math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 13

Speedup: 1.0×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{re} \cdot \sin im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.exp(re) * math.sin(im)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \sin im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.exp(re) * math.sin(im)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \sin im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.exp(re) * math.sin(im)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.995) (not (<= (exp re) 1.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.995) || !(exp(re) <= 1.0)) {
		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) <= 0.995d0) .or. (.not. (exp(re) <= 1.0d0))) 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) <= 0.995) || !(Math.exp(re) <= 1.0)) {
		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) <= 0.995) or not (math.exp(re) <= 1.0):
		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) <= 0.995) || !(exp(re) <= 1.0))
		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) <= 0.995) || ~((exp(re) <= 1.0)))
		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], 0.995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $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) < 0.994999999999999996 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.4%

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

   if 0.994999999999999996 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.995) (not (<= (exp re) 1.0)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.995) || !(exp(re) <= 1.0)) {
		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) <= 0.995d0) .or. (.not. (exp(re) <= 1.0d0))) 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) <= 0.995) || !(Math.exp(re) <= 1.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.995) or not (math.exp(re) <= 1.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.995) || !(exp(re) <= 1.0))
		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) <= 0.995) || ~((exp(re) <= 1.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.995], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.994999999999999996 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.4%

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

   if 0.994999999999999996 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 99.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.995 \lor \neg \left(e^{re} \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -180:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -180.0)
   0.0
   (if (<= re 4.9e-30)
     (sin im)
     (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -180.0) {
		tmp = 0.0;
	} else if (re <= 4.9e-30) {
		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 <= (-180.0d0)) then
        tmp = 0.0d0
    else if (re <= 4.9d-30) 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 <= -180.0) {
		tmp = 0.0;
	} else if (re <= 4.9e-30) {
		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 <= -180.0:
		tmp = 0.0
	elif re <= 4.9e-30:
		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 <= -180.0)
		tmp = 0.0;
	elseif (re <= 4.9e-30)
		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 <= -180.0)
		tmp = 0.0;
	elseif (re <= 4.9e-30)
		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, -180.0], 0.0, If[LessEqual[re, 4.9e-30], 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 3 regimes

2. if re < -180

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \log \color{blue}{1} \]

   if -180 < re < 4.89999999999999971e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 95.5%

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

   if 4.89999999999999971e-30 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 78.5%

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

   4. Taylor expanded in re around 0 48.3%

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

   5. Taylor expanded in im around 0 52.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -180:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -18:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -18.0)
   0.0
   (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -18.0) {
		tmp = 0.0;
	} 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 <= (-18.0d0)) then
        tmp = 0.0d0
    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 <= -18.0) {
		tmp = 0.0;
	} else {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -18.0:
		tmp = 0.0
	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 <= -18.0)
		tmp = 0.0;
	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 <= -18.0)
		tmp = 0.0;
	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, -18.0], 0.0, 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 < -18

   1. Initial program 100.0%

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

      1. add-log-exp 98.4%

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

      2. exp-prod 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in im around 0 98.4%

      \[\leadsto \log \color{blue}{1} \]

   if -18 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 57.7%

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

   4. Taylor expanded in re around 0 46.3%

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

   5. Taylor expanded in im around 0 47.8%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -18:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.2% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+248}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85e+248)
   (+ im (* im (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
   (* im (* (+ re 1.0) (+ 1.0 (* im (* im -0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85e+248) {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -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 (im <= 1.85d+248) then
        tmp = im + (im * (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    else
        tmp = im * ((re + 1.0d0) * (1.0d0 + (im * (im * (-0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.85e+248) {
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85e+248:
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	else:
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.85e+248)
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.85e+248)
		tmp = im + (im * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	else
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.85e+248], 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], N[(im * N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.8499999999999999e248

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.6%

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

   4. Taylor expanded in re around 0 37.4%

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

   5. Taylor expanded in im around 0 38.6%

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

   if 1.8499999999999999e248 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 47.4%

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

      1. distribute-rgt1-in 47.4%

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

      2. *-commutative 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in im around 0 19.4%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 19.4%

         \[\leadsto im \cdot \color{blue}{\left(\left(1 + re\right) + -0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)} \]

      2. +-commutative 19.4%

         \[\leadsto im \cdot \left(\color{blue}{\left(re + 1\right)} + -0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) \]

      3. distribute-rgt-in 0.7%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot im + \left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) \cdot im} \]

      4. +-commutative 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \color{blue}{\left(re + 1\right)}\right)\right) \cdot im \]

      5. *-commutative 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(-0.16666666666666666 \cdot \color{blue}{\left(\left(re + 1\right) \cdot {im}^{2}\right)}\right) \cdot im \]

      6. associate-*r* 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(re + 1\right)\right) \cdot {im}^{2}\right)} \cdot im \]

      7. distribute-lft-in 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(\color{blue}{\left(-0.16666666666666666 \cdot re + -0.16666666666666666 \cdot 1\right)} \cdot {im}^{2}\right) \cdot im \]

      8. metadata-eval 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(\left(-0.16666666666666666 \cdot re + \color{blue}{-0.16666666666666666}\right) \cdot {im}^{2}\right) \cdot im \]

      9. metadata-eval 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(\left(-0.16666666666666666 \cdot re + \color{blue}{\left(-0.16666666666666666\right)}\right) \cdot {im}^{2}\right) \cdot im \]

      10. sub-neg 0.7%

          \[\leadsto \left(re + 1\right) \cdot im + \left(\color{blue}{\left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)} \cdot {im}^{2}\right) \cdot im \]

      11. *-commutative 0.7%

          \[\leadsto \left(re + 1\right) \cdot im + \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)\right)} \cdot im \]

      12. distribute-rgt-in 19.4%

          \[\leadsto \color{blue}{im \cdot \left(\left(re + 1\right) + {im}^{2} \cdot \left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)\right)} \]

      13. *-rgt-identity 19.4%

          \[\leadsto im \cdot \left(\color{blue}{\left(re + 1\right) \cdot 1} + {im}^{2} \cdot \left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)\right) \]

      14. sub-neg 19.4%

          \[\leadsto im \cdot \left(\left(re + 1\right) \cdot 1 + {im}^{2} \cdot \color{blue}{\left(-0.16666666666666666 \cdot re + \left(-0.16666666666666666\right)\right)}\right) \]

      15. metadata-eval 19.4%

          \[\leadsto im \cdot \left(\left(re + 1\right) \cdot 1 + {im}^{2} \cdot \left(-0.16666666666666666 \cdot re + \color{blue}{-0.16666666666666666}\right)\right) \]

      16. distribute-rgt-in 18.5%

          \[\leadsto im \cdot \left(\left(re + 1\right) \cdot 1 + \color{blue}{\left(\left(-0.16666666666666666 \cdot re\right) \cdot {im}^{2} + -0.16666666666666666 \cdot {im}^{2}\right)}\right) \]

   8. Simplified 19.4%

      \[\leadsto \color{blue}{im \cdot \left(\left(re + 1\right) \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 19.4%

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

      2. associate-*r* 19.4%

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

   10. Applied egg-rr 19.4%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+248}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.9% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+248}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85e+248)
   (+ im (* re (+ im (* re (* 0.16666666666666666 (* re im))))))
   (* im (* (+ re 1.0) (+ 1.0 (* im (* im -0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85e+248) {
		tmp = im + (re * (im + (re * (0.16666666666666666 * (re * im)))));
	} else {
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -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 (im <= 1.85d+248) then
        tmp = im + (re * (im + (re * (0.16666666666666666d0 * (re * im)))))
    else
        tmp = im * ((re + 1.0d0) * (1.0d0 + (im * (im * (-0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.85e+248) {
		tmp = im + (re * (im + (re * (0.16666666666666666 * (re * im)))));
	} else {
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85e+248:
		tmp = im + (re * (im + (re * (0.16666666666666666 * (re * im)))))
	else:
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.85e+248)
		tmp = Float64(im + Float64(re * Float64(im + Float64(re * Float64(0.16666666666666666 * Float64(re * im))))));
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.85e+248)
		tmp = im + (re * (im + (re * (0.16666666666666666 * (re * im)))));
	else
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.85e+248], N[(im + N[(re * N[(im + N[(re * N[(0.16666666666666666 * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.8499999999999999e248

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.6%

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

   4. Taylor expanded in re around 0 37.4%

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

   5. Taylor expanded in re around inf 37.2%

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

   if 1.8499999999999999e248 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 47.4%

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

      1. distribute-rgt1-in 47.4%

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

      2. *-commutative 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in im around 0 19.4%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 19.4%

         \[\leadsto im \cdot \color{blue}{\left(\left(1 + re\right) + -0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)} \]

      2. +-commutative 19.4%

         \[\leadsto im \cdot \left(\color{blue}{\left(re + 1\right)} + -0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) \]

      3. distribute-rgt-in 0.7%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot im + \left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) \cdot im} \]

      4. +-commutative 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \color{blue}{\left(re + 1\right)}\right)\right) \cdot im \]

      5. *-commutative 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(-0.16666666666666666 \cdot \color{blue}{\left(\left(re + 1\right) \cdot {im}^{2}\right)}\right) \cdot im \]

      6. associate-*r* 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(re + 1\right)\right) \cdot {im}^{2}\right)} \cdot im \]

      7. distribute-lft-in 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(\color{blue}{\left(-0.16666666666666666 \cdot re + -0.16666666666666666 \cdot 1\right)} \cdot {im}^{2}\right) \cdot im \]

      8. metadata-eval 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(\left(-0.16666666666666666 \cdot re + \color{blue}{-0.16666666666666666}\right) \cdot {im}^{2}\right) \cdot im \]

      9. metadata-eval 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(\left(-0.16666666666666666 \cdot re + \color{blue}{\left(-0.16666666666666666\right)}\right) \cdot {im}^{2}\right) \cdot im \]

      10. sub-neg 0.7%

          \[\leadsto \left(re + 1\right) \cdot im + \left(\color{blue}{\left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)} \cdot {im}^{2}\right) \cdot im \]

      11. *-commutative 0.7%

          \[\leadsto \left(re + 1\right) \cdot im + \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)\right)} \cdot im \]

      12. distribute-rgt-in 19.4%

          \[\leadsto \color{blue}{im \cdot \left(\left(re + 1\right) + {im}^{2} \cdot \left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)\right)} \]

      13. *-rgt-identity 19.4%

          \[\leadsto im \cdot \left(\color{blue}{\left(re + 1\right) \cdot 1} + {im}^{2} \cdot \left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)\right) \]

      14. sub-neg 19.4%

          \[\leadsto im \cdot \left(\left(re + 1\right) \cdot 1 + {im}^{2} \cdot \color{blue}{\left(-0.16666666666666666 \cdot re + \left(-0.16666666666666666\right)\right)}\right) \]

      15. metadata-eval 19.4%

          \[\leadsto im \cdot \left(\left(re + 1\right) \cdot 1 + {im}^{2} \cdot \left(-0.16666666666666666 \cdot re + \color{blue}{-0.16666666666666666}\right)\right) \]

      16. distribute-rgt-in 18.5%

          \[\leadsto im \cdot \left(\left(re + 1\right) \cdot 1 + \color{blue}{\left(\left(-0.16666666666666666 \cdot re\right) \cdot {im}^{2} + -0.16666666666666666 \cdot {im}^{2}\right)}\right) \]

   8. Simplified 19.4%

      \[\leadsto \color{blue}{im \cdot \left(\left(re + 1\right) \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 19.4%

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

      2. associate-*r* 19.4%

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

   10. Applied egg-rr 19.4%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+248}:\\ \;\;\;\;im + re \cdot \left(im + re \cdot \left(0.16666666666666666 \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+248}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85e+248)
   (+ im (* im (* re (+ 1.0 (* re 0.5)))))
   (* im (* (+ re 1.0) (+ 1.0 (* im (* im -0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85e+248) {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	} else {
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -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 (im <= 1.85d+248) then
        tmp = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    else
        tmp = im * ((re + 1.0d0) * (1.0d0 + (im * (im * (-0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.85e+248) {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	} else {
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85e+248:
		tmp = im + (im * (re * (1.0 + (re * 0.5))))
	else:
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.85e+248)
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = Float64(im * Float64(Float64(re + 1.0) * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.85e+248)
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	else
		tmp = im * ((re + 1.0) * (1.0 + (im * (im * -0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.85e+248], N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.8499999999999999e248

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 68.6%

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

   4. Taylor expanded in re around 0 35.0%

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

   5. Taylor expanded in im around 0 36.9%

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

   if 1.8499999999999999e248 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 47.4%

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

      1. distribute-rgt1-in 47.4%

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

      2. *-commutative 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in im around 0 19.4%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 19.4%

         \[\leadsto im \cdot \color{blue}{\left(\left(1 + re\right) + -0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)} \]

      2. +-commutative 19.4%

         \[\leadsto im \cdot \left(\color{blue}{\left(re + 1\right)} + -0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) \]

      3. distribute-rgt-in 0.7%

         \[\leadsto \color{blue}{\left(re + 1\right) \cdot im + \left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right) \cdot im} \]

      4. +-commutative 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \color{blue}{\left(re + 1\right)}\right)\right) \cdot im \]

      5. *-commutative 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(-0.16666666666666666 \cdot \color{blue}{\left(\left(re + 1\right) \cdot {im}^{2}\right)}\right) \cdot im \]

      6. associate-*r* 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(re + 1\right)\right) \cdot {im}^{2}\right)} \cdot im \]

      7. distribute-lft-in 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(\color{blue}{\left(-0.16666666666666666 \cdot re + -0.16666666666666666 \cdot 1\right)} \cdot {im}^{2}\right) \cdot im \]

      8. metadata-eval 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(\left(-0.16666666666666666 \cdot re + \color{blue}{-0.16666666666666666}\right) \cdot {im}^{2}\right) \cdot im \]

      9. metadata-eval 0.7%

         \[\leadsto \left(re + 1\right) \cdot im + \left(\left(-0.16666666666666666 \cdot re + \color{blue}{\left(-0.16666666666666666\right)}\right) \cdot {im}^{2}\right) \cdot im \]

      10. sub-neg 0.7%

          \[\leadsto \left(re + 1\right) \cdot im + \left(\color{blue}{\left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)} \cdot {im}^{2}\right) \cdot im \]

      11. *-commutative 0.7%

          \[\leadsto \left(re + 1\right) \cdot im + \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)\right)} \cdot im \]

      12. distribute-rgt-in 19.4%

          \[\leadsto \color{blue}{im \cdot \left(\left(re + 1\right) + {im}^{2} \cdot \left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)\right)} \]

      13. *-rgt-identity 19.4%

          \[\leadsto im \cdot \left(\color{blue}{\left(re + 1\right) \cdot 1} + {im}^{2} \cdot \left(-0.16666666666666666 \cdot re - 0.16666666666666666\right)\right) \]

      14. sub-neg 19.4%

          \[\leadsto im \cdot \left(\left(re + 1\right) \cdot 1 + {im}^{2} \cdot \color{blue}{\left(-0.16666666666666666 \cdot re + \left(-0.16666666666666666\right)\right)}\right) \]

      15. metadata-eval 19.4%

          \[\leadsto im \cdot \left(\left(re + 1\right) \cdot 1 + {im}^{2} \cdot \left(-0.16666666666666666 \cdot re + \color{blue}{-0.16666666666666666}\right)\right) \]

      16. distribute-rgt-in 18.5%

          \[\leadsto im \cdot \left(\left(re + 1\right) \cdot 1 + \color{blue}{\left(\left(-0.16666666666666666 \cdot re\right) \cdot {im}^{2} + -0.16666666666666666 \cdot {im}^{2}\right)}\right) \]

   8. Simplified 19.4%

      \[\leadsto \color{blue}{im \cdot \left(\left(re + 1\right) \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 19.4%

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

      2. associate-*r* 19.4%

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

   10. Applied egg-rr 19.4%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+248}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.9% 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 67.3%

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

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 35.7%

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

6. Final simplification 35.7%

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

Alternative 10: 33.7% 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 67.3%

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

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 33.6%

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

   1. *-commutative 33.6%

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

   2. associate-*r* 33.6%

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

7. Simplified 33.6%

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

Alternative 11: 29.7% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.0) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.0)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 63.6%

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

   4. Taylor expanded in re around 0 32.8%

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

   if 1 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 77.6%

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

   4. Taylor expanded in re around 0 14.5%

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

   5. Taylor expanded in re around inf 14.5%

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

      1. *-commutative 14.5%

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

   7. Simplified 14.5%

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

Alternative 12: 29.6% 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 67.3%

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

4. Taylor expanded in re around 0 27.9%

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

5. Final simplification 27.9%

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

Alternative 13: 26.2% 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 67.3%

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

4. Taylor expanded in re around 0 24.9%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))