math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 18.2s

Alternatives: 16

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.5% 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: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.996:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + 0.08333333333333333 \cdot {im}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.996)
   (*
    (* 0.5 (cos re))
    (+ 2.0 (+ (* im im) (* 0.08333333333333333 (pow im 4.0)))))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.996) {
		tmp = (0.5 * cos(re)) * (2.0 + ((im * im) + (0.08333333333333333 * pow(im, 4.0))));
	} else {
		tmp = 0.5 * (exp(-im) + exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= 0.996d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + ((im * im) + (0.08333333333333333d0 * (im ** 4.0d0))))
    else
        tmp = 0.5d0 * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.996) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + ((im * im) + (0.08333333333333333 * Math.pow(im, 4.0))));
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.996:
		tmp = (0.5 * math.cos(re)) * (2.0 + ((im * im) + (0.08333333333333333 * math.pow(im, 4.0))))
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.996)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(Float64(im * im) + Float64(0.08333333333333333 * (im ^ 4.0)))));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.996)
		tmp = (0.5 * cos(re)) * (2.0 + ((im * im) + (0.08333333333333333 * (im ^ 4.0))));
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.996], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.996

   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 89.6%

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

      1. unpow2 89.6%

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

   4. Simplified 89.6%

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

   if 0.996 < (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 re around 0 99.0%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.996:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + 0.08333333333333333 \cdot {im}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 3: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.996:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.996)
   (*
    (* 0.5 (cos re))
    (+ 2.0 (* (* im im) (+ 1.0 (* im (* im 0.08333333333333333))))))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.996) {
		tmp = (0.5 * cos(re)) * (2.0 + ((im * im) * (1.0 + (im * (im * 0.08333333333333333)))));
	} else {
		tmp = 0.5 * (exp(-im) + exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= 0.996d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + ((im * im) * (1.0d0 + (im * (im * 0.08333333333333333d0)))))
    else
        tmp = 0.5d0 * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.996) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + ((im * im) * (1.0 + (im * (im * 0.08333333333333333)))));
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.996:
		tmp = (0.5 * math.cos(re)) * (2.0 + ((im * im) * (1.0 + (im * (im * 0.08333333333333333)))))
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.996)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(Float64(im * im) * Float64(1.0 + Float64(im * Float64(im * 0.08333333333333333))))));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.996)
		tmp = (0.5 * cos(re)) * (2.0 + ((im * im) * (1.0 + (im * (im * 0.08333333333333333)))));
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.996], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(im * im), $MachinePrecision] * N[(1.0 + N[(im * N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.996

   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 89.6%

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

      1. unpow2 89.6%

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

   4. Simplified 89.6%

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

      1. +-commutative 89.6%

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

      2. *-commutative 89.6%

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

      3. metadata-eval 89.6%

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

      4. pow-sqr 89.6%

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

      5. pow-prod-down 89.6%

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

      6. pow2 89.6%

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

      7. associate-*l* 89.0%

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

      8. fma-def 89.0%

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

   6. Applied egg-rr 89.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{\mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im\right)}\right) \]
   7. Step-by-step derivation

      1. fma-udef 89.0%

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

      2. *-commutative 89.0%

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

      3. *-commutative 89.0%

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

      4. distribute-lft1-in 89.0%

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

      5. *-commutative 89.0%

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

      6. associate-*l* 89.0%

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

   8. Simplified 89.0%

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

   if 0.996 < (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 re around 0 99.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.996:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 4: 81.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.999999:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + \left(im \cdot im + 0.08333333333333333 \cdot {im}^{4}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.999999)
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (* 0.5 (+ 2.0 (+ (* im im) (* 0.08333333333333333 (pow im 4.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.999999) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * (2.0 + ((im * im) + (0.08333333333333333 * 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 (cos(re) <= 0.999999d0) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else
        tmp = 0.5d0 * (2.0d0 + ((im * im) + (0.08333333333333333d0 * (im ** 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.999999) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * (2.0 + ((im * im) + (0.08333333333333333 * Math.pow(im, 4.0))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.999999:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	else:
		tmp = 0.5 * (2.0 + ((im * im) + (0.08333333333333333 * math.pow(im, 4.0))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.999999)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.5 * Float64(2.0 + Float64(Float64(im * im) + Float64(0.08333333333333333 * (im ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.999999)
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	else
		tmp = 0.5 * (2.0 + ((im * im) + (0.08333333333333333 * (im ^ 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.999999], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.999998999999999971

   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 79.1%

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

      1. unpow2 79.1%

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

   4. Simplified 79.1%

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

   if 0.999998999999999971 < (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 re around 0 100.0%

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

   3. Taylor expanded in im around 0 85.2%

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

      1. unpow2 85.2%

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

   5. Simplified 85.2%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.999999:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + \left(im \cdot im + 0.08333333333333333 \cdot {im}^{4}\right)\right)\\ \end{array} \]

Alternative 5: 87.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (* 0.5 (cos re))
  (+ 2.0 (* (* im im) (+ 1.0 (* im (* im 0.08333333333333333)))))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (2.0 + ((im * im) * (1.0 + (im * (im * 0.08333333333333333)))));
}

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) * (1.0d0 + (im * (im * 0.08333333333333333d0)))))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (2.0 + ((im * im) * (1.0 + (im * (im * 0.08333333333333333)))));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (2.0 + ((im * im) * (1.0 + (im * (im * 0.08333333333333333)))))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(Float64(im * im) * Float64(1.0 + Float64(im * Float64(im * 0.08333333333333333))))))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (2.0 + ((im * im) * (1.0 + (im * (im * 0.08333333333333333)))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 87.2%

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

   1. unpow2 87.2%

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

4. Simplified 87.2%

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

   1. +-commutative 87.2%

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

   2. *-commutative 87.2%

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

   3. metadata-eval 87.2%

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

   4. pow-sqr 87.2%

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

   5. pow-prod-down 87.2%

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

   6. pow2 87.2%

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

   7. associate-*l* 86.9%

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

   8. fma-def 86.9%

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

6. Applied egg-rr 86.9%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{\mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im\right)}\right) \]
7. Step-by-step derivation

   1. fma-udef 86.9%

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

   2. *-commutative 86.9%

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

   3. *-commutative 86.9%

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

   4. distribute-lft1-in 86.9%

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

   5. *-commutative 86.9%

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

   6. associate-*l* 86.9%

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

8. Simplified 86.9%

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

9. Final simplification 86.9%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im\right) \cdot \left(1 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\right) \]

Alternative 6: 77.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;im \leq -1.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* 0.5 (* im im)))))
   (if (<= im -1.45)
     t_0
     (if (<= im 700.0)
       (cos re)
       (if (<= im 1.35e+154) (* 2.0 (* re (* re (fma im im 2.0)))) t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -1.45) {
		tmp = t_0;
	} else if (im <= 700.0) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 2.0 * (re * (re * fma(im, im, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(0.5 * Float64(im * im)))
	tmp = 0.0
	if (im <= -1.45)
		tmp = t_0;
	elseif (im <= 700.0)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(2.0 * Float64(re * Float64(re * fma(im, im, 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.45], t$95$0, If[LessEqual[im, 700.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(2.0 * N[(re * N[(re * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.44999999999999996 or 1.35000000000000003e154 < 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 67.0%

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

      1. unpow2 67.0%

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

   4. Simplified 67.0%

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

   5. Taylor expanded in im around inf 67.0%

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

      1. unpow2 67.0%

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

      2. associate-*r* 67.0%

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

      3. *-commutative 67.0%

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

      4. associate-*r* 67.0%

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

   7. Simplified 67.0%

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

   if -1.44999999999999996 < im < 700

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 99.5%

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

      1. unpow2 99.5%

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

   4. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. metadata-eval 99.5%

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

      4. pow-sqr 99.5%

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

      5. pow-prod-down 99.5%

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

      6. pow2 99.5%

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

      7. associate-*l* 99.5%

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

      8. fma-def 99.5%

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

   6. Applied egg-rr 99.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{\mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im\right)}\right) \]
   7. Step-by-step derivation

      1. fma-udef 99.5%

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

      2. *-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-lft1-in 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*l* 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in im around 0 98.3%

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

   if 700 < im < 1.35000000000000003e154

   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 4.9%

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

      1. unpow2 4.9%

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

   4. Simplified 4.9%

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

   6. Applied egg-rr 1.4%

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

      1. *-commutative 1.4%

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

   8. Simplified 1.4%

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

   9. Taylor expanded in re around 0 35.2%

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

       1. *-commutative 35.2%

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

       2. unpow2 35.2%

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

   11. Simplified 35.2%

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

   12. Taylor expanded in re around inf 36.1%

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

       1. unpow2 36.1%

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

       2. *-commutative 36.1%

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

       3. +-commutative 36.1%

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

       4. unpow2 36.1%

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

       5. fma-udef 36.1%

          \[\leadsto 2 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)}\right) \]

       6. associate-*l* 36.1%

          \[\leadsto 2 \cdot \color{blue}{\left(re \cdot \left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right)\right)} \]

   14. Simplified 36.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.45:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(im, im, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 77.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;im \leq -1.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 720:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.05 \cdot 10^{+153}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot \left(re \cdot \left(re \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* 0.5 (* im im)))))
   (if (<= im -1.45)
     t_0
     (if (<= im 720.0)
       (cos re)
       (if (<= im 3.05e+153)
         (* (+ 2.0 (* im im)) (* 0.5 (* re (* re 4.0))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -1.45) {
		tmp = t_0;
	} else if (im <= 720.0) {
		tmp = cos(re);
	} else if (im <= 3.05e+153) {
		tmp = (2.0 + (im * im)) * (0.5 * (re * (re * 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(re) * (0.5d0 * (im * im))
    if (im <= (-1.45d0)) then
        tmp = t_0
    else if (im <= 720.0d0) then
        tmp = cos(re)
    else if (im <= 3.05d+153) then
        tmp = (2.0d0 + (im * im)) * (0.5d0 * (re * (re * 4.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -1.45) {
		tmp = t_0;
	} else if (im <= 720.0) {
		tmp = Math.cos(re);
	} else if (im <= 3.05e+153) {
		tmp = (2.0 + (im * im)) * (0.5 * (re * (re * 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (0.5 * (im * im))
	tmp = 0
	if im <= -1.45:
		tmp = t_0
	elif im <= 720.0:
		tmp = math.cos(re)
	elif im <= 3.05e+153:
		tmp = (2.0 + (im * im)) * (0.5 * (re * (re * 4.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(0.5 * Float64(im * im)))
	tmp = 0.0
	if (im <= -1.45)
		tmp = t_0;
	elseif (im <= 720.0)
		tmp = cos(re);
	elseif (im <= 3.05e+153)
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 * Float64(re * Float64(re * 4.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (0.5 * (im * im));
	tmp = 0.0;
	if (im <= -1.45)
		tmp = t_0;
	elseif (im <= 720.0)
		tmp = cos(re);
	elseif (im <= 3.05e+153)
		tmp = (2.0 + (im * im)) * (0.5 * (re * (re * 4.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.45], t$95$0, If[LessEqual[im, 720.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.05e+153], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(re * N[(re * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.44999999999999996 or 3.0499999999999999e153 < 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 67.0%

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

      1. unpow2 67.0%

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

   4. Simplified 67.0%

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

   5. Taylor expanded in im around inf 67.0%

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

      1. unpow2 67.0%

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

      2. associate-*r* 67.0%

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

      3. *-commutative 67.0%

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

      4. associate-*r* 67.0%

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

   7. Simplified 67.0%

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

   if -1.44999999999999996 < im < 720

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 99.5%

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

      1. unpow2 99.5%

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

   4. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. metadata-eval 99.5%

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

      4. pow-sqr 99.5%

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

      5. pow-prod-down 99.5%

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

      6. pow2 99.5%

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

      7. associate-*l* 99.5%

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

      8. fma-def 99.5%

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

   6. Applied egg-rr 99.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{\mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im\right)}\right) \]
   7. Step-by-step derivation

      1. fma-udef 99.5%

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

      2. *-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-lft1-in 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*l* 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in im around 0 98.3%

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

   if 720 < im < 3.0499999999999999e153

   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 4.9%

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

      1. unpow2 4.9%

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

   4. Simplified 4.9%

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

   6. Applied egg-rr 1.4%

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

      1. *-commutative 1.4%

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

   8. Simplified 1.4%

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

   9. Taylor expanded in re around 0 35.2%

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

       1. *-commutative 35.2%

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

       2. unpow2 35.2%

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

   11. Simplified 35.2%

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

   12. Taylor expanded in re around inf 36.1%

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

       1. unpow2 36.1%

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

       2. *-commutative 36.1%

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

       3. associate-*l* 36.1%

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

   14. Simplified 36.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.45:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 720:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.05 \cdot 10^{+153}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot \left(re \cdot \left(re \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 76.2% 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 78.2%

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

   1. unpow2 78.2%

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

4. Simplified 78.2%

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

5. Final simplification 78.2%

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

Alternative 9: 73.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ t_1 := t_0 \cdot \left(0.5 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{if}\;im \leq -1 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1100:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+83}:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \left(re \cdot \left(re \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im)))
        (t_1 (* t_0 (* 0.5 (+ 1.0 (* -0.5 (* re re)))))))
   (if (<= im -1e+24)
     t_1
     (if (<= im 1100.0)
       (cos re)
       (if (<= im 1.2e+83) (* t_0 (* 0.5 (* re (* re 4.0)))) t_1)))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))));
	double tmp;
	if (im <= -1e+24) {
		tmp = t_1;
	} else if (im <= 1100.0) {
		tmp = cos(re);
	} else if (im <= 1.2e+83) {
		tmp = t_0 * (0.5 * (re * (re * 4.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (im * im)
    t_1 = t_0 * (0.5d0 * (1.0d0 + ((-0.5d0) * (re * re))))
    if (im <= (-1d+24)) then
        tmp = t_1
    else if (im <= 1100.0d0) then
        tmp = cos(re)
    else if (im <= 1.2d+83) then
        tmp = t_0 * (0.5d0 * (re * (re * 4.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))));
	double tmp;
	if (im <= -1e+24) {
		tmp = t_1;
	} else if (im <= 1100.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.2e+83) {
		tmp = t_0 * (0.5 * (re * (re * 4.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	t_1 = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))))
	tmp = 0
	if im <= -1e+24:
		tmp = t_1
	elif im <= 1100.0:
		tmp = math.cos(re)
	elif im <= 1.2e+83:
		tmp = t_0 * (0.5 * (re * (re * 4.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	t_1 = Float64(t_0 * Float64(0.5 * Float64(1.0 + Float64(-0.5 * Float64(re * re)))))
	tmp = 0.0
	if (im <= -1e+24)
		tmp = t_1;
	elseif (im <= 1100.0)
		tmp = cos(re);
	elseif (im <= 1.2e+83)
		tmp = Float64(t_0 * Float64(0.5 * Float64(re * Float64(re * 4.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	t_1 = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))));
	tmp = 0.0;
	if (im <= -1e+24)
		tmp = t_1;
	elseif (im <= 1100.0)
		tmp = cos(re);
	elseif (im <= 1.2e+83)
		tmp = t_0 * (0.5 * (re * (re * 4.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(0.5 * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1e+24], t$95$1, If[LessEqual[im, 1100.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.2e+83], N[(t$95$0 * N[(0.5 * N[(re * N[(re * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.9999999999999998e23 or 1.19999999999999996e83 < 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 65.3%

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

      1. unpow2 65.3%

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

   4. Simplified 65.3%

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

   5. Taylor expanded in re around 0 55.7%

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

      1. unpow2 55.7%

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

   7. Simplified 55.7%

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

   if -9.9999999999999998e23 < im < 1100

   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 95.5%

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

      1. unpow2 95.5%

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

   4. Simplified 95.5%

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

      1. +-commutative 95.5%

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

      2. *-commutative 95.5%

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

      3. metadata-eval 95.5%

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

      4. pow-sqr 95.5%

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

      5. pow-prod-down 95.5%

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

      6. pow2 95.5%

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

      7. associate-*l* 95.5%

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

      8. fma-def 95.5%

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

   6. Applied egg-rr 95.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \color{blue}{\mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im\right)}\right) \]
   7. Step-by-step derivation

      1. fma-udef 95.5%

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

      2. *-commutative 95.5%

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

      3. *-commutative 95.5%

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

      4. distribute-lft1-in 95.5%

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

      5. *-commutative 95.5%

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

      6. associate-*l* 95.5%

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

   8. Simplified 95.5%

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

   9. Taylor expanded in im around 0 94.3%

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

   if 1100 < im < 1.19999999999999996e83

   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 3.7%

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

      1. unpow2 3.7%

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

   4. Simplified 3.7%

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

   6. Applied egg-rr 0.5%

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

      1. *-commutative 0.5%

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

   8. Simplified 0.5%

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

   9. Taylor expanded in re around 0 36.3%

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

       1. *-commutative 36.3%

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

       2. unpow2 36.3%

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

   11. Simplified 36.3%

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

   12. Taylor expanded in re around inf 37.5%

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

       1. unpow2 37.5%

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

       2. *-commutative 37.5%

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

       3. associate-*l* 37.5%

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

   14. Simplified 37.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+24}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 1100:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+83}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot \left(re \cdot \left(re \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]

Alternative 10: 49.1% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ \mathbf{if}\;im \leq 520:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+83}:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \left(re \cdot \left(re \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im))))
   (if (<= im 520.0)
     (* 0.5 t_0)
     (if (<= im 5e+83)
       (* t_0 (* 0.5 (* re (* re 4.0))))
       (* t_0 (* 0.5 (+ 1.0 (* -0.5 (* re re)))))))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double tmp;
	if (im <= 520.0) {
		tmp = 0.5 * t_0;
	} else if (im <= 5e+83) {
		tmp = t_0 * (0.5 * (re * (re * 4.0)));
	} else {
		tmp = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (im * im)
    if (im <= 520.0d0) then
        tmp = 0.5d0 * t_0
    else if (im <= 5d+83) then
        tmp = t_0 * (0.5d0 * (re * (re * 4.0d0)))
    else
        tmp = t_0 * (0.5d0 * (1.0d0 + ((-0.5d0) * (re * re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double tmp;
	if (im <= 520.0) {
		tmp = 0.5 * t_0;
	} else if (im <= 5e+83) {
		tmp = t_0 * (0.5 * (re * (re * 4.0)));
	} else {
		tmp = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	tmp = 0
	if im <= 520.0:
		tmp = 0.5 * t_0
	elif im <= 5e+83:
		tmp = t_0 * (0.5 * (re * (re * 4.0)))
	else:
		tmp = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	tmp = 0.0
	if (im <= 520.0)
		tmp = Float64(0.5 * t_0);
	elseif (im <= 5e+83)
		tmp = Float64(t_0 * Float64(0.5 * Float64(re * Float64(re * 4.0))));
	else
		tmp = Float64(t_0 * Float64(0.5 * Float64(1.0 + Float64(-0.5 * Float64(re * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	tmp = 0.0;
	if (im <= 520.0)
		tmp = 0.5 * t_0;
	elseif (im <= 5e+83)
		tmp = t_0 * (0.5 * (re * (re * 4.0)));
	else
		tmp = t_0 * (0.5 * (1.0 + (-0.5 * (re * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 520.0], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[im, 5e+83], N[(t$95$0 * N[(0.5 * N[(re * N[(re * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.5 * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $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 re around 0 59.8%

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

   3. Taylor expanded in im around 0 48.3%

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

      1. unpow2 83.4%

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

   5. Simplified 48.3%

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

   if 520 < im < 5.00000000000000029e83

   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 3.7%

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

      1. unpow2 3.7%

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

   4. Simplified 3.7%

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

   6. Applied egg-rr 0.5%

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

      1. *-commutative 0.5%

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

   8. Simplified 0.5%

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

   9. Taylor expanded in re around 0 36.3%

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

       1. *-commutative 36.3%

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

       2. unpow2 36.3%

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

   11. Simplified 36.3%

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

   12. Taylor expanded in re around inf 37.5%

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

       1. unpow2 37.5%

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

       2. *-commutative 37.5%

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

       3. associate-*l* 37.5%

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

   14. Simplified 37.5%

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

   if 5.00000000000000029e83 < 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 77.3%

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

      1. unpow2 77.3%

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

   4. Simplified 77.3%

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

   5. Taylor expanded in re around 0 65.3%

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

      1. unpow2 65.3%

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

   7. Simplified 65.3%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot \left(re \cdot \left(re \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]

Alternative 11: 48.2% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.45 \cdot 10^{+25}:\\ \;\;\;\;im \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.45e+25)
   (* im (* im (* (* re re) -0.25)))
   (* 0.5 (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.45e+25) {
		tmp = im * (im * ((re * re) * -0.25));
	} else {
		tmp = 0.5 * (2.0 + (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.45d+25)) then
        tmp = im * (im * ((re * re) * (-0.25d0)))
    else
        tmp = 0.5d0 * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.45e+25) {
		tmp = im * (im * ((re * re) * -0.25));
	} else {
		tmp = 0.5 * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.45e+25:
		tmp = im * (im * ((re * re) * -0.25))
	else:
		tmp = 0.5 * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.45e+25)
		tmp = Float64(im * Float64(im * Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.45e+25)
		tmp = im * (im * ((re * re) * -0.25));
	else
		tmp = 0.5 * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.45e+25], N[(im * N[(im * N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.4499999999999999e25

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.4%

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

      1. unpow2 78.4%

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

   4. Simplified 78.4%

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

   5. Taylor expanded in re around 0 25.9%

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

      1. unpow2 25.9%

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

   7. Simplified 25.9%

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

   8. Taylor expanded in im around inf 26.0%

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

      1. associate-*r* 26.0%

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

      2. +-commutative 26.0%

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

      3. fma-def 26.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.5, {re}^{2}, 1\right)}\right) \cdot {im}^{2} \]

      4. unpow2 26.0%

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right)\right) \cdot {im}^{2} \]

      5. *-commutative 26.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\right)} \]

      6. unpow2 26.0%

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

      7. associate-*l* 26.2%

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

      8. fma-udef 26.2%

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

      9. unpow2 26.2%

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

      10. *-commutative 26.2%

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

      11. fma-def 26.2%

          \[\leadsto im \cdot \left(im \cdot \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, -0.5, 1\right)}\right)\right) \]

      12. unpow2 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in re around inf 26.2%

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

       1. unpow2 26.2%

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

       2. associate-*r* 26.2%

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

       3. *-commutative 26.2%

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

   13. Simplified 26.2%

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

   if -3.4499999999999999e25 < 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 73.1%

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

   3. Taylor expanded in im around 0 54.2%

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

      1. unpow2 78.1%

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

   5. Simplified 54.2%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.45 \cdot 10^{+25}:\\ \;\;\;\;im \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 12: 46.7% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2000000000000 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2000000000000.0) (not (<= im 1.4))) (* 0.5 (* im im)) 1.0))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2000000000000.0) || !(im <= 1.4)) {
		tmp = 0.5 * (im * im);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-2000000000000.0d0)) .or. (.not. (im <= 1.4d0))) then
        tmp = 0.5d0 * (im * im)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2000000000000.0) || !(im <= 1.4)) {
		tmp = 0.5 * (im * im);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2000000000000.0) or not (im <= 1.4):
		tmp = 0.5 * (im * im)
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2000000000000.0) || !(im <= 1.4))
		tmp = Float64(0.5 * Float64(im * im));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2000000000000.0) || ~((im <= 1.4)))
		tmp = 0.5 * (im * im);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2000000000000.0], N[Not[LessEqual[im, 1.4]], $MachinePrecision]], N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if im < -2e12 or 1.3999999999999999 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 56.6%

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

      1. unpow2 56.6%

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

   4. Simplified 56.6%

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

   5. Taylor expanded in re around 0 48.1%

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

      1. unpow2 48.1%

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

   7. Simplified 48.1%

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

   8. Taylor expanded in im around inf 48.1%

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

      1. associate-*r* 48.1%

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

      2. +-commutative 48.1%

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

      3. fma-def 48.1%

         \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.5, {re}^{2}, 1\right)}\right) \cdot {im}^{2} \]

      4. unpow2 48.1%

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right)\right) \cdot {im}^{2} \]

      5. *-commutative 48.1%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\right)} \]

      6. unpow2 48.1%

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

      7. associate-*l* 48.1%

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

      8. fma-udef 48.1%

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

      9. unpow2 48.1%

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

      10. *-commutative 48.1%

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

      11. fma-def 48.1%

          \[\leadsto im \cdot \left(im \cdot \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, -0.5, 1\right)}\right)\right) \]

      12. unpow2 48.1%

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

   10. Simplified 48.1%

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

   11. Taylor expanded in re around 0 39.4%

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

       1. *-commutative 39.4%

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

       2. unpow2 39.4%

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

   13. Simplified 39.4%

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

   if -2e12 < im < 1.3999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 50.5%

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

   3. Taylor expanded in im around 0 49.9%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2000000000000 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 3.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

double code(double re, double im) {
	return -1.0;
}

Fortran

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

Java

public static double code(double re, double im) {
	return -1.0;
}

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0;
end

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 61.6%

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

4. Applied egg-rr 4.2%

   \[\leadsto \left(0.5 \cdot 1\right) \cdot \color{blue}{-2} \]

5. Final simplification 4.2%

   \[\leadsto -1 \]

Alternative 14: 8.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 61.6%

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

4. Applied egg-rr 8.2%

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

5. Final simplification 8.2%

   \[\leadsto 0.25 \]

Alternative 15: 9.1% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.75)

C

double code(double re, double im) {
	return 0.75;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.75d0
end function

Java

public static double code(double re, double im) {
	return 0.75;
}

Python

def code(re, im):
	return 0.75

Julia

function code(re, im)
	return 0.75
end

MATLAB

function tmp = code(re, im)
	tmp = 0.75;
end

Wolfram

code[re_, im_] := 0.75

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 61.6%

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

4. Applied egg-rr 9.2%

   \[\leadsto \left(0.5 \cdot 1\right) \cdot \color{blue}{1.5} \]

5. Final simplification 9.2%

   \[\leadsto 0.75 \]

Alternative 16: 28.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 61.6%

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

3. Taylor expanded in im around 0 27.7%

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

4. Final simplification 27.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))