math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 14.7s

Alternatives: 25

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 10^{-125}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im \cdot \left(re + \left(1 + \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1e-125)
   (exp re)
   (if (<= (exp re) 1.0)
     (*
      (cos im)
      (+ re (+ 1.0 (* (+ 0.5 (* re 0.16666666666666666)) (* re re)))))
     (* (exp re) (+ 1.0 (* im (* im -0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1e-125) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0) {
		tmp = cos(im) * (re + (1.0 + ((0.5 + (re * 0.16666666666666666)) * (re * re))));
	} else {
		tmp = exp(re) * (1.0 + (im * (im * -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 (exp(re) <= 1d-125) then
        tmp = exp(re)
    else if (exp(re) <= 1.0d0) then
        tmp = cos(im) * (re + (1.0d0 + ((0.5d0 + (re * 0.16666666666666666d0)) * (re * re))))
    else
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 1e-125) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.cos(im) * (re + (1.0 + ((0.5 + (re * 0.16666666666666666)) * (re * re))));
	} else {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 1e-125:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0:
		tmp = math.cos(im) * (re + (1.0 + ((0.5 + (re * 0.16666666666666666)) * (re * re))))
	else:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 1e-125)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = Float64(cos(im) * Float64(re + Float64(1.0 + Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * Float64(re * re)))));
	else
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 1e-125)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = cos(im) * (re + (1.0 + ((0.5 + (re * 0.16666666666666666)) * (re * re))));
	else
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 1e-125], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[(N[Cos[im], $MachinePrecision] * N[(re + N[(1.0 + N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 1.00000000000000001e-125

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 100.0%

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

   if 1.00000000000000001e-125 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.5%

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

   5. Simplified 99.5%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate-+l+ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 93.1%

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

   5. Simplified 93.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 10^{-125}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im \cdot \left(re + \left(1 + \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 10^{-125}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1e-125)
   (exp re)
   (if (<= (exp re) 1.0)
     (*
      (cos im)
      (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))
     (* (exp re) (+ 1.0 (* im (* im -0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1e-125) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0) {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = exp(re) * (1.0 + (im * (im * -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 (exp(re) <= 1d-125) then
        tmp = exp(re)
    else if (exp(re) <= 1.0d0) then
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    else
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 1e-125) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	} else {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 1e-125:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	else:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 1e-125)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))));
	else
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 1e-125)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	else
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 1e-125], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[(N[Cos[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], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 1.00000000000000001e-125

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 100.0%

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

   if 1.00000000000000001e-125 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.5%

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

   5. Simplified 99.5%

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

   if 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 93.1%

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

   5. Simplified 93.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 10^{-125}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 10^{-125}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1e-125)
   (exp re)
   (if (<= (exp re) 1.0)
     (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))
     (* (exp re) (+ 1.0 (* im (* im -0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1e-125) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0) {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = exp(re) * (1.0 + (im * (im * -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 (exp(re) <= 1d-125) then
        tmp = exp(re)
    else if (exp(re) <= 1.0d0) then
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    else
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 1e-125) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	} else {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 1e-125:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	else:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 1e-125)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	else
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 1e-125)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	else
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 1e-125], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 1.00000000000000001e-125

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 100.0%

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

   if 1.00000000000000001e-125 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.5%

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

   5. Simplified 99.5%

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

   if 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 93.1%

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

   5. Simplified 93.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 10^{-125}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 10^{-125}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 1e-125)
   (exp re)
   (if (<= (exp re) 1.0)
     (* (cos im) (+ re 1.0))
     (* (exp re) (+ 1.0 (* im (* im -0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 1e-125) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(re) * (1.0 + (im * (im * -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 (exp(re) <= 1d-125) then
        tmp = exp(re)
    else if (exp(re) <= 1.0d0) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 1e-125) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 1e-125:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 1e-125)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 1e-125)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 1e-125], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 1.00000000000000001e-125

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 100.0%

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

   if 1.00000000000000001e-125 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 99.4%

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

   5. Simplified 99.4%

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

   if 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 93.1%

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

   5. Simplified 93.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 10^{-125}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 620:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ 0.5 (* re 0.16666666666666666)) (* re re))))
   (if (<= re -0.36)
     (exp re)
     (if (<= re 620.0)
       (* (cos im) (+ re 1.0))
       (if (<= re 5e+102)
         (*
          (+ 1.0 (* im (* im -0.5)))
          (/ (- (* re re) (* t_0 t_0)) (- re t_0)))
         (*
          (* re (* re re))
          (*
           (+ 1.0 (* -0.5 (* im im)))
           (+ 0.16666666666666666 (/ 0.5 re)))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re);
	double tmp;
	if (re <= -0.36) {
		tmp = exp(re);
	} else if (re <= 620.0) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 5e+102) {
		tmp = (1.0 + (im * (im * -0.5))) * (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 + (re * 0.16666666666666666d0)) * (re * re)
    if (re <= (-0.36d0)) then
        tmp = exp(re)
    else if (re <= 620.0d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 5d+102) then
        tmp = (1.0d0 + (im * (im * (-0.5d0)))) * (((re * re) - (t_0 * t_0)) / (re - t_0))
    else
        tmp = (re * (re * re)) * ((1.0d0 + ((-0.5d0) * (im * im))) * (0.16666666666666666d0 + (0.5d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re);
	double tmp;
	if (re <= -0.36) {
		tmp = Math.exp(re);
	} else if (re <= 620.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 5e+102) {
		tmp = (1.0 + (im * (im * -0.5))) * (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re)
	tmp = 0
	if re <= -0.36:
		tmp = math.exp(re)
	elif re <= 620.0:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 5e+102:
		tmp = (1.0 + (im * (im * -0.5))) * (((re * re) - (t_0 * t_0)) / (re - t_0))
	else:
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * Float64(re * re))
	tmp = 0.0
	if (re <= -0.36)
		tmp = exp(re);
	elseif (re <= 620.0)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 5e+102)
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * -0.5))) * Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0)));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(0.16666666666666666 + Float64(0.5 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re);
	tmp = 0.0;
	if (re <= -0.36)
		tmp = exp(re);
	elseif (re <= 620.0)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 5e+102)
		tmp = (1.0 + (im * (im * -0.5))) * (((re * re) - (t_0 * t_0)) / (re - t_0));
	else
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.36], N[Exp[re], $MachinePrecision], If[LessEqual[re, 620.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+102], N[(N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.35999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 100.0%

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

   if -0.35999999999999999 < re < 620

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 99.5%

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

   5. Simplified 99.5%

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

   if 620 < re < 5e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 5.7%

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

   5. Simplified 5.7%

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

   6. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 36.0%

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

   8. Simplified 36.0%

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

   9. Taylor expanded in re around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. distribute-rgt-out N/A

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

   11. Simplified 36.0%

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

       6. metadata-eval N/A

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

       7. swap-sqr N/A

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

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re + \frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       9. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{re \cdot re - \left(\frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right) \cdot \left(\frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right)}{re - \frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(re \cdot re - \left(\frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right) \cdot \left(\frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right)\right), \left(re - \frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

   13. Applied egg-rr 82.4%

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

   if 5e102 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

       11. metadata-eval N/A

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

       12. associate-*r/ N/A

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

       13. distribute-lft-out N/A

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

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

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

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

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

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

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

       17. unpow2 N/A

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

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

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

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

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

       20. associate-*r/ N/A

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

   11. Simplified 90.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 620:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \frac{re \cdot re - \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)}{re - \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\\ t_1 := re + \left(-1 - re \cdot t\_0\right)\\ \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \frac{1}{\frac{t\_1}{\left(1 + re \cdot \left(1 + t\_0\right)\right) \cdot t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* re 0.16666666666666666))))
        (t_1 (+ re (- -1.0 (* re t_0)))))
   (if (<= re -0.36)
     (exp re)
     (if (<= re 5.8e-21)
       (cos im)
       (if (<= re 5e+102)
         (*
          (+ 1.0 (* im (* im -0.5)))
          (/ 1.0 (/ t_1 (* (+ 1.0 (* re (+ 1.0 t_0))) t_1))))
         (*
          (* re (* re re))
          (*
           (+ 1.0 (* -0.5 (* im im)))
           (+ 0.16666666666666666 (/ 0.5 re)))))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = re + (-1.0 - (re * t_0));
	double tmp;
	if (re <= -0.36) {
		tmp = exp(re);
	} else if (re <= 5.8e-21) {
		tmp = cos(im);
	} else if (re <= 5e+102) {
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)));
	} else {
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (0.5d0 + (re * 0.16666666666666666d0))
    t_1 = re + ((-1.0d0) - (re * t_0))
    if (re <= (-0.36d0)) then
        tmp = exp(re)
    else if (re <= 5.8d-21) then
        tmp = cos(im)
    else if (re <= 5d+102) then
        tmp = (1.0d0 + (im * (im * (-0.5d0)))) * (1.0d0 / (t_1 / ((1.0d0 + (re * (1.0d0 + t_0))) * t_1)))
    else
        tmp = (re * (re * re)) * ((1.0d0 + ((-0.5d0) * (im * im))) * (0.16666666666666666d0 + (0.5d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = re + (-1.0 - (re * t_0));
	double tmp;
	if (re <= -0.36) {
		tmp = Math.exp(re);
	} else if (re <= 5.8e-21) {
		tmp = Math.cos(im);
	} else if (re <= 5e+102) {
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)));
	} else {
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (re * 0.16666666666666666))
	t_1 = re + (-1.0 - (re * t_0))
	tmp = 0
	if re <= -0.36:
		tmp = math.exp(re)
	elif re <= 5.8e-21:
		tmp = math.cos(im)
	elif re <= 5e+102:
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)))
	else:
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))
	t_1 = Float64(re + Float64(-1.0 - Float64(re * t_0)))
	tmp = 0.0
	if (re <= -0.36)
		tmp = exp(re);
	elseif (re <= 5.8e-21)
		tmp = cos(im);
	elseif (re <= 5e+102)
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * -0.5))) * Float64(1.0 / Float64(t_1 / Float64(Float64(1.0 + Float64(re * Float64(1.0 + t_0))) * t_1))));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(0.16666666666666666 + Float64(0.5 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (re * 0.16666666666666666));
	t_1 = re + (-1.0 - (re * t_0));
	tmp = 0.0;
	if (re <= -0.36)
		tmp = exp(re);
	elseif (re <= 5.8e-21)
		tmp = cos(im);
	elseif (re <= 5e+102)
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)));
	else
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re + N[(-1.0 - N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.36], N[Exp[re], $MachinePrecision], If[LessEqual[re, 5.8e-21], N[Cos[im], $MachinePrecision], If[LessEqual[re, 5e+102], N[(N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$1 / N[(N[(1.0 + N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.35999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 100.0%

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

   if -0.35999999999999999 < re < 5.8e-21

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 99.1%

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

   5. Simplified 99.1%

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

   if 5.8e-21 < re < 5e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 16.2%

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

   5. Simplified 16.2%

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

   6. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 43.1%

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

   8. Simplified 43.1%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-+r+ N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 84.3%

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

   if 5e102 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

       11. metadata-eval N/A

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

       12. associate-*r/ N/A

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

       13. distribute-lft-out N/A

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

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

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

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

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

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

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

       17. unpow2 N/A

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

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

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

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

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

       20. associate-*r/ N/A

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

   11. Simplified 90.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.36:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \frac{1}{\frac{re + \left(-1 - re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re + \left(-1 - re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\\ t_1 := re + \left(-1 - re \cdot t\_0\right)\\ \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \frac{1}{\frac{t\_1}{\left(1 + re \cdot \left(1 + t\_0\right)\right) \cdot t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* re 0.16666666666666666))))
        (t_1 (+ re (- -1.0 (* re t_0)))))
   (if (<= re -102000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 5.8e-21)
       (cos im)
       (if (<= re 5e+102)
         (*
          (+ 1.0 (* im (* im -0.5)))
          (/ 1.0 (/ t_1 (* (+ 1.0 (* re (+ 1.0 t_0))) t_1))))
         (*
          (* re (* re re))
          (*
           (+ 1.0 (* -0.5 (* im im)))
           (+ 0.16666666666666666 (/ 0.5 re)))))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = re + (-1.0 - (re * t_0));
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 5.8e-21) {
		tmp = cos(im);
	} else if (re <= 5e+102) {
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)));
	} else {
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (0.5d0 + (re * 0.16666666666666666d0))
    t_1 = re + ((-1.0d0) - (re * t_0))
    if (re <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 5.8d-21) then
        tmp = cos(im)
    else if (re <= 5d+102) then
        tmp = (1.0d0 + (im * (im * (-0.5d0)))) * (1.0d0 / (t_1 / ((1.0d0 + (re * (1.0d0 + t_0))) * t_1)))
    else
        tmp = (re * (re * re)) * ((1.0d0 + ((-0.5d0) * (im * im))) * (0.16666666666666666d0 + (0.5d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = re + (-1.0 - (re * t_0));
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 5.8e-21) {
		tmp = Math.cos(im);
	} else if (re <= 5e+102) {
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)));
	} else {
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (re * 0.16666666666666666))
	t_1 = re + (-1.0 - (re * t_0))
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 5.8e-21:
		tmp = math.cos(im)
	elif re <= 5e+102:
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)))
	else:
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))
	t_1 = Float64(re + Float64(-1.0 - Float64(re * t_0)))
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 5.8e-21)
		tmp = cos(im);
	elseif (re <= 5e+102)
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * -0.5))) * Float64(1.0 / Float64(t_1 / Float64(Float64(1.0 + Float64(re * Float64(1.0 + t_0))) * t_1))));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(0.16666666666666666 + Float64(0.5 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (re * 0.16666666666666666));
	t_1 = re + (-1.0 - (re * t_0));
	tmp = 0.0;
	if (re <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 5.8e-21)
		tmp = cos(im);
	elseif (re <= 5e+102)
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)));
	else
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re + N[(-1.0 - N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.8e-21], N[Cos[im], $MachinePrecision], If[LessEqual[re, 5e+102], N[(N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$1 / N[(N[(1.0 + N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -102000

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re < 5.8e-21

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 97.7%

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

   5. Simplified 97.7%

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

   if 5.8e-21 < re < 5e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 16.2%

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

   5. Simplified 16.2%

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

   6. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 43.1%

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

   8. Simplified 43.1%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-+r+ N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 84.3%

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

   if 5e102 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

       11. metadata-eval N/A

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

       12. associate-*r/ N/A

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

       13. distribute-lft-out N/A

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

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

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

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

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

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

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

       17. unpow2 N/A

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

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

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

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

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

       20. associate-*r/ N/A

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

   11. Simplified 90.2%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \frac{1}{\frac{re + \left(-1 - re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re + \left(-1 - re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\\ t_1 := re + \left(-1 - re \cdot t\_0\right)\\ \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \frac{1}{\frac{t\_1}{\left(1 + re \cdot \left(1 + t\_0\right)\right) \cdot t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* re 0.16666666666666666))))
        (t_1 (+ re (- -1.0 (* re t_0)))))
   (if (<= re -102000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 1.6e-35)
       (+ 1.0 (* re (+ 1.0 (* re 0.5))))
       (if (<= re 5e+102)
         (*
          (+ 1.0 (* im (* im -0.5)))
          (/ 1.0 (/ t_1 (* (+ 1.0 (* re (+ 1.0 t_0))) t_1))))
         (*
          (* re (* re re))
          (*
           (+ 1.0 (* -0.5 (* im im)))
           (+ 0.16666666666666666 (/ 0.5 re)))))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = re + (-1.0 - (re * t_0));
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.6e-35) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else if (re <= 5e+102) {
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)));
	} else {
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (0.5d0 + (re * 0.16666666666666666d0))
    t_1 = re + ((-1.0d0) - (re * t_0))
    if (re <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 1.6d-35) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else if (re <= 5d+102) then
        tmp = (1.0d0 + (im * (im * (-0.5d0)))) * (1.0d0 / (t_1 / ((1.0d0 + (re * (1.0d0 + t_0))) * t_1)))
    else
        tmp = (re * (re * re)) * ((1.0d0 + ((-0.5d0) * (im * im))) * (0.16666666666666666d0 + (0.5d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = re + (-1.0 - (re * t_0));
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.6e-35) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else if (re <= 5e+102) {
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)));
	} else {
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (re * 0.16666666666666666))
	t_1 = re + (-1.0 - (re * t_0))
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 1.6e-35:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	elif re <= 5e+102:
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)))
	else:
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))
	t_1 = Float64(re + Float64(-1.0 - Float64(re * t_0)))
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 1.6e-35)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	elseif (re <= 5e+102)
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * -0.5))) * Float64(1.0 / Float64(t_1 / Float64(Float64(1.0 + Float64(re * Float64(1.0 + t_0))) * t_1))));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(0.16666666666666666 + Float64(0.5 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (re * 0.16666666666666666));
	t_1 = re + (-1.0 - (re * t_0));
	tmp = 0.0;
	if (re <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 1.6e-35)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	elseif (re <= 5e+102)
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 / (t_1 / ((1.0 + (re * (1.0 + t_0))) * t_1)));
	else
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re + N[(-1.0 - N[(re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.6e-35], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+102], N[(N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$1 / N[(N[(1.0 + N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -102000

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re < 1.5999999999999999e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 57.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 57.9%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 56.5%

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

   8. Simplified 56.5%

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

   if 1.5999999999999999e-35 < re < 5e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 20.6%

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

   5. Simplified 20.6%

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

   6. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 43.4%

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

   8. Simplified 43.4%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-+r+ N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 82.4%

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

   if 5e102 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

       11. metadata-eval N/A

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

       12. associate-*r/ N/A

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

       13. distribute-lft-out N/A

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

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

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

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

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

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

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

       17. unpow2 N/A

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

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

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

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

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

       20. associate-*r/ N/A

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

   11. Simplified 90.2%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \frac{1}{\frac{re + \left(-1 - re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re + \left(-1 - re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\\ t_1 := 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 360:\\ \;\;\;\;1 + \frac{re \cdot re - t\_1 \cdot t\_1}{re - t\_1}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ 0.5 (* re 0.16666666666666666)) (* re re)))
        (t_1 (* 0.5 (* re re))))
   (if (<= re -102000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 360.0)
       (+ 1.0 (/ (- (* re re) (* t_1 t_1)) (- re t_1)))
       (if (<= re 5e+102)
         (*
          (+ 1.0 (* im (* im -0.5)))
          (/ (- (* re re) (* t_0 t_0)) (- re t_0)))
         (*
          (* re (* re re))
          (*
           (+ 1.0 (* -0.5 (* im im)))
           (+ 0.16666666666666666 (/ 0.5 re)))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re);
	double t_1 = 0.5 * (re * re);
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 360.0) {
		tmp = 1.0 + (((re * re) - (t_1 * t_1)) / (re - t_1));
	} else if (re <= 5e+102) {
		tmp = (1.0 + (im * (im * -0.5))) * (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.5d0 + (re * 0.16666666666666666d0)) * (re * re)
    t_1 = 0.5d0 * (re * re)
    if (re <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 360.0d0) then
        tmp = 1.0d0 + (((re * re) - (t_1 * t_1)) / (re - t_1))
    else if (re <= 5d+102) then
        tmp = (1.0d0 + (im * (im * (-0.5d0)))) * (((re * re) - (t_0 * t_0)) / (re - t_0))
    else
        tmp = (re * (re * re)) * ((1.0d0 + ((-0.5d0) * (im * im))) * (0.16666666666666666d0 + (0.5d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re);
	double t_1 = 0.5 * (re * re);
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 360.0) {
		tmp = 1.0 + (((re * re) - (t_1 * t_1)) / (re - t_1));
	} else if (re <= 5e+102) {
		tmp = (1.0 + (im * (im * -0.5))) * (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re)
	t_1 = 0.5 * (re * re)
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 360.0:
		tmp = 1.0 + (((re * re) - (t_1 * t_1)) / (re - t_1))
	elif re <= 5e+102:
		tmp = (1.0 + (im * (im * -0.5))) * (((re * re) - (t_0 * t_0)) / (re - t_0))
	else:
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * Float64(re * re))
	t_1 = Float64(0.5 * Float64(re * re))
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 360.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_1 * t_1)) / Float64(re - t_1)));
	elseif (re <= 5e+102)
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * -0.5))) * Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0)));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(0.16666666666666666 + Float64(0.5 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 + (re * 0.16666666666666666)) * (re * re);
	t_1 = 0.5 * (re * re);
	tmp = 0.0;
	if (re <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 360.0)
		tmp = 1.0 + (((re * re) - (t_1 * t_1)) / (re - t_1));
	elseif (re <= 5e+102)
		tmp = (1.0 + (im * (im * -0.5))) * (((re * re) - (t_0 * t_0)) / (re - t_0));
	else
		tmp = (re * (re * re)) * ((1.0 + (-0.5 * (im * im))) * (0.16666666666666666 + (0.5 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 360.0], N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+102], N[(N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(0.5 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -102000

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re < 360

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 58.3%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 58.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 56.9%

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

   8. Simplified 56.9%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. flip-+ N/A

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

      4. *-rgt-identity N/A

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

      5. fmm-def N/A

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

      6. *-commutative N/A

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

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

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

   10. Applied egg-rr 56.9%

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

   if 360 < re < 5e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 5.7%

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

   5. Simplified 5.7%

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

   6. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 36.0%

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

   8. Simplified 36.0%

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

   9. Taylor expanded in re around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. distribute-rgt-out N/A

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

   11. Simplified 36.0%

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. *-commutative N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

       6. metadata-eval N/A

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

       7. swap-sqr N/A

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

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re + \frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       9. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{re \cdot re - \left(\frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right) \cdot \left(\frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right)}{re - \frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(re \cdot re - \left(\frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right) \cdot \left(\frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right)\right), \left(re - \frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

   13. Applied egg-rr 82.4%

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

   if 5e102 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in re around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

       11. metadata-eval N/A

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

       12. associate-*r/ N/A

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

       13. distribute-lft-out N/A

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

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

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

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

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

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

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

       17. unpow2 N/A

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

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

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

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

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

       20. associate-*r/ N/A

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

   11. Simplified 90.2%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 360:\\ \;\;\;\;1 + \frac{re \cdot re - \left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)}{re - 0.5 \cdot \left(re \cdot re\right)}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \frac{re \cdot re - \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right) \cdot \left(\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\right)}{re - \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.16666666666666666 + \frac{0.5}{re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right)\\ t_1 := 1 + im \cdot \left(im \cdot -0.5\right)\\ \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 82:\\ \;\;\;\;1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_1 \cdot \left(re \cdot \left(1 + \frac{\left(0.0625 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.0007716049382716049\right) \cdot \frac{re}{0.5 + re \cdot -0.16666666666666666}}{0.25 + \left(re \cdot re\right) \cdot 0.027777777777777776}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* re re))) (t_1 (+ 1.0 (* im (* im -0.5)))))
   (if (<= re -102000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 82.0)
       (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0)))
       (if (<= re 1.35e+154)
         (*
          t_1
          (*
           re
           (+
            1.0
            (/
             (*
              (- 0.0625 (* (* (* re re) (* re re)) 0.0007716049382716049))
              (/ re (+ 0.5 (* re -0.16666666666666666))))
             (+ 0.25 (* (* re re) 0.027777777777777776))))))
         (* t_1 (+ 1.0 (* re (+ 1.0 (* re 0.5))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double t_1 = 1.0 + (im * (im * -0.5));
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 82.0) {
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else if (re <= 1.35e+154) {
		tmp = t_1 * (re * (1.0 + (((0.0625 - (((re * re) * (re * re)) * 0.0007716049382716049)) * (re / (0.5 + (re * -0.16666666666666666)))) / (0.25 + ((re * re) * 0.027777777777777776)))));
	} else {
		tmp = t_1 * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (re * re)
    t_1 = 1.0d0 + (im * (im * (-0.5d0)))
    if (re <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 82.0d0) then
        tmp = 1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0))
    else if (re <= 1.35d+154) then
        tmp = t_1 * (re * (1.0d0 + (((0.0625d0 - (((re * re) * (re * re)) * 0.0007716049382716049d0)) * (re / (0.5d0 + (re * (-0.16666666666666666d0))))) / (0.25d0 + ((re * re) * 0.027777777777777776d0)))))
    else
        tmp = t_1 * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double t_1 = 1.0 + (im * (im * -0.5));
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 82.0) {
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else if (re <= 1.35e+154) {
		tmp = t_1 * (re * (1.0 + (((0.0625 - (((re * re) * (re * re)) * 0.0007716049382716049)) * (re / (0.5 + (re * -0.16666666666666666)))) / (0.25 + ((re * re) * 0.027777777777777776)))));
	} else {
		tmp = t_1 * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (re * re)
	t_1 = 1.0 + (im * (im * -0.5))
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 82.0:
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0))
	elif re <= 1.35e+154:
		tmp = t_1 * (re * (1.0 + (((0.0625 - (((re * re) * (re * re)) * 0.0007716049382716049)) * (re / (0.5 + (re * -0.16666666666666666)))) / (0.25 + ((re * re) * 0.027777777777777776)))))
	else:
		tmp = t_1 * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(re * re))
	t_1 = Float64(1.0 + Float64(im * Float64(im * -0.5)))
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 82.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0)));
	elseif (re <= 1.35e+154)
		tmp = Float64(t_1 * Float64(re * Float64(1.0 + Float64(Float64(Float64(0.0625 - Float64(Float64(Float64(re * re) * Float64(re * re)) * 0.0007716049382716049)) * Float64(re / Float64(0.5 + Float64(re * -0.16666666666666666)))) / Float64(0.25 + Float64(Float64(re * re) * 0.027777777777777776))))));
	else
		tmp = Float64(t_1 * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re * re);
	t_1 = 1.0 + (im * (im * -0.5));
	tmp = 0.0;
	if (re <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 82.0)
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	elseif (re <= 1.35e+154)
		tmp = t_1 * (re * (1.0 + (((0.0625 - (((re * re) * (re * re)) * 0.0007716049382716049)) * (re / (0.5 + (re * -0.16666666666666666)))) / (0.25 + ((re * re) * 0.027777777777777776)))));
	else
		tmp = t_1 * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 82.0], N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.35e+154], N[(t$95$1 * N[(re * N[(1.0 + N[(N[(N[(0.0625 - N[(N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.0007716049382716049), $MachinePrecision]), $MachinePrecision] * N[(re / N[(0.5 + N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(N[(re * re), $MachinePrecision] * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -102000

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re < 82

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 58.3%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 58.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 56.9%

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

   8. Simplified 56.9%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. flip-+ N/A

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

      4. *-rgt-identity N/A

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

      5. fmm-def N/A

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

      6. *-commutative N/A

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

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

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

   10. Applied egg-rr 56.9%

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

   if 82 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 46.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 46.1%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 52.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 52.7%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot {re}^{2}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) + \frac{1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \frac{1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \frac{1}{6} \cdot \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) + \left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right) \cdot re + \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot re\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

   11. Simplified 52.7%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + re \cdot \frac{1}{6}\right) \cdot re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)}{\frac{1}{2} - re \cdot \frac{1}{6}} \cdot re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{4} - \left(re \cdot \frac{1}{6}\right) \cdot \left(re \cdot \frac{1}{6}\right)\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       5. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot re}{\frac{1}{2} - re \cdot \frac{1}{6}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{4} - \left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot \frac{re}{\frac{1}{2} - re \cdot \frac{1}{6}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       8. flip-- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{4} \cdot \frac{1}{4} - \left(\left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{36}\right)}{\frac{1}{4} + \left(re \cdot re\right) \cdot \frac{1}{36}} \cdot \frac{re}{\frac{1}{2} - re \cdot \frac{1}{6}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{4} \cdot \frac{1}{4} - \left(\left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{36}\right)\right) \cdot \frac{re}{\frac{1}{2} - re \cdot \frac{1}{6}}}{\frac{1}{4} + \left(re \cdot re\right) \cdot \frac{1}{36}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \frac{1}{4} - \left(\left(re \cdot re\right) \cdot \frac{1}{36}\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{1}{36}\right)\right) \cdot \frac{re}{\frac{1}{2} - re \cdot \frac{1}{6}}\right), \left(\frac{1}{4} + \left(re \cdot re\right) \cdot \frac{1}{36}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

   13. Applied egg-rr 69.1%

       \[\leadsto \left(re \cdot \left(1 + \color{blue}{\frac{\left(0.0625 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.0007716049382716049\right) \cdot \frac{re}{0.5 + re \cdot -0.16666666666666666}}{0.25 + \left(re \cdot re\right) \cdot 0.027777777777777776}}\right)\right) \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]

   if 1.35000000000000003e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re} + \frac{-1}{2} \cdot \left({im}^{2} \cdot e^{re}\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{re} + \color{blue}{\frac{-1}{2}} \cdot \left({im}^{2} \cdot e^{re}\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot e^{re} + \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{e^{re}} \]

      3. distribute-rgt-in N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 96.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   5. Simplified 96.6%

      \[\leadsto \color{blue}{e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

   8. Simplified 96.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 82:\\ \;\;\;\;1 + \frac{re \cdot re - \left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)}{re - 0.5 \cdot \left(re \cdot re\right)}\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(re \cdot \left(1 + \frac{\left(0.0625 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.0007716049382716049\right) \cdot \frac{re}{0.5 + re \cdot -0.16666666666666666}}{0.25 + \left(re \cdot re\right) \cdot 0.027777777777777776}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+133}:\\ \;\;\;\;1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* re re))))
   (if (<= re -102000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 8e+133)
       (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0)))
       (*
        (* re (* re re))
        (+ 0.16666666666666666 (* (* im im) -0.08333333333333333)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 8e+133) {
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (re * re)
    if (re <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 8d+133) then
        tmp = 1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0))
    else
        tmp = (re * (re * re)) * (0.16666666666666666d0 + ((im * im) * (-0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (re * re);
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 8e+133) {
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (re * re)
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 8e+133:
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0))
	else:
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(re * re))
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 8e+133)
		tmp = Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0)));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(0.16666666666666666 + Float64(Float64(im * im) * -0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re * re);
	tmp = 0.0;
	if (re <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 8e+133)
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	else
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8e+133], N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -102000

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re < 8.0000000000000002e133

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 59.2%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 49.6%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 49.6%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(re \cdot 1 + \color{blue}{re \cdot \left(re \cdot \frac{1}{2}\right)}\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(re + \color{blue}{re} \cdot \left(re \cdot \frac{1}{2}\right)\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{\color{blue}{re - re \cdot \left(re \cdot \frac{1}{2}\right)}}\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{re \cdot 1 - \color{blue}{re} \cdot \left(re \cdot \frac{1}{2}\right)}\right)\right) \]

      5. fmm-def N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{\mathsf{fma}\left(re, \color{blue}{1}, \mathsf{neg}\left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)}{\mathsf{fma}\left(re, 1, \mathsf{neg}\left(\left(re \cdot \frac{1}{2}\right) \cdot re\right)\right)}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(re \cdot re - \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\mathsf{fma}\left(re, 1, \mathsf{neg}\left(\left(re \cdot \frac{1}{2}\right) \cdot re\right)\right)\right)}\right)\right) \]

   10. Applied egg-rr 55.8%

       \[\leadsto 1 + \color{blue}{\frac{re \cdot re - \left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)}{re - 0.5 \cdot \left(re \cdot re\right)}} \]

   if 8.0000000000000002e133 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 94.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 94.1%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \cdot \frac{1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{3}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot re\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot {re}^{2}\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left({re}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot re\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} \cdot 1 + \color{blue}{\frac{1}{6} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} + \color{blue}{\frac{1}{6}} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{6} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left({im}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{6}\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \frac{-1}{12}\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{6}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{6}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       21. metadata-eval 94.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{12}\right)\right)\right) \]

   11. Simplified 94.1%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -102000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 1.6e-35)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (*
      (+ 1.0 (* im (* im -0.5)))
      (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.6e-35) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = (1.0 + (im * (im * -0.5))) * (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 <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 1.6d-35) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else
        tmp = (1.0d0 + (im * (im * (-0.5d0)))) * (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 <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.6e-35) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 1.6e-35:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	else:
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 1.6e-35)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	else
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * -0.5))) * 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 <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 1.6e-35)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	else
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.6e-35], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if re < -102000

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re < 1.5999999999999999e-35

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 57.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 57.9%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 56.5%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]

   if 1.5999999999999999e-35 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 74.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 74.8%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 75.4%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 92000000000000:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -102000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 92000000000000.0)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (*
      (+ 1.0 (* im (* im -0.5)))
      (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 92000000000000.0) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = (1.0 + (im * (im * -0.5))) * (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 <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 92000000000000.0d0) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else
        tmp = (1.0d0 + (im * (im * (-0.5d0)))) * (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 <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 92000000000000.0) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = (1.0 + (im * (im * -0.5))) * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 92000000000000.0:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	else:
		tmp = (1.0 + (im * (im * -0.5))) * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 92000000000000.0)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	else
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * -0.5))) * 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 <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 92000000000000.0)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	else
		tmp = (1.0 + (im * (im * -0.5))) * (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 92000000000000.0], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -102000

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re < 9.2e13

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 58.3%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 58.3%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]

   if 9.2e13 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 73.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 73.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 75.0%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot {re}^{2}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) + \frac{1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \frac{1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \frac{1}{6} \cdot \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) + \left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right) \cdot re + \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot re\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

   11. Simplified 75.0%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 92000000000000:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+86}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -102000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 1.6e-35)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (if (<= re 6.2e+86)
       (* (+ 1.0 (* im (* im -0.5))) (+ re 1.0))
       (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.6e-35) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else if (re <= 6.2e+86) {
		tmp = (1.0 + (im * (im * -0.5))) * (re + 1.0);
	} else {
		tmp = 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 <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 1.6d-35) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else if (re <= 6.2d+86) then
        tmp = (1.0d0 + (im * (im * (-0.5d0)))) * (re + 1.0d0)
    else
        tmp = 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 <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.6e-35) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else if (re <= 6.2e+86) {
		tmp = (1.0 + (im * (im * -0.5))) * (re + 1.0);
	} else {
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 1.6e-35:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	elif re <= 6.2e+86:
		tmp = (1.0 + (im * (im * -0.5))) * (re + 1.0)
	else:
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 1.6e-35)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	elseif (re <= 6.2e+86)
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * -0.5))) * Float64(re + 1.0));
	else
		tmp = 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 <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 1.6e-35)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	elseif (re <= 6.2e+86)
		tmp = (1.0 + (im * (im * -0.5))) * (re + 1.0);
	else
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.6e-35], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.2e+86], N[(N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -102000

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re < 1.5999999999999999e-35

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 57.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 57.9%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 56.5%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]

   if 1.5999999999999999e-35 < re < 6.2000000000000004e86

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 24.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 24.2%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 48.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 48.6%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   if 6.2000000000000004e86 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 90.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 90.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 81.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 81.3%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot {re}^{2}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) + \frac{1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \frac{1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \frac{1}{6} \cdot \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) + \left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right) \cdot re + \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot re\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

   11. Simplified 81.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 57.4%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

   14. Simplified 57.4%

       \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+86}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 450:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+86}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -102000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 450.0)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (if (<= re 6.2e+86)
       (* re (+ 1.0 (* im (* im -0.5))))
       (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 450.0) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else if (re <= 6.2e+86) {
		tmp = re * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = 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 <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 450.0d0) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else if (re <= 6.2d+86) then
        tmp = re * (1.0d0 + (im * (im * (-0.5d0))))
    else
        tmp = 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 <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 450.0) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else if (re <= 6.2e+86) {
		tmp = re * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 450.0:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	elif re <= 6.2e+86:
		tmp = re * (1.0 + (im * (im * -0.5)))
	else:
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 450.0)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	elseif (re <= 6.2e+86)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = 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 <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 450.0)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	elseif (re <= 6.2e+86)
		tmp = re * (1.0 + (im * (im * -0.5)));
	else
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 450.0], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.2e+86], N[(re * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -102000

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re < 450

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 58.3%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 58.3%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]

   if 450 < re < 6.2000000000000004e86

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 3.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 3.5%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 39.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 39.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 39.3%

       \[\leadsto \color{blue}{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]

   if 6.2000000000000004e86 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 90.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 90.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 81.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 81.3%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(re \cdot {re}^{2}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) + \frac{1}{6}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \frac{1}{6} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \frac{1}{6} \cdot \left(re \cdot re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) \cdot {re}^{2} + \left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right) + \left(\frac{1}{6} \cdot re\right) \cdot re\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right) \cdot re + \left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot re\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

   11. Simplified 81.3%

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 57.4%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

   14. Simplified 57.4%

       \[\leadsto \color{blue}{re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 48.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -23:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 3.6:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.42 \cdot 10^{+171}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -23.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 3.6)
     (+ re 1.0)
     (if (<= re 1.42e+171)
       (* re (+ 1.0 (* im (* im -0.5))))
       (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -23.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 3.6) {
		tmp = re + 1.0;
	} else if (re <= 1.42e+171) {
		tmp = re * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-23.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 3.6d0) then
        tmp = re + 1.0d0
    else if (re <= 1.42d+171) then
        tmp = re * (1.0d0 + (im * (im * (-0.5d0))))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -23.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 3.6) {
		tmp = re + 1.0;
	} else if (re <= 1.42e+171) {
		tmp = re * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -23.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 3.6:
		tmp = re + 1.0
	elif re <= 1.42e+171:
		tmp = re * (1.0 + (im * (im * -0.5)))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -23.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 3.6)
		tmp = Float64(re + 1.0);
	elseif (re <= 1.42e+171)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -23.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 3.6)
		tmp = re + 1.0;
	elseif (re <= 1.42e+171)
		tmp = re * (1.0 + (im * (im * -0.5)));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -23.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.6], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 1.42e+171], N[(re * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -23

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.2%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.2%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 44.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 44.6%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -23 < re < 3.60000000000000009

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 57.7%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re + \color{blue}{1} \]

      2. +-lowering-+.f64 57.7%

         \[\leadsto \mathsf{+.f64}\left(re, \color{blue}{1}\right) \]

   8. Simplified 57.7%

      \[\leadsto \color{blue}{re + 1} \]

   if 3.60000000000000009 < re < 1.4199999999999999e171

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 4.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 4.0%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 28.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 28.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 28.0%

       \[\leadsto \color{blue}{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]

   if 1.4199999999999999e171 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 68.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 68.0%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 68.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 68.0%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{re}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{re} \]

       3. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       6. *-lowering-*.f64 68.0%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right) \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 48.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -20:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 460:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 1.42 \cdot 10^{+171}:\\ \;\;\;\;im \cdot \left(im \cdot \left(-0.5 + re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -20.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 460.0)
     (+ re 1.0)
     (if (<= re 1.42e+171)
       (* im (* im (+ -0.5 (* re -0.5))))
       (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -20.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 460.0) {
		tmp = re + 1.0;
	} else if (re <= 1.42e+171) {
		tmp = im * (im * (-0.5 + (re * -0.5)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-20.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 460.0d0) then
        tmp = re + 1.0d0
    else if (re <= 1.42d+171) then
        tmp = im * (im * ((-0.5d0) + (re * (-0.5d0))))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -20.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 460.0) {
		tmp = re + 1.0;
	} else if (re <= 1.42e+171) {
		tmp = im * (im * (-0.5 + (re * -0.5)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -20.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 460.0:
		tmp = re + 1.0
	elif re <= 1.42e+171:
		tmp = im * (im * (-0.5 + (re * -0.5)))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -20.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 460.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 1.42e+171)
		tmp = Float64(im * Float64(im * Float64(-0.5 + Float64(re * -0.5))));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -20.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 460.0)
		tmp = re + 1.0;
	elseif (re <= 1.42e+171)
		tmp = im * (im * (-0.5 + (re * -0.5)));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -20.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 460.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 1.42e+171], N[(im * N[(im * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -20

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.2%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.2%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 44.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 44.6%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -20 < re < 460

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 57.7%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re + \color{blue}{1} \]

      2. +-lowering-+.f64 57.7%

         \[\leadsto \mathsf{+.f64}\left(re, \color{blue}{1}\right) \]

   8. Simplified 57.7%

      \[\leadsto \color{blue}{re + 1} \]

   if 460 < re < 1.4199999999999999e171

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 4.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 4.0%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 28.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 28.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \frac{-1}{2}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + re\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{2}} \cdot \left(1 + re\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{2} \cdot \left(1 + re\right)\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{2} \cdot \left(1 + re\right)\right)\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + re\right)\right)}\right)\right) \]

       8. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} \cdot 1 + \color{blue}{\frac{-1}{2} \cdot re}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{\frac{-1}{2}} \cdot re\right)\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{-1}{2} \cdot re\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 26.7%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   11. Simplified 26.7%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(-0.5 + re \cdot -0.5\right)\right)} \]

   if 1.4199999999999999e171 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 68.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 68.0%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 68.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 68.0%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{re}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{re} \]

       3. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       6. *-lowering-*.f64 68.0%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right) \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 52.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 175:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -102000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 175.0)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (*
      (* re (* re re))
      (+ 0.16666666666666666 (* (* im im) -0.08333333333333333))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 175.0) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 175.0d0) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else
        tmp = (re * (re * re)) * (0.16666666666666666d0 + ((im * im) * (-0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 175.0) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 175.0:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	else:
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 175.0)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	else
		tmp = Float64(Float64(re * Float64(re * re)) * Float64(0.16666666666666666 + Float64(Float64(im * im) * -0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 175.0)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	else
		tmp = (re * (re * re)) * (0.16666666666666666 + ((im * im) * -0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 175.0], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -102000

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re < 175

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 58.3%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 58.3%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]

   if 175 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 73.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 73.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 75.0%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\frac{1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \cdot \frac{1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{3}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot \left(re \cdot re\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot {re}^{2}\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left({re}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(re \cdot re\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       10. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} \cdot 1 + \color{blue}{\frac{1}{6} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \left(\frac{1}{6} + \color{blue}{\frac{1}{6}} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{6} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left({im}^{2} \cdot \frac{-1}{2}\right) \cdot \frac{1}{6}\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \frac{-1}{12}\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({im}^{2} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{6}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{6}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

       21. metadata-eval 75.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{12}\right)\right)\right) \]

   11. Simplified 75.0%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \left(0.16666666666666666 + \left(im \cdot im\right) \cdot -0.08333333333333333\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 38.6% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;1\\ \mathbf{elif}\;re \leq 1.42 \cdot 10^{+171}:\\ \;\;\;\;1 + im \cdot \left(im \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.6e-35)
   1.0
   (if (<= re 1.42e+171) (+ 1.0 (* im (* im -0.5))) (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.6e-35) {
		tmp = 1.0;
	} else if (re <= 1.42e+171) {
		tmp = 1.0 + (im * (im * -0.5));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.6d-35) then
        tmp = 1.0d0
    else if (re <= 1.42d+171) then
        tmp = 1.0d0 + (im * (im * (-0.5d0)))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.6e-35) {
		tmp = 1.0;
	} else if (re <= 1.42e+171) {
		tmp = 1.0 + (im * (im * -0.5));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.6e-35:
		tmp = 1.0
	elif re <= 1.42e+171:
		tmp = 1.0 + (im * (im * -0.5))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.6e-35)
		tmp = 1.0;
	elseif (re <= 1.42e+171)
		tmp = Float64(1.0 + Float64(im * Float64(im * -0.5)));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.6e-35)
		tmp = 1.0;
	elseif (re <= 1.42e+171)
		tmp = 1.0 + (im * (im * -0.5));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.6e-35], 1.0, If[LessEqual[re, 1.42e+171], N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.5999999999999999e-35

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 71.2%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 71.2%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 41.5%

      \[\leadsto \color{blue}{1} \]

   if 1.5999999999999999e-35 < re < 1.4199999999999999e171

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 11.0%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 11.0%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 23.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   8. Simplified 23.7%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot -0.5\right)} \]

   if 1.4199999999999999e171 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 68.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 68.0%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 68.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 68.0%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{re}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{re} \]

       3. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       6. *-lowering-*.f64 68.0%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right) \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 47.8% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -85:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+31}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -85.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 1.25e+31) (+ re 1.0) (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -85.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.25e+31) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-85.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 1.25d+31) then
        tmp = re + 1.0d0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -85.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.25e+31) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -85.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 1.25e+31:
		tmp = re + 1.0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -85.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 1.25e+31)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -85.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 1.25e+31)
		tmp = re + 1.0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -85.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.25e+31], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -85

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.2%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.2%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 44.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 44.6%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -85 < re < 1.25000000000000007e31

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 56.1%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 56.1%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re + \color{blue}{1} \]

      2. +-lowering-+.f64 56.1%

         \[\leadsto \mathsf{+.f64}\left(re, \color{blue}{1}\right) \]

   8. Simplified 56.1%

      \[\leadsto \color{blue}{re + 1} \]

   if 1.25000000000000007e31 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 66.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 35.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 35.8%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{re}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{re} \]

       3. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       6. *-lowering-*.f64 35.8%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right) \]

   11. Simplified 35.8%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 47.8% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -102000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -102000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (+ 1.0 (* re (+ 1.0 (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = 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 <= (-102000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else
        tmp = 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 <= -102000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -102000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	else:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -102000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	else
		tmp = 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 <= -102000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	else
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -102000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -102000

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({im}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right) \]

       8. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right) \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -102000 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 59.2%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 50.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 50.2%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 38.3% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.25 \cdot 10^{+31}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.25e+31) 1.0 (* re (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.25e+31) {
		tmp = 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.25d+31) then
        tmp = 1.0d0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.25e+31) {
		tmp = 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.25e+31:
		tmp = 1.0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.25e+31)
		tmp = 1.0;
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.25e+31)
		tmp = 1.0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.25e+31], 1.0, N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.25000000000000007e31

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 70.2%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 70.2%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 41.2%

      \[\leadsto \color{blue}{1} \]

   if 1.25000000000000007e31 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 66.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 35.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 35.8%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(re \cdot \color{blue}{re}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{re} \]

       3. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(re, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       6. *-lowering-*.f64 35.8%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right) \]

   11. Simplified 35.8%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 29.8% accurate, 67.7× speedup

Math

\[\begin{array}{l} \\ re + 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ re 1.0))

C

double code(double re, double im) {
	return re + 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re + 1.0d0
end function

Java

public static double code(double re, double im) {
	return re + 1.0;
}

Python

def code(re, im):
	return re + 1.0

Julia

function code(re, im)
	return Float64(re + 1.0)
end

MATLAB

function tmp = code(re, im)
	tmp = re + 1.0;
end

Wolfram

code[re_, im_] := N[(re + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{e^{re}} \]
4. Step-by-step derivation

   1. exp-lowering-exp.f64 67.9%

      \[\leadsto \mathsf{exp.f64}\left(re\right) \]

5. Simplified 67.9%

   \[\leadsto \color{blue}{e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1 + re} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto re + \color{blue}{1} \]

   2. +-lowering-+.f64 33.3%

      \[\leadsto \mathsf{+.f64}\left(re, \color{blue}{1}\right) \]

8. Simplified 33.3%

   \[\leadsto \color{blue}{re + 1} \]
9. Add Preprocessing

Alternative 25: 29.3% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\cos im} \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 56.3%

      \[\leadsto \mathsf{cos.f64}\left(im\right) \]

5. Simplified 56.3%

   \[\leadsto \color{blue}{\cos im} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 33.1%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))