math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 13

Speedup: 1.0×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.02 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \sqrt[3]{0.125}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.02) (not (<= im 1.35e+154)))
   (* (* 0.5 (cos re)) (+ (* im im) 2.0))
   (* (+ (exp (- im)) (exp im)) (cbrt 0.125))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.02) || !(im <= 1.35e+154)) {
		tmp = (0.5 * cos(re)) * ((im * im) + 2.0);
	} else {
		tmp = (exp(-im) + exp(im)) * cbrt(0.125);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 0.02) || !(im <= 1.35e+154)) {
		tmp = (0.5 * Math.cos(re)) * ((im * im) + 2.0);
	} else {
		tmp = (Math.exp(-im) + Math.exp(im)) * Math.cbrt(0.125);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.02) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(im * im) + 2.0));
	else
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * cbrt(0.125));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.02], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[Power[0.125, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0200000000000000004 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 88.2%

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

      1. unpow2 88.2%

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

   4. Simplified 88.2%

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

   if 0.0200000000000000004 < 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 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 78.1%

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

      3. unpow2 78.1%

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

   4. Simplified 78.1%

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

      1. add-cbrt-cube 78.1%

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

      2. pow3 78.1%

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

      3. +-commutative 78.1%

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

      4. fma-def 78.1%

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

   6. Applied egg-rr 78.1%

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

   7. Taylor expanded in re around 0 72.6%

      \[\leadsto \left(e^{im} + e^{-im}\right) \cdot \sqrt[3]{\color{blue}{0.125}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.02 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \sqrt[3]{0.125}\\ \end{array} \]

Alternative 3: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 0.105:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, t_0, \cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \sqrt[3]{0.125}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot im + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 0.105)
     (fma (* im im) t_0 (cos re))
     (if (<= im 1.35e+154)
       (* (+ (exp (- im)) (exp im)) (cbrt 0.125))
       (* t_0 (+ (* im im) 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 0.105) {
		tmp = fma((im * im), t_0, cos(re));
	} else if (im <= 1.35e+154) {
		tmp = (exp(-im) + exp(im)) * cbrt(0.125);
	} else {
		tmp = t_0 * ((im * im) + 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 0.105)
		tmp = fma(Float64(im * im), t_0, cos(re));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * cbrt(0.125));
	else
		tmp = Float64(t_0 * Float64(Float64(im * im) + 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.105], N[(N[(im * im), $MachinePrecision] * t$95$0 + N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[Power[0.125, 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.104999999999999996

   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 86.4%

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

      1. associate-*r* 86.4%

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

      2. *-commutative 86.4%

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

      3. fma-def 86.4%

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

      4. unpow2 86.4%

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

   4. Simplified 86.4%

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

   if 0.104999999999999996 < 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 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 78.1%

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

      3. unpow2 78.1%

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

   4. Simplified 78.1%

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

      1. add-cbrt-cube 78.1%

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

      2. pow3 78.1%

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

      3. +-commutative 78.1%

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

      4. fma-def 78.1%

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

   6. Applied egg-rr 78.1%

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

   7. Taylor expanded in re around 0 72.6%

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

   if 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 0 100.0%

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

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.105:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5 \cdot \cos re, \cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \sqrt[3]{0.125}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]

Alternative 4: 84.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.017 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.017) (not (<= im 1.35e+154)))
   (* (* 0.5 (cos re)) (+ (* im im) 2.0))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.017) || !(im <= 1.35e+154)) {
		tmp = (0.5 * cos(re)) * ((im * im) + 2.0);
	} 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 ((im <= 0.017d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = (0.5d0 * cos(re)) * ((im * im) + 2.0d0)
    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 ((im <= 0.017) || !(im <= 1.35e+154)) {
		tmp = (0.5 * Math.cos(re)) * ((im * im) + 2.0);
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.017) or not (im <= 1.35e+154):
		tmp = (0.5 * math.cos(re)) * ((im * im) + 2.0)
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.017) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(im * im) + 2.0));
	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 ((im <= 0.017) || ~((im <= 1.35e+154)))
		tmp = (0.5 * cos(re)) * ((im * im) + 2.0);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.017], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $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 im < 0.017000000000000001 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 88.2%

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

      1. unpow2 88.2%

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

   4. Simplified 88.2%

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

   if 0.017000000000000001 < 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 re around 0 72.6%

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

      1. *-commutative 72.6%

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

   4. Simplified 72.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.017 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 5: 81.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot im + 2\\ t_1 := t_0 + 0.08333333333333333 \cdot {im}^{4}\\ t_2 := \left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{if}\;im \leq 7700000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+97}:\\ \;\;\;\;t_1 \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* im im) 2.0))
        (t_1 (+ t_0 (* 0.08333333333333333 (pow im 4.0))))
        (t_2 (* (* 0.5 (cos re)) t_0)))
   (if (<= im 7700000000000.0)
     t_2
     (if (<= im 1.3e+97)
       (* t_1 (+ 0.5 (* -0.25 (* re re))))
       (if (<= im 1.35e+154) (* 0.5 t_1) t_2)))))

C

double code(double re, double im) {
	double t_0 = (im * im) + 2.0;
	double t_1 = t_0 + (0.08333333333333333 * pow(im, 4.0));
	double t_2 = (0.5 * cos(re)) * t_0;
	double tmp;
	if (im <= 7700000000000.0) {
		tmp = t_2;
	} else if (im <= 1.3e+97) {
		tmp = t_1 * (0.5 + (-0.25 * (re * re)));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (im * im) + 2.0d0
    t_1 = t_0 + (0.08333333333333333d0 * (im ** 4.0d0))
    t_2 = (0.5d0 * cos(re)) * t_0
    if (im <= 7700000000000.0d0) then
        tmp = t_2
    else if (im <= 1.3d+97) then
        tmp = t_1 * (0.5d0 + ((-0.25d0) * (re * re)))
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) + 2.0;
	double t_1 = t_0 + (0.08333333333333333 * Math.pow(im, 4.0));
	double t_2 = (0.5 * Math.cos(re)) * t_0;
	double tmp;
	if (im <= 7700000000000.0) {
		tmp = t_2;
	} else if (im <= 1.3e+97) {
		tmp = t_1 * (0.5 + (-0.25 * (re * re)));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) + 2.0
	t_1 = t_0 + (0.08333333333333333 * math.pow(im, 4.0))
	t_2 = (0.5 * math.cos(re)) * t_0
	tmp = 0
	if im <= 7700000000000.0:
		tmp = t_2
	elif im <= 1.3e+97:
		tmp = t_1 * (0.5 + (-0.25 * (re * re)))
	elif im <= 1.35e+154:
		tmp = 0.5 * t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) + 2.0)
	t_1 = Float64(t_0 + Float64(0.08333333333333333 * (im ^ 4.0)))
	t_2 = Float64(Float64(0.5 * cos(re)) * t_0)
	tmp = 0.0
	if (im <= 7700000000000.0)
		tmp = t_2;
	elseif (im <= 1.3e+97)
		tmp = Float64(t_1 * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) + 2.0;
	t_1 = t_0 + (0.08333333333333333 * (im ^ 4.0));
	t_2 = (0.5 * cos(re)) * t_0;
	tmp = 0.0;
	if (im <= 7700000000000.0)
		tmp = t_2;
	elseif (im <= 1.3e+97)
		tmp = t_1 * (0.5 + (-0.25 * (re * re)));
	elseif (im <= 1.35e+154)
		tmp = 0.5 * t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[im, 7700000000000.0], t$95$2, If[LessEqual[im, 1.3e+97], N[(t$95$1 * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * t$95$1), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 7.7e12 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 87.5%

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

      1. unpow2 87.5%

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

   4. Simplified 87.5%

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

   if 7.7e12 < im < 1.3e97

   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 88.9%

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

      3. unpow2 88.9%

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

   4. Simplified 88.9%

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

   5. Taylor expanded in im around 0 52.5%

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

   7. Simplified 52.5%

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

   if 1.3e97 < 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 re around 0 75.0%

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

      1. *-commutative 75.0%

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

   4. Simplified 75.0%

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

   5. Taylor expanded in im around 0 75.0%

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

   7. Simplified 75.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7700000000000:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+97}:\\ \;\;\;\;\left(\left(im \cdot im + 2\right) + 0.08333333333333333 \cdot {im}^{4}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot im + 2\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]

Alternative 6: 80.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot im + 2\\ \mathbf{if}\;im \leq 1.85 \cdot 10^{+25} \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t_0 + 0.08333333333333333 \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* im im) 2.0)))
   (if (or (<= im 1.85e+25) (not (<= im 1.35e+154)))
     (* (* 0.5 (cos re)) t_0)
     (* 0.5 (+ t_0 (* 0.08333333333333333 (pow im 4.0)))))))

C

double code(double re, double im) {
	double t_0 = (im * im) + 2.0;
	double tmp;
	if ((im <= 1.85e+25) || !(im <= 1.35e+154)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = 0.5 * (t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = (im * im) + 2.0d0
    if ((im <= 1.85d+25) .or. (.not. (im <= 1.35d+154))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = 0.5d0 * (t_0 + (0.08333333333333333d0 * (im ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) + 2.0;
	double tmp;
	if ((im <= 1.85e+25) || !(im <= 1.35e+154)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = 0.5 * (t_0 + (0.08333333333333333 * Math.pow(im, 4.0)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) + 2.0
	tmp = 0
	if (im <= 1.85e+25) or not (im <= 1.35e+154):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = 0.5 * (t_0 + (0.08333333333333333 * math.pow(im, 4.0)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) + 2.0)
	tmp = 0.0
	if ((im <= 1.85e+25) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(0.5 * Float64(t_0 + Float64(0.08333333333333333 * (im ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) + 2.0;
	tmp = 0.0;
	if ((im <= 1.85e+25) || ~((im <= 1.35e+154)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = 0.5 * (t_0 + (0.08333333333333333 * (im ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]}, If[Or[LessEqual[im, 1.85e+25], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(0.5 * N[(t$95$0 + N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.8499999999999999e25 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 86.4%

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

      1. unpow2 86.4%

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

   4. Simplified 86.4%

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

   if 1.8499999999999999e25 < 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 re around 0 74.1%

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

      1. *-commutative 74.1%

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

   4. Simplified 74.1%

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

   5. Taylor expanded in im around 0 53.5%

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

   7. Simplified 53.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+25} \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot im + 2\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 7: 76.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 77.9%

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

   1. unpow2 77.9%

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

4. Simplified 77.9%

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

5. Final simplification 77.9%

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

Alternative 8: 62.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot im + 2\\ \mathbf{if}\;im \leq 10500000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+270} \lor \neg \left(im \leq 4 \cdot 10^{+289}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* im im) 2.0)))
   (if (<= im 10500000000000.0)
     (cos re)
     (if (or (<= im 5e+270) (not (<= im 4e+289)))
       (* t_0 (+ 0.5 (* -0.25 (* re re))))
       (* 0.5 t_0)))))

C

double code(double re, double im) {
	double t_0 = (im * im) + 2.0;
	double tmp;
	if (im <= 10500000000000.0) {
		tmp = cos(re);
	} else if ((im <= 5e+270) || !(im <= 4e+289)) {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * im) + 2.0d0
    if (im <= 10500000000000.0d0) then
        tmp = cos(re)
    else if ((im <= 5d+270) .or. (.not. (im <= 4d+289))) then
        tmp = t_0 * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = 0.5d0 * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) + 2.0;
	double tmp;
	if (im <= 10500000000000.0) {
		tmp = Math.cos(re);
	} else if ((im <= 5e+270) || !(im <= 4e+289)) {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) + 2.0
	tmp = 0
	if im <= 10500000000000.0:
		tmp = math.cos(re)
	elif (im <= 5e+270) or not (im <= 4e+289):
		tmp = t_0 * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = 0.5 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) + 2.0)
	tmp = 0.0
	if (im <= 10500000000000.0)
		tmp = cos(re);
	elseif ((im <= 5e+270) || !(im <= 4e+289))
		tmp = Float64(t_0 * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) + 2.0;
	tmp = 0.0;
	if (im <= 10500000000000.0)
		tmp = cos(re);
	elseif ((im <= 5e+270) || ~((im <= 4e+289)))
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	else
		tmp = 0.5 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[im, 10500000000000.0], N[Cos[re], $MachinePrecision], If[Or[LessEqual[im, 5e+270], N[Not[LessEqual[im, 4e+289]], $MachinePrecision]], N[(t$95$0 * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.05e13

   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 67.4%

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

   if 1.05e13 < im < 4.99999999999999976e270 or 4.0000000000000002e289 < 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 77.2%

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

      3. unpow2 77.2%

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

   4. Simplified 77.2%

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

   5. Taylor expanded in im around 0 47.5%

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

   7. Simplified 47.5%

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

   if 4.99999999999999976e270 < im < 4.0000000000000002e289

   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%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10500000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+270} \lor \neg \left(im \leq 4 \cdot 10^{+289}\right):\\ \;\;\;\;\left(im \cdot im + 2\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im + 2\right)\\ \end{array} \]

Alternative 9: 48.0% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot -0.25\right)\\ t_1 := im \cdot im + 2\\ \mathbf{if}\;re \leq 1080000000000:\\ \;\;\;\;0.5 \cdot t_1\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+165}:\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re -0.25))) (t_1 (+ (* im im) 2.0)))
   (if (<= re 1080000000000.0)
     (* 0.5 t_1)
     (if (<= re 5.5e+165) (* t_1 t_0) (* -2.0 t_0)))))

C

double code(double re, double im) {
	double t_0 = re * (re * -0.25);
	double t_1 = (im * im) + 2.0;
	double tmp;
	if (re <= 1080000000000.0) {
		tmp = 0.5 * t_1;
	} else if (re <= 5.5e+165) {
		tmp = t_1 * t_0;
	} else {
		tmp = -2.0 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (re * (-0.25d0))
    t_1 = (im * im) + 2.0d0
    if (re <= 1080000000000.0d0) then
        tmp = 0.5d0 * t_1
    else if (re <= 5.5d+165) then
        tmp = t_1 * t_0
    else
        tmp = (-2.0d0) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * -0.25);
	double t_1 = (im * im) + 2.0;
	double tmp;
	if (re <= 1080000000000.0) {
		tmp = 0.5 * t_1;
	} else if (re <= 5.5e+165) {
		tmp = t_1 * t_0;
	} else {
		tmp = -2.0 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * -0.25)
	t_1 = (im * im) + 2.0
	tmp = 0
	if re <= 1080000000000.0:
		tmp = 0.5 * t_1
	elif re <= 5.5e+165:
		tmp = t_1 * t_0
	else:
		tmp = -2.0 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * -0.25))
	t_1 = Float64(Float64(im * im) + 2.0)
	tmp = 0.0
	if (re <= 1080000000000.0)
		tmp = Float64(0.5 * t_1);
	elseif (re <= 5.5e+165)
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(-2.0 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * -0.25);
	t_1 = (im * im) + 2.0;
	tmp = 0.0;
	if (re <= 1080000000000.0)
		tmp = 0.5 * t_1;
	elseif (re <= 5.5e+165)
		tmp = t_1 * t_0;
	else
		tmp = -2.0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[re, 1080000000000.0], N[(0.5 * t$95$1), $MachinePrecision], If[LessEqual[re, 5.5e+165], N[(t$95$1 * t$95$0), $MachinePrecision], N[(-2.0 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.08e12

   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 76.9%

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

      1. *-commutative 76.9%

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

   4. Simplified 76.9%

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

   5. Taylor expanded in im around 0 56.2%

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

   7. Simplified 56.2%

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

   if 1.08e12 < re < 5.4999999999999998e165

   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 1.1%

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

      1. associate-*r* 1.1%

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

      2. distribute-rgt-out 41.1%

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

      3. unpow2 41.1%

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

   4. Simplified 41.1%

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

   5. Taylor expanded in im around 0 38.5%

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

   7. Simplified 38.5%

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

   8. Taylor expanded in re around inf 38.5%

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

      1. unpow2 2.1%

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

      2. associate-*r* 2.1%

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

      3. *-commutative 2.1%

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

   10. Simplified 38.5%

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

   if 5.4999999999999998e165 < 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 0.6%

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

      1. associate-*r* 0.6%

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

      2. distribute-rgt-out 20.6%

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

      3. unpow2 20.6%

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

   4. Simplified 20.6%

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

   6. Applied egg-rr 26.5%

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

   7. Taylor expanded in re around inf 26.5%

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

      1. unpow2 26.5%

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

      2. associate-*r* 26.5%

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

      3. *-commutative 26.5%

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

   9. Simplified 26.5%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1080000000000:\\ \;\;\;\;0.5 \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+165}:\\ \;\;\;\;\left(im \cdot im + 2\right) \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 10: 49.5% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.5 \cdot 10^{+165}:\\ \;\;\;\;\left(im \cdot im + 2\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.5e+165)
   (* (+ (* im im) 2.0) (+ 0.5 (* -0.25 (* re re))))
   (* -2.0 (* re (* re -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.5e+165) {
		tmp = ((im * im) + 2.0) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = -2.0 * (re * (re * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.5d+165) then
        tmp = ((im * im) + 2.0d0) * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = (-2.0d0) * (re * (re * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.5e+165) {
		tmp = ((im * im) + 2.0) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = -2.0 * (re * (re * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.5e+165:
		tmp = ((im * im) + 2.0) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = -2.0 * (re * (re * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.5e+165)
		tmp = Float64(Float64(Float64(im * im) + 2.0) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(-2.0 * Float64(re * Float64(re * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.5e+165)
		tmp = ((im * im) + 2.0) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = -2.0 * (re * (re * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.5e+165], N[(N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.4999999999999998e165

   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 28.3%

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

      1. associate-*r* 28.3%

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

      2. distribute-rgt-out 66.0%

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

      3. unpow2 66.0%

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

   4. Simplified 66.0%

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

   5. Taylor expanded in im around 0 51.3%

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

   7. Simplified 51.3%

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

   if 5.4999999999999998e165 < 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 0.6%

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

      1. associate-*r* 0.6%

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

      2. distribute-rgt-out 20.6%

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

      3. unpow2 20.6%

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

   4. Simplified 20.6%

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

   6. Applied egg-rr 26.5%

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

   7. Taylor expanded in re around inf 26.5%

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

      1. unpow2 26.5%

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

      2. associate-*r* 26.5%

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

      3. *-commutative 26.5%

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

   9. Simplified 26.5%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.5 \cdot 10^{+165}:\\ \;\;\;\;\left(im \cdot im + 2\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 11: 31.3% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3600000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3600000000.0) 1.0 (* -2.0 (* re (* re -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3600000000.0) {
		tmp = 1.0;
	} else {
		tmp = -2.0 * (re * (re * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3600000000.0d0) then
        tmp = 1.0d0
    else
        tmp = (-2.0d0) * (re * (re * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3600000000.0) {
		tmp = 1.0;
	} else {
		tmp = -2.0 * (re * (re * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3600000000.0:
		tmp = 1.0
	else:
		tmp = -2.0 * (re * (re * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3600000000.0)
		tmp = 1.0;
	else
		tmp = Float64(-2.0 * Float64(re * Float64(re * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3600000000.0)
		tmp = 1.0;
	else
		tmp = -2.0 * (re * (re * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3600000000.0], 1.0, N[(-2.0 * N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.6e9

   1. Initial program 100.0%

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

   3. Applied egg-rr 37.3%

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

      1. +-inverses 37.3%

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

      2. +-rgt-identity 37.3%

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

      3. *-inverses 37.3%

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

   5. Simplified 37.3%

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

   if 3.6e9 < 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 76.7%

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

      3. unpow2 76.7%

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

   4. Simplified 76.7%

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

   6. Applied egg-rr 15.9%

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

   7. Taylor expanded in re around inf 16.6%

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

      1. unpow2 16.6%

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

      2. associate-*r* 16.6%

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

      3. *-commutative 16.6%

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

   9. Simplified 16.6%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3600000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 12: 47.5% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 65.4%

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

   1. *-commutative 65.4%

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

4. Simplified 65.4%

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

5. Taylor expanded in im around 0 47.7%

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

7. Simplified 47.7%

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

8. Final simplification 47.7%

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

Alternative 13: 28.7% 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 29.1%

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

   1. +-inverses 29.1%

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

   2. +-rgt-identity 29.1%

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

   3. *-inverses 29.1%

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

5. Simplified 29.1%

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

6. Final simplification 29.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023224 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))