math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 23.4s

Alternatives: 20

Speedup: 1.0×

• 24.4% 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: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;e^{re} \leq 0.9998845436550344:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{re} \leq 1.0005:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= (exp re) 0.9998845436550344)
     t_0
     (if (<= (exp re) 1.0005) (* (sin im) (+ re 1.0)) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (exp(re) <= 0.9998845436550344) {
		tmp = t_0;
	} else if (exp(re) <= 1.0005) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = exp(re) * im
    if (exp(re) <= 0.9998845436550344d0) then
        tmp = t_0
    else if (exp(re) <= 1.0005d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (Math.exp(re) <= 0.9998845436550344) {
		tmp = t_0;
	} else if (Math.exp(re) <= 1.0005) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if math.exp(re) <= 0.9998845436550344:
		tmp = t_0
	elif math.exp(re) <= 1.0005:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (exp(re) <= 0.9998845436550344)
		tmp = t_0;
	elseif (exp(re) <= 1.0005)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (exp(re) <= 0.9998845436550344)
		tmp = t_0;
	elseif (exp(re) <= 1.0005)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[N[Exp[re], $MachinePrecision], 0.9998845436550344], t$95$0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.0005], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99988454365503443 or 1.00049999999999994 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 87.6%

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

   if 0.99988454365503443 < (exp.f64 re) < 1.00049999999999994

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 100.0%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9998845436550344:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \leq 1.0005:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;e^{re} \leq 0.9998845436550344:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{re} \leq 1.0005:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= (exp re) 0.9998845436550344)
     t_0
     (if (<= (exp re) 1.0005) (sin im) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (exp(re) <= 0.9998845436550344) {
		tmp = t_0;
	} else if (exp(re) <= 1.0005) {
		tmp = sin(im);
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = exp(re) * im
    if (exp(re) <= 0.9998845436550344d0) then
        tmp = t_0
    else if (exp(re) <= 1.0005d0) then
        tmp = sin(im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (Math.exp(re) <= 0.9998845436550344) {
		tmp = t_0;
	} else if (Math.exp(re) <= 1.0005) {
		tmp = Math.sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if math.exp(re) <= 0.9998845436550344:
		tmp = t_0
	elif math.exp(re) <= 1.0005:
		tmp = math.sin(im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (exp(re) <= 0.9998845436550344)
		tmp = t_0;
	elseif (exp(re) <= 1.0005)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (exp(re) <= 0.9998845436550344)
		tmp = t_0;
	elseif (exp(re) <= 1.0005)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[N[Exp[re], $MachinePrecision], 0.9998845436550344], t$95$0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.0005], N[Sin[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99988454365503443 or 1.00049999999999994 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 87.6%

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

   if 0.99988454365503443 < (exp.f64 re) < 1.00049999999999994

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 99.5%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 99.5%

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

Alternative 4: 95.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -0.000115:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.0005:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{+145}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))
   (if (<= re -0.000115)
     (* (exp re) im)
     (if (<= re 0.0005)
       t_0
       (if (<= re 4.1e+145)
         (* (exp re) (* im (+ 1.0 (* -0.16666666666666666 (* im im)))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.000115) {
		tmp = exp(re) * im;
	} else if (re <= 0.0005) {
		tmp = t_0;
	} else if (re <= 4.1e+145) {
		tmp = exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    if (re <= (-0.000115d0)) then
        tmp = exp(re) * im
    else if (re <= 0.0005d0) then
        tmp = t_0
    else if (re <= 4.1d+145) then
        tmp = exp(re) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.000115) {
		tmp = Math.exp(re) * im;
	} else if (re <= 0.0005) {
		tmp = t_0;
	} else if (re <= 4.1e+145) {
		tmp = Math.exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	tmp = 0
	if re <= -0.000115:
		tmp = math.exp(re) * im
	elif re <= 0.0005:
		tmp = t_0
	elif re <= 4.1e+145:
		tmp = math.exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))))
	tmp = 0.0
	if (re <= -0.000115)
		tmp = Float64(exp(re) * im);
	elseif (re <= 0.0005)
		tmp = t_0;
	elseif (re <= 4.1e+145)
		tmp = Float64(exp(re) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	tmp = 0.0;
	if (re <= -0.000115)
		tmp = exp(re) * im;
	elseif (re <= 0.0005)
		tmp = t_0;
	elseif (re <= 4.1e+145)
		tmp = exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.000115], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 0.0005], t$95$0, If[LessEqual[re, 4.1e+145], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.15e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   if -1.15e-4 < re < 5.0000000000000001e-4 or 4.1000000000000001e145 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 99.5%

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

   if 5.0000000000000001e-4 < re < 4.1000000000000001e145

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + \color{blue}{e^{re}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re} + e^{\color{blue}{re}}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot \color{blue}{e^{re}}\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{\color{blue}{re}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{e^{re}} \]

      6. *-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 73.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 73.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.000115:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.0005:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{+145}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.4 \cdot 10^{-5}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.00034:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.4e-5)
   (* (exp re) im)
   (if (<= re 0.00034)
     (* (sin im) (+ re 1.0))
     (* (exp re) (* im (+ 1.0 (* -0.16666666666666666 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.4e-5) {
		tmp = exp(re) * im;
	} else if (re <= 0.00034) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * 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 <= (-7.4d-5)) then
        tmp = exp(re) * im
    else if (re <= 0.00034d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = exp(re) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.4e-5) {
		tmp = Math.exp(re) * im;
	} else if (re <= 0.00034) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.4e-5:
		tmp = math.exp(re) * im
	elif re <= 0.00034:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.4e-5)
		tmp = Float64(exp(re) * im);
	elseif (re <= 0.00034)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(re) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.4e-5)
		tmp = exp(re) * im;
	elseif (re <= 0.00034)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.4e-5], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 0.00034], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.39999999999999962e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   if -7.39999999999999962e-5 < re < 3.4e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 100.0%

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

   if 3.4e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + \color{blue}{e^{re}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re} + e^{\color{blue}{re}}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot \color{blue}{e^{re}}\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{\color{blue}{re}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{e^{re}} \]

      6. *-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 79.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 79.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.4 \cdot 10^{-5}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.00034:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot -0.16666666666666666\right)\\ \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot t\_0\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(re + \left(1 + \frac{\left(re \cdot re\right) \cdot \left(0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629\right)}{0.25 + \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666 - 0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(1 + t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im -0.16666666666666666))))
   (if (<= re -4600.0)
     (* im t_0)
     (if (<= re 1.75e-5)
       (sin im)
       (if (<= re 7.5e+154)
         (*
          im
          (+
           re
           (+
            1.0
            (/
             (* (* re re) (+ 0.125 (* (* re (* re re)) 0.004629629629629629)))
             (+
              0.25
              (*
               (* re 0.16666666666666666)
               (- (* re 0.16666666666666666) 0.5)))))))
         (* (* 0.5 (* re re)) (* im (+ 1.0 t_0))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -4600.0) {
		tmp = im * t_0;
	} else if (re <= 1.75e-5) {
		tmp = sin(im);
	} else if (re <= 7.5e+154) {
		tmp = im * (re + (1.0 + (((re * re) * (0.125 + ((re * (re * re)) * 0.004629629629629629))) / (0.25 + ((re * 0.16666666666666666) * ((re * 0.16666666666666666) - 0.5))))));
	} else {
		tmp = (0.5 * (re * re)) * (im * (1.0 + t_0));
	}
	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) :: tmp
    t_0 = im * (im * (-0.16666666666666666d0))
    if (re <= (-4600.0d0)) then
        tmp = im * t_0
    else if (re <= 1.75d-5) then
        tmp = sin(im)
    else if (re <= 7.5d+154) then
        tmp = im * (re + (1.0d0 + (((re * re) * (0.125d0 + ((re * (re * re)) * 0.004629629629629629d0))) / (0.25d0 + ((re * 0.16666666666666666d0) * ((re * 0.16666666666666666d0) - 0.5d0))))))
    else
        tmp = (0.5d0 * (re * re)) * (im * (1.0d0 + t_0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -4600.0) {
		tmp = im * t_0;
	} else if (re <= 1.75e-5) {
		tmp = Math.sin(im);
	} else if (re <= 7.5e+154) {
		tmp = im * (re + (1.0 + (((re * re) * (0.125 + ((re * (re * re)) * 0.004629629629629629))) / (0.25 + ((re * 0.16666666666666666) * ((re * 0.16666666666666666) - 0.5))))));
	} else {
		tmp = (0.5 * (re * re)) * (im * (1.0 + t_0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * -0.16666666666666666)
	tmp = 0
	if re <= -4600.0:
		tmp = im * t_0
	elif re <= 1.75e-5:
		tmp = math.sin(im)
	elif re <= 7.5e+154:
		tmp = im * (re + (1.0 + (((re * re) * (0.125 + ((re * (re * re)) * 0.004629629629629629))) / (0.25 + ((re * 0.16666666666666666) * ((re * 0.16666666666666666) - 0.5))))))
	else:
		tmp = (0.5 * (re * re)) * (im * (1.0 + t_0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * -0.16666666666666666))
	tmp = 0.0
	if (re <= -4600.0)
		tmp = Float64(im * t_0);
	elseif (re <= 1.75e-5)
		tmp = sin(im);
	elseif (re <= 7.5e+154)
		tmp = Float64(im * Float64(re + Float64(1.0 + Float64(Float64(Float64(re * re) * Float64(0.125 + Float64(Float64(re * Float64(re * re)) * 0.004629629629629629))) / Float64(0.25 + Float64(Float64(re * 0.16666666666666666) * Float64(Float64(re * 0.16666666666666666) - 0.5)))))));
	else
		tmp = Float64(Float64(0.5 * Float64(re * re)) * Float64(im * Float64(1.0 + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * -0.16666666666666666);
	tmp = 0.0;
	if (re <= -4600.0)
		tmp = im * t_0;
	elseif (re <= 1.75e-5)
		tmp = sin(im);
	elseif (re <= 7.5e+154)
		tmp = im * (re + (1.0 + (((re * re) * (0.125 + ((re * (re * re)) * 0.004629629629629629))) / (0.25 + ((re * 0.16666666666666666) * ((re * 0.16666666666666666) - 0.5))))));
	else
		tmp = (0.5 * (re * re)) * (im * (1.0 + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4600.0], N[(im * t$95$0), $MachinePrecision], If[LessEqual[re, 1.75e-5], N[Sin[im], $MachinePrecision], If[LessEqual[re, 7.5e+154], N[(im * N[(re + N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] * N[(0.125 + N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(N[(re * 0.16666666666666666), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 36.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 36.1%

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

   if -4600 < re < 1.7499999999999998e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 98.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 98.4%

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

   if 1.7499999999999998e-5 < re < 7.5000000000000004e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 70.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 48.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 48.7%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) + 1\right), im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 \cdot re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right), 1\right)\right), im\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot re\right) \cdot re\right), 1\right)\right), im\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      12. *-lowering-*.f64 48.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), 1\right)\right), im\right) \]

   10. Applied egg-rr 48.7%

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

       1. flip3-+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{{\frac{1}{2}}^{3} + {\left(re \cdot \frac{1}{6}\right)}^{3}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)} \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

       2. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{\left({\frac{1}{2}}^{3} + {\left(re \cdot \frac{1}{6}\right)}^{3}\right) \cdot \left(re \cdot re\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)}\right), 1\right)\right), im\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left({\frac{1}{2}}^{3} + {\left(re \cdot \frac{1}{6}\right)}^{3}\right) \cdot \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\frac{1}{2}}^{3} + {\left(re \cdot \frac{1}{6}\right)}^{3}\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({\frac{1}{2}}^{3}\right), \left({\left(re \cdot \frac{1}{6}\right)}^{3}\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \left({\left(re \cdot \frac{1}{6}\right)}^{3}\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       7. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \left({re}^{3} \cdot {\frac{1}{6}}^{3}\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left({re}^{3}\right), \left({\frac{1}{6}}^{3}\right)\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot re\right)\right), \left({\frac{1}{6}}^{3}\right)\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot re\right)\right), \left({\frac{1}{6}}^{3}\right)\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left({\frac{1}{6}}^{3}\right)\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       16. distribute-rgt-out-- N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6} - \frac{1}{2}\right)\right)\right)\right), 1\right)\right), im\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(re \cdot \frac{1}{6}\right), \left(re \cdot \frac{1}{6} - \frac{1}{2}\right)\right)\right)\right), 1\right)\right), im\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \frac{1}{6}\right), \left(re \cdot \frac{1}{6} - \frac{1}{2}\right)\right)\right)\right), 1\right)\right), im\right) \]

       19. --lowering--.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \frac{1}{6}\right), \mathsf{\_.f64}\left(\left(re \cdot \frac{1}{6}\right), \frac{1}{2}\right)\right)\right)\right), 1\right)\right), im\right) \]

   12. Applied egg-rr 55.8%

       \[\leadsto \left(re + \left(\color{blue}{\frac{\left(0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629\right) \cdot \left(re \cdot re\right)}{0.25 + \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666 - 0.5\right)}} + 1\right)\right) \cdot im \]

   if 7.5000000000000004e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 87.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

   8. Simplified 87.1%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 87.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right)\right)\right) \]

   11. Simplified 87.1%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(re + \left(1 + \frac{\left(re \cdot re\right) \cdot \left(0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629\right)}{0.25 + \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666 - 0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot -0.16666666666666666\right)\\ \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot t\_0\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(re + \left(1 + \frac{\left(re \cdot re\right) \cdot \left(0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629\right)}{0.25 + \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666 - 0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(1 + t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im -0.16666666666666666))))
   (if (<= re -4600.0)
     (* im t_0)
     (if (<= re 1.4e+154)
       (*
        im
        (+
         re
         (+
          1.0
          (/
           (* (* re re) (+ 0.125 (* (* re (* re re)) 0.004629629629629629)))
           (+
            0.25
            (*
             (* re 0.16666666666666666)
             (- (* re 0.16666666666666666) 0.5)))))))
       (* (* 0.5 (* re re)) (* im (+ 1.0 t_0)))))))

C

double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -4600.0) {
		tmp = im * t_0;
	} else if (re <= 1.4e+154) {
		tmp = im * (re + (1.0 + (((re * re) * (0.125 + ((re * (re * re)) * 0.004629629629629629))) / (0.25 + ((re * 0.16666666666666666) * ((re * 0.16666666666666666) - 0.5))))));
	} else {
		tmp = (0.5 * (re * re)) * (im * (1.0 + t_0));
	}
	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) :: tmp
    t_0 = im * (im * (-0.16666666666666666d0))
    if (re <= (-4600.0d0)) then
        tmp = im * t_0
    else if (re <= 1.4d+154) then
        tmp = im * (re + (1.0d0 + (((re * re) * (0.125d0 + ((re * (re * re)) * 0.004629629629629629d0))) / (0.25d0 + ((re * 0.16666666666666666d0) * ((re * 0.16666666666666666d0) - 0.5d0))))))
    else
        tmp = (0.5d0 * (re * re)) * (im * (1.0d0 + t_0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -4600.0) {
		tmp = im * t_0;
	} else if (re <= 1.4e+154) {
		tmp = im * (re + (1.0 + (((re * re) * (0.125 + ((re * (re * re)) * 0.004629629629629629))) / (0.25 + ((re * 0.16666666666666666) * ((re * 0.16666666666666666) - 0.5))))));
	} else {
		tmp = (0.5 * (re * re)) * (im * (1.0 + t_0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * -0.16666666666666666)
	tmp = 0
	if re <= -4600.0:
		tmp = im * t_0
	elif re <= 1.4e+154:
		tmp = im * (re + (1.0 + (((re * re) * (0.125 + ((re * (re * re)) * 0.004629629629629629))) / (0.25 + ((re * 0.16666666666666666) * ((re * 0.16666666666666666) - 0.5))))))
	else:
		tmp = (0.5 * (re * re)) * (im * (1.0 + t_0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * -0.16666666666666666))
	tmp = 0.0
	if (re <= -4600.0)
		tmp = Float64(im * t_0);
	elseif (re <= 1.4e+154)
		tmp = Float64(im * Float64(re + Float64(1.0 + Float64(Float64(Float64(re * re) * Float64(0.125 + Float64(Float64(re * Float64(re * re)) * 0.004629629629629629))) / Float64(0.25 + Float64(Float64(re * 0.16666666666666666) * Float64(Float64(re * 0.16666666666666666) - 0.5)))))));
	else
		tmp = Float64(Float64(0.5 * Float64(re * re)) * Float64(im * Float64(1.0 + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * -0.16666666666666666);
	tmp = 0.0;
	if (re <= -4600.0)
		tmp = im * t_0;
	elseif (re <= 1.4e+154)
		tmp = im * (re + (1.0 + (((re * re) * (0.125 + ((re * (re * re)) * 0.004629629629629629))) / (0.25 + ((re * 0.16666666666666666) * ((re * 0.16666666666666666) - 0.5))))));
	else
		tmp = (0.5 * (re * re)) * (im * (1.0 + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4600.0], N[(im * t$95$0), $MachinePrecision], If[LessEqual[re, 1.4e+154], N[(im * N[(re + N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] * N[(0.125 + N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(N[(re * 0.16666666666666666), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -4600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 36.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 36.1%

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

   if -4600 < re < 1.4e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 63.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 59.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 59.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) + 1\right), im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 \cdot re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right), 1\right)\right), im\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot re\right) \cdot re\right), 1\right)\right), im\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      12. *-lowering-*.f64 59.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), 1\right)\right), im\right) \]

   10. Applied egg-rr 59.0%

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

       1. flip3-+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{{\frac{1}{2}}^{3} + {\left(re \cdot \frac{1}{6}\right)}^{3}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)} \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

       2. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{\left({\frac{1}{2}}^{3} + {\left(re \cdot \frac{1}{6}\right)}^{3}\right) \cdot \left(re \cdot re\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)}\right), 1\right)\right), im\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left({\frac{1}{2}}^{3} + {\left(re \cdot \frac{1}{6}\right)}^{3}\right) \cdot \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\frac{1}{2}}^{3} + {\left(re \cdot \frac{1}{6}\right)}^{3}\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({\frac{1}{2}}^{3}\right), \left({\left(re \cdot \frac{1}{6}\right)}^{3}\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \left({\left(re \cdot \frac{1}{6}\right)}^{3}\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       7. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \left({re}^{3} \cdot {\frac{1}{6}}^{3}\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left({re}^{3}\right), \left({\frac{1}{6}}^{3}\right)\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot re\right)\right), \left({\frac{1}{6}}^{3}\right)\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot re\right)\right), \left({\frac{1}{6}}^{3}\right)\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left({\frac{1}{6}}^{3}\right)\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \left(re \cdot re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)\right), 1\right)\right), im\right) \]

       16. distribute-rgt-out-- N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6} - \frac{1}{2}\right)\right)\right)\right), 1\right)\right), im\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(re \cdot \frac{1}{6}\right), \left(re \cdot \frac{1}{6} - \frac{1}{2}\right)\right)\right)\right), 1\right)\right), im\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \frac{1}{6}\right), \left(re \cdot \frac{1}{6} - \frac{1}{2}\right)\right)\right)\right), 1\right)\right), im\right) \]

       19. --lowering--.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \frac{1}{6}\right), \mathsf{\_.f64}\left(\left(re \cdot \frac{1}{6}\right), \frac{1}{2}\right)\right)\right)\right), 1\right)\right), im\right) \]

   12. Applied egg-rr 60.1%

       \[\leadsto \left(re + \left(\color{blue}{\frac{\left(0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629\right) \cdot \left(re \cdot re\right)}{0.25 + \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666 - 0.5\right)}} + 1\right)\right) \cdot im \]

   if 1.4e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

      5. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 87.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

   8. Simplified 87.1%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 87.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right)\right)\right) \]

   11. Simplified 87.1%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(re + \left(1 + \frac{\left(re \cdot re\right) \cdot \left(0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629\right)}{0.25 + \left(re \cdot 0.16666666666666666\right) \cdot \left(re \cdot 0.16666666666666666 - 0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot -0.16666666666666666\right)\\ \mathbf{if}\;re \leq -3:\\ \;\;\;\;im \cdot t\_0\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+94}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(1 + t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im -0.16666666666666666))))
   (if (<= re -3.0)
     (* im t_0)
     (if (<= re 1.9e+94)
       (* (+ re 1.0) (* im (+ 1.0 t_0)))
       (* 0.16666666666666666 (* im (* re (* re re))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -3.0) {
		tmp = im * t_0;
	} else if (re <= 1.9e+94) {
		tmp = (re + 1.0) * (im * (1.0 + t_0));
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * 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) :: tmp
    t_0 = im * (im * (-0.16666666666666666d0))
    if (re <= (-3.0d0)) then
        tmp = im * t_0
    else if (re <= 1.9d+94) then
        tmp = (re + 1.0d0) * (im * (1.0d0 + t_0))
    else
        tmp = 0.16666666666666666d0 * (im * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -3.0) {
		tmp = im * t_0;
	} else if (re <= 1.9e+94) {
		tmp = (re + 1.0) * (im * (1.0 + t_0));
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * -0.16666666666666666)
	tmp = 0
	if re <= -3.0:
		tmp = im * t_0
	elif re <= 1.9e+94:
		tmp = (re + 1.0) * (im * (1.0 + t_0))
	else:
		tmp = 0.16666666666666666 * (im * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * -0.16666666666666666))
	tmp = 0.0
	if (re <= -3.0)
		tmp = Float64(im * t_0);
	elseif (re <= 1.9e+94)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(1.0 + t_0)));
	else
		tmp = Float64(0.16666666666666666 * Float64(im * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * -0.16666666666666666);
	tmp = 0.0;
	if (re <= -3.0)
		tmp = im * t_0;
	elseif (re <= 1.9e+94)
		tmp = (re + 1.0) * (im * (1.0 + t_0));
	else
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3.0], N[(im * t$95$0), $MachinePrecision], If[LessEqual[re, 1.9e+94], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 35.5%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 35.5%

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

   if -3 < re < 1.8999999999999998e94

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 88.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 88.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 58.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

   8. Simplified 58.0%

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

   if 1.8999999999999998e94 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 80.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 80.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 80.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) + 1\right), im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 \cdot re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right), 1\right)\right), im\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot re\right) \cdot re\right), 1\right)\right), im\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      12. *-lowering-*.f64 80.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), 1\right)\right), im\right) \]

   10. Applied egg-rr 80.0%

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

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left(im \cdot {re}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left({re}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot {re}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 80.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   13. Simplified 80.0%

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

Alternative 9: 47.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+19}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4600.0)
   (* im (* im (* im -0.16666666666666666)))
   (if (<= re 6.8e+19)
     (+ im (* im (* re (+ 1.0 (* re 0.5)))))
     (* 0.16666666666666666 (* im (* re (* re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else if (re <= 6.8e+19) {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4600.0d0)) then
        tmp = im * (im * (im * (-0.16666666666666666d0)))
    else if (re <= 6.8d+19) then
        tmp = im + (im * (re * (1.0d0 + (re * 0.5d0))))
    else
        tmp = 0.16666666666666666d0 * (im * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else if (re <= 6.8e+19) {
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4600.0:
		tmp = im * (im * (im * -0.16666666666666666))
	elif re <= 6.8e+19:
		tmp = im + (im * (re * (1.0 + (re * 0.5))))
	else:
		tmp = 0.16666666666666666 * (im * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4600.0)
		tmp = Float64(im * Float64(im * Float64(im * -0.16666666666666666)));
	elseif (re <= 6.8e+19)
		tmp = Float64(im + Float64(im * Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = Float64(0.16666666666666666 * Float64(im * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4600.0)
		tmp = im * (im * (im * -0.16666666666666666));
	elseif (re <= 6.8e+19)
		tmp = im + (im * (re * (1.0 + (re * 0.5))));
	else
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4600.0], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.8e+19], N[(im + N[(im * N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 36.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 36.1%

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

   if -4600 < re < 6.8e19

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 61.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 60.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 60.8%

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

         \[\leadsto \left(re \cdot \left(1 + re \cdot \frac{1}{2}\right)\right) \cdot im + im \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(re \cdot \left(1 + re \cdot \frac{1}{2}\right)\right) \cdot im\right), \color{blue}{im}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(re \cdot \left(1 + re \cdot \frac{1}{2}\right)\right), im\right), im\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(1 + re \cdot \frac{1}{2}\right)\right), im\right), im\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right), im\right), im\right) \]

      9. *-lowering-*.f64 60.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right), im\right), im\right) \]

   10. Applied egg-rr 60.8%

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

   if 6.8e19 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 74.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 64.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) + 1\right), im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 \cdot re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right), 1\right)\right), im\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot re\right) \cdot re\right), 1\right)\right), im\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      12. *-lowering-*.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), 1\right)\right), im\right) \]

   10. Applied egg-rr 64.1%

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

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left(im \cdot {re}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left({re}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot {re}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   13. Simplified 64.1%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+19}:\\ \;\;\;\;im + im \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+19}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4600.0)
   (* im (* im (* im -0.16666666666666666)))
   (if (<= re 6.8e+19)
     (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (* 0.16666666666666666 (* im (* re (* re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else if (re <= 6.8e+19) {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4600.0d0)) then
        tmp = im * (im * (im * (-0.16666666666666666d0)))
    else if (re <= 6.8d+19) then
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else
        tmp = 0.16666666666666666d0 * (im * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else if (re <= 6.8e+19) {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4600.0:
		tmp = im * (im * (im * -0.16666666666666666))
	elif re <= 6.8e+19:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	else:
		tmp = 0.16666666666666666 * (im * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4600.0)
		tmp = Float64(im * Float64(im * Float64(im * -0.16666666666666666)));
	elseif (re <= 6.8e+19)
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = Float64(0.16666666666666666 * Float64(im * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4600.0)
		tmp = im * (im * (im * -0.16666666666666666));
	elseif (re <= 6.8e+19)
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	else
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4600.0], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.8e+19], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 36.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 36.1%

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

   if -4600 < re < 6.8e19

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 61.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 60.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 60.8%

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

   if 6.8e19 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 74.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 64.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) + 1\right), im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 \cdot re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right), 1\right)\right), im\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot re\right) \cdot re\right), 1\right)\right), im\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      12. *-lowering-*.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), 1\right)\right), im\right) \]

   10. Applied egg-rr 64.1%

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

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left(im \cdot {re}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left({re}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot {re}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   13. Simplified 64.1%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+19}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.5% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + \left(1 + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4600.0)
   (* im (* im (* im -0.16666666666666666)))
   (* im (+ re (+ 1.0 (* (* re re) (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else {
		tmp = im * (re + (1.0 + ((re * 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 <= (-4600.0d0)) then
        tmp = im * (im * (im * (-0.16666666666666666d0)))
    else
        tmp = im * (re + (1.0d0 + ((re * 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 <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else {
		tmp = im * (re + (1.0 + ((re * re) * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4600.0:
		tmp = im * (im * (im * -0.16666666666666666))
	else:
		tmp = im * (re + (1.0 + ((re * re) * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4600.0)
		tmp = Float64(im * Float64(im * Float64(im * -0.16666666666666666)));
	else
		tmp = Float64(im * Float64(re + Float64(1.0 + Float64(Float64(re * re) * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4600.0)
		tmp = im * (im * (im * -0.16666666666666666));
	else
		tmp = im * (re + (1.0 + ((re * re) * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4600.0], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re + N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 36.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 36.1%

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

   if -4600 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 65.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 61.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 61.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) + 1\right), im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 \cdot re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right), 1\right)\right), im\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot re\right) \cdot re\right), 1\right)\right), im\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      12. *-lowering-*.f64 61.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), 1\right)\right), im\right) \]

   10. Applied egg-rr 61.9%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + \left(1 + \left(re \cdot re\right) \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.5% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4600.0)
   (* im (* im (* im -0.16666666666666666)))
   (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else {
		tmp = im * (1.0 + (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 <= (-4600.0d0)) then
        tmp = im * (im * (im * (-0.16666666666666666d0)))
    else
        tmp = im * (1.0d0 + (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 <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4600.0:
		tmp = im * (im * (im * -0.16666666666666666))
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4600.0)
		tmp = Float64(im * Float64(im * Float64(im * -0.16666666666666666)));
	else
		tmp = Float64(im * Float64(1.0 + 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 <= -4600.0)
		tmp = im * (im * (im * -0.16666666666666666));
	else
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4600.0], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + 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 < -4600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 36.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 36.1%

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

   if -4600 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 65.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 61.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 61.9%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -4600:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+93}:\\ \;\;\;\;im + t\_0\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im (* im -0.16666666666666666)))))
   (if (<= re -4600.0)
     t_0
     (if (<= re 2.3e+93)
       (+ im t_0)
       (* 0.16666666666666666 (* im (* re (* re re))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * (im * -0.16666666666666666));
	double tmp;
	if (re <= -4600.0) {
		tmp = t_0;
	} else if (re <= 2.3e+93) {
		tmp = im + t_0;
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * 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) :: tmp
    t_0 = im * (im * (im * (-0.16666666666666666d0)))
    if (re <= (-4600.0d0)) then
        tmp = t_0
    else if (re <= 2.3d+93) then
        tmp = im + t_0
    else
        tmp = 0.16666666666666666d0 * (im * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * (im * -0.16666666666666666));
	double tmp;
	if (re <= -4600.0) {
		tmp = t_0;
	} else if (re <= 2.3e+93) {
		tmp = im + t_0;
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * (im * -0.16666666666666666))
	tmp = 0
	if re <= -4600.0:
		tmp = t_0
	elif re <= 2.3e+93:
		tmp = im + t_0
	else:
		tmp = 0.16666666666666666 * (im * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * Float64(im * -0.16666666666666666)))
	tmp = 0.0
	if (re <= -4600.0)
		tmp = t_0;
	elseif (re <= 2.3e+93)
		tmp = Float64(im + t_0);
	else
		tmp = Float64(0.16666666666666666 * Float64(im * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * (im * -0.16666666666666666));
	tmp = 0.0;
	if (re <= -4600.0)
		tmp = t_0;
	elseif (re <= 2.3e+93)
		tmp = im + t_0;
	else
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4600.0], t$95$0, If[LessEqual[re, 2.3e+93], N[(im + t$95$0), $MachinePrecision], N[(0.16666666666666666 * N[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -4600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 36.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 36.1%

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

   if -4600 < re < 2.3000000000000002e93

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 87.5%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 87.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 56.7%

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

      1. +-commutative N/A

         \[\leadsto im \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + \color{blue}{1}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot im + \color{blue}{1 \cdot im} \]

      3. *-lft-identity N/A

         \[\leadsto \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot im + im \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot im\right), \color{blue}{im}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right)\right), im\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \frac{-1}{6}\right)\right)\right), im\right) \]

      8. *-lowering-*.f64 56.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right), im\right) \]

   10. Applied egg-rr 56.8%

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

   if 2.3000000000000002e93 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 80.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 80.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 80.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) + 1\right), im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 \cdot re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right), 1\right)\right), im\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot re\right) \cdot re\right), 1\right)\right), im\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      12. *-lowering-*.f64 80.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), 1\right)\right), im\right) \]

   10. Applied egg-rr 80.0%

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

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left(im \cdot {re}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left({re}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot {re}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 80.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   13. Simplified 80.0%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+93}:\\ \;\;\;\;im + im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot -0.16666666666666666\right)\\ \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot t\_0\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+94}:\\ \;\;\;\;im \cdot \left(1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im -0.16666666666666666))))
   (if (<= re -4600.0)
     (* im t_0)
     (if (<= re 6.8e+94)
       (* im (+ 1.0 t_0))
       (* 0.16666666666666666 (* im (* re (* re re))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -4600.0) {
		tmp = im * t_0;
	} else if (re <= 6.8e+94) {
		tmp = im * (1.0 + t_0);
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * 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) :: tmp
    t_0 = im * (im * (-0.16666666666666666d0))
    if (re <= (-4600.0d0)) then
        tmp = im * t_0
    else if (re <= 6.8d+94) then
        tmp = im * (1.0d0 + t_0)
    else
        tmp = 0.16666666666666666d0 * (im * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -4600.0) {
		tmp = im * t_0;
	} else if (re <= 6.8e+94) {
		tmp = im * (1.0 + t_0);
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * -0.16666666666666666)
	tmp = 0
	if re <= -4600.0:
		tmp = im * t_0
	elif re <= 6.8e+94:
		tmp = im * (1.0 + t_0)
	else:
		tmp = 0.16666666666666666 * (im * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * -0.16666666666666666))
	tmp = 0.0
	if (re <= -4600.0)
		tmp = Float64(im * t_0);
	elseif (re <= 6.8e+94)
		tmp = Float64(im * Float64(1.0 + t_0));
	else
		tmp = Float64(0.16666666666666666 * Float64(im * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * -0.16666666666666666);
	tmp = 0.0;
	if (re <= -4600.0)
		tmp = im * t_0;
	elseif (re <= 6.8e+94)
		tmp = im * (1.0 + t_0);
	else
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4600.0], N[(im * t$95$0), $MachinePrecision], If[LessEqual[re, 6.8e+94], N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -4600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 36.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 36.1%

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

   if -4600 < re < 6.8000000000000004e94

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 87.5%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 87.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 56.7%

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

   if 6.8000000000000004e94 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 80.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 80.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 80.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) + 1\right), im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 \cdot re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right), 1\right)\right), im\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot re\right) \cdot re\right), 1\right)\right), im\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      12. *-lowering-*.f64 80.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), 1\right)\right), im\right) \]

   10. Applied egg-rr 80.0%

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

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left(im \cdot {re}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left({re}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot {re}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 80.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   13. Simplified 80.0%

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

Alternative 15: 47.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 2.8:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4600.0)
   (* im (* im (* im -0.16666666666666666)))
   (if (<= re 2.8)
     (* im (+ re 1.0))
     (* 0.16666666666666666 (* im (* re (* re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else if (re <= 2.8) {
		tmp = im * (re + 1.0);
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4600.0d0)) then
        tmp = im * (im * (im * (-0.16666666666666666d0)))
    else if (re <= 2.8d0) then
        tmp = im * (re + 1.0d0)
    else
        tmp = 0.16666666666666666d0 * (im * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else if (re <= 2.8) {
		tmp = im * (re + 1.0);
	} else {
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4600.0:
		tmp = im * (im * (im * -0.16666666666666666))
	elif re <= 2.8:
		tmp = im * (re + 1.0)
	else:
		tmp = 0.16666666666666666 * (im * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4600.0)
		tmp = Float64(im * Float64(im * Float64(im * -0.16666666666666666)));
	elseif (re <= 2.8)
		tmp = Float64(im * Float64(re + 1.0));
	else
		tmp = Float64(0.16666666666666666 * Float64(im * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4600.0)
		tmp = im * (im * (im * -0.16666666666666666));
	elseif (re <= 2.8)
		tmp = im * (re + 1.0);
	else
		tmp = 0.16666666666666666 * (im * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4600.0], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 36.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 36.1%

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

   if -4600 < re < 2.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 62.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), im\right) \]

      2. +-lowering-+.f64 60.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

   8. Simplified 60.6%

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

   if 2.7999999999999998 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 73.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 63.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 63.6%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) + 1\right), im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 \cdot re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(re + \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right) + 1\right), im\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re + 1\right)\right), im\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right) \cdot re\right), 1\right)\right), im\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot re\right) \cdot re\right), 1\right)\right), im\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \left(re \cdot re\right)\right), 1\right)\right), im\right) \]

      12. *-lowering-*.f64 63.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(re, re\right)\right), 1\right)\right), im\right) \]

   10. Applied egg-rr 63.6%

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

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left(im \cdot {re}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left({re}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \left(re \cdot {re}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 63.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   13. Simplified 63.6%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 2.8:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.9% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 2.75:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4600.0)
   (* im (* im (* im -0.16666666666666666)))
   (if (<= re 2.75) (* im (+ re 1.0)) (* im (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else if (re <= 2.75) {
		tmp = im * (re + 1.0);
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4600.0d0)) then
        tmp = im * (im * (im * (-0.16666666666666666d0)))
    else if (re <= 2.75d0) then
        tmp = im * (re + 1.0d0)
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4600.0) {
		tmp = im * (im * (im * -0.16666666666666666));
	} else if (re <= 2.75) {
		tmp = im * (re + 1.0);
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4600.0:
		tmp = im * (im * (im * -0.16666666666666666))
	elif re <= 2.75:
		tmp = im * (re + 1.0)
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4600.0)
		tmp = Float64(im * Float64(im * Float64(im * -0.16666666666666666)));
	elseif (re <= 2.75)
		tmp = Float64(im * Float64(re + 1.0));
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4600.0)
		tmp = im * (im * (im * -0.16666666666666666));
	elseif (re <= 2.75)
		tmp = im * (re + 1.0);
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4600.0], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.75], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4600

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 4.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 4.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 3.7%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 3.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 36.1%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 36.1%

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

   if -4600 < re < 2.75

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 62.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), im\right) \]

      2. +-lowering-+.f64 60.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

   8. Simplified 60.6%

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

   if 2.75 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 73.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 51.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 51.8%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}, im\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), im\right) \]

       3. *-lowering-*.f64 51.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), im\right) \]

   11. Simplified 51.8%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 2.75:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -4600:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 5.1 \cdot 10^{+57}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im (* im -0.16666666666666666)))))
   (if (<= re -4600.0) t_0 (if (<= re 5.1e+57) (* im (+ re 1.0)) t_0))))

C

double code(double re, double im) {
	double t_0 = im * (im * (im * -0.16666666666666666));
	double tmp;
	if (re <= -4600.0) {
		tmp = t_0;
	} else if (re <= 5.1e+57) {
		tmp = im * (re + 1.0);
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = im * (im * (im * (-0.16666666666666666d0)))
    if (re <= (-4600.0d0)) then
        tmp = t_0
    else if (re <= 5.1d+57) then
        tmp = im * (re + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * (im * -0.16666666666666666));
	double tmp;
	if (re <= -4600.0) {
		tmp = t_0;
	} else if (re <= 5.1e+57) {
		tmp = im * (re + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * (im * -0.16666666666666666))
	tmp = 0
	if re <= -4600.0:
		tmp = t_0
	elif re <= 5.1e+57:
		tmp = im * (re + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * Float64(im * -0.16666666666666666)))
	tmp = 0.0
	if (re <= -4600.0)
		tmp = t_0;
	elseif (re <= 5.1e+57)
		tmp = Float64(im * Float64(re + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * (im * -0.16666666666666666));
	tmp = 0.0;
	if (re <= -4600.0)
		tmp = t_0;
	elseif (re <= 5.1e+57)
		tmp = im * (re + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4600.0], t$95$0, If[LessEqual[re, 5.1e+57], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -4600 or 5.10000000000000023e57 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 3.6%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 3.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 11.9%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   8. Simplified 11.9%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left({im}^{2} \cdot im\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

       4. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       11. *-lowering-*.f64 29.9%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   11. Simplified 29.9%

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

   if -4600 < re < 5.10000000000000023e57

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 62.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), im\right) \]

      2. +-lowering-+.f64 57.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

   8. Simplified 57.9%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4600:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 5.1 \cdot 10^{+57}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 27.5% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{+42}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1.12e+42) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.12e+42) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.12d+42) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.12e+42) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.12e+42:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.12e+42)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.12e+42)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.12e+42], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.12e42

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 82.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 41.6%

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

   if 1.12e42 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 39.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), im\right) \]

      2. +-lowering-+.f64 6.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

   8. Simplified 6.9%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, im\right) \]
   10. Step-by-step derivation

   11. Simplified 7.2%

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

Alternative 19: 28.9% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
4. Step-by-step derivation

5. Simplified 73.6%

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

6. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, im\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), im\right) \]

   2. +-lowering-+.f64 35.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

8. Simplified 35.5%

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

9. Final simplification 35.5%

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

Alternative 20: 26.0% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
4. Step-by-step derivation

5. Simplified 73.6%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im} \]
7. Step-by-step derivation

8. Simplified 33.9%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))