math.exp on complex, imaginary part

Percentage Accurate: 99.9% → 99.9%

Time: 12.8s

Alternatives: 18

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: 99.9% 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: 99.9% 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: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.99999995)
   (* (exp re) im)
   (if (<= (exp re) 1.0)
     (* (sin im) (+ re 1.0))
     (* (exp re) (* im (+ (* -0.16666666666666666 (* im im)) 1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.99999995) {
		tmp = exp(re) * im;
	} else if (exp(re) <= 1.0) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = exp(re) * (im * ((-0.16666666666666666 * (im * im)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.99999995d0) then
        tmp = exp(re) * im
    else if (exp(re) <= 1.0d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = exp(re) * (im * (((-0.16666666666666666d0) * (im * im)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.99999995) {
		tmp = Math.exp(re) * im;
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * (im * ((-0.16666666666666666 * (im * im)) + 1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.99999995:
		tmp = math.exp(re) * im
	elif math.exp(re) <= 1.0:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * (im * ((-0.16666666666666666 * (im * im)) + 1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.99999995)
		tmp = Float64(exp(re) * im);
	elseif (exp(re) <= 1.0)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(re) * Float64(im * Float64(Float64(-0.16666666666666666 * Float64(im * im)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.99999995)
		tmp = exp(re) * im;
	elseif (exp(re) <= 1.0)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re) * (im * ((-0.16666666666666666 * (im * im)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.99999995], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.999999949999999971

   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 99.8%

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

   if 0.999999949999999971 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 100.0%

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

   5. Simplified 100.0%

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

   if 1 < (exp.f64 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%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999995:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -8.5e-8)
     t_0
     (if (<= re 1.2e-20)
       (* (sin im) (+ re 1.0))
       (if (<= re 4.5e+89)
         t_0
         (*
          (* im (+ (* -0.16666666666666666 (* im im)) 1.0))
          (+
           (* re (+ (* re (+ 0.5 (* re 0.16666666666666666))) 1.0))
           1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -8.5e-8) {
		tmp = t_0;
	} else if (re <= 1.2e-20) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 4.5e+89) {
		tmp = t_0;
	} else {
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-8.5d-8)) then
        tmp = t_0
    else if (re <= 1.2d-20) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 4.5d+89) then
        tmp = t_0
    else
        tmp = (im * (((-0.16666666666666666d0) * (im * im)) + 1.0d0)) * ((re * ((re * (0.5d0 + (re * 0.16666666666666666d0))) + 1.0d0)) + 1.0d0)
    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 <= -8.5e-8) {
		tmp = t_0;
	} else if (re <= 1.2e-20) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 4.5e+89) {
		tmp = t_0;
	} else {
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -8.5e-8:
		tmp = t_0
	elif re <= 1.2e-20:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 4.5e+89:
		tmp = t_0
	else:
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -8.5e-8)
		tmp = t_0;
	elseif (re <= 1.2e-20)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 4.5e+89)
		tmp = t_0;
	else
		tmp = Float64(Float64(im * Float64(Float64(-0.16666666666666666 * Float64(im * im)) + 1.0)) * Float64(Float64(re * Float64(Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -8.5e-8)
		tmp = t_0;
	elseif (re <= 1.2e-20)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 4.5e+89)
		tmp = t_0;
	else
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -8.5e-8], t$95$0, If[LessEqual[re, 1.2e-20], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.5e+89], t$95$0, N[(N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(re * N[(N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.49999999999999935e-8 or 1.19999999999999996e-20 < re < 4.5e89

   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 94.8%

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

   if -8.49999999999999935e-8 < re < 1.19999999999999996e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 100.0%

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

   5. Simplified 100.0%

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

   if 4.5e89 < 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 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+89}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -1.8 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -1.8e-8)
     t_0
     (if (<= re 1.2e-20)
       (sin im)
       (if (<= re 8.4e+89)
         t_0
         (*
          (* im (+ (* -0.16666666666666666 (* im im)) 1.0))
          (+
           (* re (+ (* re (+ 0.5 (* re 0.16666666666666666))) 1.0))
           1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -1.8e-8) {
		tmp = t_0;
	} else if (re <= 1.2e-20) {
		tmp = sin(im);
	} else if (re <= 8.4e+89) {
		tmp = t_0;
	} else {
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-1.8d-8)) then
        tmp = t_0
    else if (re <= 1.2d-20) then
        tmp = sin(im)
    else if (re <= 8.4d+89) then
        tmp = t_0
    else
        tmp = (im * (((-0.16666666666666666d0) * (im * im)) + 1.0d0)) * ((re * ((re * (0.5d0 + (re * 0.16666666666666666d0))) + 1.0d0)) + 1.0d0)
    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 <= -1.8e-8) {
		tmp = t_0;
	} else if (re <= 1.2e-20) {
		tmp = Math.sin(im);
	} else if (re <= 8.4e+89) {
		tmp = t_0;
	} else {
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -1.8e-8:
		tmp = t_0
	elif re <= 1.2e-20:
		tmp = math.sin(im)
	elif re <= 8.4e+89:
		tmp = t_0
	else:
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -1.8e-8)
		tmp = t_0;
	elseif (re <= 1.2e-20)
		tmp = sin(im);
	elseif (re <= 8.4e+89)
		tmp = t_0;
	else
		tmp = Float64(Float64(im * Float64(Float64(-0.16666666666666666 * Float64(im * im)) + 1.0)) * Float64(Float64(re * Float64(Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -1.8e-8)
		tmp = t_0;
	elseif (re <= 1.2e-20)
		tmp = sin(im);
	elseif (re <= 8.4e+89)
		tmp = t_0;
	else
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -1.8e-8], t$95$0, If[LessEqual[re, 1.2e-20], N[Sin[im], $MachinePrecision], If[LessEqual[re, 8.4e+89], t$95$0, N[(N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(re * N[(N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.79999999999999991e-8 or 1.19999999999999996e-20 < re < 8.39999999999999945e89

   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 94.8%

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

   if -1.79999999999999991e-8 < re < 1.19999999999999996e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 99.7%

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

   5. Simplified 99.7%

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

   if 8.39999999999999945e89 < 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 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-8}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{+89}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + re \cdot 0.16666666666666666\\ t_1 := re \cdot t\_0\\ t_2 := im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{if}\;re \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{+89}:\\ \;\;\;\;t\_2 \cdot \left(\frac{re \cdot \left(1 - re \cdot \left(t\_0 \cdot t\_1\right)\right)}{1 - t\_1} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(re \cdot \left(t\_1 + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* re 0.16666666666666666)))
        (t_1 (* re t_0))
        (t_2 (* im (+ (* -0.16666666666666666 (* im im)) 1.0))))
   (if (<= re -8.5e-8)
     (/ (* im im) (+ im (* re (* im (- -1.0 (* re 0.5))))))
     (if (<= re 1.2e-20)
       (sin im)
       (if (<= re 8.4e+89)
         (* t_2 (+ (/ (* re (- 1.0 (* re (* t_0 t_1)))) (- 1.0 t_1)) 1.0))
         (* t_2 (+ (* re (+ t_1 1.0)) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (re * 0.16666666666666666);
	double t_1 = re * t_0;
	double t_2 = im * ((-0.16666666666666666 * (im * im)) + 1.0);
	double tmp;
	if (re <= -8.5e-8) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else if (re <= 1.2e-20) {
		tmp = sin(im);
	} else if (re <= 8.4e+89) {
		tmp = t_2 * (((re * (1.0 - (re * (t_0 * t_1)))) / (1.0 - t_1)) + 1.0);
	} else {
		tmp = t_2 * ((re * (t_1 + 1.0)) + 1.0);
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 + (re * 0.16666666666666666d0)
    t_1 = re * t_0
    t_2 = im * (((-0.16666666666666666d0) * (im * im)) + 1.0d0)
    if (re <= (-8.5d-8)) then
        tmp = (im * im) / (im + (re * (im * ((-1.0d0) - (re * 0.5d0)))))
    else if (re <= 1.2d-20) then
        tmp = sin(im)
    else if (re <= 8.4d+89) then
        tmp = t_2 * (((re * (1.0d0 - (re * (t_0 * t_1)))) / (1.0d0 - t_1)) + 1.0d0)
    else
        tmp = t_2 * ((re * (t_1 + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (re * 0.16666666666666666);
	double t_1 = re * t_0;
	double t_2 = im * ((-0.16666666666666666 * (im * im)) + 1.0);
	double tmp;
	if (re <= -8.5e-8) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else if (re <= 1.2e-20) {
		tmp = Math.sin(im);
	} else if (re <= 8.4e+89) {
		tmp = t_2 * (((re * (1.0 - (re * (t_0 * t_1)))) / (1.0 - t_1)) + 1.0);
	} else {
		tmp = t_2 * ((re * (t_1 + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (re * 0.16666666666666666)
	t_1 = re * t_0
	t_2 = im * ((-0.16666666666666666 * (im * im)) + 1.0)
	tmp = 0
	if re <= -8.5e-8:
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))))
	elif re <= 1.2e-20:
		tmp = math.sin(im)
	elif re <= 8.4e+89:
		tmp = t_2 * (((re * (1.0 - (re * (t_0 * t_1)))) / (1.0 - t_1)) + 1.0)
	else:
		tmp = t_2 * ((re * (t_1 + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(re * 0.16666666666666666))
	t_1 = Float64(re * t_0)
	t_2 = Float64(im * Float64(Float64(-0.16666666666666666 * Float64(im * im)) + 1.0))
	tmp = 0.0
	if (re <= -8.5e-8)
		tmp = Float64(Float64(im * im) / Float64(im + Float64(re * Float64(im * Float64(-1.0 - Float64(re * 0.5))))));
	elseif (re <= 1.2e-20)
		tmp = sin(im);
	elseif (re <= 8.4e+89)
		tmp = Float64(t_2 * Float64(Float64(Float64(re * Float64(1.0 - Float64(re * Float64(t_0 * t_1)))) / Float64(1.0 - t_1)) + 1.0));
	else
		tmp = Float64(t_2 * Float64(Float64(re * Float64(t_1 + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (re * 0.16666666666666666);
	t_1 = re * t_0;
	t_2 = im * ((-0.16666666666666666 * (im * im)) + 1.0);
	tmp = 0.0;
	if (re <= -8.5e-8)
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	elseif (re <= 1.2e-20)
		tmp = sin(im);
	elseif (re <= 8.4e+89)
		tmp = t_2 * (((re * (1.0 - (re * (t_0 * t_1)))) / (1.0 - t_1)) + 1.0);
	else
		tmp = t_2 * ((re * (t_1 + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -8.5e-8], N[(N[(im * im), $MachinePrecision] / N[(im + N[(re * N[(im * N[(-1.0 - N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.2e-20], N[Sin[im], $MachinePrecision], If[LessEqual[re, 8.4e+89], N[(t$95$2 * N[(N[(N[(re * N[(1.0 - N[(re * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(re * N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -8.49999999999999935e-8

   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 99.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 6.1%

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

   8. Simplified 6.1%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. fmm-def N/A

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

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

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

   10. Applied egg-rr 7.6%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {im}^{2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right)}, im\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({im}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right)}, im\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(im \cdot im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)}\right), im\right)\right) \]

       5. *-lowering-*.f64 56.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)}\right), im\right)\right) \]

   13. Simplified 56.3%

       \[\leadsto \frac{\color{blue}{0 - im \cdot im}}{re \cdot \left(im \cdot \left(1 + re \cdot 0.5\right)\right) - im} \]

   if -8.49999999999999935e-8 < re < 1.19999999999999996e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 99.7%

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

   5. Simplified 99.7%

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

   if 1.19999999999999996e-20 < re < 8.39999999999999945e89

   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 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 29.9%

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

   8. Simplified 29.9%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 65.3%

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

   if 8.39999999999999945e89 < 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 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{+89}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(\frac{re \cdot \left(1 - re \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)}{1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\\ t_1 := 0.5 + re \cdot 0.16666666666666666\\ t_2 := re \cdot t\_1\\ \mathbf{if}\;re \leq -3.9:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{+89}:\\ \;\;\;\;t\_0 \cdot \left(\frac{re \cdot \left(1 - re \cdot \left(t\_1 \cdot t\_2\right)\right)}{1 - t\_2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \left(t\_2 + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ (* -0.16666666666666666 (* im im)) 1.0)))
        (t_1 (+ 0.5 (* re 0.16666666666666666)))
        (t_2 (* re t_1)))
   (if (<= re -3.9)
     (/ (* im im) (+ im (* re (* im (- -1.0 (* re 0.5))))))
     (if (<= re 8.4e+89)
       (* t_0 (+ (/ (* re (- 1.0 (* re (* t_1 t_2)))) (- 1.0 t_2)) 1.0))
       (* t_0 (+ (* re (+ t_2 1.0)) 1.0))))))

C

double code(double re, double im) {
	double t_0 = im * ((-0.16666666666666666 * (im * im)) + 1.0);
	double t_1 = 0.5 + (re * 0.16666666666666666);
	double t_2 = re * t_1;
	double tmp;
	if (re <= -3.9) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else if (re <= 8.4e+89) {
		tmp = t_0 * (((re * (1.0 - (re * (t_1 * t_2)))) / (1.0 - t_2)) + 1.0);
	} else {
		tmp = t_0 * ((re * (t_2 + 1.0)) + 1.0);
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = im * (((-0.16666666666666666d0) * (im * im)) + 1.0d0)
    t_1 = 0.5d0 + (re * 0.16666666666666666d0)
    t_2 = re * t_1
    if (re <= (-3.9d0)) then
        tmp = (im * im) / (im + (re * (im * ((-1.0d0) - (re * 0.5d0)))))
    else if (re <= 8.4d+89) then
        tmp = t_0 * (((re * (1.0d0 - (re * (t_1 * t_2)))) / (1.0d0 - t_2)) + 1.0d0)
    else
        tmp = t_0 * ((re * (t_2 + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * ((-0.16666666666666666 * (im * im)) + 1.0);
	double t_1 = 0.5 + (re * 0.16666666666666666);
	double t_2 = re * t_1;
	double tmp;
	if (re <= -3.9) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else if (re <= 8.4e+89) {
		tmp = t_0 * (((re * (1.0 - (re * (t_1 * t_2)))) / (1.0 - t_2)) + 1.0);
	} else {
		tmp = t_0 * ((re * (t_2 + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * ((-0.16666666666666666 * (im * im)) + 1.0)
	t_1 = 0.5 + (re * 0.16666666666666666)
	t_2 = re * t_1
	tmp = 0
	if re <= -3.9:
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))))
	elif re <= 8.4e+89:
		tmp = t_0 * (((re * (1.0 - (re * (t_1 * t_2)))) / (1.0 - t_2)) + 1.0)
	else:
		tmp = t_0 * ((re * (t_2 + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(Float64(-0.16666666666666666 * Float64(im * im)) + 1.0))
	t_1 = Float64(0.5 + Float64(re * 0.16666666666666666))
	t_2 = Float64(re * t_1)
	tmp = 0.0
	if (re <= -3.9)
		tmp = Float64(Float64(im * im) / Float64(im + Float64(re * Float64(im * Float64(-1.0 - Float64(re * 0.5))))));
	elseif (re <= 8.4e+89)
		tmp = Float64(t_0 * Float64(Float64(Float64(re * Float64(1.0 - Float64(re * Float64(t_1 * t_2)))) / Float64(1.0 - t_2)) + 1.0));
	else
		tmp = Float64(t_0 * Float64(Float64(re * Float64(t_2 + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * ((-0.16666666666666666 * (im * im)) + 1.0);
	t_1 = 0.5 + (re * 0.16666666666666666);
	t_2 = re * t_1;
	tmp = 0.0;
	if (re <= -3.9)
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	elseif (re <= 8.4e+89)
		tmp = t_0 * (((re * (1.0 - (re * (t_1 * t_2)))) / (1.0 - t_2)) + 1.0);
	else
		tmp = t_0 * ((re * (t_2 + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(re * t$95$1), $MachinePrecision]}, If[LessEqual[re, -3.9], N[(N[(im * im), $MachinePrecision] / N[(im + N[(re * N[(im * N[(-1.0 - N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.4e+89], N[(t$95$0 * N[(N[(N[(re * N[(1.0 - N[(re * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(re * N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.89999999999999991

   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%

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

   6. Taylor expanded in re around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 2.2%

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

   8. Simplified 2.2%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. fmm-def N/A

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

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

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

   10. Applied egg-rr 4.6%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {im}^{2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right)}, im\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({im}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right)}, im\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(im \cdot im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)}\right), im\right)\right) \]

       5. *-lowering-*.f64 56.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)}\right), im\right)\right) \]

   13. Simplified 56.3%

       \[\leadsto \frac{\color{blue}{0 - im \cdot im}}{re \cdot \left(im \cdot \left(1 + re \cdot 0.5\right)\right) - im} \]

   if -3.89999999999999991 < re < 8.39999999999999945e89

   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 51.3%

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

   5. Simplified 51.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 42.9%

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

   8. Simplified 42.9%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 47.4%

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

   if 8.39999999999999945e89 < 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 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.9:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{+89}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(\frac{re \cdot \left(1 - re \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)}{1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.6)
   (/ (* im im) (+ im (* re (* im (- -1.0 (* re 0.5))))))
   (*
    (* im (+ (* -0.16666666666666666 (* im im)) 1.0))
    (+ (* re (+ (* re (+ 0.5 (* re 0.16666666666666666))) 1.0)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.6) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else {
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.6d0)) then
        tmp = (im * im) / (im + (re * (im * ((-1.0d0) - (re * 0.5d0)))))
    else
        tmp = (im * (((-0.16666666666666666d0) * (im * im)) + 1.0d0)) * ((re * ((re * (0.5d0 + (re * 0.16666666666666666d0))) + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.6) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else {
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.6:
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))))
	else:
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.6)
		tmp = Float64(Float64(im * im) / Float64(im + Float64(re * Float64(im * Float64(-1.0 - Float64(re * 0.5))))));
	else
		tmp = Float64(Float64(im * Float64(Float64(-0.16666666666666666 * Float64(im * im)) + 1.0)) * Float64(Float64(re * Float64(Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.6)
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	else
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6], N[(N[(im * im), $MachinePrecision] / N[(im + N[(re * N[(im * N[(-1.0 - N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(re * N[(N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.6000000000000001

   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%

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

   6. Taylor expanded in re around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 2.2%

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

   8. Simplified 2.2%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. fmm-def N/A

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

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

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

   10. Applied egg-rr 4.6%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {im}^{2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right)}, im\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({im}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right)}, im\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(im \cdot im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)}\right), im\right)\right) \]

       5. *-lowering-*.f64 56.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)}\right), im\right)\right) \]

   13. Simplified 56.3%

       \[\leadsto \frac{\color{blue}{0 - im \cdot im}}{re \cdot \left(im \cdot \left(1 + re \cdot 0.5\right)\right) - im} \]

   if -1.6000000000000001 < 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 57.6%

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

   5. Simplified 57.6%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 51.2%

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

   8. Simplified 51.2%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq 270:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.5)
   (/ (* im im) (+ im (* re (* im (- -1.0 (* re 0.5))))))
   (if (<= re 270.0)
     (* im (+ (+ re 1.0) (* (+ 0.5 (* re 0.16666666666666666)) (* re re))))
     (*
      (* re (* re re))
      (*
       im
       (* (+ (* -0.16666666666666666 (* im im)) 1.0) 0.16666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.5) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else if (re <= 270.0) {
		tmp = im * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)));
	} else {
		tmp = (re * (re * re)) * (im * (((-0.16666666666666666 * (im * im)) + 1.0) * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.5d0)) then
        tmp = (im * im) / (im + (re * (im * ((-1.0d0) - (re * 0.5d0)))))
    else if (re <= 270.0d0) then
        tmp = im * ((re + 1.0d0) + ((0.5d0 + (re * 0.16666666666666666d0)) * (re * re)))
    else
        tmp = (re * (re * re)) * (im * ((((-0.16666666666666666d0) * (im * im)) + 1.0d0) * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.5) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else if (re <= 270.0) {
		tmp = im * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)));
	} else {
		tmp = (re * (re * re)) * (im * (((-0.16666666666666666 * (im * im)) + 1.0) * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.5:
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))))
	elif re <= 270.0:
		tmp = im * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)))
	else:
		tmp = (re * (re * re)) * (im * (((-0.16666666666666666 * (im * im)) + 1.0) * 0.16666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.5)
		tmp = Float64(Float64(im * im) / Float64(im + Float64(re * Float64(im * Float64(-1.0 - Float64(re * 0.5))))));
	elseif (re <= 270.0)
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * Float64(re * re))));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(im * Float64(Float64(Float64(-0.16666666666666666 * Float64(im * im)) + 1.0) * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.5)
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	elseif (re <= 270.0)
		tmp = im * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)));
	else
		tmp = (re * (re * re)) * (im * (((-0.16666666666666666 * (im * im)) + 1.0) * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.5], N[(N[(im * im), $MachinePrecision] / N[(im + N[(re * N[(im * N[(-1.0 - N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 270.0], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 2.2%

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

   8. Simplified 2.2%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. fmm-def N/A

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

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

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

   10. Applied egg-rr 4.6%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {im}^{2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right)}, im\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({im}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right)}, im\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(im \cdot im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)}\right), im\right)\right) \]

       5. *-lowering-*.f64 56.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)}\right), im\right)\right) \]

   13. Simplified 56.3%

       \[\leadsto \frac{\color{blue}{0 - im \cdot im}}{re \cdot \left(im \cdot \left(1 + re \cdot 0.5\right)\right) - im} \]

   if -1.5 < re < 270

   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 46.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 45.2%

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

   8. Simplified 45.2%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 45.2%

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

   10. Applied egg-rr 45.2%

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

   if 270 < 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 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 63.2%

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

   8. Simplified 63.2%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \left(im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{{re}^{3}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{6} \cdot \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{{re}^{3}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{6} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{3}} \]

       4. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]

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

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       18. unpow2 N/A

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

       19. *-lowering-*.f64 63.2%

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

   11. Simplified 63.2%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq 270:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -75000000:\\ \;\;\;\;im \cdot t\_0\\ \mathbf{elif}\;re \leq 105:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(\left(t\_0 + 1\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* im im))))
   (if (<= re -75000000.0)
     (* im t_0)
     (if (<= re 105.0)
       (* im (+ (+ re 1.0) (* (+ 0.5 (* re 0.16666666666666666)) (* re re))))
       (* (* re (* re re)) (* im (* (+ t_0 1.0) 0.16666666666666666)))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -75000000.0) {
		tmp = im * t_0;
	} else if (re <= 105.0) {
		tmp = im * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)));
	} else {
		tmp = (re * (re * re)) * (im * ((t_0 + 1.0) * 0.16666666666666666));
	}
	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 <= (-75000000.0d0)) then
        tmp = im * t_0
    else if (re <= 105.0d0) then
        tmp = im * ((re + 1.0d0) + ((0.5d0 + (re * 0.16666666666666666d0)) * (re * re)))
    else
        tmp = (re * (re * re)) * (im * ((t_0 + 1.0d0) * 0.16666666666666666d0))
    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 <= -75000000.0) {
		tmp = im * t_0;
	} else if (re <= 105.0) {
		tmp = im * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)));
	} else {
		tmp = (re * (re * re)) * (im * ((t_0 + 1.0) * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (im * im)
	tmp = 0
	if re <= -75000000.0:
		tmp = im * t_0
	elif re <= 105.0:
		tmp = im * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)))
	else:
		tmp = (re * (re * re)) * (im * ((t_0 + 1.0) * 0.16666666666666666))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * Float64(im * im))
	tmp = 0.0
	if (re <= -75000000.0)
		tmp = Float64(im * t_0);
	elseif (re <= 105.0)
		tmp = Float64(im * Float64(Float64(re + 1.0) + Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * Float64(re * re))));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(im * Float64(Float64(t_0 + 1.0) * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (im * im);
	tmp = 0.0;
	if (re <= -75000000.0)
		tmp = im * t_0;
	elseif (re <= 105.0)
		tmp = im * ((re + 1.0) + ((0.5 + (re * 0.16666666666666666)) * (re * re)));
	else
		tmp = (re * (re * re)) * (im * ((t_0 + 1.0) * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -75000000.0], N[(im * t$95$0), $MachinePrecision], If[LessEqual[re, 105.0], N[(im * N[(N[(re + 1.0), $MachinePrecision] + N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * N[(N[(t$95$0 + 1.0), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.5e7

   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 67.3%

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

   5. Simplified 67.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{6}, re\right)\right)\right)\right)\right)\right), \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 15.6%

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

   11. Simplified 15.6%

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

   12. Taylor expanded in re around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 36.3%

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

   14. Simplified 36.3%

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

   if -7.5e7 < re < 105

   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 46.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   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. *-lowering-*.f64 44.6%

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

   8. Simplified 44.6%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 44.6%

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

   10. Applied egg-rr 44.6%

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

   if 105 < 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 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 63.2%

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

   8. Simplified 63.2%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \left(im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{{re}^{3}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{6} \cdot \left(\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{{re}^{3}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{6} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{3}} \]

       4. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)} \]

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

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       18. unpow2 N/A

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

       19. *-lowering-*.f64 63.2%

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

   11. Simplified 63.2%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -75000000:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 105:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -190:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5 + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -190.0)
   (/ (* im im) (+ im (* re (* im (- -1.0 (* re 0.5))))))
   (*
    (* im (+ (* -0.16666666666666666 (* im im)) 1.0))
    (+ (* re (+ (* re 0.5) 1.0)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -190.0) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else {
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * 0.5) + 1.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-190.0d0)) then
        tmp = (im * im) / (im + (re * (im * ((-1.0d0) - (re * 0.5d0)))))
    else
        tmp = (im * (((-0.16666666666666666d0) * (im * im)) + 1.0d0)) * ((re * ((re * 0.5d0) + 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -190.0) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else {
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * 0.5) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -190.0:
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))))
	else:
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * 0.5) + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -190.0)
		tmp = Float64(Float64(im * im) / Float64(im + Float64(re * Float64(im * Float64(-1.0 - Float64(re * 0.5))))));
	else
		tmp = Float64(Float64(im * Float64(Float64(-0.16666666666666666 * Float64(im * im)) + 1.0)) * Float64(Float64(re * Float64(Float64(re * 0.5) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -190.0)
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	else
		tmp = (im * ((-0.16666666666666666 * (im * im)) + 1.0)) * ((re * ((re * 0.5) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -190.0], N[(N[(im * im), $MachinePrecision] / N[(im + N[(re * N[(im * N[(-1.0 - N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(re * N[(N[(re * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -190

   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%

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

   6. Taylor expanded in re around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 2.2%

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

   8. Simplified 2.2%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. fmm-def N/A

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

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

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

   10. Applied egg-rr 4.6%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {im}^{2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right)}, im\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({im}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right)}, im\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(im \cdot im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)}\right), im\right)\right) \]

       5. *-lowering-*.f64 56.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \color{blue}{\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)}\right), im\right)\right) \]

   13. Simplified 56.3%

       \[\leadsto \frac{\color{blue}{0 - im \cdot im}}{re \cdot \left(im \cdot \left(1 + re \cdot 0.5\right)\right) - im} \]

   if -190 < 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 57.6%

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

   5. Simplified 57.6%

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

   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 49.6%

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

   8. Simplified 49.6%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -190:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5 + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -1:\\ \;\;\;\;im \cdot t\_0\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+191}:\\ \;\;\;\;im \cdot \left(\left(t\_0 + 1\right) \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5 + 1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* im im))))
   (if (<= re -1.0)
     (* im t_0)
     (if (<= re 1.55e+191)
       (* im (* (+ t_0 1.0) (+ re 1.0)))
       (* im (+ (* re (+ (* re 0.5) 1.0)) 1.0))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -1.0) {
		tmp = im * t_0;
	} else if (re <= 1.55e+191) {
		tmp = im * ((t_0 + 1.0) * (re + 1.0));
	} else {
		tmp = im * ((re * ((re * 0.5) + 1.0)) + 1.0);
	}
	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 <= (-1.0d0)) then
        tmp = im * t_0
    else if (re <= 1.55d+191) then
        tmp = im * ((t_0 + 1.0d0) * (re + 1.0d0))
    else
        tmp = im * ((re * ((re * 0.5d0) + 1.0d0)) + 1.0d0)
    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 <= -1.0) {
		tmp = im * t_0;
	} else if (re <= 1.55e+191) {
		tmp = im * ((t_0 + 1.0) * (re + 1.0));
	} else {
		tmp = im * ((re * ((re * 0.5) + 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (im * im)
	tmp = 0
	if re <= -1.0:
		tmp = im * t_0
	elif re <= 1.55e+191:
		tmp = im * ((t_0 + 1.0) * (re + 1.0))
	else:
		tmp = im * ((re * ((re * 0.5) + 1.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * Float64(im * im))
	tmp = 0.0
	if (re <= -1.0)
		tmp = Float64(im * t_0);
	elseif (re <= 1.55e+191)
		tmp = Float64(im * Float64(Float64(t_0 + 1.0) * Float64(re + 1.0)));
	else
		tmp = Float64(im * Float64(Float64(re * Float64(Float64(re * 0.5) + 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (im * im);
	tmp = 0.0;
	if (re <= -1.0)
		tmp = im * t_0;
	elseif (re <= 1.55e+191)
		tmp = im * ((t_0 + 1.0) * (re + 1.0));
	else
		tmp = im * ((re * ((re * 0.5) + 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.0], N[(im * t$95$0), $MachinePrecision], If[LessEqual[re, 1.55e+191], N[(im * N[(N[(t$95$0 + 1.0), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * N[(N[(re * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   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 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{6}, re\right)\right)\right)\right)\right)\right), \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 15.1%

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

   11. Simplified 15.1%

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

   12. Taylor expanded in re around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 35.1%

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

   14. Simplified 35.1%

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

   if -1 < re < 1.54999999999999999e191

   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 54.8%

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

   5. Simplified 54.8%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft-in N/A

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt1-in N/A

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

   8. Simplified 42.8%

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

   if 1.54999999999999999e191 < 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 83.3%

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

   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 83.3%

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

   8. Simplified 83.3%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+191}:\\ \;\;\;\;im \cdot \left(\left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right) \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5 + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.9% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;re \leq -75000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 40000000000:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+191}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* -0.16666666666666666 (* im im)))))
   (if (<= re -75000000.0)
     t_0
     (if (<= re 40000000000.0)
       (* im (+ re 1.0))
       (if (<= re 1.55e+191) t_0 (* im (* 0.5 (* re re))))))))

C

double code(double re, double im) {
	double t_0 = im * (-0.16666666666666666 * (im * im));
	double tmp;
	if (re <= -75000000.0) {
		tmp = t_0;
	} else if (re <= 40000000000.0) {
		tmp = im * (re + 1.0);
	} else if (re <= 1.55e+191) {
		tmp = t_0;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * ((-0.16666666666666666d0) * (im * im))
    if (re <= (-75000000.0d0)) then
        tmp = t_0
    else if (re <= 40000000000.0d0) then
        tmp = im * (re + 1.0d0)
    else if (re <= 1.55d+191) then
        tmp = t_0
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (-0.16666666666666666 * (im * im));
	double tmp;
	if (re <= -75000000.0) {
		tmp = t_0;
	} else if (re <= 40000000000.0) {
		tmp = im * (re + 1.0);
	} else if (re <= 1.55e+191) {
		tmp = t_0;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (-0.16666666666666666 * (im * im))
	tmp = 0
	if re <= -75000000.0:
		tmp = t_0
	elif re <= 40000000000.0:
		tmp = im * (re + 1.0)
	elif re <= 1.55e+191:
		tmp = t_0
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(-0.16666666666666666 * Float64(im * im)))
	tmp = 0.0
	if (re <= -75000000.0)
		tmp = t_0;
	elseif (re <= 40000000000.0)
		tmp = Float64(im * Float64(re + 1.0));
	elseif (re <= 1.55e+191)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (-0.16666666666666666 * (im * im));
	tmp = 0.0;
	if (re <= -75000000.0)
		tmp = t_0;
	elseif (re <= 40000000000.0)
		tmp = im * (re + 1.0);
	elseif (re <= 1.55e+191)
		tmp = t_0;
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -75000000.0], t$95$0, If[LessEqual[re, 40000000000.0], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.55e+191], t$95$0, N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.5e7 or 4e10 < re < 1.54999999999999999e191

   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

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 22.1%

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

   8. Simplified 22.1%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{6}, re\right)\right)\right)\right)\right)\right), \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 27.0%

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

   11. Simplified 27.0%

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

   12. Taylor expanded in re around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 34.5%

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

   14. Simplified 34.5%

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

   if -7.5e7 < re < 4e10

   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 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 43.9%

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

   if 1.54999999999999999e191 < 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 83.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 52.3%

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

   8. Simplified 52.3%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 83.3%

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

   11. Simplified 83.3%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -75000000:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 40000000000:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+191}:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -59000:\\ \;\;\;\;im \cdot t\_0\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(t\_0 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* im im))))
   (if (<= re -59000.0)
     (* im t_0)
     (if (<= re 2.15e+105)
       (* im (+ t_0 1.0))
       (* im (* re (* re (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -59000.0) {
		tmp = im * t_0;
	} else if (re <= 2.15e+105) {
		tmp = im * (t_0 + 1.0);
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	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 <= (-59000.0d0)) then
        tmp = im * t_0
    else if (re <= 2.15d+105) then
        tmp = im * (t_0 + 1.0d0)
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    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 <= -59000.0) {
		tmp = im * t_0;
	} else if (re <= 2.15e+105) {
		tmp = im * (t_0 + 1.0);
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (im * im)
	tmp = 0
	if re <= -59000.0:
		tmp = im * t_0
	elif re <= 2.15e+105:
		tmp = im * (t_0 + 1.0)
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * Float64(im * im))
	tmp = 0.0
	if (re <= -59000.0)
		tmp = Float64(im * t_0);
	elseif (re <= 2.15e+105)
		tmp = Float64(im * Float64(t_0 + 1.0));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (im * im);
	tmp = 0.0;
	if (re <= -59000.0)
		tmp = im * t_0;
	elseif (re <= 2.15e+105)
		tmp = im * (t_0 + 1.0);
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -59000.0], N[(im * t$95$0), $MachinePrecision], If[LessEqual[re, 2.15e+105], N[(im * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -59000

   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 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{6}, re\right)\right)\right)\right)\right)\right), \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 15.4%

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

   11. Simplified 15.4%

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

   12. Taylor expanded in re around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 35.6%

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

   14. Simplified 35.6%

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

   if -59000 < re < 2.1500000000000001e105

   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 85.4%

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

   5. Simplified 85.4%

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

   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 42.0%

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

   8. Simplified 42.0%

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

   if 2.1500000000000001e105 < 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 68.9%

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

   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. *-lowering-*.f64 68.9%

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

   8. Simplified 68.9%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

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

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 68.9%

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

   11. Simplified 68.9%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -59000:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -59000:\\ \;\;\;\;im \cdot t\_0\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+191}:\\ \;\;\;\;im \cdot \left(t\_0 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* im im))))
   (if (<= re -59000.0)
     (* im t_0)
     (if (<= re 1.55e+191) (* im (+ t_0 1.0)) (* im (* 0.5 (* re re)))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -59000.0) {
		tmp = im * t_0;
	} else if (re <= 1.55e+191) {
		tmp = im * (t_0 + 1.0);
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.16666666666666666d0) * (im * im)
    if (re <= (-59000.0d0)) then
        tmp = im * t_0
    else if (re <= 1.55d+191) then
        tmp = im * (t_0 + 1.0d0)
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -59000.0) {
		tmp = im * t_0;
	} else if (re <= 1.55e+191) {
		tmp = im * (t_0 + 1.0);
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (im * im)
	tmp = 0
	if re <= -59000.0:
		tmp = im * t_0
	elif re <= 1.55e+191:
		tmp = im * (t_0 + 1.0)
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * Float64(im * im))
	tmp = 0.0
	if (re <= -59000.0)
		tmp = Float64(im * t_0);
	elseif (re <= 1.55e+191)
		tmp = Float64(im * Float64(t_0 + 1.0));
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (im * im);
	tmp = 0.0;
	if (re <= -59000.0)
		tmp = im * t_0;
	elseif (re <= 1.55e+191)
		tmp = im * (t_0 + 1.0);
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -59000.0], N[(im * t$95$0), $MachinePrecision], If[LessEqual[re, 1.55e+191], N[(im * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -59000

   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 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{6}, re\right)\right)\right)\right)\right)\right), \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 15.4%

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

   11. Simplified 15.4%

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

   12. Taylor expanded in re around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 35.6%

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

   14. Simplified 35.6%

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

   if -59000 < re < 1.54999999999999999e191

   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 78.1%

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

   5. Simplified 78.1%

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

   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 41.4%

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

   8. Simplified 41.4%

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

   if 1.54999999999999999e191 < 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 83.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 52.3%

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

   8. Simplified 52.3%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 83.3%

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

   11. Simplified 83.3%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -59000:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+191}:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.7% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -75000000:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -75000000.0)
   (* im (* -0.16666666666666666 (* im im)))
   (* im (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -75000000.0) {
		tmp = im * (-0.16666666666666666 * (im * im));
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-75000000.0d0)) then
        tmp = im * ((-0.16666666666666666d0) * (im * im))
    else
        tmp = im * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -75000000.0) {
		tmp = im * (-0.16666666666666666 * (im * im));
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -75000000.0:
		tmp = im * (-0.16666666666666666 * (im * im))
	else:
		tmp = im * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -75000000.0)
		tmp = Float64(im * Float64(-0.16666666666666666 * Float64(im * im)));
	else
		tmp = Float64(im * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -75000000.0)
		tmp = im * (-0.16666666666666666 * (im * im));
	else
		tmp = im * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -75000000.0], N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -7.5e7

   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 67.3%

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

   5. Simplified 67.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{6}, re\right)\right)\right)\right)\right)\right), \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 15.6%

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

   11. Simplified 15.6%

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

   12. Taylor expanded in re around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 36.3%

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

   14. Simplified 36.3%

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

   if -7.5e7 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   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 67.8%

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

   5. Simplified 67.8%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 35.7%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -75000000:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 28.5% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5 \cdot 10^{+44}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 5e+44) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5e+44) {
		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 <= 5d+44) 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 <= 5e+44) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5e+44:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5e+44)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5e+44)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5e+44], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.9999999999999996e44

   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 53.0%

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

   5. Simplified 53.0%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 30.0%

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

   if 4.9999999999999996e44 < im

   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 55.1%

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

   5. Simplified 55.1%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 11.8%

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

   9. Taylor expanded in re around inf

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

   11. Simplified 13.3%

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

Alternative 17: 30.1% 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 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 53.8%

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

5. Simplified 53.8%

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

6. Taylor expanded in im around 0

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

8. Simplified 28.6%

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

9. Final simplification 28.6%

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

Alternative 18: 26.8% 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 re around 0

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

   1. sin-lowering-sin.f64 53.1%

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

5. Simplified 53.1%

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

6. Taylor expanded in im around 0

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

8. Simplified 24.7%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))