math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 13.5s

Alternatives: 17

Speedup: 1.0×

• 24.6% 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: 71.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \leq 1\\ \mathbf{if}\;t\_0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (<= (exp re) 1.0))) (if t_0 (exp re) (if t_0 (cos im) (exp re)))))

C

double code(double re, double im) {
	int t_0 = exp(re) <= 1.0;
	double tmp;
	if (t_0) {
		tmp = exp(re);
	} else if (t_0) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    logical :: t_0
    real(8) :: tmp
    t_0 = exp(re) <= 1.0d0
    if (t_0) then
        tmp = exp(re)
    else if (t_0) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	boolean t_0 = Math.exp(re) <= 1.0;
	double tmp;
	if (t_0) {
		tmp = Math.exp(re);
	} else if (t_0) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) <= 1.0
	tmp = 0
	if t_0:
		tmp = math.exp(re)
	elif t_0:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	t_0 = exp(re) <= 1.0
	tmp = 0.0
	if (t_0)
		tmp = exp(re);
	elseif (t_0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) <= 1.0;
	tmp = 0.0;
	if (t_0)
		tmp = exp(re);
	elseif (t_0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = LessEqual[N[Exp[re], $MachinePrecision], 1.0]}, If[t$95$0, N[Exp[re], $MachinePrecision], If[t$95$0, N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 1 < (exp.f64 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 76.7%

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

   5. Simplified 76.7%

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

   if 1 < (exp.f64 re) < 1

   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 47.5%

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

   5. Simplified 47.5%

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

Alternative 3: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\\ t_1 := 1 + t\_0\\ t_2 := re \cdot \left(-1 - t\_0\right)\\ \mathbf{if}\;re \leq -0.44:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.051:\\ \;\;\;\;\frac{\cos im \cdot \left(1 + re \cdot \left(t\_1 \cdot t\_2\right)\right)}{1 + t\_2}\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* re 0.16666666666666666))))
        (t_1 (+ 1.0 t_0))
        (t_2 (* re (- -1.0 t_0))))
   (if (<= re -0.44)
     (exp re)
     (if (<= re 0.051)
       (/ (* (cos im) (+ 1.0 (* re (* t_1 t_2)))) (+ 1.0 t_2))
       (if (<= re 1.02e+103)
         (* (exp re) (+ 1.0 (* im (* im -0.5))))
         (* (cos im) (+ 1.0 (* re t_1))))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = 1.0 + t_0;
	double t_2 = re * (-1.0 - t_0);
	double tmp;
	if (re <= -0.44) {
		tmp = exp(re);
	} else if (re <= 0.051) {
		tmp = (cos(im) * (1.0 + (re * (t_1 * t_2)))) / (1.0 + t_2);
	} else if (re <= 1.02e+103) {
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = cos(im) * (1.0 + (re * t_1));
	}
	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) :: tmp
    t_0 = re * (0.5d0 + (re * 0.16666666666666666d0))
    t_1 = 1.0d0 + t_0
    t_2 = re * ((-1.0d0) - t_0)
    if (re <= (-0.44d0)) then
        tmp = exp(re)
    else if (re <= 0.051d0) then
        tmp = (cos(im) * (1.0d0 + (re * (t_1 * t_2)))) / (1.0d0 + t_2)
    else if (re <= 1.02d+103) then
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    else
        tmp = cos(im) * (1.0d0 + (re * t_1))
    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 = 1.0 + t_0;
	double t_2 = re * (-1.0 - t_0);
	double tmp;
	if (re <= -0.44) {
		tmp = Math.exp(re);
	} else if (re <= 0.051) {
		tmp = (Math.cos(im) * (1.0 + (re * (t_1 * t_2)))) / (1.0 + t_2);
	} else if (re <= 1.02e+103) {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = Math.cos(im) * (1.0 + (re * t_1));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (re * 0.16666666666666666))
	t_1 = 1.0 + t_0
	t_2 = re * (-1.0 - t_0)
	tmp = 0
	if re <= -0.44:
		tmp = math.exp(re)
	elif re <= 0.051:
		tmp = (math.cos(im) * (1.0 + (re * (t_1 * t_2)))) / (1.0 + t_2)
	elif re <= 1.02e+103:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	else:
		tmp = math.cos(im) * (1.0 + (re * t_1))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(re * Float64(-1.0 - t_0))
	tmp = 0.0
	if (re <= -0.44)
		tmp = exp(re);
	elseif (re <= 0.051)
		tmp = Float64(Float64(cos(im) * Float64(1.0 + Float64(re * Float64(t_1 * t_2)))) / Float64(1.0 + t_2));
	elseif (re <= 1.02e+103)
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (re * 0.16666666666666666));
	t_1 = 1.0 + t_0;
	t_2 = re * (-1.0 - t_0);
	tmp = 0.0;
	if (re <= -0.44)
		tmp = exp(re);
	elseif (re <= 0.051)
		tmp = (cos(im) * (1.0 + (re * (t_1 * t_2)))) / (1.0 + t_2);
	elseif (re <= 1.02e+103)
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	else
		tmp = cos(im) * (1.0 + (re * t_1));
	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[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(re * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.44], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.051], N[(N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.02e+103], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.440000000000000002

   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 100.0%

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

   5. Simplified 100.0%

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

   if -0.440000000000000002 < re < 0.0509999999999999967

   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 99.4%

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

   5. Simplified 99.4%

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

      1. flip-+ N/A

         \[\leadsto \frac{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)} \cdot \cos \color{blue}{im} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left(1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \cos im}{\color{blue}{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}} \]

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

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

   7. Applied egg-rr 99.4%

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

   if 0.0509999999999999967 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re} + \frac{-1}{2} \cdot \left({im}^{2} \cdot e^{re}\right)} \]
   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. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 83.3%

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

   5. Simplified 83.3%

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

   if 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 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.44:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.051:\\ \;\;\;\;\frac{\cos im \cdot \left(1 + re \cdot \left(\left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \cdot \left(re \cdot \left(-1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)\right)}{1 + re \cdot \left(-1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)}\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;re \leq -0.44:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.055:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (cos im)
          (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))
   (if (<= re -0.44)
     (exp re)
     (if (<= re 0.055)
       t_0
       (if (<= re 1.02e+103) (* (exp re) (+ 1.0 (* im (* im -0.5)))) 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 <= -0.44) {
		tmp = exp(re);
	} else if (re <= 0.055) {
		tmp = t_0;
	} else if (re <= 1.02e+103) {
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	} 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 <= (-0.44d0)) then
        tmp = exp(re)
    else if (re <= 0.055d0) then
        tmp = t_0
    else if (re <= 1.02d+103) then
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    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 <= -0.44) {
		tmp = Math.exp(re);
	} else if (re <= 0.055) {
		tmp = t_0;
	} else if (re <= 1.02e+103) {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	} 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 <= -0.44:
		tmp = math.exp(re)
	elif re <= 0.055:
		tmp = t_0
	elif re <= 1.02e+103:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	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 <= -0.44)
		tmp = exp(re);
	elseif (re <= 0.055)
		tmp = t_0;
	elseif (re <= 1.02e+103)
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	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 <= -0.44)
		tmp = exp(re);
	elseif (re <= 0.055)
		tmp = t_0;
	elseif (re <= 1.02e+103)
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	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, -0.44], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.055], t$95$0, If[LessEqual[re, 1.02e+103], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.440000000000000002

   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 100.0%

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

   5. Simplified 100.0%

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

   if -0.440000000000000002 < re < 0.0550000000000000003 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%

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

   if 0.0550000000000000003 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re} + \frac{-1}{2} \cdot \left({im}^{2} \cdot e^{re}\right)} \]
   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. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 83.3%

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

   5. Simplified 83.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.44:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.055:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -0.44:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0028:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))
   (if (<= re -0.44)
     (exp re)
     (if (<= re 0.0028) t_0 (if (<= re 1.9e+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 <= -0.44) {
		tmp = exp(re);
	} else if (re <= 0.0028) {
		tmp = t_0;
	} else if (re <= 1.9e+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 <= (-0.44d0)) then
        tmp = exp(re)
    else if (re <= 0.0028d0) then
        tmp = t_0
    else if (re <= 1.9d+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 <= -0.44) {
		tmp = Math.exp(re);
	} else if (re <= 0.0028) {
		tmp = t_0;
	} else if (re <= 1.9e+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 <= -0.44:
		tmp = math.exp(re)
	elif re <= 0.0028:
		tmp = t_0
	elif re <= 1.9e+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 <= -0.44)
		tmp = exp(re);
	elseif (re <= 0.0028)
		tmp = t_0;
	elseif (re <= 1.9e+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 <= -0.44)
		tmp = exp(re);
	elseif (re <= 0.0028)
		tmp = t_0;
	elseif (re <= 1.9e+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, -0.44], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0028], t$95$0, If[LessEqual[re, 1.9e+154], N[Exp[re], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.440000000000000002 or 0.00279999999999999997 < re < 1.8999999999999999e154

   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 95.3%

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

   5. Simplified 95.3%

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

   if -0.440000000000000002 < re < 0.00279999999999999997 or 1.8999999999999999e154 < 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 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.44:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0028:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.44:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.000105:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.44)
   (exp re)
   (if (<= re 0.000105) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.44) {
		tmp = exp(re);
	} else if (re <= 0.000105) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(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 <= (-0.44d0)) then
        tmp = exp(re)
    else if (re <= 0.000105d0) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.44) {
		tmp = Math.exp(re);
	} else if (re <= 0.000105) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.44:
		tmp = math.exp(re)
	elif re <= 0.000105:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.44)
		tmp = exp(re);
	elseif (re <= 0.000105)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.44)
		tmp = exp(re);
	elseif (re <= 0.000105)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.44], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.000105], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.440000000000000002 or 1.05e-4 < 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 92.6%

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

   5. Simplified 92.6%

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

   if -0.440000000000000002 < re < 1.05e-4

   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 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.44:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.000105:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \frac{1}{re \cdot re}\right)\right)\\ \mathbf{elif}\;re \leq 8:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{+39}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.001388888888888889 - \frac{-0.041666666666666664}{im \cdot im}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \frac{0.25 \cdot \left(0.5 - re \cdot 0.16666666666666666\right) + \left(\left(re \cdot re\right) \cdot 0.027777777777777776\right) \cdot \left(re \cdot 0.16666666666666666 - 0.5\right)}{0.25}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.32e+154)
   (* re (* re (* re (/ 1.0 (* re re)))))
   (if (<= re 8.0)
     (cos im)
     (if (<= re 1.8e+39)
       (+
        1.0
        (*
         (* im im)
         (+
          -0.5
          (*
           im
           (*
            (* im im)
            (*
             im
             (-
              -0.001388888888888889
              (/ -0.041666666666666664 (* im im)))))))))
       (+
        1.0
        (*
         re
         (+
          1.0
          (*
           re
           (/
            (+
             (* 0.25 (- 0.5 (* re 0.16666666666666666)))
             (*
              (* (* re re) 0.027777777777777776)
              (- (* re 0.16666666666666666) 0.5)))
            0.25)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.32e+154) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else if (re <= 8.0) {
		tmp = cos(im);
	} else if (re <= 1.8e+39) {
		tmp = 1.0 + ((im * im) * (-0.5 + (im * ((im * im) * (im * (-0.001388888888888889 - (-0.041666666666666664 / (im * im))))))));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (((0.25 * (0.5 - (re * 0.16666666666666666))) + (((re * re) * 0.027777777777777776) * ((re * 0.16666666666666666) - 0.5))) / 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.32d+154)) then
        tmp = re * (re * (re * (1.0d0 / (re * re))))
    else if (re <= 8.0d0) then
        tmp = cos(im)
    else if (re <= 1.8d+39) then
        tmp = 1.0d0 + ((im * im) * ((-0.5d0) + (im * ((im * im) * (im * ((-0.001388888888888889d0) - ((-0.041666666666666664d0) / (im * im))))))))
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (((0.25d0 * (0.5d0 - (re * 0.16666666666666666d0))) + (((re * re) * 0.027777777777777776d0) * ((re * 0.16666666666666666d0) - 0.5d0))) / 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.32e+154) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else if (re <= 8.0) {
		tmp = Math.cos(im);
	} else if (re <= 1.8e+39) {
		tmp = 1.0 + ((im * im) * (-0.5 + (im * ((im * im) * (im * (-0.001388888888888889 - (-0.041666666666666664 / (im * im))))))));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (((0.25 * (0.5 - (re * 0.16666666666666666))) + (((re * re) * 0.027777777777777776) * ((re * 0.16666666666666666) - 0.5))) / 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.32e+154:
		tmp = re * (re * (re * (1.0 / (re * re))))
	elif re <= 8.0:
		tmp = math.cos(im)
	elif re <= 1.8e+39:
		tmp = 1.0 + ((im * im) * (-0.5 + (im * ((im * im) * (im * (-0.001388888888888889 - (-0.041666666666666664 / (im * im))))))))
	else:
		tmp = 1.0 + (re * (1.0 + (re * (((0.25 * (0.5 - (re * 0.16666666666666666))) + (((re * re) * 0.027777777777777776) * ((re * 0.16666666666666666) - 0.5))) / 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.32e+154)
		tmp = Float64(re * Float64(re * Float64(re * Float64(1.0 / Float64(re * re)))));
	elseif (re <= 8.0)
		tmp = cos(im);
	elseif (re <= 1.8e+39)
		tmp = Float64(1.0 + Float64(Float64(im * im) * Float64(-0.5 + Float64(im * Float64(Float64(im * im) * Float64(im * Float64(-0.001388888888888889 - Float64(-0.041666666666666664 / Float64(im * im)))))))));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(Float64(Float64(0.25 * Float64(0.5 - Float64(re * 0.16666666666666666))) + Float64(Float64(Float64(re * re) * 0.027777777777777776) * Float64(Float64(re * 0.16666666666666666) - 0.5))) / 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.32e+154)
		tmp = re * (re * (re * (1.0 / (re * re))));
	elseif (re <= 8.0)
		tmp = cos(im);
	elseif (re <= 1.8e+39)
		tmp = 1.0 + ((im * im) * (-0.5 + (im * ((im * im) * (im * (-0.001388888888888889 - (-0.041666666666666664 / (im * im))))))));
	else
		tmp = 1.0 + (re * (1.0 + (re * (((0.25 * (0.5 - (re * 0.16666666666666666))) + (((re * re) * 0.027777777777777776) * ((re * 0.16666666666666666) - 0.5))) / 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.32e+154], N[(re * N[(re * N[(re * N[(1.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.0], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1.8e+39], N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.5 + N[(im * N[(N[(im * im), $MachinePrecision] * N[(im * N[(-0.001388888888888889 - N[(-0.041666666666666664 / N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(N[(N[(0.25 * N[(0.5 - N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(re * re), $MachinePrecision] * 0.027777777777777776), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.31999999999999998e154

   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 100.0%

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

   5. Simplified 100.0%

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

   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 1.6%

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

   8. Simplified 1.6%

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

   9. Taylor expanded in re around -inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left({re}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot {re}^{2}\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(re \cdot \left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto re \cdot \color{blue}{\left(\mathsf{neg}\left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right)} \]

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

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-rgt-neg-in N/A

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

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

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

       12. *-commutative N/A

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

       13. distribute-rgt-neg-in N/A

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

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

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

   11. Simplified 1.6%

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

   12. Taylor expanded in re around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 100.0%

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

   14. Simplified 100.0%

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

   if -1.31999999999999998e154 < re < 8

   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 75.4%

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

   5. Simplified 75.4%

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

   if 8 < re < 1.79999999999999992e39

   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 4.0%

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

   5. Simplified 4.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot im\right)}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 39.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 39.0%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{3} \cdot \left(\frac{1}{24} \cdot \frac{1}{{im}^{2}} - \frac{1}{720}\right)\right)}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot \frac{1}{{im}^{2}}} - \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left(\left({im}^{2} \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot \frac{1}{{im}^{2}} - \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left(\left({im}^{2} \cdot im\right) \cdot \left(\frac{1}{24} \cdot \frac{1}{{im}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{720}\right)\right)}\right)\right)\right)\right)\right)\right) \]

       4. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left(\left({im}^{2} \cdot im\right) \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left(\left({im}^{2} \cdot im\right) \cdot \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right) + \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left(\left({im}^{2} \cdot im\right) \cdot \left(\mathsf{neg}\left(\left(\frac{1}{720} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left(\left({im}^{2} \cdot im\right) \cdot \left(\mathsf{neg}\left(\left(\frac{1}{720} - \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\left(im \cdot \left(\mathsf{neg}\left(\left(\frac{1}{720} - \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right)}\right)\right)\right)\right)\right) \]

       9. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\mathsf{neg}\left(im \cdot \left(\frac{1}{720} - \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{720} - \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right) \cdot im\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{720} - \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right) \cdot im\right)\right)}\right)\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{720} - \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right) \cdot im}\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{720} - \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right) \cdot im}\right)\right)\right)\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\mathsf{neg}\left(im \cdot \left(\frac{1}{720} - \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       15. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{720} - \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 47.5%

       \[\leadsto 1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.001388888888888889 - \frac{-0.041666666666666664}{im \cdot im}\right)\right)\right)}\right) \]

   if 1.79999999999999992e39 < 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 87.8%

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

   5. Simplified 87.8%

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

   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 65.0%

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

   8. Simplified 65.0%

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

      1. flip-+ N/A

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

      2. div-sub N/A

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

      3. frac-sub N/A

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

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

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

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

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

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

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right), \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - \color{blue}{re \cdot \frac{1}{6}}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - \color{blue}{re} \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      13. swap-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(\left(re \cdot re\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \color{blue}{\frac{1}{6}}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \color{blue}{\frac{1}{6}}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 14.0%

       \[\leadsto 1 + re \cdot \left(1 + re \cdot \color{blue}{\frac{0.25 \cdot \left(0.5 - re \cdot 0.16666666666666666\right) - \left(0.5 - re \cdot 0.16666666666666666\right) \cdot \left(\left(re \cdot re\right) \cdot 0.027777777777777776\right)}{\left(0.5 - re \cdot 0.16666666666666666\right) \cdot \left(0.5 - re \cdot 0.16666666666666666\right)}}\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right)\right)\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 80.4%

       \[\leadsto 1 + re \cdot \left(1 + re \cdot \frac{0.25 \cdot \left(0.5 - re \cdot 0.16666666666666666\right) - \left(0.5 - re \cdot 0.16666666666666666\right) \cdot \left(\left(re \cdot re\right) \cdot 0.027777777777777776\right)}{\color{blue}{0.25}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \frac{1}{re \cdot re}\right)\right)\\ \mathbf{elif}\;re \leq 8:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{+39}:\\ \;\;\;\;1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.001388888888888889 - \frac{-0.041666666666666664}{im \cdot im}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \frac{0.25 \cdot \left(0.5 - re \cdot 0.16666666666666666\right) + \left(\left(re \cdot re\right) \cdot 0.027777777777777776\right) \cdot \left(re \cdot 0.16666666666666666 - 0.5\right)}{0.25}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -700:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \frac{1}{re \cdot re}\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \left(0.16666666666666666 + \frac{0.5 + \frac{1}{re}}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))))
   (if (<= re -700.0)
     (* re (* re (* re (/ 1.0 (* re re)))))
     (if (<= re 5e+102)
       (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0)))
       (*
        re
        (* re (* re (+ 0.16666666666666666 (/ (+ 0.5 (/ 1.0 re)) re)))))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -700.0) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else if (re <= 5e+102) {
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = re * (re * (re * (0.16666666666666666 + ((0.5 + (1.0 / re)) / 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 * 0.5d0)
    if (re <= (-700.0d0)) then
        tmp = re * (re * (re * (1.0d0 / (re * re))))
    else if (re <= 5d+102) then
        tmp = 1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0))
    else
        tmp = re * (re * (re * (0.16666666666666666d0 + ((0.5d0 + (1.0d0 / re)) / re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -700.0) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else if (re <= 5e+102) {
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = re * (re * (re * (0.16666666666666666 + ((0.5 + (1.0 / re)) / re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	tmp = 0
	if re <= -700.0:
		tmp = re * (re * (re * (1.0 / (re * re))))
	elif re <= 5e+102:
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0))
	else:
		tmp = re * (re * (re * (0.16666666666666666 + ((0.5 + (1.0 / re)) / re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -700.0)
		tmp = Float64(re * Float64(re * Float64(re * Float64(1.0 / Float64(re * re)))));
	elseif (re <= 5e+102)
		tmp = Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0)));
	else
		tmp = Float64(re * Float64(re * Float64(re * Float64(0.16666666666666666 + Float64(Float64(0.5 + Float64(1.0 / re)) / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -700.0)
		tmp = re * (re * (re * (1.0 / (re * re))));
	elseif (re <= 5e+102)
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	else
		tmp = re * (re * (re * (0.16666666666666666 + ((0.5 + (1.0 / re)) / re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -700.0], N[(re * N[(re * N[(re * N[(1.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+102], N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * N[(0.16666666666666666 + N[(N[(0.5 + N[(1.0 / re), $MachinePrecision]), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -700

   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 100.0%

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

   5. Simplified 100.0%

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

   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 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in re around -inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left({re}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot {re}^{2}\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(re \cdot \left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto re \cdot \color{blue}{\left(\mathsf{neg}\left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right)} \]

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

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-rgt-neg-in N/A

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

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

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

       12. *-commutative N/A

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

       13. distribute-rgt-neg-in N/A

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

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

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

   11. Simplified 1.7%

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

   12. Taylor expanded in re around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 52.6%

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

   14. Simplified 52.6%

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

   if -700 < re < 5e102

   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 62.0%

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

   5. Simplified 62.0%

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

   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 49.5%

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

   8. Simplified 49.5%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. flip-+ N/A

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

      4. *-rgt-identity N/A

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

      5. fmm-def N/A

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

      6. *-commutative N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(re \cdot re\right), \left(\left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \left(\mathsf{fma}\left(\color{blue}{re}, 1, \mathsf{neg}\left(\left(re \cdot \frac{1}{2}\right) \cdot re\right)\right)\right)\right)\right) \]

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

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

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

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

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

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. fmm-def N/A

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

      17. *-rgt-identity N/A

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

   10. Applied egg-rr 52.8%

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

   if 5e102 < 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 86.1%

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

   5. Simplified 86.1%

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

   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 86.1%

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

   8. Simplified 86.1%

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

   9. Taylor expanded in re around -inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left({re}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot {re}^{2}\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(re \cdot \left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto re \cdot \color{blue}{\left(\mathsf{neg}\left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right)} \]

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

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-rgt-neg-in N/A

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

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

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

       12. *-commutative N/A

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

       13. distribute-rgt-neg-in N/A

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

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

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

   11. Simplified 86.1%

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

Alternative 9: 55.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \frac{1}{re \cdot re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \frac{0.25 \cdot \left(0.5 - re \cdot 0.16666666666666666\right) + \left(\left(re \cdot re\right) \cdot 0.027777777777777776\right) \cdot \left(re \cdot 0.16666666666666666 - 0.5\right)}{0.25}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.2)
   (* re (* re (* re (/ 1.0 (* re re)))))
   (+
    1.0
    (*
     re
     (+
      1.0
      (*
       re
       (/
        (+
         (* 0.25 (- 0.5 (* re 0.16666666666666666)))
         (*
          (* (* re re) 0.027777777777777776)
          (- (* re 0.16666666666666666) 0.5)))
        0.25)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (((0.25 * (0.5 - (re * 0.16666666666666666))) + (((re * re) * 0.027777777777777776) * ((re * 0.16666666666666666) - 0.5))) / 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.2d0)) then
        tmp = re * (re * (re * (1.0d0 / (re * re))))
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (((0.25d0 * (0.5d0 - (re * 0.16666666666666666d0))) + (((re * re) * 0.027777777777777776d0) * ((re * 0.16666666666666666d0) - 0.5d0))) / 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.2) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (((0.25 * (0.5 - (re * 0.16666666666666666))) + (((re * re) * 0.027777777777777776) * ((re * 0.16666666666666666) - 0.5))) / 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.2:
		tmp = re * (re * (re * (1.0 / (re * re))))
	else:
		tmp = 1.0 + (re * (1.0 + (re * (((0.25 * (0.5 - (re * 0.16666666666666666))) + (((re * re) * 0.027777777777777776) * ((re * 0.16666666666666666) - 0.5))) / 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.2)
		tmp = Float64(re * Float64(re * Float64(re * Float64(1.0 / Float64(re * re)))));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(Float64(Float64(0.25 * Float64(0.5 - Float64(re * 0.16666666666666666))) + Float64(Float64(Float64(re * re) * 0.027777777777777776) * Float64(Float64(re * 0.16666666666666666) - 0.5))) / 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.2)
		tmp = re * (re * (re * (1.0 / (re * re))));
	else
		tmp = 1.0 + (re * (1.0 + (re * (((0.25 * (0.5 - (re * 0.16666666666666666))) + (((re * re) * 0.027777777777777776) * ((re * 0.16666666666666666) - 0.5))) / 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.2], N[(re * N[(re * N[(re * N[(1.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(N[(N[(0.25 * N[(0.5 - N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(re * re), $MachinePrecision] * 0.027777777777777776), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.2000000000000002

   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 100.0%

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

   5. Simplified 100.0%

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

   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 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in re around -inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left({re}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot {re}^{2}\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(re \cdot \left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto re \cdot \color{blue}{\left(\mathsf{neg}\left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right)} \]

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

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-rgt-neg-in N/A

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

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

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

       12. *-commutative N/A

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

       13. distribute-rgt-neg-in N/A

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

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

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

   11. Simplified 1.7%

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

   12. Taylor expanded in re around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 52.6%

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

   14. Simplified 52.6%

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

   if -3.2000000000000002 < 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 66.8%

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

   5. Simplified 66.8%

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

   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 57.0%

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

   8. Simplified 57.0%

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

      1. flip-+ N/A

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

      2. div-sub N/A

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

      3. frac-sub N/A

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

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

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

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

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

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

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right), \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - \color{blue}{re \cdot \frac{1}{6}}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - \color{blue}{re} \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      13. swap-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(\left(re \cdot re\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \color{blue}{\frac{1}{6}}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \color{blue}{\frac{1}{6}}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right)\right)\right), \left(\left(\frac{1}{2} - re \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{2} - re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 43.1%

       \[\leadsto 1 + re \cdot \left(1 + re \cdot \color{blue}{\frac{0.25 \cdot \left(0.5 - re \cdot 0.16666666666666666\right) - \left(0.5 - re \cdot 0.16666666666666666\right) \cdot \left(\left(re \cdot re\right) \cdot 0.027777777777777776\right)}{\left(0.5 - re \cdot 0.16666666666666666\right) \cdot \left(0.5 - re \cdot 0.16666666666666666\right)}}\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right)\right)\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 61.2%

       \[\leadsto 1 + re \cdot \left(1 + re \cdot \frac{0.25 \cdot \left(0.5 - re \cdot 0.16666666666666666\right) - \left(0.5 - re \cdot 0.16666666666666666\right) \cdot \left(\left(re \cdot re\right) \cdot 0.027777777777777776\right)}{\color{blue}{0.25}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \frac{1}{re \cdot re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \frac{0.25 \cdot \left(0.5 - re \cdot 0.16666666666666666\right) + \left(\left(re \cdot re\right) \cdot 0.027777777777777776\right) \cdot \left(re \cdot 0.16666666666666666 - 0.5\right)}{0.25}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -700:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \frac{1}{re \cdot re}\right)\right)\\ \mathbf{elif}\;re \leq 1.8:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -700.0)
   (* re (* re (* re (/ 1.0 (* re re)))))
   (if (<= re 1.8)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -700.0) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else if (re <= 1.8) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-700.0d0)) then
        tmp = re * (re * (re * (1.0d0 / (re * re))))
    else if (re <= 1.8d0) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else
        tmp = re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -700.0) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else if (re <= 1.8) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -700.0:
		tmp = re * (re * (re * (1.0 / (re * re))))
	elif re <= 1.8:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	else:
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -700.0)
		tmp = Float64(re * Float64(re * Float64(re * Float64(1.0 / Float64(re * re)))));
	elseif (re <= 1.8)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -700.0)
		tmp = re * (re * (re * (1.0 / (re * re))));
	elseif (re <= 1.8)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	else
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -700.0], N[(re * N[(re * N[(re * N[(1.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.8], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -700

   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 100.0%

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

   5. Simplified 100.0%

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

   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 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in re around -inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left({re}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot {re}^{2}\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(re \cdot \left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto re \cdot \color{blue}{\left(\mathsf{neg}\left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right)} \]

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

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-rgt-neg-in N/A

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

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

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

       12. *-commutative N/A

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

       13. distribute-rgt-neg-in N/A

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

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

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

   11. Simplified 1.7%

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

   12. Taylor expanded in re around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 52.6%

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

   14. Simplified 52.6%

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

   if -700 < re < 1.80000000000000004

   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 58.6%

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

   5. Simplified 58.6%

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

   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 58.6%

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

   8. Simplified 58.6%

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

   if 1.80000000000000004 < 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 83.3%

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

   5. Simplified 83.3%

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

   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 53.8%

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

   8. Simplified 53.8%

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

   9. Taylor expanded in re around inf

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

       1. cube-mult N/A

          \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \left(re \cdot {re}^{2}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \]

       4. associate-+r+ N/A

          \[\leadsto re \cdot \left({re}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \color{blue}{\frac{1}{{re}^{2}}}\right)\right) \]

       5. distribute-lft-in N/A

          \[\leadsto re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \color{blue}{{re}^{2} \cdot \frac{1}{{re}^{2}}}\right) \]

       6. unpow2 N/A

          \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + {\color{blue}{re}}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       7. associate-*l* N/A

          \[\leadsto re \cdot \left(re \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) + \color{blue}{{re}^{2}} \cdot \frac{1}{{re}^{2}}\right) \]

       8. +-commutative N/A

          \[\leadsto re \cdot \left(re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto re \cdot \left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right) + {re}^{\color{blue}{2}} \cdot \frac{1}{{re}^{2}}\right) \]

       10. associate-*l* N/A

           \[\leadsto re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right) + \frac{1}{6} \cdot re\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       11. lft-mult-inverse N/A

           \[\leadsto re \cdot \left(re \cdot \left(\frac{1}{2} \cdot 1 + \frac{1}{6} \cdot re\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       12. metadata-eval N/A

           \[\leadsto re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right) \]

       13. rgt-mult-inverse N/A

           \[\leadsto re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1\right) \]

       14. +-commutative N/A

           \[\leadsto re \cdot \left(1 + \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right) \]

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

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

   11. Simplified 53.8%

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

Alternative 11: 52.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -700:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \frac{1}{re \cdot re}\right)\right)\\ \mathbf{elif}\;re \leq 2.85:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -700.0)
   (* re (* re (* re (/ 1.0 (* re re)))))
   (if (<= re 2.85)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (* re (* re (+ 0.5 (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -700.0) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else if (re <= 2.85) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-700.0d0)) then
        tmp = re * (re * (re * (1.0d0 / (re * re))))
    else if (re <= 2.85d0) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else
        tmp = re * (re * (0.5d0 + (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -700.0) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else if (re <= 2.85) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -700.0:
		tmp = re * (re * (re * (1.0 / (re * re))))
	elif re <= 2.85:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	else:
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -700.0)
		tmp = Float64(re * Float64(re * Float64(re * Float64(1.0 / Float64(re * re)))));
	elseif (re <= 2.85)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	else
		tmp = Float64(re * Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -700.0)
		tmp = re * (re * (re * (1.0 / (re * re))));
	elseif (re <= 2.85)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	else
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -700.0], N[(re * N[(re * N[(re * N[(1.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.85], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -700

   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 100.0%

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

   5. Simplified 100.0%

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

   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 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in re around -inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left({re}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot {re}^{2}\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(re \cdot \left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto re \cdot \color{blue}{\left(\mathsf{neg}\left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right)} \]

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

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-rgt-neg-in N/A

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

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

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

       12. *-commutative N/A

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

       13. distribute-rgt-neg-in N/A

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

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

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

   11. Simplified 1.7%

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

   12. Taylor expanded in re around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 52.6%

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

   14. Simplified 52.6%

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

   if -700 < re < 2.85000000000000009

   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 58.6%

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

   5. Simplified 58.6%

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

   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 58.6%

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

   8. Simplified 58.6%

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

   if 2.85000000000000009 < 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 83.3%

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

   5. Simplified 83.3%

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

   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 53.8%

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

   8. Simplified 53.8%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{2} \cdot \frac{1}{re}\right) \]

       2. unpow2 N/A

          \[\leadsto \left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) \]

       3. associate-*l* N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{re} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \color{blue}{\frac{1}{6}}\right)\right) \]

       6. distribute-rgt-in N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \color{blue}{\frac{1}{6} \cdot re}\right) \]

       7. associate-*l* N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right) + \color{blue}{\frac{1}{6}} \cdot re\right) \]

       8. lft-mult-inverse N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\frac{1}{2} \cdot 1 + \frac{1}{6} \cdot re\right) \]

       9. metadata-eval N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{6}} \cdot re\right) \]

       10. associate-*r* N/A

           \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]

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

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

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

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 53.8%

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

   11. Simplified 53.8%

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

Alternative 12: 53.0% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.5:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot \frac{1}{re \cdot re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.5)
   (* re (* re (* re (/ 1.0 (* re re)))))
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.5) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.5d0)) then
        tmp = re * (re * (re * (1.0d0 / (re * re))))
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.5) {
		tmp = re * (re * (re * (1.0 / (re * re))));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.5:
		tmp = re * (re * (re * (1.0 / (re * re))))
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.5)
		tmp = Float64(re * Float64(re * Float64(re * Float64(1.0 / Float64(re * re)))));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.5)
		tmp = re * (re * (re * (1.0 / (re * re))));
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.5], N[(re * N[(re * N[(re * N[(1.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.5

   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 100.0%

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

   5. Simplified 100.0%

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

   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 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in re around -inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left({re}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(re \cdot {re}^{2}\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(re \cdot \left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto re \cdot \color{blue}{\left(\mathsf{neg}\left({re}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{re}}{re} - \frac{1}{6}\right)\right)\right)} \]

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

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-rgt-neg-in N/A

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

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

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

       12. *-commutative N/A

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

       13. distribute-rgt-neg-in N/A

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

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

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

   11. Simplified 1.7%

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

   12. Taylor expanded in re around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 52.6%

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

   14. Simplified 52.6%

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

   if -3.5 < 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 66.8%

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

   5. Simplified 66.8%

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

   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 57.0%

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

   8. Simplified 57.0%

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

Alternative 13: 40.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.86:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.86) (+ re 1.0) (* re (* re (+ 0.5 (* re 0.16666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.86) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.86d0) then
        tmp = re + 1.0d0
    else
        tmp = re * (re * (0.5d0 + (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.86) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.86:
		tmp = re + 1.0
	else:
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.86)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.86)
		tmp = re + 1.0;
	else
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.86], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.8600000000000001

   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 74.6%

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

   5. Simplified 74.6%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 36.6%

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

   8. Simplified 36.6%

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

   if 1.8600000000000001 < 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 83.3%

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

   5. Simplified 83.3%

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

   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 53.8%

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

   8. Simplified 53.8%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot re\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{2} \cdot \frac{1}{re}\right) \]

       2. unpow2 N/A

          \[\leadsto \left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) \]

       3. associate-*l* N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{re} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \color{blue}{\frac{1}{6}}\right)\right) \]

       6. distribute-rgt-in N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \color{blue}{\frac{1}{6} \cdot re}\right) \]

       7. associate-*l* N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right) + \color{blue}{\frac{1}{6}} \cdot re\right) \]

       8. lft-mult-inverse N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\frac{1}{2} \cdot 1 + \frac{1}{6} \cdot re\right) \]

       9. metadata-eval N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{6}} \cdot re\right) \]

       10. associate-*r* N/A

           \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]

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

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

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

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 53.8%

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

   11. Simplified 53.8%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.86:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.6% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.9:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.9) (+ re 1.0) (* 0.16666666666666666 (* re (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.9) {
		tmp = re + 1.0;
	} else {
		tmp = 0.16666666666666666 * (re * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.9d0) then
        tmp = re + 1.0d0
    else
        tmp = 0.16666666666666666d0 * (re * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.9) {
		tmp = re + 1.0;
	} else {
		tmp = 0.16666666666666666 * (re * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.9:
		tmp = re + 1.0
	else:
		tmp = 0.16666666666666666 * (re * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.9)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(0.16666666666666666 * Float64(re * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.9)
		tmp = re + 1.0;
	else
		tmp = 0.16666666666666666 * (re * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.9], N[(re + 1.0), $MachinePrecision], N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.89999999999999991

   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 74.6%

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

   5. Simplified 74.6%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 36.6%

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

   8. Simplified 36.6%

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

   if 2.89999999999999991 < 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 83.3%

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

   5. Simplified 83.3%

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

   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 53.8%

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

   8. Simplified 53.8%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {re}^{3}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({re}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 53.8%

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

   11. Simplified 53.8%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.9:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.6% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.8:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.8) (+ re 1.0) (* 0.5 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.8) {
		tmp = re + 1.0;
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.8d0) then
        tmp = re + 1.0d0
    else
        tmp = 0.5d0 * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.8) {
		tmp = re + 1.0;
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.8:
		tmp = re + 1.0
	else:
		tmp = 0.5 * (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.8)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(0.5 * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.8)
		tmp = re + 1.0;
	else
		tmp = 0.5 * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.8], N[(re + 1.0), $MachinePrecision], N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.7999999999999998

   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 74.6%

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

   5. Simplified 74.6%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 36.6%

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

   8. Simplified 36.6%

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

   if 2.7999999999999998 < 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 83.3%

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

   5. Simplified 83.3%

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

   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 44.0%

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

   8. Simplified 44.0%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 44.0%

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

   11. Simplified 44.0%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.8:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.5% 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 76.7%

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

5. Simplified 76.7%

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

6. Taylor expanded in re around 0

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

   1. +-lowering-+.f64 29.1%

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

8. Simplified 29.1%

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

9. Final simplification 29.1%

   \[\leadsto re + 1 \]
10. Add Preprocessing

Alternative 17: 29.0% 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 76.7%

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

5. Simplified 76.7%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 28.5%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))