math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

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. Final simplification 100.0%

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

Alternative 2: 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 3: 93.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := \sin im \cdot \left(re + 1\right)\\ \mathbf{if}\;re \leq -0.00165:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+196}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot t\_1}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)) (t_1 (* (sin im) (+ re 1.0))))
   (if (<= re -0.00165)
     t_0
     (if (<= re 4.9e-30) t_1 (if (<= re 6.2e+196) t_0 (/ (* re t_1) re))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = sin(im) * (re + 1.0);
	double tmp;
	if (re <= -0.00165) {
		tmp = t_0;
	} else if (re <= 4.9e-30) {
		tmp = t_1;
	} else if (re <= 6.2e+196) {
		tmp = t_0;
	} else {
		tmp = (re * t_1) / re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(re) * im
    t_1 = sin(im) * (re + 1.0d0)
    if (re <= (-0.00165d0)) then
        tmp = t_0
    else if (re <= 4.9d-30) then
        tmp = t_1
    else if (re <= 6.2d+196) then
        tmp = t_0
    else
        tmp = (re * t_1) / re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double t_1 = Math.sin(im) * (re + 1.0);
	double tmp;
	if (re <= -0.00165) {
		tmp = t_0;
	} else if (re <= 4.9e-30) {
		tmp = t_1;
	} else if (re <= 6.2e+196) {
		tmp = t_0;
	} else {
		tmp = (re * t_1) / re;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	t_1 = math.sin(im) * (re + 1.0)
	tmp = 0
	if re <= -0.00165:
		tmp = t_0
	elif re <= 4.9e-30:
		tmp = t_1
	elif re <= 6.2e+196:
		tmp = t_0
	else:
		tmp = (re * t_1) / re
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(sin(im) * Float64(re + 1.0))
	tmp = 0.0
	if (re <= -0.00165)
		tmp = t_0;
	elseif (re <= 4.9e-30)
		tmp = t_1;
	elseif (re <= 6.2e+196)
		tmp = t_0;
	else
		tmp = Float64(Float64(re * t_1) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	t_1 = sin(im) * (re + 1.0);
	tmp = 0.0;
	if (re <= -0.00165)
		tmp = t_0;
	elseif (re <= 4.9e-30)
		tmp = t_1;
	elseif (re <= 6.2e+196)
		tmp = t_0;
	else
		tmp = (re * t_1) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.00165], t$95$0, If[LessEqual[re, 4.9e-30], t$95$1, If[LessEqual[re, 6.2e+196], t$95$0, N[(N[(re * t$95$1), $MachinePrecision] / re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00165 or 4.89999999999999971e-30 < re < 6.2000000000000002e196

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.2%

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

   if -0.00165 < re < 4.89999999999999971e-30

   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%

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

   5. Simplified 100.0%

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

   if 6.2000000000000002e196 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 6.1%

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

      1. distribute-rgt1-in 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in re around inf 6.1%

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

   7. Taylor expanded in re around 0 6.1%

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

      1. distribute-rgt1-in 6.1%

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

      2. *-rgt-identity 6.1%

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

      3. rgt-mult-inverse 6.1%

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

      4. distribute-lft-in 6.1%

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

      5. associate-/l* 5.0%

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

      6. distribute-lft-in 5.0%

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

      7. *-rgt-identity 5.0%

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

      8. rgt-mult-inverse 5.0%

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

   9. Simplified 5.0%

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

       1. *-commutative 5.0%

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

       2. associate-*r/ 6.1%

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

       3. associate-*l/ 91.6%

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

   11. Applied egg-rr 91.6%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00165:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+196}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot \left(\sin im \cdot \left(re + 1\right)\right)}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00086 \lor \neg \left(re \leq 4.9 \cdot 10^{-30}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.00086) (not (<= re 4.9e-30)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.00086) || !(re <= 4.9e-30)) {
		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 ((re <= (-0.00086d0)) .or. (.not. (re <= 4.9d-30))) 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 ((re <= -0.00086) || !(re <= 4.9e-30)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.00086) or not (re <= 4.9e-30):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.00086) || !(re <= 4.9e-30))
		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 ((re <= -0.00086) || ~((re <= 4.9e-30)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.00086], N[Not[LessEqual[re, 4.9e-30]], $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 re < -8.59999999999999979e-4 or 4.89999999999999971e-30 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.6%

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

   if -8.59999999999999979e-4 < re < 4.89999999999999971e-30

   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%

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

   5. Simplified 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00086 \lor \neg \left(re \leq 4.9 \cdot 10^{-30}\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{+125}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.2e+125)
   (* re (/ im re))
   (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 <= -9.2e+125) {
		tmp = re * (im / re);
	} 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 <= (-9.2d+125)) then
        tmp = re * (im / re)
    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 <= -9.2e+125) {
		tmp = re * (im / re);
	} 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 <= -9.2e+125:
		tmp = re * (im / re)
	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 <= -9.2e+125)
		tmp = Float64(re * Float64(im / re));
	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 <= -9.2e+125)
		tmp = re * (im / re);
	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, -9.2e+125], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], 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 < -9.20000000000000051e125

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.5%

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

      1. distribute-rgt1-in 2.5%

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

   5. Simplified 2.5%

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

   6. Taylor expanded in re around inf 2.5%

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

   7. Taylor expanded in im around 0 2.3%

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

   8. Taylor expanded in re around 0 45.9%

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

   if -9.20000000000000051e125 < re < 4.89999999999999971e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 82.6%

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

   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%

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

      1. *-commutative 52.4%

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

   7. Simplified 52.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{+125}:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.0% accurate, 10.1× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -18.0)
   (* re (/ im re))
   (+ 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 = re * (im / re);
	} 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 = re * (im / re)
    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 = re * (im / re);
	} 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 = re * (im / re)
	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 = Float64(re * Float64(im / re));
	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 = re * (im / re);
	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], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im + N[(im * N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -18

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in re around inf 2.8%

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

   7. Taylor expanded in im around 0 2.5%

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

   8. Taylor expanded in re around 0 29.9%

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

   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%

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

      1. *-commutative 47.8%

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

   7. Simplified 47.8%

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

4. Final simplification 43.7%

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

Alternative 7: 42.7% accurate, 11.3× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (re <= -18.0) {
		tmp = re * (im / re);
	} else {
		tmp = im + (re * (im + (re * (0.16666666666666666 * (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 <= (-18.0d0)) then
        tmp = re * (im / re)
    else
        tmp = im + (re * (im + (re * (0.16666666666666666d0 * (re * im)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -18.0) {
		tmp = re * (im / re);
	} else {
		tmp = im + (re * (im + (re * (0.16666666666666666 * (re * im)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -18.0:
		tmp = re * (im / re)
	else:
		tmp = im + (re * (im + (re * (0.16666666666666666 * (re * im)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -18.0)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im + Float64(re * Float64(im + Float64(re * Float64(0.16666666666666666 * Float64(re * im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -18.0)
		tmp = re * (im / re);
	else
		tmp = im + (re * (im + (re * (0.16666666666666666 * (re * im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -18.0], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im + N[(re * N[(im + N[(re * N[(0.16666666666666666 * N[(re * im), $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. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in re around inf 2.8%

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

   7. Taylor expanded in im around 0 2.5%

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

   8. Taylor expanded in re around 0 29.9%

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

   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 re around inf 46.0%

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

4. Final simplification 42.4%

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

Alternative 8: 42.4% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.0) (* re (/ im re)) (+ im (* im (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.0) {
		tmp = re * (im / re);
	} else {
		tmp = im + (im * (re * (1.0 + (re * 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 <= (-2.0d0)) then
        tmp = re * (im / re)
    else
        tmp = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.0) {
		tmp = re * (im / re);
	} else {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.0:
		tmp = re * (im / re)
	else:
		tmp = im + (im * (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.0)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.0)
		tmp = re * (im / re);
	else
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.0], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 2.7%

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

      1. distribute-rgt1-in 2.7%

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

   5. Simplified 2.7%

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

   6. Taylor expanded in re around inf 2.7%

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

   7. Taylor expanded in im around 0 2.6%

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

   8. Taylor expanded in re around 0 28.4%

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

   if -2 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 58.6%

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

   4. Taylor expanded in re around 0 43.9%

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

      1. *-commutative 43.9%

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

   6. Simplified 43.9%

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

   7. Taylor expanded in im around 0 46.3%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.6% accurate, 20.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[re, 1.0], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], 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 re around 0 67.5%

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

      1. distribute-rgt1-in 67.5%

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

   5. Simplified 67.5%

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

   6. Taylor expanded in re around inf 66.9%

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

   7. Taylor expanded in im around 0 32.4%

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

   8. Taylor expanded in re around 0 40.6%

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

   if 1 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 4.2%

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

      1. distribute-rgt1-in 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in re around inf 4.2%

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

   7. Taylor expanded in im around 0 14.5%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.0% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -18:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -18.0) (* re (/ im re)) (+ im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -18.0) {
		tmp = re * (im / re);
	} else {
		tmp = im + (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 <= (-18.0d0)) then
        tmp = re * (im / re)
    else
        tmp = im + (re * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -18.0) {
		tmp = re * (im / re);
	} else {
		tmp = im + (re * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -18.0:
		tmp = re * (im / re)
	else:
		tmp = im + (re * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -18.0)
		tmp = Float64(re * Float64(im / re));
	else
		tmp = Float64(im + Float64(re * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -18.0)
		tmp = re * (im / re);
	else
		tmp = im + (re * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -18.0], N[(re * N[(im / re), $MachinePrecision]), $MachinePrecision], N[(im + N[(re * im), $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. Taylor expanded in re around 0 2.8%

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

      1. distribute-rgt1-in 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in re around inf 2.8%

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

   7. Taylor expanded in im around 0 2.5%

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

   8. Taylor expanded in re around 0 29.9%

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

   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 35.4%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -18:\\ \;\;\;\;re \cdot \frac{im}{re}\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.7% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 51.0%

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

   1. distribute-rgt1-in 51.0%

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

5. Simplified 51.0%

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

6. Taylor expanded in re around inf 50.5%

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

7. Taylor expanded in re around 0 50.5%

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

   1. distribute-rgt1-in 50.5%

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

   2. *-rgt-identity 50.5%

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

   3. rgt-mult-inverse 50.4%

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

   4. distribute-lft-in 50.4%

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

   5. associate-/l* 56.3%

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

   6. distribute-lft-in 56.3%

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

   7. *-rgt-identity 56.3%

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

   8. rgt-mult-inverse 56.4%

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

9. Simplified 56.4%

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

10. Taylor expanded in im around 0 33.7%

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

11. Final simplification 33.7%

    \[\leadsto re \cdot \left(\left(re + 1\right) \cdot \frac{im}{re}\right) \]
12. Add Preprocessing

Alternative 12: 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 re around 0 4.2%

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

      1. distribute-rgt1-in 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in re around inf 4.2%

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

   7. Taylor expanded in im around 0 14.5%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. 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. Final simplification 24.9%

   \[\leadsto im \]
6. Add Preprocessing

Reproduce

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