math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 13.6s

Alternatives: 25

Speedup: 1.0×

• 24.6% 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: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := \sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -0.027:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 520:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im))
        (t_1 (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))
   (if (<= re -0.027)
     t_0
     (if (<= re 520.0) t_1 (if (<= re 2.5e+149) t_0 t_1)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.027) {
		tmp = t_0;
	} else if (re <= 520.0) {
		tmp = t_1;
	} else if (re <= 2.5e+149) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(re) * im
    t_1 = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    if (re <= (-0.027d0)) then
        tmp = t_0
    else if (re <= 520.0d0) then
        tmp = t_1
    else if (re <= 2.5d+149) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double t_1 = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.027) {
		tmp = t_0;
	} else if (re <= 520.0) {
		tmp = t_1;
	} else if (re <= 2.5e+149) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	t_1 = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	tmp = 0
	if re <= -0.027:
		tmp = t_0
	elif re <= 520.0:
		tmp = t_1
	elif re <= 2.5e+149:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))))
	tmp = 0.0
	if (re <= -0.027)
		tmp = t_0;
	elseif (re <= 520.0)
		tmp = t_1;
	elseif (re <= 2.5e+149)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	t_1 = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	tmp = 0.0;
	if (re <= -0.027)
		tmp = t_0;
	elseif (re <= 520.0)
		tmp = t_1;
	elseif (re <= 2.5e+149)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.027], t$95$0, If[LessEqual[re, 520.0], t$95$1, If[LessEqual[re, 2.5e+149], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0269999999999999997 or 520 < re < 2.49999999999999995e149

   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 94.1%

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

   if -0.0269999999999999997 < re < 520 or 2.49999999999999995e149 < 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 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.027:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 520:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0022:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 8500000:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0022)
   (* (exp re) im)
   (if (<= re 8500000.0)
     (* (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 <= -0.0022) {
		tmp = exp(re) * im;
	} else if (re <= 8500000.0) {
		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 <= (-0.0022d0)) then
        tmp = exp(re) * im
    else if (re <= 8500000.0d0) 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 <= -0.0022) {
		tmp = Math.exp(re) * im;
	} else if (re <= 8500000.0) {
		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 <= -0.0022:
		tmp = math.exp(re) * im
	elif re <= 8500000.0:
		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 <= -0.0022)
		tmp = Float64(exp(re) * im);
	elseif (re <= 8500000.0)
		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 <= -0.0022)
		tmp = exp(re) * im;
	elseif (re <= 8500000.0)
		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, -0.0022], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 8500000.0], 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 < -0.00220000000000000013

   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 -0.00220000000000000013 < re < 8.5e6

   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 98.4%

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

   5. Simplified 98.4%

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

   if 8.5e6 < 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 76.2%

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

   5. Simplified 76.2%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0022:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 8500000:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.0003:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 520:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.0003)
     t_0
     (if (<= re 520.0)
       (* (sin im) (+ re 1.0))
       (if (<= re 2.5e+149)
         t_0
         (*
          (+ 1.0 (* re (+ 1.0 (* re 0.5))))
          (* im (+ 1.0 (* -0.16666666666666666 (* im im))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.0003) {
		tmp = t_0;
	} else if (re <= 520.0) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 2.5e+149) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.0003d0)) then
        tmp = t_0
    else if (re <= 520.0d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 2.5d+149) then
        tmp = t_0
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    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 (re <= -0.0003) {
		tmp = t_0;
	} else if (re <= 520.0) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 2.5e+149) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.0003:
		tmp = t_0
	elif re <= 520.0:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 2.5e+149:
		tmp = t_0
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.0003)
		tmp = t_0;
	elseif (re <= 520.0)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 2.5e+149)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.0003)
		tmp = t_0;
	elseif (re <= 520.0)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 2.5e+149)
		tmp = t_0;
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.0003], t$95$0, If[LessEqual[re, 520.0], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e+149], t$95$0, N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -2.99999999999999974e-4 or 520 < re < 2.49999999999999995e149

   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 94.1%

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

   if -2.99999999999999974e-4 < re < 520

   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 99.2%

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

   5. Simplified 99.2%

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

   if 2.49999999999999995e149 < 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 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0003:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 520:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 520:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -2e-6)
     t_0
     (if (<= re 520.0)
       (sin im)
       (if (<= re 2.5e+149)
         t_0
         (*
          (+ 1.0 (* re (+ 1.0 (* re 0.5))))
          (* im (+ 1.0 (* -0.16666666666666666 (* im im))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -2e-6) {
		tmp = t_0;
	} else if (re <= 520.0) {
		tmp = sin(im);
	} else if (re <= 2.5e+149) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (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) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-2d-6)) then
        tmp = t_0
    else if (re <= 520.0d0) then
        tmp = sin(im)
    else if (re <= 2.5d+149) then
        tmp = t_0
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (im * (1.0d0 + ((-0.16666666666666666d0) * (im * im))))
    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 (re <= -2e-6) {
		tmp = t_0;
	} else if (re <= 520.0) {
		tmp = Math.sin(im);
	} else if (re <= 2.5e+149) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -2e-6:
		tmp = t_0
	elif re <= 520.0:
		tmp = math.sin(im)
	elif re <= 2.5e+149:
		tmp = t_0
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -2e-6)
		tmp = t_0;
	elseif (re <= 520.0)
		tmp = sin(im);
	elseif (re <= 2.5e+149)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -2e-6)
		tmp = t_0;
	elseif (re <= 520.0)
		tmp = sin(im);
	elseif (re <= 2.5e+149)
		tmp = t_0;
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -2e-6], t$95$0, If[LessEqual[re, 520.0], N[Sin[im], $MachinePrecision], If[LessEqual[re, 2.5e+149], t$95$0, N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -1.99999999999999991e-6 or 520 < re < 2.49999999999999995e149

   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 94.1%

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

   if -1.99999999999999991e-6 < re < 520

   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 98.6%

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

   5. Simplified 98.6%

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

   if 2.49999999999999995e149 < 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 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.8%

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

   8. Simplified 81.8%

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

Alternative 6: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ t_1 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -400:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{0.16666666666666666}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{\left(im \cdot im\right) \cdot 0.00205026455026455}{re + 1} + \frac{0.019444444444444445}{re + 1}\right)\right)}{im}}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;t\_0 \cdot \left(1 + \frac{re \cdot re - t\_1 \cdot t\_1}{re - t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* -0.16666666666666666 (* im im)))))
        (t_1 (* re (* re 0.5))))
   (if (<= re -400.0)
     (/
      1.0
      (/
       (+
        (/ 1.0 (+ re 1.0))
        (*
         (* im im)
         (+
          (/ 0.16666666666666666 (+ re 1.0))
          (*
           (* im im)
           (+
            (/ (* (* im im) 0.00205026455026455) (+ re 1.0))
            (/ 0.019444444444444445 (+ re 1.0)))))))
       im))
     (if (<= re 1.5e+23)
       (sin im)
       (if (<= re 2.5e+149)
         (* t_0 (+ 1.0 (/ (- (* re re) (* t_1 t_1)) (- re t_1))))
         (* (+ 1.0 (* re (+ 1.0 (* re 0.5)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (1.0 + (-0.16666666666666666 * (im * im)));
	double t_1 = re * (re * 0.5);
	double tmp;
	if (re <= -400.0) {
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + ((im * im) * ((((im * im) * 0.00205026455026455) / (re + 1.0)) + (0.019444444444444445 / (re + 1.0))))))) / im);
	} else if (re <= 1.5e+23) {
		tmp = sin(im);
	} else if (re <= 2.5e+149) {
		tmp = t_0 * (1.0 + (((re * re) - (t_1 * t_1)) / (re - t_1)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * 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) :: t_1
    real(8) :: tmp
    t_0 = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    t_1 = re * (re * 0.5d0)
    if (re <= (-400.0d0)) then
        tmp = 1.0d0 / (((1.0d0 / (re + 1.0d0)) + ((im * im) * ((0.16666666666666666d0 / (re + 1.0d0)) + ((im * im) * ((((im * im) * 0.00205026455026455d0) / (re + 1.0d0)) + (0.019444444444444445d0 / (re + 1.0d0))))))) / im)
    else if (re <= 1.5d+23) then
        tmp = sin(im)
    else if (re <= 2.5d+149) then
        tmp = t_0 * (1.0d0 + (((re * re) - (t_1 * t_1)) / (re - t_1)))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (1.0 + (-0.16666666666666666 * (im * im)));
	double t_1 = re * (re * 0.5);
	double tmp;
	if (re <= -400.0) {
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + ((im * im) * ((((im * im) * 0.00205026455026455) / (re + 1.0)) + (0.019444444444444445 / (re + 1.0))))))) / im);
	} else if (re <= 1.5e+23) {
		tmp = Math.sin(im);
	} else if (re <= 2.5e+149) {
		tmp = t_0 * (1.0 + (((re * re) - (t_1 * t_1)) / (re - t_1)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (1.0 + (-0.16666666666666666 * (im * im)))
	t_1 = re * (re * 0.5)
	tmp = 0
	if re <= -400.0:
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + ((im * im) * ((((im * im) * 0.00205026455026455) / (re + 1.0)) + (0.019444444444444445 / (re + 1.0))))))) / im)
	elif re <= 1.5e+23:
		tmp = math.sin(im)
	elif re <= 2.5e+149:
		tmp = t_0 * (1.0 + (((re * re) - (t_1 * t_1)) / (re - t_1)))
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))))
	t_1 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -400.0)
		tmp = Float64(1.0 / Float64(Float64(Float64(1.0 / Float64(re + 1.0)) + Float64(Float64(im * im) * Float64(Float64(0.16666666666666666 / Float64(re + 1.0)) + Float64(Float64(im * im) * Float64(Float64(Float64(Float64(im * im) * 0.00205026455026455) / Float64(re + 1.0)) + Float64(0.019444444444444445 / Float64(re + 1.0))))))) / im));
	elseif (re <= 1.5e+23)
		tmp = sin(im);
	elseif (re <= 2.5e+149)
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_1 * t_1)) / Float64(re - t_1))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (1.0 + (-0.16666666666666666 * (im * im)));
	t_1 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -400.0)
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + ((im * im) * ((((im * im) * 0.00205026455026455) / (re + 1.0)) + (0.019444444444444445 / (re + 1.0))))))) / im);
	elseif (re <= 1.5e+23)
		tmp = sin(im);
	elseif (re <= 2.5e+149)
		tmp = t_0 * (1.0 + (((re * re) - (t_1 * t_1)) / (re - t_1)));
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -400.0], N[(1.0 / N[(N[(N[(1.0 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(im * im), $MachinePrecision] * N[(N[(0.16666666666666666 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(im * im), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * 0.00205026455026455), $MachinePrecision] / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.019444444444444445 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.5e+23], N[Sin[im], $MachinePrecision], If[LessEqual[re, 2.5e+149], N[(t$95$0 * N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -400

   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 2.7%

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

   5. Simplified 2.7%

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

      1. *-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. flip3-+ N/A

         \[\leadsto \frac{{\left(\sin im \cdot re\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}}{\color{blue}{\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\sin im \cdot re\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}\right), \color{blue}{\left(\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(re \cdot \sin im\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}\right), \left(\left(\color{blue}{\sin im} \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(re \cdot \sin im\right)}^{3}\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\color{blue}{\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right)} + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(re \cdot \sin im\right), 3\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\color{blue}{\left(\sin im \cdot re\right)} \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \sin im\right), 3\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\left(\color{blue}{\sin im} \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

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

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \left({\sin im}^{3}\right)\right), \left(\left(\sin im \cdot re\right) \cdot \left(\color{blue}{\sin im} \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \mathsf{pow.f64}\left(\sin im, 3\right)\right), \left(\left(\sin im \cdot re\right) \cdot \color{blue}{\left(\sin im \cdot re\right)} + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(im\right), 3\right)\right), \left(\left(\sin im \cdot re\right) \cdot \left(\color{blue}{\sin im} \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

   7. Applied egg-rr 4.3%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(re \cdot \sin im\right) \cdot \left(re \cdot \sin im\right) + \left(\sin im \cdot \sin im - \left(re \cdot \sin im\right) \cdot \sin im\right)}{{\left(re \cdot \sin im\right)}^{3} + {\sin im}^{3}}}} \]

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

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

      3. clear-num N/A

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

      4. flip3-+ N/A

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

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

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

      6. distribute-lft1-in N/A

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

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

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

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

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

      9. sin-lowering-sin.f64 2.7%

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

   9. Applied egg-rr 2.7%

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

   10. Taylor expanded in im around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{31}{15120} \cdot \frac{{im}^{2}}{1 + re} + \frac{7}{360} \cdot \frac{1}{1 + re}\right) + \frac{1}{6} \cdot \frac{1}{1 + re}\right) + \frac{1}{1 + re}}{im}\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{31}{15120} \cdot \frac{{im}^{2}}{1 + re} + \frac{7}{360} \cdot \frac{1}{1 + re}\right) + \frac{1}{6} \cdot \frac{1}{1 + re}\right) + \frac{1}{1 + re}\right), \color{blue}{im}\right)\right) \]

   12. Simplified 47.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{0.16666666666666666}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{\left(im \cdot im\right) \cdot 0.00205026455026455}{re + 1} + \frac{0.019444444444444445}{re + 1}\right)\right)}{im}}} \]

   if -400 < re < 1.5e23

   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 94.1%

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

   5. Simplified 94.1%

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

   if 1.5e23 < re < 2.49999999999999995e149

   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 68.0%

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

   5. Simplified 68.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 17.9%

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

   8. Simplified 17.9%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. flip-+ N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-lowering-*.f64 48.5%

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

   10. Applied egg-rr 48.5%

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

   if 2.49999999999999995e149 < 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 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -400:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{0.16666666666666666}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{\left(im \cdot im\right) \cdot 0.00205026455026455}{re + 1} + \frac{0.019444444444444445}{re + 1}\right)\right)}{im}}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;\left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(1 + \frac{re \cdot re - \left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)}{re - re \cdot \left(re \cdot 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;re \leq 1.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{0.16666666666666666}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{\left(im \cdot im\right) \cdot 0.00205026455026455}{re + 1} + \frac{0.019444444444444445}{re + 1}\right)\right)}{im}}\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;t\_1 \cdot \left(1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5)))
        (t_1 (* im (+ 1.0 (* -0.16666666666666666 (* im im))))))
   (if (<= re 1.1e-18)
     (/
      1.0
      (/
       (+
        (/ 1.0 (+ re 1.0))
        (*
         (* im im)
         (+
          (/ 0.16666666666666666 (+ re 1.0))
          (*
           (* im im)
           (+
            (/ (* (* im im) 0.00205026455026455) (+ re 1.0))
            (/ 0.019444444444444445 (+ re 1.0)))))))
       im))
     (if (<= re 2.5e+149)
       (* t_1 (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0))))
       (* (+ 1.0 (* re (+ 1.0 (* re 0.5)))) t_1)))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = im * (1.0 + (-0.16666666666666666 * (im * im)));
	double tmp;
	if (re <= 1.1e-18) {
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + ((im * im) * ((((im * im) * 0.00205026455026455) / (re + 1.0)) + (0.019444444444444445 / (re + 1.0))))))) / im);
	} else if (re <= 2.5e+149) {
		tmp = t_1 * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    if (re <= 1.1d-18) then
        tmp = 1.0d0 / (((1.0d0 / (re + 1.0d0)) + ((im * im) * ((0.16666666666666666d0 / (re + 1.0d0)) + ((im * im) * ((((im * im) * 0.00205026455026455d0) / (re + 1.0d0)) + (0.019444444444444445d0 / (re + 1.0d0))))))) / im)
    else if (re <= 2.5d+149) then
        tmp = t_1 * (1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0)))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = im * (1.0 + (-0.16666666666666666 * (im * im)));
	double tmp;
	if (re <= 1.1e-18) {
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + ((im * im) * ((((im * im) * 0.00205026455026455) / (re + 1.0)) + (0.019444444444444445 / (re + 1.0))))))) / im);
	} else if (re <= 2.5e+149) {
		tmp = t_1 * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = im * (1.0 + (-0.16666666666666666 * (im * im)))
	tmp = 0
	if re <= 1.1e-18:
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + ((im * im) * ((((im * im) * 0.00205026455026455) / (re + 1.0)) + (0.019444444444444445 / (re + 1.0))))))) / im)
	elif re <= 2.5e+149:
		tmp = t_1 * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)))
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))))
	tmp = 0.0
	if (re <= 1.1e-18)
		tmp = Float64(1.0 / Float64(Float64(Float64(1.0 / Float64(re + 1.0)) + Float64(Float64(im * im) * Float64(Float64(0.16666666666666666 / Float64(re + 1.0)) + Float64(Float64(im * im) * Float64(Float64(Float64(Float64(im * im) * 0.00205026455026455) / Float64(re + 1.0)) + Float64(0.019444444444444445 / Float64(re + 1.0))))))) / im));
	elseif (re <= 2.5e+149)
		tmp = Float64(t_1 * Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = im * (1.0 + (-0.16666666666666666 * (im * im)));
	tmp = 0.0;
	if (re <= 1.1e-18)
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + ((im * im) * ((((im * im) * 0.00205026455026455) / (re + 1.0)) + (0.019444444444444445 / (re + 1.0))))))) / im);
	elseif (re <= 2.5e+149)
		tmp = t_1 * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 1.1e-18], N[(1.0 / N[(N[(N[(1.0 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(im * im), $MachinePrecision] * N[(N[(0.16666666666666666 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(im * im), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * 0.00205026455026455), $MachinePrecision] / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.019444444444444445 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e+149], N[(t$95$1 * N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.0999999999999999e-18

   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 64.1%

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

   5. Simplified 64.1%

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

      1. *-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. flip3-+ N/A

         \[\leadsto \frac{{\left(\sin im \cdot re\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}}{\color{blue}{\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\sin im \cdot re\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}\right), \color{blue}{\left(\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(re \cdot \sin im\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}\right), \left(\left(\color{blue}{\sin im} \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(re \cdot \sin im\right)}^{3}\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\color{blue}{\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right)} + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(re \cdot \sin im\right), 3\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\color{blue}{\left(\sin im \cdot re\right)} \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \sin im\right), 3\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\left(\color{blue}{\sin im} \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

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

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \left({\sin im}^{3}\right)\right), \left(\left(\sin im \cdot re\right) \cdot \left(\color{blue}{\sin im} \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \mathsf{pow.f64}\left(\sin im, 3\right)\right), \left(\left(\sin im \cdot re\right) \cdot \color{blue}{\left(\sin im \cdot re\right)} + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(im\right), 3\right)\right), \left(\left(\sin im \cdot re\right) \cdot \left(\color{blue}{\sin im} \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

   7. Applied egg-rr 44.8%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(re \cdot \sin im\right) \cdot \left(re \cdot \sin im\right) + \left(\sin im \cdot \sin im - \left(re \cdot \sin im\right) \cdot \sin im\right)}{{\left(re \cdot \sin im\right)}^{3} + {\sin im}^{3}}}} \]

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

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

      3. clear-num N/A

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

      4. flip3-+ N/A

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

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

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

      6. distribute-lft1-in N/A

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

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

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

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

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

      9. sin-lowering-sin.f64 63.9%

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

   9. Applied egg-rr 63.9%

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

   10. Taylor expanded in im around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{31}{15120} \cdot \frac{{im}^{2}}{1 + re} + \frac{7}{360} \cdot \frac{1}{1 + re}\right) + \frac{1}{6} \cdot \frac{1}{1 + re}\right) + \frac{1}{1 + re}}{im}\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{31}{15120} \cdot \frac{{im}^{2}}{1 + re} + \frac{7}{360} \cdot \frac{1}{1 + re}\right) + \frac{1}{6} \cdot \frac{1}{1 + re}\right) + \frac{1}{1 + re}\right), \color{blue}{im}\right)\right) \]

   12. Simplified 50.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{0.16666666666666666}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{\left(im \cdot im\right) \cdot 0.00205026455026455}{re + 1} + \frac{0.019444444444444445}{re + 1}\right)\right)}{im}}} \]

   if 1.0999999999999999e-18 < re < 2.49999999999999995e149

   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 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 17.2%

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

   8. Simplified 17.2%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. flip-+ N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-lowering-*.f64 40.4%

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

   10. Applied egg-rr 40.4%

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

   if 2.49999999999999995e149 < 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 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{0.16666666666666666}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{\left(im \cdot im\right) \cdot 0.00205026455026455}{re + 1} + \frac{0.019444444444444445}{re + 1}\right)\right)}{im}}\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;\left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(1 + \frac{re \cdot re - \left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)}{re - re \cdot \left(re \cdot 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;re \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{0.16666666666666666}{re + 1} + im \cdot \left(im \cdot \frac{0.019444444444444445}{re + 1}\right)\right)}{im}}\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;t\_1 \cdot \left(1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5)))
        (t_1 (* im (+ 1.0 (* -0.16666666666666666 (* im im))))))
   (if (<= re 2.6e-18)
     (/
      1.0
      (/
       (+
        (/ 1.0 (+ re 1.0))
        (*
         (* im im)
         (+
          (/ 0.16666666666666666 (+ re 1.0))
          (* im (* im (/ 0.019444444444444445 (+ re 1.0)))))))
       im))
     (if (<= re 2.5e+149)
       (* t_1 (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0))))
       (* (+ 1.0 (* re (+ 1.0 (* re 0.5)))) t_1)))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = im * (1.0 + (-0.16666666666666666 * (im * im)));
	double tmp;
	if (re <= 2.6e-18) {
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + (im * (im * (0.019444444444444445 / (re + 1.0))))))) / im);
	} else if (re <= 2.5e+149) {
		tmp = t_1 * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    if (re <= 2.6d-18) then
        tmp = 1.0d0 / (((1.0d0 / (re + 1.0d0)) + ((im * im) * ((0.16666666666666666d0 / (re + 1.0d0)) + (im * (im * (0.019444444444444445d0 / (re + 1.0d0))))))) / im)
    else if (re <= 2.5d+149) then
        tmp = t_1 * (1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0)))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = im * (1.0 + (-0.16666666666666666 * (im * im)));
	double tmp;
	if (re <= 2.6e-18) {
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + (im * (im * (0.019444444444444445 / (re + 1.0))))))) / im);
	} else if (re <= 2.5e+149) {
		tmp = t_1 * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = im * (1.0 + (-0.16666666666666666 * (im * im)))
	tmp = 0
	if re <= 2.6e-18:
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + (im * (im * (0.019444444444444445 / (re + 1.0))))))) / im)
	elif re <= 2.5e+149:
		tmp = t_1 * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)))
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))))
	tmp = 0.0
	if (re <= 2.6e-18)
		tmp = Float64(1.0 / Float64(Float64(Float64(1.0 / Float64(re + 1.0)) + Float64(Float64(im * im) * Float64(Float64(0.16666666666666666 / Float64(re + 1.0)) + Float64(im * Float64(im * Float64(0.019444444444444445 / Float64(re + 1.0))))))) / im));
	elseif (re <= 2.5e+149)
		tmp = Float64(t_1 * Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = im * (1.0 + (-0.16666666666666666 * (im * im)));
	tmp = 0.0;
	if (re <= 2.6e-18)
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + (im * (im * (0.019444444444444445 / (re + 1.0))))))) / im);
	elseif (re <= 2.5e+149)
		tmp = t_1 * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 2.6e-18], N[(1.0 / N[(N[(N[(1.0 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(im * im), $MachinePrecision] * N[(N[(0.16666666666666666 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(im * N[(im * N[(0.019444444444444445 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e+149], N[(t$95$1 * N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 2.6e-18

   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 64.1%

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

   5. Simplified 64.1%

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

      1. *-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. flip3-+ N/A

         \[\leadsto \frac{{\left(\sin im \cdot re\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}}{\color{blue}{\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\sin im \cdot re\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}\right), \color{blue}{\left(\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(re \cdot \sin im\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}\right), \left(\left(\color{blue}{\sin im} \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(re \cdot \sin im\right)}^{3}\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\color{blue}{\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right)} + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(re \cdot \sin im\right), 3\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\color{blue}{\left(\sin im \cdot re\right)} \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \sin im\right), 3\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\left(\color{blue}{\sin im} \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

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

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \left({\sin im}^{3}\right)\right), \left(\left(\sin im \cdot re\right) \cdot \left(\color{blue}{\sin im} \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \mathsf{pow.f64}\left(\sin im, 3\right)\right), \left(\left(\sin im \cdot re\right) \cdot \color{blue}{\left(\sin im \cdot re\right)} + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(im\right), 3\right)\right), \left(\left(\sin im \cdot re\right) \cdot \left(\color{blue}{\sin im} \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

   7. Applied egg-rr 44.8%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(re \cdot \sin im\right) \cdot \left(re \cdot \sin im\right) + \left(\sin im \cdot \sin im - \left(re \cdot \sin im\right) \cdot \sin im\right)}{{\left(re \cdot \sin im\right)}^{3} + {\sin im}^{3}}}} \]

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

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

      3. clear-num N/A

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

      4. flip3-+ N/A

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

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

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

      6. distribute-lft1-in N/A

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

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

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

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

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

      9. sin-lowering-sin.f64 63.9%

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

   9. Applied egg-rr 63.9%

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

   10. Taylor expanded in im around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{{im}^{2} \cdot \left(\frac{7}{360} \cdot \frac{{im}^{2}}{1 + re} + \frac{1}{6} \cdot \frac{1}{1 + re}\right) + \frac{1}{1 + re}}{im}\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({im}^{2} \cdot \left(\frac{7}{360} \cdot \frac{{im}^{2}}{1 + re} + \frac{1}{6} \cdot \frac{1}{1 + re}\right) + \frac{1}{1 + re}\right), \color{blue}{im}\right)\right) \]

   12. Simplified 49.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{0.019444444444444445}{re + 1}\right) + \frac{0.16666666666666666}{re + 1}\right)}{im}}} \]

   if 2.6e-18 < re < 2.49999999999999995e149

   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 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 17.2%

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

   8. Simplified 17.2%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. flip-+ N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-lowering-*.f64 40.4%

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

   10. Applied egg-rr 40.4%

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

   if 2.49999999999999995e149 < 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 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{0.16666666666666666}{re + 1} + im \cdot \left(im \cdot \frac{0.019444444444444445}{re + 1}\right)\right)}{im}}\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;\left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(1 + \frac{re \cdot re - \left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)}{re - re \cdot \left(re \cdot 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.6 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{0.16666666666666666}{re + 1} + im \cdot \left(im \cdot \frac{0.019444444444444445}{re + 1}\right)\right)}{im}}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+149}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.6e-146)
   (/
    1.0
    (/
     (+
      (/ 1.0 (+ re 1.0))
      (*
       (* im im)
       (+
        (/ 0.16666666666666666 (+ re 1.0))
        (* im (* im (/ 0.019444444444444445 (+ re 1.0)))))))
     im))
   (if (<= re 2.1e+149)
     (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
     (*
      (+ 1.0 (* re (+ 1.0 (* re 0.5))))
      (* im (+ 1.0 (* -0.16666666666666666 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.6e-146) {
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + (im * (im * (0.019444444444444445 / (re + 1.0))))))) / im);
	} else if (re <= 2.1e+149) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (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 <= (-2.6d-146)) then
        tmp = 1.0d0 / (((1.0d0 / (re + 1.0d0)) + ((im * im) * ((0.16666666666666666d0 / (re + 1.0d0)) + (im * (im * (0.019444444444444445d0 / (re + 1.0d0))))))) / im)
    else if (re <= 2.1d+149) then
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (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 <= -2.6e-146) {
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + (im * (im * (0.019444444444444445 / (re + 1.0))))))) / im);
	} else if (re <= 2.1e+149) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.6e-146:
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + (im * (im * (0.019444444444444445 / (re + 1.0))))))) / im)
	elif re <= 2.1e+149:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.6e-146)
		tmp = Float64(1.0 / Float64(Float64(Float64(1.0 / Float64(re + 1.0)) + Float64(Float64(im * im) * Float64(Float64(0.16666666666666666 / Float64(re + 1.0)) + Float64(im * Float64(im * Float64(0.019444444444444445 / Float64(re + 1.0))))))) / im));
	elseif (re <= 2.1e+149)
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * 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 <= -2.6e-146)
		tmp = 1.0 / (((1.0 / (re + 1.0)) + ((im * im) * ((0.16666666666666666 / (re + 1.0)) + (im * (im * (0.019444444444444445 / (re + 1.0))))))) / im);
	elseif (re <= 2.1e+149)
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.6e-146], N[(1.0 / N[(N[(N[(1.0 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(im * im), $MachinePrecision] * N[(N[(0.16666666666666666 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(im * N[(im * N[(0.019444444444444445 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+149], 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], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -2.59999999999999987e-146

   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 31.2%

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

   5. Simplified 31.2%

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

      1. *-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. flip3-+ N/A

         \[\leadsto \frac{{\left(\sin im \cdot re\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}}{\color{blue}{\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\sin im \cdot re\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}\right), \color{blue}{\left(\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(re \cdot \sin im\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}\right), \left(\left(\color{blue}{\sin im} \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(re \cdot \sin im\right)}^{3}\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\color{blue}{\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right)} + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(re \cdot \sin im\right), 3\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\color{blue}{\left(\sin im \cdot re\right)} \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \sin im\right), 3\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\left(\color{blue}{\sin im} \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

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

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \left({\sin im}^{3}\right)\right), \left(\left(\sin im \cdot re\right) \cdot \left(\color{blue}{\sin im} \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \mathsf{pow.f64}\left(\sin im, 3\right)\right), \left(\left(\sin im \cdot re\right) \cdot \color{blue}{\left(\sin im \cdot re\right)} + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(im\right), 3\right)\right), \left(\left(\sin im \cdot re\right) \cdot \left(\color{blue}{\sin im} \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

   7. Applied egg-rr 26.2%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(re \cdot \sin im\right) \cdot \left(re \cdot \sin im\right) + \left(\sin im \cdot \sin im - \left(re \cdot \sin im\right) \cdot \sin im\right)}{{\left(re \cdot \sin im\right)}^{3} + {\sin im}^{3}}}} \]

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

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

      3. clear-num N/A

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

      4. flip3-+ N/A

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

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

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

      6. distribute-lft1-in N/A

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

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

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

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

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

      9. sin-lowering-sin.f64 31.1%

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

   9. Applied egg-rr 31.1%

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

   10. Taylor expanded in im around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{{im}^{2} \cdot \left(\frac{7}{360} \cdot \frac{{im}^{2}}{1 + re} + \frac{1}{6} \cdot \frac{1}{1 + re}\right) + \frac{1}{1 + re}}{im}\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({im}^{2} \cdot \left(\frac{7}{360} \cdot \frac{{im}^{2}}{1 + re} + \frac{1}{6} \cdot \frac{1}{1 + re}\right) + \frac{1}{1 + re}\right), \color{blue}{im}\right)\right) \]

   12. Simplified 44.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{0.019444444444444445}{re + 1}\right) + \frac{0.16666666666666666}{re + 1}\right)}{im}}} \]

   if -2.59999999999999987e-146 < re < 2.1000000000000002e149

   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 61.6%

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

   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 50.0%

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

   8. Simplified 50.0%

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

   if 2.1000000000000002e149 < 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 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.6 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{re + 1} + \left(im \cdot im\right) \cdot \left(\frac{0.16666666666666666}{re + 1} + im \cdot \left(im \cdot \frac{0.019444444444444445}{re + 1}\right)\right)}{im}}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+149}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6200:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{re + 1} + \frac{\left(im \cdot im\right) \cdot 0.16666666666666666}{re + 1}}{im}}\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6200.0)
   (/
    1.0
    (/
     (+ (/ 1.0 (+ re 1.0)) (/ (* (* im im) 0.16666666666666666) (+ re 1.0)))
     im))
   (if (<= re 2.5e+149)
     (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
     (*
      (+ 1.0 (* re (+ 1.0 (* re 0.5))))
      (* im (+ 1.0 (* -0.16666666666666666 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6200.0) {
		tmp = 1.0 / (((1.0 / (re + 1.0)) + (((im * im) * 0.16666666666666666) / (re + 1.0))) / im);
	} else if (re <= 2.5e+149) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (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 <= 6200.0d0) then
        tmp = 1.0d0 / (((1.0d0 / (re + 1.0d0)) + (((im * im) * 0.16666666666666666d0) / (re + 1.0d0))) / im)
    else if (re <= 2.5d+149) then
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (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 <= 6200.0) {
		tmp = 1.0 / (((1.0 / (re + 1.0)) + (((im * im) * 0.16666666666666666) / (re + 1.0))) / im);
	} else if (re <= 2.5e+149) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6200.0:
		tmp = 1.0 / (((1.0 / (re + 1.0)) + (((im * im) * 0.16666666666666666) / (re + 1.0))) / im)
	elif re <= 2.5e+149:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6200.0)
		tmp = Float64(1.0 / Float64(Float64(Float64(1.0 / Float64(re + 1.0)) + Float64(Float64(Float64(im * im) * 0.16666666666666666) / Float64(re + 1.0))) / im));
	elseif (re <= 2.5e+149)
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * 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 <= 6200.0)
		tmp = 1.0 / (((1.0 / (re + 1.0)) + (((im * im) * 0.16666666666666666) / (re + 1.0))) / im);
	elseif (re <= 2.5e+149)
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + (-0.16666666666666666 * (im * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6200.0], N[(1.0 / N[(N[(N[(1.0 / N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(im * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] / N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e+149], 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], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 6200

   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 64.0%

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

   5. Simplified 64.0%

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

      1. *-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. flip3-+ N/A

         \[\leadsto \frac{{\left(\sin im \cdot re\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}}{\color{blue}{\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\sin im \cdot re\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}\right), \color{blue}{\left(\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(re \cdot \sin im\right)}^{3} + {\left(\sin im \cdot 1\right)}^{3}\right), \left(\left(\color{blue}{\sin im} \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(re \cdot \sin im\right)}^{3}\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\color{blue}{\left(\sin im \cdot re\right) \cdot \left(\sin im \cdot re\right)} + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(re \cdot \sin im\right), 3\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\color{blue}{\left(\sin im \cdot re\right)} \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \sin im\right), 3\right), \left({\left(\sin im \cdot 1\right)}^{3}\right)\right), \left(\left(\color{blue}{\sin im} \cdot re\right) \cdot \left(\sin im \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

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

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \left({\sin im}^{3}\right)\right), \left(\left(\sin im \cdot re\right) \cdot \left(\color{blue}{\sin im} \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \mathsf{pow.f64}\left(\sin im, 3\right)\right), \left(\left(\sin im \cdot re\right) \cdot \color{blue}{\left(\sin im \cdot re\right)} + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, \mathsf{sin.f64}\left(im\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(im\right), 3\right)\right), \left(\left(\sin im \cdot re\right) \cdot \left(\color{blue}{\sin im} \cdot re\right) + \left(\left(\sin im \cdot 1\right) \cdot \left(\sin im \cdot 1\right) - \left(\sin im \cdot re\right) \cdot \left(\sin im \cdot 1\right)\right)\right)\right) \]

   7. Applied egg-rr 44.3%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(re \cdot \sin im\right) \cdot \left(re \cdot \sin im\right) + \left(\sin im \cdot \sin im - \left(re \cdot \sin im\right) \cdot \sin im\right)}{{\left(re \cdot \sin im\right)}^{3} + {\sin im}^{3}}}} \]

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

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

      3. clear-num N/A

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

      4. flip3-+ N/A

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

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

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

      6. distribute-lft1-in N/A

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

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

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

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

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

      9. sin-lowering-sin.f64 63.8%

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

   9. Applied egg-rr 63.8%

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

   10. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

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

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

   12. Simplified 47.7%

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

   if 6200 < re < 2.49999999999999995e149

   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 80.6%

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

   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 34.3%

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

   8. Simplified 34.3%

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

   if 2.49999999999999995e149 < 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 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6200:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{re + 1} + \frac{\left(im \cdot im\right) \cdot 0.16666666666666666}{re + 1}}{im}}\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot t\_0\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* im im))))
   (if (<= re -12000000000.0)
     (* (+ re 1.0) (* im t_0))
     (if (<= re 2.5e+149)
       (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
       (* (+ 1.0 (* re (+ 1.0 (* re 0.5)))) (* im (+ 1.0 t_0)))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -12000000000.0) {
		tmp = (re + 1.0) * (im * t_0);
	} else if (re <= 2.5e+149) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (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 = (-0.16666666666666666d0) * (im * im)
    if (re <= (-12000000000.0d0)) then
        tmp = (re + 1.0d0) * (im * t_0)
    else if (re <= 2.5d+149) then
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (im * (1.0d0 + t_0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -12000000000.0) {
		tmp = (re + 1.0) * (im * t_0);
	} else if (re <= 2.5e+149) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + t_0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (im * im)
	tmp = 0
	if re <= -12000000000.0:
		tmp = (re + 1.0) * (im * t_0)
	elif re <= 2.5e+149:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	else:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + t_0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * Float64(im * im))
	tmp = 0.0
	if (re <= -12000000000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * t_0));
	elseif (re <= 2.5e+149)
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * Float64(im * Float64(1.0 + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (im * im);
	tmp = 0.0;
	if (re <= -12000000000.0)
		tmp = (re + 1.0) * (im * t_0);
	elseif (re <= 2.5e+149)
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	else
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -12000000000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e+149], 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], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e10

   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 56.5%

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

   5. Simplified 56.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.4%

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

   8. Simplified 2.4%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{3}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. +-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 26.3%

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

   11. Simplified 26.3%

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

   if -1.2e10 < re < 2.49999999999999995e149

   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 57.7%

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

   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 47.8%

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

   8. Simplified 47.8%

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

   if 2.49999999999999995e149 < 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 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -12000000000.0)
   (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
   (if (<= re 2.5e+149)
     (* im (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
     (* im (* (* re re) (+ 0.5 (* (* im im) -0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 2.5e+149) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-12000000000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 2.5d+149) then
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    else
        tmp = im * ((re * re) * (0.5d0 + ((im * im) * (-0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 2.5e+149) {
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -12000000000.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	elif re <= 2.5e+149:
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	else:
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -12000000000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 2.5e+149)
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	else
		tmp = Float64(im * Float64(Float64(re * re) * Float64(0.5 + Float64(Float64(im * im) * -0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -12000000000.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	elseif (re <= 2.5e+149)
		tmp = im * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	else
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -12000000000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e+149], 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], N[(im * N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e10

   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 56.5%

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

   5. Simplified 56.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.4%

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

   8. Simplified 2.4%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{3}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. +-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 26.3%

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

   11. Simplified 26.3%

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

   if -1.2e10 < re < 2.49999999999999995e149

   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 57.7%

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

   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 47.8%

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

   8. Simplified 47.8%

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

   if 2.49999999999999995e149 < 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 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.8%

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

   8. Simplified 81.8%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 56.6%

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

   11. Simplified 56.6%

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

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{12} \cdot \left({im}^{2} \cdot {re}^{2}\right) + \frac{1}{2} \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{12} \cdot \left({im}^{2} \cdot {re}^{2}\right) + \frac{1}{2} \cdot {re}^{2}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{2}} \cdot {re}^{2}\right)\right) \]

       3. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2} + \frac{1}{2}\right)}\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {im}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {im}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{2}} + \frac{-1}{12} \cdot {im}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{2}} + \frac{-1}{12} \cdot {im}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{12}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 81.8%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{12}\right)\right)\right)\right) \]

   14. Simplified 81.8%

       \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+149}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -60:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -60.0)
   (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
   (if (<= re 1.55e+123)
     (*
      im
      (+
       1.0
       (*
        (* im im)
        (+ -0.16666666666666666 (* im (* im 0.008333333333333333))))))
     (* im (* (* re re) (+ 0.5 (* (* im im) -0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -60.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 1.55e+123) {
		tmp = im * (1.0 + ((im * im) * (-0.16666666666666666 + (im * (im * 0.008333333333333333)))));
	} else {
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-60.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 1.55d+123) then
        tmp = im * (1.0d0 + ((im * im) * ((-0.16666666666666666d0) + (im * (im * 0.008333333333333333d0)))))
    else
        tmp = im * ((re * re) * (0.5d0 + ((im * im) * (-0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -60.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 1.55e+123) {
		tmp = im * (1.0 + ((im * im) * (-0.16666666666666666 + (im * (im * 0.008333333333333333)))));
	} else {
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -60.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	elif re <= 1.55e+123:
		tmp = im * (1.0 + ((im * im) * (-0.16666666666666666 + (im * (im * 0.008333333333333333)))))
	else:
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -60.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 1.55e+123)
		tmp = Float64(im * Float64(1.0 + Float64(Float64(im * im) * Float64(-0.16666666666666666 + Float64(im * Float64(im * 0.008333333333333333))))));
	else
		tmp = Float64(im * Float64(Float64(re * re) * Float64(0.5 + Float64(Float64(im * im) * -0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -60.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	elseif (re <= 1.55e+123)
		tmp = im * (1.0 + ((im * im) * (-0.16666666666666666 + (im * (im * 0.008333333333333333)))));
	else
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -60.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.55e+123], N[(im * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.16666666666666666 + N[(im * N[(im * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -60

   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 57.1%

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

   5. Simplified 57.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.4%

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

   8. Simplified 2.4%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{3}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. +-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 26.0%

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

   11. Simplified 26.0%

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

   if -60 < re < 1.55000000000000003e123

   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 81.2%

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

   5. Simplified 81.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 43.9%

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot \left(0.008333333333333333 + \left(im \cdot im\right) \cdot -0.0001984126984126984\right)\right)\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 46.0%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 46.0%

       \[\leadsto \color{blue}{im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)} \]

   if 1.55000000000000003e123 < 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 81.1%

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

   5. Simplified 81.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 76.0%

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

   8. Simplified 76.0%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 53.5%

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

   11. Simplified 53.5%

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

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{12} \cdot \left({im}^{2} \cdot {re}^{2}\right) + \frac{1}{2} \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{12} \cdot \left({im}^{2} \cdot {re}^{2}\right) + \frac{1}{2} \cdot {re}^{2}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{2}} \cdot {re}^{2}\right)\right) \]

       3. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2} + \frac{1}{2}\right)}\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {im}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {im}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{2}} + \frac{-1}{12} \cdot {im}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{2}} + \frac{-1}{12} \cdot {im}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{12}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 76.0%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{12}\right)\right)\right)\right) \]

   14. Simplified 76.0%

       \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -60:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+23}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -12000000000.0)
   (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
   (if (<= re 1.95e+23)
     (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (* im (* (* re re) (+ 0.5 (* (* im im) -0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 1.95e+23) {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-12000000000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 1.95d+23) then
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else
        tmp = im * ((re * re) * (0.5d0 + ((im * im) * (-0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 1.95e+23) {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -12000000000.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	elif re <= 1.95e+23:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	else:
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -12000000000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 1.95e+23)
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = Float64(im * Float64(Float64(re * re) * Float64(0.5 + Float64(Float64(im * im) * -0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -12000000000.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	elseif (re <= 1.95e+23)
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	else
		tmp = im * ((re * re) * (0.5 + ((im * im) * -0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -12000000000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.95e+23], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * re), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e10

   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 56.5%

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

   5. Simplified 56.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.4%

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

   8. Simplified 2.4%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{3}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. +-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 26.3%

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

   11. Simplified 26.3%

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

   if -1.2e10 < re < 1.95e23

   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 52.6%

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

   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 49.6%

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

   8. Simplified 49.6%

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

   if 1.95e23 < 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 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 54.3%

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

   8. Simplified 54.3%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 39.9%

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

   11. Simplified 39.9%

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

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{12} \cdot \left({im}^{2} \cdot {re}^{2}\right) + \frac{1}{2} \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{12} \cdot \left({im}^{2} \cdot {re}^{2}\right) + \frac{1}{2} \cdot {re}^{2}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{1}{2}} \cdot {re}^{2}\right)\right) \]

       3. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2} + \frac{1}{2}\right)}\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {im}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {im}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{2}} + \frac{-1}{12} \cdot {im}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{2}} + \frac{-1}{12} \cdot {im}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{12}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 54.3%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{12}\right)\right)\right)\right) \]

   14. Simplified 54.3%

       \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+23}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -12000000000.0)
   (* re (* re (* im (* (* im im) -0.08333333333333333))))
   (if (<= re 2.2e+33)
     (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
     (* (* re re) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = re * (re * (im * ((im * im) * -0.08333333333333333)));
	} else if (re <= 2.2e+33) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-12000000000.0d0)) then
        tmp = re * (re * (im * ((im * im) * (-0.08333333333333333d0))))
    else if (re <= 2.2d+33) then
        tmp = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = re * (re * (im * ((im * im) * -0.08333333333333333)));
	} else if (re <= 2.2e+33) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -12000000000.0:
		tmp = re * (re * (im * ((im * im) * -0.08333333333333333)))
	elif re <= 2.2e+33:
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)))
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -12000000000.0)
		tmp = Float64(re * Float64(re * Float64(im * Float64(Float64(im * im) * -0.08333333333333333))));
	elseif (re <= 2.2e+33)
		tmp = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -12000000000.0)
		tmp = re * (re * (im * ((im * im) * -0.08333333333333333)));
	elseif (re <= 2.2e+33)
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -12000000000.0], N[(re * N[(re * N[(im * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.2e+33], N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e10

   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 56.5%

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

   5. Simplified 56.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 2.0%

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

   11. Simplified 2.0%

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

   12. Taylor expanded in im around inf

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{12} \cdot {im}^{3}\right)}\right)\right) \]
   13. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{12} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{12} \cdot \left({im}^{2} \cdot im\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{im}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{12}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 26.1%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{12}\right)\right)\right)\right) \]

   14. Simplified 26.1%

       \[\leadsto re \cdot \left(re \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)}\right) \]

   if -1.2e10 < re < 2.19999999999999994e33

   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 91.3%

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

   5. Simplified 91.3%

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

   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. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 49.4%

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

   8. Simplified 49.4%

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

   if 2.19999999999999994e33 < 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 74.5%

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

   5. Simplified 74.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 51.8%

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

   8. Simplified 51.8%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 36.7%

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

   11. Simplified 36.7%

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

   12. Taylor expanded in im around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

       4. unpow2 N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 45.3%

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

   14. Simplified 45.3%

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

Alternative 16: 44.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;re \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+32}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -12000000000.0)
   (* re (* im (* re (* (* im im) -0.08333333333333333))))
   (if (<= re 1.85e+32)
     (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
     (* (* re re) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = re * (im * (re * ((im * im) * -0.08333333333333333)));
	} else if (re <= 1.85e+32) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-12000000000.0d0)) then
        tmp = re * (im * (re * ((im * im) * (-0.08333333333333333d0))))
    else if (re <= 1.85d+32) then
        tmp = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = re * (im * (re * ((im * im) * -0.08333333333333333)));
	} else if (re <= 1.85e+32) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -12000000000.0:
		tmp = re * (im * (re * ((im * im) * -0.08333333333333333)))
	elif re <= 1.85e+32:
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)))
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -12000000000.0)
		tmp = Float64(re * Float64(im * Float64(re * Float64(Float64(im * im) * -0.08333333333333333))));
	elseif (re <= 1.85e+32)
		tmp = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -12000000000.0)
		tmp = re * (im * (re * ((im * im) * -0.08333333333333333)));
	elseif (re <= 1.85e+32)
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -12000000000.0], N[(re * N[(im * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.85e+32], N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e10

   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 56.5%

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

   5. Simplified 56.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 2.0%

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

   11. Simplified 2.0%

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

   12. Taylor expanded in im around inf

       \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{12} \cdot \left({im}^{3} \cdot re\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left({im}^{3} \cdot re\right) \cdot \color{blue}{\frac{-1}{12}}\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot re\right) \cdot \frac{-1}{12}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\left(im \cdot {im}^{2}\right) \cdot re\right) \cdot \frac{-1}{12}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left(im \cdot \left({im}^{2} \cdot re\right)\right) \cdot \frac{-1}{12}\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(\left({im}^{2} \cdot re\right) \cdot \frac{-1}{12}\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \left(\frac{-1}{12} \cdot \color{blue}{\left({im}^{2} \cdot re\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{12} \cdot \left({im}^{2} \cdot re\right)\right)}\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{re}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{12}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 24.7%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{12}\right)\right)\right)\right) \]

   14. Simplified 24.7%

       \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\right)} \]

   if -1.2e10 < re < 1.85e32

   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 91.3%

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

   5. Simplified 91.3%

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

   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. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 49.4%

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

   8. Simplified 49.4%

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

   if 1.85e32 < 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 74.5%

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

   5. Simplified 74.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 51.8%

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

   8. Simplified 51.8%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 36.7%

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

   11. Simplified 36.7%

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

   12. Taylor expanded in im around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot im\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       7. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right) \]

   14. Simplified 45.3%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 40.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;-0.08333333333333333 \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.26 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -12000000000.0)
   (* -0.08333333333333333 (* im (* im (* im (* re re)))))
   (if (<= re 1.26e+30)
     (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
     (* (* re re) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = -0.08333333333333333 * (im * (im * (im * (re * re))));
	} else if (re <= 1.26e+30) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-12000000000.0d0)) then
        tmp = (-0.08333333333333333d0) * (im * (im * (im * (re * re))))
    else if (re <= 1.26d+30) then
        tmp = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = -0.08333333333333333 * (im * (im * (im * (re * re))));
	} else if (re <= 1.26e+30) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -12000000000.0:
		tmp = -0.08333333333333333 * (im * (im * (im * (re * re))))
	elif re <= 1.26e+30:
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)))
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -12000000000.0)
		tmp = Float64(-0.08333333333333333 * Float64(im * Float64(im * Float64(im * Float64(re * re)))));
	elseif (re <= 1.26e+30)
		tmp = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -12000000000.0)
		tmp = -0.08333333333333333 * (im * (im * (im * (re * re))));
	elseif (re <= 1.26e+30)
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -12000000000.0], N[(-0.08333333333333333 * N[(im * N[(im * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.26e+30], N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e10

   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 56.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 2.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 2.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left({re}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{{re}^{2}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{{re}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {im}^{2}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 2.0%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

   11. Simplified 2.0%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(im \cdot \left(\left(1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot 0.5\right)\right)\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({im}^{3} \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \color{blue}{\left({im}^{3} \cdot {re}^{2}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot {\color{blue}{re}}^{2}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \left(\left(im \cdot {im}^{2}\right) \cdot {re}^{2}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \left(im \cdot \color{blue}{\left({im}^{2} \cdot {re}^{2}\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot {re}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(im, \left(\left(im \cdot im\right) \cdot {\color{blue}{re}}^{2}\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(im \cdot {re}^{2}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot {re}^{2}\right)}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 13.5%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right) \]

   14. Simplified 13.5%

       \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(re \cdot re\right)\right)\right)\right)} \]

   if -1.2e10 < re < 1.26e30

   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 91.3%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 91.3%

      \[\leadsto \color{blue}{\sin im} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 49.4%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 49.4%

      \[\leadsto \color{blue}{im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]

   if 1.26e30 < 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 74.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 74.5%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 51.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 51.8%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left({re}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{{re}^{2}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{{re}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {im}^{2}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 36.7%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

   11. Simplified 36.7%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(im \cdot \left(\left(1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot 0.5\right)\right)\right)} \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{{re}^{2}} \]

       2. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot im\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       7. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right) \]

   14. Simplified 45.3%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 46.0% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -12000000000.0)
   (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
   (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-12000000000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -12000000000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -12000000000.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -12000000000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -12000000000.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	else
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -12000000000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.2e10

   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 56.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 2.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 2.4%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({im}^{3} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{3}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. unpow3 N/A

          \[\leadsto \left(\frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot im\right)\right) \cdot \left(1 + re\right) \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot im\right)\right) \cdot \left(1 + re\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im\right) \cdot \left(\color{blue}{1} + re\right) \]

       5. *-commutative N/A

          \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + re\right), \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \cdot im\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \cdot im\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{6}\right)\right)\right) \]

       14. *-lowering-*.f64 26.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{6}\right)\right)\right) \]

   11. Simplified 26.3%

       \[\leadsto \color{blue}{\left(re + 1\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot -0.16666666666666666\right)\right)} \]

   if -1.2e10 < 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 59.3%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   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 47.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 47.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -12000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 45.9% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -13500000000:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -13500000000.0)
   (* re (* re (* im (* (* im im) -0.08333333333333333))))
   (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -13500000000.0) {
		tmp = re * (re * (im * ((im * im) * -0.08333333333333333)));
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-13500000000.0d0)) then
        tmp = re * (re * (im * ((im * im) * (-0.08333333333333333d0))))
    else
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -13500000000.0) {
		tmp = re * (re * (im * ((im * im) * -0.08333333333333333)));
	} else {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -13500000000.0:
		tmp = re * (re * (im * ((im * im) * -0.08333333333333333)))
	else:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -13500000000.0)
		tmp = Float64(re * Float64(re * Float64(im * Float64(Float64(im * im) * -0.08333333333333333))));
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -13500000000.0)
		tmp = re * (re * (im * ((im * im) * -0.08333333333333333)));
	else
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -13500000000.0], N[(re * N[(re * N[(im * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.35e10

   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 56.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 2.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 2.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left({re}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{{re}^{2}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{{re}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {im}^{2}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 2.0%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

   11. Simplified 2.0%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(im \cdot \left(\left(1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot 0.5\right)\right)\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{12} \cdot {im}^{3}\right)}\right)\right) \]
   13. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{12} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{12} \cdot \left({im}^{2} \cdot im\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{im}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2}\right)}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{12}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 26.1%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{12}\right)\right)\right)\right) \]

   14. Simplified 26.1%

       \[\leadsto re \cdot \left(re \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)}\right) \]

   if -1.35e10 < 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 59.3%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   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 47.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 47.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -13500000000:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 38.1% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.85 \cdot 10^{+32}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.85e+32)
   (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
   (* (* re re) (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.85e+32) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.85d+32) then
        tmp = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.85e+32) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.85e+32:
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)))
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.85e+32)
		tmp = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.85e+32)
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.85e+32], N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.85e32

   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 61.4%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 61.4%

      \[\leadsto \color{blue}{\sin im} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 33.6%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 33.6%

      \[\leadsto \color{blue}{im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]

   if 2.85e32 < 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 74.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 74.5%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 51.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 51.8%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left({re}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{{re}^{2}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{{re}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {im}^{2}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 36.7%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

   11. Simplified 36.7%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(im \cdot \left(\left(1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot 0.5\right)\right)\right)} \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{{re}^{2}} \]

       2. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot im\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       7. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right) \]

   14. Simplified 45.3%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 37.9% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 680:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 680.0) im (* (* re re) (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 680.0) {
		tmp = im;
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 680.0d0) then
        tmp = im
    else
        tmp = (re * re) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 680.0) {
		tmp = im;
	} else {
		tmp = (re * re) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 680.0:
		tmp = im
	else:
		tmp = (re * re) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 680.0)
		tmp = im;
	else
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 680.0)
		tmp = im;
	else
		tmp = (re * re) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 680.0], im, N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 680

   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 69.2%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 33.9%

      \[\leadsto \color{blue}{im} \]

   if 680 < 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 75.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 75.0%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 49.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 49.4%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left({re}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{{re}^{2}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{{re}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {im}^{2}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 36.4%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

   11. Simplified 36.4%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(im \cdot \left(\left(1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot 0.5\right)\right)\right)} \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{{re}^{2}} \]

       2. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot im\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       7. *-lowering-*.f64 40.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right) \]

   14. Simplified 40.6%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 34.9% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 600:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 600.0) im (* re (* re (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 600.0) {
		tmp = im;
	} else {
		tmp = re * (re * (im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 600.0d0) then
        tmp = im
    else
        tmp = re * (re * (im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 600.0) {
		tmp = im;
	} else {
		tmp = re * (re * (im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 600.0:
		tmp = im
	else:
		tmp = re * (re * (im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 600.0)
		tmp = im;
	else
		tmp = Float64(re * Float64(re * Float64(im * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 600.0)
		tmp = im;
	else
		tmp = re * (re * (im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 600.0], im, N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 600

   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 69.2%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 33.9%

      \[\leadsto \color{blue}{im} \]

   if 600 < 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 75.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 75.0%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 49.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 49.4%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left({re}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{{re}^{2}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{{re}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \frac{1}{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {im}^{2}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 36.4%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{6}\right)\right), \frac{1}{2}\right)\right)\right)\right) \]

   11. Simplified 36.4%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(im \cdot \left(\left(1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot 0.5\right)\right)\right)} \]

   12. Taylor expanded in im around 0

       \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot re\right)\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{re}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 27.6%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   14. Simplified 27.6%

       \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(im \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 28.6% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 23000000000000:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 23000000000000.0) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 23000000000000.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 23000000000000.0d0) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 23000000000000.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 23000000000000.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 23000000000000.0)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 23000000000000.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 23000000000000.0], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.3e13

   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 78.9%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 32.7%

      \[\leadsto \color{blue}{im} \]

   if 2.3e13 < 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 40.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   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 6.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

   8. Simplified 6.0%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot im \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, im\right) \]
   10. Step-by-step derivation

   11. Simplified 6.4%

       \[\leadsto \color{blue}{re} \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 30.0% 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 70.3%

   \[\leadsto e^{re} \cdot \color{blue}{im} \]

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 28.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

8. Simplified 28.6%

   \[\leadsto \color{blue}{\left(re + 1\right)} \cdot im \]

9. Final simplification 28.6%

   \[\leadsto im \cdot \left(re + 1\right) \]
10. Add Preprocessing

Alternative 25: 27.1% 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 70.3%

   \[\leadsto e^{re} \cdot \color{blue}{im} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im} \]
7. Step-by-step derivation

8. Simplified 26.0%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))