math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.9s

Alternatives: 26

Speedup: 1.0×

• 50.2% 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

\[\begin{array}{l} \\ \cos re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (cos re) (+ (* 0.5 (/ 1.0 (exp im))) (* 0.5 (exp im)))))

C

double code(double re, double im) {
	return cos(re) * ((0.5 * (1.0 / exp(im))) + (0.5 * exp(im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re) * ((0.5d0 * (1.0d0 / exp(im))) + (0.5d0 * exp(im)))
end function

Java

public static double code(double re, double im) {
	return Math.cos(re) * ((0.5 * (1.0 / Math.exp(im))) + (0.5 * Math.exp(im)));
}

Python

def code(re, im):
	return math.cos(re) * ((0.5 * (1.0 / math.exp(im))) + (0.5 * math.exp(im)))

Julia

function code(re, im)
	return Float64(cos(re) * Float64(Float64(0.5 * Float64(1.0 / exp(im))) + Float64(0.5 * exp(im))))
end

MATLAB

function tmp = code(re, im)
	tmp = cos(re) * ((0.5 * (1.0 / exp(im))) + (0.5 * exp(im)));
end

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(1.0 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Exp[im], $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. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-def 100.0%

      \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

4. Taylor expanded in re around inf 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \cos re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right) \]

Alternative 2: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.9999985:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.9999985)
   (*
    (* (cos re) 0.5)
    (+ 2.0 (+ (* im im) (* (pow im 4.0) 0.08333333333333333))))
   (+ (* 0.5 (/ 1.0 (exp im))) (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.9999985) {
		tmp = (cos(re) * 0.5) * (2.0 + ((im * im) + (pow(im, 4.0) * 0.08333333333333333)));
	} else {
		tmp = (0.5 * (1.0 / exp(im))) + (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 (cos(re) <= 0.9999985d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + ((im * im) + ((im ** 4.0d0) * 0.08333333333333333d0)))
    else
        tmp = (0.5d0 * (1.0d0 / exp(im))) + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.9999985) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + ((im * im) + (Math.pow(im, 4.0) * 0.08333333333333333)));
	} else {
		tmp = (0.5 * (1.0 / Math.exp(im))) + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.9999985:
		tmp = (math.cos(re) * 0.5) * (2.0 + ((im * im) + (math.pow(im, 4.0) * 0.08333333333333333)))
	else:
		tmp = (0.5 * (1.0 / math.exp(im))) + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.9999985)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(Float64(im * im) + Float64((im ^ 4.0) * 0.08333333333333333))));
	else
		tmp = Float64(Float64(0.5 * Float64(1.0 / exp(im))) + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.9999985)
		tmp = (cos(re) * 0.5) * (2.0 + ((im * im) + ((im ^ 4.0) * 0.08333333333333333)));
	else
		tmp = (0.5 * (1.0 / exp(im))) + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.9999985], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(1.0 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.99999850000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 90.9%

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

      1. unpow2 90.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 90.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 90.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   if 0.99999850000000001 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.9999985:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $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. Final simplification 100.0%

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

Alternative 4: 94.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{if}\;im \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 3.4:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.041666666666666664 (* (cos re) (pow im 4.0)))))
   (if (<= im -1.15e+77)
     t_0
     (if (<= im -31000000.0)
       (pow re -512.0)
       (if (<= im 3.4)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (if (<= im 1.15e+77) (+ (* 0.5 (exp im)) -1.0) t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	double tmp;
	if (im <= -1.15e+77) {
		tmp = t_0;
	} else if (im <= -31000000.0) {
		tmp = pow(re, -512.0);
	} else if (im <= 3.4) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = (0.5 * exp(im)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    if (im <= (-1.15d+77)) then
        tmp = t_0
    else if (im <= (-31000000.0d0)) then
        tmp = re ** (-512.0d0)
    else if (im <= 3.4d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.15d+77) then
        tmp = (0.5d0 * exp(im)) + (-1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	double tmp;
	if (im <= -1.15e+77) {
		tmp = t_0;
	} else if (im <= -31000000.0) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 3.4) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = (0.5 * Math.exp(im)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	tmp = 0
	if im <= -1.15e+77:
		tmp = t_0
	elif im <= -31000000.0:
		tmp = math.pow(re, -512.0)
	elif im <= 3.4:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.15e+77:
		tmp = (0.5 * math.exp(im)) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)))
	tmp = 0.0
	if (im <= -1.15e+77)
		tmp = t_0;
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 3.4)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.15e+77)
		tmp = Float64(Float64(0.5 * exp(im)) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	tmp = 0.0;
	if (im <= -1.15e+77)
		tmp = t_0;
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 3.4)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.15e+77)
		tmp = (0.5 * exp(im)) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.15e+77], t$95$0, If[LessEqual[im, -31000000.0], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 3.4], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.14999999999999997e77 or 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)} \]

   if -1.14999999999999997e77 < im < -3.1e7

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   5. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. associate-*r* 0.0%

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

      7. distribute-rgt-out 80.0%

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

      8. rec-exp 80.0%

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

      9. *-commutative 80.0%

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

      10. associate-*l* 80.0%

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

      11. metadata-eval 80.0%

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

      12. unpow2 80.0%

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

      13. associate-*l* 80.0%

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

   7. Simplified 80.0%

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

   9. Applied egg-rr 60.6%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -3.1e7 < im < 3.39999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.7%

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

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

   if 3.39999999999999991 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 82.9%

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

   6. Applied egg-rr 83.3%

      \[\leadsto 0.5 \cdot \color{blue}{-2} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around inf 83.3%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} - 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 3.4:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 5: 96.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{if}\;im \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.0185:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{elif}\;im \leq 3.9:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 10^{+77}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.041666666666666664 (* (cos re) (pow im 4.0)))))
   (if (<= im -1.15e+77)
     t_0
     (if (<= im -0.0185)
       (* 0.5 (+ (exp im) (exp (- im))))
       (if (<= im 3.9)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (if (<= im 1e+77) (+ (* 0.5 (exp im)) -1.0) t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	double tmp;
	if (im <= -1.15e+77) {
		tmp = t_0;
	} else if (im <= -0.0185) {
		tmp = 0.5 * (exp(im) + exp(-im));
	} else if (im <= 3.9) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1e+77) {
		tmp = (0.5 * exp(im)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    if (im <= (-1.15d+77)) then
        tmp = t_0
    else if (im <= (-0.0185d0)) then
        tmp = 0.5d0 * (exp(im) + exp(-im))
    else if (im <= 3.9d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1d+77) then
        tmp = (0.5d0 * exp(im)) + (-1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	double tmp;
	if (im <= -1.15e+77) {
		tmp = t_0;
	} else if (im <= -0.0185) {
		tmp = 0.5 * (Math.exp(im) + Math.exp(-im));
	} else if (im <= 3.9) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1e+77) {
		tmp = (0.5 * Math.exp(im)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	tmp = 0
	if im <= -1.15e+77:
		tmp = t_0
	elif im <= -0.0185:
		tmp = 0.5 * (math.exp(im) + math.exp(-im))
	elif im <= 3.9:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1e+77:
		tmp = (0.5 * math.exp(im)) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)))
	tmp = 0.0
	if (im <= -1.15e+77)
		tmp = t_0;
	elseif (im <= -0.0185)
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	elseif (im <= 3.9)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1e+77)
		tmp = Float64(Float64(0.5 * exp(im)) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	tmp = 0.0;
	if (im <= -1.15e+77)
		tmp = t_0;
	elseif (im <= -0.0185)
		tmp = 0.5 * (exp(im) + exp(-im));
	elseif (im <= 3.9)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1e+77)
		tmp = (0.5 * exp(im)) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.15e+77], t$95$0, If[LessEqual[im, -0.0185], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.9], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+77], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.14999999999999997e77 or 9.99999999999999983e76 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)} \]

   if -1.14999999999999997e77 < im < -0.0184999999999999991

   1. Initial program 99.7%

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

   2. Taylor expanded in re around 0 72.5%

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

   if -0.0184999999999999991 < im < 3.89999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   if 3.89999999999999991 < im < 9.99999999999999983e76

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 82.9%

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

   6. Applied egg-rr 83.3%

      \[\leadsto 0.5 \cdot \color{blue}{-2} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around inf 83.3%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} - 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -0.0185:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{elif}\;im \leq 3.9:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 10^{+77}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 6: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -0.053:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.0009765625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.15e+77)
   (* 0.041666666666666664 (* (cos re) (pow im 4.0)))
   (if (<= im -0.053)
     (* 0.5 (+ (exp im) (exp (- im))))
     (if (<= im 1.3)
       (* (* (cos re) 0.5) (+ 2.0 (* im im)))
       (* (cos re) (+ (* 0.5 (exp im)) 0.0009765625))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.15e+77) {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	} else if (im <= -0.053) {
		tmp = 0.5 * (exp(im) + exp(-im));
	} else if (im <= 1.3) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = cos(re) * ((0.5 * exp(im)) + 0.0009765625);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.15d+77)) then
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    else if (im <= (-0.053d0)) then
        tmp = 0.5d0 * (exp(im) + exp(-im))
    else if (im <= 1.3d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else
        tmp = cos(re) * ((0.5d0 * exp(im)) + 0.0009765625d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.15e+77) {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	} else if (im <= -0.053) {
		tmp = 0.5 * (Math.exp(im) + Math.exp(-im));
	} else if (im <= 1.3) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = Math.cos(re) * ((0.5 * Math.exp(im)) + 0.0009765625);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.15e+77:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	elif im <= -0.053:
		tmp = 0.5 * (math.exp(im) + math.exp(-im))
	elif im <= 1.3:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = math.cos(re) * ((0.5 * math.exp(im)) + 0.0009765625)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.15e+77)
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	elseif (im <= -0.053)
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	elseif (im <= 1.3)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5 * exp(im)) + 0.0009765625));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.15e+77)
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	elseif (im <= -0.053)
		tmp = 0.5 * (exp(im) + exp(-im));
	elseif (im <= 1.3)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	else
		tmp = cos(re) * ((0.5 * exp(im)) + 0.0009765625);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.15e+77], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -0.053], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.3], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.0009765625), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.14999999999999997e77

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)} \]

   if -1.14999999999999997e77 < im < -0.0529999999999999985

   1. Initial program 99.7%

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

   2. Taylor expanded in re around 0 72.5%

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

   if -0.0529999999999999985 < im < 1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   if 1.30000000000000004 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 99.2%

      \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{0.001953125} + 0.5 \cdot e^{im}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -0.053:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.0009765625\right)\\ \end{array} \]

Alternative 7: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot e^{im}\\ \mathbf{if}\;im \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -0.014:\\ \;\;\;\;0.5 \cdot \frac{1}{e^{im}} + t_0\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(t_0 + 0.0009765625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (exp im))))
   (if (<= im -1.15e+77)
     (* 0.041666666666666664 (* (cos re) (pow im 4.0)))
     (if (<= im -0.014)
       (+ (* 0.5 (/ 1.0 (exp im))) t_0)
       (if (<= im 1.3)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (* (cos re) (+ t_0 0.0009765625)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * exp(im);
	double tmp;
	if (im <= -1.15e+77) {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	} else if (im <= -0.014) {
		tmp = (0.5 * (1.0 / exp(im))) + t_0;
	} else if (im <= 1.3) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = cos(re) * (t_0 + 0.0009765625);
	}
	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 * exp(im)
    if (im <= (-1.15d+77)) then
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    else if (im <= (-0.014d0)) then
        tmp = (0.5d0 * (1.0d0 / exp(im))) + t_0
    else if (im <= 1.3d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else
        tmp = cos(re) * (t_0 + 0.0009765625d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.exp(im);
	double tmp;
	if (im <= -1.15e+77) {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	} else if (im <= -0.014) {
		tmp = (0.5 * (1.0 / Math.exp(im))) + t_0;
	} else if (im <= 1.3) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = Math.cos(re) * (t_0 + 0.0009765625);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.exp(im)
	tmp = 0
	if im <= -1.15e+77:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	elif im <= -0.014:
		tmp = (0.5 * (1.0 / math.exp(im))) + t_0
	elif im <= 1.3:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = math.cos(re) * (t_0 + 0.0009765625)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * exp(im))
	tmp = 0.0
	if (im <= -1.15e+77)
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	elseif (im <= -0.014)
		tmp = Float64(Float64(0.5 * Float64(1.0 / exp(im))) + t_0);
	elseif (im <= 1.3)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(cos(re) * Float64(t_0 + 0.0009765625));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * exp(im);
	tmp = 0.0;
	if (im <= -1.15e+77)
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	elseif (im <= -0.014)
		tmp = (0.5 * (1.0 / exp(im))) + t_0;
	elseif (im <= 1.3)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	else
		tmp = cos(re) * (t_0 + 0.0009765625);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.15e+77], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -0.014], N[(N[(0.5 * N[(1.0 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[im, 1.3], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(t$95$0 + 0.0009765625), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.14999999999999997e77

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)} \]

   if -1.14999999999999997e77 < im < -0.0140000000000000003

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. distribute-rgt-in 99.7%

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

      7. *-commutative 99.7%

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

      8. fma-def 99.7%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 99.9%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 99.9%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 99.9%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 72.6%

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

   if -0.0140000000000000003 < im < 1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   if 1.30000000000000004 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   6. Applied egg-rr 99.2%

      \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{0.001953125} + 0.5 \cdot e^{im}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -0.014:\\ \;\;\;\;0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.0009765625\right)\\ \end{array} \]

Alternative 8: 90.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{if}\;im \leq -3.1 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.25 \cdot 10^{+76}:\\ \;\;\;\;{im}^{4} \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.9:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* (cos re) (* 0.5 im)))))
   (if (<= im -3.1e+181)
     t_0
     (if (<= im -1.25e+76)
       (*
        (pow im 4.0)
        (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))
       (if (<= im -31000000.0)
         (pow re -512.0)
         (if (<= im 2.9)
           (* (* (cos re) 0.5) (+ 2.0 (* im im)))
           (if (<= im 1.9e+154) (+ (* 0.5 (exp im)) -1.0) t_0)))))))

C

double code(double re, double im) {
	double t_0 = im * (cos(re) * (0.5 * im));
	double tmp;
	if (im <= -3.1e+181) {
		tmp = t_0;
	} else if (im <= -1.25e+76) {
		tmp = pow(im, 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	} else if (im <= -31000000.0) {
		tmp = pow(re, -512.0);
	} else if (im <= 2.9) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.9e+154) {
		tmp = (0.5 * exp(im)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (cos(re) * (0.5d0 * im))
    if (im <= (-3.1d+181)) then
        tmp = t_0
    else if (im <= (-1.25d+76)) then
        tmp = (im ** 4.0d0) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    else if (im <= (-31000000.0d0)) then
        tmp = re ** (-512.0d0)
    else if (im <= 2.9d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.9d+154) then
        tmp = (0.5d0 * exp(im)) + (-1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (Math.cos(re) * (0.5 * im));
	double tmp;
	if (im <= -3.1e+181) {
		tmp = t_0;
	} else if (im <= -1.25e+76) {
		tmp = Math.pow(im, 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	} else if (im <= -31000000.0) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 2.9) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.9e+154) {
		tmp = (0.5 * Math.exp(im)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (math.cos(re) * (0.5 * im))
	tmp = 0
	if im <= -3.1e+181:
		tmp = t_0
	elif im <= -1.25e+76:
		tmp = math.pow(im, 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	elif im <= -31000000.0:
		tmp = math.pow(re, -512.0)
	elif im <= 2.9:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.9e+154:
		tmp = (0.5 * math.exp(im)) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(cos(re) * Float64(0.5 * im)))
	tmp = 0.0
	if (im <= -3.1e+181)
		tmp = t_0;
	elseif (im <= -1.25e+76)
		tmp = Float64((im ^ 4.0) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 2.9)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.9e+154)
		tmp = Float64(Float64(0.5 * exp(im)) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (cos(re) * (0.5 * im));
	tmp = 0.0;
	if (im <= -3.1e+181)
		tmp = t_0;
	elseif (im <= -1.25e+76)
		tmp = (im ^ 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 2.9)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.9e+154)
		tmp = (0.5 * exp(im)) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.1e+181], t$95$0, If[LessEqual[im, -1.25e+76], N[(N[Power[im, 4.0], $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -31000000.0], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 2.9], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -3.09999999999999989e181 or 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   if -3.09999999999999989e181 < im < -1.24999999999999998e76

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)} \]

   6. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right) + 0.041666666666666664 \cdot {im}^{4}} \]
   7. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.020833333333333332 \cdot {re}^{2}\right) \cdot {im}^{4}} + 0.041666666666666664 \cdot {im}^{4} \]

      2. distribute-rgt-out 92.3%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(-0.020833333333333332 \cdot {re}^{2} + 0.041666666666666664\right)} \]

      3. *-commutative 92.3%

         \[\leadsto {im}^{4} \cdot \left(\color{blue}{{re}^{2} \cdot -0.020833333333333332} + 0.041666666666666664\right) \]

      4. unpow2 92.3%

         \[\leadsto {im}^{4} \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.020833333333333332 + 0.041666666666666664\right) \]

   8. Simplified 92.3%

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\left(re \cdot re\right) \cdot -0.020833333333333332 + 0.041666666666666664\right)} \]

   if -1.24999999999999998e76 < im < -3.1e7

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   5. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. associate-*r* 0.0%

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

      7. distribute-rgt-out 80.0%

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

      8. rec-exp 80.0%

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

      9. *-commutative 80.0%

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

      10. associate-*l* 80.0%

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

      11. metadata-eval 80.0%

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

      12. unpow2 80.0%

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

      13. associate-*l* 80.0%

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

   7. Simplified 80.0%

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

   9. Applied egg-rr 60.6%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -3.1e7 < im < 2.89999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.7%

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

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

   if 2.89999999999999991 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 78.0%

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

   6. Applied egg-rr 78.2%

      \[\leadsto 0.5 \cdot \color{blue}{-2} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around inf 78.2%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} - 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.1 \cdot 10^{+181}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -1.25 \cdot 10^{+76}:\\ \;\;\;\;{im}^{4} \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.9:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 88.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{if}\;im \leq -6.2 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 1.95:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* (cos re) (* 0.5 im)))))
   (if (<= im -6.2e+176)
     t_0
     (if (<= im -31000000.0)
       (pow re -512.0)
       (if (<= im 1.95)
         (cos re)
         (if (<= im 2.05e+154) (+ (* 0.5 (exp im)) -1.0) t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (cos(re) * (0.5 * im));
	double tmp;
	if (im <= -6.2e+176) {
		tmp = t_0;
	} else if (im <= -31000000.0) {
		tmp = pow(re, -512.0);
	} else if (im <= 1.95) {
		tmp = cos(re);
	} else if (im <= 2.05e+154) {
		tmp = (0.5 * exp(im)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (cos(re) * (0.5d0 * im))
    if (im <= (-6.2d+176)) then
        tmp = t_0
    else if (im <= (-31000000.0d0)) then
        tmp = re ** (-512.0d0)
    else if (im <= 1.95d0) then
        tmp = cos(re)
    else if (im <= 2.05d+154) then
        tmp = (0.5d0 * exp(im)) + (-1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (Math.cos(re) * (0.5 * im));
	double tmp;
	if (im <= -6.2e+176) {
		tmp = t_0;
	} else if (im <= -31000000.0) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 1.95) {
		tmp = Math.cos(re);
	} else if (im <= 2.05e+154) {
		tmp = (0.5 * Math.exp(im)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (math.cos(re) * (0.5 * im))
	tmp = 0
	if im <= -6.2e+176:
		tmp = t_0
	elif im <= -31000000.0:
		tmp = math.pow(re, -512.0)
	elif im <= 1.95:
		tmp = math.cos(re)
	elif im <= 2.05e+154:
		tmp = (0.5 * math.exp(im)) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(cos(re) * Float64(0.5 * im)))
	tmp = 0.0
	if (im <= -6.2e+176)
		tmp = t_0;
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 1.95)
		tmp = cos(re);
	elseif (im <= 2.05e+154)
		tmp = Float64(Float64(0.5 * exp(im)) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (cos(re) * (0.5 * im));
	tmp = 0.0;
	if (im <= -6.2e+176)
		tmp = t_0;
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 1.95)
		tmp = cos(re);
	elseif (im <= 2.05e+154)
		tmp = (0.5 * exp(im)) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.2e+176], t$95$0, If[LessEqual[im, -31000000.0], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 1.95], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.05e+154], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -6.1999999999999998e176 or 2.05e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   if -6.1999999999999998e176 < im < -3.1e7

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   5. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. associate-*r* 0.0%

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

      7. distribute-rgt-out 88.2%

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

      8. rec-exp 88.2%

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

      9. *-commutative 88.2%

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

      10. associate-*l* 88.2%

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

      11. metadata-eval 88.2%

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

      12. unpow2 88.2%

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

      13. associate-*l* 88.2%

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

   7. Simplified 88.2%

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

   9. Applied egg-rr 65.3%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -3.1e7 < im < 1.94999999999999996

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 96.8%

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

   if 1.94999999999999996 < im < 2.05e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 78.0%

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

   6. Applied egg-rr 78.2%

      \[\leadsto 0.5 \cdot \color{blue}{-2} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around inf 78.2%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} - 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.2 \cdot 10^{+176}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 1.95:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 10: 88.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{if}\;im \leq -6.2 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 4.5:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* (cos re) (* 0.5 im)))))
   (if (<= im -6.2e+176)
     t_0
     (if (<= im -31000000.0)
       (pow re -512.0)
       (if (<= im 4.5)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (if (<= im 2.7e+154) (+ (* 0.5 (exp im)) -1.0) t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (cos(re) * (0.5 * im));
	double tmp;
	if (im <= -6.2e+176) {
		tmp = t_0;
	} else if (im <= -31000000.0) {
		tmp = pow(re, -512.0);
	} else if (im <= 4.5) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 2.7e+154) {
		tmp = (0.5 * exp(im)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (cos(re) * (0.5d0 * im))
    if (im <= (-6.2d+176)) then
        tmp = t_0
    else if (im <= (-31000000.0d0)) then
        tmp = re ** (-512.0d0)
    else if (im <= 4.5d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 2.7d+154) then
        tmp = (0.5d0 * exp(im)) + (-1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (Math.cos(re) * (0.5 * im));
	double tmp;
	if (im <= -6.2e+176) {
		tmp = t_0;
	} else if (im <= -31000000.0) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 4.5) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 2.7e+154) {
		tmp = (0.5 * Math.exp(im)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (math.cos(re) * (0.5 * im))
	tmp = 0
	if im <= -6.2e+176:
		tmp = t_0
	elif im <= -31000000.0:
		tmp = math.pow(re, -512.0)
	elif im <= 4.5:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 2.7e+154:
		tmp = (0.5 * math.exp(im)) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(cos(re) * Float64(0.5 * im)))
	tmp = 0.0
	if (im <= -6.2e+176)
		tmp = t_0;
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 4.5)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 2.7e+154)
		tmp = Float64(Float64(0.5 * exp(im)) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (cos(re) * (0.5 * im));
	tmp = 0.0;
	if (im <= -6.2e+176)
		tmp = t_0;
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 4.5)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 2.7e+154)
		tmp = (0.5 * exp(im)) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.2e+176], t$95$0, If[LessEqual[im, -31000000.0], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 4.5], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.7e+154], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -6.1999999999999998e176 or 2.70000000000000006e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   if -6.1999999999999998e176 < im < -3.1e7

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   5. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. associate-*r* 0.0%

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

      7. distribute-rgt-out 88.2%

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

      8. rec-exp 88.2%

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

      9. *-commutative 88.2%

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

      10. associate-*l* 88.2%

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

      11. metadata-eval 88.2%

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

      12. unpow2 88.2%

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

      13. associate-*l* 88.2%

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

   7. Simplified 88.2%

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

   9. Applied egg-rr 65.3%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -3.1e7 < im < 4.5

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.7%

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

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

   if 4.5 < im < 2.70000000000000006e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 78.0%

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

   6. Applied egg-rr 78.2%

      \[\leadsto 0.5 \cdot \color{blue}{-2} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around inf 78.2%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} - 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.2 \cdot 10^{+176}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 4.5:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 80.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.15 \cdot 10^{+128}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+21}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.26 \cdot 10^{+77}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.15e+128)
   (* (* im im) (+ 0.5 (* re (* re -0.25))))
   (if (<= im -31000000.0)
     (pow re -512.0)
     (if (<= im 2.1e+21)
       (cos re)
       (if (<= im 1.26e+77)
         (pow re -512.0)
         (* (pow im 4.0) 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.15e+128) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else if (im <= -31000000.0) {
		tmp = pow(re, -512.0);
	} else if (im <= 2.1e+21) {
		tmp = cos(re);
	} else if (im <= 1.26e+77) {
		tmp = pow(re, -512.0);
	} else {
		tmp = pow(im, 4.0) * 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 <= (-3.15d+128)) then
        tmp = (im * im) * (0.5d0 + (re * (re * (-0.25d0))))
    else if (im <= (-31000000.0d0)) then
        tmp = re ** (-512.0d0)
    else if (im <= 2.1d+21) then
        tmp = cos(re)
    else if (im <= 1.26d+77) then
        tmp = re ** (-512.0d0)
    else
        tmp = (im ** 4.0d0) * 0.041666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3.15e+128) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else if (im <= -31000000.0) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 2.1e+21) {
		tmp = Math.cos(re);
	} else if (im <= 1.26e+77) {
		tmp = Math.pow(re, -512.0);
	} else {
		tmp = Math.pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3.15e+128:
		tmp = (im * im) * (0.5 + (re * (re * -0.25)))
	elif im <= -31000000.0:
		tmp = math.pow(re, -512.0)
	elif im <= 2.1e+21:
		tmp = math.cos(re)
	elif im <= 1.26e+77:
		tmp = math.pow(re, -512.0)
	else:
		tmp = math.pow(im, 4.0) * 0.041666666666666664
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3.15e+128)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 2.1e+21)
		tmp = cos(re);
	elseif (im <= 1.26e+77)
		tmp = re ^ -512.0;
	else
		tmp = Float64((im ^ 4.0) * 0.041666666666666664);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3.15e+128)
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 2.1e+21)
		tmp = cos(re);
	elseif (im <= 1.26e+77)
		tmp = re ^ -512.0;
	else
		tmp = (im ^ 4.0) * 0.041666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.15e+128], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -31000000.0], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 2.1e+21], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.26e+77], N[Power[re, -512.0], $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.1499999999999999e128

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 81.2%

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

      1. unpow2 81.2%

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

   4. Simplified 81.2%

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

   5. Taylor expanded in im around inf 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-*r* 81.2%

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

      3. *-commutative 81.2%

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

      4. unpow2 81.2%

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

      5. associate-*l* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 9.7%

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

      1. associate-*r* 9.7%

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

      2. distribute-rgt-out 74.4%

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

      3. unpow2 74.4%

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

      4. unpow2 74.4%

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

      5. associate-*r* 74.4%

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

   10. Simplified 74.4%

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

   if -3.1499999999999999e128 < im < -3.1e7 or 2.1e21 < im < 1.25999999999999998e77

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   5. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. associate-*r* 0.0%

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

      7. distribute-rgt-out 83.3%

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

      8. rec-exp 83.3%

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

      9. *-commutative 83.3%

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

      10. associate-*l* 83.3%

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

      11. metadata-eval 83.3%

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

      12. unpow2 83.3%

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

      13. associate-*l* 83.3%

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

   7. Simplified 83.3%

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

   9. Applied egg-rr 67.2%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -3.1e7 < im < 2.1e21

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 94.9%

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

   if 1.25999999999999998e77 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)} \]

   6. Taylor expanded in re around 0 81.3%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.15 \cdot 10^{+128}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+21}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.26 \cdot 10^{+77}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \]

Alternative 12: 82.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.5:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -2e+131)
   (* (* im im) (+ 0.5 (* re (* re -0.25))))
   (if (<= im -31000000.0)
     (pow re -512.0)
     (if (<= im 2.5) (cos re) (+ (* 0.5 (exp im)) -1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2e+131) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else if (im <= -31000000.0) {
		tmp = pow(re, -512.0);
	} else if (im <= 2.5) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * exp(im)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-2d+131)) then
        tmp = (im * im) * (0.5d0 + (re * (re * (-0.25d0))))
    else if (im <= (-31000000.0d0)) then
        tmp = re ** (-512.0d0)
    else if (im <= 2.5d0) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * exp(im)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -2e+131) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else if (im <= -31000000.0) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 2.5) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.exp(im)) + -1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2e+131:
		tmp = (im * im) * (0.5 + (re * (re * -0.25)))
	elif im <= -31000000.0:
		tmp = math.pow(re, -512.0)
	elif im <= 2.5:
		tmp = math.cos(re)
	else:
		tmp = (0.5 * math.exp(im)) + -1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2e+131)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 2.5)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 * exp(im)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2e+131)
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 2.5)
		tmp = cos(re);
	else
		tmp = (0.5 * exp(im)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2e+131], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -31000000.0], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 2.5], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.9999999999999998e131

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 81.2%

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

      1. unpow2 81.2%

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

   4. Simplified 81.2%

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

   5. Taylor expanded in im around inf 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-*r* 81.2%

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

      3. *-commutative 81.2%

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

      4. unpow2 81.2%

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

      5. associate-*l* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 9.7%

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

      1. associate-*r* 9.7%

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

      2. distribute-rgt-out 74.4%

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

      3. unpow2 74.4%

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

      4. unpow2 74.4%

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

      5. associate-*r* 74.4%

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

   10. Simplified 74.4%

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

   if -1.9999999999999998e131 < im < -3.1e7

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   5. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. associate-*r* 0.0%

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

      7. distribute-rgt-out 87.0%

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

      8. rec-exp 87.0%

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

      9. *-commutative 87.0%

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

      10. associate-*l* 87.0%

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

      11. metadata-eval 87.0%

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

      12. unpow2 87.0%

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

      13. associate-*l* 87.0%

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

   7. Simplified 87.0%

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

   9. Applied egg-rr 65.8%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -3.1e7 < im < 2.5

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 96.8%

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

   if 2.5 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 81.5%

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

   6. Applied egg-rr 81.6%

      \[\leadsto 0.5 \cdot \color{blue}{-2} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around inf 81.6%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} - 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.5:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im} + -1\\ \end{array} \]

Alternative 13: 79.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{+131}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+21}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.26 \cdot 10^{+131}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;-0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.7e+131)
   (* (* im im) (+ 0.5 (* re (* re -0.25))))
   (if (<= im -31000000.0)
     (pow re -512.0)
     (if (<= im 2.1e+21)
       (cos re)
       (if (<= im 1.26e+131)
         (pow re -512.0)
         (+
          -0.5
          (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.7e+131) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else if (im <= -31000000.0) {
		tmp = pow(re, -512.0);
	} else if (im <= 2.1e+21) {
		tmp = cos(re);
	} else if (im <= 1.26e+131) {
		tmp = pow(re, -512.0);
	} else {
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.7d+131)) then
        tmp = (im * im) * (0.5d0 + (re * (re * (-0.25d0))))
    else if (im <= (-31000000.0d0)) then
        tmp = re ** (-512.0d0)
    else if (im <= 2.1d+21) then
        tmp = cos(re)
    else if (im <= 1.26d+131) then
        tmp = re ** (-512.0d0)
    else
        tmp = (-0.5d0) + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.7e+131) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else if (im <= -31000000.0) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 2.1e+21) {
		tmp = Math.cos(re);
	} else if (im <= 1.26e+131) {
		tmp = Math.pow(re, -512.0);
	} else {
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.7e+131:
		tmp = (im * im) * (0.5 + (re * (re * -0.25)))
	elif im <= -31000000.0:
		tmp = math.pow(re, -512.0)
	elif im <= 2.1e+21:
		tmp = math.cos(re)
	elif im <= 1.26e+131:
		tmp = math.pow(re, -512.0)
	else:
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.7e+131)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 2.1e+21)
		tmp = cos(re);
	elseif (im <= 1.26e+131)
		tmp = re ^ -512.0;
	else
		tmp = Float64(-0.5 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.7e+131)
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	elseif (im <= -31000000.0)
		tmp = re ^ -512.0;
	elseif (im <= 2.1e+21)
		tmp = cos(re);
	elseif (im <= 1.26e+131)
		tmp = re ^ -512.0;
	else
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.7e+131], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -31000000.0], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 2.1e+21], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.26e+131], N[Power[re, -512.0], $MachinePrecision], N[(-0.5 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.69999999999999993e131

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 81.2%

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

      1. unpow2 81.2%

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

   4. Simplified 81.2%

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

   5. Taylor expanded in im around inf 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-*r* 81.2%

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

      3. *-commutative 81.2%

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

      4. unpow2 81.2%

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

      5. associate-*l* 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in re around 0 9.7%

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

      1. associate-*r* 9.7%

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

      2. distribute-rgt-out 74.4%

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

      3. unpow2 74.4%

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

      4. unpow2 74.4%

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

      5. associate-*r* 74.4%

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

   10. Simplified 74.4%

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

   if -1.69999999999999993e131 < im < -3.1e7 or 2.1e21 < im < 1.26e131

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   5. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. associate-*r* 0.0%

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

      7. distribute-rgt-out 82.1%

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

      8. rec-exp 82.1%

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

      9. *-commutative 82.1%

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

      10. associate-*l* 82.1%

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

      11. metadata-eval 82.1%

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

      12. unpow2 82.1%

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

      13. associate-*l* 82.1%

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

   7. Simplified 82.1%

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

   9. Applied egg-rr 62.1%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if -3.1e7 < im < 2.1e21

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 94.9%

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

   if 1.26e131 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 84.6%

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

   6. Applied egg-rr 84.6%

      \[\leadsto 0.5 \cdot \color{blue}{-2} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around 0 84.6%

      \[\leadsto \color{blue}{\left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right) - 0.5} \]
   8. Step-by-step derivation

      1. sub-neg 84.6%

         \[\leadsto \color{blue}{\left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right) + \left(-0.5\right)} \]

      2. metadata-eval 84.6%

         \[\leadsto \left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right) + \color{blue}{-0.5} \]

      3. +-commutative 84.6%

         \[\leadsto \color{blue}{-0.5 + \left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right)} \]

      4. +-commutative 84.6%

         \[\leadsto -0.5 + \color{blue}{\left(\left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right) + 0.25 \cdot {im}^{2}\right)} \]

      5. associate-+l+ 84.6%

         \[\leadsto -0.5 + \color{blue}{\left(0.5 \cdot im + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)} \]

      6. *-commutative 84.6%

         \[\leadsto -0.5 + \left(\color{blue}{im \cdot 0.5} + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right) \]

      7. *-commutative 84.6%

         \[\leadsto -0.5 + \left(im \cdot 0.5 + \left(\color{blue}{{im}^{3} \cdot 0.08333333333333333} + 0.25 \cdot {im}^{2}\right)\right) \]

      8. cube-mult 84.6%

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

      9. unpow2 84.6%

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

      10. associate-*l* 84.6%

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

      11. *-commutative 84.6%

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

      12. unpow2 84.6%

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

      13. associate-*l* 84.6%

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

      14. distribute-lft-out 84.6%

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

      15. distribute-lft-out 84.6%

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

      16. unpow2 84.6%

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

      17. associate-*l* 84.6%

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

      18. distribute-lft-out 84.6%

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

      19. +-commutative 84.6%

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

   9. Simplified 84.6%

      \[\leadsto \color{blue}{-0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{+131}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq -31000000:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+21}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.26 \cdot 10^{+131}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;-0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]

Alternative 14: 72.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -9.6 \cdot 10^{+42}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;-0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -9.6e+42)
   (* (* im im) (+ 0.5 (* re (* re -0.25))))
   (if (<= im 3.3e+27)
     (cos re)
     (+ -0.5 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -9.6e+42) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else if (im <= 3.3e+27) {
		tmp = cos(re);
	} else {
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-9.6d+42)) then
        tmp = (im * im) * (0.5d0 + (re * (re * (-0.25d0))))
    else if (im <= 3.3d+27) then
        tmp = cos(re)
    else
        tmp = (-0.5d0) + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -9.6e+42) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else if (im <= 3.3e+27) {
		tmp = Math.cos(re);
	} else {
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -9.6e+42:
		tmp = (im * im) * (0.5 + (re * (re * -0.25)))
	elif im <= 3.3e+27:
		tmp = math.cos(re)
	else:
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -9.6e+42)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	elseif (im <= 3.3e+27)
		tmp = cos(re);
	else
		tmp = Float64(-0.5 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -9.6e+42)
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	elseif (im <= 3.3e+27)
		tmp = cos(re);
	else
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -9.6e+42], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.3e+27], N[Cos[re], $MachinePrecision], N[(-0.5 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.5999999999999994e42

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 53.9%

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

      1. unpow2 53.9%

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

   4. Simplified 53.9%

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

   5. Taylor expanded in im around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*r* 53.9%

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

      3. *-commutative 53.9%

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

      4. unpow2 53.9%

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

      5. associate-*l* 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in re around 0 13.3%

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

      1. associate-*r* 13.3%

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

      2. distribute-rgt-out 54.8%

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

      3. unpow2 54.8%

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

      4. unpow2 54.8%

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

      5. associate-*r* 54.8%

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

   10. Simplified 54.8%

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

   if -9.5999999999999994e42 < im < 3.2999999999999998e27

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in im around 0 92.4%

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

   if 3.2999999999999998e27 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 83.6%

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

   6. Applied egg-rr 83.6%

      \[\leadsto 0.5 \cdot \color{blue}{-2} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around 0 64.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right) - 0.5} \]
   8. Step-by-step derivation

      1. sub-neg 64.8%

         \[\leadsto \color{blue}{\left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right) + \left(-0.5\right)} \]

      2. metadata-eval 64.8%

         \[\leadsto \left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right) + \color{blue}{-0.5} \]

      3. +-commutative 64.8%

         \[\leadsto \color{blue}{-0.5 + \left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right)} \]

      4. +-commutative 64.8%

         \[\leadsto -0.5 + \color{blue}{\left(\left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right) + 0.25 \cdot {im}^{2}\right)} \]

      5. associate-+l+ 64.8%

         \[\leadsto -0.5 + \color{blue}{\left(0.5 \cdot im + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)} \]

      6. *-commutative 64.8%

         \[\leadsto -0.5 + \left(\color{blue}{im \cdot 0.5} + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right) \]

      7. *-commutative 64.8%

         \[\leadsto -0.5 + \left(im \cdot 0.5 + \left(\color{blue}{{im}^{3} \cdot 0.08333333333333333} + 0.25 \cdot {im}^{2}\right)\right) \]

      8. cube-mult 64.8%

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

      9. unpow2 64.8%

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

      10. associate-*l* 64.8%

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

      11. *-commutative 64.8%

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

      12. unpow2 64.8%

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

      13. associate-*l* 64.8%

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

      14. distribute-lft-out 64.8%

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

      15. distribute-lft-out 64.8%

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

      16. unpow2 64.8%

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

      17. associate-*l* 64.8%

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

      18. distribute-lft-out 64.8%

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

      19. +-commutative 64.8%

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

   9. Simplified 64.8%

      \[\leadsto \color{blue}{-0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.6 \cdot 10^{+42}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;-0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]

Alternative 15: 49.8% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+42} \lor \neg \left(im \leq 1.96 \cdot 10^{+87}\right) \land im \leq 4 \cdot 10^{+175}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -7.5e+42) (and (not (<= im 1.96e+87)) (<= im 4e+175)))
   (* (* im im) (+ 0.5 (* re (* re -0.25))))
   (* 0.5 (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -7.5e+42) || (!(im <= 1.96e+87) && (im <= 4e+175))) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 0.5 * (2.0 + (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 <= (-7.5d+42)) .or. (.not. (im <= 1.96d+87)) .and. (im <= 4d+175)) then
        tmp = (im * im) * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = 0.5d0 * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -7.5e+42) || (!(im <= 1.96e+87) && (im <= 4e+175))) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = 0.5 * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -7.5e+42) or (not (im <= 1.96e+87) and (im <= 4e+175)):
		tmp = (im * im) * (0.5 + (re * (re * -0.25)))
	else:
		tmp = 0.5 * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -7.5e+42) || (!(im <= 1.96e+87) && (im <= 4e+175)))
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -7.5e+42) || (~((im <= 1.96e+87)) && (im <= 4e+175)))
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	else
		tmp = 0.5 * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -7.5e+42], And[N[Not[LessEqual[im, 1.96e+87]], $MachinePrecision], LessEqual[im, 4e+175]]], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -7.50000000000000041e42 or 1.96e87 < im < 3.9999999999999997e175

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 47.4%

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

      1. unpow2 47.4%

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

   4. Simplified 47.4%

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

   5. Taylor expanded in im around inf 47.4%

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

      1. *-commutative 47.4%

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

      2. associate-*r* 47.4%

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

      3. *-commutative 47.4%

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

      4. unpow2 47.4%

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

      5. associate-*l* 46.3%

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

   7. Simplified 46.3%

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

   8. Taylor expanded in re around 0 16.6%

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

      1. associate-*r* 16.6%

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

      2. distribute-rgt-out 53.2%

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

      3. unpow2 53.2%

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

      4. unpow2 53.2%

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

      5. associate-*r* 53.2%

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

   10. Simplified 53.2%

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

   if -7.50000000000000041e42 < im < 1.96e87 or 3.9999999999999997e175 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 64.9%

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

   3. Taylor expanded in im around 0 56.8%

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

      1. unpow2 56.8%

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

   5. Simplified 56.8%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+42} \lor \neg \left(im \leq 1.96 \cdot 10^{+87}\right) \land im \leq 4 \cdot 10^{+175}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 16: 51.2% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+42}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 3:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -7.5e+42)
   (* (* im im) (+ 0.5 (* re (* re -0.25))))
   (if (<= im 3.0)
     (* 0.5 (+ 2.0 (* im im)))
     (+ -0.5 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -7.5e+42) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else if (im <= 3.0) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-7.5d+42)) then
        tmp = (im * im) * (0.5d0 + (re * (re * (-0.25d0))))
    else if (im <= 3.0d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else
        tmp = (-0.5d0) + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -7.5e+42) {
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	} else if (im <= 3.0) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -7.5e+42:
		tmp = (im * im) * (0.5 + (re * (re * -0.25)))
	elif im <= 3.0:
		tmp = 0.5 * (2.0 + (im * im))
	else:
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -7.5e+42)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	elseif (im <= 3.0)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(-0.5 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -7.5e+42)
		tmp = (im * im) * (0.5 + (re * (re * -0.25)));
	elseif (im <= 3.0)
		tmp = 0.5 * (2.0 + (im * im));
	else
		tmp = -0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -7.5e+42], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.0], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.50000000000000041e42

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 53.9%

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

      1. unpow2 53.9%

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

   4. Simplified 53.9%

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

   5. Taylor expanded in im around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*r* 53.9%

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

      3. *-commutative 53.9%

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

      4. unpow2 53.9%

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

      5. associate-*l* 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in re around 0 13.3%

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

      1. associate-*r* 13.3%

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

      2. distribute-rgt-out 54.8%

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

      3. unpow2 54.8%

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

      4. unpow2 54.8%

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

      5. associate-*r* 54.8%

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

   10. Simplified 54.8%

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

   if -7.50000000000000041e42 < im < 3

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 58.5%

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

   3. Taylor expanded in im around 0 54.9%

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

      1. unpow2 54.9%

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

   5. Simplified 54.9%

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

   if 3 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around 0 81.5%

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

   6. Applied egg-rr 81.6%

      \[\leadsto 0.5 \cdot \color{blue}{-2} + 0.5 \cdot e^{im} \]

   7. Taylor expanded in im around 0 61.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right) - 0.5} \]
   8. Step-by-step derivation

      1. sub-neg 61.8%

         \[\leadsto \color{blue}{\left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right) + \left(-0.5\right)} \]

      2. metadata-eval 61.8%

         \[\leadsto \left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right) + \color{blue}{-0.5} \]

      3. +-commutative 61.8%

         \[\leadsto \color{blue}{-0.5 + \left(0.25 \cdot {im}^{2} + \left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right)\right)} \]

      4. +-commutative 61.8%

         \[\leadsto -0.5 + \color{blue}{\left(\left(0.5 \cdot im + 0.08333333333333333 \cdot {im}^{3}\right) + 0.25 \cdot {im}^{2}\right)} \]

      5. associate-+l+ 61.8%

         \[\leadsto -0.5 + \color{blue}{\left(0.5 \cdot im + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)} \]

      6. *-commutative 61.8%

         \[\leadsto -0.5 + \left(\color{blue}{im \cdot 0.5} + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right) \]

      7. *-commutative 61.8%

         \[\leadsto -0.5 + \left(im \cdot 0.5 + \left(\color{blue}{{im}^{3} \cdot 0.08333333333333333} + 0.25 \cdot {im}^{2}\right)\right) \]

      8. cube-mult 61.8%

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

      9. unpow2 61.8%

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

      10. associate-*l* 61.8%

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

      11. *-commutative 61.8%

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

      12. unpow2 61.8%

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

      13. associate-*l* 61.8%

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

      14. distribute-lft-out 61.8%

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

      15. distribute-lft-out 61.8%

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

      16. unpow2 61.8%

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

      17. associate-*l* 61.8%

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

      18. distribute-lft-out 61.8%

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

      19. +-commutative 61.8%

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

   9. Simplified 61.8%

      \[\leadsto \color{blue}{-0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+42}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 3:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]

Alternative 17: 49.4% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.25 \cdot \left(im \cdot \left(re \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{if}\;re \leq -4.85 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+191}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot re - re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.25 (* im (* re (* re im))))))
   (if (<= re -4.85e+27)
     t_0
     (if (<= re 3.2e+95)
       (* 0.5 (+ 2.0 (* im im)))
       (if (<= re 1.5e+191) t_0 (- (* re re) re))))))

C

double code(double re, double im) {
	double t_0 = -0.25 * (im * (re * (re * im)));
	double tmp;
	if (re <= -4.85e+27) {
		tmp = t_0;
	} else if (re <= 3.2e+95) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (re <= 1.5e+191) {
		tmp = t_0;
	} else {
		tmp = (re * re) - re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.25d0) * (im * (re * (re * im)))
    if (re <= (-4.85d+27)) then
        tmp = t_0
    else if (re <= 3.2d+95) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else if (re <= 1.5d+191) then
        tmp = t_0
    else
        tmp = (re * re) - re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.25 * (im * (re * (re * im)));
	double tmp;
	if (re <= -4.85e+27) {
		tmp = t_0;
	} else if (re <= 3.2e+95) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (re <= 1.5e+191) {
		tmp = t_0;
	} else {
		tmp = (re * re) - re;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.25 * (im * (re * (re * im)))
	tmp = 0
	if re <= -4.85e+27:
		tmp = t_0
	elif re <= 3.2e+95:
		tmp = 0.5 * (2.0 + (im * im))
	elif re <= 1.5e+191:
		tmp = t_0
	else:
		tmp = (re * re) - re
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.25 * Float64(im * Float64(re * Float64(re * im))))
	tmp = 0.0
	if (re <= -4.85e+27)
		tmp = t_0;
	elseif (re <= 3.2e+95)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	elseif (re <= 1.5e+191)
		tmp = t_0;
	else
		tmp = Float64(Float64(re * re) - re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.25 * (im * (re * (re * im)));
	tmp = 0.0;
	if (re <= -4.85e+27)
		tmp = t_0;
	elseif (re <= 3.2e+95)
		tmp = 0.5 * (2.0 + (im * im));
	elseif (re <= 1.5e+191)
		tmp = t_0;
	else
		tmp = (re * re) - re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.25 * N[(im * N[(re * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4.85e+27], t$95$0, If[LessEqual[re, 3.2e+95], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.5e+191], t$95$0, N[(N[(re * re), $MachinePrecision] - re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.8500000000000001e27 or 3.2000000000000001e95 < re < 1.4999999999999999e191

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 79.8%

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

      1. unpow2 79.8%

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

   4. Simplified 79.8%

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

   5. Taylor expanded in im around inf 24.9%

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

      1. *-commutative 24.9%

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

      2. associate-*r* 24.9%

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

      3. *-commutative 24.9%

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

      4. unpow2 24.9%

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

      5. associate-*l* 24.9%

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

   7. Simplified 24.9%

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

   8. Taylor expanded in re around 0 12.3%

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

      1. associate-*r* 12.3%

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

      2. distribute-rgt-out 26.2%

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

      3. unpow2 26.2%

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

      4. unpow2 26.2%

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

      5. associate-*r* 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in re around inf 26.2%

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

       1. unpow2 26.2%

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

       2. associate-*r* 26.5%

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

       3. *-commutative 26.5%

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

       4. unpow2 26.5%

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

       5. associate-*l* 26.9%

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

   13. Simplified 26.9%

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

   if -4.8500000000000001e27 < re < 3.2000000000000001e95

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.5%

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

   3. Taylor expanded in im around 0 71.0%

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

      1. unpow2 71.0%

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

   5. Simplified 71.0%

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

   if 1.4999999999999999e191 < re

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   5. Taylor expanded in re around 0 0.9%

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

      1. +-commutative 0.9%

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

      2. distribute-lft-out 0.9%

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

      3. associate-*r* 0.9%

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

      4. +-commutative 0.9%

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

      5. distribute-lft-out 0.9%

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

      6. associate-*r* 0.9%

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

      7. distribute-rgt-out 12.5%

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

      8. rec-exp 12.5%

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

      9. *-commutative 12.5%

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

      10. associate-*l* 12.5%

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

      11. metadata-eval 12.5%

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

      12. unpow2 12.5%

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

      13. associate-*l* 12.5%

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

   7. Simplified 12.5%

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

   9. Applied egg-rr 35.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, -re\right)} \]
   10. Step-by-step derivation

       1. fma-neg 35.7%

          \[\leadsto \color{blue}{re \cdot re - re} \]

   11. Simplified 35.7%

       \[\leadsto \color{blue}{re \cdot re - re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.85 \cdot 10^{+27}:\\ \;\;\;\;-0.25 \cdot \left(im \cdot \left(re \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+191}:\\ \;\;\;\;-0.25 \cdot \left(im \cdot \left(re \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re - re\\ \end{array} \]

Alternative 18: 46.4% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.2 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.2) (not (<= im 1.4))) (* 0.5 (* im im)) 1.0))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.2) || !(im <= 1.4)) {
		tmp = 0.5 * (im * im);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-0.2d0)) .or. (.not. (im <= 1.4d0))) then
        tmp = 0.5d0 * (im * im)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.2) || !(im <= 1.4)) {
		tmp = 0.5 * (im * im);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.2) or not (im <= 1.4):
		tmp = 0.5 * (im * im)
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.2) || !(im <= 1.4))
		tmp = Float64(0.5 * Float64(im * im));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.2) || ~((im <= 1.4)))
		tmp = 0.5 * (im * im);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.2], N[Not[LessEqual[im, 1.4]], $MachinePrecision]], N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if im < -0.20000000000000001 or 1.3999999999999999 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 53.4%

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

      1. unpow2 53.4%

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

   4. Simplified 53.4%

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

   5. Taylor expanded in im around inf 53.2%

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

      1. *-commutative 53.2%

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

      2. associate-*r* 53.2%

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

      3. *-commutative 53.2%

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

      4. unpow2 53.2%

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

      5. associate-*l* 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in re around 0 42.3%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 42.3%

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

   10. Simplified 42.3%

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

   if -0.20000000000000001 < im < 1.3999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 57.7%

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

   3. Taylor expanded in im around 0 56.8%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.2 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19: 47.3% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8 \cdot 10^{+157}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+242}:\\ \;\;\;\;-512 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;re \cdot re - re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8e+157)
   (* 0.5 (+ 2.0 (* im im)))
   (if (<= re 1.1e+242) (- -512.0 (* re re)) (- (* re re) re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8e+157) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (re <= 1.1e+242) {
		tmp = -512.0 - (re * re);
	} else {
		tmp = (re * re) - re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 8d+157) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else if (re <= 1.1d+242) then
        tmp = (-512.0d0) - (re * re)
    else
        tmp = (re * re) - re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 8e+157) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (re <= 1.1e+242) {
		tmp = -512.0 - (re * re);
	} else {
		tmp = (re * re) - re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8e+157:
		tmp = 0.5 * (2.0 + (im * im))
	elif re <= 1.1e+242:
		tmp = -512.0 - (re * re)
	else:
		tmp = (re * re) - re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8e+157)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	elseif (re <= 1.1e+242)
		tmp = Float64(-512.0 - Float64(re * re));
	else
		tmp = Float64(Float64(re * re) - re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 8e+157)
		tmp = 0.5 * (2.0 + (im * im));
	elseif (re <= 1.1e+242)
		tmp = -512.0 - (re * re);
	else
		tmp = (re * re) - re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8e+157], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.1e+242], N[(-512.0 - N[(re * re), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] - re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 7.99999999999999987e157

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 72.6%

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

   3. Taylor expanded in im around 0 55.6%

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

      1. unpow2 55.6%

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

   5. Simplified 55.6%

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

   if 7.99999999999999987e157 < re < 1.1e242

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. *-commutative 99.9%

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

      8. fma-def 99.9%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   5. Taylor expanded in re around 0 1.2%

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

      1. +-commutative 1.2%

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

      2. distribute-lft-out 1.2%

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

      3. associate-*r* 1.2%

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

      4. +-commutative 1.2%

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

      5. distribute-lft-out 1.2%

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

      6. associate-*r* 1.2%

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

      7. distribute-rgt-out 31.2%

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

      8. rec-exp 31.2%

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

      9. *-commutative 31.2%

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

      10. associate-*l* 31.2%

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

      11. metadata-eval 31.2%

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

      12. unpow2 31.2%

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

      13. associate-*l* 31.2%

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

   7. Simplified 31.2%

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

   9. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{-512 + \left(-re\right) \cdot re} \]
   10. Step-by-step derivation

       1. cancel-sign-sub-inv 31.2%

          \[\leadsto \color{blue}{-512 - re \cdot re} \]

   11. Simplified 31.2%

       \[\leadsto \color{blue}{-512 - re \cdot re} \]

   if 1.1e242 < re

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 100.0%

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

   5. Taylor expanded in re around 0 1.1%

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

      1. +-commutative 1.1%

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

      2. distribute-lft-out 1.1%

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

      3. associate-*r* 1.1%

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

      4. +-commutative 1.1%

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

      5. distribute-lft-out 1.1%

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

      6. associate-*r* 1.1%

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

      7. distribute-rgt-out 7.0%

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

      8. rec-exp 7.0%

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

      9. *-commutative 7.0%

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

      10. associate-*l* 7.0%

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

      11. metadata-eval 7.0%

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

      12. unpow2 7.0%

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

      13. associate-*l* 7.0%

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

   7. Simplified 7.0%

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

   9. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, -re\right)} \]
   10. Step-by-step derivation

       1. fma-neg 36.4%

          \[\leadsto \color{blue}{re \cdot re - re} \]

   11. Simplified 36.4%

       \[\leadsto \color{blue}{re \cdot re - re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8 \cdot 10^{+157}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+242}:\\ \;\;\;\;-512 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;re \cdot re - re\\ \end{array} \]

Alternative 20: 2.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -512 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -512.0)

C

double code(double re, double im) {
	return -512.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -512.0d0
end function

Java

public static double code(double re, double im) {
	return -512.0;
}

Python

def code(re, im):
	return -512.0

Julia

function code(re, im)
	return -512.0
end

MATLAB

function tmp = code(re, im)
	tmp = -512.0;
end

Wolfram

code[re_, im_] := -512.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 78.2%

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

   1. unpow2 78.2%

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

4. Simplified 78.2%

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

5. Taylor expanded in im around inf 26.8%

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

   1. *-commutative 26.8%

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

   2. associate-*r* 26.8%

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

   3. *-commutative 26.8%

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

   4. unpow2 26.8%

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

   5. associate-*l* 26.5%

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

7. Simplified 26.5%

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

8. Taylor expanded in re around 0 21.5%

   \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
9. Step-by-step derivation

   1. unpow2 21.5%

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

10. Simplified 21.5%

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

12. Applied egg-rr 3.1%

    \[\leadsto \color{blue}{-512} \]

13. Final simplification 3.1%

    \[\leadsto -512 \]

Alternative 21: 3.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

double code(double re, double im) {
	return -1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.0d0
end function

Java

public static double code(double re, double im) {
	return -1.0;
}

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0;
end

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 78.2%

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

   1. unpow2 78.2%

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

4. Simplified 78.2%

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

5. Taylor expanded in im around inf 26.8%

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

   1. *-commutative 26.8%

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

   2. associate-*r* 26.8%

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

   3. *-commutative 26.8%

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

   4. unpow2 26.8%

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

   5. associate-*l* 26.5%

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

7. Simplified 26.5%

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

8. Taylor expanded in re around 0 21.5%

   \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
9. Step-by-step derivation

   1. unpow2 21.5%

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

10. Simplified 21.5%

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

12. Applied egg-rr 4.1%

    \[\leadsto \color{blue}{-1} \]

13. Final simplification 4.1%

    \[\leadsto -1 \]

Alternative 22: 6.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.015625 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.015625)

C

double code(double re, double im) {
	return 0.015625;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.015625d0
end function

Java

public static double code(double re, double im) {
	return 0.015625;
}

Python

def code(re, im):
	return 0.015625

Julia

function code(re, im)
	return 0.015625
end

MATLAB

function tmp = code(re, im)
	tmp = 0.015625;
end

Wolfram

code[re_, im_] := 0.015625

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 78.2%

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

   1. unpow2 78.2%

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

4. Simplified 78.2%

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

5. Taylor expanded in im around inf 26.8%

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

   1. *-commutative 26.8%

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

   2. associate-*r* 26.8%

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

   3. *-commutative 26.8%

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

   4. unpow2 26.8%

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

   5. associate-*l* 26.5%

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

7. Simplified 26.5%

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

8. Taylor expanded in re around 0 21.5%

   \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
9. Step-by-step derivation

   1. unpow2 21.5%

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

10. Simplified 21.5%

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

12. Applied egg-rr 7.5%

    \[\leadsto \color{blue}{0.015625} \]

13. Final simplification 7.5%

    \[\leadsto 0.015625 \]

Alternative 23: 7.6% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.125)

C

double code(double re, double im) {
	return 0.125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.125d0
end function

Java

public static double code(double re, double im) {
	return 0.125;
}

Python

def code(re, im):
	return 0.125

Julia

function code(re, im)
	return 0.125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.125;
end

Wolfram

code[re_, im_] := 0.125

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 78.2%

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

   1. unpow2 78.2%

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

4. Simplified 78.2%

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

5. Taylor expanded in im around inf 26.8%

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

   1. *-commutative 26.8%

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

   2. associate-*r* 26.8%

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

   3. *-commutative 26.8%

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

   4. unpow2 26.8%

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

   5. associate-*l* 26.5%

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

7. Simplified 26.5%

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

8. Taylor expanded in re around 0 21.5%

   \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
9. Step-by-step derivation

   1. unpow2 21.5%

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

10. Simplified 21.5%

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

12. Applied egg-rr 8.1%

    \[\leadsto \color{blue}{0.125} \]

13. Final simplification 8.1%

    \[\leadsto 0.125 \]

Alternative 24: 7.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 78.2%

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

   1. unpow2 78.2%

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

4. Simplified 78.2%

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

5. Taylor expanded in im around inf 26.8%

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

   1. *-commutative 26.8%

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

   2. associate-*r* 26.8%

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

   3. *-commutative 26.8%

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

   4. unpow2 26.8%

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

   5. associate-*l* 26.5%

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

7. Simplified 26.5%

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

8. Taylor expanded in re around 0 21.5%

   \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
9. Step-by-step derivation

   1. unpow2 21.5%

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

10. Simplified 21.5%

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

12. Applied egg-rr 8.5%

    \[\leadsto \color{blue}{0.25} \]

13. Final simplification 8.5%

    \[\leadsto 0.25 \]

Alternative 25: 9.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.75)

C

double code(double re, double im) {
	return 0.75;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.75d0
end function

Java

public static double code(double re, double im) {
	return 0.75;
}

Python

def code(re, im):
	return 0.75

Julia

function code(re, im)
	return 0.75
end

MATLAB

function tmp = code(re, im)
	tmp = 0.75;
end

Wolfram

code[re_, im_] := 0.75

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 78.2%

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

   1. unpow2 78.2%

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

4. Simplified 78.2%

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

5. Taylor expanded in im around inf 26.8%

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

   1. *-commutative 26.8%

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

   2. associate-*r* 26.8%

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

   3. *-commutative 26.8%

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

   4. unpow2 26.8%

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

   5. associate-*l* 26.5%

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

7. Simplified 26.5%

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

8. Taylor expanded in re around 0 21.5%

   \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
9. Step-by-step derivation

   1. unpow2 21.5%

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

10. Simplified 21.5%

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

12. Applied egg-rr 9.7%

    \[\leadsto \color{blue}{0.75} \]

13. Final simplification 9.7%

    \[\leadsto 0.75 \]

Alternative 26: 28.1% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 66.4%

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

3. Taylor expanded in im around 0 31.9%

   \[\leadsto 0.5 \cdot \color{blue}{2} \]

4. Final simplification 31.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023182 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))