math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 99.9%

Time: 6.7s

Alternatives: 13

Speedup: 1.0×

• 25.2% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(e^{re}\right)}^{2}} \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (cbrt (pow (exp re) 2.0)) (* (cbrt (exp re)) (sin im))))

C

double code(double re, double im) {
	return cbrt(pow(exp(re), 2.0)) * (cbrt(exp(re)) * sin(im));
}

Java

public static double code(double re, double im) {
	return Math.cbrt(Math.pow(Math.exp(re), 2.0)) * (Math.cbrt(Math.exp(re)) * Math.sin(im));
}

Julia

function code(re, im)
	return Float64(cbrt((exp(re) ^ 2.0)) * Float64(cbrt(exp(re)) * sin(im)))
end

Wolfram

code[re_, im_] := N[(N[Power[N[Power[N[Exp[re], $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[N[Exp[re], $MachinePrecision], 1/3], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 82.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)}} \]

   2. pow3 82.9%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{re} \cdot \sin im\right)}^{3}}} \]

4. Applied egg-rr 82.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(e^{re} \cdot \sin im\right)}^{3}}} \]
5. Step-by-step derivation

   1. rem-cbrt-cube 100.0%

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

   2. add-cube-cbrt 99.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right) \cdot \sqrt[3]{e^{re}}\right)} \cdot \sin im \]

   3. associate-*l* 99.9%

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right) \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right)} \]

   4. pow2 99.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{re}}\right)}^{2}} \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{re}}\right)}^{2} \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right)} \]

7. Taylor expanded in re around inf 100.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(e^{re}\right)}^{2}}} \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right) \]

8. Final simplification 100.0%

   \[\leadsto \sqrt[3]{{\left(e^{re}\right)}^{2}} \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right) \]
9. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt[3]{e^{re}} \cdot \sin im\right) \cdot \sqrt[3]{e^{re \cdot 2}} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* (cbrt (exp re)) (sin im)) (cbrt (exp (* re 2.0)))))

C

double code(double re, double im) {
	return (cbrt(exp(re)) * sin(im)) * cbrt(exp((re * 2.0)));
}

Java

public static double code(double re, double im) {
	return (Math.cbrt(Math.exp(re)) * Math.sin(im)) * Math.cbrt(Math.exp((re * 2.0)));
}

Julia

function code(re, im)
	return Float64(Float64(cbrt(exp(re)) * sin(im)) * cbrt(exp(Float64(re * 2.0))))
end

Wolfram

code[re_, im_] := N[(N[(N[Power[N[Exp[re], $MachinePrecision], 1/3], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[N[(re * 2.0), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 82.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)}} \]

   2. pow3 82.9%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{re} \cdot \sin im\right)}^{3}}} \]

4. Applied egg-rr 82.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(e^{re} \cdot \sin im\right)}^{3}}} \]
5. Step-by-step derivation

   1. rem-cbrt-cube 100.0%

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

   2. add-cube-cbrt 99.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right) \cdot \sqrt[3]{e^{re}}\right)} \cdot \sin im \]

   3. associate-*l* 99.9%

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{re}} \cdot \sqrt[3]{e^{re}}\right) \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right)} \]

   4. pow2 99.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{re}}\right)}^{2}} \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{re}}\right)}^{2} \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right)} \]

7. Taylor expanded in re around inf 100.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(e^{re}\right)}^{2}}} \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right) \]
8. Step-by-step derivation

   1. pow-exp 100.0%

      \[\leadsto \sqrt[3]{\color{blue}{e^{re \cdot 2}}} \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \sqrt[3]{\color{blue}{e^{re \cdot 2}}} \cdot \left(\sqrt[3]{e^{re}} \cdot \sin im\right) \]

10. Final simplification 100.0%

    \[\leadsto \left(\sqrt[3]{e^{re}} \cdot \sin im\right) \cdot \sqrt[3]{e^{re \cdot 2}} \]
11. Add Preprocessing

Alternative 3: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999996 \lor \neg \left(e^{re} \leq 10^{+51}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.99999996) (not (<= (exp re) 1e+51)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.99999996) || !(exp(re) <= 1e+51)) {
		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) <= 0.99999996d0) .or. (.not. (exp(re) <= 1d+51))) 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) <= 0.99999996) || !(Math.exp(re) <= 1e+51)) {
		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) <= 0.99999996) or not (math.exp(re) <= 1e+51):
		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) <= 0.99999996) || !(exp(re) <= 1e+51))
		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) <= 0.99999996) || ~((exp(re) <= 1e+51)))
		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], 0.99999996], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1e+51]], $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) < 0.99999996000000002 or 1e51 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.0%

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

   if 0.99999996000000002 < (exp.f64 re) < 1e51

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

      1. distribute-rgt1-in 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999996 \lor \neg \left(e^{re} \leq 10^{+51}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999999999995 \lor \neg \left(e^{re} \leq 10^{+51}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999999999995) (not (<= (exp re) 1e+51)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999999999995) || !(exp(re) <= 1e+51)) {
		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) <= 0.9999999999995d0) .or. (.not. (exp(re) <= 1d+51))) 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) <= 0.9999999999995) || !(Math.exp(re) <= 1e+51)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999999999995) or not (math.exp(re) <= 1e+51):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999999999995) || !(exp(re) <= 1e+51))
		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) <= 0.9999999999995) || ~((exp(re) <= 1e+51)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999999999995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1e+51]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999999999949996 or 1e51 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.2%

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

   if 0.99999999999949996 < (exp.f64 re) < 1e51

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 98.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999999999995 \lor \neg \left(e^{re} \leq 10^{+51}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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. Final simplification 100.0%

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

Alternative 6: 62.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 225:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 225.0)
   (sin im)
   (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 225.0) {
		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 <= 225.0d0) 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 <= 225.0) {
		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 <= 225.0:
		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 <= 225.0)
		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 <= 225.0)
		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, 225.0], 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 2 regimes

2. if re < 225

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 64.8%

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

   if 225 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 75.0%

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

   4. Taylor expanded in re around 0 47.1%

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

   5. Taylor expanded in im around 0 58.1%

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

      1. *-commutative 58.1%

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

   7. Simplified 58.1%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 225:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.5% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	return im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im + (im * (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
end function

Java

public static double code(double re, double im) {
	return im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
}

Python

def code(re, im):
	return im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))

Julia

function code(re, im)
	return Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(re, im)
	tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
end

Wolfram

code[re_, im_] := 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. Initial program 100.0%

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

3. Taylor expanded in im around 0 69.9%

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

4. Taylor expanded in re around 0 38.0%

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

5. Taylor expanded in im around 0 40.9%

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

   1. *-commutative 40.9%

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

7. Simplified 40.9%

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

8. Final simplification 40.9%

   \[\leadsto im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \]
9. Add Preprocessing

Alternative 8: 36.1% accurate, 18.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return im + (im * (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 + (im * (re * (1.0d0 + (re * 0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im + N[(im * 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 im around 0 69.9%

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

4. Taylor expanded in re around 0 38.0%

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

5. Taylor expanded in re around 0 33.8%

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

   1. associate-*r* 33.8%

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

   2. *-commutative 33.8%

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

   3. *-commutative 33.8%

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

7. Simplified 33.8%

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

8. Taylor expanded in im around 0 36.8%

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

9. Final simplification 36.8%

   \[\leadsto im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right) \]
10. Add Preprocessing

Alternative 9: 33.0% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im + N[(re * N[(re * N[(im * 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 69.9%

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

4. Taylor expanded in re around 0 38.0%

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

5. Taylor expanded in re around 0 33.8%

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

   1. associate-*r* 33.8%

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

   2. *-commutative 33.8%

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

   3. *-commutative 33.8%

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

7. Simplified 33.8%

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

8. Taylor expanded in re around inf 33.3%

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

   1. associate-*r* 33.3%

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

   2. *-commutative 33.3%

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

   3. *-commutative 33.3%

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

10. Simplified 33.3%

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

11. Final simplification 33.3%

    \[\leadsto im + re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) \]
12. Add Preprocessing

Alternative 10: 27.3% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3 \cdot 10^{+40}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 3e+40) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3e+40) {
		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 <= 3d+40) 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 <= 3e+40) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3e+40:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3e+40)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3e+40)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3e+40], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.0000000000000002e40

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 78.9%

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

   4. Taylor expanded in re around 0 32.2%

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

   if 3.0000000000000002e40 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 35.3%

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

   4. Taylor expanded in re around 0 7.8%

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

   5. Taylor expanded in re around inf 8.9%

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

      1. *-commutative 8.9%

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

   7. Simplified 8.9%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3 \cdot 10^{+40}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.9% 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 48.9%

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

   1. distribute-rgt1-in 48.9%

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

5. Simplified 48.9%

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

6. Taylor expanded in im around 0 29.3%

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

7. Final simplification 29.3%

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

Alternative 12: 28.9% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im + re \cdot im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 69.9%

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

4. Taylor expanded in re around 0 29.3%

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

5. Final simplification 29.3%

   \[\leadsto im + re \cdot im \]
6. Add Preprocessing

Alternative 13: 25.7% 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 69.9%

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

4. Taylor expanded in re around 0 26.1%

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

5. Final simplification 26.1%

   \[\leadsto im \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024061 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))