math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 15

Speedup: 1.0×

• 49.8% 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: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.035:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \sqrt[3]{0.125}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.035)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 1.15e+77)
     (* (+ (exp (- im)) (exp im)) (cbrt 0.125))
     (* (cos re) (* (pow im 4.0) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.035) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = (exp(-im) + exp(im)) * cbrt(0.125);
	} else {
		tmp = cos(re) * (pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.035) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = (Math.exp(-im) + Math.exp(im)) * Math.cbrt(0.125);
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.035)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.15e+77)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * cbrt(0.125));
	else
		tmp = Float64(cos(re) * Float64((im ^ 4.0) * 0.041666666666666664));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.035], 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[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[Power[0.125, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.035000000000000003

   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 88.4%

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

      1. unpow2 88.4%

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

   4. Simplified 88.4%

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

   if 0.035000000000000003 < 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 0.1%

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

      1. associate-*r* 0.1%

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

      2. distribute-rgt-out 61.2%

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

      3. unpow2 61.2%

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

   4. Simplified 61.2%

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

      1. add-cbrt-cube 61.1%

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

      2. pow3 61.1%

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

      3. +-commutative 61.1%

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

      4. fma-def 61.1%

         \[\leadsto \left(e^{im} + e^{-im}\right) \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\right)}}^{3}} \]

   6. Applied egg-rr 61.1%

      \[\leadsto \left(e^{im} + e^{-im}\right) \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\right)}^{3}}} \]

   7. Taylor expanded in re around 0 73.1%

      \[\leadsto \left(e^{im} + e^{-im}\right) \cdot \sqrt[3]{\color{blue}{0.125}} \]

   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%

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

      1. associate-+r+ 100.0%

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

      2. unpow2 100.0%

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

   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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.035:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \sqrt[3]{0.125}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 3: 81.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.982:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.982)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (+ 1.0 (* 0.5 (+ (* im im) (* (pow im 4.0) 0.08333333333333333))))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.982) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else {
		tmp = 1.0 + (0.5 * ((im * im) + (pow(im, 4.0) * 0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= 0.982d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else
        tmp = 1.0d0 + (0.5d0 * ((im * im) + ((im ** 4.0d0) * 0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.982) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else {
		tmp = 1.0 + (0.5 * ((im * im) + (Math.pow(im, 4.0) * 0.08333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.982:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	else:
		tmp = 1.0 + (0.5 * ((im * im) + (math.pow(im, 4.0) * 0.08333333333333333)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.982)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(1.0 + Float64(0.5 * Float64(Float64(im * im) + Float64((im ^ 4.0) * 0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.982)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	else
		tmp = 1.0 + (0.5 * ((im * im) + ((im ^ 4.0) * 0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.982], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[(N[(im * im), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.98199999999999998

   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 83.1%

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

      1. unpow2 83.1%

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

   4. Simplified 83.1%

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

   if 0.98199999999999998 < (cos.f64 re)

   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 88.1%

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

      1. associate-+r+ 88.1%

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

      2. unpow2 88.1%

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

   4. Simplified 88.1%

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

   5. Taylor expanded in re around 0 87.5%

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

      1. distribute-lft-in 87.5%

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

      2. metadata-eval 87.5%

         \[\leadsto \color{blue}{1} + 0.5 \cdot \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right) \]

      3. unpow2 87.5%

         \[\leadsto 1 + 0.5 \cdot \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right) \]

   7. Simplified 87.5%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(im \cdot im + 0.08333333333333333 \cdot {im}^{4}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.982:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\\ \end{array} \]

Alternative 4: 82.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 560:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.16 \cdot 10^{+64}:\\ \;\;\;\;\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 560.0)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 2.16e+64)
     (*
      (+ 2.0 (+ (* im im) (* (pow im 4.0) 0.08333333333333333)))
      (+ 0.5 (* -0.25 (* re re))))
     (* (cos re) (* (pow im 4.0) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 560.0) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 2.16e+64) {
		tmp = (2.0 + ((im * im) + (pow(im, 4.0) * 0.08333333333333333))) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = cos(re) * (pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 560.0d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 2.16d+64) then
        tmp = (2.0d0 + ((im * im) + ((im ** 4.0d0) * 0.08333333333333333d0))) * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = cos(re) * ((im ** 4.0d0) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 560.0) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 2.16e+64) {
		tmp = (2.0 + ((im * im) + (Math.pow(im, 4.0) * 0.08333333333333333))) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 560.0:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 2.16e+64:
		tmp = (2.0 + ((im * im) + (math.pow(im, 4.0) * 0.08333333333333333))) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = math.cos(re) * (math.pow(im, 4.0) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 560.0)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 2.16e+64)
		tmp = Float64(Float64(2.0 + Float64(Float64(im * im) + Float64((im ^ 4.0) * 0.08333333333333333))) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(cos(re) * Float64((im ^ 4.0) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 560.0)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 2.16e+64)
		tmp = (2.0 + ((im * im) + ((im ^ 4.0) * 0.08333333333333333))) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = cos(re) * ((im ^ 4.0) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 560.0], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.16e+64], N[(N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 560

   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 88.0%

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

      1. unpow2 88.0%

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

   4. Simplified 88.0%

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

   if 560 < im < 2.16000000000000014e64

   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 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} + e^{-im}\right) + -0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   3. 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 64.3%

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

      3. unpow2 64.3%

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

   4. Simplified 64.3%

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

   5. Taylor expanded in im around 0 30.4%

      \[\leadsto \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 30.4%

         \[\leadsto \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]

   7. Simplified 30.4%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]

   if 2.16000000000000014e64 < 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 94.3%

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

      1. associate-+r+ 94.3%

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

      2. unpow2 94.3%

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

   4. Simplified 94.3%

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

   5. Taylor expanded in im around inf 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-*l* 94.3%

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

   7. Simplified 94.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 560:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.16 \cdot 10^{+64}:\\ \;\;\;\;\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 5: 86.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.046:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.046)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 1.15e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* (pow im 4.0) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.046) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.046d0) 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 = cos(re) * ((im ** 4.0d0) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.046) {
		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 = Math.cos(re) * (Math.pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.046:
		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 = math.cos(re) * (math.pow(im, 4.0) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.046)
		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(cos(re) * Float64((im ^ 4.0) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.046)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 1.15e+77)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * ((im ^ 4.0) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.046], 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[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.045999999999999999

   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 88.4%

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

      1. unpow2 88.4%

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

   4. Simplified 88.4%

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

   if 0.045999999999999999 < 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 73.1%

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

      1. *-commutative 73.1%

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

   4. Simplified 73.1%

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

   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%

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

      1. associate-+r+ 100.0%

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

      2. unpow2 100.0%

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

   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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.046:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 6: 81.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+154}:\\ \;\;\;\;\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 520.0)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (if (<= im 2.45e+154)
     (*
      (+ 2.0 (+ (* im im) (* (pow im 4.0) 0.08333333333333333)))
      (+ 0.5 (* -0.25 (* re re))))
     (* im (* 0.5 (* (cos re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 520.0) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else if (im <= 2.45e+154) {
		tmp = (2.0 + ((im * im) + (pow(im, 4.0) * 0.08333333333333333))) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = im * (0.5 * (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 <= 520.0d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else if (im <= 2.45d+154) then
        tmp = (2.0d0 + ((im * im) + ((im ** 4.0d0) * 0.08333333333333333d0))) * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = im * (0.5d0 * (cos(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 520.0) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else if (im <= 2.45e+154) {
		tmp = (2.0 + ((im * im) + (Math.pow(im, 4.0) * 0.08333333333333333))) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = im * (0.5 * (Math.cos(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 520.0:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	elif im <= 2.45e+154:
		tmp = (2.0 + ((im * im) + (math.pow(im, 4.0) * 0.08333333333333333))) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = im * (0.5 * (math.cos(re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 520.0)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 2.45e+154)
		tmp = Float64(Float64(2.0 + Float64(Float64(im * im) + Float64((im ^ 4.0) * 0.08333333333333333))) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 520.0)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	elseif (im <= 2.45e+154)
		tmp = (2.0 + ((im * im) + ((im ^ 4.0) * 0.08333333333333333))) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = im * (0.5 * (cos(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 520.0], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.45e+154], N[(N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 520

   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 88.0%

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

      1. unpow2 88.0%

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

   4. Simplified 88.0%

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

   if 520 < im < 2.4500000000000001e154

   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 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} + e^{-im}\right) + -0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   3. 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 64.3%

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

      3. unpow2 64.3%

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

   4. Simplified 64.3%

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

   5. Taylor expanded in im around 0 40.9%

      \[\leadsto \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 40.9%

         \[\leadsto \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]

   7. Simplified 40.9%

      \[\leadsto \color{blue}{\left(2 + \left(im \cdot im + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]

   if 2.4500000000000001e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

      6. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+154}:\\ \;\;\;\;\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 65.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.7 \cdot 10^{+138}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 420.0)
   (cos re)
   (if (<= im 5.7e+138)
     (* (+ 2.0 (* im im)) (+ 0.5 (* -0.25 (* re re))))
     (* im (* 0.5 (* (cos re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = cos(re);
	} else if (im <= 5.7e+138) {
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = im * (0.5 * (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 <= 420.0d0) then
        tmp = cos(re)
    else if (im <= 5.7d+138) then
        tmp = (2.0d0 + (im * im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = im * (0.5d0 * (cos(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = Math.cos(re);
	} else if (im <= 5.7e+138) {
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = im * (0.5 * (Math.cos(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 420.0:
		tmp = math.cos(re)
	elif im <= 5.7e+138:
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = im * (0.5 * (math.cos(re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 420.0)
		tmp = cos(re);
	elseif (im <= 5.7e+138)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 420.0)
		tmp = cos(re);
	elseif (im <= 5.7e+138)
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = im * (0.5 * (cos(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 420.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5.7e+138], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 420

   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 73.5%

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

   if 420 < im < 5.69999999999999986e138

   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 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} + e^{-im}\right) + -0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   3. 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 68.0%

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

      3. unpow2 68.0%

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

   4. Simplified 68.0%

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

   5. Taylor expanded in im around 0 15.0%

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

      1. unpow2 15.0%

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

   7. Simplified 15.0%

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

   if 5.69999999999999986e138 < 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 92.7%

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

      1. unpow2 92.7%

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

   4. Simplified 92.7%

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

   5. Taylor expanded in im around inf 92.7%

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

      1. associate-*r* 92.7%

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

      2. unpow2 92.7%

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

      3. associate-*r* 92.7%

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

      4. *-commutative 92.7%

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

      5. associate-*l* 92.7%

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

      6. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.7 \cdot 10^{+138}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 75.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 (cos re)) (+ 2.0 (* im im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (2.0 + (im * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (2.0 + (im * im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (2.0 + (im * im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (2.0 + (im * im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * 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. Taylor expanded in im around 0 80.5%

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

   1. unpow2 80.5%

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

4. Simplified 80.5%

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

5. Final simplification 80.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right) \]

Alternative 9: 62.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 480.0) (cos re) (* (+ 2.0 (* im im)) (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 480.0) {
		tmp = cos(re);
	} else {
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 480.0d0) then
        tmp = cos(re)
    else
        tmp = (2.0d0 + (im * im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 480.0) {
		tmp = Math.cos(re);
	} else {
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 480.0:
		tmp = math.cos(re)
	else:
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 480.0)
		tmp = cos(re);
	else
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 480.0)
		tmp = cos(re);
	else
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 480.0], N[Cos[re], $MachinePrecision], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 480

   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 73.5%

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

   if 480 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} + e^{-im}\right) + -0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   3. 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 71.0%

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

      3. unpow2 71.0%

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

   4. Simplified 71.0%

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

   5. Taylor expanded in im around 0 48.2%

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

      1. unpow2 48.2%

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

   7. Simplified 48.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 10: 49.5% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 170:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 170.0)
   (+ 1.0 (* 0.5 (* im im)))
   (* (+ 2.0 (* im im)) (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 170.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 170.0d0) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = (2.0d0 + (im * im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 170.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 170.0:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 170.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 170.0)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 170.0], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 170

   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 88.0%

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

      1. unpow2 88.0%

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

   4. Simplified 88.0%

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

   5. Taylor expanded in re around 0 50.9%

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

      1. distribute-lft-in 50.9%

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

      2. metadata-eval 50.9%

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

      3. unpow2 50.9%

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

   7. Simplified 50.9%

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

   if 170 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} + e^{-im}\right) + -0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   3. 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 71.0%

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

      3. unpow2 71.0%

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

   4. Simplified 71.0%

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

   5. Taylor expanded in im around 0 48.2%

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

      1. unpow2 48.2%

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

   7. Simplified 48.2%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 170:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 11: 47.5% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+134}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9e+134)
   (+ 1.0 (* 0.5 (* im im)))
   (* 2.0 (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9e+134) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 2.0 * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 9d+134) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = 2.0d0 * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9e+134) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 2.0 * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9e+134:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = 2.0 * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9e+134)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(2.0 * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9e+134)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = 2.0 * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9e+134], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.9999999999999995e134

   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 79.6%

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

      1. unpow2 79.6%

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

   4. Simplified 79.6%

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

   5. Taylor expanded in re around 0 52.7%

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

      1. distribute-lft-in 52.7%

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

      2. metadata-eval 52.7%

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

      3. unpow2 52.7%

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

   7. Simplified 52.7%

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

   if 8.9999999999999995e134 < re

   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 1.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} + e^{-im}\right) + -0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   3. 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 29.6%

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

      3. unpow2 29.6%

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

   4. Simplified 29.6%

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

   5. Taylor expanded in im around 0 24.4%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+134}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 12: 47.6% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+134}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot 262144\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9e+134)
   (+ 1.0 (* 0.5 (* im im)))
   (* (+ 0.5 (* -0.25 (* re re))) 262144.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9e+134) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * 262144.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 9d+134) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = (0.5d0 + ((-0.25d0) * (re * re))) * 262144.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9e+134) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * 262144.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9e+134:
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = (0.5 + (-0.25 * (re * re))) * 262144.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9e+134)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(Float64(0.5 + Float64(-0.25 * Float64(re * re))) * 262144.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9e+134)
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = (0.5 + (-0.25 * (re * re))) * 262144.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9e+134], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 262144.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.9999999999999995e134

   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 79.6%

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

      1. unpow2 79.6%

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

   4. Simplified 79.6%

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

   5. Taylor expanded in re around 0 52.7%

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

      1. distribute-lft-in 52.7%

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

      2. metadata-eval 52.7%

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

      3. unpow2 52.7%

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

   7. Simplified 52.7%

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

   if 8.9999999999999995e134 < re

   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 1.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} + e^{-im}\right) + -0.25 \cdot \left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right)} \]
   3. 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 29.6%

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

      3. unpow2 29.6%

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

   4. Simplified 29.6%

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

   6. Applied egg-rr 24.5%

      \[\leadsto \color{blue}{262144} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+134}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot 262144\\ \end{array} \]

Alternative 13: 37.7% accurate, 43.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1.42) 1.0 (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.42) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.42d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.42) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.42:
		tmp = 1.0
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.42)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.42)
		tmp = 1.0;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.42], 1.0, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.4199999999999999

   1. Initial program 100.0%

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

   3. Applied egg-rr 40.8%

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

      1. +-inverses 40.8%

         \[\leadsto \frac{-2 \cdot \cos re}{-2 \cdot \cos re + \color{blue}{0}} \]

      2. +-rgt-identity 40.8%

         \[\leadsto \frac{-2 \cdot \cos re}{\color{blue}{-2 \cdot \cos re}} \]

      3. *-inverses 40.8%

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

   5. Simplified 40.8%

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

   if 1.4199999999999999 < 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 56.5%

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

      1. unpow2 56.5%

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

   4. Simplified 56.5%

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

   5. Taylor expanded in im around inf 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-*l* 56.5%

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

      3. *-commutative 56.5%

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

      4. unpow2 56.5%

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

   7. Simplified 56.5%

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

   8. Taylor expanded in re around 0 37.0%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 37.0%

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

   10. Simplified 37.0%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 14: 47.0% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(im \cdot im\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return 1.0 + (0.5 * (im * im));
}

Python

def code(re, im):
	return 1.0 + (0.5 * (im * im))

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 + N[(0.5 * N[(im * 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. Taylor expanded in im around 0 80.5%

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

   1. unpow2 80.5%

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

4. Simplified 80.5%

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

5. Taylor expanded in re around 0 47.6%

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

   1. distribute-lft-in 47.6%

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

   2. metadata-eval 47.6%

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

   3. unpow2 47.6%

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

7. Simplified 47.6%

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

8. Final simplification 47.6%

   \[\leadsto 1 + 0.5 \cdot \left(im \cdot im\right) \]

Alternative 15: 28.3% 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%

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

3. Applied egg-rr 31.4%

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

   1. +-inverses 31.4%

      \[\leadsto \frac{-2 \cdot \cos re}{-2 \cdot \cos re + \color{blue}{0}} \]

   2. +-rgt-identity 31.4%

      \[\leadsto \frac{-2 \cdot \cos re}{\color{blue}{-2 \cdot \cos re}} \]

   3. *-inverses 31.4%

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

5. Simplified 31.4%

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

6. Final simplification 31.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023278 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))