math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 22.2s

Alternatives: 17

Speedup: 1.0×

• 24.2% 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: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.005:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (exp re)
   (if (<= (exp re) 1.005) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = exp(re);
	} else if (exp(re) <= 1.005) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(re);
	}
	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) <= 0.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.005d0) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.005) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.005:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.005)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.005)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.005], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1.0049999999999999 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 88.9%

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

   5. Simplified 88.9%

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

   if 0.0 < (exp.f64 re) < 1.0049999999999999

   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 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.005:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.000000001:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (exp re)
   (if (<= (exp re) 1.000000001) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = exp(re);
	} else if (exp(re) <= 1.000000001) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	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) <= 0.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.000000001d0) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.000000001) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.000000001:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.000000001)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.000000001)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.000000001], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1.0000000010000001 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 88.0%

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

   5. Simplified 88.0%

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

   if 0.0 < (exp.f64 re) < 1.0000000010000001

   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 98.8%

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

   5. Simplified 98.8%

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

Alternative 4: 97.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;re \leq -0.039:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.004:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (cos im)
          (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))
   (if (<= re -0.039)
     (exp re)
     (if (<= re 0.004)
       t_0
       (if (<= re 1.02e+103) (* (exp re) (+ 1.0 (* im (* im -0.5)))) t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.039) {
		tmp = exp(re);
	} else if (re <= 0.004) {
		tmp = t_0;
	} else if (re <= 1.02e+103) {
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    if (re <= (-0.039d0)) then
        tmp = exp(re)
    else if (re <= 0.004d0) then
        tmp = t_0
    else if (re <= 1.02d+103) then
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.039) {
		tmp = Math.exp(re);
	} else if (re <= 0.004) {
		tmp = t_0;
	} else if (re <= 1.02e+103) {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	tmp = 0
	if re <= -0.039:
		tmp = math.exp(re)
	elif re <= 0.004:
		tmp = t_0
	elif re <= 1.02e+103:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
	tmp = 0.0
	if (re <= -0.039)
		tmp = exp(re);
	elseif (re <= 0.004)
		tmp = t_0;
	elseif (re <= 1.02e+103)
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	tmp = 0.0;
	if (re <= -0.039)
		tmp = exp(re);
	elseif (re <= 0.004)
		tmp = t_0;
	elseif (re <= 1.02e+103)
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$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]}, If[LessEqual[re, -0.039], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.004], t$95$0, If[LessEqual[re, 1.02e+103], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0389999999999999999

   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 -0.0389999999999999999 < re < 0.0040000000000000001 or 1.01999999999999991e103 < 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 99.9%

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

   5. Simplified 99.9%

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

   if 0.0040000000000000001 < re < 1.01999999999999991e103

   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. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      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 80.0%

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

   5. Simplified 80.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.039:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.004:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -0.028:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0037:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))
   (if (<= re -0.028)
     (exp re)
     (if (<= re 0.0037) t_0 (if (<= re 1.9e+154) (exp re) t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.028) {
		tmp = exp(re);
	} else if (re <= 0.0037) {
		tmp = t_0;
	} else if (re <= 1.9e+154) {
		tmp = exp(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    if (re <= (-0.028d0)) then
        tmp = exp(re)
    else if (re <= 0.0037d0) then
        tmp = t_0
    else if (re <= 1.9d+154) then
        tmp = exp(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.028) {
		tmp = Math.exp(re);
	} else if (re <= 0.0037) {
		tmp = t_0;
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	tmp = 0
	if re <= -0.028:
		tmp = math.exp(re)
	elif re <= 0.0037:
		tmp = t_0
	elif re <= 1.9e+154:
		tmp = math.exp(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))))
	tmp = 0.0
	if (re <= -0.028)
		tmp = exp(re);
	elseif (re <= 0.0037)
		tmp = t_0;
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	tmp = 0.0;
	if (re <= -0.028)
		tmp = exp(re);
	elseif (re <= 0.0037)
		tmp = t_0;
	elseif (re <= 1.9e+154)
		tmp = exp(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.028], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0037], t$95$0, If[LessEqual[re, 1.9e+154], N[Exp[re], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0280000000000000006 or 0.0037000000000000002 < re < 1.8999999999999999e154

   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 92.7%

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

   5. Simplified 92.7%

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

   if -0.0280000000000000006 < re < 0.0037000000000000002 or 1.8999999999999999e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   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 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.028:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0037:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0028:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0013:\\ \;\;\;\;\cos im \cdot \frac{1}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0028)
   (exp re)
   (if (<= re 0.0013) (* (cos im) (/ 1.0 (- 1.0 re))) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0028) {
		tmp = exp(re);
	} else if (re <= 0.0013) {
		tmp = cos(im) * (1.0 / (1.0 - re));
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0028d0)) then
        tmp = exp(re)
    else if (re <= 0.0013d0) then
        tmp = cos(im) * (1.0d0 / (1.0d0 - re))
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0028) {
		tmp = Math.exp(re);
	} else if (re <= 0.0013) {
		tmp = Math.cos(im) * (1.0 / (1.0 - re));
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0028:
		tmp = math.exp(re)
	elif re <= 0.0013:
		tmp = math.cos(im) * (1.0 / (1.0 - re))
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0028)
		tmp = exp(re);
	elseif (re <= 0.0013)
		tmp = Float64(cos(im) * Float64(1.0 / Float64(1.0 - re)));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0028)
		tmp = exp(re);
	elseif (re <= 0.0013)
		tmp = cos(im) * (1.0 / (1.0 - re));
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0028], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0013], N[(N[Cos[im], $MachinePrecision] * N[(1.0 / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.00279999999999999997 or 0.0012999999999999999 < 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 88.9%

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

   5. Simplified 88.9%

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

   if -0.00279999999999999997 < re < 0.0012999999999999999

   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 99.0%

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

   5. Simplified 99.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. --lowering--.f64 99.0%

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

   7. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{1 - re \cdot re}{1 - re}} \cdot \cos im \]

   8. Taylor expanded in re around 0

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

   10. Simplified 99.0%

       \[\leadsto \frac{\color{blue}{1}}{1 - re} \cdot \cos im \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0028:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0013:\\ \;\;\;\;\cos im \cdot \frac{1}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -175000:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 10^{-9}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \frac{0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629}{0.25}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -175000.0)
   (* (* -0.5 (* im im)) (+ re 1.0))
   (if (<= re 1e-9)
     (cos im)
     (+
      1.0
      (*
       re
       (+
        1.0
        (*
         re
         (/ (+ 0.125 (* (* re (* re re)) 0.004629629629629629)) 0.25))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -175000.0) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else if (re <= 1e-9) {
		tmp = cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * ((0.125 + ((re * (re * re)) * 0.004629629629629629)) / 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-175000.0d0)) then
        tmp = ((-0.5d0) * (im * im)) * (re + 1.0d0)
    else if (re <= 1d-9) then
        tmp = cos(im)
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * ((0.125d0 + ((re * (re * re)) * 0.004629629629629629d0)) / 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -175000.0) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else if (re <= 1e-9) {
		tmp = Math.cos(im);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * ((0.125 + ((re * (re * re)) * 0.004629629629629629)) / 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -175000.0:
		tmp = (-0.5 * (im * im)) * (re + 1.0)
	elif re <= 1e-9:
		tmp = math.cos(im)
	else:
		tmp = 1.0 + (re * (1.0 + (re * ((0.125 + ((re * (re * re)) * 0.004629629629629629)) / 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -175000.0)
		tmp = Float64(Float64(-0.5 * Float64(im * im)) * Float64(re + 1.0));
	elseif (re <= 1e-9)
		tmp = cos(im);
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(Float64(0.125 + Float64(Float64(re * Float64(re * re)) * 0.004629629629629629)) / 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -175000.0)
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	elseif (re <= 1e-9)
		tmp = cos(im);
	else
		tmp = 1.0 + (re * (1.0 + (re * ((0.125 + ((re * (re * re)) * 0.004629629629629629)) / 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -175000.0], N[(N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e-9], N[Cos[im], $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(N[(0.125 + N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629), $MachinePrecision]), $MachinePrecision] / 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -175000

   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. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      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 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 2.0%

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

   11. Simplified 2.0%

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

   12. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 30.5%

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

   14. Simplified 30.5%

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

   if -175000 < re < 1.00000000000000006e-9

   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 96.6%

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

   5. Simplified 96.6%

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

   if 1.00000000000000006e-9 < 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 75.6%

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

   5. Simplified 75.6%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 54.5%

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

   8. Simplified 54.5%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. unpow-prod-down N/A

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

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

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

      9. cube-mult N/A

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

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

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

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

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

      12. metadata-eval N/A

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

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

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

      14. swap-sqr N/A

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

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

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

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

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

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{2}} - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(re \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      19. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 18.6%

       \[\leadsto 1 + re \cdot \left(1 + re \cdot \color{blue}{\frac{0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629}{\left(re \cdot re\right) \cdot 0.027777777777777776 + \left(0.25 + -0.08333333333333333 \cdot re\right)}}\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 66.0%

       \[\leadsto 1 + re \cdot \left(1 + re \cdot \frac{0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629}{\color{blue}{0.25}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -175000:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 10^{-9}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \frac{0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629}{0.25}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \frac{0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629}{0.25}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-12)
   (* (* -0.5 (* im im)) (+ re 1.0))
   (+
    1.0
    (*
     re
     (+
      1.0
      (* re (/ (+ 0.125 (* (* re (* re re)) 0.004629629629629629)) 0.25)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-12) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * ((0.125 + ((re * (re * re)) * 0.004629629629629629)) / 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.5d-12)) then
        tmp = ((-0.5d0) * (im * im)) * (re + 1.0d0)
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * ((0.125d0 + ((re * (re * re)) * 0.004629629629629629d0)) / 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-12) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * ((0.125 + ((re * (re * re)) * 0.004629629629629629)) / 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5e-12:
		tmp = (-0.5 * (im * im)) * (re + 1.0)
	else:
		tmp = 1.0 + (re * (1.0 + (re * ((0.125 + ((re * (re * re)) * 0.004629629629629629)) / 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e-12)
		tmp = Float64(Float64(-0.5 * Float64(im * im)) * Float64(re + 1.0));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(Float64(0.125 + Float64(Float64(re * Float64(re * re)) * 0.004629629629629629)) / 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5e-12)
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	else
		tmp = 1.0 + (re * (1.0 + (re * ((0.125 + ((re * (re * re)) * 0.004629629629629629)) / 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e-12], N[(N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(N[(0.125 + N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629), $MachinePrecision]), $MachinePrecision] / 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.49999999999999981e-12

   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. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      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 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 1.8%

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

   8. Simplified 1.8%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 2.1%

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

   11. Simplified 2.1%

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

   12. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 28.5%

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

   14. Simplified 28.5%

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

   if -4.49999999999999981e-12 < 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 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 53.3%

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

   8. Simplified 53.3%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. unpow-prod-down N/A

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

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

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

      9. cube-mult N/A

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

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

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

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

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

      12. metadata-eval N/A

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

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

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

      14. swap-sqr N/A

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

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

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

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

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

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{2}} - \left(re \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(re \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      19. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left(re \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{36}\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 41.1%

       \[\leadsto 1 + re \cdot \left(1 + re \cdot \color{blue}{\frac{0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629}{\left(re \cdot re\right) \cdot 0.027777777777777776 + \left(0.25 + -0.08333333333333333 \cdot re\right)}}\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{216}\right)\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 57.2%

       \[\leadsto 1 + re \cdot \left(1 + re \cdot \frac{0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629}{\color{blue}{0.25}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \frac{0.125 + \left(re \cdot \left(re \cdot re\right)\right) \cdot 0.004629629629629629}{0.25}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.8:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-12)
   (* (* -0.5 (* im im)) (+ re 1.0))
   (if (<= re 1.8)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-12) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else if (re <= 1.8) {
		tmp = 1.0 + (re * (1.0 + (re * 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 <= (-4.5d-12)) then
        tmp = ((-0.5d0) * (im * im)) * (re + 1.0d0)
    else if (re <= 1.8d0) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 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 <= -4.5e-12) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else if (re <= 1.8) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5e-12:
		tmp = (-0.5 * (im * im)) * (re + 1.0)
	elif re <= 1.8:
		tmp = 1.0 + (re * (1.0 + (re * 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 <= -4.5e-12)
		tmp = Float64(Float64(-0.5 * Float64(im * im)) * Float64(re + 1.0));
	elseif (re <= 1.8)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 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 <= -4.5e-12)
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	elseif (re <= 1.8)
		tmp = 1.0 + (re * (1.0 + (re * 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, -4.5e-12], N[(N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.8], N[(1.0 + N[(re * N[(1.0 + N[(re * 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 3 regimes

2. if re < -4.49999999999999981e-12

   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. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      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 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 1.8%

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

   8. Simplified 1.8%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 2.1%

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

   11. Simplified 2.1%

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

   12. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 28.5%

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

   14. Simplified 28.5%

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

   if -4.49999999999999981e-12 < re < 1.80000000000000004

   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 53.3%

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

   5. Simplified 53.3%

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

   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 52.9%

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

   8. Simplified 52.9%

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

   if 1.80000000000000004 < 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 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 53.9%

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

   8. Simplified 53.9%

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

   9. Taylor expanded in re around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. rgt-mult-inverse N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. associate-*l* N/A

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

       12. lft-mult-inverse N/A

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

       13. metadata-eval N/A

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

       14. +-commutative N/A

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

   11. Simplified 53.9%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.8:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2.9:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-12)
   (* (* -0.5 (* im im)) (+ re 1.0))
   (if (<= re 2.9)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (* (+ 0.5 (* re 0.16666666666666666)) (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-12) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else if (re <= 2.9) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.5d-12)) then
        tmp = ((-0.5d0) * (im * im)) * (re + 1.0d0)
    else if (re <= 2.9d0) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else
        tmp = (0.5d0 + (re * 0.16666666666666666d0)) * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-12) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else if (re <= 2.9) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5e-12:
		tmp = (-0.5 * (im * im)) * (re + 1.0)
	elif re <= 2.9:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	else:
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e-12)
		tmp = Float64(Float64(-0.5 * Float64(im * im)) * Float64(re + 1.0));
	elseif (re <= 2.9)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	else
		tmp = Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5e-12)
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	elseif (re <= 2.9)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	else
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e-12], N[(N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.9], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.49999999999999981e-12

   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. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      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 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 1.8%

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

   8. Simplified 1.8%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 2.1%

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

   11. Simplified 2.1%

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

   12. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 28.5%

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

   14. Simplified 28.5%

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

   if -4.49999999999999981e-12 < re < 2.89999999999999991

   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 53.3%

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

   5. Simplified 53.3%

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

   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 52.9%

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

   8. Simplified 52.9%

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

   if 2.89999999999999991 < 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 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 53.9%

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

   8. Simplified 53.9%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. associate-*l* N/A

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

       7. lft-mult-inverse N/A

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

       8. metadata-eval N/A

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

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

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

       10. unpow2 N/A

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

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

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 53.9%

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

   11. Simplified 53.9%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2.9:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-12)
   (* (* -0.5 (* im im)) (+ re 1.0))
   (if (<= re 1.9)
     (+ re 1.0)
     (* (+ 0.5 (* re 0.16666666666666666)) (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-12) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else if (re <= 1.9) {
		tmp = re + 1.0;
	} else {
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.5d-12)) then
        tmp = ((-0.5d0) * (im * im)) * (re + 1.0d0)
    else if (re <= 1.9d0) then
        tmp = re + 1.0d0
    else
        tmp = (0.5d0 + (re * 0.16666666666666666d0)) * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-12) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else if (re <= 1.9) {
		tmp = re + 1.0;
	} else {
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5e-12:
		tmp = (-0.5 * (im * im)) * (re + 1.0)
	elif re <= 1.9:
		tmp = re + 1.0
	else:
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e-12)
		tmp = Float64(Float64(-0.5 * Float64(im * im)) * Float64(re + 1.0));
	elseif (re <= 1.9)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5e-12)
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	elseif (re <= 1.9)
		tmp = re + 1.0;
	else
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e-12], N[(N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9], N[(re + 1.0), $MachinePrecision], N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.49999999999999981e-12

   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. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      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 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 1.8%

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

   8. Simplified 1.8%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 2.1%

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

   11. Simplified 2.1%

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

   12. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 28.5%

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

   14. Simplified 28.5%

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

   if -4.49999999999999981e-12 < re < 1.8999999999999999

   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 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 52.8%

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

   8. Simplified 52.8%

      \[\leadsto \color{blue}{1 + re} \]

   if 1.8999999999999999 < 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 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 53.9%

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

   8. Simplified 53.9%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. associate-*l* N/A

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

       7. lft-mult-inverse N/A

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

       8. metadata-eval N/A

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

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

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

       10. unpow2 N/A

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

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

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 53.9%

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

   11. Simplified 53.9%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.9:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.2% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e-12)
   (* (* -0.5 (* im im)) (+ re 1.0))
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e-12) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else {
		tmp = 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 <= (-4.5d-12)) then
        tmp = ((-0.5d0) * (im * im)) * (re + 1.0d0)
    else
        tmp = 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 <= -4.5e-12) {
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5e-12:
		tmp = (-0.5 * (im * im)) * (re + 1.0)
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e-12)
		tmp = Float64(Float64(-0.5 * Float64(im * im)) * Float64(re + 1.0));
	else
		tmp = 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 <= -4.5e-12)
		tmp = (-0.5 * (im * im)) * (re + 1.0);
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e-12], N[(N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $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]]

Derivation

1. Split input into 2 regimes

2. if re < -4.49999999999999981e-12

   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. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      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 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 1.8%

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

   8. Simplified 1.8%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 2.1%

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

   11. Simplified 2.1%

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

   12. Taylor expanded in im around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 28.5%

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

   14. Simplified 28.5%

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

   if -4.49999999999999981e-12 < 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 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 53.3%

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

   8. Simplified 53.3%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;\left(-0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.9% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.9:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.9) (+ re 1.0) (* (+ 0.5 (* re 0.16666666666666666)) (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.9) {
		tmp = re + 1.0;
	} else {
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.9d0) then
        tmp = re + 1.0d0
    else
        tmp = (0.5d0 + (re * 0.16666666666666666d0)) * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.9) {
		tmp = re + 1.0;
	} else {
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.9:
		tmp = re + 1.0
	else:
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.9)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(0.5 + Float64(re * 0.16666666666666666)) * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.9)
		tmp = re + 1.0;
	else
		tmp = (0.5 + (re * 0.16666666666666666)) * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.9], N[(re + 1.0), $MachinePrecision], N[(N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.8999999999999999

   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 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 35.4%

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

   8. Simplified 35.4%

      \[\leadsto \color{blue}{1 + re} \]

   if 1.8999999999999999 < 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 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 53.9%

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

   8. Simplified 53.9%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. associate-*l* N/A

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

       7. lft-mult-inverse N/A

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

       8. metadata-eval N/A

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

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

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

       10. unpow2 N/A

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

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

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 53.9%

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

   11. Simplified 53.9%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.9:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.9% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.9:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.9) (+ re 1.0) (* re (* 0.16666666666666666 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.9) {
		tmp = re + 1.0;
	} else {
		tmp = re * (0.16666666666666666 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.9d0) then
        tmp = re + 1.0d0
    else
        tmp = re * (0.16666666666666666d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.9) {
		tmp = re + 1.0;
	} else {
		tmp = re * (0.16666666666666666 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.9:
		tmp = re + 1.0
	else:
		tmp = re * (0.16666666666666666 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.9)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(0.16666666666666666 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.9)
		tmp = re + 1.0;
	else
		tmp = re * (0.16666666666666666 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.9], N[(re + 1.0), $MachinePrecision], N[(re * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.89999999999999991

   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 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 35.4%

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

   8. Simplified 35.4%

      \[\leadsto \color{blue}{1 + re} \]

   if 2.89999999999999991 < 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 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 53.9%

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

   8. Simplified 53.9%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\frac{1}{6}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 53.9%

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

   11. Simplified 53.9%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.9:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.1% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.8:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.8) (+ re 1.0) (* 0.5 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.8) {
		tmp = re + 1.0;
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.8d0) then
        tmp = re + 1.0d0
    else
        tmp = 0.5d0 * (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.8) {
		tmp = re + 1.0;
	} else {
		tmp = 0.5 * (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.8:
		tmp = re + 1.0
	else:
		tmp = 0.5 * (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.8)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(0.5 * Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.8)
		tmp = re + 1.0;
	else
		tmp = 0.5 * (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.8], N[(re + 1.0), $MachinePrecision], N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.7999999999999998

   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 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 35.4%

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

   8. Simplified 35.4%

      \[\leadsto \color{blue}{1 + re} \]

   if 2.7999999999999998 < 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 76.3%

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

   5. Simplified 76.3%

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

   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 41.2%

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

   8. Simplified 41.2%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 41.2%

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

   11. Simplified 41.2%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.8:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.2% 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 70.2%

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

5. Simplified 70.2%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1 + re} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 28.3%

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

8. Simplified 28.3%

   \[\leadsto \color{blue}{1 + re} \]

9. Final simplification 28.3%

   \[\leadsto re + 1 \]
10. Add Preprocessing

Alternative 17: 28.8% 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 im around 0

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

   1. exp-lowering-exp.f64 70.2%

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

5. Simplified 70.2%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 27.6%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))