math.sin on complex, imaginary part

Percentage Accurate: 53.9% → 99.3%

Time: 9.1s

Alternatives: 16

Speedup: 2.9×

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

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - 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((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 53.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - 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((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+110} \lor \neg \left(t_0 \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -2e+110) (not (<= t_0 2e-10)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -2e+110) || !(t_0 <= 2e-10)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-2d+110)) .or. (.not. (t_0 <= 2d-10))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -2e+110) || !(t_0 <= 2e-10)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -2e+110) or not (t_0 <= 2e-10):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -2e+110) || !(t_0 <= 2e-10))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -2e+110) || ~((t_0 <= 2e-10)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+110], N[Not[LessEqual[t$95$0, 2e-10]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -2e110 or 2.00000000000000007e-10 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -2e110 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.00000000000000007e-10

   1. Initial program 7.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 7.3%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+110} \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 2: 94.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.00155:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;im \leq 260:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.42 \cdot 10^{+102}:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im)))
        (t_1 (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -4.1e+93)
     t_1
     (if (<= im -0.00155)
       (* 0.5 t_0)
       (if (<= im 260.0)
         (* (cos re) (- im))
         (if (<= im 1.42e+102) (* t_0 (+ 0.5 (* re (* re -0.25)))) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -4.1e+93) {
		tmp = t_1;
	} else if (im <= -0.00155) {
		tmp = 0.5 * t_0;
	} else if (im <= 260.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1.42e+102) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} 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 = exp(-im) - exp(im)
    t_1 = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-4.1d+93)) then
        tmp = t_1
    else if (im <= (-0.00155d0)) then
        tmp = 0.5d0 * t_0
    else if (im <= 260.0d0) then
        tmp = cos(re) * -im
    else if (im <= 1.42d+102) then
        tmp = t_0 * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -4.1e+93) {
		tmp = t_1;
	} else if (im <= -0.00155) {
		tmp = 0.5 * t_0;
	} else if (im <= 260.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.42e+102) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -4.1e+93:
		tmp = t_1
	elif im <= -0.00155:
		tmp = 0.5 * t_0
	elif im <= 260.0:
		tmp = math.cos(re) * -im
	elif im <= 1.42e+102:
		tmp = t_0 * (0.5 + (re * (re * -0.25)))
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -4.1e+93)
		tmp = t_1;
	elseif (im <= -0.00155)
		tmp = Float64(0.5 * t_0);
	elseif (im <= 260.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.42e+102)
		tmp = Float64(t_0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -4.1e+93)
		tmp = t_1;
	elseif (im <= -0.00155)
		tmp = 0.5 * t_0;
	elseif (im <= 260.0)
		tmp = cos(re) * -im;
	elseif (im <= 1.42e+102)
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.1e+93], t$95$1, If[LessEqual[im, -0.00155], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[im, 260.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.42e+102], N[(t$95$0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.1000000000000001e93 or 1.4200000000000001e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 98.0%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 98.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 98.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 98.0%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 98.0%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 98.0%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   if -4.1000000000000001e93 < im < -0.00154999999999999995

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 87.5%

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

   if -0.00154999999999999995 < im < 260

   1. Initial program 8.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 8.0%

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

   3. Simplified 8.0%

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

   4. Taylor expanded in im around 0 99.1%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. distribute-lft-neg-in 99.1%

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

   6. Simplified 99.1%

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

   if 260 < im < 1.4200000000000001e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 85.7%

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

      4. +-commutative 85.7%

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

      5. *-commutative 85.7%

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

      6. unpow2 85.7%

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

      7. associate-*l* 85.7%

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

   6. Simplified 85.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -0.00155:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 260:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.42 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0135:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.052:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -4.1e+93)
     t_1
     (if (<= im -0.0135)
       t_0
       (if (<= im 0.052) (* (cos re) (- im)) (if (<= im 5.6e+102) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -4.1e+93) {
		tmp = t_1;
	} else if (im <= -0.0135) {
		tmp = t_0;
	} else if (im <= 0.052) {
		tmp = cos(re) * -im;
	} else if (im <= 5.6e+102) {
		tmp = t_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 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-4.1d+93)) then
        tmp = t_1
    else if (im <= (-0.0135d0)) then
        tmp = t_0
    else if (im <= 0.052d0) then
        tmp = cos(re) * -im
    else if (im <= 5.6d+102) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -4.1e+93) {
		tmp = t_1;
	} else if (im <= -0.0135) {
		tmp = t_0;
	} else if (im <= 0.052) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 5.6e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -4.1e+93:
		tmp = t_1
	elif im <= -0.0135:
		tmp = t_0
	elif im <= 0.052:
		tmp = math.cos(re) * -im
	elif im <= 5.6e+102:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -4.1e+93)
		tmp = t_1;
	elseif (im <= -0.0135)
		tmp = t_0;
	elseif (im <= 0.052)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 5.6e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -4.1e+93)
		tmp = t_1;
	elseif (im <= -0.0135)
		tmp = t_0;
	elseif (im <= 0.052)
		tmp = cos(re) * -im;
	elseif (im <= 5.6e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.1e+93], t$95$1, If[LessEqual[im, -0.0135], t$95$0, If[LessEqual[im, 0.052], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 5.6e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.1000000000000001e93 or 5.60000000000000037e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 98.9%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 98.9%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 98.9%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 98.9%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 98.9%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   if -4.1000000000000001e93 < im < -0.0134999999999999998 or 0.0519999999999999976 < im < 5.60000000000000037e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 77.5%

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

   if -0.0134999999999999998 < im < 0.0519999999999999976

   1. Initial program 7.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 7.3%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -0.0135:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.052:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.00135 \lor \neg \left(im \leq 0.0021\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.00135) (not (<= im 0.0021)))
   (* 0.5 (- (exp (- im)) (exp im)))
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.00135) || !(im <= 0.0021)) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = cos(re) * -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-0.00135d0)) .or. (.not. (im <= 0.0021d0))) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.00135) || !(im <= 0.0021)) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.00135) or not (im <= 0.0021):
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.00135) || !(im <= 0.0021))
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.00135) || ~((im <= 0.0021)))
		tmp = 0.5 * (exp(-im) - exp(im));
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.00135], N[Not[LessEqual[im, 0.0021]], $MachinePrecision]], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -0.0013500000000000001 or 0.00209999999999999987 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 76.4%

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

   if -0.0013500000000000001 < im < 0.00209999999999999987

   1. Initial program 7.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 7.3%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00135 \lor \neg \left(im \leq 0.0021\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 5: 77.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 2100:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(im \cdot -2 + {im}^{3} \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.7e+25)
   (- (* (pow im 3.0) -0.16666666666666666) im)
   (if (<= im 2100.0)
     (* (cos re) (- im))
     (*
      (+ 0.5 (* re (* re -0.25)))
      (+ (* im -2.0) (* (pow im 3.0) -0.3333333333333333))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.7e+25) {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	} else if (im <= 2100.0) {
		tmp = cos(re) * -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * ((im * -2.0) + (pow(im, 3.0) * -0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-3.7d+25)) then
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    else if (im <= 2100.0d0) then
        tmp = cos(re) * -im
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * ((im * (-2.0d0)) + ((im ** 3.0d0) * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3.7e+25) {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	} else if (im <= 2100.0) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * ((im * -2.0) + (Math.pow(im, 3.0) * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3.7e+25:
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	elif im <= 2100.0:
		tmp = math.cos(re) * -im
	else:
		tmp = (0.5 + (re * (re * -0.25))) * ((im * -2.0) + (math.pow(im, 3.0) * -0.3333333333333333))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3.7e+25)
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 2100.0)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * Float64(Float64(im * -2.0) + Float64((im ^ 3.0) * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3.7e+25)
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	elseif (im <= 2100.0)
		tmp = cos(re) * -im;
	else
		tmp = (0.5 + (re * (re * -0.25))) * ((im * -2.0) + ((im ^ 3.0) * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.7e+25], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 2100.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.6999999999999999e25

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 84.9%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 84.9%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 84.9%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 84.9%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 84.9%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 84.9%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 66.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   if -3.6999999999999999e25 < im < 2100

   1. Initial program 13.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 13.3%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in im around 0 93.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 93.6%

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

      2. *-commutative 93.6%

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

      3. distribute-lft-neg-in 93.6%

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

   6. Simplified 93.6%

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

   if 2100 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 77.0%

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

      4. +-commutative 77.0%

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

      5. *-commutative 77.0%

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

      6. unpow2 77.0%

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

      7. associate-*l* 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in im around 0 57.0%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 2100:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(im \cdot -2 + {im}^{3} \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 6: 77.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.8e+25)
   (- (* (pow im 3.0) -0.16666666666666666) im)
   (if (<= im 700.0)
     (* (cos re) (- im))
     (* -0.3333333333333333 (* (pow im 3.0) (+ 0.5 (* -0.25 (* re re))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.8e+25) {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	} else if (im <= 700.0) {
		tmp = cos(re) * -im;
	} else {
		tmp = -0.3333333333333333 * (pow(im, 3.0) * (0.5 + (-0.25 * (re * re))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-3.8d+25)) then
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    else if (im <= 700.0d0) then
        tmp = cos(re) * -im
    else
        tmp = (-0.3333333333333333d0) * ((im ** 3.0d0) * (0.5d0 + ((-0.25d0) * (re * re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3.8e+25) {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	} else if (im <= 700.0) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = -0.3333333333333333 * (Math.pow(im, 3.0) * (0.5 + (-0.25 * (re * re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3.8e+25:
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	elif im <= 700.0:
		tmp = math.cos(re) * -im
	else:
		tmp = -0.3333333333333333 * (math.pow(im, 3.0) * (0.5 + (-0.25 * (re * re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3.8e+25)
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 700.0)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = Float64(-0.3333333333333333 * Float64((im ^ 3.0) * Float64(0.5 + Float64(-0.25 * Float64(re * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3.8e+25)
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	elseif (im <= 700.0)
		tmp = cos(re) * -im;
	else
		tmp = -0.3333333333333333 * ((im ^ 3.0) * (0.5 + (-0.25 * (re * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.8e+25], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 700.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[im, 3.0], $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.8e25

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 84.9%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 84.9%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 84.9%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 84.9%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 84.9%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 84.9%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 66.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   if -3.8e25 < im < 700

   1. Initial program 13.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 13.3%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in im around 0 93.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 93.6%

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

      2. *-commutative 93.6%

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

      3. distribute-lft-neg-in 93.6%

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

   6. Simplified 93.6%

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

   if 700 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 77.0%

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

      4. +-commutative 77.0%

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

      5. *-commutative 77.0%

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

      6. unpow2 77.0%

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

      7. associate-*l* 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in im around 0 57.0%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around inf 57.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 57.0%

         \[\leadsto -0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right)\right) \]

      2. unpow2 57.0%

         \[\leadsto -0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right)\right) \]

   10. Simplified 57.0%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]

Alternative 7: 76.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+18}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.42 \cdot 10^{+102}:\\ \;\;\;\;im \cdot \left(0.001388888888888889 \cdot {re}^{6}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -3.1e+25)
     t_0
     (if (<= im 2.25e+18)
       (* (cos re) (- im))
       (if (<= im 1.42e+102)
         (* im (* 0.001388888888888889 (pow re 6.0)))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -3.1e+25) {
		tmp = t_0;
	} else if (im <= 2.25e+18) {
		tmp = cos(re) * -im;
	} else if (im <= 1.42e+102) {
		tmp = im * (0.001388888888888889 * pow(re, 6.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-3.1d+25)) then
        tmp = t_0
    else if (im <= 2.25d+18) then
        tmp = cos(re) * -im
    else if (im <= 1.42d+102) then
        tmp = im * (0.001388888888888889d0 * (re ** 6.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -3.1e+25) {
		tmp = t_0;
	} else if (im <= 2.25e+18) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.42e+102) {
		tmp = im * (0.001388888888888889 * Math.pow(re, 6.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -3.1e+25:
		tmp = t_0
	elif im <= 2.25e+18:
		tmp = math.cos(re) * -im
	elif im <= 1.42e+102:
		tmp = im * (0.001388888888888889 * math.pow(re, 6.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -3.1e+25)
		tmp = t_0;
	elseif (im <= 2.25e+18)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.42e+102)
		tmp = Float64(im * Float64(0.001388888888888889 * (re ^ 6.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -3.1e+25)
		tmp = t_0;
	elseif (im <= 2.25e+18)
		tmp = cos(re) * -im;
	elseif (im <= 1.42e+102)
		tmp = im * (0.001388888888888889 * (re ^ 6.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -3.1e+25], t$95$0, If[LessEqual[im, 2.25e+18], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.42e+102], N[(im * N[(0.001388888888888889 * N[Power[re, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.0999999999999998e25 or 1.4200000000000001e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 90.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 90.3%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 90.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 90.3%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 90.3%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 90.3%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 90.3%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 69.4%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   if -3.0999999999999998e25 < im < 2.25e18

   1. Initial program 14.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 14.5%

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

   3. Simplified 14.5%

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

   4. Taylor expanded in im around 0 92.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 92.4%

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

      2. *-commutative 92.4%

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

      3. distribute-lft-neg-in 92.4%

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

   6. Simplified 92.4%

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

   if 2.25e18 < im < 1.4200000000000001e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 3.7%

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

      2. *-commutative 3.7%

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

      3. distribute-lft-neg-in 3.7%

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

   6. Simplified 3.7%

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

   7. Taylor expanded in re around 0 2.2%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({re}^{6} \cdot im\right) + \left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)\right)\right)} \]

   8. Taylor expanded in re around inf 27.3%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({re}^{6} \cdot im\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 27.3%

         \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot {re}^{6}\right) \cdot im} \]

   10. Simplified 27.3%

       \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot {re}^{6}\right) \cdot im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+18}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.42 \cdot 10^{+102}:\\ \;\;\;\;im \cdot \left(0.001388888888888889 \cdot {re}^{6}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 8: 59.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{+18} \lor \neg \left(im \leq 600\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.6e+18) (not (<= im 600.0)))
   (- (* 0.5 (* re (* im re))) im)
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.6e+18) || !(im <= 600.0)) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = cos(re) * -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-2.6d+18)) .or. (.not. (im <= 600.0d0))) then
        tmp = (0.5d0 * (re * (im * re))) - im
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.6e+18) || !(im <= 600.0)) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.6e+18) or not (im <= 600.0):
		tmp = (0.5 * (re * (im * re))) - im
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.6e+18) || !(im <= 600.0))
		tmp = Float64(Float64(0.5 * Float64(re * Float64(im * re))) - im);
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.6e+18) || ~((im <= 600.0)))
		tmp = (0.5 * (re * (im * re))) - im;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.6e+18], N[Not[LessEqual[im, 600.0]], $MachinePrecision]], N[(N[(0.5 * N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.6e18 or 600 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 5.6%

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

      2. *-commutative 5.6%

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

      3. distribute-lft-neg-in 5.6%

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

   6. Simplified 5.6%

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

   7. Taylor expanded in re around 0 3.8%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({re}^{6} \cdot im\right) + \left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)\right)\right)} \]

   8. Taylor expanded in re around 0 22.5%

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

      1. +-commutative 22.5%

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

      2. mul-1-neg 22.5%

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

      3. unsub-neg 22.5%

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

      4. unpow2 22.5%

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

      5. associate-*l* 22.5%

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

   10. Simplified 22.5%

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

   if -2.6e18 < im < 600

   1. Initial program 12.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 12.0%

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

   3. Simplified 12.0%

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

   4. Taylor expanded in im around 0 95.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 95.0%

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

      2. *-commutative 95.0%

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

      3. distribute-lft-neg-in 95.0%

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

   6. Simplified 95.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{+18} \lor \neg \left(im \leq 600\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 9: 60.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+18}:\\ \;\;\;\;im \cdot \left(0.001388888888888889 \cdot {re}^{6}\right)\\ \mathbf{elif}\;im \leq 2050000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -2.8e+18)
   (* im (* 0.001388888888888889 (pow re 6.0)))
   (if (<= im 2050000.0) (* (cos re) (- im)) (- (* 0.5 (* re (* im re))) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2.8e+18) {
		tmp = im * (0.001388888888888889 * pow(re, 6.0));
	} else if (im <= 2050000.0) {
		tmp = cos(re) * -im;
	} else {
		tmp = (0.5 * (re * (im * re))) - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-2.8d+18)) then
        tmp = im * (0.001388888888888889d0 * (re ** 6.0d0))
    else if (im <= 2050000.0d0) then
        tmp = cos(re) * -im
    else
        tmp = (0.5d0 * (re * (im * re))) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -2.8e+18) {
		tmp = im * (0.001388888888888889 * Math.pow(re, 6.0));
	} else if (im <= 2050000.0) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = (0.5 * (re * (im * re))) - im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2.8e+18:
		tmp = im * (0.001388888888888889 * math.pow(re, 6.0))
	elif im <= 2050000.0:
		tmp = math.cos(re) * -im
	else:
		tmp = (0.5 * (re * (im * re))) - im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2.8e+18)
		tmp = Float64(im * Float64(0.001388888888888889 * (re ^ 6.0)));
	elseif (im <= 2050000.0)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = Float64(Float64(0.5 * Float64(re * Float64(im * re))) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2.8e+18)
		tmp = im * (0.001388888888888889 * (re ^ 6.0));
	elseif (im <= 2050000.0)
		tmp = cos(re) * -im;
	else
		tmp = (0.5 * (re * (im * re))) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2.8e+18], N[(im * N[(0.001388888888888889 * N[Power[re, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2050000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(N[(0.5 * N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.8e18

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 6.1%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 6.1%

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

      2. *-commutative 6.1%

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

      3. distribute-lft-neg-in 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in re around 0 5.0%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({re}^{6} \cdot im\right) + \left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)\right)\right)} \]

   8. Taylor expanded in re around inf 21.6%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({re}^{6} \cdot im\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 21.6%

         \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot {re}^{6}\right) \cdot im} \]

   10. Simplified 21.6%

       \[\leadsto \color{blue}{\left(0.001388888888888889 \cdot {re}^{6}\right) \cdot im} \]

   if -2.8e18 < im < 2.05e6

   1. Initial program 12.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 12.0%

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

   3. Simplified 12.0%

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

   4. Taylor expanded in im around 0 95.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 95.0%

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

      2. *-commutative 95.0%

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

      3. distribute-lft-neg-in 95.0%

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

   6. Simplified 95.0%

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

   if 2.05e6 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.1%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 5.1%

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

      2. *-commutative 5.1%

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

      3. distribute-lft-neg-in 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in re around 0 2.6%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({re}^{6} \cdot im\right) + \left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)\right)\right)} \]

   8. Taylor expanded in re around 0 25.7%

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

      1. +-commutative 25.7%

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

      2. mul-1-neg 25.7%

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

      3. unsub-neg 25.7%

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

      4. unpow2 25.7%

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

      5. associate-*l* 25.7%

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

   10. Simplified 25.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+18}:\\ \;\;\;\;im \cdot \left(0.001388888888888889 \cdot {re}^{6}\right)\\ \mathbf{elif}\;im \leq 2050000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \end{array} \]

Alternative 10: 31.6% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.1 \cdot 10^{+158}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+176}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.1e+158)
   (- im)
   (if (<= re 1.85e+176) (* (* re re) 0.75) (+ 13.5 (* (* re re) -6.75)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.1e+158) {
		tmp = -im;
	} else if (re <= 1.85e+176) {
		tmp = (re * re) * 0.75;
	} else {
		tmp = 13.5 + ((re * re) * -6.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.1d+158) then
        tmp = -im
    else if (re <= 1.85d+176) then
        tmp = (re * re) * 0.75d0
    else
        tmp = 13.5d0 + ((re * re) * (-6.75d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.1e+158) {
		tmp = -im;
	} else if (re <= 1.85e+176) {
		tmp = (re * re) * 0.75;
	} else {
		tmp = 13.5 + ((re * re) * -6.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.1e+158:
		tmp = -im
	elif re <= 1.85e+176:
		tmp = (re * re) * 0.75
	else:
		tmp = 13.5 + ((re * re) * -6.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.1e+158)
		tmp = Float64(-im);
	elseif (re <= 1.85e+176)
		tmp = Float64(Float64(re * re) * 0.75);
	else
		tmp = Float64(13.5 + Float64(Float64(re * re) * -6.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.1e+158)
		tmp = -im;
	elseif (re <= 1.85e+176)
		tmp = (re * re) * 0.75;
	else
		tmp = 13.5 + ((re * re) * -6.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.1e+158], (-im), If[LessEqual[re, 1.85e+176], N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision], N[(13.5 + N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.1000000000000001e158

   1. Initial program 53.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in im around 0 52.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 52.9%

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

      2. *-commutative 52.9%

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

      3. distribute-lft-neg-in 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in re around 0 32.2%

      \[\leadsto \color{blue}{-1 \cdot im} \]
   8. Step-by-step derivation

      1. neg-mul-1 32.2%

         \[\leadsto \color{blue}{-im} \]

   9. Simplified 32.2%

      \[\leadsto \color{blue}{-im} \]

   if 1.1000000000000001e158 < re < 1.8499999999999999e176

   1. Initial program 58.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 33.4%

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

      4. +-commutative 33.4%

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

      5. *-commutative 33.4%

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

      6. unpow2 33.4%

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

      7. associate-*l* 33.4%

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

   6. Simplified 33.4%

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

   8. Applied egg-rr 51.0%

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

   9. Taylor expanded in re around inf 51.0%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 51.0%

          \[\leadsto 0.75 \cdot \color{blue}{\left(re \cdot re\right)} \]

   11. Simplified 51.0%

       \[\leadsto \color{blue}{0.75 \cdot \left(re \cdot re\right)} \]

   if 1.8499999999999999e176 < re

   1. Initial program 44.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 44.7%

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

   3. Simplified 44.7%

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

   5. Applied egg-rr 2.6%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{27} \]

   6. Taylor expanded in re around 0 24.6%

      \[\leadsto \color{blue}{13.5 + -6.75 \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 24.6%

         \[\leadsto 13.5 + \color{blue}{{re}^{2} \cdot -6.75} \]

      2. unpow2 24.6%

         \[\leadsto 13.5 + \color{blue}{\left(re \cdot re\right)} \cdot -6.75 \]

   8. Simplified 24.6%

      \[\leadsto \color{blue}{13.5 + \left(re \cdot re\right) \cdot -6.75} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.1 \cdot 10^{+158}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+176}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 11: 35.7% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{+172}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.8e+172)
   (- (* 0.5 (* re (* im re))) im)
   (+ 13.5 (* (* re re) -6.75))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.8e+172) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = 13.5 + ((re * re) * -6.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.8d+172) then
        tmp = (0.5d0 * (re * (im * re))) - im
    else
        tmp = 13.5d0 + ((re * re) * (-6.75d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.8e+172) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = 13.5 + ((re * re) * -6.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.8e+172:
		tmp = (0.5 * (re * (im * re))) - im
	else:
		tmp = 13.5 + ((re * re) * -6.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.8e+172)
		tmp = Float64(Float64(0.5 * Float64(re * Float64(im * re))) - im);
	else
		tmp = Float64(13.5 + Float64(Float64(re * re) * -6.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.8e+172)
		tmp = (0.5 * (re * (im * re))) - im;
	else
		tmp = 13.5 + ((re * re) * -6.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.8e+172], N[(N[(0.5 * N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(13.5 + N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.79999999999999987e172

   1. Initial program 53.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in im around 0 53.1%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 53.1%

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

      2. *-commutative 53.1%

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

      3. distribute-lft-neg-in 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in re around 0 29.5%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({re}^{6} \cdot im\right) + \left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)\right)\right)} \]

   8. Taylor expanded in re around 0 37.8%

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

      1. +-commutative 37.8%

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

      2. mul-1-neg 37.8%

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

      3. unsub-neg 37.8%

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

      4. unpow2 37.8%

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

      5. associate-*l* 37.8%

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

   10. Simplified 37.8%

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

   if 1.79999999999999987e172 < re

   1. Initial program 47.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 47.8%

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

   3. Simplified 47.8%

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

   5. Applied egg-rr 2.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{27} \]

   6. Taylor expanded in re around 0 23.2%

      \[\leadsto \color{blue}{13.5 + -6.75 \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 23.2%

         \[\leadsto 13.5 + \color{blue}{{re}^{2} \cdot -6.75} \]

      2. unpow2 23.2%

         \[\leadsto 13.5 + \color{blue}{\left(re \cdot re\right)} \cdot -6.75 \]

   8. Simplified 23.2%

      \[\leadsto \color{blue}{13.5 + \left(re \cdot re\right) \cdot -6.75} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{+172}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 12: 31.8% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8.1 \cdot 10^{+157}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8.1e+157) (- im) (* (* re re) 0.75)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8.1e+157) {
		tmp = -im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 8.1d+157) then
        tmp = -im
    else
        tmp = (re * re) * 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 8.1e+157) {
		tmp = -im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8.1e+157:
		tmp = -im
	else:
		tmp = (re * re) * 0.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8.1e+157)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(re * re) * 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 8.1e+157)
		tmp = -im;
	else
		tmp = (re * re) * 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8.1e+157], (-im), N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.10000000000000009e157

   1. Initial program 53.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in im around 0 52.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 52.9%

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

      2. *-commutative 52.9%

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

      3. distribute-lft-neg-in 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in re around 0 32.2%

      \[\leadsto \color{blue}{-1 \cdot im} \]
   8. Step-by-step derivation

      1. neg-mul-1 32.2%

         \[\leadsto \color{blue}{-im} \]

   9. Simplified 32.2%

      \[\leadsto \color{blue}{-im} \]

   if 8.10000000000000009e157 < re

   1. Initial program 48.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 21.8%

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

      4. +-commutative 21.8%

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

      5. *-commutative 21.8%

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

      6. unpow2 21.8%

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

      7. associate-*l* 21.8%

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

   6. Simplified 21.8%

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

   8. Applied egg-rr 27.0%

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

   9. Taylor expanded in re around inf 27.0%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 27.0%

          \[\leadsto 0.75 \cdot \color{blue}{\left(re \cdot re\right)} \]

   11. Simplified 27.0%

       \[\leadsto \color{blue}{0.75 \cdot \left(re \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8.1 \cdot 10^{+157}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \]

Alternative 13: 29.8% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

function code(re, im)
	return Float64(-im)
end

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 52.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 52.9%

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

3. Simplified 52.9%

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

4. Taylor expanded in im around 0 53.4%

   \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 53.4%

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

   2. *-commutative 53.4%

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

   3. distribute-lft-neg-in 53.4%

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

6. Simplified 53.4%

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

7. Taylor expanded in re around 0 30.4%

   \[\leadsto \color{blue}{-1 \cdot im} \]
8. Step-by-step derivation

   1. neg-mul-1 30.4%

      \[\leadsto \color{blue}{-im} \]

9. Simplified 30.4%

   \[\leadsto \color{blue}{-im} \]

10. Final simplification 30.4%

    \[\leadsto -im \]

Alternative 14: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -3.0)

C

double code(double re, double im) {
	return -3.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -3.0d0
end function

Java

public static double code(double re, double im) {
	return -3.0;
}

Python

def code(re, im):
	return -3.0

Julia

function code(re, im)
	return -3.0
end

MATLAB

function tmp = code(re, im)
	tmp = -3.0;
end

Wolfram

code[re_, im_] := -3.0

Derivation

1. Initial program 52.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 52.9%

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

3. Simplified 52.9%

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

4. Taylor expanded in im around 0 85.2%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 85.2%

      \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

   2. unsub-neg 85.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

   3. *-commutative 85.2%

      \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

   4. associate-*l* 85.2%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

   5. distribute-lft-out-- 85.2%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

6. Simplified 85.2%

   \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

8. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-3} \]

9. Final simplification 2.9%

   \[\leadsto -3 \]

Alternative 15: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -9.92290301275212 \cdot 10^{-8} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -9.92290301275212e-8)

C

double code(double re, double im) {
	return -9.92290301275212e-8;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -9.92290301275212d-8
end function

Java

public static double code(double re, double im) {
	return -9.92290301275212e-8;
}

Python

def code(re, im):
	return -9.92290301275212e-8

Julia

function code(re, im)
	return -9.92290301275212e-8
end

MATLAB

function tmp = code(re, im)
	tmp = -9.92290301275212e-8;
end

Wolfram

code[re_, im_] := -9.92290301275212e-8

Derivation

1. Initial program 52.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 52.9%

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

3. Simplified 52.9%

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

4. Taylor expanded in im around 0 85.2%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 85.2%

      \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

   2. unsub-neg 85.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

   3. *-commutative 85.2%

      \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

   4. associate-*l* 85.2%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

   5. distribute-lft-out-- 85.2%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

6. Simplified 85.2%

   \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

8. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{-9.92290301275212 \cdot 10^{-8}} \]

9. Final simplification 3.0%

   \[\leadsto -9.92290301275212 \cdot 10^{-8} \]

Alternative 16: 3.5% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 52.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 52.9%

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

3. Simplified 52.9%

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

4. Taylor expanded in im around 0 85.2%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 85.2%

      \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

   2. unsub-neg 85.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

   3. *-commutative 85.2%

      \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

   4. associate-*l* 85.2%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

   5. distribute-lft-out-- 85.2%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

6. Simplified 85.2%

   \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

8. Applied egg-rr 3.6%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 3.6%

   \[\leadsto 0 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - 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 (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023224 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))