math.sin on complex, imaginary part

Percentage Accurate: 54.7% → 99.8%

Time: 10.0s

Alternatives: 17

Speedup: 2.8×

• 50.1% 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: 54.7% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.4 \lor \neg \left(t_0 \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.4) (not (<= t_0 5e-9)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.4) || !(t_0 <= 5e-9)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - 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 <= (-0.4d0)) .or. (.not. (t_0 <= 5d-9))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - 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 <= -0.4) || !(t_0 <= 5e-9)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.4) or not (t_0 <= 5e-9):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.4) || !(t_0 <= 5e-9))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.4) || ~((t_0 <= 5e-9)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - 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, -0.4], N[Not[LessEqual[t$95$0, 5e-9]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.40000000000000002 or 5.0000000000000001e-9 < (-.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 -0.40000000000000002 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 5.0000000000000001e-9

   1. Initial program 7.7%

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

      1. sub0-neg 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. distribute-lft-out-- 99.9%

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

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.4 \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -2.5 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -11.5:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;im \leq 0.046 \lor \neg \left(im \leq 5.7 \cdot 10^{+102}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

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 -2.5e+116)
     t_1
     (if (<= im -11.5)
       (* 0.5 t_0)
       (if (or (<= im 0.046) (not (<= im 5.7e+102)))
         t_1
         (* t_0 (+ 0.5 (* re (* re -0.25)))))))))

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 <= -2.5e+116) {
		tmp = t_1;
	} else if (im <= -11.5) {
		tmp = 0.5 * t_0;
	} else if ((im <= 0.046) || !(im <= 5.7e+102)) {
		tmp = t_1;
	} else {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: 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 <= (-2.5d+116)) then
        tmp = t_1
    else if (im <= (-11.5d0)) then
        tmp = 0.5d0 * t_0
    else if ((im <= 0.046d0) .or. (.not. (im <= 5.7d+102))) then
        tmp = t_1
    else
        tmp = t_0 * (0.5d0 + (re * (re * (-0.25d0))))
    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 <= -2.5e+116) {
		tmp = t_1;
	} else if (im <= -11.5) {
		tmp = 0.5 * t_0;
	} else if ((im <= 0.046) || !(im <= 5.7e+102)) {
		tmp = t_1;
	} else {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	}
	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 <= -2.5e+116:
		tmp = t_1
	elif im <= -11.5:
		tmp = 0.5 * t_0
	elif (im <= 0.046) or not (im <= 5.7e+102):
		tmp = t_1
	else:
		tmp = t_0 * (0.5 + (re * (re * -0.25)))
	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 <= -2.5e+116)
		tmp = t_1;
	elseif (im <= -11.5)
		tmp = Float64(0.5 * t_0);
	elseif ((im <= 0.046) || !(im <= 5.7e+102))
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	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 <= -2.5e+116)
		tmp = t_1;
	elseif (im <= -11.5)
		tmp = 0.5 * t_0;
	elseif ((im <= 0.046) || ~((im <= 5.7e+102)))
		tmp = t_1;
	else
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	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, -2.5e+116], t$95$1, If[LessEqual[im, -11.5], N[(0.5 * t$95$0), $MachinePrecision], If[Or[LessEqual[im, 0.046], N[Not[LessEqual[im, 5.7e+102]], $MachinePrecision]], t$95$1, N[(t$95$0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.50000000000000013e116 or -11.5 < im < 0.045999999999999999 or 5.6999999999999999e102 < im

   1. Initial program 40.8%

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

      1. sub0-neg 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in im around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*l* 99.5%

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

      5. distribute-lft-out-- 99.5%

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

   6. Simplified 99.5%

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

   if -2.50000000000000013e116 < im < -11.5

   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 80.0%

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

   if 0.045999999999999999 < im < 5.6999999999999999e102

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 15.3%

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

      1. *-commutative 15.3%

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

      2. associate-*r* 15.3%

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

      3. distribute-rgt-out 92.2%

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

      4. +-commutative 92.2%

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

      5. *-commutative 92.2%

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

      6. unpow2 92.2%

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

      7. associate-*l* 92.2%

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

   6. Simplified 92.2%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.5 \cdot 10^{+116}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -11.5:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.046 \lor \neg \left(im \leq 5.7 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 3: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.2 \cdot 10^{+116} \lor \neg \left(im \leq -11.5 \lor \neg \left(im \leq 0.046\right) \land im \leq 5.7 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.2e+116)
         (not (or (<= im -11.5) (and (not (<= im 0.046)) (<= im 5.7e+102)))))
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* 0.5 (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.2e+116) || !((im <= -11.5) || (!(im <= 0.046) && (im <= 5.7e+102)))) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-2.2d+116)) .or. (.not. (im <= (-11.5d0)) .or. (.not. (im <= 0.046d0)) .and. (im <= 5.7d+102))) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = 0.5d0 * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.2e+116) || !((im <= -11.5) || (!(im <= 0.046) && (im <= 5.7e+102)))) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.2e+116) or not ((im <= -11.5) or (not (im <= 0.046) and (im <= 5.7e+102))):
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.2e+116) || !((im <= -11.5) || (!(im <= 0.046) && (im <= 5.7e+102))))
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.2e+116) || ~(((im <= -11.5) || (~((im <= 0.046)) && (im <= 5.7e+102)))))
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = 0.5 * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.2e+116], N[Not[Or[LessEqual[im, -11.5], And[N[Not[LessEqual[im, 0.046]], $MachinePrecision], LessEqual[im, 5.7e+102]]]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.2e116 or -11.5 < im < 0.045999999999999999 or 5.6999999999999999e102 < im

   1. Initial program 40.8%

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

      1. sub0-neg 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in im around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*l* 99.5%

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

      5. distribute-lft-out-- 99.5%

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

   6. Simplified 99.5%

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

   if -2.2e116 < im < -11.5 or 0.045999999999999999 < im < 5.6999999999999999e102

   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 79.1%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.2 \cdot 10^{+116} \lor \neg \left(im \leq -11.5 \lor \neg \left(im \leq 0.046\right) \land im \leq 5.7 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \frac{\cos re \cdot \left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -11.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0005:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1
         (/
          (* (cos re) (- 1.4551915228366852e-11 (* im im)))
          (+ im -3.814697265625e-6))))
   (if (<= im -1.32e+154)
     t_1
     (if (<= im -11.5)
       t_0
       (if (<= im 0.0005)
         (* (cos re) (- im))
         (if (<= im 1.35e+154) 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) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6);
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_1;
	} else if (im <= -11.5) {
		tmp = t_0;
	} else if (im <= 0.0005) {
		tmp = cos(re) * -im;
	} else if (im <= 1.35e+154) {
		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) * (1.4551915228366852d-11 - (im * im))) / (im + (-3.814697265625d-6))
    if (im <= (-1.32d+154)) then
        tmp = t_1
    else if (im <= (-11.5d0)) then
        tmp = t_0
    else if (im <= 0.0005d0) then
        tmp = cos(re) * -im
    else if (im <= 1.35d+154) 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) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6);
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_1;
	} else if (im <= -11.5) {
		tmp = t_0;
	} else if (im <= 0.0005) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.35e+154) {
		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) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6)
	tmp = 0
	if im <= -1.32e+154:
		tmp = t_1
	elif im <= -11.5:
		tmp = t_0
	elif im <= 0.0005:
		tmp = math.cos(re) * -im
	elif im <= 1.35e+154:
		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(Float64(cos(re) * Float64(1.4551915228366852e-11 - Float64(im * im))) / Float64(im + -3.814697265625e-6))
	tmp = 0.0
	if (im <= -1.32e+154)
		tmp = t_1;
	elseif (im <= -11.5)
		tmp = t_0;
	elseif (im <= 0.0005)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.35e+154)
		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) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6);
	tmp = 0.0;
	if (im <= -1.32e+154)
		tmp = t_1;
	elseif (im <= -11.5)
		tmp = t_0;
	elseif (im <= 0.0005)
		tmp = cos(re) * -im;
	elseif (im <= 1.35e+154)
		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[(N[Cos[re], $MachinePrecision] * N[(1.4551915228366852e-11 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.814697265625e-6), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.32e+154], t$95$1, If[LessEqual[im, -11.5], t$95$0, If[LessEqual[im, 0.0005], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.31999999999999998e154 or 1.35000000000000003e154 < 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 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.9%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3.814697265625 \cdot 10^{-6}} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.9%

         \[\leadsto \color{blue}{\left(-3.814697265625 \cdot 10^{-6} - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im}{-3.814697265625 \cdot 10^{-6} + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{1.4551915228366852 \cdot 10^{-11}} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3.814697265625 \cdot 10^{-6}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{im + -3.814697265625 \cdot 10^{-6}}} \]

   if -1.31999999999999998e154 < im < -11.5 or 5.0000000000000001e-4 < im < 1.35000000000000003e154

   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 82.4%

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

   if -11.5 < im < 5.0000000000000001e-4

   1. Initial program 8.3%

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

      1. sub0-neg 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in im around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. *-commutative 99.2%

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

      3. distribute-lft-neg-in 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{elif}\;im \leq -11.5:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.0005:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \end{array} \]

Alternative 5: 82.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := \frac{\cos re \cdot \left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -2.35 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 0.062:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0 - im\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1
         (/
          (* (cos re) (- 1.4551915228366852e-11 (* im im)))
          (+ im -3.814697265625e-6))))
   (if (<= im -1.32e+154)
     t_1
     (if (<= im -2.35e+92)
       t_0
       (if (<= im -1.55e+23)
         (- (* (* re re) (* im 0.5)) im)
         (if (<= im 0.062)
           (* (cos re) (- im))
           (if (<= im 1.35e+154) (- t_0 im) t_1)))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (cos(re) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6);
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_1;
	} else if (im <= -2.35e+92) {
		tmp = t_0;
	} else if (im <= -1.55e+23) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 0.062) {
		tmp = cos(re) * -im;
	} else if (im <= 1.35e+154) {
		tmp = t_0 - im;
	} 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 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = (cos(re) * (1.4551915228366852d-11 - (im * im))) / (im + (-3.814697265625d-6))
    if (im <= (-1.32d+154)) then
        tmp = t_1
    else if (im <= (-2.35d+92)) then
        tmp = t_0
    else if (im <= (-1.55d+23)) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else if (im <= 0.062d0) then
        tmp = cos(re) * -im
    else if (im <= 1.35d+154) then
        tmp = t_0 - im
    else
        tmp = t_1
    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;
	double t_1 = (Math.cos(re) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6);
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_1;
	} else if (im <= -2.35e+92) {
		tmp = t_0;
	} else if (im <= -1.55e+23) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 0.062) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.35e+154) {
		tmp = t_0 - im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = (math.cos(re) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6)
	tmp = 0
	if im <= -1.32e+154:
		tmp = t_1
	elif im <= -2.35e+92:
		tmp = t_0
	elif im <= -1.55e+23:
		tmp = ((re * re) * (im * 0.5)) - im
	elif im <= 0.062:
		tmp = math.cos(re) * -im
	elif im <= 1.35e+154:
		tmp = t_0 - im
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(Float64(cos(re) * Float64(1.4551915228366852e-11 - Float64(im * im))) / Float64(im + -3.814697265625e-6))
	tmp = 0.0
	if (im <= -1.32e+154)
		tmp = t_1;
	elseif (im <= -2.35e+92)
		tmp = t_0;
	elseif (im <= -1.55e+23)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	elseif (im <= 0.062)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.35e+154)
		tmp = Float64(t_0 - im);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = (cos(re) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6);
	tmp = 0.0;
	if (im <= -1.32e+154)
		tmp = t_1;
	elseif (im <= -2.35e+92)
		tmp = t_0;
	elseif (im <= -1.55e+23)
		tmp = ((re * re) * (im * 0.5)) - im;
	elseif (im <= 0.062)
		tmp = cos(re) * -im;
	elseif (im <= 1.35e+154)
		tmp = t_0 - im;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(1.4551915228366852e-11 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.814697265625e-6), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.32e+154], t$95$1, If[LessEqual[im, -2.35e+92], t$95$0, If[LessEqual[im, -1.55e+23], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 0.062], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(t$95$0 - im), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -1.31999999999999998e154 or 1.35000000000000003e154 < 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 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.9%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3.814697265625 \cdot 10^{-6}} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.9%

         \[\leadsto \color{blue}{\left(-3.814697265625 \cdot 10^{-6} - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im}{-3.814697265625 \cdot 10^{-6} + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{1.4551915228366852 \cdot 10^{-11}} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3.814697265625 \cdot 10^{-6}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{im + -3.814697265625 \cdot 10^{-6}}} \]

   if -1.31999999999999998e154 < im < -2.35e92

   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 92.4%

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

      1. mul-1-neg 92.4%

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

      2. unsub-neg 92.4%

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

      3. *-commutative 92.4%

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

      4. associate-*l* 92.4%

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

      5. distribute-lft-out-- 92.4%

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

   6. Simplified 92.4%

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

   7. Taylor expanded in re around 0 84.0%

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

   8. Taylor expanded in im around inf 84.0%

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

   if -2.35e92 < im < -1.54999999999999985e23

   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 23.4%

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

      1. neg-mul-1 23.4%

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

      2. +-commutative 23.4%

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

      3. unsub-neg 23.4%

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

      4. *-commutative 23.4%

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

      5. associate-*l* 23.4%

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

      6. unpow2 23.4%

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

   9. Simplified 23.4%

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

   if -1.54999999999999985e23 < im < 0.062

   1. Initial program 13.5%

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

      1. sub0-neg 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in im around 0 93.8%

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

      1. mul-1-neg 93.8%

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

      2. *-commutative 93.8%

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

      3. distribute-lft-neg-in 93.8%

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

   6. Simplified 93.8%

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

   if 0.062 < im < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in im around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. unsub-neg 48.7%

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

      3. *-commutative 48.7%

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

      4. associate-*l* 48.7%

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

      5. distribute-lft-out-- 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in re around 0 43.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{elif}\;im \leq -2.35 \cdot 10^{+92}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 0.062:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \end{array} \]

Alternative 6: 82.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := \frac{\cos re \cdot \left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ t_2 := t_0 - im\\ \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -5 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.2 \cdot 10^{+23}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot t_2\\ \mathbf{elif}\;im \leq 0.0027:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1
         (/
          (* (cos re) (- 1.4551915228366852e-11 (* im im)))
          (+ im -3.814697265625e-6)))
        (t_2 (- t_0 im)))
   (if (<= im -1.32e+154)
     t_1
     (if (<= im -5e+128)
       t_0
       (if (<= im -2.2e+23)
         (* (+ (* -0.5 (* re re)) 1.0) t_2)
         (if (<= im 0.0027)
           (* (cos re) (- im))
           (if (<= im 5e+153) t_2 t_1)))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (cos(re) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6);
	double t_2 = t_0 - im;
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_1;
	} else if (im <= -5e+128) {
		tmp = t_0;
	} else if (im <= -2.2e+23) {
		tmp = ((-0.5 * (re * re)) + 1.0) * t_2;
	} else if (im <= 0.0027) {
		tmp = cos(re) * -im;
	} else if (im <= 5e+153) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = (cos(re) * (1.4551915228366852d-11 - (im * im))) / (im + (-3.814697265625d-6))
    t_2 = t_0 - im
    if (im <= (-1.32d+154)) then
        tmp = t_1
    else if (im <= (-5d+128)) then
        tmp = t_0
    else if (im <= (-2.2d+23)) then
        tmp = (((-0.5d0) * (re * re)) + 1.0d0) * t_2
    else if (im <= 0.0027d0) then
        tmp = cos(re) * -im
    else if (im <= 5d+153) then
        tmp = t_2
    else
        tmp = t_1
    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;
	double t_1 = (Math.cos(re) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6);
	double t_2 = t_0 - im;
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_1;
	} else if (im <= -5e+128) {
		tmp = t_0;
	} else if (im <= -2.2e+23) {
		tmp = ((-0.5 * (re * re)) + 1.0) * t_2;
	} else if (im <= 0.0027) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 5e+153) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = (math.cos(re) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6)
	t_2 = t_0 - im
	tmp = 0
	if im <= -1.32e+154:
		tmp = t_1
	elif im <= -5e+128:
		tmp = t_0
	elif im <= -2.2e+23:
		tmp = ((-0.5 * (re * re)) + 1.0) * t_2
	elif im <= 0.0027:
		tmp = math.cos(re) * -im
	elif im <= 5e+153:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(Float64(cos(re) * Float64(1.4551915228366852e-11 - Float64(im * im))) / Float64(im + -3.814697265625e-6))
	t_2 = Float64(t_0 - im)
	tmp = 0.0
	if (im <= -1.32e+154)
		tmp = t_1;
	elseif (im <= -5e+128)
		tmp = t_0;
	elseif (im <= -2.2e+23)
		tmp = Float64(Float64(Float64(-0.5 * Float64(re * re)) + 1.0) * t_2);
	elseif (im <= 0.0027)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 5e+153)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = (cos(re) * (1.4551915228366852e-11 - (im * im))) / (im + -3.814697265625e-6);
	t_2 = t_0 - im;
	tmp = 0.0;
	if (im <= -1.32e+154)
		tmp = t_1;
	elseif (im <= -5e+128)
		tmp = t_0;
	elseif (im <= -2.2e+23)
		tmp = ((-0.5 * (re * re)) + 1.0) * t_2;
	elseif (im <= 0.0027)
		tmp = cos(re) * -im;
	elseif (im <= 5e+153)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(1.4551915228366852e-11 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.814697265625e-6), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - im), $MachinePrecision]}, If[LessEqual[im, -1.32e+154], t$95$1, If[LessEqual[im, -5e+128], t$95$0, If[LessEqual[im, -2.2e+23], N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[im, 0.0027], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 5e+153], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -1.31999999999999998e154 or 5.00000000000000018e153 < 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 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.9%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3.814697265625 \cdot 10^{-6}} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.9%

         \[\leadsto \color{blue}{\left(-3.814697265625 \cdot 10^{-6} - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im}{-3.814697265625 \cdot 10^{-6} + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{1.4551915228366852 \cdot 10^{-11}} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3.814697265625 \cdot 10^{-6}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{im + -3.814697265625 \cdot 10^{-6}}} \]

   if -1.31999999999999998e154 < im < -5e128

   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 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

   if -5e128 < im < -2.20000000000000008e23

   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 18.7%

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

      1. mul-1-neg 18.7%

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

      2. unsub-neg 18.7%

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

      3. *-commutative 18.7%

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

      4. associate-*l* 18.7%

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

      5. distribute-lft-out-- 18.7%

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

   6. Simplified 18.7%

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

   7. Taylor expanded in re around 0 20.8%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. associate--l+ 20.8%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      2. associate-*r* 20.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      3. distribute-lft1-in 34.6%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      4. unpow2 34.6%

         \[\leadsto \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

   9. Simplified 34.6%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

   if -2.20000000000000008e23 < im < 0.0027000000000000001

   1. Initial program 13.5%

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

      1. sub0-neg 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in im around 0 93.8%

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

      1. mul-1-neg 93.8%

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

      2. *-commutative 93.8%

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

      3. distribute-lft-neg-in 93.8%

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

   6. Simplified 93.8%

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

   if 0.0027000000000000001 < im < 5.00000000000000018e153

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in im around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. unsub-neg 48.7%

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

      3. *-commutative 48.7%

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

      4. associate-*l* 48.7%

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

      5. distribute-lft-out-- 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in re around 0 43.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{elif}\;im \leq -5 \cdot 10^{+128}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -2.2 \cdot 10^{+23}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 0.0027:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+153}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \end{array} \]

Alternative 7: 55.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ t_1 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.35 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (/
          (* (- 1.4551915228366852e-11 (* im im)) (+ (* -0.5 (* re re)) 1.0))
          (+ im -3.814697265625e-6)))
        (t_1 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -1.6e+154)
     t_0
     (if (<= im -2.35e+92)
       t_1
       (if (<= im -1.55e+23)
         (- (* (* re re) (* im 0.5)) im)
         (if (<= im 2.15e-7) (- im) (if (<= im 4.3e+95) t_0 t_1)))))))

C

double code(double re, double im) {
	double t_0 = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	double t_1 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -1.6e+154) {
		tmp = t_0;
	} else if (im <= -2.35e+92) {
		tmp = t_1;
	} else if (im <= -1.55e+23) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 2.15e-7) {
		tmp = -im;
	} else if (im <= 4.3e+95) {
		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 = ((1.4551915228366852d-11 - (im * im)) * (((-0.5d0) * (re * re)) + 1.0d0)) / (im + (-3.814697265625d-6))
    t_1 = (im ** 3.0d0) * (-0.16666666666666666d0)
    if (im <= (-1.6d+154)) then
        tmp = t_0
    else if (im <= (-2.35d+92)) then
        tmp = t_1
    else if (im <= (-1.55d+23)) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else if (im <= 2.15d-7) then
        tmp = -im
    else if (im <= 4.3d+95) 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 = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	double t_1 = Math.pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -1.6e+154) {
		tmp = t_0;
	} else if (im <= -2.35e+92) {
		tmp = t_1;
	} else if (im <= -1.55e+23) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 2.15e-7) {
		tmp = -im;
	} else if (im <= 4.3e+95) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6)
	t_1 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -1.6e+154:
		tmp = t_0
	elif im <= -2.35e+92:
		tmp = t_1
	elif im <= -1.55e+23:
		tmp = ((re * re) * (im * 0.5)) - im
	elif im <= 2.15e-7:
		tmp = -im
	elif im <= 4.3e+95:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(1.4551915228366852e-11 - Float64(im * im)) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0)) / Float64(im + -3.814697265625e-6))
	t_1 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -1.6e+154)
		tmp = t_0;
	elseif (im <= -2.35e+92)
		tmp = t_1;
	elseif (im <= -1.55e+23)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	elseif (im <= 2.15e-7)
		tmp = Float64(-im);
	elseif (im <= 4.3e+95)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	t_1 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -1.6e+154)
		tmp = t_0;
	elseif (im <= -2.35e+92)
		tmp = t_1;
	elseif (im <= -1.55e+23)
		tmp = ((re * re) * (im * 0.5)) - im;
	elseif (im <= 2.15e-7)
		tmp = -im;
	elseif (im <= 4.3e+95)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(1.4551915228366852e-11 - N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(im + -3.814697265625e-6), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -1.6e+154], t$95$0, If[LessEqual[im, -2.35e+92], t$95$1, If[LessEqual[im, -1.55e+23], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 2.15e-7], (-im), If[LessEqual[im, 4.3e+95], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.6e154 or 2.1500000000000001e-7 < im < 4.3e95

   1. Initial program 99.3%

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

      1. sub0-neg 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in im around 0 74.7%

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

      1. mul-1-neg 74.7%

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

      2. unsub-neg 74.7%

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

      3. *-commutative 74.7%

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

      4. associate-*l* 74.7%

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

      5. distribute-lft-out-- 74.7%

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

   6. Simplified 74.7%

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

   8. Applied egg-rr 7.1%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3.814697265625 \cdot 10^{-6}} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.1%

         \[\leadsto \color{blue}{\left(-3.814697265625 \cdot 10^{-6} - im\right) \cdot \cos re} \]

      2. flip-- 72.0%

         \[\leadsto \color{blue}{\frac{-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im}{-3.814697265625 \cdot 10^{-6} + im}} \cdot \cos re \]

      3. associate-*l/ 72.0%

         \[\leadsto \color{blue}{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im}} \]

      4. metadata-eval 72.0%

         \[\leadsto \frac{\left(\color{blue}{1.4551915228366852 \cdot 10^{-11}} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im} \]

      5. +-commutative 72.0%

         \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3.814697265625 \cdot 10^{-6}}} \]

   10. Applied egg-rr 72.0%

       \[\leadsto \color{blue}{\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{im + -3.814697265625 \cdot 10^{-6}}} \]

   11. Taylor expanded in re around 0 8.6%

       \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} + -0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right)\right) - {im}^{2}}}{im + -3.814697265625 \cdot 10^{-6}} \]
   12. Step-by-step derivation

       1. sub-neg 8.6%

          \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} + -0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right)\right) + \left(-{im}^{2}\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       2. +-commutative 8.6%

          \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) + 1.4551915228366852 \cdot 10^{-11}\right)} + \left(-{im}^{2}\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       3. associate-+l+ 8.6%

          \[\leadsto \frac{\color{blue}{-0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       4. *-commutative 8.6%

          \[\leadsto \frac{\color{blue}{\left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) \cdot -0.5} + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       5. unpow2 8.6%

          \[\leadsto \frac{\left(\left(1.4551915228366852 \cdot 10^{-11} - \color{blue}{im \cdot im}\right) \cdot {re}^{2}\right) \cdot -0.5 + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       6. associate-*l* 8.6%

          \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right)} + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       7. sub-neg 8.6%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       8. unpow2 8.6%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \left(1.4551915228366852 \cdot 10^{-11} - \color{blue}{im \cdot im}\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       9. *-rgt-identity 8.6%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot 1}}{im + -3.814697265625 \cdot 10^{-6}} \]

       10. distribute-lft-out 69.1%

           \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5 + 1\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       11. unpow2 69.1%

           \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5 + 1\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

   13. Simplified 69.1%

       \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

   if -1.6e154 < im < -2.35e92 or 4.3e95 < 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 96.1%

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

      1. mul-1-neg 96.1%

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

      2. unsub-neg 96.1%

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

      3. *-commutative 96.1%

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

      4. associate-*l* 96.1%

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

      5. distribute-lft-out-- 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in re around 0 81.2%

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

   8. Taylor expanded in im around inf 81.2%

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

   if -2.35e92 < im < -1.54999999999999985e23

   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 23.4%

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

      1. neg-mul-1 23.4%

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

      2. +-commutative 23.4%

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

      3. unsub-neg 23.4%

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

      4. *-commutative 23.4%

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

      5. associate-*l* 23.4%

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

      6. unpow2 23.4%

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

   9. Simplified 23.4%

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

   if -1.54999999999999985e23 < im < 2.1500000000000001e-7

   1. Initial program 13.0%

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

      1. sub0-neg 13.0%

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

   3. Simplified 13.0%

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

   4. Taylor expanded in im around 0 93.9%

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

      1. mul-1-neg 93.9%

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

      2. unsub-neg 93.9%

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

      3. *-commutative 93.9%

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

      4. associate-*l* 93.9%

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

      5. distribute-lft-out-- 93.9%

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

   6. Simplified 93.9%

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

   7. Taylor expanded in re around 0 56.1%

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

   8. Taylor expanded in im around 0 56.0%

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

      1. neg-mul-1 56.0%

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

   10. Simplified 56.0%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{elif}\;im \leq -2.35 \cdot 10^{+92}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+95}:\\ \;\;\;\;\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 8: 75.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{elif}\;im \leq -2.7 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+24}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -2e+155)
     (/
      (* (- 1.4551915228366852e-11 (* im im)) (+ (* -0.5 (* re re)) 1.0))
      (+ im -3.814697265625e-6))
     (if (<= im -2.7e+91)
       t_0
       (if (<= im -8e+24)
         (- (* (* re re) (* im 0.5)) im)
         (if (<= im 7e-6) (* (cos re) (- im)) (- t_0 im)))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -2e+155) {
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	} else if (im <= -2.7e+91) {
		tmp = t_0;
	} else if (im <= -8e+24) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 7e-6) {
		tmp = cos(re) * -im;
	} else {
		tmp = t_0 - 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 = (im ** 3.0d0) * (-0.16666666666666666d0)
    if (im <= (-2d+155)) then
        tmp = ((1.4551915228366852d-11 - (im * im)) * (((-0.5d0) * (re * re)) + 1.0d0)) / (im + (-3.814697265625d-6))
    else if (im <= (-2.7d+91)) then
        tmp = t_0
    else if (im <= (-8d+24)) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else if (im <= 7d-6) then
        tmp = cos(re) * -im
    else
        tmp = t_0 - im
    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;
	double tmp;
	if (im <= -2e+155) {
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	} else if (im <= -2.7e+91) {
		tmp = t_0;
	} else if (im <= -8e+24) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 7e-6) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -2e+155:
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6)
	elif im <= -2.7e+91:
		tmp = t_0
	elif im <= -8e+24:
		tmp = ((re * re) * (im * 0.5)) - im
	elif im <= 7e-6:
		tmp = math.cos(re) * -im
	else:
		tmp = t_0 - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -2e+155)
		tmp = Float64(Float64(Float64(1.4551915228366852e-11 - Float64(im * im)) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0)) / Float64(im + -3.814697265625e-6));
	elseif (im <= -2.7e+91)
		tmp = t_0;
	elseif (im <= -8e+24)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	elseif (im <= 7e-6)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -2e+155)
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	elseif (im <= -2.7e+91)
		tmp = t_0;
	elseif (im <= -8e+24)
		tmp = ((re * re) * (im * 0.5)) - im;
	elseif (im <= 7e-6)
		tmp = cos(re) * -im;
	else
		tmp = t_0 - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -2e+155], N[(N[(N[(1.4551915228366852e-11 - N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(im + -3.814697265625e-6), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -2.7e+91], t$95$0, If[LessEqual[im, -8e+24], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 7e-6], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(t$95$0 - im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -2.00000000000000001e155

   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 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.9%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3.814697265625 \cdot 10^{-6}} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.9%

         \[\leadsto \color{blue}{\left(-3.814697265625 \cdot 10^{-6} - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im}{-3.814697265625 \cdot 10^{-6} + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{1.4551915228366852 \cdot 10^{-11}} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3.814697265625 \cdot 10^{-6}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{im + -3.814697265625 \cdot 10^{-6}}} \]

   11. Taylor expanded in re around 0 0.0%

       \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} + -0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right)\right) - {im}^{2}}}{im + -3.814697265625 \cdot 10^{-6}} \]
   12. Step-by-step derivation

       1. sub-neg 0.0%

          \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} + -0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right)\right) + \left(-{im}^{2}\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       2. +-commutative 0.0%

          \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) + 1.4551915228366852 \cdot 10^{-11}\right)} + \left(-{im}^{2}\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       3. associate-+l+ 0.0%

          \[\leadsto \frac{\color{blue}{-0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       4. *-commutative 0.0%

          \[\leadsto \frac{\color{blue}{\left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) \cdot -0.5} + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       5. unpow2 0.0%

          \[\leadsto \frac{\left(\left(1.4551915228366852 \cdot 10^{-11} - \color{blue}{im \cdot im}\right) \cdot {re}^{2}\right) \cdot -0.5 + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       6. associate-*l* 0.0%

          \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right)} + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       7. sub-neg 0.0%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       8. unpow2 0.0%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \left(1.4551915228366852 \cdot 10^{-11} - \color{blue}{im \cdot im}\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       9. *-rgt-identity 0.0%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot 1}}{im + -3.814697265625 \cdot 10^{-6}} \]

       10. distribute-lft-out 86.7%

           \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5 + 1\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       11. unpow2 86.7%

           \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5 + 1\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

   13. Simplified 86.7%

       \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

   if -2.00000000000000001e155 < im < -2.7e91

   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 92.4%

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

      1. mul-1-neg 92.4%

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

      2. unsub-neg 92.4%

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

      3. *-commutative 92.4%

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

      4. associate-*l* 92.4%

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

      5. distribute-lft-out-- 92.4%

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

   6. Simplified 92.4%

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

   7. Taylor expanded in re around 0 84.0%

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

   8. Taylor expanded in im around inf 84.0%

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

   if -2.7e91 < im < -7.9999999999999999e24

   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 23.4%

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

      1. neg-mul-1 23.4%

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

      2. +-commutative 23.4%

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

      3. unsub-neg 23.4%

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

      4. *-commutative 23.4%

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

      5. associate-*l* 23.4%

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

      6. unpow2 23.4%

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

   9. Simplified 23.4%

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

   if -7.9999999999999999e24 < im < 6.99999999999999989e-6

   1. Initial program 13.5%

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

      1. sub0-neg 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in im around 0 93.8%

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

      1. mul-1-neg 93.8%

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

      2. *-commutative 93.8%

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

      3. distribute-lft-neg-in 93.8%

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

   6. Simplified 93.8%

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

   if 6.99999999999999989e-6 < 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 74.9%

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

      1. mul-1-neg 74.9%

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

      2. unsub-neg 74.9%

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

      3. *-commutative 74.9%

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

      4. associate-*l* 74.9%

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

      5. distribute-lft-out-- 74.9%

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

   6. Simplified 74.9%

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

   7. Taylor expanded in re around 0 61.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{elif}\;im \leq -2.7 \cdot 10^{+91}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+24}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 9: 75.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{elif}\;im \leq -1.3 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -3e+154)
     (/
      (* (- 1.4551915228366852e-11 (* im im)) (+ (* -0.5 (* re re)) 1.0))
      (+ im -3.814697265625e-6))
     (if (<= im -1.3e+92)
       t_0
       (if (<= im -1.55e+23)
         (- (* (* re re) (* im 0.5)) im)
         (if (<= im 1.05e+85) (* (cos re) (- im)) t_0))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -3e+154) {
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	} else if (im <= -1.3e+92) {
		tmp = t_0;
	} else if (im <= -1.55e+23) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 1.05e+85) {
		tmp = cos(re) * -im;
	} 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)
    if (im <= (-3d+154)) then
        tmp = ((1.4551915228366852d-11 - (im * im)) * (((-0.5d0) * (re * re)) + 1.0d0)) / (im + (-3.814697265625d-6))
    else if (im <= (-1.3d+92)) then
        tmp = t_0
    else if (im <= (-1.55d+23)) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else if (im <= 1.05d+85) then
        tmp = cos(re) * -im
    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;
	double tmp;
	if (im <= -3e+154) {
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	} else if (im <= -1.3e+92) {
		tmp = t_0;
	} else if (im <= -1.55e+23) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (im <= 1.05e+85) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -3e+154:
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6)
	elif im <= -1.3e+92:
		tmp = t_0
	elif im <= -1.55e+23:
		tmp = ((re * re) * (im * 0.5)) - im
	elif im <= 1.05e+85:
		tmp = math.cos(re) * -im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -3e+154)
		tmp = Float64(Float64(Float64(1.4551915228366852e-11 - Float64(im * im)) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0)) / Float64(im + -3.814697265625e-6));
	elseif (im <= -1.3e+92)
		tmp = t_0;
	elseif (im <= -1.55e+23)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	elseif (im <= 1.05e+85)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -3e+154)
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	elseif (im <= -1.3e+92)
		tmp = t_0;
	elseif (im <= -1.55e+23)
		tmp = ((re * re) * (im * 0.5)) - im;
	elseif (im <= 1.05e+85)
		tmp = cos(re) * -im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -3e+154], N[(N[(N[(1.4551915228366852e-11 - N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(im + -3.814697265625e-6), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -1.3e+92], t$95$0, If[LessEqual[im, -1.55e+23], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 1.05e+85], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.00000000000000026e154

   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 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.9%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3.814697265625 \cdot 10^{-6}} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.9%

         \[\leadsto \color{blue}{\left(-3.814697265625 \cdot 10^{-6} - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im}{-3.814697265625 \cdot 10^{-6} + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{1.4551915228366852 \cdot 10^{-11}} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3.814697265625 \cdot 10^{-6}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{im + -3.814697265625 \cdot 10^{-6}}} \]

   11. Taylor expanded in re around 0 0.0%

       \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} + -0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right)\right) - {im}^{2}}}{im + -3.814697265625 \cdot 10^{-6}} \]
   12. Step-by-step derivation

       1. sub-neg 0.0%

          \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} + -0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right)\right) + \left(-{im}^{2}\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       2. +-commutative 0.0%

          \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) + 1.4551915228366852 \cdot 10^{-11}\right)} + \left(-{im}^{2}\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       3. associate-+l+ 0.0%

          \[\leadsto \frac{\color{blue}{-0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       4. *-commutative 0.0%

          \[\leadsto \frac{\color{blue}{\left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) \cdot -0.5} + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       5. unpow2 0.0%

          \[\leadsto \frac{\left(\left(1.4551915228366852 \cdot 10^{-11} - \color{blue}{im \cdot im}\right) \cdot {re}^{2}\right) \cdot -0.5 + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       6. associate-*l* 0.0%

          \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right)} + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       7. sub-neg 0.0%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       8. unpow2 0.0%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \left(1.4551915228366852 \cdot 10^{-11} - \color{blue}{im \cdot im}\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       9. *-rgt-identity 0.0%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot 1}}{im + -3.814697265625 \cdot 10^{-6}} \]

       10. distribute-lft-out 86.7%

           \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5 + 1\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       11. unpow2 86.7%

           \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5 + 1\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

   13. Simplified 86.7%

       \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

   if -3.00000000000000026e154 < im < -1.2999999999999999e92 or 1.05000000000000005e85 < 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 96.1%

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

      1. mul-1-neg 96.1%

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

      2. unsub-neg 96.1%

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

      3. *-commutative 96.1%

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

      4. associate-*l* 96.1%

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

      5. distribute-lft-out-- 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in re around 0 81.2%

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

   8. Taylor expanded in im around inf 81.2%

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

   if -1.2999999999999999e92 < im < -1.54999999999999985e23

   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 23.4%

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

      1. neg-mul-1 23.4%

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

      2. +-commutative 23.4%

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

      3. unsub-neg 23.4%

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

      4. *-commutative 23.4%

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

      5. associate-*l* 23.4%

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

      6. unpow2 23.4%

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

   9. Simplified 23.4%

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

   if -1.54999999999999985e23 < im < 1.05000000000000005e85

   1. Initial program 20.1%

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

      1. sub0-neg 20.1%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in im around 0 87.1%

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

      1. mul-1-neg 87.1%

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

      2. *-commutative 87.1%

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

      3. distribute-lft-neg-in 87.1%

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

   6. Simplified 87.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{elif}\;im \leq -1.3 \cdot 10^{+92}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 10: 52.1% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.55 \cdot 10^{+23} \lor \neg \left(im \leq 2.15 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.55e+23) (not (<= im 2.15e-7)))
   (/
    (* (- 1.4551915228366852e-11 (* im im)) (+ (* -0.5 (* re re)) 1.0))
    (+ im -3.814697265625e-6))
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.55e+23) || !(im <= 2.15e-7)) {
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	} else {
		tmp = -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.55d+23)) .or. (.not. (im <= 2.15d-7))) then
        tmp = ((1.4551915228366852d-11 - (im * im)) * (((-0.5d0) * (re * re)) + 1.0d0)) / (im + (-3.814697265625d-6))
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.55e+23) || !(im <= 2.15e-7)) {
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.55e+23) or not (im <= 2.15e-7):
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6)
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.55e+23) || !(im <= 2.15e-7))
		tmp = Float64(Float64(Float64(1.4551915228366852e-11 - Float64(im * im)) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0)) / Float64(im + -3.814697265625e-6));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.55e+23) || ~((im <= 2.15e-7)))
		tmp = ((1.4551915228366852e-11 - (im * im)) * ((-0.5 * (re * re)) + 1.0)) / (im + -3.814697265625e-6);
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.55e+23], N[Not[LessEqual[im, 2.15e-7]], $MachinePrecision]], N[(N[(N[(1.4551915228366852e-11 - N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(im + -3.814697265625e-6), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -1.54999999999999985e23 or 2.1500000000000001e-7 < im

   1. Initial program 99.7%

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

      1. sub0-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in im around 0 69.0%

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

      1. mul-1-neg 69.0%

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

      2. unsub-neg 69.0%

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

      3. *-commutative 69.0%

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

      4. associate-*l* 69.0%

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

      5. distribute-lft-out-- 69.0%

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

   6. Simplified 69.0%

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

   8. Applied egg-rr 5.8%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3.814697265625 \cdot 10^{-6}} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 5.8%

         \[\leadsto \color{blue}{\left(-3.814697265625 \cdot 10^{-6} - im\right) \cdot \cos re} \]

      2. flip-- 49.9%

         \[\leadsto \color{blue}{\frac{-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im}{-3.814697265625 \cdot 10^{-6} + im}} \cdot \cos re \]

      3. associate-*l/ 49.9%

         \[\leadsto \color{blue}{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im}} \]

      4. metadata-eval 49.9%

         \[\leadsto \frac{\left(\color{blue}{1.4551915228366852 \cdot 10^{-11}} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im} \]

      5. +-commutative 49.9%

         \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3.814697265625 \cdot 10^{-6}}} \]

   10. Applied egg-rr 49.9%

       \[\leadsto \color{blue}{\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{im + -3.814697265625 \cdot 10^{-6}}} \]

   11. Taylor expanded in re around 0 9.6%

       \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} + -0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right)\right) - {im}^{2}}}{im + -3.814697265625 \cdot 10^{-6}} \]
   12. Step-by-step derivation

       1. sub-neg 9.6%

          \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} + -0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right)\right) + \left(-{im}^{2}\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       2. +-commutative 9.6%

          \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) + 1.4551915228366852 \cdot 10^{-11}\right)} + \left(-{im}^{2}\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       3. associate-+l+ 9.6%

          \[\leadsto \frac{\color{blue}{-0.5 \cdot \left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       4. *-commutative 9.6%

          \[\leadsto \frac{\color{blue}{\left(\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right) \cdot {re}^{2}\right) \cdot -0.5} + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       5. unpow2 9.6%

          \[\leadsto \frac{\left(\left(1.4551915228366852 \cdot 10^{-11} - \color{blue}{im \cdot im}\right) \cdot {re}^{2}\right) \cdot -0.5 + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       6. associate-*l* 9.6%

          \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right)} + \left(1.4551915228366852 \cdot 10^{-11} + \left(-{im}^{2}\right)\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       7. sub-neg 9.6%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - {im}^{2}\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       8. unpow2 9.6%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \left(1.4551915228366852 \cdot 10^{-11} - \color{blue}{im \cdot im}\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

       9. *-rgt-identity 9.6%

          \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5\right) + \color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot 1}}{im + -3.814697265625 \cdot 10^{-6}} \]

       10. distribute-lft-out 48.2%

           \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left({re}^{2} \cdot -0.5 + 1\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

       11. unpow2 48.2%

           \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5 + 1\right)}{im + -3.814697265625 \cdot 10^{-6}} \]

   13. Simplified 48.2%

       \[\leadsto \frac{\color{blue}{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)}}{im + -3.814697265625 \cdot 10^{-6}} \]

   if -1.54999999999999985e23 < im < 2.1500000000000001e-7

   1. Initial program 13.0%

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

      1. sub0-neg 13.0%

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

   3. Simplified 13.0%

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

   4. Taylor expanded in im around 0 93.9%

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

      1. mul-1-neg 93.9%

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

      2. unsub-neg 93.9%

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

      3. *-commutative 93.9%

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

      4. associate-*l* 93.9%

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

      5. distribute-lft-out-- 93.9%

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

   6. Simplified 93.9%

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

   7. Taylor expanded in re around 0 56.1%

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

   8. Taylor expanded in im around 0 56.0%

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

      1. neg-mul-1 56.0%

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

   10. Simplified 56.0%

       \[\leadsto \color{blue}{-im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.55 \cdot 10^{+23} \lor \neg \left(im \leq 2.15 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 11: 50.3% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1.4551915228366852 \cdot 10^{-11} - im \cdot im}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{if}\;im \leq -6.5 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -520:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+156}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (/ (- 1.4551915228366852e-11 (* im im)) (+ im -3.814697265625e-6))))
   (if (<= im -6.5e+150)
     t_0
     (if (<= im -520.0)
       (* (* re re) 0.75)
       (if (<= im 2.15e-7)
         (- im)
         (if (<= im 2.9e+156) (- (* (* re re) (* im 0.5)) im) t_0))))))

C

double code(double re, double im) {
	double t_0 = (1.4551915228366852e-11 - (im * im)) / (im + -3.814697265625e-6);
	double tmp;
	if (im <= -6.5e+150) {
		tmp = t_0;
	} else if (im <= -520.0) {
		tmp = (re * re) * 0.75;
	} else if (im <= 2.15e-7) {
		tmp = -im;
	} else if (im <= 2.9e+156) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} 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 = (1.4551915228366852d-11 - (im * im)) / (im + (-3.814697265625d-6))
    if (im <= (-6.5d+150)) then
        tmp = t_0
    else if (im <= (-520.0d0)) then
        tmp = (re * re) * 0.75d0
    else if (im <= 2.15d-7) then
        tmp = -im
    else if (im <= 2.9d+156) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (1.4551915228366852e-11 - (im * im)) / (im + -3.814697265625e-6);
	double tmp;
	if (im <= -6.5e+150) {
		tmp = t_0;
	} else if (im <= -520.0) {
		tmp = (re * re) * 0.75;
	} else if (im <= 2.15e-7) {
		tmp = -im;
	} else if (im <= 2.9e+156) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (1.4551915228366852e-11 - (im * im)) / (im + -3.814697265625e-6)
	tmp = 0
	if im <= -6.5e+150:
		tmp = t_0
	elif im <= -520.0:
		tmp = (re * re) * 0.75
	elif im <= 2.15e-7:
		tmp = -im
	elif im <= 2.9e+156:
		tmp = ((re * re) * (im * 0.5)) - im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(1.4551915228366852e-11 - Float64(im * im)) / Float64(im + -3.814697265625e-6))
	tmp = 0.0
	if (im <= -6.5e+150)
		tmp = t_0;
	elseif (im <= -520.0)
		tmp = Float64(Float64(re * re) * 0.75);
	elseif (im <= 2.15e-7)
		tmp = Float64(-im);
	elseif (im <= 2.9e+156)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (1.4551915228366852e-11 - (im * im)) / (im + -3.814697265625e-6);
	tmp = 0.0;
	if (im <= -6.5e+150)
		tmp = t_0;
	elseif (im <= -520.0)
		tmp = (re * re) * 0.75;
	elseif (im <= 2.15e-7)
		tmp = -im;
	elseif (im <= 2.9e+156)
		tmp = ((re * re) * (im * 0.5)) - im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(1.4551915228366852e-11 - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(im + -3.814697265625e-6), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.5e+150], t$95$0, If[LessEqual[im, -520.0], N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision], If[LessEqual[im, 2.15e-7], (-im), If[LessEqual[im, 2.9e+156], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -6.50000000000000033e150 or 2.9000000000000001e156 < 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 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.9%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3.814697265625 \cdot 10^{-6}} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.9%

         \[\leadsto \color{blue}{\left(-3.814697265625 \cdot 10^{-6} - im\right) \cdot \cos re} \]

      2. flip-- 98.2%

         \[\leadsto \color{blue}{\frac{-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im}{-3.814697265625 \cdot 10^{-6} + im}} \cdot \cos re \]

      3. associate-*l/ 98.2%

         \[\leadsto \color{blue}{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot -3.814697265625 \cdot 10^{-6} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im}} \]

      4. metadata-eval 98.2%

         \[\leadsto \frac{\left(\color{blue}{1.4551915228366852 \cdot 10^{-11}} - im \cdot im\right) \cdot \cos re}{-3.814697265625 \cdot 10^{-6} + im} \]

      5. +-commutative 98.2%

         \[\leadsto \frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3.814697265625 \cdot 10^{-6}}} \]

   10. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\frac{\left(1.4551915228366852 \cdot 10^{-11} - im \cdot im\right) \cdot \cos re}{im + -3.814697265625 \cdot 10^{-6}}} \]

   11. Taylor expanded in re around 0 64.9%

       \[\leadsto \frac{\color{blue}{1.4551915228366852 \cdot 10^{-11} - {im}^{2}}}{im + -3.814697265625 \cdot 10^{-6}} \]
   12. Step-by-step derivation

       1. unpow2 64.9%

          \[\leadsto \frac{1.4551915228366852 \cdot 10^{-11} - \color{blue}{im \cdot im}}{im + -3.814697265625 \cdot 10^{-6}} \]

   13. Simplified 64.9%

       \[\leadsto \frac{\color{blue}{1.4551915228366852 \cdot 10^{-11} - im \cdot im}}{im + -3.814697265625 \cdot 10^{-6}} \]

   if -6.50000000000000033e150 < im < -520

   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 73.8%

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

      4. +-commutative 73.8%

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

      5. *-commutative 73.8%

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

      6. unpow2 73.8%

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

      7. associate-*l* 73.8%

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

   6. Simplified 73.8%

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

   8. Applied egg-rr 15.4%

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

   9. Taylor expanded in re around inf 16.1%

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

       1. unpow2 16.1%

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

   11. Simplified 16.1%

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

   if -520 < im < 2.1500000000000001e-7

   1. Initial program 8.5%

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

      1. sub0-neg 8.5%

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

   3. Simplified 8.5%

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

   4. Taylor expanded in im around 0 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*l* 98.6%

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

      5. distribute-lft-out-- 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in re around 0 58.8%

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

   8. Taylor expanded in im around 0 58.8%

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

      1. neg-mul-1 58.8%

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

   10. Simplified 58.8%

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

   if 2.1500000000000001e-7 < im < 2.9000000000000001e156

   1. Initial program 98.8%

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

      1. sub0-neg 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in im around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. *-commutative 9.0%

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

      3. distribute-lft-neg-in 9.0%

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

   6. Simplified 9.0%

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

   7. Taylor expanded in re around 0 23.9%

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

      1. neg-mul-1 23.9%

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

      2. +-commutative 23.9%

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

      3. unsub-neg 23.9%

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

      4. *-commutative 23.9%

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

      5. associate-*l* 23.9%

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

      6. unpow2 23.9%

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

   9. Simplified 23.9%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{1.4551915228366852 \cdot 10^{-11} - im \cdot im}{im + -3.814697265625 \cdot 10^{-6}}\\ \mathbf{elif}\;im \leq -520:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+156}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;\frac{1.4551915228366852 \cdot 10^{-11} - im \cdot im}{im + -3.814697265625 \cdot 10^{-6}}\\ \end{array} \]

Alternative 12: 37.3% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8 \cdot 10^{+166}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+273}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8e+166)
   (- (* (* re re) (* im 0.5)) im)
   (if (<= re 1.45e+273) (* (* re re) 0.75) (+ 13.5 (* (* re re) -6.75)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8e+166) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (re <= 1.45e+273) {
		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 <= 8d+166) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else if (re <= 1.45d+273) 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 <= 8e+166) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else if (re <= 1.45e+273) {
		tmp = (re * re) * 0.75;
	} else {
		tmp = 13.5 + ((re * re) * -6.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8e+166:
		tmp = ((re * re) * (im * 0.5)) - im
	elif re <= 1.45e+273:
		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 <= 8e+166)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	elseif (re <= 1.45e+273)
		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 <= 8e+166)
		tmp = ((re * re) * (im * 0.5)) - im;
	elseif (re <= 1.45e+273)
		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, 8e+166], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[re, 1.45e+273], 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 < 7.99999999999999952e166

   1. Initial program 50.8%

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

      1. sub0-neg 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in im around 0 55.5%

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

      1. mul-1-neg 55.5%

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

      2. *-commutative 55.5%

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

      3. distribute-lft-neg-in 55.5%

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

   6. Simplified 55.5%

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

   7. Taylor expanded in re around 0 41.7%

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

      1. neg-mul-1 41.7%

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

      2. +-commutative 41.7%

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

      3. unsub-neg 41.7%

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

      4. *-commutative 41.7%

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

      5. associate-*l* 41.7%

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

      6. unpow2 41.7%

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

   9. Simplified 41.7%

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

   if 7.99999999999999952e166 < re < 1.4499999999999999e273

   1. Initial program 57.0%

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

      1. sub0-neg 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in re around 0 0.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   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 19.1%

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

      4. +-commutative 19.1%

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

      5. *-commutative 19.1%

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

      6. unpow2 19.1%

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

      7. associate-*l* 19.1%

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

   6. Simplified 19.1%

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

   8. Applied egg-rr 39.1%

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

   9. Taylor expanded in re around inf 39.1%

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

       1. unpow2 39.1%

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

   11. Simplified 39.1%

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

   if 1.4499999999999999e273 < re

   1. Initial program 59.0%

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

      1. sub0-neg 59.0%

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

   3. Simplified 59.0%

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

   5. Applied egg-rr 2.5%

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

   6. Taylor expanded in re around 0 40.7%

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

      1. *-commutative 40.7%

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

      2. unpow2 40.7%

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

   8. Simplified 40.7%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8 \cdot 10^{+166}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+273}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 13: 35.0% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -600 \lor \neg \left(im \leq 24000000000\right):\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -600.0) (not (<= im 24000000000.0)))
   (* (* re re) 0.75)
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -600.0) || !(im <= 24000000000.0)) {
		tmp = (re * re) * 0.75;
	} else {
		tmp = -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 <= (-600.0d0)) .or. (.not. (im <= 24000000000.0d0))) then
        tmp = (re * re) * 0.75d0
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -600.0) || !(im <= 24000000000.0)) {
		tmp = (re * re) * 0.75;
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -600.0) or not (im <= 24000000000.0):
		tmp = (re * re) * 0.75
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -600.0) || !(im <= 24000000000.0))
		tmp = Float64(Float64(re * re) * 0.75);
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -600.0) || ~((im <= 24000000000.0)))
		tmp = (re * re) * 0.75;
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -600.0], N[Not[LessEqual[im, 24000000000.0]], $MachinePrecision]], N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -600 or 2.4e10 < 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 77.8%

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

      4. +-commutative 77.8%

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

      5. *-commutative 77.8%

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

      6. unpow2 77.8%

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

      7. associate-*l* 77.8%

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

   6. Simplified 77.8%

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

   8. Applied egg-rr 13.3%

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

   9. Taylor expanded in re around inf 13.4%

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

       1. unpow2 13.4%

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

   11. Simplified 13.4%

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

   if -600 < im < 2.4e10

   1. Initial program 11.0%

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

      1. sub0-neg 11.0%

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

   3. Simplified 11.0%

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

   4. Taylor expanded in im around 0 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

      3. *-commutative 96.9%

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

      4. associate-*l* 96.9%

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

      5. distribute-lft-out-- 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in re around 0 57.6%

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

   8. Taylor expanded in im around 0 57.5%

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

      1. neg-mul-1 57.5%

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

   10. Simplified 57.5%

       \[\leadsto \color{blue}{-im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -600 \lor \neg \left(im \leq 24000000000\right):\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 14: 34.5% accurate, 33.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{+160}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \mathbf{elif}\;re \leq 7.8 \cdot 10^{+167}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8e+160)
   (+ 13.5 (* (* re re) -6.75))
   (if (<= re 7.8e+167) (- im) (* (* re re) 0.75))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8e+160) {
		tmp = 13.5 + ((re * re) * -6.75);
	} else if (re <= 7.8e+167) {
		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 <= (-8d+160)) then
        tmp = 13.5d0 + ((re * re) * (-6.75d0))
    else if (re <= 7.8d+167) 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 <= -8e+160) {
		tmp = 13.5 + ((re * re) * -6.75);
	} else if (re <= 7.8e+167) {
		tmp = -im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8e+160:
		tmp = 13.5 + ((re * re) * -6.75)
	elif re <= 7.8e+167:
		tmp = -im
	else:
		tmp = (re * re) * 0.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8e+160)
		tmp = Float64(13.5 + Float64(Float64(re * re) * -6.75));
	elseif (re <= 7.8e+167)
		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 <= -8e+160)
		tmp = 13.5 + ((re * re) * -6.75);
	elseif (re <= 7.8e+167)
		tmp = -im;
	else
		tmp = (re * re) * 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8e+160], N[(13.5 + N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.8e+167], (-im), N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.00000000000000005e160

   1. Initial program 54.6%

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

      1. sub0-neg 54.6%

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

   3. Simplified 54.6%

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

   5. Applied egg-rr 2.5%

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

   6. Taylor expanded in re around 0 33.9%

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

      1. *-commutative 33.9%

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

      2. unpow2 33.9%

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

   8. Simplified 33.9%

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

   if -8.00000000000000005e160 < re < 7.7999999999999996e167

   1. Initial program 50.0%

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

      1. sub0-neg 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in im around 0 83.7%

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

      1. mul-1-neg 83.7%

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

      2. unsub-neg 83.7%

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

      3. *-commutative 83.7%

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

      4. associate-*l* 83.7%

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

      5. distribute-lft-out-- 83.7%

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

   6. Simplified 83.7%

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

   7. Taylor expanded in re around 0 60.0%

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

   8. Taylor expanded in im around 0 39.2%

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

      1. neg-mul-1 39.2%

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

   10. Simplified 39.2%

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

   if 7.7999999999999996e167 < re

   1. Initial program 61.3%

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

      1. sub0-neg 61.3%

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

   3. Simplified 61.3%

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

   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 20.8%

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

      4. +-commutative 20.8%

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

      5. *-commutative 20.8%

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

      6. unpow2 20.8%

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

      7. associate-*l* 20.8%

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

   6. Simplified 20.8%

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

   8. Applied egg-rr 32.0%

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

   9. Taylor expanded in re around inf 32.0%

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

       1. unpow2 32.0%

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

   11. Simplified 32.0%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{+160}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \mathbf{elif}\;re \leq 7.8 \cdot 10^{+167}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \]

Alternative 15: 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 51.7%

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

   1. sub0-neg 51.7%

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

3. Simplified 51.7%

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

4. Taylor expanded in im around 0 82.8%

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

   1. mul-1-neg 82.8%

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

   2. unsub-neg 82.8%

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

   3. *-commutative 82.8%

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

   4. associate-*l* 82.8%

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

   5. distribute-lft-out-- 82.8%

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

6. Simplified 82.8%

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

7. Taylor expanded in re around 0 53.0%

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

8. Taylor expanded in im around 0 33.0%

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

   1. neg-mul-1 33.0%

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

10. Simplified 33.0%

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

11. Final simplification 33.0%

    \[\leadsto -im \]

Alternative 16: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.5)

C

double code(double re, double im) {
	return -1.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.5d0
end function

Java

public static double code(double re, double im) {
	return -1.5;
}

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

function tmp = code(re, im)
	tmp = -1.5;
end

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 51.7%

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

   1. sub0-neg 51.7%

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

3. Simplified 51.7%

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

4. Taylor expanded in re around 0 3.5%

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

   1. *-commutative 3.5%

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

   2. associate-*r* 3.5%

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

   3. distribute-rgt-out 39.5%

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

   4. +-commutative 39.5%

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

   5. *-commutative 39.5%

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

   6. unpow2 39.5%

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

   7. associate-*l* 39.5%

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

6. Simplified 39.5%

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

8. Applied egg-rr 7.8%

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

9. Taylor expanded in re around 0 2.9%

   \[\leadsto \color{blue}{-1.5} \]

10. Final simplification 2.9%

    \[\leadsto -1.5 \]

Alternative 17: 2.8% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 13.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 13.5)

C

double code(double re, double im) {
	return 13.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 13.5d0
end function

Java

public static double code(double re, double im) {
	return 13.5;
}

Python

def code(re, im):
	return 13.5

Julia

function code(re, im)
	return 13.5
end

MATLAB

function tmp = code(re, im)
	tmp = 13.5;
end

Wolfram

code[re_, im_] := 13.5

Derivation

1. Initial program 51.7%

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

   1. sub0-neg 51.7%

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

3. Simplified 51.7%

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

5. Applied egg-rr 3.1%

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

6. Taylor expanded in re around 0 3.0%

   \[\leadsto \color{blue}{13.5} \]

7. Final simplification 3.0%

   \[\leadsto 13.5 \]

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 2023193 
(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))))