= re -600000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (*
      (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
      (+ 1.0 (* -0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re));
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (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 <= (-1.34d+154)) then
        tmp = (1.0d0 + ((im * im) * ((-0.5d0) + ((im * im) * 0.041666666666666664d0)))) / ((re * re) + (1.0d0 - re))
    else if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))) * (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 <= -1.34e+154) {
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re));
	} else if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.34e+154:
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re))
	elif re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	else:
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.34e+154)
		tmp = Float64(Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(Float64(im * im) * 0.041666666666666664)))) / Float64(Float64(re * re) + Float64(1.0 - re)));
	elseif (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))) * 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 <= -1.34e+154)
		tmp = (1.0 + ((im * im) * (-0.5 + ((im * im) * 0.041666666666666664)))) / ((re * re) + (1.0 - re));
	elseif (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	else
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.34e+154], N[(N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(re * re), $MachinePrecision] + N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.34000000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 1.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 1.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 1.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 1.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \color{blue}{\left(re + 1\right)} \]

      2. flip3-+ N/A

         \[\leadsto \left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \frac{{re}^{3} + {1}^{3}}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)}{\color{blue}{re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + im \cdot \left(im \cdot \left(\frac{-1}{2} + \left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left({re}^{3} + {1}^{3}\right)\right), \color{blue}{\left(re \cdot re + \left(1 \cdot 1 - re \cdot 1\right)\right)}\right) \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot re\right)\right)}{re \cdot re + \left(1 - re\right)}} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, re\right)}, \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{re}\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

       12. *-lowering-*.f64 52.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{\_.f64}\left(1, re\right)\right)\right) \]

   13. Simplified 52.4%

       \[\leadsto \frac{\color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}}{re \cdot re + \left(1 - re\right)} \]

   if -1.34000000000000001e154 < re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 61.7%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 61.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 59.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 49.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 49.5%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(im \cdot im\right) \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)}{re \cdot re + \left(1 - re\right)}\\ \mathbf{elif}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+46}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -600000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 6.8e+46)
     (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
     (*
      (+ 1.0 (* -0.5 (* im im)))
      (* (* re re) (* re (+ 0.16666666666666666 (/ 0.5 re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 6.8e+46) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / 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 <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 6.8d+46) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((re * re) * (re * (0.16666666666666666d0 + (0.5d0 / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 6.8e+46) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 6.8e+46:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 6.8e+46)
		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(-0.5 * Float64(im * im))) * Float64(Float64(re * re) * Float64(re * Float64(0.16666666666666666 + Float64(0.5 / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 6.8e+46)
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	else
		tmp = (1.0 + (-0.5 * (im * im))) * ((re * re) * (re * (0.16666666666666666 + (0.5 / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.8e+46], 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[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 41.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 41.3%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 41.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 6.7999999999999996e46

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 56.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 56.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 44.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 44.4%

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

   if 6.7999999999999996e46 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 70.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 70.4%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

       11. /-lowering-/.f64 70.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   11. Simplified 70.4%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+46}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -600000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (*
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
    (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (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 <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))) * (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 <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	else:
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))) * 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 <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	else
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 41.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 41.3%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 41.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 59.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 49.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 49.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+46}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -600000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 6.8e+46)
     (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
     (*
      (* re (* re re))
      (+ 0.16666666666666666 (* (* im im) -0.08333333333333333))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 6.8e+46) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 6.8d+46) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    else
        tmp = (re * (re * re)) * (0.16666666666666666d0 + ((im * im) * (-0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 6.8e+46) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 6.8e+46:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	else:
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 6.8e+46)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(0.16666666666666666 + Float64(Float64(im * im) * -0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 6.8e+46)
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	else
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.8e+46], 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[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 41.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 41.3%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 41.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 6.7999999999999996e46

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 56.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 56.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 44.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 44.4%

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

   if 6.7999999999999996e46 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 70.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 70.4%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \cdot \frac{1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{3}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot re\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot {re}^{2}\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left({re}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot re\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} \cdot 1 + \color{blue}{\frac{1}{6} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} + \color{blue}{\frac{1}{6}} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{6} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left({im}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{6}\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \frac{-1}{12}\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{6}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{6}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       21. metadata-eval 70.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{12}\right)\right)\right) \]

   11. Simplified 70.4%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+46}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 51.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+46}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -600000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 6.8e+46)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (*
      (* re (* re re))
      (+ 0.16666666666666666 (* (* im im) -0.08333333333333333))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 6.8e+46) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 6.8d+46) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else
        tmp = (re * (re * re)) * (0.16666666666666666d0 + ((im * im) * (-0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 6.8e+46) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 6.8e+46:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	else:
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 6.8e+46)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(0.16666666666666666 + Float64(Float64(im * im) * -0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 6.8e+46)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	else
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.8e+46], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 41.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 41.3%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 41.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re < 6.7999999999999996e46

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 56.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 56.9%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 44.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 44.3%

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

   if 6.7999999999999996e46 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   5. Simplified 83.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 70.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]

   8. Simplified 70.4%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \cdot \frac{1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{3}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot re\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot {re}^{2}\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left({re}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot re\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} \cdot 1 + \color{blue}{\frac{1}{6} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} + \color{blue}{\frac{1}{6}} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{6} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left({im}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{6}\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \frac{-1}{12}\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{6}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{6}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       21. metadata-eval 70.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{12}\right)\right)\right) \]

   11. Simplified 70.4%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+46}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 38.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -48000000000000:\\ \;\;\;\;\left(re + 1\right) \cdot t\_0\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{+48}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* im im))))
   (if (<= re -48000000000000.0)
     (* (+ re 1.0) t_0)
     (if (<= re 2.65e+48) (+ re 1.0) (* re (+ 1.0 t_0))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -48000000000000.0) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 2.65e+48) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (im * im)
    if (re <= (-48000000000000.0d0)) then
        tmp = (re + 1.0d0) * t_0
    else if (re <= 2.65d+48) then
        tmp = re + 1.0d0
    else
        tmp = re * (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * (im * im);
	double tmp;
	if (re <= -48000000000000.0) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 2.65e+48) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * (im * im)
	tmp = 0
	if re <= -48000000000000.0:
		tmp = (re + 1.0) * t_0
	elif re <= 2.65e+48:
		tmp = re + 1.0
	else:
		tmp = re * (1.0 + t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(im * im))
	tmp = 0.0
	if (re <= -48000000000000.0)
		tmp = Float64(Float64(re + 1.0) * t_0);
	elseif (re <= 2.65e+48)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * (im * im);
	tmp = 0.0;
	if (re <= -48000000000000.0)
		tmp = (re + 1.0) * t_0;
	elseif (re <= 2.65e+48)
		tmp = re + 1.0;
	else
		tmp = re * (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -48000000000000.0], N[(N[(re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[re, 2.65e+48], N[(re + 1.0), $MachinePrecision], N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.8e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 2.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + re\right), \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \left(\color{blue}{\frac{-1}{2}} \cdot {im}^{2}\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(\color{blue}{\frac{-1}{2}} \cdot {im}^{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 29.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 29.2%

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

   if -4.8e13 < re < 2.65e48

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 57.8%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 57.8%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 43.1%

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{re}\right) \]

   8. Simplified 43.1%

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

   if 2.65e48 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 6.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 6.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 32.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 32.2%

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

   9. Taylor expanded in re around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 32.2%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 32.2%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -48000000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{+48}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 46.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -600000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (+ 1.0 (* re (+ 1.0 (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (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 <= (-600000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (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 <= -600000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -600000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	else:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * 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 <= -600000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	else
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -600000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot im\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 2.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(-0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 41.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 41.3%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 41.8%

       \[\leadsto \color{blue}{1} \cdot \left(0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \]

   if -6e5 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 59.2%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 59.2%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 42.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 42.3%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 43.3% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -48000000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(-0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -48000000000000.0)
   (* (+ 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 <= -48000000000000.0) {
		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 <= (-48000000000000.0d0)) 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 <= -48000000000000.0) {
		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 <= -48000000000000.0:
		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 <= -48000000000000.0)
		tmp = Float64(Float64(re + 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 <= -48000000000000.0)
		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, -48000000000000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.8e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 2.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + re\right), \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \left(\color{blue}{\frac{-1}{2}} \cdot {im}^{2}\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(\color{blue}{\frac{-1}{2}} \cdot {im}^{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 29.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 29.2%

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

   if -4.8e13 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 59.6%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 59.6%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 41.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 41.8%

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

Alternative 23: 33.4% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7.2 \cdot 10^{+46}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7.2e+46) 1.0 (* re (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7.2e+46) {
		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 <= 7.2d+46) 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 <= 7.2e+46) {
		tmp = 1.0;
	} else {
		tmp = re * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7.2e+46:
		tmp = 1.0
	else:
		tmp = re * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7.2e+46)
		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 <= 7.2e+46)
		tmp = 1.0;
	else
		tmp = re * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7.2e+46], 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 < 7.1999999999999997e46

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 72.5%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 72.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 29.0%

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

   if 7.1999999999999997e46 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 5.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 5.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 31.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 31.6%

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

   9. Taylor expanded in re around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 31.6%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 31.6%

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

Alternative 24: 31.2% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.7e-174) 1.0 (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.7e-174) {
		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 <= 1.7d-174) 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 <= 1.7e-174) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.7e-174:
		tmp = 1.0
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.7e-174)
		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 <= 1.7e-174)
		tmp = 1.0;
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.7e-174], 1.0, N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.7000000000000001e-174

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 73.5%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 73.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 27.0%

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

   if 1.7000000000000001e-174 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 32.4%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 32.4%

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

   6. Taylor expanded in im around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right) \]

      4. *-lowering-*.f64 30.6%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   8. Simplified 30.6%

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

Alternative 25: 28.8% 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 im around 0

   \[\leadsto \color{blue}{e^{re}} \]
4. Step-by-step derivation

   1. exp-lowering-exp.f64 71.1%

      \[\leadsto \mathsf{exp.f64}\left(re\right) \]

5. Simplified 71.1%

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

6. Taylor expanded in re around 0

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

   1. +-lowering-+.f64 24.1%

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{re}\right) \]

8. Simplified 24.1%

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

9. Final simplification 24.1%

   \[\leadsto re + 1 \]
10. Add Preprocessing

Alternative 26: 28.3% 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 im around 0

   \[\leadsto \color{blue}{e^{re}} \]
4. Step-by-step derivation

   1. exp-lowering-exp.f64 71.1%

      \[\leadsto \mathsf{exp.f64}\left(re\right) \]

5. Simplified 71.1%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 23.9%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))