math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 17

Speedup: 1.0×

• 49.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2000000000000:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2000000000000.0)
   (*
    (* 0.5 (cos re))
    (+ (+ 2.0 (* im im)) (* 0.08333333333333333 (pow im 4.0))))
   (if (<= im 1.15e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2000000000000.0) {
		tmp = (0.5 * cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * pow(im, 4.0)));
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2000000000000.0d0) then
        tmp = (0.5d0 * cos(re)) * ((2.0d0 + (im * im)) + (0.08333333333333333d0 * (im ** 4.0d0)))
    else if (im <= 1.15d+77) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2000000000000.0) {
		tmp = (0.5 * Math.cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * Math.pow(im, 4.0)));
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2000000000000.0:
		tmp = (0.5 * math.cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * math.pow(im, 4.0)))
	elif im <= 1.15e+77:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2000000000000.0)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64(0.08333333333333333 * (im ^ 4.0))));
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2000000000000.0)
		tmp = (0.5 * cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * (im ^ 4.0)));
	elseif (im <= 1.15e+77)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2000000000000.0], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2e12

   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 90.8%

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

   4. Simplified 90.8%

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

   if 2e12 < 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 re around 0 77.8%

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

   if 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%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2000000000000:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 3: 82.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.55 \cdot 10^{+14}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{+73}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.020833333333333332 \cdot {re}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.55e+14)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 5.3e+73)
     (* (* im im) (* 0.020833333333333332 (pow re 4.0)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.55e+14) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 5.3e+73) {
		tmp = (im * im) * (0.020833333333333332 * pow(re, 4.0));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.55d+14) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 5.3d+73) then
        tmp = (im * im) * (0.020833333333333332d0 * (re ** 4.0d0))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.55e+14) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 5.3e+73) {
		tmp = (im * im) * (0.020833333333333332 * Math.pow(re, 4.0));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.55e+14:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 5.3e+73:
		tmp = (im * im) * (0.020833333333333332 * math.pow(re, 4.0))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.55e+14)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 5.3e+73)
		tmp = Float64(Float64(im * im) * Float64(0.020833333333333332 * (re ^ 4.0)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.55e+14)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 5.3e+73)
		tmp = (im * im) * (0.020833333333333332 * (re ^ 4.0));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.55e+14], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.3e+73], N[(N[(im * im), $MachinePrecision] * N[(0.020833333333333332 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.55e14

   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 80.8%

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

   4. Simplified 80.8%

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

   if 1.55e14 < im < 5.29999999999999996e73

   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 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around inf 3.9%

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

      1. *-commutative 3.9%

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

      2. unpow2 3.9%

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

      3. associate-*l* 3.9%

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

      4. associate-*r* 3.9%

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

      5. *-commutative 3.9%

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

   7. Simplified 3.9%

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

   8. Taylor expanded in re around 0 1.5%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2} + \left(0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right) + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 1.5%

         \[\leadsto \color{blue}{{im}^{2} \cdot 0.5} + \left(0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right) + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right) \]

      2. +-commutative 1.5%

         \[\leadsto {im}^{2} \cdot 0.5 + \color{blue}{\left(-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) + 0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)\right)} \]

      3. associate-*r* 1.5%

         \[\leadsto {im}^{2} \cdot 0.5 + \left(\color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} + 0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)\right) \]

      4. associate-*r* 1.5%

         \[\leadsto {im}^{2} \cdot 0.5 + \left(\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2} + \color{blue}{\left(0.020833333333333332 \cdot {re}^{4}\right) \cdot {im}^{2}}\right) \]

      5. distribute-rgt-out 7.4%

         \[\leadsto {im}^{2} \cdot 0.5 + \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + 0.020833333333333332 \cdot {re}^{4}\right)} \]

      6. distribute-lft-out 7.4%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 + \left(-0.25 \cdot {re}^{2} + 0.020833333333333332 \cdot {re}^{4}\right)\right)} \]

      7. unpow2 7.4%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + \left(-0.25 \cdot {re}^{2} + 0.020833333333333332 \cdot {re}^{4}\right)\right) \]

      8. +-commutative 7.4%

         \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \color{blue}{\left(0.020833333333333332 \cdot {re}^{4} + -0.25 \cdot {re}^{2}\right)}\right) \]

      9. *-commutative 7.4%

         \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left(\color{blue}{{re}^{4} \cdot 0.020833333333333332} + -0.25 \cdot {re}^{2}\right)\right) \]

      10. unpow2 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

      11. associate-*r* 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + \color{blue}{\left(-0.25 \cdot re\right) \cdot re}\right)\right) \]

      12. *-commutative 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + \color{blue}{re \cdot \left(-0.25 \cdot re\right)}\right)\right) \]

      13. *-commutative 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + re \cdot \color{blue}{\left(re \cdot -0.25\right)}\right)\right) \]

   10. Simplified 7.4%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + re \cdot \left(re \cdot -0.25\right)\right)\right)} \]

   11. Taylor expanded in re around inf 42.1%

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

       1. *-commutative 42.1%

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

       2. unpow2 42.1%

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

       3. *-commutative 42.1%

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

       4. associate-*l* 42.1%

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

       5. *-commutative 42.1%

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

   13. Simplified 42.1%

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

   if 5.29999999999999996e73 < 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%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.55 \cdot 10^{+14}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{+73}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.020833333333333332 \cdot {re}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 4: 86.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2000000000000:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2000000000000.0)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 1.15e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2000000000000.0) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2000000000000.0d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 1.15d+77) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2000000000000.0) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2000000000000.0:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 1.15e+77:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2000000000000.0)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2000000000000.0)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 1.15e+77)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2000000000000.0], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2e12

   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%

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

   4. Simplified 81.2%

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

   if 2e12 < 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 re around 0 77.8%

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

   if 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%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2000000000000:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 5: 81.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{if}\;im \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.4 \cdot 10^{+73}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.020833333333333332 \cdot {re}^{4}\right)\\ \mathbf{elif}\;im \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ 2.0 (* im im)))))
   (if (<= im 1.6e+14)
     t_0
     (if (<= im 7.4e+73)
       (* (* im im) (* 0.020833333333333332 (pow re 4.0)))
       (if (<= im 1.28e+154) (* (pow im 4.0) 0.041666666666666664) t_0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (2.0 + (im * im));
	double tmp;
	if (im <= 1.6e+14) {
		tmp = t_0;
	} else if (im <= 7.4e+73) {
		tmp = (im * im) * (0.020833333333333332 * pow(re, 4.0));
	} else if (im <= 1.28e+154) {
		tmp = pow(im, 4.0) * 0.041666666666666664;
	} 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 = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    if (im <= 1.6d+14) then
        tmp = t_0
    else if (im <= 7.4d+73) then
        tmp = (im * im) * (0.020833333333333332d0 * (re ** 4.0d0))
    else if (im <= 1.28d+154) then
        tmp = (im ** 4.0d0) * 0.041666666666666664d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	double tmp;
	if (im <= 1.6e+14) {
		tmp = t_0;
	} else if (im <= 7.4e+73) {
		tmp = (im * im) * (0.020833333333333332 * Math.pow(re, 4.0));
	} else if (im <= 1.28e+154) {
		tmp = Math.pow(im, 4.0) * 0.041666666666666664;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (2.0 + (im * im))
	tmp = 0
	if im <= 1.6e+14:
		tmp = t_0
	elif im <= 7.4e+73:
		tmp = (im * im) * (0.020833333333333332 * math.pow(re, 4.0))
	elif im <= 1.28e+154:
		tmp = math.pow(im, 4.0) * 0.041666666666666664
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)))
	tmp = 0.0
	if (im <= 1.6e+14)
		tmp = t_0;
	elseif (im <= 7.4e+73)
		tmp = Float64(Float64(im * im) * Float64(0.020833333333333332 * (re ^ 4.0)));
	elseif (im <= 1.28e+154)
		tmp = Float64((im ^ 4.0) * 0.041666666666666664);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * cos(re)) * (2.0 + (im * im));
	tmp = 0.0;
	if (im <= 1.6e+14)
		tmp = t_0;
	elseif (im <= 7.4e+73)
		tmp = (im * im) * (0.020833333333333332 * (re ^ 4.0));
	elseif (im <= 1.28e+154)
		tmp = (im ^ 4.0) * 0.041666666666666664;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.6e+14], t$95$0, If[LessEqual[im, 7.4e+73], N[(N[(im * im), $MachinePrecision] * N[(0.020833333333333332 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.28e+154], N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.6e14 or 1.2800000000000001e154 < 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 82.4%

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

   4. Simplified 82.4%

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

   if 1.6e14 < im < 7.39999999999999947e73

   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 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around inf 3.9%

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

      1. *-commutative 3.9%

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

      2. unpow2 3.9%

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

      3. associate-*l* 3.9%

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

      4. associate-*r* 3.9%

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

      5. *-commutative 3.9%

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

   7. Simplified 3.9%

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

   8. Taylor expanded in re around 0 1.5%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2} + \left(0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right) + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 1.5%

         \[\leadsto \color{blue}{{im}^{2} \cdot 0.5} + \left(0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right) + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right) \]

      2. +-commutative 1.5%

         \[\leadsto {im}^{2} \cdot 0.5 + \color{blue}{\left(-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) + 0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)\right)} \]

      3. associate-*r* 1.5%

         \[\leadsto {im}^{2} \cdot 0.5 + \left(\color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} + 0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)\right) \]

      4. associate-*r* 1.5%

         \[\leadsto {im}^{2} \cdot 0.5 + \left(\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2} + \color{blue}{\left(0.020833333333333332 \cdot {re}^{4}\right) \cdot {im}^{2}}\right) \]

      5. distribute-rgt-out 7.4%

         \[\leadsto {im}^{2} \cdot 0.5 + \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + 0.020833333333333332 \cdot {re}^{4}\right)} \]

      6. distribute-lft-out 7.4%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 + \left(-0.25 \cdot {re}^{2} + 0.020833333333333332 \cdot {re}^{4}\right)\right)} \]

      7. unpow2 7.4%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + \left(-0.25 \cdot {re}^{2} + 0.020833333333333332 \cdot {re}^{4}\right)\right) \]

      8. +-commutative 7.4%

         \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \color{blue}{\left(0.020833333333333332 \cdot {re}^{4} + -0.25 \cdot {re}^{2}\right)}\right) \]

      9. *-commutative 7.4%

         \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left(\color{blue}{{re}^{4} \cdot 0.020833333333333332} + -0.25 \cdot {re}^{2}\right)\right) \]

      10. unpow2 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

      11. associate-*r* 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + \color{blue}{\left(-0.25 \cdot re\right) \cdot re}\right)\right) \]

      12. *-commutative 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + \color{blue}{re \cdot \left(-0.25 \cdot re\right)}\right)\right) \]

      13. *-commutative 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + re \cdot \color{blue}{\left(re \cdot -0.25\right)}\right)\right) \]

   10. Simplified 7.4%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + re \cdot \left(re \cdot -0.25\right)\right)\right)} \]

   11. Taylor expanded in re around inf 42.1%

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

       1. *-commutative 42.1%

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

       2. unpow2 42.1%

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

       3. *-commutative 42.1%

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

       4. associate-*l* 42.1%

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

       5. *-commutative 42.1%

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

   13. Simplified 42.1%

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

   if 7.39999999999999947e73 < im < 1.2800000000000001e154

   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%

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

   4. Simplified 100.0%

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

   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 55.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 7.4 \cdot 10^{+73}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.020833333333333332 \cdot {re}^{4}\right)\\ \mathbf{elif}\;im \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 6: 81.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 370:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot \left(0.041666666666666664 + re \cdot \left(re \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \left(\cos re \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 370.0)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 3e+154)
     (*
      (pow im 4.0)
      (+ 0.041666666666666664 (* re (* re -0.020833333333333332))))
     (* (* 0.5 im) (* (cos re) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 370.0) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 3e+154) {
		tmp = pow(im, 4.0) * (0.041666666666666664 + (re * (re * -0.020833333333333332)));
	} else {
		tmp = (0.5 * im) * (cos(re) * 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 <= 370.0d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 3d+154) then
        tmp = (im ** 4.0d0) * (0.041666666666666664d0 + (re * (re * (-0.020833333333333332d0))))
    else
        tmp = (0.5d0 * im) * (cos(re) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 370.0) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 3e+154) {
		tmp = Math.pow(im, 4.0) * (0.041666666666666664 + (re * (re * -0.020833333333333332)));
	} else {
		tmp = (0.5 * im) * (Math.cos(re) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 370.0:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 3e+154:
		tmp = math.pow(im, 4.0) * (0.041666666666666664 + (re * (re * -0.020833333333333332)))
	else:
		tmp = (0.5 * im) * (math.cos(re) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 370.0)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 3e+154)
		tmp = Float64((im ^ 4.0) * Float64(0.041666666666666664 + Float64(re * Float64(re * -0.020833333333333332))));
	else
		tmp = Float64(Float64(0.5 * im) * Float64(cos(re) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 370.0)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 3e+154)
		tmp = (im ^ 4.0) * (0.041666666666666664 + (re * (re * -0.020833333333333332)));
	else
		tmp = (0.5 * im) * (cos(re) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 370.0], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3e+154], N[(N[Power[im, 4.0], $MachinePrecision] * N[(0.041666666666666664 + N[(re * N[(re * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 370

   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 81.6%

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

   4. Simplified 81.6%

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

   if 370 < im < 3.00000000000000026e154

   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 54.9%

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

   4. Simplified 54.9%

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

   5. Taylor expanded in im around inf 54.9%

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

   6. Taylor expanded in re around 0 8.4%

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

      1. associate-*r* 8.4%

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

      2. distribute-rgt-out 50.9%

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

      3. +-commutative 50.9%

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

      4. unpow2 50.9%

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

      5. associate-*r* 50.9%

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

      6. *-commutative 50.9%

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

      7. *-commutative 50.9%

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

   8. Simplified 50.9%

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

   if 3.00000000000000026e154 < 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%

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

   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. unpow2 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 370:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot \left(0.041666666666666664 + re \cdot \left(re \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \left(\cos re \cdot im\right)\\ \end{array} \]

Alternative 7: 68.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.9 \cdot 10^{+73}:\\ \;\;\;\;-2 \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \left(\cos re \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.9e+14)
   (cos re)
   (if (<= im 4.9e+73)
     (* -2.0 (+ 0.5 (* (* re re) -0.25)))
     (if (<= im 1.28e+154)
       (* (pow im 4.0) 0.041666666666666664)
       (* (* 0.5 im) (* (cos re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+14) {
		tmp = cos(re);
	} else if (im <= 4.9e+73) {
		tmp = -2.0 * (0.5 + ((re * re) * -0.25));
	} else if (im <= 1.28e+154) {
		tmp = pow(im, 4.0) * 0.041666666666666664;
	} else {
		tmp = (0.5 * im) * (cos(re) * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.9d+14) then
        tmp = cos(re)
    else if (im <= 4.9d+73) then
        tmp = (-2.0d0) * (0.5d0 + ((re * re) * (-0.25d0)))
    else if (im <= 1.28d+154) then
        tmp = (im ** 4.0d0) * 0.041666666666666664d0
    else
        tmp = (0.5d0 * im) * (cos(re) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.9e+14) {
		tmp = Math.cos(re);
	} else if (im <= 4.9e+73) {
		tmp = -2.0 * (0.5 + ((re * re) * -0.25));
	} else if (im <= 1.28e+154) {
		tmp = Math.pow(im, 4.0) * 0.041666666666666664;
	} else {
		tmp = (0.5 * im) * (Math.cos(re) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.9e+14:
		tmp = math.cos(re)
	elif im <= 4.9e+73:
		tmp = -2.0 * (0.5 + ((re * re) * -0.25))
	elif im <= 1.28e+154:
		tmp = math.pow(im, 4.0) * 0.041666666666666664
	else:
		tmp = (0.5 * im) * (math.cos(re) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.9e+14)
		tmp = cos(re);
	elseif (im <= 4.9e+73)
		tmp = Float64(-2.0 * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	elseif (im <= 1.28e+154)
		tmp = Float64((im ^ 4.0) * 0.041666666666666664);
	else
		tmp = Float64(Float64(0.5 * im) * Float64(cos(re) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.9e+14)
		tmp = cos(re);
	elseif (im <= 4.9e+73)
		tmp = -2.0 * (0.5 + ((re * re) * -0.25));
	elseif (im <= 1.28e+154)
		tmp = (im ^ 4.0) * 0.041666666666666664;
	else
		tmp = (0.5 * im) * (cos(re) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.9e+14], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4.9e+73], N[(-2.0 * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.28e+154], N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 4.9e14

   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 68.8%

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

   if 4.9e14 < im < 4.8999999999999999e73

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt-out 52.9%

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

      3. *-commutative 52.9%

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

      4. unpow2 52.9%

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

   6. Simplified 52.9%

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

   8. Applied egg-rr 36.4%

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

   if 4.8999999999999999e73 < im < 1.2800000000000001e154

   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%

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

   4. Simplified 100.0%

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

   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 55.0%

      \[\leadsto 0.041666666666666664 \cdot \color{blue}{{im}^{4}} \]

   if 1.2800000000000001e154 < 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 96.4%

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

   4. Simplified 96.4%

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

   5. Taylor expanded in im around inf 96.4%

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

      1. *-commutative 96.4%

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

      2. unpow2 96.4%

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

      3. associate-*l* 96.4%

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

      4. associate-*r* 96.4%

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

      5. *-commutative 96.4%

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

   7. Simplified 96.4%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.9 \cdot 10^{+73}:\\ \;\;\;\;-2 \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \left(\cos re \cdot im\right)\\ \end{array} \]

Alternative 8: 69.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 7.4 \cdot 10^{+73}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.020833333333333332 \cdot {re}^{4}\right)\\ \mathbf{elif}\;im \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \left(\cos re \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.95e+14)
   (cos re)
   (if (<= im 7.4e+73)
     (* (* im im) (* 0.020833333333333332 (pow re 4.0)))
     (if (<= im 1.28e+154)
       (* (pow im 4.0) 0.041666666666666664)
       (* (* 0.5 im) (* (cos re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.95e+14) {
		tmp = cos(re);
	} else if (im <= 7.4e+73) {
		tmp = (im * im) * (0.020833333333333332 * pow(re, 4.0));
	} else if (im <= 1.28e+154) {
		tmp = pow(im, 4.0) * 0.041666666666666664;
	} else {
		tmp = (0.5 * im) * (cos(re) * 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 <= 1.95d+14) then
        tmp = cos(re)
    else if (im <= 7.4d+73) then
        tmp = (im * im) * (0.020833333333333332d0 * (re ** 4.0d0))
    else if (im <= 1.28d+154) then
        tmp = (im ** 4.0d0) * 0.041666666666666664d0
    else
        tmp = (0.5d0 * im) * (cos(re) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.95e+14) {
		tmp = Math.cos(re);
	} else if (im <= 7.4e+73) {
		tmp = (im * im) * (0.020833333333333332 * Math.pow(re, 4.0));
	} else if (im <= 1.28e+154) {
		tmp = Math.pow(im, 4.0) * 0.041666666666666664;
	} else {
		tmp = (0.5 * im) * (Math.cos(re) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.95e+14:
		tmp = math.cos(re)
	elif im <= 7.4e+73:
		tmp = (im * im) * (0.020833333333333332 * math.pow(re, 4.0))
	elif im <= 1.28e+154:
		tmp = math.pow(im, 4.0) * 0.041666666666666664
	else:
		tmp = (0.5 * im) * (math.cos(re) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.95e+14)
		tmp = cos(re);
	elseif (im <= 7.4e+73)
		tmp = Float64(Float64(im * im) * Float64(0.020833333333333332 * (re ^ 4.0)));
	elseif (im <= 1.28e+154)
		tmp = Float64((im ^ 4.0) * 0.041666666666666664);
	else
		tmp = Float64(Float64(0.5 * im) * Float64(cos(re) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.95e+14)
		tmp = cos(re);
	elseif (im <= 7.4e+73)
		tmp = (im * im) * (0.020833333333333332 * (re ^ 4.0));
	elseif (im <= 1.28e+154)
		tmp = (im ^ 4.0) * 0.041666666666666664;
	else
		tmp = (0.5 * im) * (cos(re) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.95e+14], N[Cos[re], $MachinePrecision], If[LessEqual[im, 7.4e+73], N[(N[(im * im), $MachinePrecision] * N[(0.020833333333333332 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.28e+154], N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.95e14

   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 68.8%

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

   if 1.95e14 < im < 7.39999999999999947e73

   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 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around inf 3.9%

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

      1. *-commutative 3.9%

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

      2. unpow2 3.9%

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

      3. associate-*l* 3.9%

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

      4. associate-*r* 3.9%

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

      5. *-commutative 3.9%

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

   7. Simplified 3.9%

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

   8. Taylor expanded in re around 0 1.5%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2} + \left(0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right) + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 1.5%

         \[\leadsto \color{blue}{{im}^{2} \cdot 0.5} + \left(0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right) + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right) \]

      2. +-commutative 1.5%

         \[\leadsto {im}^{2} \cdot 0.5 + \color{blue}{\left(-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) + 0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)\right)} \]

      3. associate-*r* 1.5%

         \[\leadsto {im}^{2} \cdot 0.5 + \left(\color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} + 0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)\right) \]

      4. associate-*r* 1.5%

         \[\leadsto {im}^{2} \cdot 0.5 + \left(\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2} + \color{blue}{\left(0.020833333333333332 \cdot {re}^{4}\right) \cdot {im}^{2}}\right) \]

      5. distribute-rgt-out 7.4%

         \[\leadsto {im}^{2} \cdot 0.5 + \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + 0.020833333333333332 \cdot {re}^{4}\right)} \]

      6. distribute-lft-out 7.4%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 + \left(-0.25 \cdot {re}^{2} + 0.020833333333333332 \cdot {re}^{4}\right)\right)} \]

      7. unpow2 7.4%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + \left(-0.25 \cdot {re}^{2} + 0.020833333333333332 \cdot {re}^{4}\right)\right) \]

      8. +-commutative 7.4%

         \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \color{blue}{\left(0.020833333333333332 \cdot {re}^{4} + -0.25 \cdot {re}^{2}\right)}\right) \]

      9. *-commutative 7.4%

         \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left(\color{blue}{{re}^{4} \cdot 0.020833333333333332} + -0.25 \cdot {re}^{2}\right)\right) \]

      10. unpow2 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

      11. associate-*r* 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + \color{blue}{\left(-0.25 \cdot re\right) \cdot re}\right)\right) \]

      12. *-commutative 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + \color{blue}{re \cdot \left(-0.25 \cdot re\right)}\right)\right) \]

      13. *-commutative 7.4%

          \[\leadsto \left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + re \cdot \color{blue}{\left(re \cdot -0.25\right)}\right)\right) \]

   10. Simplified 7.4%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 + \left({re}^{4} \cdot 0.020833333333333332 + re \cdot \left(re \cdot -0.25\right)\right)\right)} \]

   11. Taylor expanded in re around inf 42.1%

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

       1. *-commutative 42.1%

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

       2. unpow2 42.1%

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

       3. *-commutative 42.1%

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

       4. associate-*l* 42.1%

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

       5. *-commutative 42.1%

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

   13. Simplified 42.1%

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

   if 7.39999999999999947e73 < im < 1.2800000000000001e154

   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%

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

   4. Simplified 100.0%

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

   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 55.0%

      \[\leadsto 0.041666666666666664 \cdot \color{blue}{{im}^{4}} \]

   if 1.2800000000000001e154 < 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 96.4%

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

   4. Simplified 96.4%

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

   5. Taylor expanded in im around inf 96.4%

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

      1. *-commutative 96.4%

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

      2. unpow2 96.4%

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

      3. associate-*l* 96.4%

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

      4. associate-*r* 96.4%

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

      5. *-commutative 96.4%

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

   7. Simplified 96.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 7.4 \cdot 10^{+73}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.020833333333333332 \cdot {re}^{4}\right)\\ \mathbf{elif}\;im \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot \left(\cos re \cdot im\right)\\ \end{array} \]

Alternative 9: 65.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \left(re \cdot re\right) \cdot -0.25\\ \mathbf{if}\;im \leq 3.3 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 7.4 \cdot 10^{+73}:\\ \;\;\;\;-2 \cdot t_0\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+153}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \mathbf{elif}\;im \leq 10^{+178}:\\ \;\;\;\;\left(im \cdot im\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* (* re re) -0.25))))
   (if (<= im 3.3e+14)
     (cos re)
     (if (<= im 7.4e+73)
       (* -2.0 t_0)
       (if (<= im 5e+153)
         (* (pow im 4.0) 0.041666666666666664)
         (if (<= im 1e+178) (* (* im im) t_0) (* im (* 0.5 im))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + ((re * re) * -0.25);
	double tmp;
	if (im <= 3.3e+14) {
		tmp = cos(re);
	} else if (im <= 7.4e+73) {
		tmp = -2.0 * t_0;
	} else if (im <= 5e+153) {
		tmp = pow(im, 4.0) * 0.041666666666666664;
	} else if (im <= 1e+178) {
		tmp = (im * im) * t_0;
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + ((re * re) * (-0.25d0))
    if (im <= 3.3d+14) then
        tmp = cos(re)
    else if (im <= 7.4d+73) then
        tmp = (-2.0d0) * t_0
    else if (im <= 5d+153) then
        tmp = (im ** 4.0d0) * 0.041666666666666664d0
    else if (im <= 1d+178) then
        tmp = (im * im) * t_0
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + ((re * re) * -0.25);
	double tmp;
	if (im <= 3.3e+14) {
		tmp = Math.cos(re);
	} else if (im <= 7.4e+73) {
		tmp = -2.0 * t_0;
	} else if (im <= 5e+153) {
		tmp = Math.pow(im, 4.0) * 0.041666666666666664;
	} else if (im <= 1e+178) {
		tmp = (im * im) * t_0;
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + ((re * re) * -0.25)
	tmp = 0
	if im <= 3.3e+14:
		tmp = math.cos(re)
	elif im <= 7.4e+73:
		tmp = -2.0 * t_0
	elif im <= 5e+153:
		tmp = math.pow(im, 4.0) * 0.041666666666666664
	elif im <= 1e+178:
		tmp = (im * im) * t_0
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(Float64(re * re) * -0.25))
	tmp = 0.0
	if (im <= 3.3e+14)
		tmp = cos(re);
	elseif (im <= 7.4e+73)
		tmp = Float64(-2.0 * t_0);
	elseif (im <= 5e+153)
		tmp = Float64((im ^ 4.0) * 0.041666666666666664);
	elseif (im <= 1e+178)
		tmp = Float64(Float64(im * im) * t_0);
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + ((re * re) * -0.25);
	tmp = 0.0;
	if (im <= 3.3e+14)
		tmp = cos(re);
	elseif (im <= 7.4e+73)
		tmp = -2.0 * t_0;
	elseif (im <= 5e+153)
		tmp = (im ^ 4.0) * 0.041666666666666664;
	elseif (im <= 1e+178)
		tmp = (im * im) * t_0;
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.3e+14], N[Cos[re], $MachinePrecision], If[LessEqual[im, 7.4e+73], N[(-2.0 * t$95$0), $MachinePrecision], If[LessEqual[im, 5e+153], N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision], If[LessEqual[im, 1e+178], N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 3.3e14

   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 68.8%

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

   if 3.3e14 < im < 7.39999999999999947e73

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt-out 52.9%

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

      3. *-commutative 52.9%

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

      4. unpow2 52.9%

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

   6. Simplified 52.9%

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

   8. Applied egg-rr 36.4%

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

   if 7.39999999999999947e73 < im < 5.00000000000000018e153

   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%

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

   4. Simplified 100.0%

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

   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 57.9%

      \[\leadsto 0.041666666666666664 \cdot \color{blue}{{im}^{4}} \]

   if 5.00000000000000018e153 < im < 1.0000000000000001e178

   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 58.3%

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

   4. Simplified 58.3%

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

   5. Taylor expanded in im around inf 58.3%

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

      1. *-commutative 58.3%

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

      2. unpow2 58.3%

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

      3. associate-*l* 58.3%

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

      4. associate-*r* 58.3%

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

      5. *-commutative 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in re around 0 50.0%

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

      1. associate-*r* 50.0%

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

      2. distribute-rgt-out 75.0%

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

      3. unpow2 75.0%

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

      4. *-commutative 75.0%

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

      5. unpow2 75.0%

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

   10. Simplified 75.0%

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

   if 1.0000000000000001e178 < 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%

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

   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. unpow2 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 80.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.3 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 7.4 \cdot 10^{+73}:\\ \;\;\;\;-2 \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+153}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \mathbf{elif}\;im \leq 10^{+178}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 10: 63.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot im\right)\\ t_1 := re \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot -0.25\right)\right)\\ t_2 := 0.5 + \left(re \cdot re\right) \cdot -0.25\\ \mathbf{if}\;im \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{+85}:\\ \;\;\;\;-2 \cdot t_2\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_0 \cdot t_0 - t_1 \cdot t_1}{t_0 - t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 im)))
        (t_1 (* re (* (* im im) (* re -0.25))))
        (t_2 (+ 0.5 (* (* re re) -0.25))))
   (if (<= im 2.7e+14)
     (cos re)
     (if (<= im 5.3e+85)
       (* -2.0 t_2)
       (if (<= im 2e+143)
         (/ (- (* t_0 t_0) (* t_1 t_1)) (- t_0 t_1))
         (* (* im im) t_2))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double t_1 = re * ((im * im) * (re * -0.25));
	double t_2 = 0.5 + ((re * re) * -0.25);
	double tmp;
	if (im <= 2.7e+14) {
		tmp = cos(re);
	} else if (im <= 5.3e+85) {
		tmp = -2.0 * t_2;
	} else if (im <= 2e+143) {
		tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
	} else {
		tmp = (im * im) * t_2;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = im * (0.5d0 * im)
    t_1 = re * ((im * im) * (re * (-0.25d0)))
    t_2 = 0.5d0 + ((re * re) * (-0.25d0))
    if (im <= 2.7d+14) then
        tmp = cos(re)
    else if (im <= 5.3d+85) then
        tmp = (-2.0d0) * t_2
    else if (im <= 2d+143) then
        tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)
    else
        tmp = (im * im) * t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double t_1 = re * ((im * im) * (re * -0.25));
	double t_2 = 0.5 + ((re * re) * -0.25);
	double tmp;
	if (im <= 2.7e+14) {
		tmp = Math.cos(re);
	} else if (im <= 5.3e+85) {
		tmp = -2.0 * t_2;
	} else if (im <= 2e+143) {
		tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
	} else {
		tmp = (im * im) * t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * im)
	t_1 = re * ((im * im) * (re * -0.25))
	t_2 = 0.5 + ((re * re) * -0.25)
	tmp = 0
	if im <= 2.7e+14:
		tmp = math.cos(re)
	elif im <= 5.3e+85:
		tmp = -2.0 * t_2
	elif im <= 2e+143:
		tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)
	else:
		tmp = (im * im) * t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * im))
	t_1 = Float64(re * Float64(Float64(im * im) * Float64(re * -0.25)))
	t_2 = Float64(0.5 + Float64(Float64(re * re) * -0.25))
	tmp = 0.0
	if (im <= 2.7e+14)
		tmp = cos(re);
	elseif (im <= 5.3e+85)
		tmp = Float64(-2.0 * t_2);
	elseif (im <= 2e+143)
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(t_1 * t_1)) / Float64(t_0 - t_1));
	else
		tmp = Float64(Float64(im * im) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * im);
	t_1 = re * ((im * im) * (re * -0.25));
	t_2 = 0.5 + ((re * re) * -0.25);
	tmp = 0.0;
	if (im <= 2.7e+14)
		tmp = cos(re);
	elseif (im <= 5.3e+85)
		tmp = -2.0 * t_2;
	elseif (im <= 2e+143)
		tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
	else
		tmp = (im * im) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(N[(im * im), $MachinePrecision] * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 2.7e+14], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5.3e+85], N[(-2.0 * t$95$2), $MachinePrecision], If[LessEqual[im, 2e+143], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 2.7e14

   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 68.8%

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

   if 2.7e14 < im < 5.2999999999999999e85

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt-out 55.6%

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

      3. *-commutative 55.6%

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

      4. unpow2 55.6%

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

   6. Simplified 55.6%

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

   8. Applied egg-rr 34.4%

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

   if 5.2999999999999999e85 < im < 2e143

   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 6.6%

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

   4. Simplified 6.6%

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

   5. Taylor expanded in im around inf 6.6%

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

      1. *-commutative 6.6%

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

      2. unpow2 6.6%

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

      3. associate-*l* 6.6%

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

      4. associate-*r* 6.6%

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

      5. *-commutative 6.6%

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

   7. Simplified 6.6%

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

   8. Taylor expanded in re around 0 32.8%

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

      1. associate-*r* 32.8%

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

      2. distribute-rgt-out 32.8%

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

      3. unpow2 32.8%

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

      4. *-commutative 32.8%

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

      5. unpow2 32.8%

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

   10. Simplified 32.8%

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

       1. distribute-lft-in 32.8%

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

       2. flip-+ 42.9%

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

       3. associate-*l* 42.9%

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

       4. associate-*l* 42.9%

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

       5. associate-*r* 42.9%

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

       6. *-commutative 42.9%

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

       7. associate-*l* 42.9%

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

       8. associate-*r* 42.9%

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

       9. *-commutative 42.9%

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

       10. associate-*l* 42.9%

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

       11. associate-*l* 42.9%

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

       12. associate-*r* 42.9%

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

       13. *-commutative 42.9%

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

       14. associate-*l* 42.9%

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

   12. Applied egg-rr 42.9%

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

   if 2e143 < 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 81.2%

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

   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. unpow2 81.2%

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

      3. associate-*l* 81.2%

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

      4. associate-*r* 81.2%

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

      5. *-commutative 81.2%

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

   7. Simplified 81.2%

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

   8. Taylor expanded in re around 0 14.7%

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

      1. associate-*r* 14.7%

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

      2. distribute-rgt-out 68.3%

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

      3. unpow2 68.3%

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

      4. *-commutative 68.3%

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

      5. unpow2 68.3%

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

   10. Simplified 68.3%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{+85}:\\ \;\;\;\;-2 \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(im \cdot \left(0.5 \cdot im\right)\right) \cdot \left(im \cdot \left(0.5 \cdot im\right)\right) - \left(re \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot -0.25\right)\right)\right)}{im \cdot \left(0.5 \cdot im\right) - re \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot -0.25\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 11: 49.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \left(re \cdot re\right) \cdot -0.25\\ t_1 := im \cdot \left(0.5 \cdot im\right)\\ t_2 := re \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{if}\;im \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+88}:\\ \;\;\;\;-2 \cdot t_0\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+142}:\\ \;\;\;\;\frac{t_1 \cdot t_1 - t_2 \cdot t_2}{t_1 - t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* (* re re) -0.25)))
        (t_1 (* im (* 0.5 im)))
        (t_2 (* re (* (* im im) (* re -0.25)))))
   (if (<= im 1.95e+14)
     (+ 1.0 (* 0.5 (* im im)))
     (if (<= im 2.2e+88)
       (* -2.0 t_0)
       (if (<= im 1.15e+142)
         (/ (- (* t_1 t_1) (* t_2 t_2)) (- t_1 t_2))
         (* (* im im) t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + ((re * re) * -0.25);
	double t_1 = im * (0.5 * im);
	double t_2 = re * ((im * im) * (re * -0.25));
	double tmp;
	if (im <= 1.95e+14) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if (im <= 2.2e+88) {
		tmp = -2.0 * t_0;
	} else if (im <= 1.15e+142) {
		tmp = ((t_1 * t_1) - (t_2 * t_2)) / (t_1 - t_2);
	} else {
		tmp = (im * im) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 + ((re * re) * (-0.25d0))
    t_1 = im * (0.5d0 * im)
    t_2 = re * ((im * im) * (re * (-0.25d0)))
    if (im <= 1.95d+14) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else if (im <= 2.2d+88) then
        tmp = (-2.0d0) * t_0
    else if (im <= 1.15d+142) then
        tmp = ((t_1 * t_1) - (t_2 * t_2)) / (t_1 - t_2)
    else
        tmp = (im * im) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + ((re * re) * -0.25);
	double t_1 = im * (0.5 * im);
	double t_2 = re * ((im * im) * (re * -0.25));
	double tmp;
	if (im <= 1.95e+14) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if (im <= 2.2e+88) {
		tmp = -2.0 * t_0;
	} else if (im <= 1.15e+142) {
		tmp = ((t_1 * t_1) - (t_2 * t_2)) / (t_1 - t_2);
	} else {
		tmp = (im * im) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + ((re * re) * -0.25)
	t_1 = im * (0.5 * im)
	t_2 = re * ((im * im) * (re * -0.25))
	tmp = 0
	if im <= 1.95e+14:
		tmp = 1.0 + (0.5 * (im * im))
	elif im <= 2.2e+88:
		tmp = -2.0 * t_0
	elif im <= 1.15e+142:
		tmp = ((t_1 * t_1) - (t_2 * t_2)) / (t_1 - t_2)
	else:
		tmp = (im * im) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(Float64(re * re) * -0.25))
	t_1 = Float64(im * Float64(0.5 * im))
	t_2 = Float64(re * Float64(Float64(im * im) * Float64(re * -0.25)))
	tmp = 0.0
	if (im <= 1.95e+14)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	elseif (im <= 2.2e+88)
		tmp = Float64(-2.0 * t_0);
	elseif (im <= 1.15e+142)
		tmp = Float64(Float64(Float64(t_1 * t_1) - Float64(t_2 * t_2)) / Float64(t_1 - t_2));
	else
		tmp = Float64(Float64(im * im) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + ((re * re) * -0.25);
	t_1 = im * (0.5 * im);
	t_2 = re * ((im * im) * (re * -0.25));
	tmp = 0.0;
	if (im <= 1.95e+14)
		tmp = 1.0 + (0.5 * (im * im));
	elseif (im <= 2.2e+88)
		tmp = -2.0 * t_0;
	elseif (im <= 1.15e+142)
		tmp = ((t_1 * t_1) - (t_2 * t_2)) / (t_1 - t_2);
	else
		tmp = (im * im) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(re * N[(N[(im * im), $MachinePrecision] * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.95e+14], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.2e+88], N[(-2.0 * t$95$0), $MachinePrecision], If[LessEqual[im, 1.15e+142], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.95e14

   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 80.8%

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

   4. Simplified 80.8%

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

   5. Taylor expanded in re around 0 47.5%

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

      1. distribute-lft-in 47.5%

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

      2. metadata-eval 47.5%

         \[\leadsto \color{blue}{1} + 0.5 \cdot {im}^{2} \]

      3. unpow2 47.5%

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

   7. Simplified 47.5%

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

   if 1.95e14 < im < 2.20000000000000009e88

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt-out 55.6%

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

      3. *-commutative 55.6%

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

      4. unpow2 55.6%

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

   6. Simplified 55.6%

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

   8. Applied egg-rr 34.4%

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

   if 2.20000000000000009e88 < im < 1.15000000000000001e142

   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 6.6%

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

   4. Simplified 6.6%

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

   5. Taylor expanded in im around inf 6.6%

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

      1. *-commutative 6.6%

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

      2. unpow2 6.6%

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

      3. associate-*l* 6.6%

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

      4. associate-*r* 6.6%

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

      5. *-commutative 6.6%

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

   7. Simplified 6.6%

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

   8. Taylor expanded in re around 0 32.8%

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

      1. associate-*r* 32.8%

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

      2. distribute-rgt-out 32.8%

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

      3. unpow2 32.8%

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

      4. *-commutative 32.8%

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

      5. unpow2 32.8%

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

   10. Simplified 32.8%

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

       1. distribute-lft-in 32.8%

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

       2. flip-+ 42.9%

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

       3. associate-*l* 42.9%

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

       4. associate-*l* 42.9%

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

       5. associate-*r* 42.9%

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

       6. *-commutative 42.9%

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

       7. associate-*l* 42.9%

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

       8. associate-*r* 42.9%

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

       9. *-commutative 42.9%

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

       10. associate-*l* 42.9%

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

       11. associate-*l* 42.9%

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

       12. associate-*r* 42.9%

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

       13. *-commutative 42.9%

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

       14. associate-*l* 42.9%

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

   12. Applied egg-rr 42.9%

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

   if 1.15000000000000001e142 < 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 81.2%

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

   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. unpow2 81.2%

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

      3. associate-*l* 81.2%

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

      4. associate-*r* 81.2%

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

      5. *-commutative 81.2%

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

   7. Simplified 81.2%

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

   8. Taylor expanded in re around 0 14.7%

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

      1. associate-*r* 14.7%

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

      2. distribute-rgt-out 68.3%

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

      3. unpow2 68.3%

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

      4. *-commutative 68.3%

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

      5. unpow2 68.3%

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

   10. Simplified 68.3%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+88}:\\ \;\;\;\;-2 \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+142}:\\ \;\;\;\;\frac{\left(im \cdot \left(0.5 \cdot im\right)\right) \cdot \left(im \cdot \left(0.5 \cdot im\right)\right) - \left(re \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot -0.25\right)\right)\right)}{im \cdot \left(0.5 \cdot im\right) - re \cdot \left(\left(im \cdot im\right) \cdot \left(re \cdot -0.25\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 12: 48.3% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.16 \cdot 10^{+14}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+230}:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot im\right) \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.16e+14)
   (+ 1.0 (* 0.5 (* im im)))
   (if (<= re 2.1e+230)
     (* -0.25 (* (* re im) (* re im)))
     (* -2.0 (+ 0.5 (* (* re re) -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.16e+14) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if (re <= 2.1e+230) {
		tmp = -0.25 * ((re * im) * (re * im));
	} else {
		tmp = -2.0 * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.16d+14) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else if (re <= 2.1d+230) then
        tmp = (-0.25d0) * ((re * im) * (re * im))
    else
        tmp = (-2.0d0) * (0.5d0 + ((re * re) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.16e+14) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if (re <= 2.1e+230) {
		tmp = -0.25 * ((re * im) * (re * im));
	} else {
		tmp = -2.0 * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.16e+14:
		tmp = 1.0 + (0.5 * (im * im))
	elif re <= 2.1e+230:
		tmp = -0.25 * ((re * im) * (re * im))
	else:
		tmp = -2.0 * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.16e+14)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	elseif (re <= 2.1e+230)
		tmp = Float64(-0.25 * Float64(Float64(re * im) * Float64(re * im)));
	else
		tmp = Float64(-2.0 * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.16e+14)
		tmp = 1.0 + (0.5 * (im * im));
	elseif (re <= 2.1e+230)
		tmp = -0.25 * ((re * im) * (re * im));
	else
		tmp = -2.0 * (0.5 + ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.16e+14], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+230], N[(-0.25 * N[(N[(re * im), $MachinePrecision] * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.16e14

   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 71.7%

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

   4. Simplified 71.7%

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

   5. Taylor expanded in re around 0 54.4%

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

      1. distribute-lft-in 54.4%

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

      2. metadata-eval 54.4%

         \[\leadsto \color{blue}{1} + 0.5 \cdot {im}^{2} \]

      3. unpow2 54.4%

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

   7. Simplified 54.4%

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

   if 1.16e14 < re < 2.09999999999999993e230

   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 69.9%

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

   4. Simplified 69.9%

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

   5. Taylor expanded in im around inf 20.9%

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

      1. *-commutative 20.9%

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

      2. unpow2 20.9%

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

      3. associate-*l* 20.9%

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

      4. associate-*r* 20.9%

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

      5. *-commutative 20.9%

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

   7. Simplified 20.9%

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

   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 27.9%

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

      3. unpow2 27.9%

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

      4. *-commutative 27.9%

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

      5. unpow2 27.9%

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

   10. Simplified 27.9%

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

   11. Taylor expanded in re around inf 27.9%

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

       1. unpow2 27.9%

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

       2. unpow2 27.9%

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

       3. unswap-sqr 28.6%

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

   13. Simplified 28.6%

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

   if 2.09999999999999993e230 < re

   1. Initial program 100.0%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in re around 0 0.9%

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

      1. associate-*r* 0.9%

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

      2. distribute-rgt-out 15.2%

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

      3. *-commutative 15.2%

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

      4. unpow2 15.2%

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

   6. Simplified 15.2%

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

   8. Applied egg-rr 29.8%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.16 \cdot 10^{+14}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+230}:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot im\right) \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 13: 40.5% accurate, 23.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.6e-8) 1.0 (* (* im im) (+ 0.5 (* (* re re) -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.6e-8) {
		tmp = 1.0;
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.6d-8) then
        tmp = 1.0d0
    else
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.6e-8) {
		tmp = 1.0;
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.6e-8:
		tmp = 1.0
	else:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.6e-8)
		tmp = 1.0;
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.6e-8)
		tmp = 1.0;
	else
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.6e-8], 1.0, N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.6000000000000002e-8

   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 69.2%

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

   3. Taylor expanded in re around 0 40.4%

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

   if 4.6000000000000002e-8 < 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 41.3%

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

   4. Simplified 41.3%

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

   5. Taylor expanded in im around inf 38.5%

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

      1. *-commutative 38.5%

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

      2. unpow2 38.5%

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

      3. associate-*l* 38.5%

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

      4. associate-*r* 38.5%

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

      5. *-commutative 38.5%

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

   7. Simplified 38.5%

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

   8. Taylor expanded in re around 0 19.0%

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

      1. associate-*r* 19.0%

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

      2. distribute-rgt-out 42.4%

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

      3. unpow2 42.4%

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

      4. *-commutative 42.4%

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

      5. unpow2 42.4%

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

   10. Simplified 42.4%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 14: 39.1% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+163}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.6e-8)
   1.0
   (if (<= im 5.4e+163) (+ 1.0 (* (* re re) -0.5)) (* im (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.6e-8) {
		tmp = 1.0;
	} else if (im <= 5.4e+163) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.6d-8) then
        tmp = 1.0d0
    else if (im <= 5.4d+163) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.6e-8) {
		tmp = 1.0;
	} else if (im <= 5.4e+163) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.6e-8:
		tmp = 1.0
	elif im <= 5.4e+163:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.6e-8)
		tmp = 1.0;
	elseif (im <= 5.4e+163)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.6e-8)
		tmp = 1.0;
	elseif (im <= 5.4e+163)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.6e-8], 1.0, If[LessEqual[im, 5.4e+163], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.6000000000000002e-8

   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 69.2%

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

   3. Taylor expanded in re around 0 40.4%

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

   if 4.6000000000000002e-8 < im < 5.39999999999999998e163

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   5. Applied egg-rr 7.4%

      \[\leadsto \color{blue}{\log \left(e^{\cos re}\right)} \]

   6. Taylor expanded in re around 0 14.5%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 14.5%

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

   8. Simplified 14.5%

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

   if 5.39999999999999998e163 < 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%

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

   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. unpow2 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 77.3%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+163}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 15: 48.2% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.05e+153)
   (+ 1.0 (* 0.5 (* im im)))
   (* -2.0 (+ 0.5 (* (* re re) -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.05e+153) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = -2.0 * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.05d+153) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = (-2.0d0) * (0.5d0 + ((re * re) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.05e+153) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = -2.0 * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.05e+153:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = -2.0 * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.05e+153)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(-2.0 * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.05e+153)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = -2.0 * (0.5 + ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.05e+153], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.05000000000000008e153

   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 72.7%

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

   4. Simplified 72.7%

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

   5. Taylor expanded in re around 0 48.6%

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

      1. distribute-lft-in 48.6%

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

      2. metadata-eval 48.6%

         \[\leadsto \color{blue}{1} + 0.5 \cdot {im}^{2} \]

      3. unpow2 48.6%

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

   7. Simplified 48.6%

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

   if 1.05000000000000008e153 < re

   1. Initial program 100.0%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in re around 0 1.0%

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

      1. associate-*r* 1.0%

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

      2. distribute-rgt-out 25.3%

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

      3. *-commutative 25.3%

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

      4. unpow2 25.3%

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

   6. Simplified 25.3%

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

   8. Applied egg-rr 25.2%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 16: 38.2% accurate, 43.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 4.6e-8) 1.0 (* im (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.6e-8) {
		tmp = 1.0;
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.6d-8) then
        tmp = 1.0d0
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.6e-8) {
		tmp = 1.0;
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.6e-8:
		tmp = 1.0
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.6e-8)
		tmp = 1.0;
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.6e-8)
		tmp = 1.0;
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.6e-8], 1.0, N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.6000000000000002e-8

   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 69.2%

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

   3. Taylor expanded in re around 0 40.4%

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

   if 4.6000000000000002e-8 < 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 41.3%

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

   4. Simplified 41.3%

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

   5. Taylor expanded in im around inf 38.5%

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

      1. *-commutative 38.5%

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

      2. unpow2 38.5%

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

      3. associate-*l* 38.5%

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

      4. associate-*r* 38.5%

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

      5. *-commutative 38.5%

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

   7. Simplified 38.5%

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

   8. Taylor expanded in re around 0 28.8%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 17: 29.1% 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 im around 0 53.4%

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

3. Taylor expanded in re around 0 30.9%

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

4. Final simplification 30.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023243 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))