math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 11

Speedup: 1.0×

• 23.8% 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: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 20000000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 20000000.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 20000000.0)) {
		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) <= 1.0d0) .or. (.not. (exp(re) <= 20000000.0d0))) 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) <= 1.0) || !(Math.exp(re) <= 20000000.0)) {
		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) <= 1.0) or not (math.exp(re) <= 20000000.0):
		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) <= 1.0) || !(exp(re) <= 20000000.0))
		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) <= 1.0) || ~((exp(re) <= 20000000.0)))
		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], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 20000000.0]], $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) < 1 or 2e7 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 72.3%

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

   if 1 < (exp.f64 re) < 2e7

   1. Initial program 99.7%

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

   3. Taylor expanded in re around 0 74.9%

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

      1. distribute-rgt1-in 74.9%

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

   5. Simplified 74.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 20000000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 20000000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 20000000.0)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 20000000.0)) {
		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) <= 1.0d0) .or. (.not. (exp(re) <= 20000000.0d0))) 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) <= 1.0) || !(Math.exp(re) <= 20000000.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 20000000.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 20000000.0))
		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) <= 1.0) || ~((exp(re) <= 20000000.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 20000000.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 2e7 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 72.3%

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

   if 1 < (exp.f64 re) < 2e7

   1. Initial program 99.7%

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

   3. Taylor expanded in re around 0 50.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 20000000\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 215000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im - im \cdot \left(re \cdot \left(-1 + re \cdot \left(re \cdot -0.16666666666666666 - 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 215000.0)
   (sin im)
   (- im (* im (* re (+ -1.0 (* re (- (* re -0.16666666666666666) 0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 215000.0) {
		tmp = sin(im);
	} else {
		tmp = im - (im * (re * (-1.0 + (re * ((re * -0.16666666666666666) - 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 <= 215000.0d0) then
        tmp = sin(im)
    else
        tmp = im - (im * (re * ((-1.0d0) + (re * ((re * (-0.16666666666666666d0)) - 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 215000.0) {
		tmp = Math.sin(im);
	} else {
		tmp = im - (im * (re * (-1.0 + (re * ((re * -0.16666666666666666) - 0.5)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 215000.0:
		tmp = math.sin(im)
	else:
		tmp = im - (im * (re * (-1.0 + (re * ((re * -0.16666666666666666) - 0.5)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 215000.0)
		tmp = sin(im);
	else
		tmp = Float64(im - Float64(im * Float64(re * Float64(-1.0 + Float64(re * Float64(Float64(re * -0.16666666666666666) - 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 215000.0)
		tmp = sin(im);
	else
		tmp = im - (im * (re * (-1.0 + (re * ((re * -0.16666666666666666) - 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 215000.0], N[Sin[im], $MachinePrecision], N[(im - N[(im * N[(re * N[(-1.0 + N[(re * N[(N[(re * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 215000

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 66.3%

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

   if 215000 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 77.8%

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

   4. Taylor expanded in re around 0 50.3%

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

   5. Taylor expanded in im around -inf 53.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 215000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im - im \cdot \left(re \cdot \left(-1 + re \cdot \left(re \cdot -0.16666666666666666 - 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.4% accurate, 13.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return im + (re * (im + (im * (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 + (re * (im + (im * (re * (0.5d0 + (re * 0.16666666666666666d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Taylor expanded in im around 0 71.7%

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

4. Taylor expanded in re around 0 40.5%

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

5. Taylor expanded in im around 0 40.9%

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

6. Final simplification 40.9%

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

Alternative 6: 38.6% accurate, 13.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return im - (im * (re * (-1.0 + (re * ((re * -0.16666666666666666) - 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 * ((re * (-0.16666666666666666d0)) - 0.5d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im - N[(im * N[(re * N[(-1.0 + N[(re * N[(N[(re * -0.16666666666666666), $MachinePrecision] - 0.5), $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 71.7%

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

4. Taylor expanded in re around 0 40.5%

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

5. Taylor expanded in im around -inf 41.3%

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

6. Final simplification 41.3%

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

Alternative 7: 36.2% 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 71.7%

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

4. Taylor expanded in re around 0 37.0%

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

   1. associate-*r* 37.0%

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

   2. *-commutative 37.0%

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

6. Simplified 37.0%

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

7. Taylor expanded in im around 0 40.2%

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

   1. *-commutative 40.2%

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

9. Simplified 40.2%

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

10. Final simplification 40.2%

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

Alternative 8: 33.1% 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 71.7%

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

4. Taylor expanded in re around 0 37.0%

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

   1. associate-*r* 37.0%

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

   2. *-commutative 37.0%

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

6. Simplified 37.0%

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

7. Taylor expanded in re around inf 36.5%

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

   1. *-commutative 36.5%

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

   2. associate-*r* 36.5%

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

9. Simplified 36.5%

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

10. Final simplification 36.5%

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

Alternative 9: 29.5% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 280:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 280.0) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 280.0) {
		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 <= 280.0d0) 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 <= 280.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 280.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 280.0)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 280.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 280.0], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 280

   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 37.6%

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

   if 280 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.1%

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

      1. distribute-rgt1-in 4.1%

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

   5. Simplified 4.1%

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

   6. Taylor expanded in re around inf 4.1%

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

   7. Taylor expanded in im around 0 12.1%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 280:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.4% 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 71.7%

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

4. Taylor expanded in re around 0 31.3%

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

5. Final simplification 31.3%

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

Alternative 11: 26.6% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 71.7%

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

4. Taylor expanded in re around 0 28.7%

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

5. Final simplification 28.7%

   \[\leadsto im \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))