math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 12

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: 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 2: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.998 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.998) (not (<= (exp re) 2.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.998) || !(exp(re) <= 2.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.998d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.998) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.998) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.998) || !(exp(re) <= 2.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.998) || ~((exp(re) <= 2.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.998 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.0%

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

   if 0.998 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.6%

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

      1. distribute-rgt1-in 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.998 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.998 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.998) (not (<= (exp re) 2.0)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.998) || !(exp(re) <= 2.0)) {
		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) <= 0.998d0) .or. (.not. (exp(re) <= 2.0d0))) 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) <= 0.998) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.998) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.998) || !(exp(re) <= 2.0))
		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) <= 0.998) || ~((exp(re) <= 2.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.998], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.998 or 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.0%

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

   if 0.998 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.998 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% 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. Add Preprocessing
   3. 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 77.8%

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

      3. pow-to-exp 77.8%

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

      4. pow3 77.8%

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

      5. log-pow 77.8%

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

      6. log-prod 54.0%

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

      7. add-log-exp 54.0%

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

   4. Applied egg-rr 54.0%

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

   5. Taylor expanded in re around inf 77.8%

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

   if 0.0 < (exp.f64 re) < 2

   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 88.3%

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

Alternative 5: 60.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.58 \lor \neg \left(re \leq 1.4 \cdot 10^{-42}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im + im \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.58) (not (<= re 1.4e-42)))
   (exp re)
   (+ im (* re (+ im (* im (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -1.58) || !(re <= 1.4e-42)) {
		tmp = exp(re);
	} else {
		tmp = im + (re * (im + (im * (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-1.58d0)) .or. (.not. (re <= 1.4d-42))) then
        tmp = exp(re)
    else
        tmp = im + (re * (im + (im * (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -1.58) || !(re <= 1.4e-42)) {
		tmp = Math.exp(re);
	} else {
		tmp = im + (re * (im + (im * (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -1.58) or not (re <= 1.4e-42):
		tmp = math.exp(re)
	else:
		tmp = im + (re * (im + (im * (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -1.58) || !(re <= 1.4e-42))
		tmp = exp(re);
	else
		tmp = Float64(im + Float64(re * Float64(im + Float64(im * Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -1.58) || ~((re <= 1.4e-42)))
		tmp = exp(re);
	else
		tmp = im + (re * (im + (im * (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -1.58], N[Not[LessEqual[re, 1.4e-42]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(im + N[(re * N[(im + N[(im * 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 < -1.5800000000000001 or 1.39999999999999999e-42 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. 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 76.3%

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

      3. pow-to-exp 76.3%

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

      4. pow3 76.3%

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

      5. log-pow 76.3%

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

      6. log-prod 53.4%

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

      7. add-log-exp 53.4%

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

   4. Applied egg-rr 53.4%

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

   5. Taylor expanded in re around inf 75.2%

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

   if -1.5800000000000001 < re < 1.39999999999999999e-42

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 51.6%

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

   4. Taylor expanded in re around 0 51.4%

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

   5. Taylor expanded in im around 0 51.4%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.58 \lor \neg \left(re \leq 1.4 \cdot 10^{-42}\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot \left(im + im \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.8% 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 67.0%

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

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 40.2%

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

6. Final simplification 40.2%

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

Alternative 7: 37.8% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im + N[(re * N[(im + N[(re * N[(im * N[(re * 0.16666666666666666), $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 67.0%

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

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 inf 37.9%

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

   1. *-commutative 37.9%

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

   2. associate-*r* 37.9%

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

7. Simplified 37.9%

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

8. Final simplification 37.9%

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

Alternative 8: 37.3% 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 67.0%

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

4. Taylor expanded in re around 0 33.9%

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

5. Taylor expanded in im around 0 36.9%

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

6. Final simplification 36.9%

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

Alternative 9: 33.9% 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 67.0%

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

4. Taylor expanded in re around 0 33.9%

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

5. Taylor expanded in re around inf 33.8%

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

   1. *-commutative 33.8%

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

   2. associate-*r* 33.8%

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

7. Simplified 33.8%

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

8. Final simplification 33.8%

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

Alternative 10: 27.6% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.7 \cdot 10^{+92}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 4.7e+92) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.7e+92) {
		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 <= 4.7d+92) 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 <= 4.7e+92) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.7e+92:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.7e+92)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.7e+92)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.7e+92], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.7e92

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.3%

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

   4. Taylor expanded in re around 0 32.2%

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

   if 4.7e92 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 36.3%

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

   4. Taylor expanded in re around 0 6.6%

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

   5. Taylor expanded in re around inf 7.3%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.7 \cdot 10^{+92}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.3% 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 67.0%

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

4. Taylor expanded in re around 0 28.7%

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

5. Final simplification 28.7%

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

Alternative 12: 26.0% 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 67.0%

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

4. Taylor expanded in re around 0 26.5%

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

5. Final simplification 26.5%

   \[\leadsto im \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))