math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

Alternatives: 14

Speedup: 1.0×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;e^{re} \leq 0.995:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= (exp re) 0.995)
     t_0
     (if (<= (exp re) 1.0)
       (* (sin im) (+ re 1.0))
       (if (<= (exp re) 2.0) (+ im (* re im)) t_0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (exp(re) <= 0.995) {
		tmp = t_0;
	} else if (exp(re) <= 1.0) {
		tmp = sin(im) * (re + 1.0);
	} else if (exp(re) <= 2.0) {
		tmp = im + (re * im);
	} 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 = exp(re) * im
    if (exp(re) <= 0.995d0) then
        tmp = t_0
    else if (exp(re) <= 1.0d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (exp(re) <= 2.0d0) then
        tmp = im + (re * im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (Math.exp(re) <= 0.995) {
		tmp = t_0;
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (Math.exp(re) <= 2.0) {
		tmp = im + (re * im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if math.exp(re) <= 0.995:
		tmp = t_0
	elif math.exp(re) <= 1.0:
		tmp = math.sin(im) * (re + 1.0)
	elif math.exp(re) <= 2.0:
		tmp = im + (re * im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (exp(re) <= 0.995)
		tmp = t_0;
	elseif (exp(re) <= 1.0)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (exp(re) <= 2.0)
		tmp = Float64(im + Float64(re * im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (exp(re) <= 0.995)
		tmp = t_0;
	elseif (exp(re) <= 1.0)
		tmp = sin(im) * (re + 1.0);
	elseif (exp(re) <= 2.0)
		tmp = im + (re * im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[N[Exp[re], $MachinePrecision], 0.995], t$95$0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.994999999999999996 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.3%

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

   if 0.994999999999999996 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   5. Simplified 100.0%

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

   if 1 < (exp.f64 re) < 2

   1. Initial program 98.4%

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

   3. Taylor expanded in im around 0 98.4%

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

   4. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{im + im \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.995:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;e^{re} \leq 0.995:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= (exp re) 0.995)
     t_0
     (if (<= (exp re) 1.0)
       (sin im)
       (if (<= (exp re) 2.0) (+ im (* re im)) t_0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (exp(re) <= 0.995) {
		tmp = t_0;
	} else if (exp(re) <= 1.0) {
		tmp = sin(im);
	} else if (exp(re) <= 2.0) {
		tmp = im + (re * im);
	} 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 = exp(re) * im
    if (exp(re) <= 0.995d0) then
        tmp = t_0
    else if (exp(re) <= 1.0d0) then
        tmp = sin(im)
    else if (exp(re) <= 2.0d0) then
        tmp = im + (re * im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (Math.exp(re) <= 0.995) {
		tmp = t_0;
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.sin(im);
	} else if (Math.exp(re) <= 2.0) {
		tmp = im + (re * im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if math.exp(re) <= 0.995:
		tmp = t_0
	elif math.exp(re) <= 1.0:
		tmp = math.sin(im)
	elif math.exp(re) <= 2.0:
		tmp = im + (re * im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (exp(re) <= 0.995)
		tmp = t_0;
	elseif (exp(re) <= 1.0)
		tmp = sin(im);
	elseif (exp(re) <= 2.0)
		tmp = Float64(im + Float64(re * im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (exp(re) <= 0.995)
		tmp = t_0;
	elseif (exp(re) <= 1.0)
		tmp = sin(im);
	elseif (exp(re) <= 2.0)
		tmp = im + (re * im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[N[Exp[re], $MachinePrecision], 0.995], t$95$0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Sin[im], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.994999999999999996 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.3%

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

   if 0.994999999999999996 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.9%

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

   if 1 < (exp.f64 re) < 2

   1. Initial program 98.4%

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

   3. Taylor expanded in im around 0 98.4%

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

   4. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{im + im \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.995:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;im + re \cdot im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\\ t_1 := im + re \cdot \left(im + im \cdot t\_0\right)\\ t_2 := im + im \cdot \left(re \cdot \left(1 + t\_0\right)\right)\\ t_3 := im + re \cdot \left(im + re \cdot \left(\left(re \cdot im\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -4.5 \cdot 10^{+126}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 2050000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 19000000:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 22000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+47}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+49}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+60}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+67}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{+67}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 9.2 \cdot 10^{+72}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{+72}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+92}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+158} \lor \neg \left(re \leq 5 \cdot 10^{+197}\right) \land \left(re \leq 3.2 \cdot 10^{+203} \lor \neg \left(re \leq 2.3 \cdot 10^{+246}\right) \land \left(re \leq 2.35 \cdot 10^{+246} \lor \neg \left(re \leq 1.08 \cdot 10^{+248}\right) \land \left(re \leq 1.8 \cdot 10^{+252} \lor \neg \left(re \leq 7.2 \cdot 10^{+299}\right) \land re \leq 7.5 \cdot 10^{+299}\right)\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* re 0.16666666666666666))))
        (t_1 (+ im (* re (+ im (* im t_0)))))
        (t_2 (+ im (* im (* re (+ 1.0 t_0)))))
        (t_3 (+ im (* re (+ im (* re (* (* re im) 0.16666666666666666)))))))
   (if (<= re -4.5e+126)
     (* re (/ im re))
     (if (<= re 4.9e-30)
       (sin im)
       (if (<= re 2050000.0)
         t_1
         (if (<= re 19000000.0)
           (sin im)
           (if (<= re 22000000.0)
             t_1
             (if (<= re 2e+29)
               t_2
               (if (<= re 3.8e+47)
                 t_3
                 (if (<= re 1.4e+49)
                   (sin im)
                   (if (<= re 1e+59)
                     t_1
                     (if (<= re 2e+60)
                       t_3
                       (if (<= re 5e+60)
                         t_1
                         (if (<= re 9e+67)
                           t_3
                           (if (<= re 9.5e+67)
                             (sin im)
                             (if (<= re 9.2e+72)
                               t_3
                               (if (<= re 9.5e+72)
                                 (sin im)
                                 (if (<= re 7.2e+91)
                                   t_1
                                   (if (<= re 7.2e+92)
                                     (sin im)
                                     (if (<= re 1.65e+133)
                                       t_2
                                       (if (<= re 1.7e+133)
                                         (sin im)
                                         (if (<= re 6.2e+153)
                                           t_2
                                           (if (or (<= re 5e+158)
                                                   (and (not (<= re 5e+197))
                                                        (or (<= re 3.2e+203)
                                                            (and (not
                                                                  (<=
                                                                   re
                                                                   2.3e+246))
                                                                 (or (<=
                                                                      re
                                                                      2.35e+246)
                                                                     (and (not
                                                                           (<=
                                                                            re
                                                                            1.08e+248))
                                                                          (or (<=
                                                                               re
                                                                               1.8e+252)
                                                                              (and (not
                                                                                    (<=
                                                                                     re
                                                                                     7.2e+299))
                                                                                   (<=
                                                                                    re
                                                                                    7.5e+299)))))))))
                                             (sin im)
                                             (+
                                              im
                                              (*
                                               im
                                               (*
                                                re
                                                (+
                                                 1.0
                                                 (*
                                                  re
                                                  0.5))))))))))))))))))))))))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = im + (re * (im + (im * t_0)));
	double t_2 = im + (im * (re * (1.0 + t_0)));
	double t_3 = im + (re * (im + (re * ((re * im) * 0.16666666666666666))));
	double tmp;
	if (re <= -4.5e+126) {
		tmp = re * (im / re);
	} else if (re <= 4.9e-30) {
		tmp = sin(im);
	} else if (re <= 2050000.0) {
		tmp = t_1;
	} else if (re <= 19000000.0) {
		tmp = sin(im);
	} else if (re <= 22000000.0) {
		tmp = t_1;
	} else if (re <= 2e+29) {
		tmp = t_2;
	} else if (re <= 3.8e+47) {
		tmp = t_3;
	} else if (re <= 1.4e+49) {
		tmp = sin(im);
	} else if (re <= 1e+59) {
		tmp = t_1;
	} else if (re <= 2e+60) {
		tmp = t_3;
	} else if (re <= 5e+60) {
		tmp = t_1;
	} else if (re <= 9e+67) {
		tmp = t_3;
	} else if (re <= 9.5e+67) {
		tmp = sin(im);
	} else if (re <= 9.2e+72) {
		tmp = t_3;
	} else if (re <= 9.5e+72) {
		tmp = sin(im);
	} else if (re <= 7.2e+91) {
		tmp = t_1;
	} else if (re <= 7.2e+92) {
		tmp = sin(im);
	} else if (re <= 1.65e+133) {
		tmp = t_2;
	} else if (re <= 1.7e+133) {
		tmp = sin(im);
	} else if (re <= 6.2e+153) {
		tmp = t_2;
	} else if ((re <= 5e+158) || (!(re <= 5e+197) && ((re <= 3.2e+203) || (!(re <= 2.3e+246) && ((re <= 2.35e+246) || (!(re <= 1.08e+248) && ((re <= 1.8e+252) || (!(re <= 7.2e+299) && (re <= 7.5e+299))))))))) {
		tmp = sin(im);
	} else {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = re * (0.5d0 + (re * 0.16666666666666666d0))
    t_1 = im + (re * (im + (im * t_0)))
    t_2 = im + (im * (re * (1.0d0 + t_0)))
    t_3 = im + (re * (im + (re * ((re * im) * 0.16666666666666666d0))))
    if (re <= (-4.5d+126)) then
        tmp = re * (im / re)
    else if (re <= 4.9d-30) then
        tmp = sin(im)
    else if (re <= 2050000.0d0) then
        tmp = t_1
    else if (re <= 19000000.0d0) then
        tmp = sin(im)
    else if (re <= 22000000.0d0) then
        tmp = t_1
    else if (re <= 2d+29) then
        tmp = t_2
    else if (re <= 3.8d+47) then
        tmp = t_3
    else if (re <= 1.4d+49) then
        tmp = sin(im)
    else if (re <= 1d+59) then
        tmp = t_1
    else if (re <= 2d+60) then
        tmp = t_3
    else if (re <= 5d+60) then
        tmp = t_1
    else if (re <= 9d+67) then
        tmp = t_3
    else if (re <= 9.5d+67) then
        tmp = sin(im)
    else if (re <= 9.2d+72) then
        tmp = t_3
    else if (re <= 9.5d+72) then
        tmp = sin(im)
    else if (re <= 7.2d+91) then
        tmp = t_1
    else if (re <= 7.2d+92) then
        tmp = sin(im)
    else if (re <= 1.65d+133) then
        tmp = t_2
    else if (re <= 1.7d+133) then
        tmp = sin(im)
    else if (re <= 6.2d+153) then
        tmp = t_2
    else if ((re <= 5d+158) .or. (.not. (re <= 5d+197)) .and. (re <= 3.2d+203) .or. (.not. (re <= 2.3d+246)) .and. (re <= 2.35d+246) .or. (.not. (re <= 1.08d+248)) .and. (re <= 1.8d+252) .or. (.not. (re <= 7.2d+299)) .and. (re <= 7.5d+299)) then
        tmp = sin(im)
    else
        tmp = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = im + (re * (im + (im * t_0)));
	double t_2 = im + (im * (re * (1.0 + t_0)));
	double t_3 = im + (re * (im + (re * ((re * im) * 0.16666666666666666))));
	double tmp;
	if (re <= -4.5e+126) {
		tmp = re * (im / re);
	} else if (re <= 4.9e-30) {
		tmp = Math.sin(im);
	} else if (re <= 2050000.0) {
		tmp = t_1;
	} else if (re <= 19000000.0) {
		tmp = Math.sin(im);
	} else if (re <= 22000000.0) {
		tmp = t_1;
	} else if (re <= 2e+29) {
		tmp = t_2;
	} else if (re <= 3.8e+47) {
		tmp = t_3;
	} else if (re <= 1.4e+49) {
		tmp = Math.sin(im);
	} else if (re <= 1e+59) {
		tmp = t_1;
	} else if (re <= 2e+60) {
		tmp = t_3;
	} else if (re <= 5e+60) {
		tmp = t_1;
	} else if (re <= 9e+67) {
		tmp = t_3;
	} else if (re <= 9.5e+67) {
		tmp = Math.sin(im);
	} else if (re <= 9.2e+72) {
		tmp = t_3;
	} else if (re <= 9.5e+72) {
		tmp = Math.sin(im);
	} else if (re <= 7.2e+91) {
		tmp = t_1;
	} else if (re <= 7.2e+92) {
		tmp = Math.sin(im);
	} else if (re <= 1.65e+133) {
		tmp = t_2;
	} else if (re <= 1.7e+133) {
		tmp = Math.sin(im);
	} else if (re <= 6.2e+153) {
		tmp = t_2;
	} else if ((re <= 5e+158) || (!(re <= 5e+197) && ((re <= 3.2e+203) || (!(re <= 2.3e+246) && ((re <= 2.35e+246) || (!(re <= 1.08e+248) && ((re <= 1.8e+252) || (!(re <= 7.2e+299) && (re <= 7.5e+299))))))))) {
		tmp = Math.sin(im);
	} else {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (re * 0.16666666666666666))
	t_1 = im + (re * (im + (im * t_0)))
	t_2 = im + (im * (re * (1.0 + t_0)))
	t_3 = im + (re * (im + (re * ((re * im) * 0.16666666666666666))))
	tmp = 0
	if re <= -4.5e+126:
		tmp = re * (im / re)
	elif re <= 4.9e-30:
		tmp = math.sin(im)
	elif re <= 2050000.0:
		tmp = t_1
	elif re <= 19000000.0:
		tmp = math.sin(im)
	elif re <= 22000000.0:
		tmp = t_1
	elif re <= 2e+29:
		tmp = t_2
	elif re <= 3.8e+47:
		tmp = t_3
	elif re <= 1.4e+49:
		tmp = math.sin(im)
	elif re <= 1e+59:
		tmp = t_1
	elif re <= 2e+60:
		tmp = t_3
	elif re <= 5e+60:
		tmp = t_1
	elif re <= 9e+67:
		tmp = t_3
	elif re <= 9.5e+67:
		tmp = math.sin(im)
	elif re <= 9.2e+72:
		tmp = t_3
	elif re <= 9.5e+72:
		tmp = math.sin(im)
	elif re <= 7.2e+91:
		tmp = t_1
	elif re <= 7.2e+92:
		tmp = math.sin(im)
	elif re <= 1.65e+133:
		tmp = t_2
	elif re <= 1.7e+133:
		tmp = math.sin(im)
	elif re <= 6.2e+153:
		tmp = t_2
	elif (re <= 5e+158) or (not (re <= 5e+197) and ((re <= 3.2e+203) or (not (re <= 2.3e+246) and ((re <= 2.35e+246) or (not (re <= 1.08e+248) and ((re <= 1.8e+252) or (not (re <= 7.2e+299) and (re <= 7.5e+299)))))))):
		tmp = math.sin(im)
	else:
		tmp = im + (im * (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))
	t_1 = Float64(im + Float64(re * Float64(im + Float64(im * t_0))))
	t_2 = Float64(im + Float64(im * Float64(re * Float64(1.0 + t_0))))
	t_3 = Float64(im + Float64(re * Float64(im + Float64(re * Float64(Float64(re * im) * 0.16666666666666666)))))
	tmp = 0.0
	if (re <= -4.5e+126)
		tmp = Float64(re * Float64(im / re));
	elseif (re <= 4.9e-30)
		tmp = sin(im);
	elseif (re <= 2050000.0)
		tmp = t_1;
	elseif (re <= 19000000.0)
		tmp = sin(im);
	elseif (re <= 22000000.0)
		tmp = t_1;
	elseif (re <= 2e+29)
		tmp = t_2;
	elseif (re <= 3.8e+47)
		tmp = t_3;
	elseif (re <= 1.4e+49)
		tmp = sin(im);
	elseif (re <= 1e+59)
		tmp = t_1;
	elseif (re <= 2e+60)
		tmp = t_3;
	elseif (re <= 5e+60)
		tmp = t_1;
	elseif (re <= 9e+67)
		tmp = t_3;
	elseif (re <= 9.5e+67)
		tmp = sin(im);
	elseif (re <= 9.2e+72)
		tmp = t_3;
	elseif (re <= 9.5e+72)
		tmp = sin(im);
	elseif (re <= 7.2e+91)
		tmp = t_1;
	elseif (re <= 7.2e+92)
		tmp = sin(im);
	elseif (re <= 1.65e+133)
		tmp = t_2;
	elseif (re <= 1.7e+133)
		tmp = sin(im);
	elseif (re <= 6.2e+153)
		tmp = t_2;
	elseif ((re <= 5e+158) || (!(re <= 5e+197) && ((re <= 3.2e+203) || (!(re <= 2.3e+246) && ((re <= 2.35e+246) || (!(re <= 1.08e+248) && ((re <= 1.8e+252) || (!(re <= 7.2e+299) && (re <= 7.5e+299)))))))))
		tmp = sin(im);
	else
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (re * 0.16666666666666666));
	t_1 = im + (re * (im + (im * t_0)));
	t_2 = im + (im * (re * (1.0 + t_0)));
	t_3 = im + (re * (im + (re * ((re * im) * 0.16666666666666666))));
	tmp = 0.0;
	if (re <= -4.5e+126)
		tmp = re * (im / re);
	elseif (re <= 4.9e-30)
		tmp = sin(im);
	elseif (re <= 2050000.0)
		tmp = t_1;
	elseif (re <= 19000000.0)
		tmp = sin(im);
	elseif (re <= 22000000.0)
		tmp = t_1;
	elseif (re <= 2e+29)
		tmp = t_2;
	elseif (re <= 3.8e+47)
		tmp = t_3;
	elseif (re <= 1.4e+49)
		tmp = sin(im);
	elseif (re <= 1e+59)
		tmp = t_1;
	elseif (re <= 2e+60)
		tmp = t_3;
	elseif (re <= 5e+60)
		tmp = t_1;
	elseif (re <= 9e+67)
		tmp = t_3;
	elseif (re <= 9.5e+67)
		tmp = sin(im);
	elseif (re <= 9.2e+72)
		tmp = t_3;
	elseif (re <= 9.5e+72)
		tmp = sin(im);
	elseif (re <= 7.2e+91)
		tmp = t_1;
	elseif (re <= 7.2e+92)
		tmp = sin(im);
	elseif (re <= 1.65e+133)
		tmp = t_2;
	elseif (re <= 1.7e+133)
		tmp = sin(im);
	elseif (re <= 6.2e+153)
		tmp = t_2;
	elseif ((re <= 5e+158) || (~((re <= 5e+197)) && ((re <= 3.2e+203) || (~((re <= 2.3e+246)) && ((re <= 2.35e+246) || (~((re <= 1.08e+248)) && ((re <= 1.8e+252) || (~((re <= 7.2e+299)) && (re <= 7.5e+299)))))))))
		tmp = sin(im);
	else
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	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[(im + N[(re * N[(im + N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im + N[(im * N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(im + N[(re * N[(im + N[(re * N[(N[(re * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4.5e+126], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.9e-30], N[Sin[im], $MachinePrecision], If[LessEqual[re, 2050000.0], t$95$1, If[LessEqual[re, 19000000.0], N[Sin[im], $MachinePrecision], If[LessEqual[re, 22000000.0], t$95$1, If[LessEqual[re, 2e+29], t$95$2, If[LessEqual[re, 3.8e+47], t$95$3, If[LessEqual[re, 1.4e+49], N[Sin[im], $MachinePrecision], If[LessEqual[re, 1e+59], t$95$1, If[LessEqual[re, 2e+60], t$95$3, If[LessEqual[re, 5e+60], t$95$1, If[LessEqual[re, 9e+67], t$95$3, If[LessEqual[re, 9.5e+67], N[Sin[im], $MachinePrecision], If[LessEqual[re, 9.2e+72], t$95$3, If[LessEqual[re, 9.5e+72], N[Sin[im], $MachinePrecision], If[LessEqual[re, 7.2e+91], t$95$1, If[LessEqual[re, 7.2e+92], N[Sin[im], $MachinePrecision], If[LessEqual[re, 1.65e+133], t$95$2, If[LessEqual[re, 1.7e+133], N[Sin[im], $MachinePrecision], If[LessEqual[re, 6.2e+153], t$95$2, If[Or[LessEqual[re, 5e+158], And[N[Not[LessEqual[re, 5e+197]], $MachinePrecision], Or[LessEqual[re, 3.2e+203], And[N[Not[LessEqual[re, 2.3e+246]], $MachinePrecision], Or[LessEqual[re, 2.35e+246], And[N[Not[LessEqual[re, 1.08e+248]], $MachinePrecision], Or[LessEqual[re, 1.8e+252], And[N[Not[LessEqual[re, 7.2e+299]], $MachinePrecision], LessEqual[re, 7.5e+299]]]]]]]]], N[Sin[im], $MachinePrecision], N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if re < -4.49999999999999974e126

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.5%

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

      1. distribute-rgt1-in 2.5%

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

   5. Simplified 2.5%

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

   6. Taylor expanded in re around inf 2.5%

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

   7. Taylor expanded in im around 0 2.3%

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

   8. Taylor expanded in re around 0 45.9%

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

   if -4.49999999999999974e126 < re < 4.89999999999999971e-30 or 2.05e6 < re < 1.9e7 or 3.8000000000000003e47 < re < 1.3999999999999999e49 or 8.9999999999999997e67 < re < 9.5000000000000002e67 or 9.199999999999999e72 < re < 9.50000000000000054e72 or 7.2e91 < re < 7.2e92 or 1.65e133 < re < 1.69999999999999994e133 or 6.2e153 < re < 4.9999999999999996e158 or 5.00000000000000009e197 < re < 3.1999999999999997e203 or 2.30000000000000014e246 < re < 2.34999999999999988e246 or 1.08e248 < re < 1.7999999999999999e252 or 7.20000000000000008e299 < re < 7.50000000000000039e299

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 75.4%

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

   if 4.89999999999999971e-30 < re < 2.05e6 or 1.9e7 < re < 2.2e7 or 1.3999999999999999e49 < re < 9.99999999999999972e58 or 1.9999999999999999e60 < re < 4.99999999999999975e60 or 9.50000000000000054e72 < re < 7.2e91

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0 99.9%

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

   4. Taylor expanded in re around 0 44.7%

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

   5. Taylor expanded in im around 0 44.7%

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

      1. *-commutative 44.7%

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

   7. Simplified 44.7%

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

   if 2.2e7 < re < 1.99999999999999983e29 or 7.2e92 < re < 1.65e133 or 1.69999999999999994e133 < re < 6.2e153

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 43.6%

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

   5. Taylor expanded in im around 0 67.5%

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

      1. *-commutative 67.5%

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

   7. Simplified 67.5%

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

   if 1.99999999999999983e29 < re < 3.8000000000000003e47 or 9.99999999999999972e58 < re < 1.9999999999999999e60 or 4.99999999999999975e60 < re < 8.9999999999999997e67 or 9.5000000000000002e67 < re < 9.199999999999999e72

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 4.2%

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

   5. Taylor expanded in re around inf 4.2%

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

   if 4.9999999999999996e158 < re < 5.00000000000000009e197 or 3.1999999999999997e203 < re < 2.30000000000000014e246 or 2.34999999999999988e246 < re < 1.08e248 or 1.7999999999999999e252 < re < 7.20000000000000008e299 or 7.50000000000000039e299 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 78.7%

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

      1. *-commutative 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in im around 0 100.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+126}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 2050000:\\ \;\;\;\;im + re \cdot \left(im + im \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 19000000:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 22000000:\\ \;\;\;\;im + re \cdot \left(im + im \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+29}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+47}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(\left(re \cdot im\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+49}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 10^{+59}:\\ \;\;\;\;im + re \cdot \left(im + im \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+60}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(\left(re \cdot im\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+60}:\\ \;\;\;\;im + re \cdot \left(im + im \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+67}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(\left(re \cdot im\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{+67}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 9.2 \cdot 10^{+72}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(\left(re \cdot im\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{+72}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+91}:\\ \;\;\;\;im + re \cdot \left(im + im \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+92}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+158} \lor \neg \left(re \leq 5 \cdot 10^{+197}\right) \land \left(re \leq 3.2 \cdot 10^{+203} \lor \neg \left(re \leq 2.3 \cdot 10^{+246}\right) \land \left(re \leq 2.35 \cdot 10^{+246} \lor \neg \left(re \leq 1.08 \cdot 10^{+248}\right) \land \left(re \leq 1.8 \cdot 10^{+252} \lor \neg \left(re \leq 7.2 \cdot 10^{+299}\right) \land re \leq 7.5 \cdot 10^{+299}\right)\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \frac{im}{re}\\ t_1 := re \cdot \left(im + \frac{im}{re}\right)\\ t_2 := im + re \cdot im\\ t_3 := re \cdot \left(im \cdot \left(1 + \frac{1}{re}\right)\right)\\ \mathbf{if}\;re \leq -18:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 10^{+209}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+211}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 10^{+214}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+246}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.08 \cdot 10^{+248}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{+252}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 10^{+268}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+272}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+278}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+299}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+299}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (/ im re)))
        (t_1 (* re (+ im (/ im re))))
        (t_2 (+ im (* re im)))
        (t_3 (* re (* im (+ 1.0 (/ 1.0 re))))))
   (if (<= re -18.0)
     t_0
     (if (<= re 5e+197)
       t_2
       (if (<= re 3.2e+203)
         t_0
         (if (<= re 1e+209)
           t_2
           (if (<= re 2e+211)
             (* re im)
             (if (<= re 1e+214)
               t_2
               (if (<= re 2.3e+246)
                 t_3
                 (if (<= re 2.35e+246)
                   im
                   (if (<= re 1.08e+248)
                     t_1
                     (if (<= re 1.8e+252)
                       im
                       (if (<= re 1e+268)
                         t_3
                         (if (<= re 5e+272)
                           (* re im)
                           (if (<= re 5e+278)
                             t_1
                             (if (<= re 7.2e+299)
                               t_3
                               (if (<= re 7.5e+299) im t_1)))))))))))))))))

C

double code(double re, double im) {
	double t_0 = re * (im / re);
	double t_1 = re * (im + (im / re));
	double t_2 = im + (re * im);
	double t_3 = re * (im * (1.0 + (1.0 / re)));
	double tmp;
	if (re <= -18.0) {
		tmp = t_0;
	} else if (re <= 5e+197) {
		tmp = t_2;
	} else if (re <= 3.2e+203) {
		tmp = t_0;
	} else if (re <= 1e+209) {
		tmp = t_2;
	} else if (re <= 2e+211) {
		tmp = re * im;
	} else if (re <= 1e+214) {
		tmp = t_2;
	} else if (re <= 2.3e+246) {
		tmp = t_3;
	} else if (re <= 2.35e+246) {
		tmp = im;
	} else if (re <= 1.08e+248) {
		tmp = t_1;
	} else if (re <= 1.8e+252) {
		tmp = im;
	} else if (re <= 1e+268) {
		tmp = t_3;
	} else if (re <= 5e+272) {
		tmp = re * im;
	} else if (re <= 5e+278) {
		tmp = t_1;
	} else if (re <= 7.2e+299) {
		tmp = t_3;
	} else if (re <= 7.5e+299) {
		tmp = im;
	} else {
		tmp = 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) :: t_3
    real(8) :: tmp
    t_0 = re * (im / re)
    t_1 = re * (im + (im / re))
    t_2 = im + (re * im)
    t_3 = re * (im * (1.0d0 + (1.0d0 / re)))
    if (re <= (-18.0d0)) then
        tmp = t_0
    else if (re <= 5d+197) then
        tmp = t_2
    else if (re <= 3.2d+203) then
        tmp = t_0
    else if (re <= 1d+209) then
        tmp = t_2
    else if (re <= 2d+211) then
        tmp = re * im
    else if (re <= 1d+214) then
        tmp = t_2
    else if (re <= 2.3d+246) then
        tmp = t_3
    else if (re <= 2.35d+246) then
        tmp = im
    else if (re <= 1.08d+248) then
        tmp = t_1
    else if (re <= 1.8d+252) then
        tmp = im
    else if (re <= 1d+268) then
        tmp = t_3
    else if (re <= 5d+272) then
        tmp = re * im
    else if (re <= 5d+278) then
        tmp = t_1
    else if (re <= 7.2d+299) then
        tmp = t_3
    else if (re <= 7.5d+299) then
        tmp = im
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (im / re);
	double t_1 = re * (im + (im / re));
	double t_2 = im + (re * im);
	double t_3 = re * (im * (1.0 + (1.0 / re)));
	double tmp;
	if (re <= -18.0) {
		tmp = t_0;
	} else if (re <= 5e+197) {
		tmp = t_2;
	} else if (re <= 3.2e+203) {
		tmp = t_0;
	} else if (re <= 1e+209) {
		tmp = t_2;
	} else if (re <= 2e+211) {
		tmp = re * im;
	} else if (re <= 1e+214) {
		tmp = t_2;
	} else if (re <= 2.3e+246) {
		tmp = t_3;
	} else if (re <= 2.35e+246) {
		tmp = im;
	} else if (re <= 1.08e+248) {
		tmp = t_1;
	} else if (re <= 1.8e+252) {
		tmp = im;
	} else if (re <= 1e+268) {
		tmp = t_3;
	} else if (re <= 5e+272) {
		tmp = re * im;
	} else if (re <= 5e+278) {
		tmp = t_1;
	} else if (re <= 7.2e+299) {
		tmp = t_3;
	} else if (re <= 7.5e+299) {
		tmp = im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (im / re)
	t_1 = re * (im + (im / re))
	t_2 = im + (re * im)
	t_3 = re * (im * (1.0 + (1.0 / re)))
	tmp = 0
	if re <= -18.0:
		tmp = t_0
	elif re <= 5e+197:
		tmp = t_2
	elif re <= 3.2e+203:
		tmp = t_0
	elif re <= 1e+209:
		tmp = t_2
	elif re <= 2e+211:
		tmp = re * im
	elif re <= 1e+214:
		tmp = t_2
	elif re <= 2.3e+246:
		tmp = t_3
	elif re <= 2.35e+246:
		tmp = im
	elif re <= 1.08e+248:
		tmp = t_1
	elif re <= 1.8e+252:
		tmp = im
	elif re <= 1e+268:
		tmp = t_3
	elif re <= 5e+272:
		tmp = re * im
	elif re <= 5e+278:
		tmp = t_1
	elif re <= 7.2e+299:
		tmp = t_3
	elif re <= 7.5e+299:
		tmp = im
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(im / re))
	t_1 = Float64(re * Float64(im + Float64(im / re)))
	t_2 = Float64(im + Float64(re * im))
	t_3 = Float64(re * Float64(im * Float64(1.0 + Float64(1.0 / re))))
	tmp = 0.0
	if (re <= -18.0)
		tmp = t_0;
	elseif (re <= 5e+197)
		tmp = t_2;
	elseif (re <= 3.2e+203)
		tmp = t_0;
	elseif (re <= 1e+209)
		tmp = t_2;
	elseif (re <= 2e+211)
		tmp = Float64(re * im);
	elseif (re <= 1e+214)
		tmp = t_2;
	elseif (re <= 2.3e+246)
		tmp = t_3;
	elseif (re <= 2.35e+246)
		tmp = im;
	elseif (re <= 1.08e+248)
		tmp = t_1;
	elseif (re <= 1.8e+252)
		tmp = im;
	elseif (re <= 1e+268)
		tmp = t_3;
	elseif (re <= 5e+272)
		tmp = Float64(re * im);
	elseif (re <= 5e+278)
		tmp = t_1;
	elseif (re <= 7.2e+299)
		tmp = t_3;
	elseif (re <= 7.5e+299)
		tmp = im;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (im / re);
	t_1 = re * (im + (im / re));
	t_2 = im + (re * im);
	t_3 = re * (im * (1.0 + (1.0 / re)));
	tmp = 0.0;
	if (re <= -18.0)
		tmp = t_0;
	elseif (re <= 5e+197)
		tmp = t_2;
	elseif (re <= 3.2e+203)
		tmp = t_0;
	elseif (re <= 1e+209)
		tmp = t_2;
	elseif (re <= 2e+211)
		tmp = re * im;
	elseif (re <= 1e+214)
		tmp = t_2;
	elseif (re <= 2.3e+246)
		tmp = t_3;
	elseif (re <= 2.35e+246)
		tmp = im;
	elseif (re <= 1.08e+248)
		tmp = t_1;
	elseif (re <= 1.8e+252)
		tmp = im;
	elseif (re <= 1e+268)
		tmp = t_3;
	elseif (re <= 5e+272)
		tmp = re * im;
	elseif (re <= 5e+278)
		tmp = t_1;
	elseif (re <= 7.2e+299)
		tmp = t_3;
	elseif (re <= 7.5e+299)
		tmp = im;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(im + N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(re * N[(im * N[(1.0 + N[(1.0 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -18.0], t$95$0, If[LessEqual[re, 5e+197], t$95$2, If[LessEqual[re, 3.2e+203], t$95$0, If[LessEqual[re, 1e+209], t$95$2, If[LessEqual[re, 2e+211], N[(re * im), $MachinePrecision], If[LessEqual[re, 1e+214], t$95$2, If[LessEqual[re, 2.3e+246], t$95$3, If[LessEqual[re, 2.35e+246], im, If[LessEqual[re, 1.08e+248], t$95$1, If[LessEqual[re, 1.8e+252], im, If[LessEqual[re, 1e+268], t$95$3, If[LessEqual[re, 5e+272], N[(re * im), $MachinePrecision], If[LessEqual[re, 5e+278], t$95$1, If[LessEqual[re, 7.2e+299], t$95$3, If[LessEqual[re, 7.5e+299], im, t$95$1]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if re < -18 or 5.00000000000000009e197 < re < 3.1999999999999997e203

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in re around inf 2.9%

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

   7. Taylor expanded in im around 0 2.5%

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

   8. Taylor expanded in re around 0 28.9%

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

   if -18 < re < 5.00000000000000009e197 or 3.1999999999999997e203 < re < 1.0000000000000001e209 or 1.9999999999999999e211 < re < 9.9999999999999995e213

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 56.1%

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

   4. Taylor expanded in re around 0 37.0%

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

   if 1.0000000000000001e209 < re < 1.9999999999999999e211 or 9.9999999999999997e267 < re < 4.99999999999999973e272

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.7%

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

      1. distribute-rgt1-in 4.7%

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

   5. Simplified 4.7%

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

   6. Taylor expanded in re around inf 4.7%

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

   7. Taylor expanded in im around 0 4.7%

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

   if 9.9999999999999995e213 < re < 2.30000000000000014e246 or 1.7999999999999999e252 < re < 9.9999999999999997e267 or 5.00000000000000029e278 < re < 7.20000000000000008e299

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.6%

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

      1. distribute-rgt1-in 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in re around inf 5.6%

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

   7. Taylor expanded in im around 0 40.5%

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

   if 2.30000000000000014e246 < re < 2.34999999999999988e246 or 1.08e248 < re < 1.7999999999999999e252 or 7.20000000000000008e299 < re < 7.50000000000000039e299

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 0.0%

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

   4. Taylor expanded in re around 0 0.2%

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

   if 2.34999999999999988e246 < re < 1.08e248 or 4.99999999999999973e272 < re < 5.00000000000000029e278 or 7.50000000000000039e299 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 6.7%

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

      1. distribute-rgt1-in 6.7%

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

   5. Simplified 6.7%

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

   6. Taylor expanded in re around inf 6.7%

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

   7. Taylor expanded in im around 0 23.9%

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

      1. distribute-rgt-in 23.9%

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

      2. *-lft-identity 23.9%

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

      3. associate-*l/ 23.9%

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

      4. *-lft-identity 23.9%

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

   9. Simplified 23.9%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -18:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 10^{+209}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+211}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 10^{+214}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246}:\\ \;\;\;\;re \cdot \left(im \cdot \left(1 + \frac{1}{re}\right)\right)\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+246}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.08 \cdot 10^{+248}:\\ \;\;\;\;re \cdot \left(im + \frac{im}{re}\right)\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{+252}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 10^{+268}:\\ \;\;\;\;re \cdot \left(im \cdot \left(1 + \frac{1}{re}\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+272}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+278}:\\ \;\;\;\;re \cdot \left(im + \frac{im}{re}\right)\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+299}:\\ \;\;\;\;re \cdot \left(im \cdot \left(1 + \frac{1}{re}\right)\right)\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+299}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im + \frac{im}{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ t_1 := re \cdot \frac{im}{re}\\ t_2 := im + re \cdot \left(im + re \cdot \left(\left(re \cdot im\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+158}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246} \lor \neg \left(re \leq 2.35 \cdot 10^{+246}\right) \land \left(re \leq 1.08 \cdot 10^{+248} \lor \neg \left(re \leq 1.8 \cdot 10^{+252}\right) \land \left(re \leq 7.2 \cdot 10^{+299} \lor \neg \left(re \leq 7.5 \cdot 10^{+299}\right)\right)\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ im (* im (* re (+ 1.0 (* re 0.5))))))
        (t_1 (* re (/ im re)))
        (t_2 (+ im (* re (+ im (* re (* (* re im) 0.16666666666666666)))))))
   (if (<= re -2.0)
     t_1
     (if (<= re 2.4e+112)
       t_0
       (if (<= re 1.65e+133)
         t_2
         (if (<= re 1.7e+133)
           im
           (if (<= re 6.2e+153)
             t_2
             (if (<= re 5e+158)
               im
               (if (<= re 5e+197)
                 t_0
                 (if (<= re 3.2e+203)
                   t_1
                   (if (or (<= re 2.3e+246)
                           (and (not (<= re 2.35e+246))
                                (or (<= re 1.08e+248)
                                    (and (not (<= re 1.8e+252))
                                         (or (<= re 7.2e+299)
                                             (not (<= re 7.5e+299)))))))
                     t_0
                     im)))))))))))

C

double code(double re, double im) {
	double t_0 = im + (im * (re * (1.0 + (re * 0.5))));
	double t_1 = re * (im / re);
	double t_2 = im + (re * (im + (re * ((re * im) * 0.16666666666666666))));
	double tmp;
	if (re <= -2.0) {
		tmp = t_1;
	} else if (re <= 2.4e+112) {
		tmp = t_0;
	} else if (re <= 1.65e+133) {
		tmp = t_2;
	} else if (re <= 1.7e+133) {
		tmp = im;
	} else if (re <= 6.2e+153) {
		tmp = t_2;
	} else if (re <= 5e+158) {
		tmp = im;
	} else if (re <= 5e+197) {
		tmp = t_0;
	} else if (re <= 3.2e+203) {
		tmp = t_1;
	} else if ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299)))))) {
		tmp = t_0;
	} else {
		tmp = im;
	}
	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 = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    t_1 = re * (im / re)
    t_2 = im + (re * (im + (re * ((re * im) * 0.16666666666666666d0))))
    if (re <= (-2.0d0)) then
        tmp = t_1
    else if (re <= 2.4d+112) then
        tmp = t_0
    else if (re <= 1.65d+133) then
        tmp = t_2
    else if (re <= 1.7d+133) then
        tmp = im
    else if (re <= 6.2d+153) then
        tmp = t_2
    else if (re <= 5d+158) then
        tmp = im
    else if (re <= 5d+197) then
        tmp = t_0
    else if (re <= 3.2d+203) then
        tmp = t_1
    else if ((re <= 2.3d+246) .or. (.not. (re <= 2.35d+246)) .and. (re <= 1.08d+248) .or. (.not. (re <= 1.8d+252)) .and. (re <= 7.2d+299) .or. (.not. (re <= 7.5d+299))) then
        tmp = t_0
    else
        tmp = im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im + (im * (re * (1.0 + (re * 0.5))));
	double t_1 = re * (im / re);
	double t_2 = im + (re * (im + (re * ((re * im) * 0.16666666666666666))));
	double tmp;
	if (re <= -2.0) {
		tmp = t_1;
	} else if (re <= 2.4e+112) {
		tmp = t_0;
	} else if (re <= 1.65e+133) {
		tmp = t_2;
	} else if (re <= 1.7e+133) {
		tmp = im;
	} else if (re <= 6.2e+153) {
		tmp = t_2;
	} else if (re <= 5e+158) {
		tmp = im;
	} else if (re <= 5e+197) {
		tmp = t_0;
	} else if (re <= 3.2e+203) {
		tmp = t_1;
	} else if ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299)))))) {
		tmp = t_0;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im + (im * (re * (1.0 + (re * 0.5))))
	t_1 = re * (im / re)
	t_2 = im + (re * (im + (re * ((re * im) * 0.16666666666666666))))
	tmp = 0
	if re <= -2.0:
		tmp = t_1
	elif re <= 2.4e+112:
		tmp = t_0
	elif re <= 1.65e+133:
		tmp = t_2
	elif re <= 1.7e+133:
		tmp = im
	elif re <= 6.2e+153:
		tmp = t_2
	elif re <= 5e+158:
		tmp = im
	elif re <= 5e+197:
		tmp = t_0
	elif re <= 3.2e+203:
		tmp = t_1
	elif (re <= 2.3e+246) or (not (re <= 2.35e+246) and ((re <= 1.08e+248) or (not (re <= 1.8e+252) and ((re <= 7.2e+299) or not (re <= 7.5e+299))))):
		tmp = t_0
	else:
		tmp = im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))))
	t_1 = Float64(re * Float64(im / re))
	t_2 = Float64(im + Float64(re * Float64(im + Float64(re * Float64(Float64(re * im) * 0.16666666666666666)))))
	tmp = 0.0
	if (re <= -2.0)
		tmp = t_1;
	elseif (re <= 2.4e+112)
		tmp = t_0;
	elseif (re <= 1.65e+133)
		tmp = t_2;
	elseif (re <= 1.7e+133)
		tmp = im;
	elseif (re <= 6.2e+153)
		tmp = t_2;
	elseif (re <= 5e+158)
		tmp = im;
	elseif (re <= 5e+197)
		tmp = t_0;
	elseif (re <= 3.2e+203)
		tmp = t_1;
	elseif ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299))))))
		tmp = t_0;
	else
		tmp = im;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im + (im * (re * (1.0 + (re * 0.5))));
	t_1 = re * (im / re);
	t_2 = im + (re * (im + (re * ((re * im) * 0.16666666666666666))));
	tmp = 0.0;
	if (re <= -2.0)
		tmp = t_1;
	elseif (re <= 2.4e+112)
		tmp = t_0;
	elseif (re <= 1.65e+133)
		tmp = t_2;
	elseif (re <= 1.7e+133)
		tmp = im;
	elseif (re <= 6.2e+153)
		tmp = t_2;
	elseif (re <= 5e+158)
		tmp = im;
	elseif (re <= 5e+197)
		tmp = t_0;
	elseif (re <= 3.2e+203)
		tmp = t_1;
	elseif ((re <= 2.3e+246) || (~((re <= 2.35e+246)) && ((re <= 1.08e+248) || (~((re <= 1.8e+252)) && ((re <= 7.2e+299) || ~((re <= 7.5e+299)))))))
		tmp = t_0;
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im + N[(re * N[(im + N[(re * N[(N[(re * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.0], t$95$1, If[LessEqual[re, 2.4e+112], t$95$0, If[LessEqual[re, 1.65e+133], t$95$2, If[LessEqual[re, 1.7e+133], im, If[LessEqual[re, 6.2e+153], t$95$2, If[LessEqual[re, 5e+158], im, If[LessEqual[re, 5e+197], t$95$0, If[LessEqual[re, 3.2e+203], t$95$1, If[Or[LessEqual[re, 2.3e+246], And[N[Not[LessEqual[re, 2.35e+246]], $MachinePrecision], Or[LessEqual[re, 1.08e+248], And[N[Not[LessEqual[re, 1.8e+252]], $MachinePrecision], Or[LessEqual[re, 7.2e+299], N[Not[LessEqual[re, 7.5e+299]], $MachinePrecision]]]]]], t$95$0, im]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2 or 5.00000000000000009e197 < re < 3.1999999999999997e203

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in re around inf 2.8%

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

   7. Taylor expanded in im around 0 2.5%

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

   8. Taylor expanded in re around 0 27.5%

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

   if -2 < re < 2.4e112 or 4.9999999999999996e158 < re < 5.00000000000000009e197 or 3.1999999999999997e203 < re < 2.30000000000000014e246 or 2.34999999999999988e246 < re < 1.08e248 or 1.7999999999999999e252 < re < 7.20000000000000008e299 or 7.50000000000000039e299 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 60.1%

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

   4. Taylor expanded in re around 0 46.3%

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

      1. *-commutative 46.3%

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

   6. Simplified 46.3%

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

   7. Taylor expanded in im around 0 48.9%

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

   if 2.4e112 < re < 1.65e133 or 1.69999999999999994e133 < re < 6.2e153

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 52.3%

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

   5. Taylor expanded in re around inf 52.3%

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

   if 1.65e133 < re < 1.69999999999999994e133 or 6.2e153 < re < 4.9999999999999996e158 or 2.30000000000000014e246 < re < 2.34999999999999988e246 or 1.08e248 < re < 1.7999999999999999e252 or 7.20000000000000008e299 < re < 7.50000000000000039e299

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 0.0%

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

   4. Taylor expanded in re around 0 0.2%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+112}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(\left(re \cdot im\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(\left(re \cdot im\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+158}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246} \lor \neg \left(re \leq 2.35 \cdot 10^{+246}\right) \land \left(re \leq 1.08 \cdot 10^{+248} \lor \neg \left(re \leq 1.8 \cdot 10^{+252}\right) \land \left(re \leq 7.2 \cdot 10^{+299} \lor \neg \left(re \leq 7.5 \cdot 10^{+299}\right)\right)\right):\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ t_1 := re \cdot \frac{im}{re}\\ t_2 := im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;re \leq -18:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+158}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246} \lor \neg \left(re \leq 2.35 \cdot 10^{+246}\right) \land \left(re \leq 1.08 \cdot 10^{+248} \lor \neg \left(re \leq 1.8 \cdot 10^{+252}\right) \land \left(re \leq 7.2 \cdot 10^{+299} \lor \neg \left(re \leq 7.5 \cdot 10^{+299}\right)\right)\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ im (* im (* re (+ 1.0 (* re 0.5))))))
        (t_1 (* re (/ im re)))
        (t_2
         (+
          im
          (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))
   (if (<= re -18.0)
     t_1
     (if (<= re 1.65e+133)
       t_2
       (if (<= re 1.7e+133)
         im
         (if (<= re 6.2e+153)
           t_2
           (if (<= re 5e+158)
             im
             (if (<= re 5e+197)
               t_0
               (if (<= re 3.2e+203)
                 t_1
                 (if (or (<= re 2.3e+246)
                         (and (not (<= re 2.35e+246))
                              (or (<= re 1.08e+248)
                                  (and (not (<= re 1.8e+252))
                                       (or (<= re 7.2e+299)
                                           (not (<= re 7.5e+299)))))))
                   t_0
                   im))))))))))

C

double code(double re, double im) {
	double t_0 = im + (im * (re * (1.0 + (re * 0.5))));
	double t_1 = re * (im / re);
	double t_2 = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -18.0) {
		tmp = t_1;
	} else if (re <= 1.65e+133) {
		tmp = t_2;
	} else if (re <= 1.7e+133) {
		tmp = im;
	} else if (re <= 6.2e+153) {
		tmp = t_2;
	} else if (re <= 5e+158) {
		tmp = im;
	} else if (re <= 5e+197) {
		tmp = t_0;
	} else if (re <= 3.2e+203) {
		tmp = t_1;
	} else if ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299)))))) {
		tmp = t_0;
	} else {
		tmp = im;
	}
	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 = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    t_1 = re * (im / re)
    t_2 = im + (im * (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    if (re <= (-18.0d0)) then
        tmp = t_1
    else if (re <= 1.65d+133) then
        tmp = t_2
    else if (re <= 1.7d+133) then
        tmp = im
    else if (re <= 6.2d+153) then
        tmp = t_2
    else if (re <= 5d+158) then
        tmp = im
    else if (re <= 5d+197) then
        tmp = t_0
    else if (re <= 3.2d+203) then
        tmp = t_1
    else if ((re <= 2.3d+246) .or. (.not. (re <= 2.35d+246)) .and. (re <= 1.08d+248) .or. (.not. (re <= 1.8d+252)) .and. (re <= 7.2d+299) .or. (.not. (re <= 7.5d+299))) then
        tmp = t_0
    else
        tmp = im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im + (im * (re * (1.0 + (re * 0.5))));
	double t_1 = re * (im / re);
	double t_2 = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -18.0) {
		tmp = t_1;
	} else if (re <= 1.65e+133) {
		tmp = t_2;
	} else if (re <= 1.7e+133) {
		tmp = im;
	} else if (re <= 6.2e+153) {
		tmp = t_2;
	} else if (re <= 5e+158) {
		tmp = im;
	} else if (re <= 5e+197) {
		tmp = t_0;
	} else if (re <= 3.2e+203) {
		tmp = t_1;
	} else if ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299)))))) {
		tmp = t_0;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im + (im * (re * (1.0 + (re * 0.5))))
	t_1 = re * (im / re)
	t_2 = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	tmp = 0
	if re <= -18.0:
		tmp = t_1
	elif re <= 1.65e+133:
		tmp = t_2
	elif re <= 1.7e+133:
		tmp = im
	elif re <= 6.2e+153:
		tmp = t_2
	elif re <= 5e+158:
		tmp = im
	elif re <= 5e+197:
		tmp = t_0
	elif re <= 3.2e+203:
		tmp = t_1
	elif (re <= 2.3e+246) or (not (re <= 2.35e+246) and ((re <= 1.08e+248) or (not (re <= 1.8e+252) and ((re <= 7.2e+299) or not (re <= 7.5e+299))))):
		tmp = t_0
	else:
		tmp = im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))))
	t_1 = Float64(re * Float64(im / re))
	t_2 = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
	tmp = 0.0
	if (re <= -18.0)
		tmp = t_1;
	elseif (re <= 1.65e+133)
		tmp = t_2;
	elseif (re <= 1.7e+133)
		tmp = im;
	elseif (re <= 6.2e+153)
		tmp = t_2;
	elseif (re <= 5e+158)
		tmp = im;
	elseif (re <= 5e+197)
		tmp = t_0;
	elseif (re <= 3.2e+203)
		tmp = t_1;
	elseif ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299))))))
		tmp = t_0;
	else
		tmp = im;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im + (im * (re * (1.0 + (re * 0.5))));
	t_1 = re * (im / re);
	t_2 = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	tmp = 0.0;
	if (re <= -18.0)
		tmp = t_1;
	elseif (re <= 1.65e+133)
		tmp = t_2;
	elseif (re <= 1.7e+133)
		tmp = im;
	elseif (re <= 6.2e+153)
		tmp = t_2;
	elseif (re <= 5e+158)
		tmp = im;
	elseif (re <= 5e+197)
		tmp = t_0;
	elseif (re <= 3.2e+203)
		tmp = t_1;
	elseif ((re <= 2.3e+246) || (~((re <= 2.35e+246)) && ((re <= 1.08e+248) || (~((re <= 1.8e+252)) && ((re <= 7.2e+299) || ~((re <= 7.5e+299)))))))
		tmp = t_0;
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im + N[(im * N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -18.0], t$95$1, If[LessEqual[re, 1.65e+133], t$95$2, If[LessEqual[re, 1.7e+133], im, If[LessEqual[re, 6.2e+153], t$95$2, If[LessEqual[re, 5e+158], im, If[LessEqual[re, 5e+197], t$95$0, If[LessEqual[re, 3.2e+203], t$95$1, If[Or[LessEqual[re, 2.3e+246], And[N[Not[LessEqual[re, 2.35e+246]], $MachinePrecision], Or[LessEqual[re, 1.08e+248], And[N[Not[LessEqual[re, 1.8e+252]], $MachinePrecision], Or[LessEqual[re, 7.2e+299], N[Not[LessEqual[re, 7.5e+299]], $MachinePrecision]]]]]], t$95$0, im]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -18 or 5.00000000000000009e197 < re < 3.1999999999999997e203

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in re around inf 2.9%

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

   7. Taylor expanded in im around 0 2.5%

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

   8. Taylor expanded in re around 0 28.9%

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

   if -18 < re < 1.65e133 or 1.69999999999999994e133 < re < 6.2e153

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 55.2%

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

   4. Taylor expanded in re around 0 41.8%

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

   5. Taylor expanded in im around 0 43.5%

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

      1. *-commutative 43.5%

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

   7. Simplified 43.5%

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

   if 1.65e133 < re < 1.69999999999999994e133 or 6.2e153 < re < 4.9999999999999996e158 or 2.30000000000000014e246 < re < 2.34999999999999988e246 or 1.08e248 < re < 1.7999999999999999e252 or 7.20000000000000008e299 < re < 7.50000000000000039e299

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 0.0%

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

   4. Taylor expanded in re around 0 0.2%

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

   if 4.9999999999999996e158 < re < 5.00000000000000009e197 or 3.1999999999999997e203 < re < 2.30000000000000014e246 or 2.34999999999999988e246 < re < 1.08e248 or 1.7999999999999999e252 < re < 7.20000000000000008e299 or 7.50000000000000039e299 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 78.7%

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

      1. *-commutative 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in im around 0 100.0%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -18:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+158}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246} \lor \neg \left(re \leq 2.35 \cdot 10^{+246}\right) \land \left(re \leq 1.08 \cdot 10^{+248} \lor \neg \left(re \leq 1.8 \cdot 10^{+252}\right) \land \left(re \leq 7.2 \cdot 10^{+299} \lor \neg \left(re \leq 7.5 \cdot 10^{+299}\right)\right)\right):\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \frac{im}{re}\\ t_1 := im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+158}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246} \lor \neg \left(re \leq 2.35 \cdot 10^{+246}\right) \land \left(re \leq 1.08 \cdot 10^{+248} \lor \neg \left(re \leq 1.8 \cdot 10^{+252}\right) \land \left(re \leq 7.2 \cdot 10^{+299} \lor \neg \left(re \leq 7.5 \cdot 10^{+299}\right)\right)\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (/ im re))) (t_1 (+ im (* im (* re (+ 1.0 (* re 0.5)))))))
   (if (<= re -2.0)
     t_0
     (if (<= re 1.65e+133)
       t_1
       (if (<= re 1.7e+133)
         im
         (if (<= re 6.2e+153)
           t_1
           (if (<= re 5e+158)
             im
             (if (<= re 5e+197)
               t_1
               (if (<= re 3.2e+203)
                 t_0
                 (if (or (<= re 2.3e+246)
                         (and (not (<= re 2.35e+246))
                              (or (<= re 1.08e+248)
                                  (and (not (<= re 1.8e+252))
                                       (or (<= re 7.2e+299)
                                           (not (<= re 7.5e+299)))))))
                   t_1
                   im))))))))))

C

double code(double re, double im) {
	double t_0 = re * (im / re);
	double t_1 = im + (im * (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -2.0) {
		tmp = t_0;
	} else if (re <= 1.65e+133) {
		tmp = t_1;
	} else if (re <= 1.7e+133) {
		tmp = im;
	} else if (re <= 6.2e+153) {
		tmp = t_1;
	} else if (re <= 5e+158) {
		tmp = im;
	} else if (re <= 5e+197) {
		tmp = t_1;
	} else if (re <= 3.2e+203) {
		tmp = t_0;
	} else if ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299)))))) {
		tmp = t_1;
	} else {
		tmp = im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (im / re)
    t_1 = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    if (re <= (-2.0d0)) then
        tmp = t_0
    else if (re <= 1.65d+133) then
        tmp = t_1
    else if (re <= 1.7d+133) then
        tmp = im
    else if (re <= 6.2d+153) then
        tmp = t_1
    else if (re <= 5d+158) then
        tmp = im
    else if (re <= 5d+197) then
        tmp = t_1
    else if (re <= 3.2d+203) then
        tmp = t_0
    else if ((re <= 2.3d+246) .or. (.not. (re <= 2.35d+246)) .and. (re <= 1.08d+248) .or. (.not. (re <= 1.8d+252)) .and. (re <= 7.2d+299) .or. (.not. (re <= 7.5d+299))) then
        tmp = t_1
    else
        tmp = im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (im / re);
	double t_1 = im + (im * (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -2.0) {
		tmp = t_0;
	} else if (re <= 1.65e+133) {
		tmp = t_1;
	} else if (re <= 1.7e+133) {
		tmp = im;
	} else if (re <= 6.2e+153) {
		tmp = t_1;
	} else if (re <= 5e+158) {
		tmp = im;
	} else if (re <= 5e+197) {
		tmp = t_1;
	} else if (re <= 3.2e+203) {
		tmp = t_0;
	} else if ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299)))))) {
		tmp = t_1;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (im / re)
	t_1 = im + (im * (re * (1.0 + (re * 0.5))))
	tmp = 0
	if re <= -2.0:
		tmp = t_0
	elif re <= 1.65e+133:
		tmp = t_1
	elif re <= 1.7e+133:
		tmp = im
	elif re <= 6.2e+153:
		tmp = t_1
	elif re <= 5e+158:
		tmp = im
	elif re <= 5e+197:
		tmp = t_1
	elif re <= 3.2e+203:
		tmp = t_0
	elif (re <= 2.3e+246) or (not (re <= 2.35e+246) and ((re <= 1.08e+248) or (not (re <= 1.8e+252) and ((re <= 7.2e+299) or not (re <= 7.5e+299))))):
		tmp = t_1
	else:
		tmp = im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(im / re))
	t_1 = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))))
	tmp = 0.0
	if (re <= -2.0)
		tmp = t_0;
	elseif (re <= 1.65e+133)
		tmp = t_1;
	elseif (re <= 1.7e+133)
		tmp = im;
	elseif (re <= 6.2e+153)
		tmp = t_1;
	elseif (re <= 5e+158)
		tmp = im;
	elseif (re <= 5e+197)
		tmp = t_1;
	elseif (re <= 3.2e+203)
		tmp = t_0;
	elseif ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299))))))
		tmp = t_1;
	else
		tmp = im;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (im / re);
	t_1 = im + (im * (re * (1.0 + (re * 0.5))));
	tmp = 0.0;
	if (re <= -2.0)
		tmp = t_0;
	elseif (re <= 1.65e+133)
		tmp = t_1;
	elseif (re <= 1.7e+133)
		tmp = im;
	elseif (re <= 6.2e+153)
		tmp = t_1;
	elseif (re <= 5e+158)
		tmp = im;
	elseif (re <= 5e+197)
		tmp = t_1;
	elseif (re <= 3.2e+203)
		tmp = t_0;
	elseif ((re <= 2.3e+246) || (~((re <= 2.35e+246)) && ((re <= 1.08e+248) || (~((re <= 1.8e+252)) && ((re <= 7.2e+299) || ~((re <= 7.5e+299)))))))
		tmp = t_1;
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.0], t$95$0, If[LessEqual[re, 1.65e+133], t$95$1, If[LessEqual[re, 1.7e+133], im, If[LessEqual[re, 6.2e+153], t$95$1, If[LessEqual[re, 5e+158], im, If[LessEqual[re, 5e+197], t$95$1, If[LessEqual[re, 3.2e+203], t$95$0, If[Or[LessEqual[re, 2.3e+246], And[N[Not[LessEqual[re, 2.35e+246]], $MachinePrecision], Or[LessEqual[re, 1.08e+248], And[N[Not[LessEqual[re, 1.8e+252]], $MachinePrecision], Or[LessEqual[re, 7.2e+299], N[Not[LessEqual[re, 7.5e+299]], $MachinePrecision]]]]]], t$95$1, im]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -2 or 5.00000000000000009e197 < re < 3.1999999999999997e203

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in re around inf 2.8%

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

   7. Taylor expanded in im around 0 2.5%

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

   8. Taylor expanded in re around 0 27.5%

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

   if -2 < re < 1.65e133 or 1.69999999999999994e133 < re < 6.2e153 or 4.9999999999999996e158 < re < 5.00000000000000009e197 or 3.1999999999999997e203 < re < 2.30000000000000014e246 or 2.34999999999999988e246 < re < 1.08e248 or 1.7999999999999999e252 < re < 7.20000000000000008e299 or 7.50000000000000039e299 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 61.4%

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

   4. Taylor expanded in re around 0 46.0%

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

      1. *-commutative 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in im around 0 48.5%

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

   if 1.65e133 < re < 1.69999999999999994e133 or 6.2e153 < re < 4.9999999999999996e158 or 2.30000000000000014e246 < re < 2.34999999999999988e246 or 1.08e248 < re < 1.7999999999999999e252 or 7.20000000000000008e299 < re < 7.50000000000000039e299

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 0.0%

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

   4. Taylor expanded in re around 0 0.2%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+158}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246} \lor \neg \left(re \leq 2.35 \cdot 10^{+246}\right) \land \left(re \leq 1.08 \cdot 10^{+248} \lor \neg \left(re \leq 1.8 \cdot 10^{+252}\right) \land \left(re \leq 7.2 \cdot 10^{+299} \lor \neg \left(re \leq 7.5 \cdot 10^{+299}\right)\right)\right):\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(im + \frac{im}{re}\right)\\ t_1 := re \cdot \frac{im}{re}\\ t_2 := im + re \cdot im\\ \mathbf{if}\;re \leq -18:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 10^{+209}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+211}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+246}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.08 \cdot 10^{+248}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{+252}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+272}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+299} \lor \neg \left(re \leq 7.5 \cdot 10^{+299}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ im (/ im re))))
        (t_1 (* re (/ im re)))
        (t_2 (+ im (* re im))))
   (if (<= re -18.0)
     t_1
     (if (<= re 5e+197)
       t_2
       (if (<= re 3.2e+203)
         t_1
         (if (<= re 1e+209)
           t_2
           (if (<= re 2e+211)
             (* re im)
             (if (<= re 2.3e+246)
               t_2
               (if (<= re 2.35e+246)
                 im
                 (if (<= re 1.08e+248)
                   t_0
                   (if (<= re 1.8e+252)
                     im
                     (if (<= re 5e+272)
                       (* re im)
                       (if (or (<= re 7.2e+299) (not (<= re 7.5e+299)))
                         t_0
                         im)))))))))))))

C

double code(double re, double im) {
	double t_0 = re * (im + (im / re));
	double t_1 = re * (im / re);
	double t_2 = im + (re * im);
	double tmp;
	if (re <= -18.0) {
		tmp = t_1;
	} else if (re <= 5e+197) {
		tmp = t_2;
	} else if (re <= 3.2e+203) {
		tmp = t_1;
	} else if (re <= 1e+209) {
		tmp = t_2;
	} else if (re <= 2e+211) {
		tmp = re * im;
	} else if (re <= 2.3e+246) {
		tmp = t_2;
	} else if (re <= 2.35e+246) {
		tmp = im;
	} else if (re <= 1.08e+248) {
		tmp = t_0;
	} else if (re <= 1.8e+252) {
		tmp = im;
	} else if (re <= 5e+272) {
		tmp = re * im;
	} else if ((re <= 7.2e+299) || !(re <= 7.5e+299)) {
		tmp = t_0;
	} else {
		tmp = im;
	}
	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 * (im + (im / re))
    t_1 = re * (im / re)
    t_2 = im + (re * im)
    if (re <= (-18.0d0)) then
        tmp = t_1
    else if (re <= 5d+197) then
        tmp = t_2
    else if (re <= 3.2d+203) then
        tmp = t_1
    else if (re <= 1d+209) then
        tmp = t_2
    else if (re <= 2d+211) then
        tmp = re * im
    else if (re <= 2.3d+246) then
        tmp = t_2
    else if (re <= 2.35d+246) then
        tmp = im
    else if (re <= 1.08d+248) then
        tmp = t_0
    else if (re <= 1.8d+252) then
        tmp = im
    else if (re <= 5d+272) then
        tmp = re * im
    else if ((re <= 7.2d+299) .or. (.not. (re <= 7.5d+299))) then
        tmp = t_0
    else
        tmp = im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (im + (im / re));
	double t_1 = re * (im / re);
	double t_2 = im + (re * im);
	double tmp;
	if (re <= -18.0) {
		tmp = t_1;
	} else if (re <= 5e+197) {
		tmp = t_2;
	} else if (re <= 3.2e+203) {
		tmp = t_1;
	} else if (re <= 1e+209) {
		tmp = t_2;
	} else if (re <= 2e+211) {
		tmp = re * im;
	} else if (re <= 2.3e+246) {
		tmp = t_2;
	} else if (re <= 2.35e+246) {
		tmp = im;
	} else if (re <= 1.08e+248) {
		tmp = t_0;
	} else if (re <= 1.8e+252) {
		tmp = im;
	} else if (re <= 5e+272) {
		tmp = re * im;
	} else if ((re <= 7.2e+299) || !(re <= 7.5e+299)) {
		tmp = t_0;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (im + (im / re))
	t_1 = re * (im / re)
	t_2 = im + (re * im)
	tmp = 0
	if re <= -18.0:
		tmp = t_1
	elif re <= 5e+197:
		tmp = t_2
	elif re <= 3.2e+203:
		tmp = t_1
	elif re <= 1e+209:
		tmp = t_2
	elif re <= 2e+211:
		tmp = re * im
	elif re <= 2.3e+246:
		tmp = t_2
	elif re <= 2.35e+246:
		tmp = im
	elif re <= 1.08e+248:
		tmp = t_0
	elif re <= 1.8e+252:
		tmp = im
	elif re <= 5e+272:
		tmp = re * im
	elif (re <= 7.2e+299) or not (re <= 7.5e+299):
		tmp = t_0
	else:
		tmp = im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(im + Float64(im / re)))
	t_1 = Float64(re * Float64(im / re))
	t_2 = Float64(im + Float64(re * im))
	tmp = 0.0
	if (re <= -18.0)
		tmp = t_1;
	elseif (re <= 5e+197)
		tmp = t_2;
	elseif (re <= 3.2e+203)
		tmp = t_1;
	elseif (re <= 1e+209)
		tmp = t_2;
	elseif (re <= 2e+211)
		tmp = Float64(re * im);
	elseif (re <= 2.3e+246)
		tmp = t_2;
	elseif (re <= 2.35e+246)
		tmp = im;
	elseif (re <= 1.08e+248)
		tmp = t_0;
	elseif (re <= 1.8e+252)
		tmp = im;
	elseif (re <= 5e+272)
		tmp = Float64(re * im);
	elseif ((re <= 7.2e+299) || !(re <= 7.5e+299))
		tmp = t_0;
	else
		tmp = im;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (im + (im / re));
	t_1 = re * (im / re);
	t_2 = im + (re * im);
	tmp = 0.0;
	if (re <= -18.0)
		tmp = t_1;
	elseif (re <= 5e+197)
		tmp = t_2;
	elseif (re <= 3.2e+203)
		tmp = t_1;
	elseif (re <= 1e+209)
		tmp = t_2;
	elseif (re <= 2e+211)
		tmp = re * im;
	elseif (re <= 2.3e+246)
		tmp = t_2;
	elseif (re <= 2.35e+246)
		tmp = im;
	elseif (re <= 1.08e+248)
		tmp = t_0;
	elseif (re <= 1.8e+252)
		tmp = im;
	elseif (re <= 5e+272)
		tmp = re * im;
	elseif ((re <= 7.2e+299) || ~((re <= 7.5e+299)))
		tmp = t_0;
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(im + N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -18.0], t$95$1, If[LessEqual[re, 5e+197], t$95$2, If[LessEqual[re, 3.2e+203], t$95$1, If[LessEqual[re, 1e+209], t$95$2, If[LessEqual[re, 2e+211], N[(re * im), $MachinePrecision], If[LessEqual[re, 2.3e+246], t$95$2, If[LessEqual[re, 2.35e+246], im, If[LessEqual[re, 1.08e+248], t$95$0, If[LessEqual[re, 1.8e+252], im, If[LessEqual[re, 5e+272], N[(re * im), $MachinePrecision], If[Or[LessEqual[re, 7.2e+299], N[Not[LessEqual[re, 7.5e+299]], $MachinePrecision]], t$95$0, im]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -18 or 5.00000000000000009e197 < re < 3.1999999999999997e203

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in re around inf 2.9%

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

   7. Taylor expanded in im around 0 2.5%

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

   8. Taylor expanded in re around 0 28.9%

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

   if -18 < re < 5.00000000000000009e197 or 3.1999999999999997e203 < re < 1.0000000000000001e209 or 1.9999999999999999e211 < re < 2.30000000000000014e246

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 57.0%

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

   4. Taylor expanded in re around 0 36.8%

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

   if 1.0000000000000001e209 < re < 1.9999999999999999e211 or 1.7999999999999999e252 < re < 4.99999999999999973e272

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 5.7%

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

      1. distribute-rgt1-in 5.7%

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

   5. Simplified 5.7%

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

   6. Taylor expanded in re around inf 5.7%

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

   7. Taylor expanded in im around 0 42.9%

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

   if 2.30000000000000014e246 < re < 2.34999999999999988e246 or 1.08e248 < re < 1.7999999999999999e252 or 7.20000000000000008e299 < re < 7.50000000000000039e299

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 0.0%

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

   4. Taylor expanded in re around 0 0.2%

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

   if 2.34999999999999988e246 < re < 1.08e248 or 4.99999999999999973e272 < re < 7.20000000000000008e299 or 7.50000000000000039e299 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 6.3%

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

      1. distribute-rgt1-in 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in re around inf 6.3%

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

   7. Taylor expanded in im around 0 20.7%

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

      1. distribute-rgt-in 20.7%

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

      2. *-lft-identity 20.7%

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

      3. associate-*l/ 20.7%

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

      4. *-lft-identity 20.7%

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

   9. Simplified 20.7%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -18:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 10^{+209}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+211}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+246}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.08 \cdot 10^{+248}:\\ \;\;\;\;re \cdot \left(im + \frac{im}{re}\right)\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{+252}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+272}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+299} \lor \neg \left(re \leq 7.5 \cdot 10^{+299}\right):\\ \;\;\;\;re \cdot \left(im + \frac{im}{re}\right)\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \frac{im}{re}\\ t_1 := im + re \cdot im\\ \mathbf{if}\;re \leq -18:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 10^{+209}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+211}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+246} \lor \neg \left(re \leq 1.08 \cdot 10^{+248}\right) \land \left(re \leq 1.8 \cdot 10^{+252} \lor \neg \left(re \leq 7.2 \cdot 10^{+299}\right) \land re \leq 7.5 \cdot 10^{+299}\right):\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (/ im re))) (t_1 (+ im (* re im))))
   (if (<= re -18.0)
     t_0
     (if (<= re 5e+197)
       t_1
       (if (<= re 3.2e+203)
         t_0
         (if (<= re 1e+209)
           t_1
           (if (<= re 2e+211)
             (* re im)
             (if (<= re 2.3e+246)
               t_1
               (if (or (<= re 2.35e+246)
                       (and (not (<= re 1.08e+248))
                            (or (<= re 1.8e+252)
                                (and (not (<= re 7.2e+299))
                                     (<= re 7.5e+299)))))
                 im
                 (* re im))))))))))

C

double code(double re, double im) {
	double t_0 = re * (im / re);
	double t_1 = im + (re * im);
	double tmp;
	if (re <= -18.0) {
		tmp = t_0;
	} else if (re <= 5e+197) {
		tmp = t_1;
	} else if (re <= 3.2e+203) {
		tmp = t_0;
	} else if (re <= 1e+209) {
		tmp = t_1;
	} else if (re <= 2e+211) {
		tmp = re * im;
	} else if (re <= 2.3e+246) {
		tmp = t_1;
	} else if ((re <= 2.35e+246) || (!(re <= 1.08e+248) && ((re <= 1.8e+252) || (!(re <= 7.2e+299) && (re <= 7.5e+299))))) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (im / re)
    t_1 = im + (re * im)
    if (re <= (-18.0d0)) then
        tmp = t_0
    else if (re <= 5d+197) then
        tmp = t_1
    else if (re <= 3.2d+203) then
        tmp = t_0
    else if (re <= 1d+209) then
        tmp = t_1
    else if (re <= 2d+211) then
        tmp = re * im
    else if (re <= 2.3d+246) then
        tmp = t_1
    else if ((re <= 2.35d+246) .or. (.not. (re <= 1.08d+248)) .and. (re <= 1.8d+252) .or. (.not. (re <= 7.2d+299)) .and. (re <= 7.5d+299)) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (im / re);
	double t_1 = im + (re * im);
	double tmp;
	if (re <= -18.0) {
		tmp = t_0;
	} else if (re <= 5e+197) {
		tmp = t_1;
	} else if (re <= 3.2e+203) {
		tmp = t_0;
	} else if (re <= 1e+209) {
		tmp = t_1;
	} else if (re <= 2e+211) {
		tmp = re * im;
	} else if (re <= 2.3e+246) {
		tmp = t_1;
	} else if ((re <= 2.35e+246) || (!(re <= 1.08e+248) && ((re <= 1.8e+252) || (!(re <= 7.2e+299) && (re <= 7.5e+299))))) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (im / re)
	t_1 = im + (re * im)
	tmp = 0
	if re <= -18.0:
		tmp = t_0
	elif re <= 5e+197:
		tmp = t_1
	elif re <= 3.2e+203:
		tmp = t_0
	elif re <= 1e+209:
		tmp = t_1
	elif re <= 2e+211:
		tmp = re * im
	elif re <= 2.3e+246:
		tmp = t_1
	elif (re <= 2.35e+246) or (not (re <= 1.08e+248) and ((re <= 1.8e+252) or (not (re <= 7.2e+299) and (re <= 7.5e+299)))):
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(im / re))
	t_1 = Float64(im + Float64(re * im))
	tmp = 0.0
	if (re <= -18.0)
		tmp = t_0;
	elseif (re <= 5e+197)
		tmp = t_1;
	elseif (re <= 3.2e+203)
		tmp = t_0;
	elseif (re <= 1e+209)
		tmp = t_1;
	elseif (re <= 2e+211)
		tmp = Float64(re * im);
	elseif (re <= 2.3e+246)
		tmp = t_1;
	elseif ((re <= 2.35e+246) || (!(re <= 1.08e+248) && ((re <= 1.8e+252) || (!(re <= 7.2e+299) && (re <= 7.5e+299)))))
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (im / re);
	t_1 = im + (re * im);
	tmp = 0.0;
	if (re <= -18.0)
		tmp = t_0;
	elseif (re <= 5e+197)
		tmp = t_1;
	elseif (re <= 3.2e+203)
		tmp = t_0;
	elseif (re <= 1e+209)
		tmp = t_1;
	elseif (re <= 2e+211)
		tmp = re * im;
	elseif (re <= 2.3e+246)
		tmp = t_1;
	elseif ((re <= 2.35e+246) || (~((re <= 1.08e+248)) && ((re <= 1.8e+252) || (~((re <= 7.2e+299)) && (re <= 7.5e+299)))))
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im + N[(re * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -18.0], t$95$0, If[LessEqual[re, 5e+197], t$95$1, If[LessEqual[re, 3.2e+203], t$95$0, If[LessEqual[re, 1e+209], t$95$1, If[LessEqual[re, 2e+211], N[(re * im), $MachinePrecision], If[LessEqual[re, 2.3e+246], t$95$1, If[Or[LessEqual[re, 2.35e+246], And[N[Not[LessEqual[re, 1.08e+248]], $MachinePrecision], Or[LessEqual[re, 1.8e+252], And[N[Not[LessEqual[re, 7.2e+299]], $MachinePrecision], LessEqual[re, 7.5e+299]]]]], im, N[(re * im), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -18 or 5.00000000000000009e197 < re < 3.1999999999999997e203

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.9%

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

      1. distribute-rgt1-in 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in re around inf 2.9%

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

   7. Taylor expanded in im around 0 2.5%

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

   8. Taylor expanded in re around 0 28.9%

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

   if -18 < re < 5.00000000000000009e197 or 3.1999999999999997e203 < re < 1.0000000000000001e209 or 1.9999999999999999e211 < re < 2.30000000000000014e246

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 57.0%

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

   4. Taylor expanded in re around 0 36.8%

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

   if 1.0000000000000001e209 < re < 1.9999999999999999e211 or 2.34999999999999988e246 < re < 1.08e248 or 1.7999999999999999e252 < re < 7.20000000000000008e299 or 7.50000000000000039e299 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 6.0%

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

      1. distribute-rgt1-in 6.0%

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

   5. Simplified 6.0%

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

   6. Taylor expanded in re around inf 6.0%

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

   7. Taylor expanded in im around 0 30.8%

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

   if 2.30000000000000014e246 < re < 2.34999999999999988e246 or 1.08e248 < re < 1.7999999999999999e252 or 7.20000000000000008e299 < re < 7.50000000000000039e299

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 0.0%

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

   4. Taylor expanded in re around 0 0.2%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -18:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 10^{+209}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+211}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246}:\\ \;\;\;\;im + re \cdot im\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+246} \lor \neg \left(re \leq 1.08 \cdot 10^{+248}\right) \land \left(re \leq 1.8 \cdot 10^{+252} \lor \neg \left(re \leq 7.2 \cdot 10^{+299}\right) \land re \leq 7.5 \cdot 10^{+299}\right):\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \frac{im}{re}\\ \mathbf{if}\;re \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246} \lor \neg \left(re \leq 2.35 \cdot 10^{+246}\right) \land \left(re \leq 1.08 \cdot 10^{+248} \lor \neg \left(re \leq 1.8 \cdot 10^{+252}\right) \land \left(re \leq 7.2 \cdot 10^{+299} \lor \neg \left(re \leq 7.5 \cdot 10^{+299}\right)\right)\right):\\ \;\;\;\;re \cdot im\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (/ im re))))
   (if (<= re 1.0)
     t_0
     (if (<= re 5e+197)
       (* re im)
       (if (<= re 3.2e+203)
         t_0
         (if (or (<= re 2.3e+246)
                 (and (not (<= re 2.35e+246))
                      (or (<= re 1.08e+248)
                          (and (not (<= re 1.8e+252))
                               (or (<= re 7.2e+299) (not (<= re 7.5e+299)))))))
           (* re im)
           im))))))

C

double code(double re, double im) {
	double t_0 = re * (im / re);
	double tmp;
	if (re <= 1.0) {
		tmp = t_0;
	} else if (re <= 5e+197) {
		tmp = re * im;
	} else if (re <= 3.2e+203) {
		tmp = t_0;
	} else if ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299)))))) {
		tmp = re * im;
	} else {
		tmp = im;
	}
	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 * (im / re)
    if (re <= 1.0d0) then
        tmp = t_0
    else if (re <= 5d+197) then
        tmp = re * im
    else if (re <= 3.2d+203) then
        tmp = t_0
    else if ((re <= 2.3d+246) .or. (.not. (re <= 2.35d+246)) .and. (re <= 1.08d+248) .or. (.not. (re <= 1.8d+252)) .and. (re <= 7.2d+299) .or. (.not. (re <= 7.5d+299))) then
        tmp = re * im
    else
        tmp = im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (im / re);
	double tmp;
	if (re <= 1.0) {
		tmp = t_0;
	} else if (re <= 5e+197) {
		tmp = re * im;
	} else if (re <= 3.2e+203) {
		tmp = t_0;
	} else if ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299)))))) {
		tmp = re * im;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (im / re)
	tmp = 0
	if re <= 1.0:
		tmp = t_0
	elif re <= 5e+197:
		tmp = re * im
	elif re <= 3.2e+203:
		tmp = t_0
	elif (re <= 2.3e+246) or (not (re <= 2.35e+246) and ((re <= 1.08e+248) or (not (re <= 1.8e+252) and ((re <= 7.2e+299) or not (re <= 7.5e+299))))):
		tmp = re * im
	else:
		tmp = im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(im / re))
	tmp = 0.0
	if (re <= 1.0)
		tmp = t_0;
	elseif (re <= 5e+197)
		tmp = Float64(re * im);
	elseif (re <= 3.2e+203)
		tmp = t_0;
	elseif ((re <= 2.3e+246) || (!(re <= 2.35e+246) && ((re <= 1.08e+248) || (!(re <= 1.8e+252) && ((re <= 7.2e+299) || !(re <= 7.5e+299))))))
		tmp = Float64(re * im);
	else
		tmp = im;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (im / re);
	tmp = 0.0;
	if (re <= 1.0)
		tmp = t_0;
	elseif (re <= 5e+197)
		tmp = re * im;
	elseif (re <= 3.2e+203)
		tmp = t_0;
	elseif ((re <= 2.3e+246) || (~((re <= 2.35e+246)) && ((re <= 1.08e+248) || (~((re <= 1.8e+252)) && ((re <= 7.2e+299) || ~((re <= 7.5e+299)))))))
		tmp = re * im;
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 1.0], t$95$0, If[LessEqual[re, 5e+197], N[(re * im), $MachinePrecision], If[LessEqual[re, 3.2e+203], t$95$0, If[Or[LessEqual[re, 2.3e+246], And[N[Not[LessEqual[re, 2.35e+246]], $MachinePrecision], Or[LessEqual[re, 1.08e+248], And[N[Not[LessEqual[re, 1.8e+252]], $MachinePrecision], Or[LessEqual[re, 7.2e+299], N[Not[LessEqual[re, 7.5e+299]], $MachinePrecision]]]]]], N[(re * im), $MachinePrecision], im]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 1 or 5.00000000000000009e197 < re < 3.1999999999999997e203

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 66.9%

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

      1. distribute-rgt1-in 66.9%

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

   5. Simplified 66.9%

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

   6. Taylor expanded in re around inf 66.3%

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

   7. Taylor expanded in im around 0 32.0%

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

   8. Taylor expanded in re around 0 40.1%

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

   if 1 < re < 5.00000000000000009e197 or 3.1999999999999997e203 < re < 2.30000000000000014e246 or 2.34999999999999988e246 < re < 1.08e248 or 1.7999999999999999e252 < re < 7.20000000000000008e299 or 7.50000000000000039e299 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.9%

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

      1. distribute-rgt1-in 3.9%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in re around inf 3.9%

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

   7. Taylor expanded in im around 0 15.9%

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

   if 2.30000000000000014e246 < re < 2.34999999999999988e246 or 1.08e248 < re < 1.7999999999999999e252 or 7.20000000000000008e299 < re < 7.50000000000000039e299

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 0.0%

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

   4. Taylor expanded in re around 0 0.2%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+197}:\\ \;\;\;\;re \cdot im\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+203}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+246} \lor \neg \left(re \leq 2.35 \cdot 10^{+246}\right) \land \left(re \leq 1.08 \cdot 10^{+248} \lor \neg \left(re \leq 1.8 \cdot 10^{+252}\right) \land \left(re \leq 7.2 \cdot 10^{+299} \lor \neg \left(re \leq 7.5 \cdot 10^{+299}\right)\right)\right):\\ \;\;\;\;re \cdot im\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1 \lor \neg \left(re \leq 2.3 \cdot 10^{+246}\right) \land \left(re \leq 2.35 \cdot 10^{+246} \lor \neg \left(re \leq 1.08 \cdot 10^{+248}\right) \land \left(re \leq 1.8 \cdot 10^{+252} \lor \neg \left(re \leq 7.2 \cdot 10^{+299}\right) \land re \leq 7.5 \cdot 10^{+299}\right)\right):\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 1.0)
         (and (not (<= re 2.3e+246))
              (or (<= re 2.35e+246)
                  (and (not (<= re 1.08e+248))
                       (or (<= re 1.8e+252)
                           (and (not (<= re 7.2e+299)) (<= re 7.5e+299)))))))
   im
   (* re im)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 1.0) || (!(re <= 2.3e+246) && ((re <= 2.35e+246) || (!(re <= 1.08e+248) && ((re <= 1.8e+252) || (!(re <= 7.2e+299) && (re <= 7.5e+299))))))) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= 1.0d0) .or. (.not. (re <= 2.3d+246)) .and. (re <= 2.35d+246) .or. (.not. (re <= 1.08d+248)) .and. (re <= 1.8d+252) .or. (.not. (re <= 7.2d+299)) .and. (re <= 7.5d+299)) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 1.0) || (!(re <= 2.3e+246) && ((re <= 2.35e+246) || (!(re <= 1.08e+248) && ((re <= 1.8e+252) || (!(re <= 7.2e+299) && (re <= 7.5e+299))))))) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 1.0) or (not (re <= 2.3e+246) and ((re <= 2.35e+246) or (not (re <= 1.08e+248) and ((re <= 1.8e+252) or (not (re <= 7.2e+299) and (re <= 7.5e+299)))))):
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 1.0) || (!(re <= 2.3e+246) && ((re <= 2.35e+246) || (!(re <= 1.08e+248) && ((re <= 1.8e+252) || (!(re <= 7.2e+299) && (re <= 7.5e+299)))))))
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 1.0) || (~((re <= 2.3e+246)) && ((re <= 2.35e+246) || (~((re <= 1.08e+248)) && ((re <= 1.8e+252) || (~((re <= 7.2e+299)) && (re <= 7.5e+299)))))))
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 1.0], And[N[Not[LessEqual[re, 2.3e+246]], $MachinePrecision], Or[LessEqual[re, 2.35e+246], And[N[Not[LessEqual[re, 1.08e+248]], $MachinePrecision], Or[LessEqual[re, 1.8e+252], And[N[Not[LessEqual[re, 7.2e+299]], $MachinePrecision], LessEqual[re, 7.5e+299]]]]]]], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1 or 2.30000000000000014e246 < re < 2.34999999999999988e246 or 1.08e248 < re < 1.7999999999999999e252 or 7.20000000000000008e299 < re < 7.50000000000000039e299

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 62.3%

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

   4. Taylor expanded in re around 0 32.1%

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

   if 1 < re < 2.30000000000000014e246 or 2.34999999999999988e246 < re < 1.08e248 or 1.7999999999999999e252 < re < 7.20000000000000008e299 or 7.50000000000000039e299 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.0%

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

      1. distribute-rgt1-in 4.0%

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

   5. Simplified 4.0%

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

   6. Taylor expanded in re around inf 4.0%

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

   7. Taylor expanded in im around 0 15.4%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1 \lor \neg \left(re \leq 2.3 \cdot 10^{+246}\right) \land \left(re \leq 2.35 \cdot 10^{+246} \lor \neg \left(re \leq 1.08 \cdot 10^{+248}\right) \land \left(re \leq 1.8 \cdot 10^{+252} \lor \neg \left(re \leq 7.2 \cdot 10^{+299}\right) \land re \leq 7.5 \cdot 10^{+299}\right)\right):\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.7% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(\left(re + 1\right) \cdot \frac{im}{re}\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re (* (+ re 1.0) (/ im re))))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re * ((re + 1.0) * (im / re))

Julia

function code(re, im)
	return Float64(re * Float64(Float64(re + 1.0) * Float64(im / re)))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(re * N[(N[(re + 1.0), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 51.0%

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

   1. distribute-rgt1-in 51.0%

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

5. Simplified 51.0%

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

6. Taylor expanded in re around inf 50.5%

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

7. Taylor expanded in re around 0 50.5%

   \[\leadsto re \cdot \color{blue}{\frac{\sin im + re \cdot \sin im}{re}} \]
8. Step-by-step derivation

   1. distribute-rgt1-in 50.5%

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

   2. *-rgt-identity 50.5%

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

   3. rgt-mult-inverse 50.4%

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

   4. distribute-lft-in 50.4%

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

   5. associate-/l* 56.3%

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

   6. distribute-lft-in 56.3%

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

   7. *-rgt-identity 56.3%

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

   8. rgt-mult-inverse 56.4%

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

9. Simplified 56.4%

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

10. Taylor expanded in im around 0 33.7%

    \[\leadsto re \cdot \left(\left(re + 1\right) \cdot \color{blue}{\frac{im}{re}}\right) \]
11. Add Preprocessing

Alternative 14: 26.2% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 67.3%

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

4. Taylor expanded in re around 0 24.9%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))