math.sin on complex, imaginary part

Percentage Accurate: 54.9% → 99.8%

Time: 8.9s

Alternatives: 11

Speedup: 2.8×

• 50.4% 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: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -10000000 \lor \neg \left(t_0 \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right) - im \cdot \cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -10000000.0) (not (<= t_0 0.001)))
     (* (* 0.5 (cos re)) t_0)
     (- (* (pow im 3.0) (* (cos re) -0.16666666666666666)) (* im (cos re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -10000000.0) || !(t_0 <= 0.001)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = (pow(im, 3.0) * (cos(re) * -0.16666666666666666)) - (im * cos(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-10000000.0d0)) .or. (.not. (t_0 <= 0.001d0))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = ((im ** 3.0d0) * (cos(re) * (-0.16666666666666666d0))) - (im * cos(re))
    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 <= -10000000.0) || !(t_0 <= 0.001)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = (Math.pow(im, 3.0) * (Math.cos(re) * -0.16666666666666666)) - (im * Math.cos(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -10000000.0) or not (t_0 <= 0.001):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = (math.pow(im, 3.0) * (math.cos(re) * -0.16666666666666666)) - (im * math.cos(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -10000000.0) || !(t_0 <= 0.001))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(Float64((im ^ 3.0) * Float64(cos(re) * -0.16666666666666666)) - Float64(im * cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -10000000.0) || ~((t_0 <= 0.001)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = ((im ^ 3.0) * (cos(re) * -0.16666666666666666)) - (im * cos(re));
	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, -10000000.0], N[Not[LessEqual[t$95$0, 0.001]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -1e7 or 1e-3 < (-.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. neg-sub0 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 -1e7 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1e-3

   1. Initial program 8.8%

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

      1. neg-sub0 8.8%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out-- 99.8%

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

      6. *-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-rgt-in 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. unsub-neg 99.9%

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

      7. associate-*r* 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -10000000 \lor \neg \left(e^{-im} - e^{im} \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right) - im \cdot \cos re\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -10000000 \lor \neg \left(t_0 \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -10000000.0) (not (<= t_0 0.001)))
     (* (* 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 <= -10000000.0) || !(t_0 <= 0.001)) {
		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 <= (-10000000.0d0)) .or. (.not. (t_0 <= 0.001d0))) 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 <= -10000000.0) || !(t_0 <= 0.001)) {
		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 <= -10000000.0) or not (t_0 <= 0.001):
		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 <= -10000000.0) || !(t_0 <= 0.001))
		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 <= -10000000.0) || ~((t_0 <= 0.001)))
		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, -10000000.0], N[Not[LessEqual[t$95$0, 0.001]], $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)) < -1e7 or 1e-3 < (-.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. neg-sub0 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 -1e7 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1e-3

   1. Initial program 8.8%

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

      1. neg-sub0 8.8%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out-- 99.8%

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

      6. *-commutative 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -10000000 \lor \neg \left(e^{-im} - e^{im} \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -1.7 \cdot 10^{+52}:\\ \;\;\;\;\cos re \cdot \sqrt{{im}^{6} \cdot 0.027777777777777776}\\ \mathbf{elif}\;im \leq -48000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4000:\\ \;\;\;\;\cos re \cdot \left(t_1 - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -1.7e+52)
     (* (cos re) (sqrt (* (pow im 6.0) 0.027777777777777776)))
     (if (<= im -48000000000.0)
       t_0
       (if (<= im 4000.0)
         (* (cos re) (- t_1 im))
         (if (<= im 5.6e+102) t_0 (* (cos re) t_1)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -1.7e+52) {
		tmp = cos(re) * sqrt((pow(im, 6.0) * 0.027777777777777776));
	} else if (im <= -48000000000.0) {
		tmp = t_0;
	} else if (im <= 4000.0) {
		tmp = cos(re) * (t_1 - im);
	} else if (im <= 5.6e+102) {
		tmp = t_0;
	} else {
		tmp = cos(re) * 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 = (im ** 3.0d0) * (-0.16666666666666666d0)
    if (im <= (-1.7d+52)) then
        tmp = cos(re) * sqrt(((im ** 6.0d0) * 0.027777777777777776d0))
    else if (im <= (-48000000000.0d0)) then
        tmp = t_0
    else if (im <= 4000.0d0) then
        tmp = cos(re) * (t_1 - im)
    else if (im <= 5.6d+102) then
        tmp = t_0
    else
        tmp = cos(re) * 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.pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -1.7e+52) {
		tmp = Math.cos(re) * Math.sqrt((Math.pow(im, 6.0) * 0.027777777777777776));
	} else if (im <= -48000000000.0) {
		tmp = t_0;
	} else if (im <= 4000.0) {
		tmp = Math.cos(re) * (t_1 - im);
	} else if (im <= 5.6e+102) {
		tmp = t_0;
	} else {
		tmp = Math.cos(re) * t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -1.7e+52:
		tmp = math.cos(re) * math.sqrt((math.pow(im, 6.0) * 0.027777777777777776))
	elif im <= -48000000000.0:
		tmp = t_0
	elif im <= 4000.0:
		tmp = math.cos(re) * (t_1 - im)
	elif im <= 5.6e+102:
		tmp = t_0
	else:
		tmp = math.cos(re) * t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -1.7e+52)
		tmp = Float64(cos(re) * sqrt(Float64((im ^ 6.0) * 0.027777777777777776)));
	elseif (im <= -48000000000.0)
		tmp = t_0;
	elseif (im <= 4000.0)
		tmp = Float64(cos(re) * Float64(t_1 - im));
	elseif (im <= 5.6e+102)
		tmp = t_0;
	else
		tmp = Float64(cos(re) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -1.7e+52)
		tmp = cos(re) * sqrt(((im ^ 6.0) * 0.027777777777777776));
	elseif (im <= -48000000000.0)
		tmp = t_0;
	elseif (im <= 4000.0)
		tmp = cos(re) * (t_1 - im);
	elseif (im <= 5.6e+102)
		tmp = t_0;
	else
		tmp = cos(re) * 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[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -1.7e+52], N[(N[Cos[re], $MachinePrecision] * N[Sqrt[N[(N[Power[im, 6.0], $MachinePrecision] * 0.027777777777777776), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -48000000000.0], t$95$0, If[LessEqual[im, 4000.0], N[(N[Cos[re], $MachinePrecision] * N[(t$95$1 - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e+102], t$95$0, N[(N[Cos[re], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.7e52

   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. neg-sub0 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 79.9%

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

      1. +-commutative 79.9%

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

      2. mul-1-neg 79.9%

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

      3. unsub-neg 79.9%

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

      4. associate-*r* 79.9%

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

      5. distribute-rgt-out-- 79.9%

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

      6. *-commutative 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in im around inf 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*r* 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-*r* 79.9%

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

   9. Simplified 79.9%

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

       1. *-commutative 79.9%

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

       2. add-sqr-sqrt 79.9%

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

       3. sqrt-unprod 100.0%

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

       4. swap-sqr 100.0%

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

       5. pow-prod-up 100.0%

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

       6. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \sqrt{{im}^{\color{blue}{6}} \cdot \left(-0.16666666666666666 \cdot -0.16666666666666666\right)} \]

       7. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \sqrt{{im}^{6} \cdot \color{blue}{0.027777777777777776}} \]

   11. Applied egg-rr 100.0%

       \[\leadsto \cos re \cdot \color{blue}{\sqrt{{im}^{6} \cdot 0.027777777777777776}} \]

   if -1.7e52 < im < -4.8e10 or 4e3 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

      1. neg-sub0 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 79.5%

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

   if -4.8e10 < im < 4e3

   1. Initial program 12.5%

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

      1. neg-sub0 12.5%

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

   3. Simplified 12.5%

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

   4. Taylor expanded in im around 0 96.0%

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

      1. +-commutative 96.0%

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

      2. mul-1-neg 96.0%

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

      3. unsub-neg 96.0%

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

      4. associate-*r* 96.0%

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

      5. distribute-rgt-out-- 96.0%

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

      6. *-commutative 96.0%

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

   6. Simplified 96.0%

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

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

      1. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{+52}:\\ \;\;\;\;\cos re \cdot \sqrt{{im}^{6} \cdot 0.027777777777777776}\\ \mathbf{elif}\;im \leq -48000000000:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 4000:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 4: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{if}\;im \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* (cos re) (* (pow im 3.0) -0.16666666666666666))))
   (if (<= im -5.5e+102)
     t_1
     (if (<= im -0.0017)
       t_0
       (if (<= im 4000.0)
         (* im (- (cos re)))
         (if (<= im 5.6e+102) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = cos(re) * (pow(im, 3.0) * -0.16666666666666666);
	double tmp;
	if (im <= -5.5e+102) {
		tmp = t_1;
	} else if (im <= -0.0017) {
		tmp = t_0;
	} else if (im <= 4000.0) {
		tmp = im * -cos(re);
	} else if (im <= 5.6e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = cos(re) * ((im ** 3.0d0) * (-0.16666666666666666d0))
    if (im <= (-5.5d+102)) then
        tmp = t_1
    else if (im <= (-0.0017d0)) then
        tmp = t_0
    else if (im <= 4000.0d0) then
        tmp = im * -cos(re)
    else if (im <= 5.6d+102) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = Math.cos(re) * (Math.pow(im, 3.0) * -0.16666666666666666);
	double tmp;
	if (im <= -5.5e+102) {
		tmp = t_1;
	} else if (im <= -0.0017) {
		tmp = t_0;
	} else if (im <= 4000.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 5.6e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = math.cos(re) * (math.pow(im, 3.0) * -0.16666666666666666)
	tmp = 0
	if im <= -5.5e+102:
		tmp = t_1
	elif im <= -0.0017:
		tmp = t_0
	elif im <= 4000.0:
		tmp = im * -math.cos(re)
	elif im <= 5.6e+102:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(cos(re) * Float64((im ^ 3.0) * -0.16666666666666666))
	tmp = 0.0
	if (im <= -5.5e+102)
		tmp = t_1;
	elseif (im <= -0.0017)
		tmp = t_0;
	elseif (im <= 4000.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 5.6e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = cos(re) * ((im ^ 3.0) * -0.16666666666666666);
	tmp = 0.0;
	if (im <= -5.5e+102)
		tmp = t_1;
	elseif (im <= -0.0017)
		tmp = t_0;
	elseif (im <= 4000.0)
		tmp = im * -cos(re);
	elseif (im <= 5.6e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.5e+102], t$95$1, If[LessEqual[im, -0.0017], t$95$0, If[LessEqual[im, 4000.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 5.6e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.49999999999999981e102 or 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

      1. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

   9. Simplified 100.0%

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

   if -5.49999999999999981e102 < im < -0.00169999999999999991 or 4e3 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

      1. neg-sub0 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 73.4%

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

   if -0.00169999999999999991 < im < 4e3

   1. Initial program 11.1%

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

      1. neg-sub0 11.1%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in im around 0 96.9%

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

      1. associate-*r* 96.9%

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

      2. neg-mul-1 96.9%

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

   6. Simplified 96.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -0.0017:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 4000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 5: 94.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{3} \cdot -0.16666666666666666\\ t_2 := \cos re \cdot t_1\\ \mathbf{if}\;im \leq -5 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -48000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4000:\\ \;\;\;\;\cos re \cdot \left(t_1 - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* (pow im 3.0) -0.16666666666666666))
        (t_2 (* (cos re) t_1)))
   (if (<= im -5e+103)
     t_2
     (if (<= im -48000000000.0)
       t_0
       (if (<= im 4000.0)
         (* (cos re) (- t_1 im))
         (if (<= im 5.6e+102) t_0 t_2))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = pow(im, 3.0) * -0.16666666666666666;
	double t_2 = cos(re) * t_1;
	double tmp;
	if (im <= -5e+103) {
		tmp = t_2;
	} else if (im <= -48000000000.0) {
		tmp = t_0;
	} else if (im <= 4000.0) {
		tmp = cos(re) * (t_1 - im);
	} else if (im <= 5.6e+102) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_2 = cos(re) * t_1
    if (im <= (-5d+103)) then
        tmp = t_2
    else if (im <= (-48000000000.0d0)) then
        tmp = t_0
    else if (im <= 4000.0d0) then
        tmp = cos(re) * (t_1 - im)
    else if (im <= 5.6d+102) then
        tmp = t_0
    else
        tmp = t_2
    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.pow(im, 3.0) * -0.16666666666666666;
	double t_2 = Math.cos(re) * t_1;
	double tmp;
	if (im <= -5e+103) {
		tmp = t_2;
	} else if (im <= -48000000000.0) {
		tmp = t_0;
	} else if (im <= 4000.0) {
		tmp = Math.cos(re) * (t_1 - im);
	} else if (im <= 5.6e+102) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = math.pow(im, 3.0) * -0.16666666666666666
	t_2 = math.cos(re) * t_1
	tmp = 0
	if im <= -5e+103:
		tmp = t_2
	elif im <= -48000000000.0:
		tmp = t_0
	elif im <= 4000.0:
		tmp = math.cos(re) * (t_1 - im)
	elif im <= 5.6e+102:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_2 = Float64(cos(re) * t_1)
	tmp = 0.0
	if (im <= -5e+103)
		tmp = t_2;
	elseif (im <= -48000000000.0)
		tmp = t_0;
	elseif (im <= 4000.0)
		tmp = Float64(cos(re) * Float64(t_1 - im));
	elseif (im <= 5.6e+102)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = (im ^ 3.0) * -0.16666666666666666;
	t_2 = cos(re) * t_1;
	tmp = 0.0;
	if (im <= -5e+103)
		tmp = t_2;
	elseif (im <= -48000000000.0)
		tmp = t_0;
	elseif (im <= 4000.0)
		tmp = cos(re) * (t_1 - im);
	elseif (im <= 5.6e+102)
		tmp = t_0;
	else
		tmp = t_2;
	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[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[re], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[im, -5e+103], t$95$2, If[LessEqual[im, -48000000000.0], t$95$0, If[LessEqual[im, 4000.0], N[(N[Cos[re], $MachinePrecision] * N[(t$95$1 - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e+102], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5e103 or 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

      1. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

   9. Simplified 100.0%

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

   if -5e103 < im < -4.8e10 or 4e3 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 76.0%

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

   if -4.8e10 < im < 4e3

   1. Initial program 12.5%

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

      1. neg-sub0 12.5%

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

   3. Simplified 12.5%

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

   4. Taylor expanded in im around 0 96.0%

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

      1. +-commutative 96.0%

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

      2. mul-1-neg 96.0%

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

      3. unsub-neg 96.0%

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

      4. associate-*r* 96.0%

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

      5. distribute-rgt-out-- 96.0%

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

      6. *-commutative 96.0%

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

   6. Simplified 96.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+103}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -48000000000:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 4000:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 6: 85.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.014 \lor \neg \left(im \leq 4000\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.014) (not (<= im 4000.0)))
   (* 0.5 (- (exp (- im)) (exp im)))
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.014) || !(im <= 4000.0)) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-0.014d0)) .or. (.not. (im <= 4000.0d0))) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.014) || !(im <= 4000.0)) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.014) or not (im <= 4000.0):
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.014) || !(im <= 4000.0))
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.014) || ~((im <= 4000.0)))
		tmp = 0.5 * (exp(-im) - exp(im));
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.014], N[Not[LessEqual[im, 4000.0]], $MachinePrecision]], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -0.0140000000000000003 or 4e3 < 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. neg-sub0 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 73.1%

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

   if -0.0140000000000000003 < im < 4e3

   1. Initial program 11.1%

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

      1. neg-sub0 11.1%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in im around 0 96.9%

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

      1. associate-*r* 96.9%

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

      2. neg-mul-1 96.9%

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

   6. Simplified 96.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.014 \lor \neg \left(im \leq 4000\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 7: 75.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -8.5 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 620:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+96}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -8.5e+97)
     t_0
     (if (<= im 620.0)
       (* im (- (cos re)))
       (if (<= im 8.5e+96) (- (* (pow re 2.0) (* im 0.5)) im) t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -8.5e+97) {
		tmp = t_0;
	} else if (im <= 620.0) {
		tmp = im * -cos(re);
	} else if (im <= 8.5e+96) {
		tmp = (pow(re, 2.0) * (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 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-8.5d+97)) then
        tmp = t_0
    else if (im <= 620.0d0) then
        tmp = im * -cos(re)
    else if (im <= 8.5d+96) then
        tmp = ((re ** 2.0d0) * (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 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -8.5e+97) {
		tmp = t_0;
	} else if (im <= 620.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 8.5e+96) {
		tmp = (Math.pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -8.5e+97:
		tmp = t_0
	elif im <= 620.0:
		tmp = im * -math.cos(re)
	elif im <= 8.5e+96:
		tmp = (math.pow(re, 2.0) * (im * 0.5)) - im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -8.5e+97)
		tmp = t_0;
	elseif (im <= 620.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 8.5e+96)
		tmp = Float64(Float64((re ^ 2.0) * Float64(im * 0.5)) - im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -8.5e+97)
		tmp = t_0;
	elseif (im <= 620.0)
		tmp = im * -cos(re);
	elseif (im <= 8.5e+96)
		tmp = ((re ^ 2.0) * (im * 0.5)) - im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -8.5e+97], t$95$0, If[LessEqual[im, 620.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 8.5e+96], N[(N[(N[Power[re, 2.0], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.4999999999999993e97 or 8.50000000000000025e96 < 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. neg-sub0 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 94.2%

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

      1. +-commutative 94.2%

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

      2. mul-1-neg 94.2%

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

      3. unsub-neg 94.2%

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

      4. associate-*r* 94.2%

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

      5. distribute-rgt-out-- 94.2%

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

      6. *-commutative 94.2%

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

   6. Simplified 94.2%

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

   7. Taylor expanded in re around 0 68.9%

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

   if -8.4999999999999993e97 < im < 620

   1. Initial program 26.5%

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

      1. neg-sub0 26.5%

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

   3. Simplified 26.5%

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

   4. Taylor expanded in im around 0 80.7%

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

      1. associate-*r* 80.7%

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

      2. neg-mul-1 80.7%

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

   6. Simplified 80.7%

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

   if 620 < im < 8.50000000000000025e96

   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. neg-sub0 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 3.4%

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

      1. associate-*r* 3.4%

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

      2. neg-mul-1 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in re around 0 26.1%

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

      1. +-commutative 26.1%

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

      2. mul-1-neg 26.1%

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

      3. unsub-neg 26.1%

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

      4. *-commutative 26.1%

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

      5. *-commutative 26.1%

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

      6. associate-*r* 26.1%

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

   9. Simplified 26.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.5 \cdot 10^{+97}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 620:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+96}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 8: 79.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.2 \lor \neg \left(im \leq 3.65 \cdot 10^{+25}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.016666666666666666 \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -6.2) (not (<= im 3.65e+25)))
   (* 0.5 (+ (* im -2.0) (* -0.016666666666666666 (pow im 5.0))))
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -6.2) || !(im <= 3.65e+25)) {
		tmp = 0.5 * ((im * -2.0) + (-0.016666666666666666 * pow(im, 5.0)));
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-6.2d0)) .or. (.not. (im <= 3.65d+25))) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((-0.016666666666666666d0) * (im ** 5.0d0)))
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -6.2) || !(im <= 3.65e+25)) {
		tmp = 0.5 * ((im * -2.0) + (-0.016666666666666666 * Math.pow(im, 5.0)));
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -6.2) or not (im <= 3.65e+25):
		tmp = 0.5 * ((im * -2.0) + (-0.016666666666666666 * math.pow(im, 5.0)))
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -6.2) || !(im <= 3.65e+25))
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(-0.016666666666666666 * (im ^ 5.0))));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -6.2) || ~((im <= 3.65e+25)))
		tmp = 0.5 * ((im * -2.0) + (-0.016666666666666666 * (im ^ 5.0)));
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -6.2], N[Not[LessEqual[im, 3.65e+25]], $MachinePrecision]], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -6.20000000000000018 or 3.6499999999999998e25 < 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. neg-sub0 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 74.4%

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

   5. Taylor expanded in im around 0 57.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   6. Taylor expanded in im around inf 57.4%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.016666666666666666 \cdot {im}^{5}}\right) \]

   if -6.20000000000000018 < im < 3.6499999999999998e25

   1. Initial program 17.3%

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

      1. neg-sub0 17.3%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in im around 0 90.3%

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

      1. associate-*r* 90.3%

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

      2. neg-mul-1 90.3%

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

   6. Simplified 90.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.2 \lor \neg \left(im \leq 3.65 \cdot 10^{+25}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.016666666666666666 \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 9: 74.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8.5 \cdot 10^{+97} \lor \neg \left(im \leq 1.52 \cdot 10^{+25}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -8.5e+97) (not (<= im 1.52e+25)))
   (- (* (pow im 3.0) -0.16666666666666666) im)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -8.5e+97) || !(im <= 1.52e+25)) {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-8.5d+97)) .or. (.not. (im <= 1.52d+25))) then
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -8.5e+97) || !(im <= 1.52e+25)) {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -8.5e+97) or not (im <= 1.52e+25):
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -8.5e+97) || !(im <= 1.52e+25))
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -8.5e+97) || ~((im <= 1.52e+25)))
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -8.5e+97], N[Not[LessEqual[im, 1.52e+25]], $MachinePrecision]], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -8.4999999999999993e97 or 1.52000000000000006e25 < 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. neg-sub0 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 85.4%

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

      1. +-commutative 85.4%

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

      2. mul-1-neg 85.4%

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

      3. unsub-neg 85.4%

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

      4. associate-*r* 85.4%

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

      5. distribute-rgt-out-- 85.4%

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

      6. *-commutative 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in re around 0 62.4%

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

   if -8.4999999999999993e97 < im < 1.52000000000000006e25

   1. Initial program 31.7%

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

      1. neg-sub0 31.7%

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

   3. Simplified 31.7%

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

   4. Taylor expanded in im around 0 75.2%

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

      1. associate-*r* 75.2%

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

      2. neg-mul-1 75.2%

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

   6. Simplified 75.2%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.5 \cdot 10^{+97} \lor \neg \left(im \leq 1.52 \cdot 10^{+25}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 10: 51.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.7%

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

   1. neg-sub0 58.7%

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

3. Simplified 58.7%

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

4. Taylor expanded in im around 0 47.8%

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

   1. associate-*r* 47.8%

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

   2. neg-mul-1 47.8%

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

6. Simplified 47.8%

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

7. Final simplification 47.8%

   \[\leadsto im \cdot \left(-\cos re\right) \]

Alternative 11: 29.2% 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 58.7%

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

   1. neg-sub0 58.7%

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

3. Simplified 58.7%

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

4. Taylor expanded in im around 0 47.8%

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

   1. associate-*r* 47.8%

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

   2. neg-mul-1 47.8%

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

6. Simplified 47.8%

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

7. Taylor expanded in re around 0 37.6%

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

   1. +-commutative 37.6%

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

   2. mul-1-neg 37.6%

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

   3. unsub-neg 37.6%

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

   4. *-commutative 37.6%

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

   5. *-commutative 37.6%

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

   6. associate-*r* 37.6%

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

9. Simplified 37.6%

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

10. Taylor expanded in re around 0 29.2%

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

    1. mul-1-neg 29.2%

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

12. Simplified 29.2%

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

13. Final simplification 29.2%

    \[\leadsto -im \]

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 2023320 
(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))))