math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 22.1s

Alternatives: 18

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999999999:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.99999999999)
   (exp re)
   (if (<= (exp re) 1.0) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.99999999999) {
		tmp = exp(re);
	} else if (exp(re) <= 1.0) {
		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) <= 0.99999999999d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.0d0) 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) <= 0.99999999999) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.0) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.99999999999:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.0:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.99999999999)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.99999999999)
		tmp = exp(re);
	elseif (exp(re) <= 1.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.99999999999], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.0], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999999999 or 1 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 89.5%

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

   5. Simplified 89.5%

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

   if 0.99999999999 < (exp.f64 re) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 99.9%

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

   5. Simplified 99.9%

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

Alternative 3: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\\ t_1 := re \cdot \left(1 + t\_0\right)\\ t_2 := re \cdot t\_0\\ \mathbf{if}\;re \leq -0.105:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cos im \cdot \left(1 + t\_1 \cdot \left(t\_1 \cdot t\_1\right)\right)}{1 + t\_1 \cdot \left(t\_1 + -1\right)}\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;1 + \frac{re \cdot re - t\_2 \cdot t\_2}{re - t\_2}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* re 0.16666666666666666))))
        (t_1 (* re (+ 1.0 t_0)))
        (t_2 (* re t_0)))
   (if (<= re -0.105)
     (exp re)
     (if (<= re 4.4e+51)
       (/
        (* (cos im) (+ 1.0 (* t_1 (* t_1 t_1))))
        (+ 1.0 (* t_1 (+ t_1 -1.0))))
       (if (<= re 1e+103)
         (+ 1.0 (/ (- (* re re) (* t_2 t_2)) (- re t_2)))
         (* (cos im) (+ 1.0 t_1)))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = re * (1.0 + t_0);
	double t_2 = re * t_0;
	double tmp;
	if (re <= -0.105) {
		tmp = exp(re);
	} else if (re <= 4.4e+51) {
		tmp = (cos(im) * (1.0 + (t_1 * (t_1 * t_1)))) / (1.0 + (t_1 * (t_1 + -1.0)));
	} else if (re <= 1e+103) {
		tmp = 1.0 + (((re * re) - (t_2 * t_2)) / (re - t_2));
	} else {
		tmp = cos(im) * (1.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = re * (0.5d0 + (re * 0.16666666666666666d0))
    t_1 = re * (1.0d0 + t_0)
    t_2 = re * t_0
    if (re <= (-0.105d0)) then
        tmp = exp(re)
    else if (re <= 4.4d+51) then
        tmp = (cos(im) * (1.0d0 + (t_1 * (t_1 * t_1)))) / (1.0d0 + (t_1 * (t_1 + (-1.0d0))))
    else if (re <= 1d+103) then
        tmp = 1.0d0 + (((re * re) - (t_2 * t_2)) / (re - t_2))
    else
        tmp = cos(im) * (1.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double t_1 = re * (1.0 + t_0);
	double t_2 = re * t_0;
	double tmp;
	if (re <= -0.105) {
		tmp = Math.exp(re);
	} else if (re <= 4.4e+51) {
		tmp = (Math.cos(im) * (1.0 + (t_1 * (t_1 * t_1)))) / (1.0 + (t_1 * (t_1 + -1.0)));
	} else if (re <= 1e+103) {
		tmp = 1.0 + (((re * re) - (t_2 * t_2)) / (re - t_2));
	} else {
		tmp = Math.cos(im) * (1.0 + t_1);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (re * 0.16666666666666666))
	t_1 = re * (1.0 + t_0)
	t_2 = re * t_0
	tmp = 0
	if re <= -0.105:
		tmp = math.exp(re)
	elif re <= 4.4e+51:
		tmp = (math.cos(im) * (1.0 + (t_1 * (t_1 * t_1)))) / (1.0 + (t_1 * (t_1 + -1.0)))
	elif re <= 1e+103:
		tmp = 1.0 + (((re * re) - (t_2 * t_2)) / (re - t_2))
	else:
		tmp = math.cos(im) * (1.0 + t_1)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))
	t_1 = Float64(re * Float64(1.0 + t_0))
	t_2 = Float64(re * t_0)
	tmp = 0.0
	if (re <= -0.105)
		tmp = exp(re);
	elseif (re <= 4.4e+51)
		tmp = Float64(Float64(cos(im) * Float64(1.0 + Float64(t_1 * Float64(t_1 * t_1)))) / Float64(1.0 + Float64(t_1 * Float64(t_1 + -1.0))));
	elseif (re <= 1e+103)
		tmp = Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_2 * t_2)) / Float64(re - t_2)));
	else
		tmp = Float64(cos(im) * Float64(1.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (re * 0.16666666666666666));
	t_1 = re * (1.0 + t_0);
	t_2 = re * t_0;
	tmp = 0.0;
	if (re <= -0.105)
		tmp = exp(re);
	elseif (re <= 4.4e+51)
		tmp = (cos(im) * (1.0 + (t_1 * (t_1 * t_1)))) / (1.0 + (t_1 * (t_1 + -1.0)));
	elseif (re <= 1e+103)
		tmp = 1.0 + (((re * re) - (t_2 * t_2)) / (re - t_2));
	else
		tmp = cos(im) * (1.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(re * t$95$0), $MachinePrecision]}, If[LessEqual[re, -0.105], N[Exp[re], $MachinePrecision], If[LessEqual[re, 4.4e+51], N[(N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.104999999999999996

   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 -0.104999999999999996 < re < 4.39999999999999984e51

   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 92.7%

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

   5. Simplified 92.7%

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

      1. flip3-+ N/A

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

      2. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 96.8%

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

   if 4.39999999999999984e51 < re < 1e103

   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 81.8%

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

   5. Simplified 81.8%

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

   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 6.3%

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

   8. Simplified 6.3%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. flip-+ N/A

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

      4. *-lft-identity N/A

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

      5. fmm-def N/A

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

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

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

   10. Applied egg-rr 81.8%

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

   if 1e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.105:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cos im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)\right)}{1 + \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;1 + \frac{re \cdot re - \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)}{re - re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\\ \mathbf{if}\;re \leq -0.066:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.11:\\ \;\;\;\;\cos im \cdot \left(re \cdot t\_0 + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (+ 0.5 (* re 0.16666666666666666)))))
   (if (<= re -0.066)
     (exp re)
     (if (<= re 0.11)
       (* (cos im) (+ (* re t_0) (+ re 1.0)))
       (if (<= re 1e+103) (exp re) (* (cos im) (+ 1.0 (* re (+ 1.0 t_0)))))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double tmp;
	if (re <= -0.066) {
		tmp = exp(re);
	} else if (re <= 0.11) {
		tmp = cos(im) * ((re * t_0) + (re + 1.0));
	} else if (re <= 1e+103) {
		tmp = exp(re);
	} else {
		tmp = cos(im) * (1.0 + (re * (1.0 + 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 = re * (0.5d0 + (re * 0.16666666666666666d0))
    if (re <= (-0.066d0)) then
        tmp = exp(re)
    else if (re <= 0.11d0) then
        tmp = cos(im) * ((re * t_0) + (re + 1.0d0))
    else if (re <= 1d+103) then
        tmp = exp(re)
    else
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + t_0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (0.5 + (re * 0.16666666666666666));
	double tmp;
	if (re <= -0.066) {
		tmp = Math.exp(re);
	} else if (re <= 0.11) {
		tmp = Math.cos(im) * ((re * t_0) + (re + 1.0));
	} else if (re <= 1e+103) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + t_0)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 + (re * 0.16666666666666666))
	tmp = 0
	if re <= -0.066:
		tmp = math.exp(re)
	elif re <= 0.11:
		tmp = math.cos(im) * ((re * t_0) + (re + 1.0))
	elif re <= 1e+103:
		tmp = math.exp(re)
	else:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + t_0)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))
	tmp = 0.0
	if (re <= -0.066)
		tmp = exp(re);
	elseif (re <= 0.11)
		tmp = Float64(cos(im) * Float64(Float64(re * t_0) + Float64(re + 1.0)));
	elseif (re <= 1e+103)
		tmp = exp(re);
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 + (re * 0.16666666666666666));
	tmp = 0.0;
	if (re <= -0.066)
		tmp = exp(re);
	elseif (re <= 0.11)
		tmp = cos(im) * ((re * t_0) + (re + 1.0));
	elseif (re <= 1e+103)
		tmp = exp(re);
	else
		tmp = cos(im) * (1.0 + (re * (1.0 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.066], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.11], N[(N[Cos[im], $MachinePrecision] * N[(N[(re * t$95$0), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.066000000000000003 or 0.110000000000000001 < re < 1e103

   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 93.3%

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

   5. Simplified 93.3%

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

   if -0.066000000000000003 < re < 0.110000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.5%

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

   5. Simplified 99.5%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+r+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 1e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.066:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.11:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;re \leq -0.07:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.065:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (cos im)
          (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))
   (if (<= re -0.07)
     (exp re)
     (if (<= re 0.065) t_0 (if (<= re 1e+103) (exp re) 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 <= -0.07) {
		tmp = exp(re);
	} else if (re <= 0.065) {
		tmp = t_0;
	} else if (re <= 1e+103) {
		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 + (re * 0.16666666666666666d0))))))
    if (re <= (-0.07d0)) then
        tmp = exp(re)
    else if (re <= 0.065d0) then
        tmp = t_0
    else if (re <= 1d+103) 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 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.07) {
		tmp = Math.exp(re);
	} else if (re <= 0.065) {
		tmp = t_0;
	} else if (re <= 1e+103) {
		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 + (re * 0.16666666666666666))))))
	tmp = 0
	if re <= -0.07:
		tmp = math.exp(re)
	elif re <= 0.065:
		tmp = t_0
	elif re <= 1e+103:
		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 * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
	tmp = 0.0
	if (re <= -0.07)
		tmp = exp(re);
	elseif (re <= 0.065)
		tmp = t_0;
	elseif (re <= 1e+103)
		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 + (re * 0.16666666666666666))))));
	tmp = 0.0;
	if (re <= -0.07)
		tmp = exp(re);
	elseif (re <= 0.065)
		tmp = t_0;
	elseif (re <= 1e+103)
		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 * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.07], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.065], t$95$0, If[LessEqual[re, 1e+103], N[Exp[re], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.070000000000000007 or 0.065000000000000002 < re < 1e103

   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 93.3%

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

   5. Simplified 93.3%

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

   if -0.070000000000000007 < re < 0.065000000000000002 or 1e103 < 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 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.07:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.065:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -0.066:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0115:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))
   (if (<= re -0.066)
     (exp re)
     (if (<= re 0.0115) 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 <= -0.066) {
		tmp = exp(re);
	} else if (re <= 0.0115) {
		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 <= (-0.066d0)) then
        tmp = exp(re)
    else if (re <= 0.0115d0) 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 <= -0.066) {
		tmp = Math.exp(re);
	} else if (re <= 0.0115) {
		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 <= -0.066:
		tmp = math.exp(re)
	elif re <= 0.0115:
		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 <= -0.066)
		tmp = exp(re);
	elseif (re <= 0.0115)
		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 <= -0.066)
		tmp = exp(re);
	elseif (re <= 0.0115)
		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, -0.066], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0115], 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 < -0.066000000000000003 or 0.0115 < 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 92.9%

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

   5. Simplified 92.9%

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

   if -0.066000000000000003 < re < 0.0115 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 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.066:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0115:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.013:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0036:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.013)
   (exp re)
   (if (<= re 0.0036) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.013) {
		tmp = exp(re);
	} else if (re <= 0.0036) {
		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 (re <= (-0.013d0)) then
        tmp = exp(re)
    else if (re <= 0.0036d0) 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 (re <= -0.013) {
		tmp = Math.exp(re);
	} else if (re <= 0.0036) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.013:
		tmp = math.exp(re)
	elif re <= 0.0036:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.013)
		tmp = exp(re);
	elseif (re <= 0.0036)
		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 (re <= -0.013)
		tmp = exp(re);
	elseif (re <= 0.0036)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.013], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0036], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0129999999999999994 or 0.0035999999999999999 < 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 90.6%

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

   5. Simplified 90.6%

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

   if -0.0129999999999999994 < re < 0.0035999999999999999

   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 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.013:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0036:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -17000000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-22}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re (+ 0.5 (* re 0.16666666666666666))))))
   (if (<= re -17000000000000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 5.8e-22)
       (cos im)
       (if (<= re 1e+103)
         (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0)))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = re * (re * (0.5 + (re * 0.16666666666666666)));
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 5.8e-22) {
		tmp = cos(im);
	} else if (re <= 1e+103) {
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (re * (0.5d0 + (re * 0.16666666666666666d0)))
    if (re <= (-17000000000000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 5.8d-22) then
        tmp = cos(im)
    else if (re <= 1d+103) then
        tmp = 1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * (0.5 + (re * 0.16666666666666666)));
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 5.8e-22) {
		tmp = Math.cos(im);
	} else if (re <= 1e+103) {
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * (0.5 + (re * 0.16666666666666666)))
	tmp = 0
	if re <= -17000000000000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 5.8e-22:
		tmp = math.cos(im)
	elif re <= 1e+103:
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))
	tmp = 0.0
	if (re <= -17000000000000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 5.8e-22)
		tmp = cos(im);
	elseif (re <= 1e+103)
		tmp = Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * (0.5 + (re * 0.16666666666666666)));
	tmp = 0.0;
	if (re <= -17000000000000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 5.8e-22)
		tmp = cos(im);
	elseif (re <= 1e+103)
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -17000000000000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.8e-22], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1e+103], N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.7e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 2.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.4%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right)\right)\right)} \]

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 36.2%

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

   14. Simplified 36.2%

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

   if -1.7e13 < re < 5.8000000000000003e-22

   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 95.4%

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

   5. Simplified 95.4%

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

   if 5.8000000000000003e-22 < re < 1e103

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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 72.8%

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

   5. Simplified 72.8%

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

   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 16.6%

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

   8. Simplified 16.6%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. flip-+ N/A

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

      4. *-lft-identity N/A

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

      5. fmm-def N/A

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

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

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

   10. Applied egg-rr 49.8%

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

   if 1e103 < 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 84.2%

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

   5. Simplified 84.2%

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

   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 84.2%

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

   8. Simplified 84.2%

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

   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. unpow2 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. associate-*l* N/A

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

       8. lft-mult-inverse N/A

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

       9. metadata-eval N/A

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

       10. associate-*r* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 84.2%

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

   11. Simplified 84.2%

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

Alternative 9: 52.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -17000000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;1 + \frac{re \cdot re - t\_0 \cdot t\_0}{re - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re (+ 0.5 (* re 0.16666666666666666))))))
   (if (<= re -17000000000000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 1e+103)
       (+ 1.0 (/ (- (* re re) (* t_0 t_0)) (- re t_0)))
       t_0))))

C

double code(double re, double im) {
	double t_0 = re * (re * (0.5 + (re * 0.16666666666666666)));
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1e+103) {
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (re * (0.5d0 + (re * 0.16666666666666666d0)))
    if (re <= (-17000000000000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 1d+103) then
        tmp = 1.0d0 + (((re * re) - (t_0 * t_0)) / (re - t_0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * (0.5 + (re * 0.16666666666666666)));
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1e+103) {
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * (0.5 + (re * 0.16666666666666666)))
	tmp = 0
	if re <= -17000000000000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 1e+103:
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))
	tmp = 0.0
	if (re <= -17000000000000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 1e+103)
		tmp = Float64(1.0 + Float64(Float64(Float64(re * re) - Float64(t_0 * t_0)) / Float64(re - t_0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * (0.5 + (re * 0.16666666666666666)));
	tmp = 0.0;
	if (re <= -17000000000000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 1e+103)
		tmp = 1.0 + (((re * re) - (t_0 * t_0)) / (re - t_0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -17000000000000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[(1.0 + N[(N[(N[(re * re), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 2.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.4%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right)\right)\right)} \]

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 36.2%

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

   14. Simplified 36.2%

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

   if -1.7e13 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 56.0%

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

   5. Simplified 56.0%

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

   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 43.8%

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

   8. Simplified 43.8%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. flip-+ N/A

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

      4. *-lft-identity N/A

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

      5. fmm-def N/A

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

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

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

   10. Applied egg-rr 49.2%

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

   if 1e103 < 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 84.2%

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

   5. Simplified 84.2%

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

   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 84.2%

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

   8. Simplified 84.2%

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

   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. unpow2 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. associate-*l* N/A

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

       8. lft-mult-inverse N/A

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

       9. metadata-eval N/A

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

       10. associate-*r* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 84.2%

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

   11. Simplified 84.2%

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

Alternative 10: 52.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + re \cdot 0.16666666666666666\\ t_1 := re \cdot t\_0\\ t_2 := re \cdot t\_1\\ \mathbf{if}\;re \leq -17000000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+120}:\\ \;\;\;\;1 + \frac{re \cdot \left(1 - t\_0 \cdot t\_2\right)}{1 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* re 0.16666666666666666)))
        (t_1 (* re t_0))
        (t_2 (* re t_1)))
   (if (<= re -17000000000000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 5e+120)
       (+ 1.0 (/ (* re (- 1.0 (* t_0 t_2))) (- 1.0 t_1)))
       t_2))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (re * 0.16666666666666666);
	double t_1 = re * t_0;
	double t_2 = re * t_1;
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 5e+120) {
		tmp = 1.0 + ((re * (1.0 - (t_0 * t_2))) / (1.0 - t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 + (re * 0.16666666666666666d0)
    t_1 = re * t_0
    t_2 = re * t_1
    if (re <= (-17000000000000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 5d+120) then
        tmp = 1.0d0 + ((re * (1.0d0 - (t_0 * t_2))) / (1.0d0 - t_1))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (re * 0.16666666666666666);
	double t_1 = re * t_0;
	double t_2 = re * t_1;
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 5e+120) {
		tmp = 1.0 + ((re * (1.0 - (t_0 * t_2))) / (1.0 - t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (re * 0.16666666666666666)
	t_1 = re * t_0
	t_2 = re * t_1
	tmp = 0
	if re <= -17000000000000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 5e+120:
		tmp = 1.0 + ((re * (1.0 - (t_0 * t_2))) / (1.0 - t_1))
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(re * 0.16666666666666666))
	t_1 = Float64(re * t_0)
	t_2 = Float64(re * t_1)
	tmp = 0.0
	if (re <= -17000000000000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 5e+120)
		tmp = Float64(1.0 + Float64(Float64(re * Float64(1.0 - Float64(t_0 * t_2))) / Float64(1.0 - t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (re * 0.16666666666666666);
	t_1 = re * t_0;
	t_2 = re * t_1;
	tmp = 0.0;
	if (re <= -17000000000000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 5e+120)
		tmp = 1.0 + ((re * (1.0 - (t_0 * t_2))) / (1.0 - t_1));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(re * t$95$1), $MachinePrecision]}, If[LessEqual[re, -17000000000000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+120], N[(1.0 + N[(N[(re * N[(1.0 - N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 2.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.4%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right)\right)\right)} \]

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 36.2%

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

   14. Simplified 36.2%

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

   if -1.7e13 < re < 5.00000000000000019e120

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. exp-lowering-exp.f64 56.8%

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

   5. Simplified 56.8%

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

   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 45.0%

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

   8. Simplified 45.0%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 49.0%

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

   if 5.00000000000000019e120 < 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 84.8%

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

   5. Simplified 84.8%

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

   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 84.8%

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

   8. Simplified 84.8%

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

   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. unpow2 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. associate-*l* N/A

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

       8. lft-mult-inverse N/A

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

       9. metadata-eval N/A

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

       10. associate-*r* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 84.8%

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

   11. Simplified 84.8%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -17000000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+120}:\\ \;\;\;\;1 + \frac{re \cdot \left(1 - \left(0.5 + re \cdot 0.16666666666666666\right) \cdot \left(re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\right)}{1 - re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -17000000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.9:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -17000000000000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 2.9)
     (+ 1.0 (* re (+ 1.0 (* re 0.5))))
     (* re (* re (+ 0.5 (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 2.9) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = re * (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 <= (-17000000000000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 2.9d0) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    else
        tmp = re * (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 <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 2.9) {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	} else {
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -17000000000000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 2.9:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	else:
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -17000000000000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 2.9)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	else
		tmp = Float64(re * 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 <= -17000000000000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 2.9)
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	else
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -17000000000000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.9], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 2.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.4%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right)\right)\right)} \]

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 36.2%

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

   14. Simplified 36.2%

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

   if -1.7e13 < 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 53.6%

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

   5. Simplified 53.6%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 50.0%

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

   8. Simplified 50.0%

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

   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 79.7%

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

   5. Simplified 79.7%

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

   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 55.8%

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

   8. Simplified 55.8%

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

   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. unpow2 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. associate-*l* N/A

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

       8. lft-mult-inverse N/A

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

       9. metadata-eval N/A

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

       10. associate-*r* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 55.8%

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

   11. Simplified 55.8%

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

Alternative 12: 49.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -17000000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.86:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -17000000000000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 1.86)
     (+ re 1.0)
     (* re (* re (+ 0.5 (* re 0.16666666666666666)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.86) {
		tmp = re + 1.0;
	} else {
		tmp = re * (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 <= (-17000000000000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 1.86d0) then
        tmp = re + 1.0d0
    else
        tmp = re * (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 <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 1.86) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -17000000000000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 1.86:
		tmp = re + 1.0
	else:
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -17000000000000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 1.86)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * 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 <= -17000000000000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 1.86)
		tmp = re + 1.0;
	else
		tmp = re * (re * (0.5 + (re * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -17000000000000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.86], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 2.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.4%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right)\right)\right)} \]

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 36.2%

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

   14. Simplified 36.2%

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

   if -1.7e13 < re < 1.8600000000000001

   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 53.6%

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

   5. Simplified 53.6%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 50.0%

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

   8. Simplified 50.0%

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

   if 1.8600000000000001 < 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 79.7%

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

   5. Simplified 79.7%

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

   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 55.8%

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

   8. Simplified 55.8%

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

   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. unpow2 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. associate-*l* N/A

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

       8. lft-mult-inverse N/A

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

       9. metadata-eval N/A

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

       10. associate-*r* N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 55.8%

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

   11. Simplified 55.8%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -17000000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.86:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -17000000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -17000000000000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} 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 <= (-17000000000000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    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 <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -17000000000000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	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 <= -17000000000000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	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 <= -17000000000000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	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, -17000000000000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.7e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 2.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.4%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right)\right)\right)} \]

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 36.2%

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

   14. Simplified 36.2%

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

   if -1.7e13 < 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 61.6%

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

   5. Simplified 61.6%

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

   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 51.8%

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

   8. Simplified 51.8%

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

Alternative 14: 49.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -17000000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.9:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -17000000000000.0)
   (* 0.041666666666666664 (* (* im im) (* im im)))
   (if (<= re 2.9) (+ re 1.0) (* re (* re (* re 0.16666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 2.9) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * (re * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-17000000000000.0d0)) then
        tmp = 0.041666666666666664d0 * ((im * im) * (im * im))
    else if (re <= 2.9d0) then
        tmp = re + 1.0d0
    else
        tmp = re * (re * (re * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -17000000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 2.9) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * (re * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -17000000000000.0:
		tmp = 0.041666666666666664 * ((im * im) * (im * im))
	elif re <= 2.9:
		tmp = re + 1.0
	else:
		tmp = re * (re * (re * 0.16666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -17000000000000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 2.9)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * Float64(re * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -17000000000000.0)
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	elseif (re <= 2.9)
		tmp = re + 1.0;
	else
		tmp = re * (re * (re * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -17000000000000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.9], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7e13

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 2.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.4%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot -0.001388888888888889\right)\right)\right)} \]

   9. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in im around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 36.2%

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

   14. Simplified 36.2%

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

   if -1.7e13 < 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 53.6%

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

   5. Simplified 53.6%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 50.0%

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

   8. Simplified 50.0%

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

   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 79.7%

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

   5. Simplified 79.7%

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

   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 55.8%

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

   8. Simplified 55.8%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 55.8%

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

   11. Simplified 55.8%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -17000000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.9:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.7% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.0) (+ re 1.0) (* re (* re (* re 0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * (re * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.0d0) then
        tmp = re + 1.0d0
    else
        tmp = re * (re * (re * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * (re * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.0:
		tmp = re + 1.0
	else:
		tmp = re * (re * (re * 0.16666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * Float64(re * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.0)
		tmp = re + 1.0;
	else
		tmp = re * (re * (re * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.0], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3

   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 68.5%

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

   5. Simplified 68.5%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 34.7%

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

   8. Simplified 34.7%

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

   if 3 < 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 79.7%

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

   5. Simplified 79.7%

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

   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 55.8%

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

   8. Simplified 55.8%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 55.8%

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

   11. Simplified 55.8%

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

4. Final simplification 39.6%

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

Alternative 16: 37.9% accurate, 20.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < 2.7000000000000002

   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 68.5%

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

   5. Simplified 68.5%

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

   6. Taylor expanded in re around 0

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

      1. +-lowering-+.f64 34.7%

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

   8. Simplified 34.7%

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

   if 2.7000000000000002 < 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 79.7%

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

   5. Simplified 79.7%

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

   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 42.7%

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

   8. Simplified 42.7%

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

   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 42.7%

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

   11. Simplified 42.7%

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

4. Final simplification 36.6%

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

Alternative 17: 28.9% 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 71.0%

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

5. Simplified 71.0%

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

6. Taylor expanded in re around 0

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

   1. +-lowering-+.f64 27.8%

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

8. Simplified 27.8%

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

9. Final simplification 27.8%

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

Alternative 18: 28.4% 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 71.0%

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

5. Simplified 71.0%

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

6. Taylor expanded in re around 0

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

8. Simplified 27.3%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))