math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 10.3s

Alternatives: 21

Speedup: 1.0×

• 24.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;e^{re} \leq 0.1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{re} \leq 20:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= (exp re) 0.1) t_0 (if (<= (exp re) 20.0) (sin im) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (exp(re) <= 0.1) {
		tmp = t_0;
	} else if (exp(re) <= 20.0) {
		tmp = sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (exp(re) <= 0.1d0) then
        tmp = t_0
    else if (exp(re) <= 20.0d0) then
        tmp = sin(im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (Math.exp(re) <= 0.1) {
		tmp = t_0;
	} else if (Math.exp(re) <= 20.0) {
		tmp = Math.sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if math.exp(re) <= 0.1:
		tmp = t_0
	elif math.exp(re) <= 20.0:
		tmp = math.sin(im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (exp(re) <= 0.1)
		tmp = t_0;
	elseif (exp(re) <= 20.0)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (exp(re) <= 0.1)
		tmp = t_0;
	elseif (exp(re) <= 20.0)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[N[Exp[re], $MachinePrecision], 0.1], t$95$0, If[LessEqual[N[Exp[re], $MachinePrecision], 20.0], N[Sin[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.10000000000000001 or 20 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 90.6%

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

   if 0.10000000000000001 < (exp.f64 re) < 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 98.0%

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

   5. Simplified 98.0%

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

Alternative 3: 97.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -0.175:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2700000:\\ \;\;\;\;\sin im \cdot \left(1 + t\_0\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
   (if (<= re -0.175)
     (* (exp re) im)
     (if (<= re 2700000.0)
       (* (sin im) (+ 1.0 t_0))
       (if (<= re 1e+103)
         (* (exp re) (* im (+ 1.0 (* im (* im -0.16666666666666666)))))
         (* (sin im) t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	double tmp;
	if (re <= -0.175) {
		tmp = exp(re) * im;
	} else if (re <= 2700000.0) {
		tmp = sin(im) * (1.0 + t_0);
	} else if (re <= 1e+103) {
		tmp = exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	} else {
		tmp = sin(im) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))
    if (re <= (-0.175d0)) then
        tmp = exp(re) * im
    else if (re <= 2700000.0d0) then
        tmp = sin(im) * (1.0d0 + t_0)
    else if (re <= 1d+103) then
        tmp = exp(re) * (im * (1.0d0 + (im * (im * (-0.16666666666666666d0)))))
    else
        tmp = sin(im) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	double tmp;
	if (re <= -0.175) {
		tmp = Math.exp(re) * im;
	} else if (re <= 2700000.0) {
		tmp = Math.sin(im) * (1.0 + t_0);
	} else if (re <= 1e+103) {
		tmp = Math.exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	} else {
		tmp = Math.sin(im) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))
	tmp = 0
	if re <= -0.175:
		tmp = math.exp(re) * im
	elif re <= 2700000.0:
		tmp = math.sin(im) * (1.0 + t_0)
	elif re <= 1e+103:
		tmp = math.exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))))
	else:
		tmp = math.sin(im) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))
	tmp = 0.0
	if (re <= -0.175)
		tmp = Float64(exp(re) * im);
	elseif (re <= 2700000.0)
		tmp = Float64(sin(im) * Float64(1.0 + t_0));
	elseif (re <= 1e+103)
		tmp = Float64(exp(re) * Float64(im * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666)))));
	else
		tmp = Float64(sin(im) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	tmp = 0.0;
	if (re <= -0.175)
		tmp = exp(re) * im;
	elseif (re <= 2700000.0)
		tmp = sin(im) * (1.0 + t_0);
	elseif (re <= 1e+103)
		tmp = exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	else
		tmp = sin(im) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.175], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 2700000.0], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.17499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   if -0.17499999999999999 < re < 2.7e6

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 98.6%

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

   5. Simplified 98.6%

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

   if 2.7e6 < re < 1e103

   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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 89.5%

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

   5. Simplified 89.5%

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

   if 1e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. distribute-lft-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

   8. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.175:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2700000:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0095:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2700000:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0095)
   (* (exp re) im)
   (if (<= re 2700000.0)
     (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (if (<= re 1e+103)
       (* (exp re) (* im (+ 1.0 (* im (* im -0.16666666666666666)))))
       (*
        (sin im)
        (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0095) {
		tmp = exp(re) * im;
	} else if (re <= 2700000.0) {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 1e+103) {
		tmp = exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	} else {
		tmp = sin(im) * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0095d0)) then
        tmp = exp(re) * im
    else if (re <= 2700000.0d0) then
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else if (re <= 1d+103) then
        tmp = exp(re) * (im * (1.0d0 + (im * (im * (-0.16666666666666666d0)))))
    else
        tmp = sin(im) * (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0095) {
		tmp = Math.exp(re) * im;
	} else if (re <= 2700000.0) {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 1e+103) {
		tmp = Math.exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	} else {
		tmp = Math.sin(im) * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0095:
		tmp = math.exp(re) * im
	elif re <= 2700000.0:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	elif re <= 1e+103:
		tmp = math.exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))))
	else:
		tmp = math.sin(im) * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0095)
		tmp = Float64(exp(re) * im);
	elseif (re <= 2700000.0)
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	elseif (re <= 1e+103)
		tmp = Float64(exp(re) * Float64(im * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666)))));
	else
		tmp = Float64(sin(im) * Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0095)
		tmp = exp(re) * im;
	elseif (re <= 2700000.0)
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	elseif (re <= 1e+103)
		tmp = exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	else
		tmp = sin(im) * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0095], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 2700000.0], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.00949999999999999976

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   if -0.00949999999999999976 < re < 2.7e6

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 98.6%

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

   5. Simplified 98.6%

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

   if 2.7e6 < re < 1e103

   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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 89.5%

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

   5. Simplified 89.5%

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

   if 1e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. distribute-lft-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

   8. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0095:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2700000:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.000112:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 2.8:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(0.16666666666666666 \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.000112)
     t_0
     (if (<= re 2.8)
       (* (sin im) (+ re 1.0))
       (if (<= re 7e+94)
         t_0
         (if (<= re 1.9e+154)
           (*
            (* re (* re re))
            (*
             im
             (*
              0.16666666666666666
              (+ 1.0 (* im (* im -0.16666666666666666))))))
           (* (sin im) (* re (* re 0.5)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.000112) {
		tmp = t_0;
	} else if (re <= 2.8) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 7e+94) {
		tmp = t_0;
	} else if (re <= 1.9e+154) {
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))));
	} else {
		tmp = sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.000112d0)) then
        tmp = t_0
    else if (re <= 2.8d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 7d+94) then
        tmp = t_0
    else if (re <= 1.9d+154) then
        tmp = (re * (re * re)) * (im * (0.16666666666666666d0 * (1.0d0 + (im * (im * (-0.16666666666666666d0))))))
    else
        tmp = sin(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.000112) {
		tmp = t_0;
	} else if (re <= 2.8) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 7e+94) {
		tmp = t_0;
	} else if (re <= 1.9e+154) {
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))));
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.000112:
		tmp = t_0
	elif re <= 2.8:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 7e+94:
		tmp = t_0
	elif re <= 1.9e+154:
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))))
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.000112)
		tmp = t_0;
	elseif (re <= 2.8)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 7e+94)
		tmp = t_0;
	elseif (re <= 1.9e+154)
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(im * Float64(0.16666666666666666 * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666))))));
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.000112)
		tmp = t_0;
	elseif (re <= 2.8)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 7e+94)
		tmp = t_0;
	elseif (re <= 1.9e+154)
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))));
	else
		tmp = sin(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.000112], t$95$0, If[LessEqual[re, 2.8], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7e+94], t$95$0, If[LessEqual[re, 1.9e+154], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * N[(0.16666666666666666 * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.11999999999999998e-4 or 2.7999999999999998 < re < 6.9999999999999994e94

   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 97.7%

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

   if -1.11999999999999998e-4 < re < 2.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 99.2%

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

   5. Simplified 99.2%

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

   if 6.9999999999999994e94 < re < 1.8999999999999999e154

   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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 91.7%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 75.7%

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

   8. Simplified 75.7%

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

   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. *-commutative N/A

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

       18. unpow2 N/A

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

       19. associate-*l* N/A

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

       20. *-commutative N/A

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

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

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

       22. *-commutative N/A

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

       23. *-lowering-*.f64 83.7%

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

   11. Simplified 83.7%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.000112:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2.8:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+94}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(0.16666666666666666 \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0072:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2700000:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0072)
   (* (exp re) im)
   (if (<= re 2700000.0)
     (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (if (<= re 1.9e+154)
       (* (exp re) (* im (+ 1.0 (* im (* im -0.16666666666666666)))))
       (* (sin im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0072) {
		tmp = exp(re) * im;
	} else if (re <= 2700000.0) {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.9e+154) {
		tmp = exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	} else {
		tmp = sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0072d0)) then
        tmp = exp(re) * im
    else if (re <= 2700000.0d0) then
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else if (re <= 1.9d+154) then
        tmp = exp(re) * (im * (1.0d0 + (im * (im * (-0.16666666666666666d0)))))
    else
        tmp = sin(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0072) {
		tmp = Math.exp(re) * im;
	} else if (re <= 2700000.0) {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0072:
		tmp = math.exp(re) * im
	elif re <= 2700000.0:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	elif re <= 1.9e+154:
		tmp = math.exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))))
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0072)
		tmp = Float64(exp(re) * im);
	elseif (re <= 2700000.0)
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	elseif (re <= 1.9e+154)
		tmp = Float64(exp(re) * Float64(im * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666)))));
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0072)
		tmp = exp(re) * im;
	elseif (re <= 2700000.0)
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	elseif (re <= 1.9e+154)
		tmp = exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	else
		tmp = sin(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0072], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 2700000.0], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.0071999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   if -0.0071999999999999998 < re < 2.7e6

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 98.6%

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

   5. Simplified 98.6%

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

   if 2.7e6 < re < 1.8999999999999999e154

   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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 88.9%

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

   5. Simplified 88.9%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0072:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2700000:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0085:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2700000:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0085)
   (* (exp re) im)
   (if (<= re 2700000.0)
     (* (sin im) (+ re 1.0))
     (if (<= re 1.9e+154)
       (* (exp re) (* im (+ 1.0 (* im (* im -0.16666666666666666)))))
       (* (sin im) (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0085) {
		tmp = exp(re) * im;
	} else if (re <= 2700000.0) {
		tmp = sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	} else {
		tmp = sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0085d0)) then
        tmp = exp(re) * im
    else if (re <= 2700000.0d0) then
        tmp = sin(im) * (re + 1.0d0)
    else if (re <= 1.9d+154) then
        tmp = exp(re) * (im * (1.0d0 + (im * (im * (-0.16666666666666666d0)))))
    else
        tmp = sin(im) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0085) {
		tmp = Math.exp(re) * im;
	} else if (re <= 2700000.0) {
		tmp = Math.sin(im) * (re + 1.0);
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	} else {
		tmp = Math.sin(im) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0085:
		tmp = math.exp(re) * im
	elif re <= 2700000.0:
		tmp = math.sin(im) * (re + 1.0)
	elif re <= 1.9e+154:
		tmp = math.exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))))
	else:
		tmp = math.sin(im) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0085)
		tmp = Float64(exp(re) * im);
	elseif (re <= 2700000.0)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = Float64(exp(re) * Float64(im * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666)))));
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0085)
		tmp = exp(re) * im;
	elseif (re <= 2700000.0)
		tmp = sin(im) * (re + 1.0);
	elseif (re <= 1.9e+154)
		tmp = exp(re) * (im * (1.0 + (im * (im * -0.16666666666666666))));
	else
		tmp = sin(im) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0085], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 2700000.0], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.0085000000000000006

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   if -0.0085000000000000006 < re < 2.7e6

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 98.4%

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

   5. Simplified 98.4%

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

   if 2.7e6 < re < 1.8999999999999999e154

   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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 88.9%

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

   5. Simplified 88.9%

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

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0085:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2700000:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.000285:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 2.8:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.000285) t_0 (if (<= re 2.8) (* (sin im) (+ re 1.0)) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.000285) {
		tmp = t_0;
	} else if (re <= 2.8) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * im
    if (re <= (-0.000285d0)) then
        tmp = t_0
    else if (re <= 2.8d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double tmp;
	if (re <= -0.000285) {
		tmp = t_0;
	} else if (re <= 2.8) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	tmp = 0
	if re <= -0.000285:
		tmp = t_0
	elif re <= 2.8:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -0.000285)
		tmp = t_0;
	elseif (re <= 2.8)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	tmp = 0.0;
	if (re <= -0.000285)
		tmp = t_0;
	elseif (re <= 2.8)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -0.000285], t$95$0, If[LessEqual[re, 2.8], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -2.8499999999999999e-4 or 2.7999999999999998 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 90.6%

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

   if -2.8499999999999999e-4 < re < 2.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 99.2%

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

   5. Simplified 99.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.000285:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 2.8:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -120000000000:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2700000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(0.16666666666666666 \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -120000000000.0)
   (* im (* (+ re 1.0) (* -0.16666666666666666 (* im im))))
   (if (<= re 2700000.0)
     (sin im)
     (*
      (* re (* re re))
      (*
       im
       (* 0.16666666666666666 (+ 1.0 (* im (* im -0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -120000000000.0) {
		tmp = im * ((re + 1.0) * (-0.16666666666666666 * (im * im)));
	} else if (re <= 2700000.0) {
		tmp = sin(im);
	} else {
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-120000000000.0d0)) then
        tmp = im * ((re + 1.0d0) * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 2700000.0d0) then
        tmp = sin(im)
    else
        tmp = (re * (re * re)) * (im * (0.16666666666666666d0 * (1.0d0 + (im * (im * (-0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -120000000000.0) {
		tmp = im * ((re + 1.0) * (-0.16666666666666666 * (im * im)));
	} else if (re <= 2700000.0) {
		tmp = Math.sin(im);
	} else {
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -120000000000.0:
		tmp = im * ((re + 1.0) * (-0.16666666666666666 * (im * im)))
	elif re <= 2700000.0:
		tmp = math.sin(im)
	else:
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -120000000000.0)
		tmp = Float64(im * Float64(Float64(re + 1.0) * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 2700000.0)
		tmp = sin(im);
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(im * Float64(0.16666666666666666 * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -120000000000.0)
		tmp = im * ((re + 1.0) * (-0.16666666666666666 * (im * im)));
	elseif (re <= 2700000.0)
		tmp = sin(im);
	else
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -120000000000.0], N[(im * N[(N[(re + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2700000.0], N[Sin[im], $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * N[(0.16666666666666666 * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e11

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in im around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

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

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

      16. +-commutative N/A

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

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

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

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

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 2.5%

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 30.8%

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

   11. Simplified 30.8%

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

   if -1.2e11 < re < 2.7e6

   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 95.2%

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

   5. Simplified 95.2%

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

   if 2.7e6 < 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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 85.5%

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

   5. Simplified 85.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 62.5%

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

   8. Simplified 62.5%

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

   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. *-commutative N/A

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

       18. unpow2 N/A

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

       19. associate-*l* N/A

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

       20. *-commutative N/A

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

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

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

       22. *-commutative N/A

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

       23. *-lowering-*.f64 64.3%

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

   11. Simplified 64.3%

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

Alternative 10: 46.3% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6], N[(im * N[(N[(re + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in im around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

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

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

      16. +-commutative N/A

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

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

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

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

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 2.5%

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 29.6%

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

   11. Simplified 29.6%

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

   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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 58.4%

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

   5. Simplified 58.4%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 51.6%

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

   8. Simplified 51.6%

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

Alternative 11: 46.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -120000000000:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 3400000:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(0.16666666666666666 \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -120000000000.0)
   (* im (* (+ re 1.0) (* -0.16666666666666666 (* im im))))
   (if (<= re 3400000.0)
     (* im (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (*
      (* re (* re re))
      (*
       im
       (* 0.16666666666666666 (+ 1.0 (* im (* im -0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -120000000000.0) {
		tmp = im * ((re + 1.0) * (-0.16666666666666666 * (im * im)));
	} else if (re <= 3400000.0) {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-120000000000.0d0)) then
        tmp = im * ((re + 1.0d0) * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 3400000.0d0) then
        tmp = im * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else
        tmp = (re * (re * re)) * (im * (0.16666666666666666d0 * (1.0d0 + (im * (im * (-0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -120000000000.0) {
		tmp = im * ((re + 1.0) * (-0.16666666666666666 * (im * im)));
	} else if (re <= 3400000.0) {
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -120000000000.0:
		tmp = im * ((re + 1.0) * (-0.16666666666666666 * (im * im)))
	elif re <= 3400000.0:
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))))
	else:
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -120000000000.0)
		tmp = Float64(im * Float64(Float64(re + 1.0) * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 3400000.0)
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(im * Float64(0.16666666666666666 * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -120000000000.0)
		tmp = im * ((re + 1.0) * (-0.16666666666666666 * (im * im)));
	elseif (re <= 3400000.0)
		tmp = im * (1.0 + (re * (1.0 + (re * 0.5))));
	else
		tmp = (re * (re * re)) * (im * (0.16666666666666666 * (1.0 + (im * (im * -0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -120000000000.0], N[(im * N[(N[(re + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3400000.0], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * N[(0.16666666666666666 * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e11

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in im around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

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

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

      16. +-commutative N/A

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

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

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

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

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 2.5%

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 30.8%

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

   11. Simplified 30.8%

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

   if -1.2e11 < re < 3.4e6

   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 49.3%

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

   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 46.4%

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

   8. Simplified 46.4%

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

   if 3.4e6 < 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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 85.5%

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

   5. Simplified 85.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 62.5%

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

   8. Simplified 62.5%

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

   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. *-commutative N/A

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

       18. unpow2 N/A

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

       19. associate-*l* N/A

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

       20. *-commutative N/A

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

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

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

       22. *-commutative N/A

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

       23. *-lowering-*.f64 64.3%

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

   11. Simplified 64.3%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -120000000000:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 3400000:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot \left(0.16666666666666666 \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -1:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot t\_0\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+122}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(1 + t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* im im))))
   (if (<= re -1.0)
     (* im (* (+ re 1.0) t_0))
     (if (<= re 1.4e+122)
       (* im (* (+ re 1.0) (+ 1.0 t_0)))
       (* im (* (+ 0.5 (* re 0.16666666666666666)) (* re re)))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -1.0) {
		tmp = im * ((re + 1.0) * t_0);
	} else if (re <= 1.4e+122) {
		tmp = im * ((re + 1.0) * (1.0 + t_0));
	} else {
		tmp = im * ((0.5 + (re * 0.16666666666666666)) * (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 <= (-1.0d0)) then
        tmp = im * ((re + 1.0d0) * t_0)
    else if (re <= 1.4d+122) then
        tmp = im * ((re + 1.0d0) * (1.0d0 + t_0))
    else
        tmp = im * ((0.5d0 + (re * 0.16666666666666666d0)) * (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 <= -1.0) {
		tmp = im * ((re + 1.0) * t_0);
	} else if (re <= 1.4e+122) {
		tmp = im * ((re + 1.0) * (1.0 + t_0));
	} else {
		tmp = im * ((0.5 + (re * 0.16666666666666666)) * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (im * im)
	tmp = 0
	if re <= -1.0:
		tmp = im * ((re + 1.0) * t_0)
	elif re <= 1.4e+122:
		tmp = im * ((re + 1.0) * (1.0 + t_0))
	else:
		tmp = im * ((0.5 + (re * 0.16666666666666666)) * (re * re))
	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 * Float64(Float64(re + 1.0) * t_0));
	elseif (re <= 1.4e+122)
		tmp = Float64(im * Float64(Float64(re + 1.0) * Float64(1.0 + t_0)));
	else
		tmp = Float64(im * Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * 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 <= -1.0)
		tmp = im * ((re + 1.0) * t_0);
	elseif (re <= 1.4e+122)
		tmp = im * ((re + 1.0) * (1.0 + t_0));
	else
		tmp = im * ((0.5 + (re * 0.16666666666666666)) * (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, -1.0], N[(im * N[(N[(re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e+122], N[(im * N[(N[(re + 1.0), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $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 re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in im around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

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

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

      16. +-commutative N/A

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

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

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

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

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 2.5%

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 29.6%

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

   11. Simplified 29.6%

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

   if -1 < re < 1.4e122

   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 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in im around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

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

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

      16. +-commutative N/A

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

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

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

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

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 42.8%

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

   8. Simplified 42.8%

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

   if 1.4e122 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 87.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 87.5%

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

   8. Simplified 87.5%

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

   9. Taylor expanded in re around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. lft-mult-inverse N/A

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

       10. *-lft-identity N/A

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

       11. unpow2 N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. *-commutative N/A

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

       15. distribute-lft-in N/A

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

       16. +-commutative N/A

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

       17. associate-*r* N/A

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

       18. unpow2 N/A

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

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

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

   11. Simplified 87.5%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+122}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -120000000000:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot t\_0\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+122}:\\ \;\;\;\;im \cdot \left(1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* im im))))
   (if (<= re -120000000000.0)
     (* im (* (+ re 1.0) t_0))
     (if (<= re 1.4e+122)
       (* im (+ 1.0 t_0))
       (* im (* (+ 0.5 (* re 0.16666666666666666)) (* re re)))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -120000000000.0) {
		tmp = im * ((re + 1.0) * t_0);
	} else if (re <= 1.4e+122) {
		tmp = im * (1.0 + t_0);
	} else {
		tmp = im * ((0.5 + (re * 0.16666666666666666)) * (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 <= (-120000000000.0d0)) then
        tmp = im * ((re + 1.0d0) * t_0)
    else if (re <= 1.4d+122) then
        tmp = im * (1.0d0 + t_0)
    else
        tmp = im * ((0.5d0 + (re * 0.16666666666666666d0)) * (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 <= -120000000000.0) {
		tmp = im * ((re + 1.0) * t_0);
	} else if (re <= 1.4e+122) {
		tmp = im * (1.0 + t_0);
	} else {
		tmp = im * ((0.5 + (re * 0.16666666666666666)) * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (im * im)
	tmp = 0
	if re <= -120000000000.0:
		tmp = im * ((re + 1.0) * t_0)
	elif re <= 1.4e+122:
		tmp = im * (1.0 + t_0)
	else:
		tmp = im * ((0.5 + (re * 0.16666666666666666)) * (re * re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * Float64(im * im))
	tmp = 0.0
	if (re <= -120000000000.0)
		tmp = Float64(im * Float64(Float64(re + 1.0) * t_0));
	elseif (re <= 1.4e+122)
		tmp = Float64(im * Float64(1.0 + t_0));
	else
		tmp = Float64(im * Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * 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 <= -120000000000.0)
		tmp = im * ((re + 1.0) * t_0);
	elseif (re <= 1.4e+122)
		tmp = im * (1.0 + t_0);
	else
		tmp = im * ((0.5 + (re * 0.16666666666666666)) * (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, -120000000000.0], N[(im * N[(N[(re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e+122], N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e11

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in im around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

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

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

      16. +-commutative N/A

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

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

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

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

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 2.5%

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 30.8%

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

   11. Simplified 30.8%

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

   if -1.2e11 < re < 1.4e122

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 81.5%

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

   5. Simplified 81.5%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. *-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 41.6%

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

   8. Simplified 41.6%

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

   if 1.4e122 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 87.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 87.5%

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

   8. Simplified 87.5%

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

   9. Taylor expanded in re around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. lft-mult-inverse N/A

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

       10. *-lft-identity N/A

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

       11. unpow2 N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. *-commutative N/A

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

       15. distribute-lft-in N/A

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

       16. +-commutative N/A

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

       17. associate-*r* N/A

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

       18. unpow2 N/A

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

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

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

   11. Simplified 87.5%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -120000000000:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+122}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.5% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in im around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

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

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

      16. +-commutative N/A

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

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

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

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

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 2.5%

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 29.6%

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

   11. Simplified 29.6%

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

   if -1.6000000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 57.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 49.2%

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

   8. Simplified 49.2%

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

4. Final simplification 43.7%

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

Alternative 15: 44.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -120000000000:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot t\_0\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+122}:\\ \;\;\;\;im \cdot \left(1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* im im))))
   (if (<= re -120000000000.0)
     (* im (* (+ re 1.0) t_0))
     (if (<= re 1.4e+122)
       (* im (+ 1.0 t_0))
       (* im (* re (* 0.16666666666666666 (* re re))))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -120000000000.0) {
		tmp = im * ((re + 1.0) * t_0);
	} else if (re <= 1.4e+122) {
		tmp = im * (1.0 + t_0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (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 <= (-120000000000.0d0)) then
        tmp = im * ((re + 1.0d0) * t_0)
    else if (re <= 1.4d+122) then
        tmp = im * (1.0d0 + t_0)
    else
        tmp = im * (re * (0.16666666666666666d0 * (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 <= -120000000000.0) {
		tmp = im * ((re + 1.0) * t_0);
	} else if (re <= 1.4e+122) {
		tmp = im * (1.0 + t_0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (im * im)
	tmp = 0
	if re <= -120000000000.0:
		tmp = im * ((re + 1.0) * t_0)
	elif re <= 1.4e+122:
		tmp = im * (1.0 + t_0)
	else:
		tmp = im * (re * (0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * Float64(im * im))
	tmp = 0.0
	if (re <= -120000000000.0)
		tmp = Float64(im * Float64(Float64(re + 1.0) * t_0));
	elseif (re <= 1.4e+122)
		tmp = Float64(im * Float64(1.0 + t_0));
	else
		tmp = Float64(im * Float64(re * Float64(0.16666666666666666 * 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 <= -120000000000.0)
		tmp = im * ((re + 1.0) * t_0);
	elseif (re <= 1.4e+122)
		tmp = im * (1.0 + t_0);
	else
		tmp = im * (re * (0.16666666666666666 * (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, -120000000000.0], N[(im * N[(N[(re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e+122], N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e11

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in im around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. distribute-lft-in N/A

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

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

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

      16. +-commutative N/A

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

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

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

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

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 2.5%

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 30.8%

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

   11. Simplified 30.8%

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

   if -1.2e11 < re < 1.4e122

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 81.5%

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

   5. Simplified 81.5%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. *-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 41.6%

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

   8. Simplified 41.6%

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

   if 1.4e122 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 87.5%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 87.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]

   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. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{6}\right)\right), im\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot \left(re \cdot \frac{1}{6}\right)\right)\right), im\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), im\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(re \cdot re\right)\right)\right), im\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot {re}^{2}\right)\right), im\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{6}, \left({re}^{2}\right)\right)\right), im\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{6}, \left(re \cdot re\right)\right)\right), im\right) \]

       14. *-lowering-*.f64 87.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(re, re\right)\right)\right), im\right) \]

   11. Simplified 87.5%

       \[\leadsto \color{blue}{\left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -120000000000:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+122}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -120000000000:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(-0.16666666666666666 + re \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+122}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -120000000000.0)
   (* im (* im (* im (+ -0.16666666666666666 (* re -0.16666666666666666)))))
   (if (<= re 1.4e+122)
     (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
     (* im (* re (* 0.16666666666666666 (* re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -120000000000.0) {
		tmp = im * (im * (im * (-0.16666666666666666 + (re * -0.16666666666666666))));
	} else if (re <= 1.4e+122) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-120000000000.0d0)) then
        tmp = im * (im * (im * ((-0.16666666666666666d0) + (re * (-0.16666666666666666d0)))))
    else if (re <= 1.4d+122) then
        tmp = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = im * (re * (0.16666666666666666d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -120000000000.0) {
		tmp = im * (im * (im * (-0.16666666666666666 + (re * -0.16666666666666666))));
	} else if (re <= 1.4e+122) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -120000000000.0:
		tmp = im * (im * (im * (-0.16666666666666666 + (re * -0.16666666666666666))))
	elif re <= 1.4e+122:
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)))
	else:
		tmp = im * (re * (0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -120000000000.0)
		tmp = Float64(im * Float64(im * Float64(im * Float64(-0.16666666666666666 + Float64(re * -0.16666666666666666)))));
	elseif (re <= 1.4e+122)
		tmp = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(re * Float64(0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -120000000000.0)
		tmp = im * (im * (im * (-0.16666666666666666 + (re * -0.16666666666666666))));
	elseif (re <= 1.4e+122)
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	else
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -120000000000.0], N[(im * N[(im * N[(im * N[(-0.16666666666666666 + N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e+122], N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2e11

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{sin.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{sin.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.8%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto im \cdot \left(\left(1 + re\right) + \color{blue}{\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto im \cdot \left(1 + re\right) + \color{blue}{im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto im \cdot \left(1 + re\right) + im \cdot \left(\left({im}^{2} \cdot \left(1 + re\right)\right) \cdot \color{blue}{\frac{-1}{6}}\right) \]

      4. associate-*r* N/A

         \[\leadsto im \cdot \left(1 + re\right) + im \cdot \left({im}^{2} \cdot \color{blue}{\left(\left(1 + re\right) \cdot \frac{-1}{6}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto im \cdot \left(1 + re\right) + im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(1 + re\right)}\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(1 + re\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\left(1 + re\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(1 + re\right) + \left(\frac{-1}{6} \cdot \left(1 + re\right)\right) \cdot \color{blue}{{im}^{2}}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(1 + re \cdot 1\right) + \left(\frac{-1}{6} \cdot \color{blue}{\left(1 + re\right)}\right) \cdot {im}^{2}\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(re + 1\right) \cdot 1 + \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + re\right)\right)} \cdot {im}^{2}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(1 + re\right) \cdot 1 + \left(\color{blue}{\frac{-1}{6}} \cdot \left(1 + re\right)\right) \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(1 + re\right) \cdot 1 + \left(\left(1 + re\right) \cdot \frac{-1}{6}\right) \cdot {\color{blue}{im}}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(1 + re\right) \cdot 1 + \left(1 + re\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(1 + re\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(1 + re\right), \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(re + 1\right), \left(\color{blue}{1} + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(\color{blue}{1} + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 2.5%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   8. Simplified 2.5%

      \[\leadsto \color{blue}{im \cdot \left(\left(re + 1\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + re\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left({im}^{2} \cdot \frac{-1}{6}\right) \cdot \left(\color{blue}{1} + re\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + re\right)\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{6}} \cdot \left(1 + re\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + re\right)\right)}\right)\right)\right) \]

       8. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot 1 + \color{blue}{\frac{-1}{6} \cdot re}\right)\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{6} + \color{blue}{\frac{-1}{6}} \cdot re\right)\right)\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 29.7%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{re}\right)\right)\right)\right)\right) \]

   11. Simplified 29.7%

       \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(-0.16666666666666666 + -0.16666666666666666 \cdot re\right)\right)\right)} \]

   if -1.2e11 < re < 1.4e122

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 81.5%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\sin im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      3. *-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 41.6%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 41.6%

      \[\leadsto \color{blue}{im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]

   if 1.4e122 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 87.5%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 87.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]

   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. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{6}\right)\right), im\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot \left(re \cdot \frac{1}{6}\right)\right)\right), im\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), im\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(re \cdot re\right)\right)\right), im\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot {re}^{2}\right)\right), im\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{6}, \left({re}^{2}\right)\right)\right), im\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{6}, \left(re \cdot re\right)\right)\right), im\right) \]

       14. *-lowering-*.f64 87.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(re, re\right)\right)\right), im\right) \]

   11. Simplified 87.5%

       \[\leadsto \color{blue}{\left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -120000000000:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(-0.16666666666666666 + re \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+122}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+122}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.4e+122)
   (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
   (* im (* re (* 0.16666666666666666 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.4e+122) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.4d+122) then
        tmp = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = im * (re * (0.16666666666666666d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.4e+122) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.4e+122:
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)))
	else:
		tmp = im * (re * (0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.4e+122)
		tmp = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(re * Float64(0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.4e+122)
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	else
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.4e+122], N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.4e122

   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 57.7%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\sin im} \]

   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 29.9%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 29.9%

      \[\leadsto \color{blue}{im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]

   if 1.4e122 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 87.5%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 87.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]

   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. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{6}\right)\right), im\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot \left(re \cdot \frac{1}{6}\right)\right)\right), im\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), im\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(re \cdot re\right)\right)\right), im\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot {re}^{2}\right)\right), im\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{6}, \left({re}^{2}\right)\right)\right), im\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{6}, \left(re \cdot re\right)\right)\right), im\right) \]

       14. *-lowering-*.f64 87.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(re, re\right)\right)\right), im\right) \]

   11. Simplified 87.5%

       \[\leadsto \color{blue}{\left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+122}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.5% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{+123}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e+123)
   (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
   (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e+123) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1d+123) then
        tmp = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1e+123) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1e+123:
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)))
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e+123)
		tmp = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1e+123)
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e+123], N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.99999999999999978e122

   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 57.7%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\sin im} \]

   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 29.9%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 29.9%

      \[\leadsto \color{blue}{im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]

   if 9.99999999999999978e122 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
   4. Step-by-step derivation

   5. Simplified 87.5%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   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 78.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 78.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot im \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}, im\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), im\right) \]

       3. *-lowering-*.f64 78.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), im\right) \]

   11. Simplified 78.5%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right)} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 10^{+123}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.9% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.7e-6) im (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.7e-6) {
		tmp = im;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.7d-6) then
        tmp = im
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.7e-6) {
		tmp = im;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.7e-6:
		tmp = im
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.7e-6)
		tmp = im;
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.7e-6)
		tmp = im;
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.7e-6], im, N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.70000000000000003e-6

   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 67.1%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 31.9%

      \[\leadsto \color{blue}{im} \]

   if 1.70000000000000003e-6 < 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 76.1%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, im\right) \]
   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 46.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), im\right) \]

   8. Simplified 46.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot im \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}, im\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), im\right) \]

       3. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), im\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right)} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.9% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(re + 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ re 1.0)))

C

double code(double re, double im) {
	return im * (re + 1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * (re + 1.0d0)
end function

Java

public static double code(double re, double im) {
	return im * (re + 1.0);
}

Python

def code(re, im):
	return im * (re + 1.0)

Julia

function code(re, im)
	return Float64(im * Float64(re + 1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (re + 1.0);
end

Wolfram

code[re_, im_] := N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
4. Step-by-step derivation

5. Simplified 69.1%

   \[\leadsto e^{re} \cdot \color{blue}{im} \]

6. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, im\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), im\right) \]

   2. +-lowering-+.f64 26.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

8. Simplified 26.3%

   \[\leadsto \color{blue}{\left(re + 1\right)} \cdot im \]

9. Final simplification 26.3%

   \[\leadsto im \cdot \left(re + 1\right) \]
10. Add Preprocessing

Alternative 21: 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 im around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
4. Step-by-step derivation

5. Simplified 69.1%

   \[\leadsto e^{re} \cdot \color{blue}{im} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im} \]
7. Step-by-step derivation

8. Simplified 25.2%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))