math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.4s

Alternatives: 18

Speedup: 1.0×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.99995:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right) + 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.99995)
   (*
    (* 0.5 (cos re))
    (+
     2.0
     (*
      im
      (*
       im
       (+
        (*
         im
         (* im (+ (* im (* im 0.002777777777777778)) 0.08333333333333333)))
        1.0)))))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.99995) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * (im * ((im * (im * ((im * (im * 0.002777777777777778)) + 0.08333333333333333))) + 1.0))));
	} else {
		tmp = 0.5 * (exp(-im) + exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= 0.99995d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * (im * ((im * (im * ((im * (im * 0.002777777777777778d0)) + 0.08333333333333333d0))) + 1.0d0))))
    else
        tmp = 0.5d0 * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.99995) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * (im * ((im * (im * ((im * (im * 0.002777777777777778)) + 0.08333333333333333))) + 1.0))));
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.99995:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * (im * ((im * (im * ((im * (im * 0.002777777777777778)) + 0.08333333333333333))) + 1.0))))
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.99995)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * Float64(im * Float64(Float64(im * Float64(im * Float64(Float64(im * Float64(im * 0.002777777777777778)) + 0.08333333333333333))) + 1.0)))));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.99995)
		tmp = (0.5 * cos(re)) * (2.0 + (im * (im * ((im * (im * ((im * (im * 0.002777777777777778)) + 0.08333333333333333))) + 1.0))));
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.99995], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(im * N[(N[(im * N[(im * N[(N[(im * N[(im * 0.002777777777777778), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.999950000000000006

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 94.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 94.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \]

      2. unpow2 94.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \]

      3. *-commutative 94.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{{im}^{2} \cdot 0.002777777777777778}\right)\right)\right) \]

      4. unpow2 94.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right)} \cdot 0.002777777777777778\right)\right)\right) \]

      5. associate-*l* 94.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \]

   5. Simplified 94.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l* 94.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)}\right) \]

      2. +-commutative 94.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right) + 1\right)}\right)\right) \]

      3. associate-*l* 94.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(im \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)} + 1\right)\right)\right) \]

      4. +-commutative 94.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)}\right) + 1\right)\right)\right) \]

   7. Applied egg-rr 94.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right) + 1\right)\right)}\right) \]

   if 0.999950000000000006 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 99.6%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.99995:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right) + 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 6.5:\\ \;\;\;\;t\_0 \cdot \left(2 + im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right) + 1\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 6.5)
     (*
      t_0
      (+
       2.0
       (*
        im
        (*
         im
         (+
          (*
           im
           (* im (+ (* im (* im 0.002777777777777778)) 0.08333333333333333)))
          1.0)))))
     (if (<= im 2.4e+51)
       (+ 0.5 (* 0.5 (exp im)))
       (*
        t_0
        (+
         2.0
         (* 0.002777777777777778 (* (* im im) (* (* im im) (* im im))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 6.5) {
		tmp = t_0 * (2.0 + (im * (im * ((im * (im * ((im * (im * 0.002777777777777778)) + 0.08333333333333333))) + 1.0))));
	} else if (im <= 2.4e+51) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = t_0 * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    if (im <= 6.5d0) then
        tmp = t_0 * (2.0d0 + (im * (im * ((im * (im * ((im * (im * 0.002777777777777778d0)) + 0.08333333333333333d0))) + 1.0d0))))
    else if (im <= 2.4d+51) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = t_0 * (2.0d0 + (0.002777777777777778d0 * ((im * im) * ((im * im) * (im * im)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 6.5) {
		tmp = t_0 * (2.0 + (im * (im * ((im * (im * ((im * (im * 0.002777777777777778)) + 0.08333333333333333))) + 1.0))));
	} else if (im <= 2.4e+51) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = t_0 * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 6.5:
		tmp = t_0 * (2.0 + (im * (im * ((im * (im * ((im * (im * 0.002777777777777778)) + 0.08333333333333333))) + 1.0))))
	elif im <= 2.4e+51:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = t_0 * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 6.5)
		tmp = Float64(t_0 * Float64(2.0 + Float64(im * Float64(im * Float64(Float64(im * Float64(im * Float64(Float64(im * Float64(im * 0.002777777777777778)) + 0.08333333333333333))) + 1.0)))));
	elseif (im <= 2.4e+51)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(t_0 * Float64(2.0 + Float64(0.002777777777777778 * Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(im * im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 6.5)
		tmp = t_0 * (2.0 + (im * (im * ((im * (im * ((im * (im * 0.002777777777777778)) + 0.08333333333333333))) + 1.0))));
	elseif (im <= 2.4e+51)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = t_0 * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 6.5], N[(t$95$0 * N[(2.0 + N[(im * N[(im * N[(N[(im * N[(im * N[(N[(im * N[(im * 0.002777777777777778), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.4e+51], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 + N[(0.002777777777777778 * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 97.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 97.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \]

      2. unpow2 97.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \]

      3. *-commutative 97.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{{im}^{2} \cdot 0.002777777777777778}\right)\right)\right) \]

      4. unpow2 97.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right)} \cdot 0.002777777777777778\right)\right)\right) \]

      5. associate-*l* 97.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \]

   5. Simplified 97.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l* 97.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{im \cdot \left(im \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)}\right) \]

      2. +-commutative 97.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right) + 1\right)}\right)\right) \]

      3. associate-*l* 97.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(im \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)} + 1\right)\right)\right) \]

      4. +-commutative 97.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)}\right) + 1\right)\right)\right) \]

   7. Applied egg-rr 97.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right) + 1\right)\right)}\right) \]

   if 6.5 < im < 2.3999999999999999e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 66.7%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 66.7%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 66.7%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 66.7%

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

      6. associate-*r* 66.7%

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

      7. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 66.7%

      \[\leadsto \left(e^{im} + \color{blue}{1}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   7. Taylor expanded in re around 0 77.0%

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

      1. distribute-lft-in 77.0%

         \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 77.0%

         \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{im} \]

   9. Simplified 77.0%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot e^{im}} \]

   if 2.3999999999999999e51 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \]

      2. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \]

      3. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{{im}^{2} \cdot 0.002777777777777778}\right)\right)\right) \]

      4. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right)} \cdot 0.002777777777777778\right)\right)\right) \]

      5. associate-*l* 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{0.002777777777777778 \cdot {im}^{6}}\right) \]
   7. Step-by-step derivation

      1. metadata-eval 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

      2. pow-sqr 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

      3. cube-prod 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

      4. cube-unmult 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   8. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.4:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.4)
   (*
    (cos re)
    (+ 1.0 (* im (* im (+ 0.5 (* im (* im 0.041666666666666664)))))))
   (if (<= im 2.4e+51)
     (+ 0.5 (* 0.5 (exp im)))
     (*
      (* 0.5 (cos re))
      (+
       2.0
       (* 0.002777777777777778 (* (* im im) (* (* im im) (* im im)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.4) {
		tmp = cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664))))));
	} else if (im <= 2.4e+51) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = (0.5 * cos(re)) * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.4d0) then
        tmp = cos(re) * (1.0d0 + (im * (im * (0.5d0 + (im * (im * 0.041666666666666664d0))))))
    else if (im <= 2.4d+51) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (0.002777777777777778d0 * ((im * im) * ((im * im) * (im * im)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.4) {
		tmp = Math.cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664))))));
	} else if (im <= 2.4e+51) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.4:
		tmp = math.cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664))))))
	elif im <= 2.4e+51:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = (0.5 * math.cos(re)) * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.4)
		tmp = Float64(cos(re) * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.041666666666666664)))))));
	elseif (im <= 2.4e+51)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(0.002777777777777778 * Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(im * im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.4)
		tmp = cos(re) * (1.0 + (im * (im * (0.5 + (im * (im * 0.041666666666666664))))));
	elseif (im <= 2.4e+51)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = (0.5 * cos(re)) * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.4], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.4e+51], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(0.002777777777777778 * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.4000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 95.7%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 95.7%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 95.7%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 95.7%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 95.7%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 95.7%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 95.7%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 95.7%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 95.7%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 95.7%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 95.7%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 95.7%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 95.7%

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

   if 4.4000000000000004 < im < 2.3999999999999999e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 66.7%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 66.7%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 66.7%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 66.7%

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

      6. associate-*r* 66.7%

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

      7. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 66.7%

      \[\leadsto \left(e^{im} + \color{blue}{1}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   7. Taylor expanded in re around 0 77.0%

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

      1. distribute-lft-in 77.0%

         \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 77.0%

         \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{im} \]

   9. Simplified 77.0%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot e^{im}} \]

   if 2.3999999999999999e51 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \]

      2. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \]

      3. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{{im}^{2} \cdot 0.002777777777777778}\right)\right)\right) \]

      4. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right)} \cdot 0.002777777777777778\right)\right)\right) \]

      5. associate-*l* 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{0.002777777777777778 \cdot {im}^{6}}\right) \]
   7. Step-by-step derivation

      1. metadata-eval 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

      2. pow-sqr 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

      3. cube-prod 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

      4. cube-unmult 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   8. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.4:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot 0.041666666666666664\right)\\ \mathbf{if}\;im \leq 3.6:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + t\_0\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im 0.041666666666666664))))
   (if (<= im 3.6)
     (* (cos re) (+ 1.0 (* im (* im (+ 0.5 t_0)))))
     (if (<= im 2.6e+77)
       (+ 0.5 (* 0.5 (exp im)))
       (* (cos re) (+ 1.0 (* im (* im t_0))))))))

C

double code(double re, double im) {
	double t_0 = im * (im * 0.041666666666666664);
	double tmp;
	if (im <= 3.6) {
		tmp = cos(re) * (1.0 + (im * (im * (0.5 + t_0))));
	} else if (im <= 2.6e+77) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = cos(re) * (1.0 + (im * (im * 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 = im * (im * 0.041666666666666664d0)
    if (im <= 3.6d0) then
        tmp = cos(re) * (1.0d0 + (im * (im * (0.5d0 + t_0))))
    else if (im <= 2.6d+77) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = cos(re) * (1.0d0 + (im * (im * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * 0.041666666666666664);
	double tmp;
	if (im <= 3.6) {
		tmp = Math.cos(re) * (1.0 + (im * (im * (0.5 + t_0))));
	} else if (im <= 2.6e+77) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = Math.cos(re) * (1.0 + (im * (im * t_0)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * 0.041666666666666664)
	tmp = 0
	if im <= 3.6:
		tmp = math.cos(re) * (1.0 + (im * (im * (0.5 + t_0))))
	elif im <= 2.6e+77:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = math.cos(re) * (1.0 + (im * (im * t_0)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * 0.041666666666666664))
	tmp = 0.0
	if (im <= 3.6)
		tmp = Float64(cos(re) * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + t_0)))));
	elseif (im <= 2.6e+77)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(im * Float64(im * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * 0.041666666666666664);
	tmp = 0.0;
	if (im <= 3.6)
		tmp = cos(re) * (1.0 + (im * (im * (0.5 + t_0))));
	elseif (im <= 2.6e+77)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = cos(re) * (1.0 + (im * (im * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.6], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.60000000000000009

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 95.7%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 95.7%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 95.7%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 95.7%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 95.7%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 95.7%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 95.7%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 95.7%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 95.7%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 95.7%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 95.7%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 95.7%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 95.7%

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

   if 3.60000000000000009 < im < 2.6000000000000002e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 68.8%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 68.8%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 68.8%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 68.8%

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

      6. associate-*r* 68.8%

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

      7. *-commutative 68.8%

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

   5. Simplified 68.8%

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

   6. Taylor expanded in im around 0 68.8%

      \[\leadsto \left(e^{im} + \color{blue}{1}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   7. Taylor expanded in re around 0 76.5%

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

      1. distribute-lft-in 76.5%

         \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 76.5%

         \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{im} \]

   9. Simplified 76.5%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot e^{im}} \]

   if 2.6000000000000002e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 100.0%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 100.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 100.0%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 100.0%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 100.0%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \cos re \cdot \left(im \cdot \color{blue}{\left(0.041666666666666664 \cdot {im}^{3}\right)} + 1\right) \]
   7. Step-by-step derivation

      1. cube-unmult 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \cos re \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} + 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.8:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.8)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 2.6e+77)
     (+ 0.5 (* 0.5 (exp im)))
     (* (cos re) (+ 1.0 (* im (* im (* im (* im 0.041666666666666664)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.8) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 2.6e+77) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = cos(re) * (1.0 + (im * (im * (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.8d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 2.6d+77) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = cos(re) * (1.0d0 + (im * (im * (im * (im * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.8) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 2.6e+77) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = Math.cos(re) * (1.0 + (im * (im * (im * (im * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.8:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 2.6e+77:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = math.cos(re) * (1.0 + (im * (im * (im * (im * 0.041666666666666664)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.8)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 2.6e+77)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(im * Float64(im * Float64(im * Float64(im * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.8)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 2.6e+77)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = cos(re) * (1.0 + (im * (im * (im * (im * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.8], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7999999999999998

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 86.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 86.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{im \cdot im}\right) \]

   5. Simplified 86.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   if 3.7999999999999998 < im < 2.6000000000000002e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 68.8%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 68.8%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 68.8%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 68.8%

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

      6. associate-*r* 68.8%

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

      7. *-commutative 68.8%

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

   5. Simplified 68.8%

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

   6. Taylor expanded in im around 0 68.8%

      \[\leadsto \left(e^{im} + \color{blue}{1}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   7. Taylor expanded in re around 0 76.5%

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

      1. distribute-lft-in 76.5%

         \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 76.5%

         \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{im} \]

   9. Simplified 76.5%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot e^{im}} \]

   if 2.6000000000000002e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 100.0%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 100.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 100.0%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 100.0%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 100.0%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \cos re \cdot \left(im \cdot \color{blue}{\left(0.041666666666666664 \cdot {im}^{3}\right)} + 1\right) \]
   7. Step-by-step derivation

      1. cube-unmult 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \cos re \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} + 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.2 \lor \neg \left(im \leq 3.7 \cdot 10^{+151}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 4.2) (not (<= im 3.7e+151)))
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (+ 0.5 (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 4.2) || !(im <= 3.7e+151)) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.5 + (0.5 * exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 4.2d0) .or. (.not. (im <= 3.7d+151))) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else
        tmp = 0.5d0 + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 4.2) || !(im <= 3.7e+151)) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 4.2) or not (im <= 3.7e+151):
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	else:
		tmp = 0.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 4.2) || !(im <= 3.7e+151))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 4.2) || ~((im <= 3.7e+151)))
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 4.2], N[Not[LessEqual[im, 3.7e+151]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.20000000000000018 or 3.6999999999999997e151 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 88.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 88.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{im \cdot im}\right) \]

   5. Simplified 88.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   if 4.20000000000000018 < im < 3.6999999999999997e151

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 66.7%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 66.7%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 66.7%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 66.7%

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

      6. associate-*r* 66.7%

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

      7. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 66.7%

      \[\leadsto \left(e^{im} + \color{blue}{1}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   7. Taylor expanded in re around 0 78.6%

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

      1. distribute-lft-in 78.6%

         \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 78.6%

         \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{im} \]

   9. Simplified 78.6%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot e^{im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \lor \neg \left(im \leq 3.7 \cdot 10^{+151}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.82:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+151}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.82)
   (cos re)
   (if (<= im 1.95e+151)
     (+ 0.5 (* 0.5 (exp im)))
     (if (<= im 1.6e+227)
       (* (+ 2.0 (* im im)) (+ 0.5 (* re (* re -0.25))))
       (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.82) {
		tmp = cos(re);
	} else if (im <= 1.95e+151) {
		tmp = 0.5 + (0.5 * exp(im));
	} else if (im <= 1.6e+227) {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.82d0) then
        tmp = cos(re)
    else if (im <= 1.95d+151) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else if (im <= 1.6d+227) then
        tmp = (2.0d0 + (im * im)) * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.82) {
		tmp = Math.cos(re);
	} else if (im <= 1.95e+151) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else if (im <= 1.6e+227) {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.82:
		tmp = math.cos(re)
	elif im <= 1.95e+151:
		tmp = 0.5 + (0.5 * math.exp(im))
	elif im <= 1.6e+227:
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)))
	else:
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.82)
		tmp = cos(re);
	elseif (im <= 1.95e+151)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	elseif (im <= 1.6e+227)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.82)
		tmp = cos(re);
	elseif (im <= 1.95e+151)
		tmp = 0.5 + (0.5 * exp(im));
	elseif (im <= 1.6e+227)
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	else
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.82], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.95e+151], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.6e+227], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.82000000000000006

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 60.9%

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

   if 1.82000000000000006 < im < 1.94999999999999988e151

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 66.7%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 66.7%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 66.7%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 66.7%

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

      6. associate-*r* 66.7%

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

      7. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 66.7%

      \[\leadsto \left(e^{im} + \color{blue}{1}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   7. Taylor expanded in re around 0 78.6%

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

      1. distribute-lft-in 78.6%

         \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{im}} \]

      2. metadata-eval 78.6%

         \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{im} \]

   9. Simplified 78.6%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot e^{im}} \]

   if 1.94999999999999988e151 < im < 1.59999999999999994e227

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 90.5%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 90.5%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 90.5%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 90.5%

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

      6. associate-*r* 90.5%

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

      7. *-commutative 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in im around 0 90.5%

      \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 90.5%

         \[\leadsto \left(2 + \color{blue}{im \cdot im}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 90.5%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   if 1.59999999999999994e227 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 100.0%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 100.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 100.0%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 100.0%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 100.0%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 82.4%

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 82.4%

         \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \]

      2. +-commutative 82.4%

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} \]

      3. unpow2 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r* 82.4%

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

      6. associate-*r* 82.4%

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

      7. +-commutative 82.4%

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

      8. associate-*r* 82.4%

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

      9. *-commutative 82.4%

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

   10. Simplified 82.4%

       \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.82:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+151}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(im \cdot im\right)\\ t_1 := 0.5 + re \cdot \left(re \cdot -0.25\right)\\ t_2 := im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\\ \mathbf{if}\;im \leq 64000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+51}:\\ \;\;\;\;t\_1 \cdot \left(2 + \frac{t\_0 \cdot \left(1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(t\_2 \cdot t\_2\right)\right)\right)}{\left(im \cdot im\right) \cdot \left(1 - \left(im \cdot im\right) \cdot t\_2\right)}\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot t\_0\right)\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (* im im)))
        (t_1 (+ 0.5 (* re (* re -0.25))))
        (t_2 (+ (* im (* im 0.002777777777777778)) 0.08333333333333333)))
   (if (<= im 64000000.0)
     (cos re)
     (if (<= im 3e+51)
       (*
        t_1
        (+
         2.0
         (/
          (* t_0 (- 1.0 (* (* im im) (* (* im im) (* t_2 t_2)))))
          (* (* im im) (- 1.0 (* (* im im) t_2))))))
       (if (<= im 3.7e+151)
         (* 0.5 (+ 2.0 (* 0.002777777777777778 (* (* im im) t_0))))
         (if (<= im 1.8e+227)
           (* (+ 2.0 (* im im)) t_1)
           (+
            1.0
            (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664)))))))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (im * im);
	double t_1 = 0.5 + (re * (re * -0.25));
	double t_2 = (im * (im * 0.002777777777777778)) + 0.08333333333333333;
	double tmp;
	if (im <= 64000000.0) {
		tmp = cos(re);
	} else if (im <= 3e+51) {
		tmp = t_1 * (2.0 + ((t_0 * (1.0 - ((im * im) * ((im * im) * (t_2 * t_2))))) / ((im * im) * (1.0 - ((im * im) * t_2)))));
	} else if (im <= 3.7e+151) {
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * t_0)));
	} else if (im <= 1.8e+227) {
		tmp = (2.0 + (im * im)) * t_1;
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	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 = (im * im) * (im * im)
    t_1 = 0.5d0 + (re * (re * (-0.25d0)))
    t_2 = (im * (im * 0.002777777777777778d0)) + 0.08333333333333333d0
    if (im <= 64000000.0d0) then
        tmp = cos(re)
    else if (im <= 3d+51) then
        tmp = t_1 * (2.0d0 + ((t_0 * (1.0d0 - ((im * im) * ((im * im) * (t_2 * t_2))))) / ((im * im) * (1.0d0 - ((im * im) * t_2)))))
    else if (im <= 3.7d+151) then
        tmp = 0.5d0 * (2.0d0 + (0.002777777777777778d0 * ((im * im) * t_0)))
    else if (im <= 1.8d+227) then
        tmp = (2.0d0 + (im * im)) * t_1
    else
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (im * im);
	double t_1 = 0.5 + (re * (re * -0.25));
	double t_2 = (im * (im * 0.002777777777777778)) + 0.08333333333333333;
	double tmp;
	if (im <= 64000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 3e+51) {
		tmp = t_1 * (2.0 + ((t_0 * (1.0 - ((im * im) * ((im * im) * (t_2 * t_2))))) / ((im * im) * (1.0 - ((im * im) * t_2)))));
	} else if (im <= 3.7e+151) {
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * t_0)));
	} else if (im <= 1.8e+227) {
		tmp = (2.0 + (im * im)) * t_1;
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (im * im)
	t_1 = 0.5 + (re * (re * -0.25))
	t_2 = (im * (im * 0.002777777777777778)) + 0.08333333333333333
	tmp = 0
	if im <= 64000000.0:
		tmp = math.cos(re)
	elif im <= 3e+51:
		tmp = t_1 * (2.0 + ((t_0 * (1.0 - ((im * im) * ((im * im) * (t_2 * t_2))))) / ((im * im) * (1.0 - ((im * im) * t_2)))))
	elif im <= 3.7e+151:
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * t_0)))
	elif im <= 1.8e+227:
		tmp = (2.0 + (im * im)) * t_1
	else:
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(im * im))
	t_1 = Float64(0.5 + Float64(re * Float64(re * -0.25)))
	t_2 = Float64(Float64(im * Float64(im * 0.002777777777777778)) + 0.08333333333333333)
	tmp = 0.0
	if (im <= 64000000.0)
		tmp = cos(re);
	elseif (im <= 3e+51)
		tmp = Float64(t_1 * Float64(2.0 + Float64(Float64(t_0 * Float64(1.0 - Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(t_2 * t_2))))) / Float64(Float64(im * im) * Float64(1.0 - Float64(Float64(im * im) * t_2))))));
	elseif (im <= 3.7e+151)
		tmp = Float64(0.5 * Float64(2.0 + Float64(0.002777777777777778 * Float64(Float64(im * im) * t_0))));
	elseif (im <= 1.8e+227)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * t_1);
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (im * im);
	t_1 = 0.5 + (re * (re * -0.25));
	t_2 = (im * (im * 0.002777777777777778)) + 0.08333333333333333;
	tmp = 0.0;
	if (im <= 64000000.0)
		tmp = cos(re);
	elseif (im <= 3e+51)
		tmp = t_1 * (2.0 + ((t_0 * (1.0 - ((im * im) * ((im * im) * (t_2 * t_2))))) / ((im * im) * (1.0 - ((im * im) * t_2)))));
	elseif (im <= 3.7e+151)
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * t_0)));
	elseif (im <= 1.8e+227)
		tmp = (2.0 + (im * im)) * t_1;
	else
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(im * N[(im * 0.002777777777777778), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]}, If[LessEqual[im, 64000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3e+51], N[(t$95$1 * N[(2.0 + N[(N[(t$95$0 * N[(1.0 - N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(im * im), $MachinePrecision] * N[(1.0 - N[(N[(im * im), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.7e+151], N[(0.5 * N[(2.0 + N[(0.002777777777777778 * N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.8e+227], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 6.4e7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 60.0%

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

   if 6.4e7 < im < 3e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 88.9%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 88.9%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 88.9%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 88.9%

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

      6. associate-*r* 88.9%

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

      7. *-commutative 88.9%

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

   5. Simplified 88.9%

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

   6. Taylor expanded in im around 0 26.1%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 26.1%

         \[\leadsto \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. unpow2 26.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      3. unpow2 26.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      4. *-commutative 26.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right) \cdot 0.002777777777777778}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      5. associate-*r* 26.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 26.1%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 26.1%

         \[\leadsto \left(2 + \color{blue}{\left(1 \cdot \left(im \cdot im\right) + \left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(im \cdot im\right)\right)}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. *-un-lft-identity 26.1%

         \[\leadsto \left(2 + \left(\color{blue}{im \cdot im} + \left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      3. flip-+ 67.5%

         \[\leadsto \left(2 + \color{blue}{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(\left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(im \cdot im\right)\right)}{im \cdot im - \left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(im \cdot im\right)}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   10. Applied egg-rr 67.5%

       \[\leadsto \left(2 + \color{blue}{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot im\right)\right)}{im \cdot im - \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot im\right)}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 67.5%

       \[\leadsto \left(2 + \color{blue}{\frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)\right)}{\left(\left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right) \cdot \left(im \cdot \left(im \cdot -1\right)\right) + 1\right) \cdot \left(im \cdot im\right)}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   if 3e51 < im < 3.6999999999999997e151

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 66.7%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 66.7%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 66.7%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 66.7%

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

      6. associate-*r* 66.7%

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

      7. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 66.7%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 66.7%

         \[\leadsto \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. unpow2 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      3. unpow2 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      4. *-commutative 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right) \cdot 0.002777777777777778}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      5. associate-*r* 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 66.7%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   9. Taylor expanded in im around inf 66.7%

      \[\leadsto \left(2 + \color{blue}{0.002777777777777778 \cdot {im}^{6}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

       2. pow-sqr 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

       3. cube-prod 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

       4. cube-unmult 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   11. Simplified 66.7%

       \[\leadsto \left(2 + \color{blue}{0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   12. Taylor expanded in re around 0 80.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{6}\right)} \]
   13. Step-by-step derivation

       1. metadata-eval 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

       2. pow-sqr 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

       3. cube-prod 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

       4. cube-mult 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   14. Simplified 80.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

   if 3.6999999999999997e151 < im < 1.79999999999999996e227

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 90.5%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 90.5%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 90.5%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 90.5%

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

      6. associate-*r* 90.5%

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

      7. *-commutative 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in im around 0 90.5%

      \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 90.5%

         \[\leadsto \left(2 + \color{blue}{im \cdot im}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 90.5%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   if 1.79999999999999996e227 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 100.0%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 100.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 100.0%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 100.0%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 100.0%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 82.4%

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 82.4%

         \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \]

      2. +-commutative 82.4%

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} \]

      3. unpow2 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r* 82.4%

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

      6. associate-*r* 82.4%

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

      7. +-commutative 82.4%

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

      8. associate-*r* 82.4%

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

      9. *-commutative 82.4%

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

   10. Simplified 82.4%

       \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 64000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(2 + \frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right)\right)\right)}{\left(im \cdot im\right) \cdot \left(1 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right)}\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\\ t_1 := 0.5 + re \cdot \left(re \cdot -0.25\right)\\ t_2 := 1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ t_3 := \left(im \cdot im\right) \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 4800000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;t\_1 \cdot \left(2 + \frac{t\_3 \cdot \left(1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(t\_0 \cdot t\_0\right)\right)\right)}{\left(im \cdot im\right) \cdot \left(1 - \left(im \cdot im\right) \cdot t\_0\right)}\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot t\_3\right)\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* im (* im 0.002777777777777778)) 0.08333333333333333))
        (t_1 (+ 0.5 (* re (* re -0.25))))
        (t_2 (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))
        (t_3 (* (* im im) (* im im))))
   (if (<= im 4800000000.0)
     t_2
     (if (<= im 2.4e+51)
       (*
        t_1
        (+
         2.0
         (/
          (* t_3 (- 1.0 (* (* im im) (* (* im im) (* t_0 t_0)))))
          (* (* im im) (- 1.0 (* (* im im) t_0))))))
       (if (<= im 3.7e+151)
         (* 0.5 (+ 2.0 (* 0.002777777777777778 (* (* im im) t_3))))
         (if (<= im 1.8e+227) (* (+ 2.0 (* im im)) t_1) t_2))))))

C

double code(double re, double im) {
	double t_0 = (im * (im * 0.002777777777777778)) + 0.08333333333333333;
	double t_1 = 0.5 + (re * (re * -0.25));
	double t_2 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	double t_3 = (im * im) * (im * im);
	double tmp;
	if (im <= 4800000000.0) {
		tmp = t_2;
	} else if (im <= 2.4e+51) {
		tmp = t_1 * (2.0 + ((t_3 * (1.0 - ((im * im) * ((im * im) * (t_0 * t_0))))) / ((im * im) * (1.0 - ((im * im) * t_0)))));
	} else if (im <= 3.7e+151) {
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * t_3)));
	} else if (im <= 1.8e+227) {
		tmp = (2.0 + (im * im)) * 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) :: t_3
    real(8) :: tmp
    t_0 = (im * (im * 0.002777777777777778d0)) + 0.08333333333333333d0
    t_1 = 0.5d0 + (re * (re * (-0.25d0)))
    t_2 = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    t_3 = (im * im) * (im * im)
    if (im <= 4800000000.0d0) then
        tmp = t_2
    else if (im <= 2.4d+51) then
        tmp = t_1 * (2.0d0 + ((t_3 * (1.0d0 - ((im * im) * ((im * im) * (t_0 * t_0))))) / ((im * im) * (1.0d0 - ((im * im) * t_0)))))
    else if (im <= 3.7d+151) then
        tmp = 0.5d0 * (2.0d0 + (0.002777777777777778d0 * ((im * im) * t_3)))
    else if (im <= 1.8d+227) then
        tmp = (2.0d0 + (im * im)) * t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * (im * 0.002777777777777778)) + 0.08333333333333333;
	double t_1 = 0.5 + (re * (re * -0.25));
	double t_2 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	double t_3 = (im * im) * (im * im);
	double tmp;
	if (im <= 4800000000.0) {
		tmp = t_2;
	} else if (im <= 2.4e+51) {
		tmp = t_1 * (2.0 + ((t_3 * (1.0 - ((im * im) * ((im * im) * (t_0 * t_0))))) / ((im * im) * (1.0 - ((im * im) * t_0)))));
	} else if (im <= 3.7e+151) {
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * t_3)));
	} else if (im <= 1.8e+227) {
		tmp = (2.0 + (im * im)) * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * (im * 0.002777777777777778)) + 0.08333333333333333
	t_1 = 0.5 + (re * (re * -0.25))
	t_2 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	t_3 = (im * im) * (im * im)
	tmp = 0
	if im <= 4800000000.0:
		tmp = t_2
	elif im <= 2.4e+51:
		tmp = t_1 * (2.0 + ((t_3 * (1.0 - ((im * im) * ((im * im) * (t_0 * t_0))))) / ((im * im) * (1.0 - ((im * im) * t_0)))))
	elif im <= 3.7e+151:
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * t_3)))
	elif im <= 1.8e+227:
		tmp = (2.0 + (im * im)) * t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * Float64(im * 0.002777777777777778)) + 0.08333333333333333)
	t_1 = Float64(0.5 + Float64(re * Float64(re * -0.25)))
	t_2 = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	t_3 = Float64(Float64(im * im) * Float64(im * im))
	tmp = 0.0
	if (im <= 4800000000.0)
		tmp = t_2;
	elseif (im <= 2.4e+51)
		tmp = Float64(t_1 * Float64(2.0 + Float64(Float64(t_3 * Float64(1.0 - Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(t_0 * t_0))))) / Float64(Float64(im * im) * Float64(1.0 - Float64(Float64(im * im) * t_0))))));
	elseif (im <= 3.7e+151)
		tmp = Float64(0.5 * Float64(2.0 + Float64(0.002777777777777778 * Float64(Float64(im * im) * t_3))));
	elseif (im <= 1.8e+227)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * (im * 0.002777777777777778)) + 0.08333333333333333;
	t_1 = 0.5 + (re * (re * -0.25));
	t_2 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	t_3 = (im * im) * (im * im);
	tmp = 0.0;
	if (im <= 4800000000.0)
		tmp = t_2;
	elseif (im <= 2.4e+51)
		tmp = t_1 * (2.0 + ((t_3 * (1.0 - ((im * im) * ((im * im) * (t_0 * t_0))))) / ((im * im) * (1.0 - ((im * im) * t_0)))));
	elseif (im <= 3.7e+151)
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * t_3)));
	elseif (im <= 1.8e+227)
		tmp = (2.0 + (im * im)) * t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * N[(im * 0.002777777777777778), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4800000000.0], t$95$2, If[LessEqual[im, 2.4e+51], N[(t$95$1 * N[(2.0 + N[(N[(t$95$3 * N[(1.0 - N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(im * im), $MachinePrecision] * N[(1.0 - N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.7e+151], N[(0.5 * N[(2.0 + N[(0.002777777777777778 * N[(N[(im * im), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.8e+227], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 4.8e9 or 1.79999999999999996e227 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 94.7%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 94.7%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 94.7%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 94.7%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 94.7%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 94.7%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 94.7%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 94.7%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 94.7%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 94.7%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 94.7%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 94.7%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 94.7%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 94.7%

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

      2. +-commutative 94.7%

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

      3. associate-*r* 94.7%

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

   7. Applied egg-rr 94.7%

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

   8. Taylor expanded in re around 0 60.2%

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 60.2%

         \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \]

      2. +-commutative 60.2%

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} \]

      3. unpow2 60.2%

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

      4. *-commutative 60.2%

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

      5. associate-*r* 60.2%

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

      6. associate-*r* 60.2%

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

      7. +-commutative 60.2%

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

      8. associate-*r* 60.2%

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

      9. *-commutative 60.2%

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

   10. Simplified 60.2%

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

   if 4.8e9 < im < 2.3999999999999999e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 88.9%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 88.9%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 88.9%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 88.9%

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

      6. associate-*r* 88.9%

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

      7. *-commutative 88.9%

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

   5. Simplified 88.9%

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

   6. Taylor expanded in im around 0 26.1%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 26.1%

         \[\leadsto \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. unpow2 26.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      3. unpow2 26.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      4. *-commutative 26.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right) \cdot 0.002777777777777778}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      5. associate-*r* 26.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 26.1%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 26.1%

         \[\leadsto \left(2 + \color{blue}{\left(1 \cdot \left(im \cdot im\right) + \left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(im \cdot im\right)\right)}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. *-un-lft-identity 26.1%

         \[\leadsto \left(2 + \left(\color{blue}{im \cdot im} + \left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      3. flip-+ 67.5%

         \[\leadsto \left(2 + \color{blue}{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(\left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(im \cdot im\right)\right)}{im \cdot im - \left(\left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(im \cdot im\right)}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   10. Applied egg-rr 67.5%

       \[\leadsto \left(2 + \color{blue}{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot im\right)\right)}{im \cdot im - \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right)\right) \cdot \left(im \cdot im\right)}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 67.5%

       \[\leadsto \left(2 + \color{blue}{\frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)\right)}{\left(\left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right) \cdot \left(im \cdot \left(im \cdot -1\right)\right) + 1\right) \cdot \left(im \cdot im\right)}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   if 2.3999999999999999e51 < im < 3.6999999999999997e151

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 66.7%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 66.7%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 66.7%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 66.7%

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

      6. associate-*r* 66.7%

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

      7. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 66.7%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 66.7%

         \[\leadsto \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. unpow2 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      3. unpow2 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      4. *-commutative 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right) \cdot 0.002777777777777778}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      5. associate-*r* 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 66.7%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   9. Taylor expanded in im around inf 66.7%

      \[\leadsto \left(2 + \color{blue}{0.002777777777777778 \cdot {im}^{6}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

       2. pow-sqr 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

       3. cube-prod 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

       4. cube-unmult 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   11. Simplified 66.7%

       \[\leadsto \left(2 + \color{blue}{0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   12. Taylor expanded in re around 0 80.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{6}\right)} \]
   13. Step-by-step derivation

       1. metadata-eval 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

       2. pow-sqr 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

       3. cube-prod 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

       4. cube-mult 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   14. Simplified 80.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

   if 3.6999999999999997e151 < im < 1.79999999999999996e227

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 90.5%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 90.5%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 90.5%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 90.5%

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

      6. associate-*r* 90.5%

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

      7. *-commutative 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in im around 0 90.5%

      \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 90.5%

         \[\leadsto \left(2 + \color{blue}{im \cdot im}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 90.5%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4800000000:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(2 + \frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right)\right)\right)}{\left(im \cdot im\right) \cdot \left(1 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right) + 0.08333333333333333\right)\right)}\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + re \cdot \left(re \cdot -0.25\right)\\ t_1 := 1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ t_2 := \left(im \cdot im\right) \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 112000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+77}:\\ \;\;\;\;t\_0 \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{t\_2 \cdot \left(0.006944444444444444 - t\_2 \cdot 7.71604938271605 \cdot 10^{-6}\right)}{\left(im \cdot im\right) \cdot \left(0.08333333333333333 - im \cdot \left(im \cdot 0.002777777777777778\right)\right)}\right)\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+151} \lor \neg \left(im \leq 1.75 \cdot 10^{+227}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* re (* re -0.25))))
        (t_1 (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))
        (t_2 (* (* im im) (* im im))))
   (if (<= im 112000000.0)
     t_1
     (if (<= im 5e+77)
       (*
        t_0
        (+
         2.0
         (*
          (* im im)
          (+
           1.0
           (/
            (* t_2 (- 0.006944444444444444 (* t_2 7.71604938271605e-6)))
            (*
             (* im im)
             (- 0.08333333333333333 (* im (* im 0.002777777777777778)))))))))
       (if (or (<= im 3.7e+151) (not (<= im 1.75e+227)))
         t_1
         (* (+ 2.0 (* im im)) t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (re * (re * -0.25));
	double t_1 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	double t_2 = (im * im) * (im * im);
	double tmp;
	if (im <= 112000000.0) {
		tmp = t_1;
	} else if (im <= 5e+77) {
		tmp = t_0 * (2.0 + ((im * im) * (1.0 + ((t_2 * (0.006944444444444444 - (t_2 * 7.71604938271605e-6))) / ((im * im) * (0.08333333333333333 - (im * (im * 0.002777777777777778))))))));
	} else if ((im <= 3.7e+151) || !(im <= 1.75e+227)) {
		tmp = t_1;
	} else {
		tmp = (2.0 + (im * im)) * 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 + (re * (re * (-0.25d0)))
    t_1 = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    t_2 = (im * im) * (im * im)
    if (im <= 112000000.0d0) then
        tmp = t_1
    else if (im <= 5d+77) then
        tmp = t_0 * (2.0d0 + ((im * im) * (1.0d0 + ((t_2 * (0.006944444444444444d0 - (t_2 * 7.71604938271605d-6))) / ((im * im) * (0.08333333333333333d0 - (im * (im * 0.002777777777777778d0))))))))
    else if ((im <= 3.7d+151) .or. (.not. (im <= 1.75d+227))) then
        tmp = t_1
    else
        tmp = (2.0d0 + (im * im)) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (re * (re * -0.25));
	double t_1 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	double t_2 = (im * im) * (im * im);
	double tmp;
	if (im <= 112000000.0) {
		tmp = t_1;
	} else if (im <= 5e+77) {
		tmp = t_0 * (2.0 + ((im * im) * (1.0 + ((t_2 * (0.006944444444444444 - (t_2 * 7.71604938271605e-6))) / ((im * im) * (0.08333333333333333 - (im * (im * 0.002777777777777778))))))));
	} else if ((im <= 3.7e+151) || !(im <= 1.75e+227)) {
		tmp = t_1;
	} else {
		tmp = (2.0 + (im * im)) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (re * (re * -0.25))
	t_1 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	t_2 = (im * im) * (im * im)
	tmp = 0
	if im <= 112000000.0:
		tmp = t_1
	elif im <= 5e+77:
		tmp = t_0 * (2.0 + ((im * im) * (1.0 + ((t_2 * (0.006944444444444444 - (t_2 * 7.71604938271605e-6))) / ((im * im) * (0.08333333333333333 - (im * (im * 0.002777777777777778))))))))
	elif (im <= 3.7e+151) or not (im <= 1.75e+227):
		tmp = t_1
	else:
		tmp = (2.0 + (im * im)) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(re * Float64(re * -0.25)))
	t_1 = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	t_2 = Float64(Float64(im * im) * Float64(im * im))
	tmp = 0.0
	if (im <= 112000000.0)
		tmp = t_1;
	elseif (im <= 5e+77)
		tmp = Float64(t_0 * Float64(2.0 + Float64(Float64(im * im) * Float64(1.0 + Float64(Float64(t_2 * Float64(0.006944444444444444 - Float64(t_2 * 7.71604938271605e-6))) / Float64(Float64(im * im) * Float64(0.08333333333333333 - Float64(im * Float64(im * 0.002777777777777778)))))))));
	elseif ((im <= 3.7e+151) || !(im <= 1.75e+227))
		tmp = t_1;
	else
		tmp = Float64(Float64(2.0 + Float64(im * im)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (re * (re * -0.25));
	t_1 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	t_2 = (im * im) * (im * im);
	tmp = 0.0;
	if (im <= 112000000.0)
		tmp = t_1;
	elseif (im <= 5e+77)
		tmp = t_0 * (2.0 + ((im * im) * (1.0 + ((t_2 * (0.006944444444444444 - (t_2 * 7.71604938271605e-6))) / ((im * im) * (0.08333333333333333 - (im * (im * 0.002777777777777778))))))));
	elseif ((im <= 3.7e+151) || ~((im <= 1.75e+227)))
		tmp = t_1;
	else
		tmp = (2.0 + (im * im)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 112000000.0], t$95$1, If[LessEqual[im, 5e+77], N[(t$95$0 * N[(2.0 + N[(N[(im * im), $MachinePrecision] * N[(1.0 + N[(N[(t$95$2 * N[(0.006944444444444444 - N[(t$95$2 * 7.71604938271605e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(im * im), $MachinePrecision] * N[(0.08333333333333333 - N[(im * N[(im * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 3.7e+151], N[Not[LessEqual[im, 1.75e+227]], $MachinePrecision]], t$95$1, N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.12e8 or 5.00000000000000004e77 < im < 3.6999999999999997e151 or 1.75e227 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 95.0%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 95.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 95.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 95.0%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 95.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 95.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 95.0%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 95.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 95.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 95.0%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 95.0%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 95.0%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 95.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 95.0%

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

      2. +-commutative 95.0%

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

      3. associate-*r* 95.0%

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

   7. Applied egg-rr 95.0%

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

   8. Taylor expanded in re around 0 61.3%

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 61.3%

         \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \]

      2. +-commutative 61.3%

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} \]

      3. unpow2 61.3%

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

      4. *-commutative 61.3%

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

      5. associate-*r* 61.3%

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

      6. associate-*r* 61.3%

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

      7. +-commutative 61.3%

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

      8. associate-*r* 61.3%

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

      9. *-commutative 61.3%

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

   10. Simplified 61.3%

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

   if 1.12e8 < im < 5.00000000000000004e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 84.6%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 84.6%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 84.6%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 84.6%

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

      6. associate-*r* 84.6%

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

      7. *-commutative 84.6%

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

   5. Simplified 84.6%

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

   6. Taylor expanded in im around 0 41.1%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 41.1%

         \[\leadsto \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. unpow2 41.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      3. unpow2 41.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      4. *-commutative 41.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right) \cdot 0.002777777777777778}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      5. associate-*r* 41.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 41.1%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in 41.1%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(\left(im \cdot im\right) \cdot 0.08333333333333333 + \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. flip-+ 55.2%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(im \cdot im\right) \cdot 0.08333333333333333\right) \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right) - \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)}{\left(im \cdot im\right) \cdot 0.08333333333333333 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)}}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   10. Applied egg-rr 55.2%

       \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(im \cdot im\right) \cdot 0.08333333333333333\right) \cdot \left(\left(im \cdot im\right) \cdot 0.08333333333333333\right) - \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)}{\left(im \cdot im\right) \cdot 0.08333333333333333 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)}}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   11. Step-by-step derivation

       1. swap-sqr 55.2%

          \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.08333333333333333 \cdot 0.08333333333333333\right)} - \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)}{\left(im \cdot im\right) \cdot 0.08333333333333333 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

       2. swap-sqr 55.2%

          \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.08333333333333333 \cdot 0.08333333333333333\right) - \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(im \cdot \left(im \cdot 0.002777777777777778\right)\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)}}{\left(im \cdot im\right) \cdot 0.08333333333333333 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

       3. distribute-lft-out-- 55.2%

          \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.08333333333333333 \cdot 0.08333333333333333 - \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)}}{\left(im \cdot im\right) \cdot 0.08333333333333333 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

       4. metadata-eval 55.2%

          \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{0.006944444444444444} - \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)}{\left(im \cdot im\right) \cdot 0.08333333333333333 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

       5. associate-*r* 55.2%

          \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.006944444444444444 - \color{blue}{\left(\left(im \cdot im\right) \cdot 0.002777777777777778\right)} \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)}{\left(im \cdot im\right) \cdot 0.08333333333333333 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

       6. associate-*r* 55.2%

          \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.006944444444444444 - \left(\left(im \cdot im\right) \cdot 0.002777777777777778\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot 0.002777777777777778\right)}\right)}{\left(im \cdot im\right) \cdot 0.08333333333333333 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

       7. swap-sqr 55.2%

          \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.006944444444444444 - \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.002777777777777778 \cdot 0.002777777777777778\right)}\right)}{\left(im \cdot im\right) \cdot 0.08333333333333333 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

       8. metadata-eval 55.2%

          \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.006944444444444444 - \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{7.71604938271605 \cdot 10^{-6}}\right)}{\left(im \cdot im\right) \cdot 0.08333333333333333 - \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot 0.002777777777777778\right)\right)}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

       9. distribute-lft-out-- 55.2%

          \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.006944444444444444 - \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 7.71604938271605 \cdot 10^{-6}\right)}{\color{blue}{\left(im \cdot im\right) \cdot \left(0.08333333333333333 - im \cdot \left(im \cdot 0.002777777777777778\right)\right)}}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   12. Simplified 55.2%

       \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.006944444444444444 - \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 7.71604938271605 \cdot 10^{-6}\right)}{\left(im \cdot im\right) \cdot \left(0.08333333333333333 - im \cdot \left(im \cdot 0.002777777777777778\right)\right)}}\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   if 3.6999999999999997e151 < im < 1.75e227

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 90.5%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 90.5%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 90.5%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 90.5%

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

      6. associate-*r* 90.5%

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

      7. *-commutative 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in im around 0 90.5%

      \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 90.5%

         \[\leadsto \left(2 + \color{blue}{im \cdot im}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 90.5%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 112000000:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \frac{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.006944444444444444 - \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 7.71604938271605 \cdot 10^{-6}\right)}{\left(im \cdot im\right) \cdot \left(0.08333333333333333 - im \cdot \left(im \cdot 0.002777777777777778\right)\right)}\right)\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+151} \lor \neg \left(im \leq 1.75 \cdot 10^{+227}\right):\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ t_1 := 2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{if}\;im \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(t\_1 \cdot \left(-0.25 + \frac{0.5}{re \cdot re}\right)\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot t\_1\\ \mathbf{elif}\;im \leq 1.72 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))
        (t_1
         (+
          2.0
          (* 0.002777777777777778 (* (* im im) (* (* im im) (* im im)))))))
   (if (<= im 5.8e+15)
     t_0
     (if (<= im 2.4e+51)
       (* (* re re) (* t_1 (+ -0.25 (/ 0.5 (* re re)))))
       (if (<= im 3.7e+151)
         (* 0.5 t_1)
         (if (<= im 1.72e+227)
           (* (+ 2.0 (* im im)) (+ 0.5 (* re (* re -0.25))))
           t_0))))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	double t_1 = 2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im))));
	double tmp;
	if (im <= 5.8e+15) {
		tmp = t_0;
	} else if (im <= 2.4e+51) {
		tmp = (re * re) * (t_1 * (-0.25 + (0.5 / (re * re))));
	} else if (im <= 3.7e+151) {
		tmp = 0.5 * t_1;
	} else if (im <= 1.72e+227) {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    t_1 = 2.0d0 + (0.002777777777777778d0 * ((im * im) * ((im * im) * (im * im))))
    if (im <= 5.8d+15) then
        tmp = t_0
    else if (im <= 2.4d+51) then
        tmp = (re * re) * (t_1 * ((-0.25d0) + (0.5d0 / (re * re))))
    else if (im <= 3.7d+151) then
        tmp = 0.5d0 * t_1
    else if (im <= 1.72d+227) then
        tmp = (2.0d0 + (im * im)) * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	double t_1 = 2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im))));
	double tmp;
	if (im <= 5.8e+15) {
		tmp = t_0;
	} else if (im <= 2.4e+51) {
		tmp = (re * re) * (t_1 * (-0.25 + (0.5 / (re * re))));
	} else if (im <= 3.7e+151) {
		tmp = 0.5 * t_1;
	} else if (im <= 1.72e+227) {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	t_1 = 2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im))))
	tmp = 0
	if im <= 5.8e+15:
		tmp = t_0
	elif im <= 2.4e+51:
		tmp = (re * re) * (t_1 * (-0.25 + (0.5 / (re * re))))
	elif im <= 3.7e+151:
		tmp = 0.5 * t_1
	elif im <= 1.72e+227:
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	t_1 = Float64(2.0 + Float64(0.002777777777777778 * Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(im * im)))))
	tmp = 0.0
	if (im <= 5.8e+15)
		tmp = t_0;
	elseif (im <= 2.4e+51)
		tmp = Float64(Float64(re * re) * Float64(t_1 * Float64(-0.25 + Float64(0.5 / Float64(re * re)))));
	elseif (im <= 3.7e+151)
		tmp = Float64(0.5 * t_1);
	elseif (im <= 1.72e+227)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	t_1 = 2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im))));
	tmp = 0.0;
	if (im <= 5.8e+15)
		tmp = t_0;
	elseif (im <= 2.4e+51)
		tmp = (re * re) * (t_1 * (-0.25 + (0.5 / (re * re))));
	elseif (im <= 3.7e+151)
		tmp = 0.5 * t_1;
	elseif (im <= 1.72e+227)
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(0.002777777777777778 * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 5.8e+15], t$95$0, If[LessEqual[im, 2.4e+51], N[(N[(re * re), $MachinePrecision] * N[(t$95$1 * N[(-0.25 + N[(0.5 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.7e+151], N[(0.5 * t$95$1), $MachinePrecision], If[LessEqual[im, 1.72e+227], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 5.8e15 or 1.71999999999999995e227 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 93.9%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 93.9%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 93.9%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 93.9%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 93.9%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 93.9%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 93.9%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 93.9%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 93.9%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 93.9%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 93.9%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 93.9%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 93.9%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 93.9%

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

      2. +-commutative 93.9%

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

      3. associate-*r* 93.9%

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

   7. Applied egg-rr 93.9%

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

   8. Taylor expanded in re around 0 59.7%

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 59.7%

         \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \]

      2. +-commutative 59.7%

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} \]

      3. unpow2 59.7%

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

      4. *-commutative 59.7%

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

      5. associate-*r* 59.7%

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

      6. associate-*r* 59.7%

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

      7. +-commutative 59.7%

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

      8. associate-*r* 59.7%

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

      9. *-commutative 59.7%

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

   10. Simplified 59.7%

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

   if 5.8e15 < im < 2.3999999999999999e51

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 85.7%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 85.7%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 85.7%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 85.7%

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

      6. associate-*r* 85.7%

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

      7. *-commutative 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in im around 0 32.5%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 32.5%

         \[\leadsto \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. unpow2 32.5%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      3. unpow2 32.5%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      4. *-commutative 32.5%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right) \cdot 0.002777777777777778}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      5. associate-*r* 32.5%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 32.5%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   9. Taylor expanded in im around inf 32.5%

      \[\leadsto \left(2 + \color{blue}{0.002777777777777778 \cdot {im}^{6}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval 6.4%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

       2. pow-sqr 6.4%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

       3. cube-prod 6.4%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

       4. cube-unmult 6.4%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   11. Simplified 32.5%

       \[\leadsto \left(2 + \color{blue}{0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   12. Taylor expanded in re around inf 57.8%

       \[\leadsto \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{6}\right) + 0.5 \cdot \frac{2 + 0.002777777777777778 \cdot {im}^{6}}{{re}^{2}}\right)} \]
   13. Step-by-step derivation

       1. unpow2 57.8%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(-0.25 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{6}\right) + 0.5 \cdot \frac{2 + 0.002777777777777778 \cdot {im}^{6}}{{re}^{2}}\right) \]

       2. *-commutative 57.8%

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{\left(2 + 0.002777777777777778 \cdot {im}^{6}\right) \cdot -0.25} + 0.5 \cdot \frac{2 + 0.002777777777777778 \cdot {im}^{6}}{{re}^{2}}\right) \]

       3. +-commutative 57.8%

          \[\leadsto \left(re \cdot re\right) \cdot \left(\color{blue}{\left(0.002777777777777778 \cdot {im}^{6} + 2\right)} \cdot -0.25 + 0.5 \cdot \frac{2 + 0.002777777777777778 \cdot {im}^{6}}{{re}^{2}}\right) \]

       4. *-commutative 57.8%

          \[\leadsto \left(re \cdot re\right) \cdot \left(\left(\color{blue}{{im}^{6} \cdot 0.002777777777777778} + 2\right) \cdot -0.25 + 0.5 \cdot \frac{2 + 0.002777777777777778 \cdot {im}^{6}}{{re}^{2}}\right) \]

       5. metadata-eval 57.8%

          \[\leadsto \left(re \cdot re\right) \cdot \left(\left({im}^{\color{blue}{\left(2 \cdot 3\right)}} \cdot 0.002777777777777778 + 2\right) \cdot -0.25 + 0.5 \cdot \frac{2 + 0.002777777777777778 \cdot {im}^{6}}{{re}^{2}}\right) \]

       6. pow-sqr 57.8%

          \[\leadsto \left(re \cdot re\right) \cdot \left(\left(\color{blue}{\left({im}^{3} \cdot {im}^{3}\right)} \cdot 0.002777777777777778 + 2\right) \cdot -0.25 + 0.5 \cdot \frac{2 + 0.002777777777777778 \cdot {im}^{6}}{{re}^{2}}\right) \]

       7. cube-unmult 57.8%

          \[\leadsto \left(re \cdot re\right) \cdot \left(\left(\left(\color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot {im}^{3}\right) \cdot 0.002777777777777778 + 2\right) \cdot -0.25 + 0.5 \cdot \frac{2 + 0.002777777777777778 \cdot {im}^{6}}{{re}^{2}}\right) \]

       8. cube-unmult 57.8%

          \[\leadsto \left(re \cdot re\right) \cdot \left(\left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) \cdot 0.002777777777777778 + 2\right) \cdot -0.25 + 0.5 \cdot \frac{2 + 0.002777777777777778 \cdot {im}^{6}}{{re}^{2}}\right) \]

       9. *-commutative 57.8%

          \[\leadsto \left(re \cdot re\right) \cdot \left(\left(\color{blue}{0.002777777777777778 \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)} + 2\right) \cdot -0.25 + 0.5 \cdot \frac{2 + 0.002777777777777778 \cdot {im}^{6}}{{re}^{2}}\right) \]

       10. associate-*r/ 57.8%

           \[\leadsto \left(re \cdot re\right) \cdot \left(\left(0.002777777777777778 \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + 2\right) \cdot -0.25 + \color{blue}{\frac{0.5 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{6}\right)}{{re}^{2}}}\right) \]

       11. *-commutative 57.8%

           \[\leadsto \left(re \cdot re\right) \cdot \left(\left(0.002777777777777778 \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + 2\right) \cdot -0.25 + \frac{\color{blue}{\left(2 + 0.002777777777777778 \cdot {im}^{6}\right) \cdot 0.5}}{{re}^{2}}\right) \]

       12. associate-/l* 57.8%

           \[\leadsto \left(re \cdot re\right) \cdot \left(\left(0.002777777777777778 \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + 2\right) \cdot -0.25 + \color{blue}{\left(2 + 0.002777777777777778 \cdot {im}^{6}\right) \cdot \frac{0.5}{{re}^{2}}}\right) \]

   14. Simplified 57.8%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(\left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \left(-0.25 + \frac{0.5}{re \cdot re}\right)\right)} \]

   if 2.3999999999999999e51 < im < 3.6999999999999997e151

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 66.7%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 66.7%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 66.7%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 66.7%

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

      6. associate-*r* 66.7%

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

      7. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 66.7%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 66.7%

         \[\leadsto \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. unpow2 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      3. unpow2 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      4. *-commutative 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right) \cdot 0.002777777777777778}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      5. associate-*r* 66.7%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 66.7%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   9. Taylor expanded in im around inf 66.7%

      \[\leadsto \left(2 + \color{blue}{0.002777777777777778 \cdot {im}^{6}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

       2. pow-sqr 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

       3. cube-prod 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

       4. cube-unmult 100.0%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   11. Simplified 66.7%

       \[\leadsto \left(2 + \color{blue}{0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   12. Taylor expanded in re around 0 80.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{6}\right)} \]
   13. Step-by-step derivation

       1. metadata-eval 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

       2. pow-sqr 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

       3. cube-prod 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

       4. cube-mult 80.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   14. Simplified 80.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

   if 3.6999999999999997e151 < im < 1.71999999999999995e227

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 90.5%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 90.5%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 90.5%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 90.5%

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

      6. associate-*r* 90.5%

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

      7. *-commutative 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in im around 0 90.5%

      \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 90.5%

         \[\leadsto \left(2 + \color{blue}{im \cdot im}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 90.5%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \left(-0.25 + \frac{0.5}{re \cdot re}\right)\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.72 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;1 + 0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e+151)
   (+
    1.0
    (*
     0.5
     (* (* im im) (+ 1.0 (* im (* 0.002777777777777778 (* im (* im im))))))))
   (if (<= im 1.6e+227)
     (* (+ 2.0 (* im im)) (+ 0.5 (* re (* re -0.25))))
     (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e+151) {
		tmp = 1.0 + (0.5 * ((im * im) * (1.0 + (im * (0.002777777777777778 * (im * (im * im)))))));
	} else if (im <= 1.6e+227) {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d+151) then
        tmp = 1.0d0 + (0.5d0 * ((im * im) * (1.0d0 + (im * (0.002777777777777778d0 * (im * (im * im)))))))
    else if (im <= 1.6d+227) then
        tmp = (2.0d0 + (im * im)) * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e+151) {
		tmp = 1.0 + (0.5 * ((im * im) * (1.0 + (im * (0.002777777777777778 * (im * (im * im)))))));
	} else if (im <= 1.6e+227) {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e+151:
		tmp = 1.0 + (0.5 * ((im * im) * (1.0 + (im * (0.002777777777777778 * (im * (im * im)))))))
	elif im <= 1.6e+227:
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)))
	else:
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e+151)
		tmp = Float64(1.0 + Float64(0.5 * Float64(Float64(im * im) * Float64(1.0 + Float64(im * Float64(0.002777777777777778 * Float64(im * Float64(im * im))))))));
	elseif (im <= 1.6e+227)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e+151)
		tmp = 1.0 + (0.5 * ((im * im) * (1.0 + (im * (0.002777777777777778 * (im * (im * im)))))));
	elseif (im <= 1.6e+227)
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	else
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e+151], N[(1.0 + N[(0.5 * N[(N[(im * im), $MachinePrecision] * N[(1.0 + N[(im * N[(0.002777777777777778 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.6e+227], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.6999999999999997e151

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 92.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 92.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \]

      2. unpow2 92.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \]

      3. *-commutative 92.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{{im}^{2} \cdot 0.002777777777777778}\right)\right)\right) \]

      4. unpow2 92.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right)} \cdot 0.002777777777777778\right)\right)\right) \]

      5. associate-*l* 92.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \]

   5. Simplified 92.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \]

   6. Taylor expanded in im around inf 92.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{0.002777777777777778 \cdot {im}^{4}}\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval 92.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      2. pow-sqr 92.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      3. unpow2 92.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + 0.002777777777777778 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right)\right) \]

      4. unpow2 92.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \]

   8. Simplified 92.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}\right)\right) \]

   9. Taylor expanded in re around 0 59.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 + {im}^{2} \cdot \left(1 + 0.002777777777777778 \cdot {im}^{4}\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-in 59.2%

          \[\leadsto \color{blue}{2 \cdot 0.5 + \left({im}^{2} \cdot \left(1 + 0.002777777777777778 \cdot {im}^{4}\right)\right) \cdot 0.5} \]

       2. metadata-eval 59.2%

          \[\leadsto \color{blue}{1} + \left({im}^{2} \cdot \left(1 + 0.002777777777777778 \cdot {im}^{4}\right)\right) \cdot 0.5 \]

       3. unpow2 59.2%

          \[\leadsto 1 + \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + 0.002777777777777778 \cdot {im}^{4}\right)\right) \cdot 0.5 \]

       4. metadata-eval 59.2%

          \[\leadsto 1 + \left(\left(im \cdot im\right) \cdot \left(1 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(3 + 1\right)}}\right)\right) \cdot 0.5 \]

       5. pow-plus 59.2%

          \[\leadsto 1 + \left(\left(im \cdot im\right) \cdot \left(1 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot im\right)}\right)\right) \cdot 0.5 \]

       6. cube-unmult 59.2%

          \[\leadsto 1 + \left(\left(im \cdot im\right) \cdot \left(1 + 0.002777777777777778 \cdot \left(\color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot im\right)\right)\right) \cdot 0.5 \]

       7. associate-*l* 59.2%

          \[\leadsto 1 + \left(\left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot im}\right)\right) \cdot 0.5 \]

       8. *-commutative 59.2%

          \[\leadsto 1 + \left(\left(im \cdot im\right) \cdot \left(1 + \color{blue}{im \cdot \left(0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)}\right)\right) \cdot 0.5 \]

   11. Simplified 59.2%

       \[\leadsto \color{blue}{1 + \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right) \cdot 0.5} \]

   if 3.6999999999999997e151 < im < 1.59999999999999994e227

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 90.5%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 90.5%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 90.5%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 90.5%

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

      6. associate-*r* 90.5%

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

      7. *-commutative 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in im around 0 90.5%

      \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 90.5%

         \[\leadsto \left(2 + \color{blue}{im \cdot im}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 90.5%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   if 1.59999999999999994e227 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 100.0%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 100.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 100.0%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 100.0%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 100.0%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 82.4%

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 82.4%

         \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \]

      2. +-commutative 82.4%

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} \]

      3. unpow2 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r* 82.4%

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

      6. associate-*r* 82.4%

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

      7. +-commutative 82.4%

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

      8. associate-*r* 82.4%

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

      9. *-commutative 82.4%

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

   10. Simplified 82.4%

       \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;1 + 0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(1 + im \cdot \left(0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{+151} \lor \neg \left(im \leq 1.6 \cdot 10^{+227}\right):\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 3.7e+151) (not (<= im 1.6e+227)))
   (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664)))))
   (* (+ 2.0 (* im im)) (+ 0.5 (* re (* re -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 3.7e+151) || !(im <= 1.6e+227)) {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 3.7d+151) .or. (.not. (im <= 1.6d+227))) then
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = (2.0d0 + (im * im)) * (0.5d0 + (re * (re * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 3.7e+151) || !(im <= 1.6e+227)) {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 3.7e+151) or not (im <= 1.6e+227):
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 3.7e+151) || !(im <= 1.6e+227))
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 3.7e+151) || ~((im <= 1.6e+227)))
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 3.7e+151], N[Not[LessEqual[im, 1.6e+227]], $MachinePrecision]], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.6999999999999997e151 or 1.59999999999999994e227 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 90.0%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 90.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 90.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 90.0%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 90.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 90.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 90.0%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 90.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 90.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 90.0%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 90.0%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 90.0%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 90.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 90.0%

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

      2. +-commutative 90.0%

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

      3. associate-*r* 90.0%

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

   7. Applied egg-rr 90.0%

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

   8. Taylor expanded in re around 0 58.1%

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 58.1%

         \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \]

      2. +-commutative 58.1%

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} \]

      3. unpow2 58.1%

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

      4. *-commutative 58.1%

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

      5. associate-*r* 58.1%

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

      6. associate-*r* 58.1%

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

      7. +-commutative 58.1%

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

      8. associate-*r* 58.1%

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

      9. *-commutative 58.1%

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

   10. Simplified 58.1%

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

   if 3.6999999999999997e151 < im < 1.59999999999999994e227

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 90.5%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 90.5%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 90.5%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 90.5%

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

      6. associate-*r* 90.5%

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

      7. *-commutative 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in im around 0 90.5%

      \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 90.5%

         \[\leadsto \left(2 + \color{blue}{im \cdot im}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 90.5%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{+151} \lor \neg \left(im \leq 1.6 \cdot 10^{+227}\right):\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e+151)
   (*
    0.5
    (+ 2.0 (* 0.002777777777777778 (* (* im im) (* (* im im) (* im im))))))
   (if (<= im 1.65e+227)
     (* (+ 2.0 (* im im)) (+ 0.5 (* re (* re -0.25))))
     (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e+151) {
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))));
	} else if (im <= 1.65e+227) {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d+151) then
        tmp = 0.5d0 * (2.0d0 + (0.002777777777777778d0 * ((im * im) * ((im * im) * (im * im)))))
    else if (im <= 1.65d+227) then
        tmp = (2.0d0 + (im * im)) * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = 1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e+151) {
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))));
	} else if (im <= 1.65e+227) {
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e+151:
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))))
	elif im <= 1.65e+227:
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)))
	else:
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e+151)
		tmp = Float64(0.5 * Float64(2.0 + Float64(0.002777777777777778 * Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(im * im))))));
	elseif (im <= 1.65e+227)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e+151)
		tmp = 0.5 * (2.0 + (0.002777777777777778 * ((im * im) * ((im * im) * (im * im)))));
	elseif (im <= 1.65e+227)
		tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
	else
		tmp = 1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e+151], N[(0.5 * N[(2.0 + N[(0.002777777777777778 * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.65e+227], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.6999999999999997e151

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 26.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 26.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 57.6%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 57.6%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 57.6%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 57.6%

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

      6. associate-*r* 57.6%

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

      7. *-commutative 57.6%

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

   5. Simplified 57.6%

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

   6. Taylor expanded in im around 0 54.2%

      \[\leadsto \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 54.2%

         \[\leadsto \left(2 + \color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      2. unpow2 54.2%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {im}^{2}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      3. unpow2 54.2%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      4. *-commutative 54.2%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{\left(im \cdot im\right) \cdot 0.002777777777777778}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

      5. associate-*r* 54.2%

         \[\leadsto \left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \color{blue}{im \cdot \left(im \cdot 0.002777777777777778\right)}\right)\right)\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 54.2%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + im \cdot \left(im \cdot 0.002777777777777778\right)\right)\right)\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   9. Taylor expanded in im around inf 53.9%

      \[\leadsto \left(2 + \color{blue}{0.002777777777777778 \cdot {im}^{6}}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval 92.1%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

       2. pow-sqr 92.1%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

       3. cube-prod 92.1%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

       4. cube-unmult 92.1%

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   11. Simplified 53.9%

       \[\leadsto \left(2 + \color{blue}{0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   12. Taylor expanded in re around 0 59.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{6}\right)} \]
   13. Step-by-step derivation

       1. metadata-eval 59.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot {im}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

       2. pow-sqr 59.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \]

       3. cube-prod 59.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{{\left(im \cdot im\right)}^{3}}\right) \]

       4. cube-mult 59.0%

          \[\leadsto 0.5 \cdot \left(2 + 0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]

   14. Simplified 59.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)} \]

   if 3.6999999999999997e151 < im < 1.6499999999999999e227

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

      2. distribute-rgt-out 90.5%

         \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. exp-neg 90.5%

         \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

      4. +-commutative 90.5%

         \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. unpow2 90.5%

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

      6. associate-*r* 90.5%

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

      7. *-commutative 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in im around 0 90.5%

      \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 90.5%

         \[\leadsto \left(2 + \color{blue}{im \cdot im}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   8. Simplified 90.5%

      \[\leadsto \color{blue}{\left(2 + im \cdot im\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

   if 1.6499999999999999e227 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) + \cos re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({im}^{2} \cdot \cos re\right) + 0.5 \cdot \cos re\right) \cdot {im}^{2}} + \cos re \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \cos re} + 0.5 \cdot \cos re\right) \cdot {im}^{2} + \cos re \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)\right)} \cdot {im}^{2} + \cos re \]

      5. associate-*l* 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right)} + \cos re \]

      6. *-rgt-identity 100.0%

         \[\leadsto \cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2}\right) + \color{blue}{\cos re \cdot 1} \]

      7. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right) \cdot {im}^{2} + 1\right)} \]

      8. *-commutative 100.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} + 1\right) \]

      9. +-commutative 100.0%

         \[\leadsto \cos re \cdot \left({im}^{2} \cdot \color{blue}{\left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} + 1\right) \]

      10. distribute-lft-out 100.0%

          \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{2} \cdot 0.5 + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right)} + 1\right) \]

      11. *-commutative 100.0%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{0.5 \cdot {im}^{2}} + {im}^{2} \cdot \left(0.041666666666666664 \cdot {im}^{2}\right)\right) + 1\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right) + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 82.4%

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 82.4%

         \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \]

      2. +-commutative 82.4%

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)} \]

      3. unpow2 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r* 82.4%

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

      6. associate-*r* 82.4%

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

      7. +-commutative 82.4%

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

      8. associate-*r* 82.4%

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

      9. *-commutative 82.4%

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

   10. Simplified 82.4%

       \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+227}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.3% accurate, 23.7× speedup

Math

\[\begin{array}{l} \\ \left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (+ 2.0 (* im im)) (+ 0.5 (* re (* re -0.25)))))

C

double code(double re, double im) {
	return (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (2.0d0 + (im * im)) * (0.5d0 + (re * (re * (-0.25d0))))
end function

Java

public static double code(double re, double im) {
	return (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
}

Python

def code(re, im):
	return (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)))

Julia

function code(re, im)
	return Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(re * Float64(re * -0.25))))
end

MATLAB

function tmp = code(re, im)
	tmp = (2.0 + (im * im)) * (0.5 + (re * (re * -0.25)));
end

Wolfram

code[re_, im_] := N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0 22.1%

   \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
4. Step-by-step derivation

   1. associate-*r* 22.1%

      \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

   2. distribute-rgt-out 61.2%

      \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

   3. exp-neg 61.2%

      \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

   4. +-commutative 61.2%

      \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

   5. unpow2 61.2%

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

   6. associate-*r* 61.2%

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

   7. *-commutative 61.2%

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

5. Simplified 61.2%

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

6. Taylor expanded in im around 0 49.7%

   \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]
7. Step-by-step derivation

   1. unpow2 49.7%

      \[\leadsto \left(2 + \color{blue}{im \cdot im}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

8. Simplified 49.7%

   \[\leadsto \color{blue}{\left(2 + im \cdot im\right)} \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

9. Final simplification 49.7%

   \[\leadsto \left(2 + im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
10. Add Preprocessing

Alternative 17: 35.4% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(im + 2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (+ 0.5 (* re (* re -0.25))) (+ im 2.0)))

C

double code(double re, double im) {
	return (0.5 + (re * (re * -0.25))) * (im + 2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 + (re * (re * (-0.25d0)))) * (im + 2.0d0)
end function

Java

public static double code(double re, double im) {
	return (0.5 + (re * (re * -0.25))) * (im + 2.0);
}

Python

def code(re, im):
	return (0.5 + (re * (re * -0.25))) * (im + 2.0)

Julia

function code(re, im)
	return Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * Float64(im + 2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 + (re * (re * -0.25))) * (im + 2.0);
end

Wolfram

code[re_, im_] := N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0 22.1%

   \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
4. Step-by-step derivation

   1. associate-*r* 22.1%

      \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

   2. distribute-rgt-out 61.2%

      \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

   3. exp-neg 61.2%

      \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

   4. +-commutative 61.2%

      \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

   5. unpow2 61.2%

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

   6. associate-*r* 61.2%

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

   7. *-commutative 61.2%

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

5. Simplified 61.2%

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

6. Taylor expanded in im around 0 45.4%

   \[\leadsto \left(e^{im} + \color{blue}{1}\right) \cdot \left(0.5 + re \cdot \left(-0.25 \cdot re\right)\right) \]

7. Taylor expanded in im around 0 31.9%

   \[\leadsto \color{blue}{2 \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right) + im \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-out 35.9%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right) \cdot \left(2 + im\right)} \]

   2. *-commutative 35.9%

      \[\leadsto \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \cdot \left(2 + im\right) \]

   3. unpow2 35.9%

      \[\leadsto \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \cdot \left(2 + im\right) \]

   4. associate-*r* 35.9%

      \[\leadsto \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \cdot \left(2 + im\right) \]

   5. *-commutative 35.9%

      \[\leadsto \color{blue}{\left(2 + im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   6. +-commutative 35.9%

      \[\leadsto \color{blue}{\left(im + 2\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

9. Simplified 35.9%

   \[\leadsto \color{blue}{\left(im + 2\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

10. Final simplification 35.9%

    \[\leadsto \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(im + 2\right) \]
11. Add Preprocessing

Alternative 18: 32.6% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0 22.1%

   \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) + 0.5 \cdot \left(e^{im} + e^{-im}\right)} \]
4. Step-by-step derivation

   1. associate-*r* 22.1%

      \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{im} + e^{-im}\right)} + 0.5 \cdot \left(e^{im} + e^{-im}\right) \]

   2. distribute-rgt-out 61.2%

      \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

   3. exp-neg 61.2%

      \[\leadsto \left(e^{im} + \color{blue}{\frac{1}{e^{im}}}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right) \]

   4. +-commutative 61.2%

      \[\leadsto \left(e^{im} + \frac{1}{e^{im}}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

   5. unpow2 61.2%

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

   6. associate-*r* 61.2%

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

   7. *-commutative 61.2%

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

5. Simplified 61.2%

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

6. Taylor expanded in im around 0 30.2%

   \[\leadsto \color{blue}{2 \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]
7. Step-by-step derivation

   1. distribute-rgt-in 30.2%

      \[\leadsto \color{blue}{0.5 \cdot 2 + \left(-0.25 \cdot {re}^{2}\right) \cdot 2} \]

   2. metadata-eval 30.2%

      \[\leadsto \color{blue}{1} + \left(-0.25 \cdot {re}^{2}\right) \cdot 2 \]

   3. unpow2 30.2%

      \[\leadsto 1 + \left(-0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot 2 \]

8. Simplified 30.2%

   \[\leadsto \color{blue}{1 + \left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot 2} \]
9. Step-by-step derivation

   1. *-commutative 30.2%

      \[\leadsto 1 + \color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot 2 \]

   2. associate-*l* 30.2%

      \[\leadsto 1 + \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot 2\right)} \]

   3. metadata-eval 30.2%

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

10. Applied egg-rr 30.2%

    \[\leadsto 1 + \color{blue}{\left(re \cdot re\right) \cdot -0.5} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))