math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 8

Speedup: 1.0×

• 25.0% 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. Final simplification 100.0%

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

Alternative 2: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999996:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.99999996)
   (* (exp re) im)
   (if (<= (exp re) 2.0) (* (sin im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.99999996) {
		tmp = exp(re) * im;
	} else if (exp(re) <= 2.0) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.99999996d0) then
        tmp = exp(re) * im
    else if (exp(re) <= 2.0d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.99999996) {
		tmp = Math.exp(re) * im;
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.99999996:
		tmp = math.exp(re) * im
	elif math.exp(re) <= 2.0:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.99999996)
		tmp = Float64(exp(re) * im);
	elseif (exp(re) <= 2.0)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.99999996)
		tmp = exp(re) * im;
	elseif (exp(re) <= 2.0)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.99999996], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.99999996000000002

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 98.6%

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

   if 0.99999996000000002 < (exp.f64 re) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   4. Simplified 100.0%

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

   if 2 < (exp.f64 re)

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 45.5%

         \[\leadsto \color{blue}{{\left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 45.5%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 45.5%

         \[\leadsto e^{\log \color{blue}{\left({\left(e^{re} \cdot \sin im\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 45.5%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(e^{re} \cdot \sin im\right)\right)} \cdot 0.3333333333333333} \]

      6. log-prod 45.5%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{re}\right) + \log \sin im\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 45.5%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{re} + \log \sin im\right)\right) \cdot 0.3333333333333333} \]

   3. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(re + \log \sin im\right)\right) \cdot 0.3333333333333333}} \]

   4. Taylor expanded in re around inf 45.5%

      \[\leadsto e^{\color{blue}{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999996:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0))) (exp re) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} 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) <= 2.0d0))) then
        tmp = exp(re)
    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) <= 2.0)) {
		tmp = Math.exp(re);
	} 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) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0))
		tmp = exp(re);
	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) <= 2.0)))
		tmp = exp(re);
	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], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 73.5%

         \[\leadsto \color{blue}{{\left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 73.5%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 73.5%

         \[\leadsto e^{\log \color{blue}{\left({\left(e^{re} \cdot \sin im\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 73.5%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(e^{re} \cdot \sin im\right)\right)} \cdot 0.3333333333333333} \]

      6. log-prod 43.4%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{re}\right) + \log \sin im\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 43.4%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{re} + \log \sin im\right)\right) \cdot 0.3333333333333333} \]

   3. Applied egg-rr 43.4%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(re + \log \sin im\right)\right) \cdot 0.3333333333333333}} \]

   4. Taylor expanded in re around inf 73.5%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 98.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 4: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999996:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.99999996)
   (* (exp re) im)
   (if (<= (exp re) 2.0) (sin im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.99999996) {
		tmp = exp(re) * im;
	} else if (exp(re) <= 2.0) {
		tmp = sin(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.99999996d0) then
        tmp = exp(re) * im
    else if (exp(re) <= 2.0d0) then
        tmp = sin(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.99999996) {
		tmp = Math.exp(re) * im;
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.sin(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.99999996:
		tmp = math.exp(re) * im
	elif math.exp(re) <= 2.0:
		tmp = math.sin(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.99999996)
		tmp = Float64(exp(re) * im);
	elseif (exp(re) <= 2.0)
		tmp = sin(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.99999996)
		tmp = exp(re) * im;
	elseif (exp(re) <= 2.0)
		tmp = sin(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.99999996], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[Sin[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.99999996000000002

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 98.6%

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

   if 0.99999996000000002 < (exp.f64 re) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 99.4%

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

   if 2 < (exp.f64 re)

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 45.5%

         \[\leadsto \color{blue}{{\left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 45.5%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 45.5%

         \[\leadsto e^{\log \color{blue}{\left({\left(e^{re} \cdot \sin im\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 45.5%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(e^{re} \cdot \sin im\right)\right)} \cdot 0.3333333333333333} \]

      6. log-prod 45.5%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{re}\right) + \log \sin im\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 45.5%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{re} + \log \sin im\right)\right) \cdot 0.3333333333333333} \]

   3. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(re + \log \sin im\right)\right) \cdot 0.3333333333333333}} \]

   4. Taylor expanded in re around inf 45.5%

      \[\leadsto e^{\color{blue}{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999996:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 5: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -11 \lor \neg \left(re \leq 1.8 \cdot 10^{-13}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -11.0) (not (<= re 1.8e-13))) (exp re) (* im (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -11.0) || !(re <= 1.8e-13)) {
		tmp = exp(re);
	} else {
		tmp = 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 <= (-11.0d0)) .or. (.not. (re <= 1.8d-13))) then
        tmp = exp(re)
    else
        tmp = im * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -11.0) || !(re <= 1.8e-13)) {
		tmp = Math.exp(re);
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -11.0) or not (re <= 1.8e-13):
		tmp = math.exp(re)
	else:
		tmp = im * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -11.0) || !(re <= 1.8e-13))
		tmp = exp(re);
	else
		tmp = Float64(im * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -11.0) || ~((re <= 1.8e-13)))
		tmp = exp(re);
	else
		tmp = im * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -11.0], N[Not[LessEqual[re, 1.8e-13]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -11 or 1.7999999999999999e-13 < re

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 72.5%

         \[\leadsto \color{blue}{{\left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 72.5%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 72.5%

         \[\leadsto e^{\log \color{blue}{\left({\left(e^{re} \cdot \sin im\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 72.5%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(e^{re} \cdot \sin im\right)\right)} \cdot 0.3333333333333333} \]

      6. log-prod 42.8%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{re}\right) + \log \sin im\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 42.8%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{re} + \log \sin im\right)\right) \cdot 0.3333333333333333} \]

   3. Applied egg-rr 42.8%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(re + \log \sin im\right)\right) \cdot 0.3333333333333333}} \]

   4. Taylor expanded in re around inf 72.5%

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

   if -11 < re < 1.7999999999999999e-13

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 99.1%

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

      1. distribute-rgt1-in 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in im around 0 53.1%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -11 \lor \neg \left(re \leq 1.8 \cdot 10^{-13}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 6: 30.1% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{-13}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.8e-13) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.8e-13) {
		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 <= 1.8d-13) 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 <= 1.8e-13) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.8e-13:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.8e-13)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.8e-13)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.8e-13], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.7999999999999999e-13

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 63.3%

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

      1. distribute-rgt1-in 63.3%

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

   4. Simplified 63.3%

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

   5. Taylor expanded in im around 0 34.3%

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

   6. Taylor expanded in re around 0 34.6%

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

   if 1.7999999999999999e-13 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 7.2%

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

      1. distribute-rgt1-in 7.2%

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

   4. Simplified 7.2%

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

   5. Taylor expanded in im around 0 11.6%

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

   6. Taylor expanded in re around inf 11.6%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{-13}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 7: 30.3% 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. Taylor expanded in re around 0 48.4%

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

   1. distribute-rgt1-in 48.4%

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

4. Simplified 48.4%

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

5. Taylor expanded in im around 0 28.3%

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

6. Final simplification 28.3%

   \[\leadsto im \cdot \left(re + 1\right) \]

Alternative 8: 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. Taylor expanded in re around 0 48.4%

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

   1. distribute-rgt1-in 48.4%

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

4. Simplified 48.4%

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

5. Taylor expanded in im around 0 28.3%

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

6. Taylor expanded in re around 0 26.0%

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

7. Final simplification 26.0%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023334 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))