math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 21.4s

Alternatives: 18

Speedup: 1.0×

• 24.8% 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: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{re} \leq 1.000002:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= (exp re) 0.0) t_0 (if (<= (exp re) 1.000002) (sin im) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = t_0;
	} else if (exp(re) <= 1.000002) {
		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) <= 0.0d0) then
        tmp = t_0
    else if (exp(re) <= 1.000002d0) 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) <= 0.0) {
		tmp = t_0;
	} else if (Math.exp(re) <= 1.000002) {
		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) <= 0.0:
		tmp = t_0
	elif math.exp(re) <= 1.000002:
		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) <= 0.0)
		tmp = t_0;
	elseif (exp(re) <= 1.000002)
		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) <= 0.0)
		tmp = t_0;
	elseif (exp(re) <= 1.000002)
		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], 0.0], t$95$0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.000002], N[Sin[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1.00000200000000006 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 90.0%

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

   if 0.0 < (exp.f64 re) < 1.00000200000000006

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 94.9%

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

   5. Simplified 94.9%

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

Alternative 3: 97.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := \sin im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \mathbf{if}\;re \leq -0.118:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.0055:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im))
        (t_1
         (*
          (sin im)
          (+ (* re (+ (* re (+ 0.5 (* re 0.16666666666666666))) 1.0)) 1.0))))
   (if (<= re -0.118)
     t_0
     (if (<= re 0.0055) 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) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	double tmp;
	if (re <= -0.118) {
		tmp = t_0;
	} else if (re <= 0.0055) {
		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) * ((re * ((re * (0.5d0 + (re * 0.16666666666666666d0))) + 1.0d0)) + 1.0d0)
    if (re <= (-0.118d0)) then
        tmp = t_0
    else if (re <= 0.0055d0) 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) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	double tmp;
	if (re <= -0.118) {
		tmp = t_0;
	} else if (re <= 0.0055) {
		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) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0)
	tmp = 0
	if re <= -0.118:
		tmp = t_0
	elif re <= 0.0055:
		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(Float64(re * Float64(Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))) + 1.0)) + 1.0))
	tmp = 0.0
	if (re <= -0.118)
		tmp = t_0;
	elseif (re <= 0.0055)
		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) * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
	tmp = 0.0;
	if (re <= -0.118)
		tmp = t_0;
	elseif (re <= 0.0055)
		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[(N[(re * N[(N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.118], t$95$0, If[LessEqual[re, 0.0055], 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 < -0.11799999999999999 or 0.0054999999999999997 < 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 96.3%

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

   if -0.11799999999999999 < re < 0.0054999999999999997 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 99.2%

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

   5. Simplified 99.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.118:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.0055:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := \sin im \cdot \left(1 - re \cdot \left(-1 - re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -54:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.0055:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -54.0)
     t_0
     (if (<= re 0.0055) t_1 (if (<= re 1.9e+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 <= -54.0) {
		tmp = t_0;
	} else if (re <= 0.0055) {
		tmp = t_1;
	} else if (re <= 1.9e+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 <= (-54.0d0)) then
        tmp = t_0
    else if (re <= 0.0055d0) then
        tmp = t_1
    else if (re <= 1.9d+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 <= -54.0) {
		tmp = t_0;
	} else if (re <= 0.0055) {
		tmp = t_1;
	} else if (re <= 1.9e+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 <= -54.0:
		tmp = t_0
	elif re <= 0.0055:
		tmp = t_1
	elif re <= 1.9e+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 <= -54.0)
		tmp = t_0;
	elseif (re <= 0.0055)
		tmp = t_1;
	elseif (re <= 1.9e+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 <= -54.0)
		tmp = t_0;
	elseif (re <= 0.0055)
		tmp = t_1;
	elseif (re <= 1.9e+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, -54.0], t$95$0, If[LessEqual[re, 0.0055], t$95$1, If[LessEqual[re, 1.9e+154], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if re < -54 or 0.0054999999999999997 < re < 1.8999999999999999e154

   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 95.6%

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

   if -54 < re < 0.0054999999999999997 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -54:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.0055:\\ \;\;\;\;\sin im \cdot \left(1 - re \cdot \left(-1 - re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 - re \cdot \left(-1 - re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\\ t_1 := re \cdot \left(t\_0 \cdot -0.16666666666666666\right)\\ \mathbf{if}\;re \leq -2.35 \cdot 10^{+165}:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;im \cdot \left(\frac{1 - re \cdot re}{1 - re \cdot \left(re \cdot re\right)} \cdot \left(re \cdot \left(re + 1\right) + 1\right)\right)\\ \mathbf{elif}\;re \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;\left(-0.16666666666666666 + t\_1\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+14}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(t\_1 + \frac{re \cdot t\_0}{im \cdot im}\right)\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* re (+ 0.5 (* re 0.16666666666666666))) 1.0))
        (t_1 (* re (* t_0 -0.16666666666666666))))
   (if (<= re -2.35e+165)
     (/ (* im im) (+ im (* re (* im (- -1.0 (* re 0.5))))))
     (if (<= re -5.6e+102)
       (*
        im
        (*
         (/ (- 1.0 (* re re)) (- 1.0 (* re (* re re))))
         (+ (* re (+ re 1.0)) 1.0)))
       (if (<= re -6.5e+20)
         (* (+ -0.16666666666666666 t_1) (* im (* im im)))
         (if (<= re 4.7e+14)
           (sin im)
           (if (<= re 1.02e+103)
             (*
              im
              (+
               (*
                (* im im)
                (+ -0.16666666666666666 (+ t_1 (/ (* re t_0) (* im im)))))
               1.0))
             (* im (* re (* 0.16666666666666666 (* re re)))))))))))

C

double code(double re, double im) {
	double t_0 = (re * (0.5 + (re * 0.16666666666666666))) + 1.0;
	double t_1 = re * (t_0 * -0.16666666666666666);
	double tmp;
	if (re <= -2.35e+165) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else if (re <= -5.6e+102) {
		tmp = im * (((1.0 - (re * re)) / (1.0 - (re * (re * re)))) * ((re * (re + 1.0)) + 1.0));
	} else if (re <= -6.5e+20) {
		tmp = (-0.16666666666666666 + t_1) * (im * (im * im));
	} else if (re <= 4.7e+14) {
		tmp = sin(im);
	} else if (re <= 1.02e+103) {
		tmp = im * (((im * im) * (-0.16666666666666666 + (t_1 + ((re * t_0) / (im * im))))) + 1.0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (re * (0.5d0 + (re * 0.16666666666666666d0))) + 1.0d0
    t_1 = re * (t_0 * (-0.16666666666666666d0))
    if (re <= (-2.35d+165)) then
        tmp = (im * im) / (im + (re * (im * ((-1.0d0) - (re * 0.5d0)))))
    else if (re <= (-5.6d+102)) then
        tmp = im * (((1.0d0 - (re * re)) / (1.0d0 - (re * (re * re)))) * ((re * (re + 1.0d0)) + 1.0d0))
    else if (re <= (-6.5d+20)) then
        tmp = ((-0.16666666666666666d0) + t_1) * (im * (im * im))
    else if (re <= 4.7d+14) then
        tmp = sin(im)
    else if (re <= 1.02d+103) then
        tmp = im * (((im * im) * ((-0.16666666666666666d0) + (t_1 + ((re * t_0) / (im * im))))) + 1.0d0)
    else
        tmp = im * (re * (0.16666666666666666d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * (0.5 + (re * 0.16666666666666666))) + 1.0;
	double t_1 = re * (t_0 * -0.16666666666666666);
	double tmp;
	if (re <= -2.35e+165) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else if (re <= -5.6e+102) {
		tmp = im * (((1.0 - (re * re)) / (1.0 - (re * (re * re)))) * ((re * (re + 1.0)) + 1.0));
	} else if (re <= -6.5e+20) {
		tmp = (-0.16666666666666666 + t_1) * (im * (im * im));
	} else if (re <= 4.7e+14) {
		tmp = Math.sin(im);
	} else if (re <= 1.02e+103) {
		tmp = im * (((im * im) * (-0.16666666666666666 + (t_1 + ((re * t_0) / (im * im))))) + 1.0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * (0.5 + (re * 0.16666666666666666))) + 1.0
	t_1 = re * (t_0 * -0.16666666666666666)
	tmp = 0
	if re <= -2.35e+165:
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))))
	elif re <= -5.6e+102:
		tmp = im * (((1.0 - (re * re)) / (1.0 - (re * (re * re)))) * ((re * (re + 1.0)) + 1.0))
	elif re <= -6.5e+20:
		tmp = (-0.16666666666666666 + t_1) * (im * (im * im))
	elif re <= 4.7e+14:
		tmp = math.sin(im)
	elif re <= 1.02e+103:
		tmp = im * (((im * im) * (-0.16666666666666666 + (t_1 + ((re * t_0) / (im * im))))) + 1.0)
	else:
		tmp = im * (re * (0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))) + 1.0)
	t_1 = Float64(re * Float64(t_0 * -0.16666666666666666))
	tmp = 0.0
	if (re <= -2.35e+165)
		tmp = Float64(Float64(im * im) / Float64(im + Float64(re * Float64(im * Float64(-1.0 - Float64(re * 0.5))))));
	elseif (re <= -5.6e+102)
		tmp = Float64(im * Float64(Float64(Float64(1.0 - Float64(re * re)) / Float64(1.0 - Float64(re * Float64(re * re)))) * Float64(Float64(re * Float64(re + 1.0)) + 1.0)));
	elseif (re <= -6.5e+20)
		tmp = Float64(Float64(-0.16666666666666666 + t_1) * Float64(im * Float64(im * im)));
	elseif (re <= 4.7e+14)
		tmp = sin(im);
	elseif (re <= 1.02e+103)
		tmp = Float64(im * Float64(Float64(Float64(im * im) * Float64(-0.16666666666666666 + Float64(t_1 + Float64(Float64(re * t_0) / Float64(im * im))))) + 1.0));
	else
		tmp = Float64(im * Float64(re * Float64(0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * (0.5 + (re * 0.16666666666666666))) + 1.0;
	t_1 = re * (t_0 * -0.16666666666666666);
	tmp = 0.0;
	if (re <= -2.35e+165)
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	elseif (re <= -5.6e+102)
		tmp = im * (((1.0 - (re * re)) / (1.0 - (re * (re * re)))) * ((re * (re + 1.0)) + 1.0));
	elseif (re <= -6.5e+20)
		tmp = (-0.16666666666666666 + t_1) * (im * (im * im));
	elseif (re <= 4.7e+14)
		tmp = sin(im);
	elseif (re <= 1.02e+103)
		tmp = im * (((im * im) * (-0.16666666666666666 + (t_1 + ((re * t_0) / (im * im))))) + 1.0);
	else
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(t$95$0 * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.35e+165], N[(N[(im * im), $MachinePrecision] / N[(im + N[(re * N[(im * N[(-1.0 - N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -5.6e+102], N[(im * N[(N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -6.5e+20], N[(N[(-0.16666666666666666 + t$95$1), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.7e+14], N[Sin[im], $MachinePrecision], If[LessEqual[re, 1.02e+103], N[(im * N[(N[(N[(im * im), $MachinePrecision] * N[(-0.16666666666666666 + N[(t$95$1 + N[(N[(re * t$95$0), $MachinePrecision] / N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if re < -2.35000000000000008e165

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 1.7%

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

   8. Simplified 1.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

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

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

   10. Applied egg-rr 0.5%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 91.3%

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

   13. Simplified 91.3%

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

   if -2.35000000000000008e165 < re < -5.60000000000000037e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.5%

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

   5. Simplified 2.5%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 2.1%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. metadata-eval N/A

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

      4. flip3-- N/A

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

      5. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

      11. cube-unmult N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

      17. distribute-rgt-out N/A

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

      18. +-commutative N/A

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

      19. flip-+ N/A

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

      20. metadata-eval N/A

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

      21. un-div-inv N/A

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

   10. Applied egg-rr 80.0%

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

   if -5.60000000000000037e102 < re < -6.5e20

   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 2.6%

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

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   8. Simplified 2.3%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. sub-neg N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. distribute-lft1-in N/A

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

       6. distribute-lft-in N/A

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

       7. sum3-define N/A

          \[\leadsto \left(\mathsf{sum3}\left(\left(re \cdot 1\right), \left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right), 1\right) \cdot \frac{-1}{6}\right) \cdot {im}^{3} \]

       8. *-rgt-identity N/A

          \[\leadsto \left(\mathsf{sum3}\left(re, \left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right), 1\right) \cdot \frac{-1}{6}\right) \cdot {im}^{3} \]

       9. associate-*r* N/A

          \[\leadsto \left(\mathsf{sum3}\left(re, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right), 1\right) \cdot \frac{-1}{6}\right) \cdot {im}^{3} \]

       10. unpow2 N/A

           \[\leadsto \left(\mathsf{sum3}\left(re, \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right), 1\right) \cdot \frac{-1}{6}\right) \cdot {im}^{3} \]

       11. sum3-define N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

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

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

   11. Simplified 39.2%

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

   if -6.5e20 < re < 4.7e14

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 91.1%

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

   5. Simplified 91.1%

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

   if 4.7e14 < re < 1.01999999999999991e103

   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 4.3%

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

   5. Simplified 4.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   8. Simplified 15.7%

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

   9. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

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

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

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

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

   11. Simplified 60.2%

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

   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 76.9%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 67.3%

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

   8. Simplified 67.3%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

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

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 76.9%

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

   11. Simplified 76.9%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.35 \cdot 10^{+165}:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;im \cdot \left(\frac{1 - re \cdot re}{1 - re \cdot \left(re \cdot re\right)} \cdot \left(re \cdot \left(re + 1\right) + 1\right)\right)\\ \mathbf{elif}\;re \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;\left(-0.16666666666666666 + re \cdot \left(\left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{+14}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(re \cdot \left(\left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) \cdot -0.16666666666666666\right) + \frac{re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right)}{im \cdot im}\right)\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -0.032:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.0052:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -0.032) t_0 (if (<= re 0.0052) (* (sin im) (+ re 1.0)) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -0.032) {
		tmp = t_0;
	} else if (re <= 0.0052) {
		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 <= (-0.032d0)) then
        tmp = t_0
    else if (re <= 0.0052d0) 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 <= -0.032) {
		tmp = t_0;
	} else if (re <= 0.0052) {
		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 <= -0.032:
		tmp = t_0
	elif re <= 0.0052:
		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 <= -0.032)
		tmp = t_0;
	elseif (re <= 0.0052)
		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 <= -0.032)
		tmp = t_0;
	elseif (re <= 0.0052)
		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, -0.032], t$95$0, If[LessEqual[re, 0.0052], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.032000000000000001 or 0.0051999999999999998 < 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 89.9%

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

   if -0.032000000000000001 < re < 0.0051999999999999998

   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 97.4%

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

   5. Simplified 97.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.032:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.0052:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\\ \mathbf{if}\;re \leq -2.35 \cdot 10^{+165}:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{+101}:\\ \;\;\;\;im \cdot \left(\frac{1 - re \cdot re}{1 - re \cdot \left(re \cdot re\right)} \cdot \left(re \cdot \left(re + 1\right) + 1\right)\right)\\ \mathbf{elif}\;re \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;\left(-0.16666666666666666 + re \cdot \left(t\_0 \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot t\_0 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* re (+ 0.5 (* re 0.16666666666666666))) 1.0)))
   (if (<= re -2.35e+165)
     (/ (* im im) (+ im (* re (* im (- -1.0 (* re 0.5))))))
     (if (<= re -6.8e+101)
       (*
        im
        (*
         (/ (- 1.0 (* re re)) (- 1.0 (* re (* re re))))
         (+ (* re (+ re 1.0)) 1.0)))
       (if (<= re -6.5e+20)
         (*
          (+ -0.16666666666666666 (* re (* t_0 -0.16666666666666666)))
          (* im (* im im)))
         (* im (+ (* re t_0) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = (re * (0.5 + (re * 0.16666666666666666))) + 1.0;
	double tmp;
	if (re <= -2.35e+165) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else if (re <= -6.8e+101) {
		tmp = im * (((1.0 - (re * re)) / (1.0 - (re * (re * re)))) * ((re * (re + 1.0)) + 1.0));
	} else if (re <= -6.5e+20) {
		tmp = (-0.16666666666666666 + (re * (t_0 * -0.16666666666666666))) * (im * (im * im));
	} else {
		tmp = im * ((re * t_0) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * (0.5d0 + (re * 0.16666666666666666d0))) + 1.0d0
    if (re <= (-2.35d+165)) then
        tmp = (im * im) / (im + (re * (im * ((-1.0d0) - (re * 0.5d0)))))
    else if (re <= (-6.8d+101)) then
        tmp = im * (((1.0d0 - (re * re)) / (1.0d0 - (re * (re * re)))) * ((re * (re + 1.0d0)) + 1.0d0))
    else if (re <= (-6.5d+20)) then
        tmp = ((-0.16666666666666666d0) + (re * (t_0 * (-0.16666666666666666d0)))) * (im * (im * im))
    else
        tmp = im * ((re * t_0) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * (0.5 + (re * 0.16666666666666666))) + 1.0;
	double tmp;
	if (re <= -2.35e+165) {
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	} else if (re <= -6.8e+101) {
		tmp = im * (((1.0 - (re * re)) / (1.0 - (re * (re * re)))) * ((re * (re + 1.0)) + 1.0));
	} else if (re <= -6.5e+20) {
		tmp = (-0.16666666666666666 + (re * (t_0 * -0.16666666666666666))) * (im * (im * im));
	} else {
		tmp = im * ((re * t_0) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * (0.5 + (re * 0.16666666666666666))) + 1.0
	tmp = 0
	if re <= -2.35e+165:
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))))
	elif re <= -6.8e+101:
		tmp = im * (((1.0 - (re * re)) / (1.0 - (re * (re * re)))) * ((re * (re + 1.0)) + 1.0))
	elif re <= -6.5e+20:
		tmp = (-0.16666666666666666 + (re * (t_0 * -0.16666666666666666))) * (im * (im * im))
	else:
		tmp = im * ((re * t_0) + 1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))) + 1.0)
	tmp = 0.0
	if (re <= -2.35e+165)
		tmp = Float64(Float64(im * im) / Float64(im + Float64(re * Float64(im * Float64(-1.0 - Float64(re * 0.5))))));
	elseif (re <= -6.8e+101)
		tmp = Float64(im * Float64(Float64(Float64(1.0 - Float64(re * re)) / Float64(1.0 - Float64(re * Float64(re * re)))) * Float64(Float64(re * Float64(re + 1.0)) + 1.0)));
	elseif (re <= -6.5e+20)
		tmp = Float64(Float64(-0.16666666666666666 + Float64(re * Float64(t_0 * -0.16666666666666666))) * Float64(im * Float64(im * im)));
	else
		tmp = Float64(im * Float64(Float64(re * t_0) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * (0.5 + (re * 0.16666666666666666))) + 1.0;
	tmp = 0.0;
	if (re <= -2.35e+165)
		tmp = (im * im) / (im + (re * (im * (-1.0 - (re * 0.5)))));
	elseif (re <= -6.8e+101)
		tmp = im * (((1.0 - (re * re)) / (1.0 - (re * (re * re)))) * ((re * (re + 1.0)) + 1.0));
	elseif (re <= -6.5e+20)
		tmp = (-0.16666666666666666 + (re * (t_0 * -0.16666666666666666))) * (im * (im * im));
	else
		tmp = im * ((re * t_0) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[re, -2.35e+165], N[(N[(im * im), $MachinePrecision] / N[(im + N[(re * N[(im * N[(-1.0 - N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -6.8e+101], N[(im * N[(N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * N[(re + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -6.5e+20], N[(N[(-0.16666666666666666 + N[(re * N[(t$95$0 * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.35000000000000008e165

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 1.7%

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

   8. Simplified 1.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

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

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

   10. Applied egg-rr 0.5%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 91.3%

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

   13. Simplified 91.3%

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

   if -2.35000000000000008e165 < re < -6.80000000000000034e101

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 2.5%

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

   5. Simplified 2.5%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 2.1%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. metadata-eval N/A

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

      4. flip3-- N/A

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

      5. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

      11. cube-unmult N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

      17. distribute-rgt-out N/A

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

      18. +-commutative N/A

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

      19. flip-+ N/A

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

      20. metadata-eval N/A

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

      21. un-div-inv N/A

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

   10. Applied egg-rr 80.0%

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

   if -6.80000000000000034e101 < re < -6.5e20

   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 2.6%

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

   5. Simplified 2.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   8. Simplified 2.3%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. sub-neg N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. distribute-lft1-in N/A

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

       6. distribute-lft-in N/A

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

       7. sum3-define N/A

          \[\leadsto \left(\mathsf{sum3}\left(\left(re \cdot 1\right), \left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right), 1\right) \cdot \frac{-1}{6}\right) \cdot {im}^{3} \]

       8. *-rgt-identity N/A

          \[\leadsto \left(\mathsf{sum3}\left(re, \left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right), 1\right) \cdot \frac{-1}{6}\right) \cdot {im}^{3} \]

       9. associate-*r* N/A

          \[\leadsto \left(\mathsf{sum3}\left(re, \left(\left(re \cdot re\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right), 1\right) \cdot \frac{-1}{6}\right) \cdot {im}^{3} \]

       10. unpow2 N/A

           \[\leadsto \left(\mathsf{sum3}\left(re, \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right), 1\right) \cdot \frac{-1}{6}\right) \cdot {im}^{3} \]

       11. sum3-define N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

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

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

   11. Simplified 39.2%

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

   if -6.5e20 < 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 89.4%

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

   5. Simplified 89.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 46.8%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.35 \cdot 10^{+165}:\\ \;\;\;\;\frac{im \cdot im}{im + re \cdot \left(im \cdot \left(-1 - re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{+101}:\\ \;\;\;\;im \cdot \left(\frac{1 - re \cdot re}{1 - re \cdot \left(re \cdot re\right)} \cdot \left(re \cdot \left(re + 1\right) + 1\right)\right)\\ \mathbf{elif}\;re \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;\left(-0.16666666666666666 + re \cdot \left(\left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) \cdot -0.16666666666666666\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.7% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 98.4%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 2.0%

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

   8. Simplified 2.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

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

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

   10. Applied egg-rr 1.2%

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

   11. Taylor expanded in re around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 52.5%

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

   13. Simplified 52.5%

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

   if -1.6000000000000001 < 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 90.7%

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

   5. Simplified 90.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 47.5%

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

4. Final simplification 48.7%

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

Alternative 9: 39.9% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.5 \cdot 10^{+47}:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot 0.008333333333333333\right)\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9.5e+47)
   (*
    im
    (+
     (* im (* im (+ -0.16666666666666666 (* (* im im) 0.008333333333333333))))
     1.0))
   (* im (* re (* 0.16666666666666666 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9.5e+47) {
		tmp = im * ((im * (im * (-0.16666666666666666 + ((im * im) * 0.008333333333333333)))) + 1.0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 9.5d+47) then
        tmp = im * ((im * (im * ((-0.16666666666666666d0) + ((im * im) * 0.008333333333333333d0)))) + 1.0d0)
    else
        tmp = im * (re * (0.16666666666666666d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9.5e+47) {
		tmp = im * ((im * (im * (-0.16666666666666666 + ((im * im) * 0.008333333333333333)))) + 1.0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9.5e+47:
		tmp = im * ((im * (im * (-0.16666666666666666 + ((im * im) * 0.008333333333333333)))) + 1.0)
	else:
		tmp = im * (re * (0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9.5e+47)
		tmp = Float64(im * Float64(Float64(im * Float64(im * Float64(-0.16666666666666666 + Float64(Float64(im * im) * 0.008333333333333333)))) + 1.0));
	else
		tmp = Float64(im * Float64(re * Float64(0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9.5e+47)
		tmp = im * ((im * (im * (-0.16666666666666666 + ((im * im) * 0.008333333333333333)))) + 1.0);
	else
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9.5e+47], N[(im * N[(N[(im * N[(im * N[(-0.16666666666666666 + N[(N[(im * im), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.50000000000000001e47

   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 63.7%

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

   5. Simplified 63.7%

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

   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. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 29.7%

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

   8. Simplified 29.7%

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

   if 9.50000000000000001e47 < 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 78.7%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 60.6%

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

   8. Simplified 60.6%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

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

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 68.6%

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

   11. Simplified 68.6%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9.5 \cdot 10^{+47}:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot 0.008333333333333333\right)\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.8% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.65 \cdot 10^{+53}:\\ \;\;\;\;\left(im \cdot \left(im \cdot -0.16666666666666666\right) + 1\right) \cdot \left(im \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.65e+53)
   (* (+ (* im (* im -0.16666666666666666)) 1.0) (* im (+ re 1.0)))
   (* im (* re (* 0.16666666666666666 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.65e+53) {
		tmp = ((im * (im * -0.16666666666666666)) + 1.0) * (im * (re + 1.0));
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.65d+53) then
        tmp = ((im * (im * (-0.16666666666666666d0))) + 1.0d0) * (im * (re + 1.0d0))
    else
        tmp = im * (re * (0.16666666666666666d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.65e+53) {
		tmp = ((im * (im * -0.16666666666666666)) + 1.0) * (im * (re + 1.0));
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.65e+53:
		tmp = ((im * (im * -0.16666666666666666)) + 1.0) * (im * (re + 1.0))
	else:
		tmp = im * (re * (0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.65e+53)
		tmp = Float64(Float64(Float64(im * Float64(im * -0.16666666666666666)) + 1.0) * Float64(im * Float64(re + 1.0)));
	else
		tmp = Float64(im * Float64(re * Float64(0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.65e+53)
		tmp = ((im * (im * -0.16666666666666666)) + 1.0) * (im * (re + 1.0));
	else
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.65e+53], N[(N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.6500000000000001e53

   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 65.7%

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

   5. Simplified 65.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   8. Simplified 29.1%

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

      1. *-commutative N/A

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

      2. flip3-+ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. cube-mult N/A

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

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

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

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

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      20. *-lowering-*.f64 29.0%

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

   10. Applied egg-rr 29.0%

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

   11. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-out N/A

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

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

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

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

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. *-rgt-identity N/A

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

       14. distribute-lft-in N/A

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

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

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

       16. +-commutative N/A

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

       17. +-lowering-+.f64 29.3%

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

   13. Simplified 29.3%

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

   if 1.6500000000000001e53 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 80.4%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 61.9%

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

   8. Simplified 61.9%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

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

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 70.1%

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

   11. Simplified 70.1%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.65 \cdot 10^{+53}:\\ \;\;\;\;\left(im \cdot \left(im \cdot -0.16666666666666666\right) + 1\right) \cdot \left(im \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.7% accurate, 11.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e+48)
   (* im (+ (* im (* im (* (* im im) 0.008333333333333333))) 1.0))
   (* im (* re (* 0.16666666666666666 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e+48) {
		tmp = im * ((im * (im * ((im * im) * 0.008333333333333333))) + 1.0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1d+48) then
        tmp = im * ((im * (im * ((im * im) * 0.008333333333333333d0))) + 1.0d0)
    else
        tmp = im * (re * (0.16666666666666666d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1e+48) {
		tmp = im * ((im * (im * ((im * im) * 0.008333333333333333))) + 1.0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1e+48:
		tmp = im * ((im * (im * ((im * im) * 0.008333333333333333))) + 1.0)
	else:
		tmp = im * (re * (0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e+48)
		tmp = Float64(im * Float64(Float64(im * Float64(im * Float64(Float64(im * im) * 0.008333333333333333))) + 1.0));
	else
		tmp = Float64(im * Float64(re * Float64(0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1e+48)
		tmp = im * ((im * (im * ((im * im) * 0.008333333333333333))) + 1.0);
	else
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e+48], N[(im * N[(N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.00000000000000004e48

   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 63.7%

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

   5. Simplified 63.7%

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

   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. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 29.7%

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

   8. Simplified 29.7%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{3}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left({im}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 29.3%

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

   11. Simplified 29.3%

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

   if 1.00000000000000004e48 < 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 78.7%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 60.6%

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

   8. Simplified 60.6%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

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

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 68.6%

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

   11. Simplified 68.6%

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

4. Final simplification 36.5%

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

Alternative 12: 39.6% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* im (+ (* re (+ (* re (+ 0.5 (* re 0.16666666666666666))) 1.0)) 1.0)))

C

double code(double re, double im) {
	return im * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * ((re * ((re * (0.5d0 + (re * 0.16666666666666666d0))) + 1.0d0)) + 1.0d0)
end function

Java

public static double code(double re, double im) {
	return im * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
}

Python

def code(re, im):
	return im * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0)

Julia

function code(re, im)
	return Float64(im * Float64(Float64(re * Float64(Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))) + 1.0)) + 1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = im * ((re * ((re * (0.5 + (re * 0.16666666666666666))) + 1.0)) + 1.0);
end

Wolfram

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

Derivation

1. 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 69.2%

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

5. Simplified 69.2%

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

6. Taylor expanded in im around 0

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

8. Simplified 36.5%

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

9. Final simplification 36.5%

   \[\leadsto im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right) + 1\right) + 1\right) \]
10. Add Preprocessing

Alternative 13: 39.8% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{+52}:\\ \;\;\;\;im \cdot \left(\left(im \cdot im\right) \cdot -0.16666666666666666 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.2e+52)
   (* im (+ (* (* im im) -0.16666666666666666) 1.0))
   (* im (* re (* 0.16666666666666666 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.2e+52) {
		tmp = im * (((im * im) * -0.16666666666666666) + 1.0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.2d+52) then
        tmp = im * (((im * im) * (-0.16666666666666666d0)) + 1.0d0)
    else
        tmp = im * (re * (0.16666666666666666d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.2e+52) {
		tmp = im * (((im * im) * -0.16666666666666666) + 1.0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.2e+52:
		tmp = im * (((im * im) * -0.16666666666666666) + 1.0)
	else:
		tmp = im * (re * (0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.2e+52)
		tmp = Float64(im * Float64(Float64(Float64(im * im) * -0.16666666666666666) + 1.0));
	else
		tmp = Float64(im * Float64(re * Float64(0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.2e+52)
		tmp = im * (((im * im) * -0.16666666666666666) + 1.0);
	else
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.2e+52], N[(im * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.2e52

   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 63.4%

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

   5. Simplified 63.4%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 29.1%

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

   8. Simplified 29.1%

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

   if 1.2e52 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 80.4%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 61.9%

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

   8. Simplified 61.9%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

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

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 70.1%

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

   11. Simplified 70.1%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{+52}:\\ \;\;\;\;im \cdot \left(\left(im \cdot im\right) \cdot -0.16666666666666666 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.8:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.8)
   (* im (+ re 1.0))
   (* im (* re (* 0.16666666666666666 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.8) {
		tmp = im * (re + 1.0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.8d0) then
        tmp = im * (re + 1.0d0)
    else
        tmp = im * (re * (0.16666666666666666d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.8) {
		tmp = im * (re + 1.0);
	} else {
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.8:
		tmp = im * (re + 1.0)
	else:
		tmp = im * (re * (0.16666666666666666 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.8)
		tmp = Float64(im * Float64(re + 1.0));
	else
		tmp = Float64(im * Float64(re * Float64(0.16666666666666666 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.8)
		tmp = im * (re + 1.0);
	else
		tmp = im * (re * (0.16666666666666666 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.8], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 67.8%

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

   5. Simplified 67.8%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 29.9%

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

   if 2.7999999999999998 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 80.4%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 53.1%

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

   8. Simplified 53.1%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

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

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 59.8%

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

   11. Simplified 59.8%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.8:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.0% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.7:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.7) (* im (+ re 1.0)) (* 0.5 (* im (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.7) {
		tmp = im * (re + 1.0);
	} else {
		tmp = 0.5 * (im * (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 <= 2.7d0) then
        tmp = im * (re + 1.0d0)
    else
        tmp = 0.5d0 * (im * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.7) {
		tmp = im * (re + 1.0);
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.7:
		tmp = im * (re + 1.0)
	else:
		tmp = 0.5 * (im * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.7)
		tmp = Float64(im * Float64(re + 1.0));
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.7)
		tmp = im * (re + 1.0);
	else
		tmp = 0.5 * (im * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.7], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.7000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 67.8%

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

   5. Simplified 67.8%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 29.9%

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

   if 2.7000000000000002 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 80.4%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 37.8%

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

   8. Simplified 37.8%

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

   9. Taylor expanded in re around inf

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 51.2%

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

   11. Simplified 51.2%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.7:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 28.2% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.08 \cdot 10^{+80}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1.08e+80) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.08e+80) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.08d+80) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.08e+80) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.08e+80:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.08e+80)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.08e+80)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.08e+80], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.08e80

   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 51.0%

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

   5. Simplified 51.0%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 27.9%

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

   if 1.08e80 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 11.5%

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

   9. Taylor expanded in re around inf

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

   11. Simplified 13.0%

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

Alternative 17: 29.6% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(re + 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ re 1.0)))

C

double code(double re, double im) {
	return im * (re + 1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * (re + 1.0d0)
end function

Java

public static double code(double re, double im) {
	return im * (re + 1.0);
}

Python

def code(re, im):
	return im * (re + 1.0)

Julia

function code(re, im)
	return Float64(im * Float64(re + 1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (re + 1.0);
end

Wolfram

code[re_, im_] := N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 53.9%

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

5. Simplified 53.9%

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

6. Taylor expanded in im around 0

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

8. Simplified 27.4%

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

9. Final simplification 27.4%

   \[\leadsto im \cdot \left(re + 1\right) \]
10. Add Preprocessing

Alternative 18: 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 re around 0

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

   1. sin-lowering-sin.f64 52.5%

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

5. Simplified 52.5%

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

6. Taylor expanded in im around 0

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

8. Simplified 23.7%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024140 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))