math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 11.2s

Alternatives: 12

Speedup: 1.0×

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

Specification

Math

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

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: 92.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.2%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.6%

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

      1. distribute-rgt1-in 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 91.3%

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

Alternative 3: 92.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.2%

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

   if 0.0 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.9%

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

4. Final simplification 91.0%

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

Alternative 4: 68.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 0.000146:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+160)
   (* re (/ im re))
   (if (<= re 0.000146)
     (sin im)
     (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else if (re <= 0.000146) {
		tmp = sin(im);
	} else {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.8d+160)) then
        tmp = re * (im / re)
    else if (re <= 0.000146d0) then
        tmp = sin(im)
    else
        tmp = im + (im * (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else if (re <= 0.000146) {
		tmp = Math.sin(im);
	} else {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+160:
		tmp = re * (im / re)
	elif re <= 0.000146:
		tmp = math.sin(im)
	else:
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+160)
		tmp = Float64(re * Float64(im / re));
	elseif (re <= 0.000146)
		tmp = sin(im);
	else
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e+160)
		tmp = re * (im / re);
	elseif (re <= 0.000146)
		tmp = sin(im);
	else
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+160], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.000146], N[Sin[im], $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.80000000000000011e160

   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 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in re around 0 46.6%

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

   if -1.80000000000000011e160 < re < 1.45999999999999998e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 76.9%

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

   if 1.45999999999999998e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 70.1%

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

   4. Taylor expanded in re around 0 44.0%

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

   5. Taylor expanded in im around 0 49.4%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 0.000146:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{+231} \lor \neg \left(re \leq 1.08 \cdot 10^{+279}\right):\\ \;\;\;\;im + re \cdot \left(im \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + \left(re + re \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+160)
   (* re (/ im re))
   (if (or (<= re 3.1e+231) (not (<= re 1.08e+279)))
     (+ im (* re (* im (* re (+ 0.5 (* re 0.16666666666666666))))))
     (* im (+ 1.0 (+ re (* re (* im (* im -0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else if ((re <= 3.1e+231) || !(re <= 1.08e+279)) {
		tmp = im + (re * (im * (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.8d+160)) then
        tmp = re * (im / re)
    else if ((re <= 3.1d+231) .or. (.not. (re <= 1.08d+279))) then
        tmp = im + (re * (im * (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    else
        tmp = im * (1.0d0 + (re + (re * (im * (im * (-0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else if ((re <= 3.1e+231) || !(re <= 1.08e+279)) {
		tmp = im + (re * (im * (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+160:
		tmp = re * (im / re)
	elif (re <= 3.1e+231) or not (re <= 1.08e+279):
		tmp = im + (re * (im * (re * (0.5 + (re * 0.16666666666666666)))))
	else:
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+160)
		tmp = Float64(re * Float64(im / re));
	elseif ((re <= 3.1e+231) || !(re <= 1.08e+279))
		tmp = Float64(im + Float64(re * Float64(im * Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	else
		tmp = Float64(im * Float64(1.0 + Float64(re + Float64(re * Float64(im * Float64(im * -0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e+160)
		tmp = re * (im / re);
	elseif ((re <= 3.1e+231) || ~((re <= 1.08e+279)))
		tmp = im + (re * (im * (re * (0.5 + (re * 0.16666666666666666)))));
	else
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+160], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 3.1e+231], N[Not[LessEqual[re, 1.08e+279]], $MachinePrecision]], N[(im + N[(re * N[(im * N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + N[(re + N[(re * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.80000000000000011e160

   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 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in re around 0 46.6%

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

   if -1.80000000000000011e160 < re < 3.0999999999999999e231 or 1.08000000000000004e279 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 64.4%

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

   4. Taylor expanded in re around 0 40.0%

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

      1. pow1 40.0%

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

      2. associate-*r* 40.0%

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

      3. *-commutative 40.0%

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

   6. Applied egg-rr 40.0%

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

      1. unpow1 40.0%

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

      2. *-commutative 40.0%

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

   8. Simplified 40.0%

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

      1. distribute-rgt-in 39.9%

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

      2. associate-*r* 39.9%

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

      3. *-commutative 39.9%

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

   10. Applied egg-rr 39.9%

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

   11. Taylor expanded in re around inf 41.4%

       \[\leadsto im + \color{blue}{{re}^{3} \cdot \left(0.16666666666666666 \cdot im + 0.5 \cdot \frac{im}{re}\right)} \]
   12. Step-by-step derivation

       1. cube-mult 41.4%

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

       2. unpow2 41.4%

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

       3. associate-*l* 39.7%

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

       4. distribute-lft-in 34.9%

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

       5. unpow2 34.9%

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

       6. *-commutative 34.9%

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

       7. associate-*r* 34.9%

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

       8. *-commutative 34.9%

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

       9. associate-*r* 34.9%

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

       10. associate-*r* 34.9%

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

       11. *-commutative 34.9%

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

       12. associate-*r* 34.9%

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

       13. unpow2 34.9%

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

       14. associate-*l* 39.8%

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

       15. associate-*r/ 39.9%

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

       16. *-commutative 39.9%

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

       17. associate-/l* 39.9%

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

       18. *-rgt-identity 39.9%

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

       19. associate-*r/ 39.9%

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

       20. rgt-mult-inverse 39.9%

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

       21. *-rgt-identity 39.9%

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

   13. Simplified 40.0%

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

   14. Taylor expanded in im around 0 40.0%

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

   if 3.0999999999999999e231 < re < 1.08000000000000004e279

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 6.9%

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

   4. Taylor expanded in im around 0 75.7%

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

   5. Taylor expanded in re around inf 75.7%

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

      1. distribute-lft-out 75.7%

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

      2. unpow2 75.7%

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

      3. associate-*r/ 75.7%

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

      4. unpow2 75.7%

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

      5. distribute-lft-out 75.7%

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

      6. +-commutative 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in im around 0 75.7%

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

      1. +-commutative 75.7%

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

      2. distribute-rgt-in 75.7%

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

      3. associate-*l/ 75.7%

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

      4. *-lft-identity 75.7%

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

      5. *-lft-identity 75.7%

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

      6. distribute-lft-out 75.7%

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

      7. unpow2 75.7%

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

      8. associate-*r/ 75.7%

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

      9. associate-*r* 75.7%

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

      10. unpow2 75.7%

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

      11. associate-*r* 75.7%

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

      12. distribute-lft-in 75.7%

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

      13. +-commutative 75.7%

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

      14. *-commutative 75.7%

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

      15. associate-*l* 75.7%

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

   10. Simplified 75.7%

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

   11. Taylor expanded in re around inf 75.7%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{+231} \lor \neg \left(re \leq 1.08 \cdot 10^{+279}\right):\\ \;\;\;\;im + re \cdot \left(im \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + \left(re + re \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\\ \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{+231}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;re \leq 1.08 \cdot 10^{+279}:\\ \;\;\;\;im \cdot \left(1 + \left(re + re \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* re 0.16666666666666666)))))
   (if (<= re -1.8e+160)
     (* re (/ im re))
     (if (<= re 3.1e+231)
       (+ im (* im (* re (+ 1.0 t_0))))
       (if (<= re 1.08e+279)
         (* im (+ 1.0 (+ re (* re (* im (* im -0.16666666666666666))))))
         (+ im (* re (* im t_0))))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else if (re <= 3.1e+231) {
		tmp = im + (im * (re * (1.0 + t_0)));
	} else if (re <= 1.08e+279) {
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))));
	} else {
		tmp = im + (re * (im * 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 = re * (0.5d0 + (re * 0.16666666666666666d0))
    if (re <= (-1.8d+160)) then
        tmp = re * (im / re)
    else if (re <= 3.1d+231) then
        tmp = im + (im * (re * (1.0d0 + t_0)))
    else if (re <= 1.08d+279) then
        tmp = im * (1.0d0 + (re + (re * (im * (im * (-0.16666666666666666d0))))))
    else
        tmp = im + (re * (im * t_0))
    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 tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else if (re <= 3.1e+231) {
		tmp = im + (im * (re * (1.0 + t_0)));
	} else if (re <= 1.08e+279) {
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))));
	} else {
		tmp = im + (re * (im * t_0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (re * 0.16666666666666666))
	tmp = 0
	if re <= -1.8e+160:
		tmp = re * (im / re)
	elif re <= 3.1e+231:
		tmp = im + (im * (re * (1.0 + t_0)))
	elif re <= 1.08e+279:
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))))
	else:
		tmp = im + (re * (im * t_0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))
	tmp = 0.0
	if (re <= -1.8e+160)
		tmp = Float64(re * Float64(im / re));
	elseif (re <= 3.1e+231)
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + t_0))));
	elseif (re <= 1.08e+279)
		tmp = Float64(im * Float64(1.0 + Float64(re + Float64(re * Float64(im * Float64(im * -0.16666666666666666))))));
	else
		tmp = Float64(im + Float64(re * Float64(im * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (re * 0.16666666666666666));
	tmp = 0.0;
	if (re <= -1.8e+160)
		tmp = re * (im / re);
	elseif (re <= 3.1e+231)
		tmp = im + (im * (re * (1.0 + t_0)));
	elseif (re <= 1.08e+279)
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))));
	else
		tmp = im + (re * (im * t_0));
	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]}, If[LessEqual[re, -1.8e+160], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.1e+231], N[(im + N[(im * N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.08e+279], N[(im * N[(1.0 + N[(re + N[(re * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im + N[(re * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.80000000000000011e160

   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 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in re around 0 46.6%

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

   if -1.80000000000000011e160 < re < 3.0999999999999999e231

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 63.8%

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

   4. Taylor expanded in re around 0 38.7%

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

   5. Taylor expanded in im around 0 40.5%

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

   if 3.0999999999999999e231 < re < 1.08000000000000004e279

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 6.9%

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

   4. Taylor expanded in im around 0 75.7%

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

   5. Taylor expanded in re around inf 75.7%

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

      1. distribute-lft-out 75.7%

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

      2. unpow2 75.7%

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

      3. associate-*r/ 75.7%

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

      4. unpow2 75.7%

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

      5. distribute-lft-out 75.7%

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

      6. +-commutative 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in im around 0 75.7%

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

      1. +-commutative 75.7%

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

      2. distribute-rgt-in 75.7%

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

      3. associate-*l/ 75.7%

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

      4. *-lft-identity 75.7%

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

      5. *-lft-identity 75.7%

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

      6. distribute-lft-out 75.7%

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

      7. unpow2 75.7%

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

      8. associate-*r/ 75.7%

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

      9. associate-*r* 75.7%

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

      10. unpow2 75.7%

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

      11. associate-*r* 75.7%

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

      12. distribute-lft-in 75.7%

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

      13. +-commutative 75.7%

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

      14. *-commutative 75.7%

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

      15. associate-*l* 75.7%

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

   10. Simplified 75.7%

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

   11. Taylor expanded in re around inf 75.7%

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

   if 1.08000000000000004e279 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 83.3%

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

   4. Taylor expanded in re around 0 83.3%

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

      1. pow1 83.3%

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

      2. associate-*r* 83.3%

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

      3. *-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. unpow1 83.3%

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

      2. *-commutative 83.3%

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

   8. Simplified 83.3%

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

      1. distribute-rgt-in 83.3%

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

      2. associate-*r* 83.3%

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

      3. *-commutative 83.3%

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

   10. Applied egg-rr 83.3%

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

   11. Taylor expanded in re around inf 83.3%

       \[\leadsto im + \color{blue}{{re}^{3} \cdot \left(0.16666666666666666 \cdot im + 0.5 \cdot \frac{im}{re}\right)} \]
   12. Step-by-step derivation

       1. cube-mult 83.3%

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

       2. unpow2 83.3%

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

       3. associate-*l* 83.3%

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

       4. distribute-lft-in 16.7%

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

       5. unpow2 16.7%

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

       6. *-commutative 16.7%

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

       7. associate-*r* 16.7%

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

       8. *-commutative 16.7%

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

       9. associate-*r* 16.7%

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

       10. associate-*r* 16.7%

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

       11. *-commutative 16.7%

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

       12. associate-*r* 16.7%

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

       13. unpow2 16.7%

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

       14. associate-*l* 83.3%

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

       15. associate-*r/ 83.3%

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

       16. *-commutative 83.3%

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

       17. associate-/l* 83.3%

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

       18. *-rgt-identity 83.3%

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

       19. associate-*r/ 83.3%

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

       20. rgt-mult-inverse 83.3%

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

       21. *-rgt-identity 83.3%

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

   13. Simplified 83.3%

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

   14. Taylor expanded in im around 0 83.3%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{+231}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.08 \cdot 10^{+279}:\\ \;\;\;\;im \cdot \left(1 + \left(re + re \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{+231}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.08 \cdot 10^{+279}:\\ \;\;\;\;im \cdot \left(1 + \left(re + re \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+160)
   (* re (/ im re))
   (if (<= re 3.1e+231)
     (+ im (* im (* re (+ 1.0 (* re 0.5)))))
     (if (<= re 1.08e+279)
       (* im (+ 1.0 (+ re (* re (* im (* im -0.16666666666666666))))))
       (+ im (* re (* re (* im 0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else if (re <= 3.1e+231) {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.08e+279) {
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))));
	} else {
		tmp = im + (re * (re * (im * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.8d+160)) then
        tmp = re * (im / re)
    else if (re <= 3.1d+231) then
        tmp = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    else if (re <= 1.08d+279) then
        tmp = im * (1.0d0 + (re + (re * (im * (im * (-0.16666666666666666d0))))))
    else
        tmp = im + (re * (re * (im * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else if (re <= 3.1e+231) {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.08e+279) {
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))));
	} else {
		tmp = im + (re * (re * (im * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+160:
		tmp = re * (im / re)
	elif re <= 3.1e+231:
		tmp = im + (im * (re * (1.0 + (re * 0.5))))
	elif re <= 1.08e+279:
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))))
	else:
		tmp = im + (re * (re * (im * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+160)
		tmp = Float64(re * Float64(im / re));
	elseif (re <= 3.1e+231)
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	elseif (re <= 1.08e+279)
		tmp = Float64(im * Float64(1.0 + Float64(re + Float64(re * Float64(im * Float64(im * -0.16666666666666666))))));
	else
		tmp = Float64(im + Float64(re * Float64(re * Float64(im * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e+160)
		tmp = re * (im / re);
	elseif (re <= 3.1e+231)
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	elseif (re <= 1.08e+279)
		tmp = im * (1.0 + (re + (re * (im * (im * -0.16666666666666666)))));
	else
		tmp = im + (re * (re * (im * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+160], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.1e+231], N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.08e+279], N[(im * N[(1.0 + N[(re + N[(re * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im + N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.80000000000000011e160

   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 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in re around 0 46.6%

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

   if -1.80000000000000011e160 < re < 3.0999999999999999e231

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 63.8%

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

   4. Taylor expanded in re around 0 35.1%

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

      1. *-commutative 35.1%

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

   6. Simplified 35.1%

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

   7. Taylor expanded in im around 0 37.8%

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

   if 3.0999999999999999e231 < re < 1.08000000000000004e279

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 6.9%

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

   4. Taylor expanded in im around 0 75.7%

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

   5. Taylor expanded in re around inf 75.7%

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

      1. distribute-lft-out 75.7%

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

      2. unpow2 75.7%

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

      3. associate-*r/ 75.7%

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

      4. unpow2 75.7%

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

      5. distribute-lft-out 75.7%

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

      6. +-commutative 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in im around 0 75.7%

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

      1. +-commutative 75.7%

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

      2. distribute-rgt-in 75.7%

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

      3. associate-*l/ 75.7%

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

      4. *-lft-identity 75.7%

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

      5. *-lft-identity 75.7%

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

      6. distribute-lft-out 75.7%

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

      7. unpow2 75.7%

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

      8. associate-*r/ 75.7%

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

      9. associate-*r* 75.7%

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

      10. unpow2 75.7%

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

      11. associate-*r* 75.7%

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

      12. distribute-lft-in 75.7%

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

      13. +-commutative 75.7%

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

      14. *-commutative 75.7%

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

      15. associate-*l* 75.7%

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

   10. Simplified 75.7%

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

   11. Taylor expanded in re around inf 75.7%

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

   if 1.08000000000000004e279 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 83.3%

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

   4. Taylor expanded in re around 0 83.3%

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

      1. pow1 83.3%

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

      2. associate-*r* 83.3%

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

      3. *-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. unpow1 83.3%

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

      2. *-commutative 83.3%

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

   8. Simplified 83.3%

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

      1. distribute-rgt-in 83.3%

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

      2. associate-*r* 83.3%

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

      3. *-commutative 83.3%

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

   10. Applied egg-rr 83.3%

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

   11. Taylor expanded in re around inf 83.3%

       \[\leadsto im + \color{blue}{{re}^{3} \cdot \left(0.16666666666666666 \cdot im + 0.5 \cdot \frac{im}{re}\right)} \]
   12. Step-by-step derivation

       1. cube-mult 83.3%

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

       2. unpow2 83.3%

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

       3. associate-*l* 83.3%

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

       4. distribute-lft-in 16.7%

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

       5. unpow2 16.7%

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

       6. *-commutative 16.7%

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

       7. associate-*r* 16.7%

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

       8. *-commutative 16.7%

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

       9. associate-*r* 16.7%

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

       10. associate-*r* 16.7%

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

       11. *-commutative 16.7%

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

       12. associate-*r* 16.7%

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

       13. unpow2 16.7%

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

       14. associate-*l* 83.3%

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

       15. associate-*r/ 83.3%

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

       16. *-commutative 83.3%

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

       17. associate-/l* 83.3%

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

       18. *-rgt-identity 83.3%

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

       19. associate-*r/ 83.3%

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

       20. rgt-mult-inverse 83.3%

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

       21. *-rgt-identity 83.3%

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

   13. Simplified 83.3%

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

   14. Taylor expanded in re around 0 83.3%

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

       1. associate-*r* 83.3%

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

       2. *-commutative 83.3%

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

       3. *-commutative 83.3%

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

   16. Simplified 83.3%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{+231}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.08 \cdot 10^{+279}:\\ \;\;\;\;im \cdot \left(1 + \left(re + re \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.9% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+160)
   (* re (/ im re))
   (+ im (* im (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} 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) :: tmp
    if (re <= (-1.8d+160)) then
        tmp = re * (im / re)
    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 tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+160:
		tmp = re * (im / re)
	else:
		tmp = im + (im * (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+160)
		tmp = Float64(re * Float64(im / re));
	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)
	tmp = 0.0;
	if (re <= -1.8e+160)
		tmp = re * (im / re);
	else
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+160], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.80000000000000011e160

   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 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in re around 0 46.6%

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

   if -1.80000000000000011e160 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 63.0%

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

   4. Taylor expanded in re around 0 36.0%

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

      1. *-commutative 36.0%

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

   6. Simplified 36.0%

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

   7. Taylor expanded in im around 0 38.6%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.8% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+160}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+160) (* re (/ im re)) (+ im (* re (* re (* im 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else {
		tmp = im + (re * (re * (im * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.8d+160)) then
        tmp = re * (im / re)
    else
        tmp = im + (re * (re * (im * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+160) {
		tmp = re * (im / re);
	} else {
		tmp = im + (re * (re * (im * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+160:
		tmp = re * (im / re)
	else:
		tmp = im + (re * (re * (im * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+160)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im + Float64(re * Float64(re * Float64(im * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e+160)
		tmp = re * (im / re);
	else
		tmp = im + (re * (re * (im * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+160], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im + N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.80000000000000011e160

   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 2.3%

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

   5. Taylor expanded in re around inf 2.3%

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

   6. Taylor expanded in re around 0 46.6%

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

   if -1.80000000000000011e160 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 63.0%

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

   4. Taylor expanded in re around 0 39.4%

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

      1. pow1 39.4%

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

      2. associate-*r* 39.4%

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

      3. *-commutative 39.4%

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

   6. Applied egg-rr 39.4%

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

      1. unpow1 39.4%

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

      2. *-commutative 39.4%

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

   8. Simplified 39.4%

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

      1. distribute-rgt-in 39.4%

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

      2. associate-*r* 39.4%

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

      3. *-commutative 39.4%

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

   10. Applied egg-rr 39.4%

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

   11. Taylor expanded in re around inf 40.8%

       \[\leadsto im + \color{blue}{{re}^{3} \cdot \left(0.16666666666666666 \cdot im + 0.5 \cdot \frac{im}{re}\right)} \]
   12. Step-by-step derivation

       1. cube-mult 40.8%

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

       2. unpow2 40.8%

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

       3. associate-*l* 39.1%

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

       4. distribute-lft-in 34.1%

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

       5. unpow2 34.1%

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

       6. *-commutative 34.1%

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

       7. associate-*r* 34.1%

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

       8. *-commutative 34.1%

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

       9. associate-*r* 34.1%

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

       10. associate-*r* 34.1%

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

       11. *-commutative 34.1%

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

       12. associate-*r* 34.1%

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

       13. unpow2 34.1%

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

       14. associate-*l* 39.3%

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

       15. associate-*r/ 39.4%

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

       16. *-commutative 39.4%

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

       17. associate-/l* 39.4%

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

       18. *-rgt-identity 39.4%

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

       19. associate-*r/ 39.4%

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

       20. rgt-mult-inverse 39.4%

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

       21. *-rgt-identity 39.4%

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

   13. Simplified 39.4%

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

   14. Taylor expanded in re around 0 36.0%

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

       1. associate-*r* 36.0%

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

       2. *-commutative 36.0%

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

       3. *-commutative 36.0%

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

   16. Simplified 36.0%

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

Alternative 10: 33.6% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.056) (* re (/ im re)) (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = re * (im / re);
	} 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 (im <= 0.056d0) then
        tmp = re * (im / re)
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = re * (im / re);
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.056:
		tmp = re * (im / re)
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.056)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.056)
		tmp = re * (im / re);
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.056], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0560000000000000012

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.6%

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

   4. Taylor expanded in re around 0 32.8%

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

   5. Taylor expanded in re around inf 32.7%

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

   6. Taylor expanded in re around 0 38.7%

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

   if 0.0560000000000000012 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 48.7%

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

      1. distribute-rgt1-in 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in re around inf 3.7%

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

   7. Taylor expanded in im around 0 9.0%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.1% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.056) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		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 (im <= 0.056d0) 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 (im <= 0.056) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.056:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.056)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.056)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.056], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0560000000000000012

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.6%

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

   4. Taylor expanded in re around 0 30.9%

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

   if 0.0560000000000000012 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 48.7%

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

      1. distribute-rgt1-in 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in re around inf 3.7%

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

   7. Taylor expanded in im around 0 9.0%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.6% 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 67.7%

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

4. Taylor expanded in re around 0 23.6%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))