math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

Alternatives: 11

Speedup: 1.0×

• 24.4% 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: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 50\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 50.0))) (* (exp re) im) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 50.0)) {
		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) <= 50.0d0))) 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) <= 50.0)) {
		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) <= 50.0):
		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) <= 50.0))
		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) <= 50.0)))
		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], 50.0]], $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 50 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 92.2%

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

   if 0.0 < (exp.f64 re) < 50

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 97.5%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 50\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.023 \lor \neg \left(re \leq 3.9\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.023) (and (not (<= re 3.9)) (<= re 1.9e+154)))
   (* (exp re) im)
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.023) || (!(re <= 3.9) && (re <= 1.9e+154))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.023d0)) .or. (.not. (re <= 3.9d0)) .and. (re <= 1.9d+154)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.023) || (!(re <= 3.9) && (re <= 1.9e+154))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.023) or (not (re <= 3.9) and (re <= 1.9e+154)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.023) || (!(re <= 3.9) && (re <= 1.9e+154)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.023) || (~((re <= 3.9)) && (re <= 1.9e+154)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.023], And[N[Not[LessEqual[re, 3.9]], $MachinePrecision], LessEqual[re, 1.9e+154]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.023 or 3.89999999999999991 < re < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 95.3%

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

   if -0.023 < re < 3.89999999999999991 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 89.9%

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

      1. *-rgt-identity 89.9%

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

      2. distribute-lft-in 89.9%

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

      3. associate-*r* 89.9%

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

      4. associate-*r* 99.3%

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

      5. distribute-rgt-out 99.3%

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

      6. distribute-lft-out 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. *-commutative 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.023 \lor \neg \left(re \leq 3.9\right) \land re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00084 \lor \neg \left(re \leq 3.9\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00084) (not (<= re 3.9)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00084) || !(re <= 3.9)) {
		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 ((re <= (-0.00084d0)) .or. (.not. (re <= 3.9d0))) 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 ((re <= -0.00084) || !(re <= 3.9)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00084) or not (re <= 3.9):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00084) || !(re <= 3.9))
		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 ((re <= -0.00084) || ~((re <= 3.9)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00084], N[Not[LessEqual[re, 3.9]], $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 re < -8.4000000000000003e-4 or 3.89999999999999991 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 92.2%

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

   if -8.4000000000000003e-4 < re < 3.89999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.8%

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

      1. distribute-rgt1-in 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00084 \lor \neg \left(re \leq 3.9\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -38:\\ \;\;\;\;t\_0 \cdot 0\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+25}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re (+ 1.0 (* re 0.5))))))
   (if (<= re -38.0) (* t_0 0.0) (if (<= re 6.5e+25) (sin im) (* im t_0)))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	double tmp;
	if (re <= -38.0) {
		tmp = t_0 * 0.0;
	} else if (re <= 6.5e+25) {
		tmp = sin(im);
	} else {
		tmp = 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 = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    if (re <= (-38.0d0)) then
        tmp = t_0 * 0.0d0
    else if (re <= 6.5d+25) then
        tmp = sin(im)
    else
        tmp = im * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	double tmp;
	if (re <= -38.0) {
		tmp = t_0 * 0.0;
	} else if (re <= 6.5e+25) {
		tmp = Math.sin(im);
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * (1.0 + (re * 0.5)))
	tmp = 0
	if re <= -38.0:
		tmp = t_0 * 0.0
	elif re <= 6.5e+25:
		tmp = math.sin(im)
	else:
		tmp = im * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))))
	tmp = 0.0
	if (re <= -38.0)
		tmp = Float64(t_0 * 0.0);
	elseif (re <= 6.5e+25)
		tmp = sin(im);
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	tmp = 0.0;
	if (re <= -38.0)
		tmp = t_0 * 0.0;
	elseif (re <= 6.5e+25)
		tmp = sin(im);
	else
		tmp = im * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -38.0], N[(t$95$0 * 0.0), $MachinePrecision], If[LessEqual[re, 6.5e+25], N[Sin[im], $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -38

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.1%

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

      1. *-rgt-identity 2.1%

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

      2. distribute-lft-in 2.1%

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

      3. associate-*r* 2.1%

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

      4. associate-*r* 2.0%

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

      5. distribute-rgt-out 2.0%

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

      6. distribute-lft-out 2.0%

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

      7. *-rgt-identity 2.0%

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

      8. distribute-lft-out 2.0%

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

      9. *-commutative 2.0%

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

   5. Simplified 2.0%

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

      1. expm1-log1p-u 2.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      2. expm1-undefine 21.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sin im\right)} - 1\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      3. log1p-undefine 21.7%

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

      4. rem-exp-log 21.7%

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

   7. Applied egg-rr 21.7%

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

   8. Taylor expanded in im around 0 50.0%

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

   if -38 < re < 6.50000000000000005e25

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 93.6%

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

   if 6.50000000000000005e25 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 36.2%

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

      1. *-rgt-identity 36.2%

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

      2. distribute-lft-in 36.2%

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

      3. associate-*r* 36.2%

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

      4. associate-*r* 60.2%

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

      5. distribute-rgt-out 60.2%

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

      6. distribute-lft-out 60.2%

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

      7. *-rgt-identity 60.2%

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

      8. distribute-lft-out 60.2%

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

      9. *-commutative 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in im around 0 56.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -38:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot 0\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+25}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -1.08 \cdot 10^{-10}:\\ \;\;\;\;t\_0 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re (+ 1.0 (* re 0.5))))))
   (if (<= re -1.08e-10) (* t_0 0.0) (* im t_0))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	double tmp;
	if (re <= -1.08e-10) {
		tmp = t_0 * 0.0;
	} else {
		tmp = 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 = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    if (re <= (-1.08d-10)) then
        tmp = t_0 * 0.0d0
    else
        tmp = im * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	double tmp;
	if (re <= -1.08e-10) {
		tmp = t_0 * 0.0;
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * (1.0 + (re * 0.5)))
	tmp = 0
	if re <= -1.08e-10:
		tmp = t_0 * 0.0
	else:
		tmp = im * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))))
	tmp = 0.0
	if (re <= -1.08e-10)
		tmp = Float64(t_0 * 0.0);
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * (1.0 + (re * 0.5)));
	tmp = 0.0;
	if (re <= -1.08e-10)
		tmp = t_0 * 0.0;
	else
		tmp = im * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.08e-10], N[(t$95$0 * 0.0), $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -1.08000000000000002e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 3.3%

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

      1. *-rgt-identity 3.3%

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

      2. distribute-lft-in 3.3%

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

      3. associate-*r* 3.3%

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

      4. associate-*r* 3.3%

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

      5. distribute-rgt-out 3.3%

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

      6. distribute-lft-out 3.3%

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

      7. *-rgt-identity 3.3%

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

      8. distribute-lft-out 3.3%

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

      9. *-commutative 3.3%

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

   5. Simplified 3.3%

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

      1. expm1-log1p-u 3.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin im\right)\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      2. expm1-undefine 22.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sin im\right)} - 1\right)} \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \]

      3. log1p-undefine 22.6%

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

      4. rem-exp-log 22.6%

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

   7. Applied egg-rr 22.6%

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

   8. Taylor expanded in im around 0 49.4%

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

   if -1.08000000000000002e-10 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 75.7%

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

      1. *-rgt-identity 75.7%

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

      2. distribute-lft-in 75.7%

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

      3. associate-*r* 75.7%

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

      4. associate-*r* 83.6%

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

      5. distribute-rgt-out 83.6%

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

      6. distribute-lft-out 83.6%

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

      7. *-rgt-identity 83.6%

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

      8. distribute-lft-out 83.6%

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

      9. *-commutative 83.6%

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

   5. Simplified 83.6%

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

   6. Taylor expanded in im around 0 53.5%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{-10}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.5% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ 1.0 (* re (+ 1.0 (* re 0.5))))))

C

double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * 0.5))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
end function

Java

public static double code(double re, double im) {
	return im * (1.0 + (re * (1.0 + (re * 0.5))));
}

Python

def code(re, im):
	return im * (1.0 + (re * (1.0 + (re * 0.5))))

Julia

function code(re, im)
	return Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 53.4%

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

   1. *-rgt-identity 53.4%

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

   2. distribute-lft-in 53.4%

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

   3. associate-*r* 53.4%

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

   4. associate-*r* 58.8%

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

   5. distribute-rgt-out 58.8%

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

   6. distribute-lft-out 58.8%

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

   7. *-rgt-identity 58.8%

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

   8. distribute-lft-out 58.8%

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

   9. *-commutative 58.8%

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

5. Simplified 58.8%

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

6. Taylor expanded in im around 0 37.6%

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

Alternative 8: 34.1% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ im + re \cdot \left(im \cdot \left(re \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ im (* re (* im (* re 0.5)))))

C

double code(double re, double im) {
	return im + (re * (im * (re * 0.5)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im + (re * (im * (re * 0.5d0)))
end function

Java

public static double code(double re, double im) {
	return im + (re * (im * (re * 0.5)));
}

Python

def code(re, im):
	return im + (re * (im * (re * 0.5)))

Julia

function code(re, im)
	return Float64(im + Float64(re * Float64(im * Float64(re * 0.5))))
end

MATLAB

function tmp = code(re, im)
	tmp = im + (re * (im * (re * 0.5)));
end

Wolfram

code[re_, im_] := N[(im + N[(re * N[(im * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 75.1%

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

4. Taylor expanded in re around 0 32.1%

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

   1. associate-*r* 32.1%

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

   2. *-commutative 32.1%

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

6. Simplified 32.1%

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

7. Taylor expanded in re around inf 32.0%

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

   1. *-commutative 32.0%

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

   2. associate-*r* 32.0%

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

9. Simplified 32.0%

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

Alternative 9: 29.0% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.12 \cdot 10^{-27}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.12e-27) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.12e-27) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.12d-27) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.12e-27) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.12e-27:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.12e-27)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.12e-27)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.12e-27], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.1199999999999999e-27

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.9%

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

   4. Taylor expanded in re around 0 34.3%

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

   if 1.1199999999999999e-27 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 10.6%

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

      1. distribute-rgt1-in 10.5%

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

   5. Simplified 10.5%

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

   6. Taylor expanded in im around 0 11.2%

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

   7. Taylor expanded in re around inf 11.2%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.12 \cdot 10^{-27}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.7% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 46.2%

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

   1. distribute-rgt1-in 46.2%

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

5. Simplified 46.2%

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

6. Taylor expanded in im around 0 27.9%

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

7. Final simplification 27.9%

   \[\leadsto im \cdot \left(re + 1\right) \]
8. Add Preprocessing

Alternative 11: 27.0% 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 75.1%

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

4. Taylor expanded in re around 0 25.7%

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

Reproduce

herbie shell --seed 2024106 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))