math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 20.8s

Alternatives: 24

Speedup: 1.0×

• 25.8% 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: 92.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;e^{re} \leq 0.9995:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{re} \leq 1.05:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= (exp re) 0.9995) t_0 (if (<= (exp re) 1.05) (sin im) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (exp(re) <= 0.9995) {
		tmp = t_0;
	} else if (exp(re) <= 1.05) {
		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) <= 0.9995d0) then
        tmp = t_0
    else if (exp(re) <= 1.05d0) 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) <= 0.9995) {
		tmp = t_0;
	} else if (Math.exp(re) <= 1.05) {
		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) <= 0.9995:
		tmp = t_0
	elif math.exp(re) <= 1.05:
		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) <= 0.9995)
		tmp = t_0;
	elseif (exp(re) <= 1.05)
		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) <= 0.9995)
		tmp = t_0;
	elseif (exp(re) <= 1.05)
		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], 0.9995], t$95$0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.05], N[Sin[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99950000000000006 or 1.05000000000000004 < (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 92.1%

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

   if 0.99950000000000006 < (exp.f64 re) < 1.05000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 99.1%

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

   5. Simplified 99.1%

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

Alternative 3: 97.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := \sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;re \leq -0.038:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.047:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im))
        (t_1
         (*
          (sin im)
          (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))
   (if (<= re -0.038)
     t_0
     (if (<= re 0.047) t_1 (if (<= re 1.02e+103) t_0 t_1)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.038) {
		tmp = t_0;
	} else if (re <= 0.047) {
		tmp = t_1;
	} else if (re <= 1.02e+103) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(re) * im
    t_1 = sin(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    if (re <= (-0.038d0)) then
        tmp = t_0
    else if (re <= 0.047d0) then
        tmp = t_1
    else if (re <= 1.02d+103) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double t_1 = Math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.038) {
		tmp = t_0;
	} else if (re <= 0.047) {
		tmp = t_1;
	} else if (re <= 1.02e+103) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	t_1 = math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	tmp = 0
	if re <= -0.038:
		tmp = t_0
	elif re <= 0.047:
		tmp = t_1
	elif re <= 1.02e+103:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
	tmp = 0.0
	if (re <= -0.038)
		tmp = t_0;
	elseif (re <= 0.047)
		tmp = t_1;
	elseif (re <= 1.02e+103)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	t_1 = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	tmp = 0.0;
	if (re <= -0.038)
		tmp = t_0;
	elseif (re <= 0.047)
		tmp = t_1;
	elseif (re <= 1.02e+103)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.038], t$95$0, If[LessEqual[re, 0.047], t$95$1, If[LessEqual[re, 1.02e+103], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0379999999999999991 or 0.047 < re < 1.01999999999999991e103

   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 95.1%

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

   if -0.0379999999999999991 < re < 0.047 or 1.01999999999999991e103 < 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 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.038:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.047:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := \sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -0.015:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.034:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im))
        (t_1 (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))
   (if (<= re -0.015)
     t_0
     (if (<= re 0.034) t_1 (if (<= re 1.85e+154) t_0 t_1)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.015) {
		tmp = t_0;
	} else if (re <= 0.034) {
		tmp = t_1;
	} else if (re <= 1.85e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(re) * im
    t_1 = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    if (re <= (-0.015d0)) then
        tmp = t_0
    else if (re <= 0.034d0) then
        tmp = t_1
    else if (re <= 1.85d+154) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * im;
	double t_1 = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.015) {
		tmp = t_0;
	} else if (re <= 0.034) {
		tmp = t_1;
	} else if (re <= 1.85e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * im
	t_1 = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	tmp = 0
	if re <= -0.015:
		tmp = t_0
	elif re <= 0.034:
		tmp = t_1
	elif re <= 1.85e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))))
	tmp = 0.0
	if (re <= -0.015)
		tmp = t_0;
	elseif (re <= 0.034)
		tmp = t_1;
	elseif (re <= 1.85e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * im;
	t_1 = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	tmp = 0.0;
	if (re <= -0.015)
		tmp = t_0;
	elseif (re <= 0.034)
		tmp = t_1;
	elseif (re <= 1.85e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.015], t$95$0, If[LessEqual[re, 0.034], t$95$1, If[LessEqual[re, 1.85e+154], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.014999999999999999 or 0.034000000000000002 < re < 1.84999999999999997e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 94.8%

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

   if -0.014999999999999999 < re < 0.034000000000000002 or 1.84999999999999997e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.015:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.034:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.00036:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.024:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.00036) t_0 (if (<= re 0.024) (* (sin im) (+ re 1.0)) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.00036) {
		tmp = t_0;
	} else if (re <= 0.024) {
		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 <= (-0.00036d0)) then
        tmp = t_0
    else if (re <= 0.024d0) 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 <= -0.00036) {
		tmp = t_0;
	} else if (re <= 0.024) {
		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 <= -0.00036:
		tmp = t_0
	elif re <= 0.024:
		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 <= -0.00036)
		tmp = t_0;
	elseif (re <= 0.024)
		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 <= -0.00036)
		tmp = t_0;
	elseif (re <= 0.024)
		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, -0.00036], t$95$0, If[LessEqual[re, 0.024], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -3.60000000000000023e-4 or 0.024 < 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 92.1%

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

   if -3.60000000000000023e-4 < re < 0.024

   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 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00036:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.024:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + re \cdot 0.16666666666666666\\ t_1 := 1 + re \cdot t\_0\\ t_2 := re \cdot t\_1\\ t_3 := t\_0 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -48000000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 0.024:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{im \cdot \left(1 + t\_2 \cdot \left(\left(re \cdot re\right) \cdot \left(t\_1 \cdot t\_1\right)\right)\right)}{1 + t\_2 \cdot \left(t\_2 + -1\right)}\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot re - t\_3 \cdot t\_3}{re - t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* re 0.16666666666666666)))
        (t_1 (+ 1.0 (* re t_0)))
        (t_2 (* re t_1))
        (t_3 (* t_0 (* re re))))
   (if (<= re -48000000000000.0)
     (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
     (if (<= re 0.024)
       (sin im)
       (if (<= re 1.15e+51)
         (/
          (* im (+ 1.0 (* t_2 (* (* re re) (* t_1 t_1)))))
          (+ 1.0 (* t_2 (+ t_2 -1.0))))
         (if (<= re 1.02e+103)
           (* im (+ 1.0 (/ (- (* re re) (* t_3 t_3)) (- re t_3))))
           (* im (* re (* re (* re 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (re * 0.16666666666666666);
	double t_1 = 1.0 + (re * t_0);
	double t_2 = re * t_1;
	double t_3 = t_0 * (re * re);
	double tmp;
	if (re <= -48000000000000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 0.024) {
		tmp = sin(im);
	} else if (re <= 1.15e+51) {
		tmp = (im * (1.0 + (t_2 * ((re * re) * (t_1 * t_1))))) / (1.0 + (t_2 * (t_2 + -1.0)));
	} else if (re <= 1.02e+103) {
		tmp = im * (1.0 + (((re * re) - (t_3 * t_3)) / (re - t_3)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 + (re * 0.16666666666666666d0)
    t_1 = 1.0d0 + (re * t_0)
    t_2 = re * t_1
    t_3 = t_0 * (re * re)
    if (re <= (-48000000000000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 0.024d0) then
        tmp = sin(im)
    else if (re <= 1.15d+51) then
        tmp = (im * (1.0d0 + (t_2 * ((re * re) * (t_1 * t_1))))) / (1.0d0 + (t_2 * (t_2 + (-1.0d0))))
    else if (re <= 1.02d+103) then
        tmp = im * (1.0d0 + (((re * re) - (t_3 * t_3)) / (re - t_3)))
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (re * 0.16666666666666666);
	double t_1 = 1.0 + (re * t_0);
	double t_2 = re * t_1;
	double t_3 = t_0 * (re * re);
	double tmp;
	if (re <= -48000000000000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 0.024) {
		tmp = Math.sin(im);
	} else if (re <= 1.15e+51) {
		tmp = (im * (1.0 + (t_2 * ((re * re) * (t_1 * t_1))))) / (1.0 + (t_2 * (t_2 + -1.0)));
	} else if (re <= 1.02e+103) {
		tmp = im * (1.0 + (((re * re) - (t_3 * t_3)) / (re - t_3)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (re * 0.16666666666666666)
	t_1 = 1.0 + (re * t_0)
	t_2 = re * t_1
	t_3 = t_0 * (re * re)
	tmp = 0
	if re <= -48000000000000.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	elif re <= 0.024:
		tmp = math.sin(im)
	elif re <= 1.15e+51:
		tmp = (im * (1.0 + (t_2 * ((re * re) * (t_1 * t_1))))) / (1.0 + (t_2 * (t_2 + -1.0)))
	elif re <= 1.02e+103:
		tmp = im * (1.0 + (((re * re) - (t_3 * t_3)) / (re - t_3)))
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(re * 0.16666666666666666))
	t_1 = Float64(1.0 + Float64(re * t_0))
	t_2 = Float64(re * t_1)
	t_3 = Float64(t_0 * Float64(re * re))
	tmp = 0.0
	if (re <= -48000000000000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 0.024)
		tmp = sin(im);
	elseif (re <= 1.15e+51)
		tmp = Float64(Float64(im * Float64(1.0 + Float64(t_2 * Float64(Float64(re * re) * Float64(t_1 * t_1))))) / Float64(1.0 + Float64(t_2 * Float64(t_2 + -1.0))));
	elseif (re <= 1.02e+103)
		tmp = Float64(im * Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_3 * t_3)) / Float64(re - t_3))));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (re * 0.16666666666666666);
	t_1 = 1.0 + (re * t_0);
	t_2 = re * t_1;
	t_3 = t_0 * (re * re);
	tmp = 0.0;
	if (re <= -48000000000000.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	elseif (re <= 0.024)
		tmp = sin(im);
	elseif (re <= 1.15e+51)
		tmp = (im * (1.0 + (t_2 * ((re * re) * (t_1 * t_1))))) / (1.0 + (t_2 * (t_2 + -1.0)));
	elseif (re <= 1.02e+103)
		tmp = im * (1.0 + (((re * re) - (t_3 * t_3)) / (re - t_3)));
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(re * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -48000000000000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.024], N[Sin[im], $MachinePrecision], If[LessEqual[re, 1.15e+51], N[(N[(im * N[(1.0 + N[(t$95$2 * N[(N[(re * re), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 * N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.02e+103], N[(im * N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -4.8e13

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 65.8%

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

   5. Simplified 65.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 40.9%

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

   11. Simplified 40.9%

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

   if -4.8e13 < re < 0.024

   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 95.3%

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

   5. Simplified 95.3%

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

   if 0.024 < re < 1.15000000000000003e51

   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 72.2%

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

   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 7.4%

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

   8. Simplified 7.4%

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

      1. flip3-+ N/A

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

      2. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 39.3%

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

   if 1.15000000000000003e51 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   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 28.9%

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

   8. Simplified 28.9%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. flip-+ N/A

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

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

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

   10. Applied egg-rr 100.0%

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

   if 1.01999999999999991e103 < 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 85.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 85.0%

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

   8. Simplified 85.0%

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

   9. 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. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 85.0%

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

   11. Simplified 85.0%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -48000000000000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 0.024:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(\left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)\right)}{1 + \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot re - \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)}{re - \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + re \cdot 0.16666666666666666\\ t_1 := 1 + re \cdot t\_0\\ t_2 := re \cdot t\_1\\ t_3 := t\_0 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{im \cdot \left(1 + t\_2 \cdot \left(\left(re \cdot re\right) \cdot \left(t\_1 \cdot t\_1\right)\right)\right)}{1 + t\_2 \cdot \left(t\_2 + -1\right)}\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot re - t\_3 \cdot t\_3}{re - t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* re 0.16666666666666666)))
        (t_1 (+ 1.0 (* re t_0)))
        (t_2 (* re t_1))
        (t_3 (* t_0 (* re re))))
   (if (<= re -600000.0)
     (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
     (if (<= re 1.15e+51)
       (/
        (* im (+ 1.0 (* t_2 (* (* re re) (* t_1 t_1)))))
        (+ 1.0 (* t_2 (+ t_2 -1.0))))
       (if (<= re 1.02e+103)
         (* im (+ 1.0 (/ (- (* re re) (* t_3 t_3)) (- re t_3))))
         (* im (* re (* re (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (re * 0.16666666666666666);
	double t_1 = 1.0 + (re * t_0);
	double t_2 = re * t_1;
	double t_3 = t_0 * (re * re);
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 1.15e+51) {
		tmp = (im * (1.0 + (t_2 * ((re * re) * (t_1 * t_1))))) / (1.0 + (t_2 * (t_2 + -1.0)));
	} else if (re <= 1.02e+103) {
		tmp = im * (1.0 + (((re * re) - (t_3 * t_3)) / (re - t_3)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 + (re * 0.16666666666666666d0)
    t_1 = 1.0d0 + (re * t_0)
    t_2 = re * t_1
    t_3 = t_0 * (re * re)
    if (re <= (-600000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 1.15d+51) then
        tmp = (im * (1.0d0 + (t_2 * ((re * re) * (t_1 * t_1))))) / (1.0d0 + (t_2 * (t_2 + (-1.0d0))))
    else if (re <= 1.02d+103) then
        tmp = im * (1.0d0 + (((re * re) - (t_3 * t_3)) / (re - t_3)))
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (re * 0.16666666666666666);
	double t_1 = 1.0 + (re * t_0);
	double t_2 = re * t_1;
	double t_3 = t_0 * (re * re);
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 1.15e+51) {
		tmp = (im * (1.0 + (t_2 * ((re * re) * (t_1 * t_1))))) / (1.0 + (t_2 * (t_2 + -1.0)));
	} else if (re <= 1.02e+103) {
		tmp = im * (1.0 + (((re * re) - (t_3 * t_3)) / (re - t_3)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (re * 0.16666666666666666)
	t_1 = 1.0 + (re * t_0)
	t_2 = re * t_1
	t_3 = t_0 * (re * re)
	tmp = 0
	if re <= -600000.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	elif re <= 1.15e+51:
		tmp = (im * (1.0 + (t_2 * ((re * re) * (t_1 * t_1))))) / (1.0 + (t_2 * (t_2 + -1.0)))
	elif re <= 1.02e+103:
		tmp = im * (1.0 + (((re * re) - (t_3 * t_3)) / (re - t_3)))
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(re * 0.16666666666666666))
	t_1 = Float64(1.0 + Float64(re * t_0))
	t_2 = Float64(re * t_1)
	t_3 = Float64(t_0 * Float64(re * re))
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 1.15e+51)
		tmp = Float64(Float64(im * Float64(1.0 + Float64(t_2 * Float64(Float64(re * re) * Float64(t_1 * t_1))))) / Float64(1.0 + Float64(t_2 * Float64(t_2 + -1.0))));
	elseif (re <= 1.02e+103)
		tmp = Float64(im * Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_3 * t_3)) / Float64(re - t_3))));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (re * 0.16666666666666666);
	t_1 = 1.0 + (re * t_0);
	t_2 = re * t_1;
	t_3 = t_0 * (re * re);
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	elseif (re <= 1.15e+51)
		tmp = (im * (1.0 + (t_2 * ((re * re) * (t_1 * t_1))))) / (1.0 + (t_2 * (t_2 + -1.0)));
	elseif (re <= 1.02e+103)
		tmp = im * (1.0 + (((re * re) - (t_3 * t_3)) / (re - t_3)));
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(re * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -600000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.15e+51], N[(N[(im * N[(1.0 + N[(t$95$2 * N[(N[(re * re), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 * N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.02e+103], N[(im * N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 39.9%

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

   11. Simplified 39.9%

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

   if -6e5 < re < 1.15000000000000003e51

   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 51.5%

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

   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 41.3%

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

   8. Simplified 41.3%

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

      1. flip3-+ N/A

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

      2. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 45.6%

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

   if 1.15000000000000003e51 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   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 28.9%

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

   8. Simplified 28.9%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. flip-+ N/A

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

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

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

   10. Applied egg-rr 100.0%

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

   if 1.01999999999999991e103 < 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 85.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 85.0%

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

   8. Simplified 85.0%

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

   9. 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. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 85.0%

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

   11. Simplified 85.0%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(\left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)\right)}{1 + \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot re - \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)}{re - \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\left(1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))))
   (if (<= re -600000.0)
     (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
     (if (<= re 2e+97)
       (*
        (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0)))
        (*
         im
         (+
          1.0
          (*
           im
           (*
            im
            (+ -0.16666666666666666 (* im (* im 0.008333333333333333))))))))
       (* im (* re (* re (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 2e+97) {
		tmp = (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0))) * (im * (1.0 + (im * (im * (-0.16666666666666666 + (im * (im * 0.008333333333333333)))))));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    if (re <= (-600000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 2d+97) then
        tmp = (1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0))) * (im * (1.0d0 + (im * (im * ((-0.16666666666666666d0) + (im * (im * 0.008333333333333333d0)))))))
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 2e+97) {
		tmp = (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0))) * (im * (1.0 + (im * (im * (-0.16666666666666666 + (im * (im * 0.008333333333333333)))))));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	tmp = 0
	if re <= -600000.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	elif re <= 2e+97:
		tmp = (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0))) * (im * (1.0 + (im * (im * (-0.16666666666666666 + (im * (im * 0.008333333333333333)))))))
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 2e+97)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0))) * Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(-0.16666666666666666 + Float64(im * Float64(im * 0.008333333333333333))))))));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	elseif (re <= 2e+97)
		tmp = (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0))) * (im * (1.0 + (im * (im * (-0.16666666666666666 + (im * (im * 0.008333333333333333)))))));
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -600000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2e+97], N[(N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + N[(im * N[(im * N[(-0.16666666666666666 + N[(im * N[(im * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 39.9%

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

   11. Simplified 39.9%

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

   if -6e5 < re < 2.0000000000000001e97

   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 80.7%

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

   5. Simplified 80.7%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 42.7%

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

   8. Simplified 42.7%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. flip-+ N/A

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

      4. *-lft-identity N/A

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

      5. fmm-def N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. fmm-def N/A

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

      17. *-lft-identity N/A

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

   10. Applied egg-rr 45.4%

       \[\leadsto \left(1 + \color{blue}{\frac{re \cdot re - \left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)}{re - re \cdot \left(re \cdot 0.5\right)}}\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\right) \]

   if 2.0000000000000001e97 < 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 85.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 85.0%

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

   8. Simplified 85.0%

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

   9. 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. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 85.0%

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

   11. Simplified 85.0%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\left(1 + \frac{re \cdot re - \left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)}{re - re \cdot \left(re \cdot 0.5\right)}\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ 0.5 (* re 0.16666666666666666)) (* re re))))
   (if (<= re -600000.0)
     (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
     (if (<= re 1.02e+103)
       (* im (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0))))
       (* im (* re (* re (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re);
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 1.02e+103) {
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 + (re * 0.16666666666666666d0)) * (re * re)
    if (re <= (-600000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 1.02d+103) then
        tmp = im * (1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0)))
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re);
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 1.02e+103) {
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re)
	tmp = 0
	if re <= -600000.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	elif re <= 1.02e+103:
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)))
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * Float64(re * re))
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 1.02e+103)
		tmp = Float64(im * Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0))));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re);
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	elseif (re <= 1.02e+103)
		tmp = im * (1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0)));
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -600000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.02e+103], N[(im * N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 39.9%

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

   11. Simplified 39.9%

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

   if -6e5 < re < 1.01999999999999991e103

   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 54.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   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 40.6%

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

   8. Simplified 40.6%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. flip-+ N/A

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

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

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

   10. Applied egg-rr 44.6%

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

   if 1.01999999999999991e103 < 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 85.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 85.0%

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

   8. Simplified 85.0%

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

   9. 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. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 85.0%

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

   11. Simplified 85.0%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot re - \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)}{re - \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + re \cdot 0.16666666666666666\\ \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+153}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot \left(1 - t\_0 \cdot \left(t\_0 \cdot \left(re \cdot re\right)\right)\right)}{1 - re \cdot t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* re 0.16666666666666666))))
   (if (<= re -600000.0)
     (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
     (if (<= re 5e+153)
       (*
        im
        (+
         1.0
         (/ (* re (- 1.0 (* t_0 (* t_0 (* re re))))) (- 1.0 (* re t_0)))))
       (* im (* 0.5 (* re re)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (re * 0.16666666666666666);
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 5e+153) {
		tmp = im * (1.0 + ((re * (1.0 - (t_0 * (t_0 * (re * re))))) / (1.0 - (re * t_0))));
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (re * 0.16666666666666666d0)
    if (re <= (-600000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 5d+153) then
        tmp = im * (1.0d0 + ((re * (1.0d0 - (t_0 * (t_0 * (re * re))))) / (1.0d0 - (re * t_0))))
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (re * 0.16666666666666666);
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 5e+153) {
		tmp = im * (1.0 + ((re * (1.0 - (t_0 * (t_0 * (re * re))))) / (1.0 - (re * t_0))));
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (re * 0.16666666666666666)
	tmp = 0
	if re <= -600000.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	elif re <= 5e+153:
		tmp = im * (1.0 + ((re * (1.0 - (t_0 * (t_0 * (re * re))))) / (1.0 - (re * t_0))))
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(re * 0.16666666666666666))
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 5e+153)
		tmp = Float64(im * Float64(1.0 + Float64(Float64(re * Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(re * re))))) / Float64(1.0 - Float64(re * t_0)))));
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (re * 0.16666666666666666);
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	elseif (re <= 5e+153)
		tmp = im * (1.0 + ((re * (1.0 - (t_0 * (t_0 * (re * re))))) / (1.0 - (re * t_0))));
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -600000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+153], N[(im * N[(1.0 + N[(N[(re * N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 39.9%

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

   11. Simplified 39.9%

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

   if -6e5 < re < 5.00000000000000018e153

   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 57.2%

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

   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 44.6%

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

   8. Simplified 44.6%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 47.7%

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

   if 5.00000000000000018e153 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 82.1%

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

   8. Simplified 82.1%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\right)} \]

   9. Taylor expanded in re around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 82.1%

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

   11. Simplified 82.1%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\right) \]

   12. Taylor expanded in im around 0

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

   14. Simplified 82.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+153}:\\ \;\;\;\;im \cdot \left(1 + \frac{re \cdot \left(1 - \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\right)}{1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2.45 \cdot 10^{+48}:\\ \;\;\;\;im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -600000.0)
   (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
   (if (<= re 2.45e+48)
     (*
      im
      (+
       1.0
       (*
        im
        (* im (+ -0.16666666666666666 (* im (* im 0.008333333333333333)))))))
     (* im (* re (* re (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 2.45e+48) {
		tmp = im * (1.0 + (im * (im * (-0.16666666666666666 + (im * (im * 0.008333333333333333))))));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-600000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else if (re <= 2.45d+48) then
        tmp = im * (1.0d0 + (im * (im * ((-0.16666666666666666d0) + (im * (im * 0.008333333333333333d0))))))
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else if (re <= 2.45e+48) {
		tmp = im * (1.0 + (im * (im * (-0.16666666666666666 + (im * (im * 0.008333333333333333))))));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -600000.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	elif re <= 2.45e+48:
		tmp = im * (1.0 + (im * (im * (-0.16666666666666666 + (im * (im * 0.008333333333333333))))))
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	elseif (re <= 2.45e+48)
		tmp = Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(-0.16666666666666666 + Float64(im * Float64(im * 0.008333333333333333)))))));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	elseif (re <= 2.45e+48)
		tmp = im * (1.0 + (im * (im * (-0.16666666666666666 + (im * (im * 0.008333333333333333))))));
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -600000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.45e+48], N[(im * N[(1.0 + N[(im * N[(im * N[(-0.16666666666666666 + N[(im * N[(im * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 39.9%

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

   11. Simplified 39.9%

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

   if -6e5 < re < 2.45000000000000015e48

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 85.1%

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

   5. Simplified 85.1%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 42.7%

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

   8. Simplified 42.7%

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

   if 2.45000000000000015e48 < 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 87.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-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 74.2%

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

   8. Simplified 74.2%

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

   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. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 74.2%

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

   11. Simplified 74.2%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 2.45 \cdot 10^{+48}:\\ \;\;\;\;im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot t\_0\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+46}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* -0.16666666666666666 (* im im)))))
   (if (<= re -600000.0)
     (* (+ re 1.0) t_0)
     (if (<= re 3e+46)
       (* (+ re 1.0) (+ im t_0))
       (* im (* re (* re (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double t_0 = im * (-0.16666666666666666 * (im * im));
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 3e+46) {
		tmp = (re + 1.0) * (im + t_0);
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * ((-0.16666666666666666d0) * (im * im))
    if (re <= (-600000.0d0)) then
        tmp = (re + 1.0d0) * t_0
    else if (re <= 3d+46) then
        tmp = (re + 1.0d0) * (im + t_0)
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (-0.16666666666666666 * (im * im));
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 3e+46) {
		tmp = (re + 1.0) * (im + t_0);
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (-0.16666666666666666 * (im * im))
	tmp = 0
	if re <= -600000.0:
		tmp = (re + 1.0) * t_0
	elif re <= 3e+46:
		tmp = (re + 1.0) * (im + t_0)
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(-0.16666666666666666 * Float64(im * im)))
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(Float64(re + 1.0) * t_0);
	elseif (re <= 3e+46)
		tmp = Float64(Float64(re + 1.0) * Float64(im + t_0));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (-0.16666666666666666 * (im * im));
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (re + 1.0) * t_0;
	elseif (re <= 3e+46)
		tmp = (re + 1.0) * (im + t_0);
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -600000.0], N[(N[(re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[re, 3e+46], N[(N[(re + 1.0), $MachinePrecision] * N[(im + t$95$0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 39.9%

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

   11. Simplified 39.9%

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

   if -6e5 < re < 3.00000000000000023e46

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 51.6%

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

   5. Simplified 51.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.8%

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

   8. Simplified 42.8%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 42.8%

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

   10. Applied egg-rr 42.8%

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

   if 3.00000000000000023e46 < 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 88.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 72.8%

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

   8. Simplified 72.8%

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

   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. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 72.8%

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

   11. Simplified 72.8%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+46}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im + im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot t\_0\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+46}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(1 + t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* im im))))
   (if (<= re -600000.0)
     (* (+ re 1.0) (* im t_0))
     (if (<= re 1.85e+46)
       (* (+ re 1.0) (* im (+ 1.0 t_0)))
       (* im (* re (* re (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * t_0);
	} else if (re <= 1.85e+46) {
		tmp = (re + 1.0) * (im * (1.0 + t_0));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.16666666666666666d0) * (im * im)
    if (re <= (-600000.0d0)) then
        tmp = (re + 1.0d0) * (im * t_0)
    else if (re <= 1.85d+46) then
        tmp = (re + 1.0d0) * (im * (1.0d0 + t_0))
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (im * im);
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * t_0);
	} else if (re <= 1.85e+46) {
		tmp = (re + 1.0) * (im * (1.0 + t_0));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (im * im)
	tmp = 0
	if re <= -600000.0:
		tmp = (re + 1.0) * (im * t_0)
	elif re <= 1.85e+46:
		tmp = (re + 1.0) * (im * (1.0 + t_0))
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * Float64(im * im))
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * t_0));
	elseif (re <= 1.85e+46)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(1.0 + t_0)));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (im * im);
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (re + 1.0) * (im * t_0);
	elseif (re <= 1.85e+46)
		tmp = (re + 1.0) * (im * (1.0 + t_0));
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -600000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.85e+46], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 39.9%

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

   11. Simplified 39.9%

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

   if -6e5 < re < 1.84999999999999995e46

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 51.6%

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

   5. Simplified 51.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.8%

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

   8. Simplified 42.8%

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

   if 1.84999999999999995e46 < 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 88.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 72.8%

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

   8. Simplified 72.8%

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

   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. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 72.8%

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

   11. Simplified 72.8%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+46}:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -600000.0)
   (* (+ re 1.0) (* im (* -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 <= -600000.0) {
		tmp = (re + 1.0) * (im * (-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 <= (-600000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-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 <= -600000.0) {
		tmp = (re + 1.0) * (im * (-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 <= -600000.0:
		tmp = (re + 1.0) * (im * (-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 <= -600000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (re + 1.0) * (im * (-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, -600000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 39.9%

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

   11. Simplified 39.9%

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

   if -6e5 < 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 61.1%

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

   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 50.4%

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

   8. Simplified 50.4%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.3% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(\left(re \cdot re\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -600000.0)
   (* (+ re 1.0) (* im (* -0.16666666666666666 (* im im))))
   (* im (+ 1.0 (* re (* (* re re) (+ 0.16666666666666666 (/ 0.5 re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (1.0 + (re * ((re * re) * (0.16666666666666666 + (0.5 / 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 <= (-600000.0d0)) then
        tmp = (re + 1.0d0) * (im * ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = im * (1.0d0 + (re * ((re * re) * (0.16666666666666666d0 + (0.5d0 / re)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (1.0 + (re * ((re * re) * (0.16666666666666666 + (0.5 / re)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -600000.0:
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)))
	else:
		tmp = im * (1.0 + (re * ((re * re) * (0.16666666666666666 + (0.5 / re)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(1.0 + Float64(re * Float64(Float64(re * re) * Float64(0.16666666666666666 + Float64(0.5 / re))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (re + 1.0) * (im * (-0.16666666666666666 * (im * im)));
	else
		tmp = im * (1.0 + (re * ((re * re) * (0.16666666666666666 + (0.5 / re)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -600000.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 + N[(re * N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 65.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(\frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot im\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 39.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 39.9%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   if -6e5 < 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 61.1%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   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 50.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 50.4%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}\right)\right), im\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), im\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right), im\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{2} \cdot \frac{1}{re}\right)\right)\right)\right)\right), im\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} \cdot 1}{re}\right)\right)\right)\right)\right), im\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2}}{re}\right)\right)\right)\right)\right), im\right) \]

       7. /-lowering-/.f64 50.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{1}{2}, re\right)\right)\right)\right)\right), im\right) \]

   11. Simplified 50.0%

       \[\leadsto \left(1 + re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)}\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(\left(re \cdot re\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot t\_0\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+46}:\\ \;\;\;\;im + t\_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* -0.16666666666666666 (* im im)))))
   (if (<= re -600000.0)
     (* (+ re 1.0) t_0)
     (if (<= re 1.55e+46)
       (+ im t_0)
       (* im (* re (* re (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double t_0 = im * (-0.16666666666666666 * (im * im));
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 1.55e+46) {
		tmp = im + t_0;
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * ((-0.16666666666666666d0) * (im * im))
    if (re <= (-600000.0d0)) then
        tmp = (re + 1.0d0) * t_0
    else if (re <= 1.55d+46) then
        tmp = im + t_0
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (-0.16666666666666666 * (im * im));
	double tmp;
	if (re <= -600000.0) {
		tmp = (re + 1.0) * t_0;
	} else if (re <= 1.55e+46) {
		tmp = im + t_0;
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (-0.16666666666666666 * (im * im))
	tmp = 0
	if re <= -600000.0:
		tmp = (re + 1.0) * t_0
	elif re <= 1.55e+46:
		tmp = im + t_0
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(-0.16666666666666666 * Float64(im * im)))
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(Float64(re + 1.0) * t_0);
	elseif (re <= 1.55e+46)
		tmp = Float64(im + t_0);
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (-0.16666666666666666 * (im * im));
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (re + 1.0) * t_0;
	elseif (re <= 1.55e+46)
		tmp = im + t_0;
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -600000.0], N[(N[(re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[re, 1.55e+46], N[(im + t$95$0), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + \color{blue}{e^{re}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re} + e^{\color{blue}{re}}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot \color{blue}{e^{re}}\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{\color{blue}{re}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{e^{re}} \]

      6. *-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 65.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(\frac{-1}{6} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot im\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 39.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   11. Simplified 39.9%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   if -6e5 < re < 1.54999999999999988e46

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + \color{blue}{e^{re}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re} + e^{\color{blue}{re}}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot \color{blue}{e^{re}}\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{\color{blue}{re}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{e^{re}} \]

      6. *-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 51.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 51.6%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 42.8%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + \color{blue}{1}\right)\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \color{blue}{im \cdot 1}\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) + im\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right)\right), \color{blue}{im}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right)\right), im\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot im\right)\right)\right), im\right)\right) \]

      7. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right), im\right)\right) \]

   10. Applied egg-rr 42.8%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) + im\right)} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right), im\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 42.5%

       \[\leadsto \color{blue}{1} \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) + im\right) \]

   if 1.54999999999999988e46 < 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 88.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 72.8%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]

   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. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \left(re \cdot \left(re \cdot re\right)\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \left(re \cdot {re}^{2}\right)\right), im\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot re\right) \cdot {re}^{2}\right), im\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(\frac{1}{6} \cdot re\right)\right), im\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot re\right) \cdot \left(\frac{1}{6} \cdot re\right)\right), im\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(re \cdot \frac{1}{6}\right)\right)\right), im\right) \]

       10. *-lowering-*.f64 72.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), im\right) \]

   11. Simplified 72.8%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+46}:\\ \;\;\;\;im + im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.16666666666666666 + re \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+46}:\\ \;\;\;\;im + im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -600000.0)
   (* (* im im) (* im (+ -0.16666666666666666 (* re -0.16666666666666666))))
   (if (<= re 1.95e+46)
     (+ im (* im (* -0.16666666666666666 (* im im))))
     (* im (* re (* re (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = (im * im) * (im * (-0.16666666666666666 + (re * -0.16666666666666666)));
	} else if (re <= 1.95e+46) {
		tmp = im + (im * (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-600000.0d0)) then
        tmp = (im * im) * (im * ((-0.16666666666666666d0) + (re * (-0.16666666666666666d0))))
    else if (re <= 1.95d+46) then
        tmp = im + (im * ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -600000.0) {
		tmp = (im * im) * (im * (-0.16666666666666666 + (re * -0.16666666666666666)));
	} else if (re <= 1.95e+46) {
		tmp = im + (im * (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -600000.0:
		tmp = (im * im) * (im * (-0.16666666666666666 + (re * -0.16666666666666666)))
	elif re <= 1.95e+46:
		tmp = im + (im * (-0.16666666666666666 * (im * im)))
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -600000.0)
		tmp = Float64(Float64(im * im) * Float64(im * Float64(-0.16666666666666666 + Float64(re * -0.16666666666666666))));
	elseif (re <= 1.95e+46)
		tmp = Float64(im + Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -600000.0)
		tmp = (im * im) * (im * (-0.16666666666666666 + (re * -0.16666666666666666)));
	elseif (re <= 1.95e+46)
		tmp = im + (im * (-0.16666666666666666 * (im * im)));
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -600000.0], N[(N[(im * im), $MachinePrecision] * N[(im * N[(-0.16666666666666666 + N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.95e+46], N[(im + N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6e5

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + \color{blue}{e^{re}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re} + e^{\color{blue}{re}}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot \color{blue}{e^{re}}\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{\color{blue}{re}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{e^{re}} \]

      6. *-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 65.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({im}^{3} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{3}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{3} \cdot \frac{-1}{6}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{3} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + re\right)\right)} \]

       4. unpow3 N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{6}} \cdot \left(1 + re\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot im\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{im} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{im} \cdot \left(\frac{-1}{6} \cdot \left(1 + re\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + re\right)\right)}\right)\right) \]

       11. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \left(1 \cdot \frac{-1}{6} + \color{blue}{re \cdot \frac{-1}{6}}\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{6} + \color{blue}{re} \cdot \frac{-1}{6}\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(re \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

       14. *-lowering-*.f64 34.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 34.8%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.16666666666666666 + re \cdot -0.16666666666666666\right)\right)} \]

   if -6e5 < re < 1.94999999999999997e46

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + \color{blue}{e^{re}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re} + e^{\color{blue}{re}}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot \color{blue}{e^{re}}\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{\color{blue}{re}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{e^{re}} \]

      6. *-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 51.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 51.6%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 42.8%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + \color{blue}{1}\right)\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \color{blue}{im \cdot 1}\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) + im\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right)\right), \color{blue}{im}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right)\right), im\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot im\right)\right)\right), im\right)\right) \]

      7. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right), im\right)\right) \]

   10. Applied egg-rr 42.8%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) + im\right)} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right), im\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 42.5%

       \[\leadsto \color{blue}{1} \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) + im\right) \]

   if 1.94999999999999997e46 < 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 88.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 72.8%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]

   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. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \left(re \cdot \left(re \cdot re\right)\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \left(re \cdot {re}^{2}\right)\right), im\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot re\right) \cdot {re}^{2}\right), im\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(\frac{1}{6} \cdot re\right)\right), im\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot re\right) \cdot \left(\frac{1}{6} \cdot re\right)\right), im\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(re \cdot \frac{1}{6}\right)\right)\right), im\right) \]

       10. *-lowering-*.f64 72.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), im\right) \]

   11. Simplified 72.8%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -600000:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.16666666666666666 + re \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+46}:\\ \;\;\;\;im + im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+46}:\\ \;\;\;\;im + im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.3e+46)
   (+ im (* im (* -0.16666666666666666 (* im im))))
   (* im (* re (* re (* re 0.16666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.3e+46) {
		tmp = im + (im * (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.3d+46) then
        tmp = im + (im * ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.3e+46) {
		tmp = im + (im * (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.3e+46:
		tmp = im + (im * (-0.16666666666666666 * (im * im)))
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.3e+46)
		tmp = Float64(im + Float64(im * Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.3e+46)
		tmp = im + (im * (-0.16666666666666666 * (im * im)));
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.3e+46], N[(im + N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.2999999999999998e46

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right) + \color{blue}{e^{re}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{re} + e^{\color{blue}{re}}\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot \color{blue}{e^{re}}\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot e^{\color{blue}{re}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{e^{re}} \]

      6. *-commutative N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{im} \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 56.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right) \]

   5. Simplified 56.6%

      \[\leadsto \color{blue}{e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 28.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   8. Simplified 28.2%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + \color{blue}{1}\right)\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \color{blue}{im \cdot 1}\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) + im\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\left(im \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right)\right), \color{blue}{im}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right)\right), im\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot im\right)\right)\right), im\right)\right) \]

      7. *-lowering-*.f64 28.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right), im\right)\right) \]

   10. Applied egg-rr 28.2%

       \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) + im\right)} \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right)\right), im\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 28.3%

       \[\leadsto \color{blue}{1} \cdot \left(im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right) + im\right) \]

   if 3.2999999999999998e46 < 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 88.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 72.8%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]

   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. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \left(re \cdot \left(re \cdot re\right)\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \left(re \cdot {re}^{2}\right)\right), im\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot re\right) \cdot {re}^{2}\right), im\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(\frac{1}{6} \cdot re\right)\right), im\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot re\right) \cdot \left(\frac{1}{6} \cdot re\right)\right), im\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(re \cdot \frac{1}{6}\right)\right)\right), im\right) \]

       10. *-lowering-*.f64 72.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), im\right) \]

   11. Simplified 72.8%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+46}:\\ \;\;\;\;im + im \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 40.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.25 \cdot 10^{+46}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.25e+46)
   (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
   (* im (* re (* re (* re 0.16666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.25e+46) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.25d+46) then
        tmp = im * (1.0d0 + ((-0.16666666666666666d0) * (im * im)))
    else
        tmp = im * (re * (re * (re * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.25e+46) {
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	} else {
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.25e+46:
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)))
	else:
		tmp = im * (re * (re * (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.25e+46)
		tmp = Float64(im * Float64(1.0 + Float64(-0.16666666666666666 * Float64(im * im))));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.25e+46)
		tmp = im * (1.0 + (-0.16666666666666666 * (im * im)));
	else
		tmp = im * (re * (re * (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.25e+46], N[(im * N[(1.0 + N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.25000000000000005e46

   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 56.0%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 56.0%

      \[\leadsto \color{blue}{\sin im} \]

   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 28.3%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 28.3%

      \[\leadsto \color{blue}{im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]

   if 2.25000000000000005e46 < 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 88.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, im\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), im\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), im\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

      7. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), im\right) \]

   8. Simplified 72.8%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]

   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. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \left(re \cdot \left(re \cdot re\right)\right)\right), im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \left(re \cdot {re}^{2}\right)\right), im\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{6} \cdot re\right) \cdot {re}^{2}\right), im\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2} \cdot \left(\frac{1}{6} \cdot re\right)\right), im\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot re\right) \cdot \left(\frac{1}{6} \cdot re\right)\right), im\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot re\right)\right)\right), im\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(re \cdot \frac{1}{6}\right)\right)\right), im\right) \]

       10. *-lowering-*.f64 72.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right), im\right) \]

   11. Simplified 72.8%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.25 \cdot 10^{+46}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 38.0% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+46}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.4e+46)
   (* im (+ 1.0 (* -0.16666666666666666 (* im im))))
   (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.4e+46) {
		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 <= 1.4d+46) 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 <= 1.4e+46) {
		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 <= 1.4e+46:
		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 <= 1.4e+46)
		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 <= 1.4e+46)
		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, 1.4e+46], 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 < 1.40000000000000009e46

   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 56.0%

         \[\leadsto \mathsf{sin.f64}\left(im\right) \]

   5. Simplified 56.0%

      \[\leadsto \color{blue}{\sin im} \]

   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 28.3%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 28.3%

      \[\leadsto \color{blue}{im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]

   if 1.40000000000000009e46 < 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 58.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 58.3%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 61.3%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 61.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\right) \]

   12. Taylor expanded in im around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \color{blue}{im}\right) \]
   13. Step-by-step derivation

   14. Simplified 59.3%

       \[\leadsto \left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+46}:\\ \;\;\;\;im \cdot \left(1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 37.7% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.4:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.4) im (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.4) {
		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 <= 1.4d0) 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 <= 1.4) {
		tmp = im;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.4:
		tmp = im
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.4)
		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 <= 1.4)
		tmp = im;
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.4], im, N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.3999999999999999

   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 69.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 29.4%

      \[\leadsto \color{blue}{im} \]

   if 1.3999999999999999 < 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 46.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(im\right)\right) \]

   5. Simplified 46.2%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \sin im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 51.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 51.3%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 51.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 51.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right)} \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot 0.008333333333333333\right)\right)\right)\right)\right) \]

   12. Taylor expanded in im around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \color{blue}{im}\right) \]
   13. Step-by-step derivation

   14. Simplified 46.8%

       \[\leadsto \left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 29.9% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 290:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 290.0) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 290.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 290.0d0) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 290.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 290.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 290.0)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 290.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 290.0], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 290

   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 69.0%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 29.4%

      \[\leadsto \color{blue}{im} \]

   if 290 < 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 82.8%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]

   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 24.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

   8. Simplified 24.2%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot im \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, im\right) \]
   10. Step-by-step derivation

   11. Simplified 24.2%

       \[\leadsto \color{blue}{re} \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 29.8% 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 72.5%

   \[\leadsto e^{re} \cdot \color{blue}{im} \]

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 27.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

8. Simplified 27.9%

   \[\leadsto \color{blue}{\left(re + 1\right)} \cdot im \]

9. Final simplification 27.9%

   \[\leadsto im \cdot \left(re + 1\right) \]
10. Add Preprocessing

Alternative 24: 26.5% 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 72.5%

   \[\leadsto e^{re} \cdot \color{blue}{im} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im} \]
7. Step-by-step derivation

8. Simplified 22.9%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))