math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 24.4s

Alternatives: 17

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: 69.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 2.0)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 1.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 1.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 2.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 2.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 2 < (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 72.1%

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

   if 1 < (exp.f64 re) < 2

   1. Initial program 99.7%

      \[e^{re} \cdot \sin im \]
   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 91.0%

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

   5. Simplified 91.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 1.0) (not (<= (exp re) 2.0))) (* (exp re) im) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 1.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 1.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 1.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 1.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 1.0) || !(exp(re) <= 2.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 1.0) || ~((exp(re) <= 2.0)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1 or 2 < (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 72.1%

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

   if 1 < (exp.f64 re) < 2

   1. Initial program 99.7%

      \[e^{re} \cdot \sin im \]
   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 72.4%

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

   5. Simplified 72.4%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 1 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.05 \lor \neg \left(re \leq 0.037\right) \land re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.05) (and (not (<= re 0.037)) (<= re 1e+103)))
   (* (exp re) im)
   (*
    (sin im)
    (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.05) || (!(re <= 0.037) && (re <= 1e+103))) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.05d0)) .or. (.not. (re <= 0.037d0)) .and. (re <= 1d+103)) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.05) || (!(re <= 0.037) && (re <= 1e+103))) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.05) or (not (re <= 0.037) and (re <= 1e+103)):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.05) || (!(re <= 0.037) && (re <= 1e+103)))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.05) || (~((re <= 0.037)) && (re <= 1e+103)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.05], And[N[Not[LessEqual[re, 0.037]], $MachinePrecision], LessEqual[re, 1e+103]]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.050000000000000003 or 0.0369999999999999982 < re < 1e103

   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 95.5%

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

   if -0.050000000000000003 < re < 0.0369999999999999982 or 1e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.05 \lor \neg \left(re \leq 0.037\right) \land re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{-5} \lor \neg \left(re \leq 0.0105\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.3e-5) (not (<= re 0.0105)))
   (* (exp re) im)
   (* (sin im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -1.3e-5) || !(re <= 0.0105)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-1.3d-5)) .or. (.not. (re <= 0.0105d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -1.3e-5) || !(re <= 0.0105)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -1.3e-5) or not (re <= 0.0105):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -1.3e-5) || !(re <= 0.0105))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -1.3e-5) || ~((re <= 0.0105)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -1.3e-5], N[Not[LessEqual[re, 0.0105]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.29999999999999992e-5 or 0.0105000000000000007 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 92.4%

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

   if -1.29999999999999992e-5 < re < 0.0105000000000000007

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{-5} \lor \neg \left(re \leq 0.0105\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot 0.5\\ t_1 := re \cdot t\_0\\ \mathbf{if}\;re \leq -68:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 145:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+77}:\\ \;\;\;\;\frac{im \cdot \left(1 + t\_1 \cdot \left(re \cdot \left(t\_0 \cdot t\_1\right)\right)\right)}{1 + t\_1 \cdot \left(t\_1 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re 0.5))) (t_1 (* re t_0)))
   (if (<= re -68.0)
     (* (+ re 1.0) (* im (* im (* im -0.16666666666666666))))
     (if (<= re 145.0)
       (sin im)
       (if (<= re 1.65e+77)
         (/
          (* im (+ 1.0 (* t_1 (* re (* t_0 t_1)))))
          (+ 1.0 (* t_1 (+ t_1 -1.0))))
         (* im (* 0.16666666666666666 (* re (* re re)))))))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * 0.5);
	double t_1 = re * t_0;
	double tmp;
	if (re <= -68.0) {
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)));
	} else if (re <= 145.0) {
		tmp = sin(im);
	} else if (re <= 1.65e+77) {
		tmp = (im * (1.0 + (t_1 * (re * (t_0 * t_1))))) / (1.0 + (t_1 * (t_1 + -1.0)));
	} else {
		tmp = im * (0.16666666666666666 * (re * (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 = 1.0d0 + (re * 0.5d0)
    t_1 = re * t_0
    if (re <= (-68.0d0)) then
        tmp = (re + 1.0d0) * (im * (im * (im * (-0.16666666666666666d0))))
    else if (re <= 145.0d0) then
        tmp = sin(im)
    else if (re <= 1.65d+77) then
        tmp = (im * (1.0d0 + (t_1 * (re * (t_0 * t_1))))) / (1.0d0 + (t_1 * (t_1 + (-1.0d0))))
    else
        tmp = im * (0.16666666666666666d0 * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * 0.5);
	double t_1 = re * t_0;
	double tmp;
	if (re <= -68.0) {
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)));
	} else if (re <= 145.0) {
		tmp = Math.sin(im);
	} else if (re <= 1.65e+77) {
		tmp = (im * (1.0 + (t_1 * (re * (t_0 * t_1))))) / (1.0 + (t_1 * (t_1 + -1.0)));
	} else {
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * 0.5)
	t_1 = re * t_0
	tmp = 0
	if re <= -68.0:
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)))
	elif re <= 145.0:
		tmp = math.sin(im)
	elif re <= 1.65e+77:
		tmp = (im * (1.0 + (t_1 * (re * (t_0 * t_1))))) / (1.0 + (t_1 * (t_1 + -1.0)))
	else:
		tmp = im * (0.16666666666666666 * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * 0.5))
	t_1 = Float64(re * t_0)
	tmp = 0.0
	if (re <= -68.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(im * Float64(im * -0.16666666666666666))));
	elseif (re <= 145.0)
		tmp = sin(im);
	elseif (re <= 1.65e+77)
		tmp = Float64(Float64(im * Float64(1.0 + Float64(t_1 * Float64(re * Float64(t_0 * t_1))))) / Float64(1.0 + Float64(t_1 * Float64(t_1 + -1.0))));
	else
		tmp = Float64(im * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * 0.5);
	t_1 = re * t_0;
	tmp = 0.0;
	if (re <= -68.0)
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)));
	elseif (re <= 145.0)
		tmp = sin(im);
	elseif (re <= 1.65e+77)
		tmp = (im * (1.0 + (t_1 * (re * (t_0 * t_1))))) / (1.0 + (t_1 * (t_1 + -1.0)));
	else
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * t$95$0), $MachinePrecision]}, If[LessEqual[re, -68.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 145.0], N[Sin[im], $MachinePrecision], If[LessEqual[re, 1.65e+77], N[(N[(im * N[(1.0 + N[(t$95$1 * N[(re * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -68

   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 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 2.8%

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

   8. Simplified 2.8%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 41.3%

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

   11. Simplified 41.3%

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

   if -68 < re < 145

   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 97.8%

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

   5. Simplified 97.8%

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

   if 145 < re < 1.6499999999999999e77

   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 81.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 3.0%

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

   8. Simplified 3.0%

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

      1. flip3-+ N/A

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

      2. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 43.9%

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

   if 1.6499999999999999e77 < 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 86.0%

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

   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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 75.1%

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

   8. Simplified 75.1%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. associate-+r+ N/A

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

       8. +-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

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

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

       13. *-commutative N/A

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

       14. associate-*r/ N/A

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

       15. *-commutative N/A

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

       16. associate-/l* N/A

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

       17. metadata-eval N/A

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

       18. associate-*r/ N/A

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

       19. distribute-lft-out N/A

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

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

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

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

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

       22. associate-*r/ N/A

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

       23. metadata-eval N/A

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

       24. /-lowering-/.f64 83.8%

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

   11. Simplified 83.8%

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

   12. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

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

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 83.8%

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

   14. Simplified 83.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -68:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 145:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+77}:\\ \;\;\;\;\frac{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(\left(1 + re \cdot 0.5\right) \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\right)\right)\right)}{1 + \left(re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(1 + re \cdot 0.5\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot 0.5\\ t_1 := re \cdot t\_0\\ \mathbf{if}\;re \leq -33:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+77}:\\ \;\;\;\;\frac{im \cdot \left(1 + t\_1 \cdot \left(re \cdot \left(t\_0 \cdot t\_1\right)\right)\right)}{1 + t\_1 \cdot \left(t\_1 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re 0.5))) (t_1 (* re t_0)))
   (if (<= re -33.0)
     (* (+ re 1.0) (* im (* im (* im -0.16666666666666666))))
     (if (<= re 1.65e+77)
       (/
        (* im (+ 1.0 (* t_1 (* re (* t_0 t_1)))))
        (+ 1.0 (* t_1 (+ t_1 -1.0))))
       (* im (* 0.16666666666666666 (* re (* re re))))))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * 0.5);
	double t_1 = re * t_0;
	double tmp;
	if (re <= -33.0) {
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)));
	} else if (re <= 1.65e+77) {
		tmp = (im * (1.0 + (t_1 * (re * (t_0 * t_1))))) / (1.0 + (t_1 * (t_1 + -1.0)));
	} else {
		tmp = im * (0.16666666666666666 * (re * (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 = 1.0d0 + (re * 0.5d0)
    t_1 = re * t_0
    if (re <= (-33.0d0)) then
        tmp = (re + 1.0d0) * (im * (im * (im * (-0.16666666666666666d0))))
    else if (re <= 1.65d+77) then
        tmp = (im * (1.0d0 + (t_1 * (re * (t_0 * t_1))))) / (1.0d0 + (t_1 * (t_1 + (-1.0d0))))
    else
        tmp = im * (0.16666666666666666d0 * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * 0.5);
	double t_1 = re * t_0;
	double tmp;
	if (re <= -33.0) {
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)));
	} else if (re <= 1.65e+77) {
		tmp = (im * (1.0 + (t_1 * (re * (t_0 * t_1))))) / (1.0 + (t_1 * (t_1 + -1.0)));
	} else {
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * 0.5)
	t_1 = re * t_0
	tmp = 0
	if re <= -33.0:
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)))
	elif re <= 1.65e+77:
		tmp = (im * (1.0 + (t_1 * (re * (t_0 * t_1))))) / (1.0 + (t_1 * (t_1 + -1.0)))
	else:
		tmp = im * (0.16666666666666666 * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * 0.5))
	t_1 = Float64(re * t_0)
	tmp = 0.0
	if (re <= -33.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(im * Float64(im * -0.16666666666666666))));
	elseif (re <= 1.65e+77)
		tmp = Float64(Float64(im * Float64(1.0 + Float64(t_1 * Float64(re * Float64(t_0 * t_1))))) / Float64(1.0 + Float64(t_1 * Float64(t_1 + -1.0))));
	else
		tmp = Float64(im * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * 0.5);
	t_1 = re * t_0;
	tmp = 0.0;
	if (re <= -33.0)
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)));
	elseif (re <= 1.65e+77)
		tmp = (im * (1.0 + (t_1 * (re * (t_0 * t_1))))) / (1.0 + (t_1 * (t_1 + -1.0)));
	else
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * t$95$0), $MachinePrecision]}, If[LessEqual[re, -33.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.65e+77], N[(N[(im * N[(1.0 + N[(t$95$1 * N[(re * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -33

   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 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 2.8%

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

   8. Simplified 2.8%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 41.3%

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

   11. Simplified 41.3%

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

   if -33 < re < 1.6499999999999999e77

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 54.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 42.4%

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

   8. Simplified 42.4%

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

      1. flip3-+ N/A

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

      2. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 48.2%

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

   if 1.6499999999999999e77 < 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 86.0%

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

   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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 75.1%

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

   8. Simplified 75.1%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. associate-+r+ N/A

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

       8. +-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

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

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

       13. *-commutative N/A

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

       14. associate-*r/ N/A

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

       15. *-commutative N/A

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

       16. associate-/l* N/A

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

       17. metadata-eval N/A

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

       18. associate-*r/ N/A

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

       19. distribute-lft-out N/A

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

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

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

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

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

       22. associate-*r/ N/A

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

       23. metadata-eval N/A

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

       24. /-lowering-/.f64 83.8%

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

   11. Simplified 83.8%

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

   12. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

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

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 83.8%

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

   14. Simplified 83.8%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -33:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+77}:\\ \;\;\;\;\frac{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(\left(1 + re \cdot 0.5\right) \cdot \left(re \cdot \left(1 + re \cdot 0.5\right)\right)\right)\right)\right)}{1 + \left(re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(re \cdot \left(1 + re \cdot 0.5\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot -0.16666666666666666\right)\\ \mathbf{if}\;re \leq -75:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot t\_0\right)\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+56}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot 0.008333333333333333\right) \cdot \left(0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im -0.16666666666666666))))
   (if (<= re -75.0)
     (* (+ re 1.0) (* im t_0))
     (if (<= re 4.5e+56)
       (* (+ 1.0 (* re (+ 1.0 (* re 0.5)))) (* im (+ 1.0 t_0)))
       (*
        im
        (*
         (* re (* re re))
         (+
          0.16666666666666666
          (*
           (+ -0.16666666666666666 (* (* im im) 0.008333333333333333))
           (* 0.16666666666666666 (* im im))))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -75.0) {
		tmp = (re + 1.0) * (im * t_0);
	} else if (re <= 4.5e+56) {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + t_0));
	} else {
		tmp = im * ((re * (re * re)) * (0.16666666666666666 + ((-0.16666666666666666 + ((im * im) * 0.008333333333333333)) * (0.16666666666666666 * (im * im)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im * (-0.16666666666666666d0))
    if (re <= (-75.0d0)) then
        tmp = (re + 1.0d0) * (im * t_0)
    else if (re <= 4.5d+56) then
        tmp = (1.0d0 + (re * (1.0d0 + (re * 0.5d0)))) * (im * (1.0d0 + t_0))
    else
        tmp = im * ((re * (re * re)) * (0.16666666666666666d0 + (((-0.16666666666666666d0) + ((im * im) * 0.008333333333333333d0)) * (0.16666666666666666d0 * (im * im)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -75.0) {
		tmp = (re + 1.0) * (im * t_0);
	} else if (re <= 4.5e+56) {
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + t_0));
	} else {
		tmp = im * ((re * (re * re)) * (0.16666666666666666 + ((-0.16666666666666666 + ((im * im) * 0.008333333333333333)) * (0.16666666666666666 * (im * im)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * -0.16666666666666666)
	tmp = 0
	if re <= -75.0:
		tmp = (re + 1.0) * (im * t_0)
	elif re <= 4.5e+56:
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + t_0))
	else:
		tmp = im * ((re * (re * re)) * (0.16666666666666666 + ((-0.16666666666666666 + ((im * im) * 0.008333333333333333)) * (0.16666666666666666 * (im * im)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * -0.16666666666666666))
	tmp = 0.0
	if (re <= -75.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * t_0));
	elseif (re <= 4.5e+56)
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))) * Float64(im * Float64(1.0 + t_0)));
	else
		tmp = Float64(im * Float64(Float64(re * Float64(re * re)) * Float64(0.16666666666666666 + Float64(Float64(-0.16666666666666666 + Float64(Float64(im * im) * 0.008333333333333333)) * Float64(0.16666666666666666 * Float64(im * im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * -0.16666666666666666);
	tmp = 0.0;
	if (re <= -75.0)
		tmp = (re + 1.0) * (im * t_0);
	elseif (re <= 4.5e+56)
		tmp = (1.0 + (re * (1.0 + (re * 0.5)))) * (im * (1.0 + t_0));
	else
		tmp = im * ((re * (re * re)) * (0.16666666666666666 + ((-0.16666666666666666 + ((im * im) * 0.008333333333333333)) * (0.16666666666666666 * (im * im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -75.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.5e+56], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(-0.16666666666666666 + N[(N[(im * im), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -75

   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 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 2.8%

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

   8. Simplified 2.8%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 41.3%

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

   11. Simplified 41.3%

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

   if -75 < re < 4.5000000000000003e56

   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 90.2%

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

   5. Simplified 90.2%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 46.8%

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

   8. Simplified 46.8%

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

   if 4.5000000000000003e56 < 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 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

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

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

   8. Simplified 33.8%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{im \cdot \left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({re}^{3}\right), \color{blue}{\left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)}\right)\right) \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

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

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

       14. +-commutative N/A

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

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

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

       16. *-commutative N/A

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

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

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

       18. unpow2 N/A

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

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

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

       20. *-commutative N/A

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

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

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

       22. unpow2 N/A

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

       23. *-lowering-*.f64 75.0%

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

   11. Simplified 75.0%

       \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot 0.008333333333333333\right) \cdot \left(\left(im \cdot im\right) \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -75:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+56}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right) \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot 0.008333333333333333\right) \cdot \left(0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.9% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 2.8%

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

   8. Simplified 2.8%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 40.8%

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

   11. Simplified 40.8%

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

   if -1.6000000000000001 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 61.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 52.5%

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

   8. Simplified 52.5%

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

4. Final simplification 49.5%

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

Alternative 10: 47.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot -0.16666666666666666\right)\\ \mathbf{if}\;re \leq -50:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot t\_0\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+56}:\\ \;\;\;\;im \cdot \left(1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im -0.16666666666666666))))
   (if (<= re -50.0)
     (* (+ re 1.0) (* im t_0))
     (if (<= re 5.8e+56)
       (* im (+ 1.0 t_0))
       (* im (* 0.16666666666666666 (* re (* re re))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -50.0) {
		tmp = (re + 1.0) * (im * t_0);
	} else if (re <= 5.8e+56) {
		tmp = im * (1.0 + t_0);
	} else {
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im * (-0.16666666666666666d0))
    if (re <= (-50.0d0)) then
        tmp = (re + 1.0d0) * (im * t_0)
    else if (re <= 5.8d+56) then
        tmp = im * (1.0d0 + t_0)
    else
        tmp = im * (0.16666666666666666d0 * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * -0.16666666666666666);
	double tmp;
	if (re <= -50.0) {
		tmp = (re + 1.0) * (im * t_0);
	} else if (re <= 5.8e+56) {
		tmp = im * (1.0 + t_0);
	} else {
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * -0.16666666666666666)
	tmp = 0
	if re <= -50.0:
		tmp = (re + 1.0) * (im * t_0)
	elif re <= 5.8e+56:
		tmp = im * (1.0 + t_0)
	else:
		tmp = im * (0.16666666666666666 * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * -0.16666666666666666))
	tmp = 0.0
	if (re <= -50.0)
		tmp = Float64(Float64(re + 1.0) * Float64(im * t_0));
	elseif (re <= 5.8e+56)
		tmp = Float64(im * Float64(1.0 + t_0));
	else
		tmp = Float64(im * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * -0.16666666666666666);
	tmp = 0.0;
	if (re <= -50.0)
		tmp = (re + 1.0) * (im * t_0);
	elseif (re <= 5.8e+56)
		tmp = im * (1.0 + t_0);
	else
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -50.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.8e+56], N[(im * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -50

   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 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 2.8%

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

   8. Simplified 2.8%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 41.3%

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

   11. Simplified 41.3%

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

   if -50 < re < 5.80000000000000014e56

   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 89.0%

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

   5. Simplified 89.0%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 46.2%

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

   8. Simplified 46.2%

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

   if 5.80000000000000014e56 < 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 86.3%

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

   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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. associate-+r+ N/A

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

       8. +-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

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

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

       13. *-commutative N/A

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

       14. associate-*r/ N/A

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

       15. *-commutative N/A

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

       16. associate-/l* N/A

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

       17. metadata-eval N/A

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

       18. associate-*r/ N/A

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

       19. distribute-lft-out N/A

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

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

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

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

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

       22. associate-*r/ N/A

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

       23. metadata-eval N/A

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

       24. /-lowering-/.f64 73.2%

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

   11. Simplified 73.2%

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

   12. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

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

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 73.2%

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

   14. Simplified 73.2%

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

Alternative 11: 47.6% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8:\\ \;\;\;\;\left(re + 1\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8)
   (* (+ re 1.0) (* im (* im (* im -0.16666666666666666))))
   (+ im (* im (* 0.16666666666666666 (* re (* re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8) {
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)));
	} else {
		tmp = im + (im * (0.16666666666666666 * (re * (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 <= (-1.8d0)) then
        tmp = (re + 1.0d0) * (im * (im * (im * (-0.16666666666666666d0))))
    else
        tmp = im + (im * (0.16666666666666666d0 * (re * (re * re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8) {
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)));
	} else {
		tmp = im + (im * (0.16666666666666666 * (re * (re * re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8:
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)))
	else:
		tmp = im + (im * (0.16666666666666666 * (re * (re * re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8)
		tmp = Float64(Float64(re + 1.0) * Float64(im * Float64(im * Float64(im * -0.16666666666666666))));
	else
		tmp = Float64(im + Float64(im * Float64(0.16666666666666666 * Float64(re * Float64(re * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8)
		tmp = (re + 1.0) * (im * (im * (im * -0.16666666666666666)));
	else
		tmp = im + (im * (0.16666666666666666 * (re * (re * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im + N[(im * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.80000000000000004

   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 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 2.8%

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

   8. Simplified 2.8%

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

   9. Taylor expanded in im around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 40.8%

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

   11. Simplified 40.8%

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

   if -1.80000000000000004 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 61.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im + re \cdot \left(im + re \cdot \left(\frac{1}{6} \cdot \left(im \cdot re\right) + \frac{1}{2} \cdot im\right)\right)} \]
   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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 50.5%

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

   8. Simplified 50.5%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. associate-+r+ N/A

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

       8. +-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

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

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

       13. *-commutative N/A

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

       14. associate-*r/ N/A

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

       15. *-commutative N/A

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

       16. associate-/l* N/A

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

       17. metadata-eval N/A

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

       18. associate-*r/ N/A

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

       19. distribute-lft-out N/A

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

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

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

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

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

       22. associate-*r/ N/A

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

       23. metadata-eval N/A

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

       24. /-lowering-/.f64 37.8%

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

   11. Simplified 37.8%

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

   12. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

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

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 52.0%

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

   14. Simplified 52.0%

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

Alternative 12: 39.9% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+56}:\\ \;\;\;\;im \cdot \left(1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9e+56)
   (* im (+ 1.0 (* im (* im -0.16666666666666666))))
   (* im (* 0.16666666666666666 (* re (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9e+56) {
		tmp = im * (1.0 + (im * (im * -0.16666666666666666)));
	} else {
		tmp = im * (0.16666666666666666 * (re * (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 <= 9d+56) then
        tmp = im * (1.0d0 + (im * (im * (-0.16666666666666666d0))))
    else
        tmp = im * (0.16666666666666666d0 * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9e+56) {
		tmp = im * (1.0 + (im * (im * -0.16666666666666666)));
	} else {
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9e+56:
		tmp = im * (1.0 + (im * (im * -0.16666666666666666)))
	else:
		tmp = im * (0.16666666666666666 * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9e+56)
		tmp = Float64(im * Float64(1.0 + Float64(im * Float64(im * -0.16666666666666666))));
	else
		tmp = Float64(im * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9e+56)
		tmp = im * (1.0 + (im * (im * -0.16666666666666666)));
	else
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9e+56], N[(im * N[(1.0 + N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.0000000000000006e56

   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 62.2%

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

   5. Simplified 62.2%

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

   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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 32.8%

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

   8. Simplified 32.8%

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

   if 9.0000000000000006e56 < 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 86.3%

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

   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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. associate-+r+ N/A

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

       8. +-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

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

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

       13. *-commutative N/A

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

       14. associate-*r/ N/A

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

       15. *-commutative N/A

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

       16. associate-/l* N/A

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

       17. metadata-eval N/A

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

       18. associate-*r/ N/A

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

       19. distribute-lft-out N/A

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

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

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

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

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

       22. associate-*r/ N/A

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

       23. metadata-eval N/A

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

       24. /-lowering-/.f64 73.2%

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

   11. Simplified 73.2%

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

   12. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

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

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 73.2%

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

   14. Simplified 73.2%

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

Alternative 13: 39.7% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.000235:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.000235) im (* im (* 0.16666666666666666 (* re (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.000235) {
		tmp = im;
	} else {
		tmp = im * (0.16666666666666666 * (re * (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 <= 0.000235d0) then
        tmp = im
    else
        tmp = im * (0.16666666666666666d0 * (re * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.000235) {
		tmp = im;
	} else {
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.000235:
		tmp = im
	else:
		tmp = im * (0.16666666666666666 * (re * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.000235)
		tmp = im;
	else
		tmp = Float64(im * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.000235)
		tmp = im;
	else
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.000235], im, N[(im * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.34999999999999993e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 67.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 33.4%

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

   if 2.34999999999999993e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 83.1%

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

   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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 52.2%

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

   8. Simplified 52.2%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. associate-+r+ N/A

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

       8. +-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

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

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

       13. *-commutative N/A

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

       14. associate-*r/ N/A

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

       15. *-commutative N/A

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

       16. associate-/l* N/A

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

       17. metadata-eval N/A

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

       18. associate-*r/ N/A

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

       19. distribute-lft-out N/A

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

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

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

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

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

       22. associate-*r/ N/A

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

       23. metadata-eval N/A

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

       24. /-lowering-/.f64 58.0%

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

   11. Simplified 58.0%

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

   12. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

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

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 58.0%

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

   14. Simplified 58.0%

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

Alternative 14: 37.0% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.000235:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.000235) im (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.000235) {
		tmp = im;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 0.000235d0) then
        tmp = im
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 0.000235) {
		tmp = im;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.000235:
		tmp = im
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.000235)
		tmp = im;
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 0.000235)
		tmp = im;
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.000235], im, N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.34999999999999993e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 67.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 33.4%

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

   if 2.34999999999999993e-4 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 83.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 50.5%

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

   8. Simplified 50.5%

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

   9. Taylor expanded in re around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 50.5%

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

   11. Simplified 50.5%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.000235:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.8% accurate, 25.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 7.5e+14) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.5e+14) {
		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 <= 7.5d+14) 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 <= 7.5e+14) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.5e+14:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.5e+14)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.5e+14)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.5e+14], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.5e14

   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 79.3%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 32.6%

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

   if 7.5e14 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 43.6%

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

   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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 15.5%

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

   8. Simplified 15.5%

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

   9. Taylor expanded in re around 0

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

   11. Simplified 10.7%

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

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{im \cdot re} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 11.0%

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

   14. Simplified 11.0%

       \[\leadsto \color{blue}{im \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.3% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

5. Simplified 71.1%

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

6. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, im\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), im\right) \]

   2. +-lowering-+.f64 29.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), im\right) \]

8. Simplified 29.8%

   \[\leadsto \color{blue}{\left(re + 1\right)} \cdot im \]

9. Final simplification 29.8%

   \[\leadsto im \cdot \left(re + 1\right) \]
10. Add Preprocessing

Alternative 17: 26.2% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 im)

C

double code(double re, double im) {
	return im;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im
end function

Java

public static double code(double re, double im) {
	return im;
}

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

function tmp = code(re, im)
	tmp = im;
end

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \color{blue}{im}\right) \]
4. Step-by-step derivation

5. Simplified 71.1%

   \[\leadsto e^{re} \cdot \color{blue}{im} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im} \]
7. Step-by-step derivation

8. Simplified 25.7%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))