math.cos on complex, imaginary part

Percentage Accurate: 65.6% → 99.4%

Time: 8.3s

Alternatives: 10

Speedup: 2.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+110} \lor \neg \left(t_0 \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -2e+110) (not (<= t_0 2e-10)))
     (* t_0 (* 0.5 (sin re)))
     (* (sin 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 <= -2e+110) || !(t_0 <= 2e-10)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(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 <= (-2d+110)) .or. (.not. (t_0 <= 2d-10))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(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 <= -2e+110) || !(t_0 <= 2e-10)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(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 <= -2e+110) or not (t_0 <= 2e-10):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(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 <= -2e+110) || !(t_0 <= 2e-10))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(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 <= -2e+110) || ~((t_0 <= 2e-10)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(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, -2e+110], N[Not[LessEqual[t$95$0, 2e-10]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[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 (neg.f64 im)) (exp.f64 im)) < -2e110 or 2.00000000000000007e-10 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

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

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

   1. Initial program 29.4%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+110} \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0014:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 260:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 3.35 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -4.1e+93)
     t_1
     (if (<= im -0.0014)
       t_0
       (if (<= im 260.0)
         (* im (- (sin re)))
         (if (<= im 3.35e+100) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	double t_1 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -4.1e+93) {
		tmp = t_1;
	} else if (im <= -0.0014) {
		tmp = t_0;
	} else if (im <= 260.0) {
		tmp = im * -sin(re);
	} else if (im <= 3.35e+100) {
		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)) * re)
    t_1 = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    if (im <= (-4.1d+93)) then
        tmp = t_1
    else if (im <= (-0.0014d0)) then
        tmp = t_0
    else if (im <= 260.0d0) then
        tmp = im * -sin(re)
    else if (im <= 3.35d+100) 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)) * re);
	double t_1 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -4.1e+93) {
		tmp = t_1;
	} else if (im <= -0.0014) {
		tmp = t_0;
	} else if (im <= 260.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 3.35e+100) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	t_1 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -4.1e+93:
		tmp = t_1
	elif im <= -0.0014:
		tmp = t_0
	elif im <= 260.0:
		tmp = im * -math.sin(re)
	elif im <= 3.35e+100:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -4.1e+93)
		tmp = t_1;
	elseif (im <= -0.0014)
		tmp = t_0;
	elseif (im <= 260.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 3.35e+100)
		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)) * re);
	t_1 = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -4.1e+93)
		tmp = t_1;
	elseif (im <= -0.0014)
		tmp = t_0;
	elseif (im <= 260.0)
		tmp = im * -sin(re);
	elseif (im <= 3.35e+100)
		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[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.1e+93], t$95$1, If[LessEqual[im, -0.0014], t$95$0, If[LessEqual[im, 260.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 3.35e+100], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.1000000000000001e93 or 3.3499999999999998e100 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.0%

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

      1. mul-1-neg 97.0%

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

      2. unsub-neg 97.0%

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

      3. *-commutative 97.0%

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

      4. associate-*l* 97.0%

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

      5. distribute-lft-out-- 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in im around inf 97.0%

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

   if -4.1000000000000001e93 < im < -0.00139999999999999999 or 260 < im < 3.3499999999999998e100

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 86.1%

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

   if -0.00139999999999999999 < im < 260

   1. Initial program 29.9%

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

   2. Taylor expanded in im around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. distribute-rgt-neg-in 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -0.0014:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 260:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 3.35 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 3: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.31:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 260:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 3.35 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -4.1e+93)
     t_1
     (if (<= im -0.31)
       t_0
       (if (<= im 260.0)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 3.35e+100) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	double t_1 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -4.1e+93) {
		tmp = t_1;
	} else if (im <= -0.31) {
		tmp = t_0;
	} else if (im <= 260.0) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 3.35e+100) {
		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)) * re)
    t_1 = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    if (im <= (-4.1d+93)) then
        tmp = t_1
    else if (im <= (-0.31d0)) then
        tmp = t_0
    else if (im <= 260.0d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 3.35d+100) 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)) * re);
	double t_1 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -4.1e+93) {
		tmp = t_1;
	} else if (im <= -0.31) {
		tmp = t_0;
	} else if (im <= 260.0) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 3.35e+100) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	t_1 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -4.1e+93:
		tmp = t_1
	elif im <= -0.31:
		tmp = t_0
	elif im <= 260.0:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 3.35e+100:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -4.1e+93)
		tmp = t_1;
	elseif (im <= -0.31)
		tmp = t_0;
	elseif (im <= 260.0)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 3.35e+100)
		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)) * re);
	t_1 = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -4.1e+93)
		tmp = t_1;
	elseif (im <= -0.31)
		tmp = t_0;
	elseif (im <= 260.0)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 3.35e+100)
		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[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.1e+93], t$95$1, If[LessEqual[im, -0.31], t$95$0, If[LessEqual[im, 260.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.35e+100], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.1000000000000001e93 or 3.3499999999999998e100 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.0%

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

      1. mul-1-neg 97.0%

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

      2. unsub-neg 97.0%

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

      3. *-commutative 97.0%

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

      4. associate-*l* 97.0%

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

      5. distribute-lft-out-- 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in im around inf 97.0%

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

   if -4.1000000000000001e93 < im < -0.309999999999999998 or 260 < im < 3.3499999999999998e100

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 86.1%

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

   if -0.309999999999999998 < im < 260

   1. Initial program 29.9%

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

   2. Taylor expanded in im around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*l* 99.1%

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

      5. distribute-lft-out-- 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -0.31:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 260:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 3.35 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 4: 84.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -2.4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+18}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+99}:\\ \;\;\;\;-2.25 \cdot {re}^{3} + re \cdot 13.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -2.4)
     t_0
     (if (<= im 3e+18)
       (* im (- (sin re)))
       (if (<= im 6e+99) (+ (* -2.25 (pow re 3.0)) (* re 13.5)) t_0)))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -2.4) {
		tmp = t_0;
	} else if (im <= 3e+18) {
		tmp = im * -sin(re);
	} else if (im <= 6e+99) {
		tmp = (-2.25 * pow(re, 3.0)) + (re * 13.5);
	} 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 = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    if (im <= (-2.4d0)) then
        tmp = t_0
    else if (im <= 3d+18) then
        tmp = im * -sin(re)
    else if (im <= 6d+99) then
        tmp = ((-2.25d0) * (re ** 3.0d0)) + (re * 13.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -2.4) {
		tmp = t_0;
	} else if (im <= 3e+18) {
		tmp = im * -Math.sin(re);
	} else if (im <= 6e+99) {
		tmp = (-2.25 * Math.pow(re, 3.0)) + (re * 13.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -2.4:
		tmp = t_0
	elif im <= 3e+18:
		tmp = im * -math.sin(re)
	elif im <= 6e+99:
		tmp = (-2.25 * math.pow(re, 3.0)) + (re * 13.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -2.4)
		tmp = t_0;
	elseif (im <= 3e+18)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 6e+99)
		tmp = Float64(Float64(-2.25 * (re ^ 3.0)) + Float64(re * 13.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -2.4)
		tmp = t_0;
	elseif (im <= 3e+18)
		tmp = im * -sin(re);
	elseif (im <= 6e+99)
		tmp = (-2.25 * (re ^ 3.0)) + (re * 13.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.4], t$95$0, If[LessEqual[im, 3e+18], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 6e+99], N[(N[(-2.25 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] + N[(re * 13.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.39999999999999991 or 6.00000000000000029e99 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 82.9%

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

      1. mul-1-neg 82.9%

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

      2. unsub-neg 82.9%

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

      3. *-commutative 82.9%

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

      4. associate-*l* 82.9%

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

      5. distribute-lft-out-- 82.9%

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

   4. Simplified 82.9%

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

   5. Taylor expanded in im around inf 82.9%

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

   if -2.39999999999999991 < im < 3e18

   1. Initial program 31.0%

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

   2. Taylor expanded in im around 0 97.6%

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

      1. mul-1-neg 97.6%

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

      2. *-commutative 97.6%

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

      3. distribute-rgt-neg-in 97.6%

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

   4. Simplified 97.6%

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

   if 3e18 < im < 6.00000000000000029e99

   1. Initial program 100.0%

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

   3. Applied egg-rr 0.9%

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

   4. Taylor expanded in re around 0 33.9%

      \[\leadsto \color{blue}{-2.25 \cdot {re}^{3} + 13.5 \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.4:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+18}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+99}:\\ \;\;\;\;-2.25 \cdot {re}^{3} + re \cdot 13.5\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 5: 77.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+25}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq 2150000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.3e+25)
   (* -0.16666666666666666 (* re (pow im 3.0)))
   (if (<= im 2150000000.0)
     (* im (- (sin re)))
     (* re (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.3e+25) {
		tmp = -0.16666666666666666 * (re * pow(im, 3.0));
	} else if (im <= 2150000000.0) {
		tmp = im * -sin(re);
	} else {
		tmp = 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) :: tmp
    if (im <= (-1.3d+25)) then
        tmp = (-0.16666666666666666d0) * (re * (im ** 3.0d0))
    else if (im <= 2150000000.0d0) then
        tmp = im * -sin(re)
    else
        tmp = re * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.3e+25) {
		tmp = -0.16666666666666666 * (re * Math.pow(im, 3.0));
	} else if (im <= 2150000000.0) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = re * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.3e+25:
		tmp = -0.16666666666666666 * (re * math.pow(im, 3.0))
	elif im <= 2150000000.0:
		tmp = im * -math.sin(re)
	else:
		tmp = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.3e+25)
		tmp = Float64(-0.16666666666666666 * Float64(re * (im ^ 3.0)));
	elseif (im <= 2150000000.0)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.3e+25)
		tmp = -0.16666666666666666 * (re * (im ^ 3.0));
	elseif (im <= 2150000000.0)
		tmp = im * -sin(re);
	else
		tmp = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.3e+25], N[(-0.16666666666666666 * N[(re * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2150000000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.2999999999999999e25

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 83.2%

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

      1. mul-1-neg 83.2%

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

      2. unsub-neg 83.2%

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

      3. *-commutative 83.2%

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

      4. associate-*l* 83.2%

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

      5. distribute-lft-out-- 83.2%

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

   4. Simplified 83.2%

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

   5. Taylor expanded in im around inf 83.2%

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

   6. Taylor expanded in re around 0 61.9%

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

      1. *-commutative 61.9%

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

   8. Simplified 61.9%

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

   if -1.2999999999999999e25 < im < 2.15e9

   1. Initial program 33.5%

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

   2. Taylor expanded in im around 0 94.3%

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

      1. mul-1-neg 94.3%

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

      2. *-commutative 94.3%

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

      3. distribute-rgt-neg-in 94.3%

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

   4. Simplified 94.3%

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

   if 2.15e9 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 65.9%

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

      1. mul-1-neg 65.9%

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

      2. unsub-neg 65.9%

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

      3. *-commutative 65.9%

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

      4. associate-*l* 65.9%

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

      5. distribute-lft-out-- 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in re around 0 61.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+25}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq 2150000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 6: 77.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+25} \lor \neg \left(im \leq 540000000\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.1e+25) (not (<= im 540000000.0)))
   (* -0.16666666666666666 (* re (pow im 3.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.1e+25) || !(im <= 540000000.0)) {
		tmp = -0.16666666666666666 * (re * pow(im, 3.0));
	} else {
		tmp = im * -sin(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 <= (-1.1d+25)) .or. (.not. (im <= 540000000.0d0))) then
        tmp = (-0.16666666666666666d0) * (re * (im ** 3.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.1e+25) || !(im <= 540000000.0)) {
		tmp = -0.16666666666666666 * (re * Math.pow(im, 3.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.1e+25) or not (im <= 540000000.0):
		tmp = -0.16666666666666666 * (re * math.pow(im, 3.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.1e+25) || !(im <= 540000000.0))
		tmp = Float64(-0.16666666666666666 * Float64(re * (im ^ 3.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.1e+25) || ~((im <= 540000000.0)))
		tmp = -0.16666666666666666 * (re * (im ^ 3.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.1e+25], N[Not[LessEqual[im, 540000000.0]], $MachinePrecision]], N[(-0.16666666666666666 * N[(re * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.1e25 or 5.4e8 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 74.3%

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

      1. mul-1-neg 74.3%

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

      2. unsub-neg 74.3%

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

      3. *-commutative 74.3%

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

      4. associate-*l* 74.3%

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

      5. distribute-lft-out-- 74.3%

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

   4. Simplified 74.3%

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

   5. Taylor expanded in im around inf 74.3%

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

   6. Taylor expanded in re around 0 61.7%

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

      1. *-commutative 61.7%

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

   8. Simplified 61.7%

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

   if -1.1e25 < im < 5.4e8

   1. Initial program 33.5%

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

   2. Taylor expanded in im around 0 94.3%

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

      1. mul-1-neg 94.3%

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

      2. *-commutative 94.3%

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

      3. distribute-rgt-neg-in 94.3%

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

   4. Simplified 94.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+25} \lor \neg \left(im \leq 540000000\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 7: 58.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1350:\\ \;\;\;\;im \cdot \left(0.0001984126984126984 \cdot {re}^{7}\right)\\ \mathbf{elif}\;im \leq 1220000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1350.0)
   (* im (* 0.0001984126984126984 (pow re 7.0)))
   (if (<= im 1220000.0) (* im (- (sin re))) (* im (- re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1350.0) {
		tmp = im * (0.0001984126984126984 * pow(re, 7.0));
	} else if (im <= 1220000.0) {
		tmp = im * -sin(re);
	} else {
		tmp = im * -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 <= (-1350.0d0)) then
        tmp = im * (0.0001984126984126984d0 * (re ** 7.0d0))
    else if (im <= 1220000.0d0) then
        tmp = im * -sin(re)
    else
        tmp = im * -re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1350.0) {
		tmp = im * (0.0001984126984126984 * Math.pow(re, 7.0));
	} else if (im <= 1220000.0) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1350.0:
		tmp = im * (0.0001984126984126984 * math.pow(re, 7.0))
	elif im <= 1220000.0:
		tmp = im * -math.sin(re)
	else:
		tmp = im * -re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1350.0)
		tmp = Float64(im * Float64(0.0001984126984126984 * (re ^ 7.0)));
	elseif (im <= 1220000.0)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = Float64(im * Float64(-re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1350.0)
		tmp = im * (0.0001984126984126984 * (re ^ 7.0));
	elseif (im <= 1220000.0)
		tmp = im * -sin(re);
	else
		tmp = im * -re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1350.0], N[(im * N[(0.0001984126984126984 * N[Power[re, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1220000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(im * (-re)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1350

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.7%

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

      1. mul-1-neg 4.7%

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

      2. *-commutative 4.7%

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

      3. distribute-rgt-neg-in 4.7%

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

   4. Simplified 4.7%

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

   5. Taylor expanded in re around 0 3.4%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({re}^{5} \cdot im\right) + \left(-1 \cdot \left(re \cdot im\right) + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot im\right) + 0.0001984126984126984 \cdot \left({re}^{7} \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 3.4%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left({re}^{5} \cdot im\right) + -1 \cdot \left(re \cdot im\right)\right) + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot im\right) + 0.0001984126984126984 \cdot \left({re}^{7} \cdot im\right)\right)} \]

      2. +-commutative 3.4%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot im\right) + 0.0001984126984126984 \cdot \left({re}^{7} \cdot im\right)\right) + \left(-0.008333333333333333 \cdot \left({re}^{5} \cdot im\right) + -1 \cdot \left(re \cdot im\right)\right)} \]

      3. associate-*r* 3.4%

         \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot im} + 0.0001984126984126984 \cdot \left({re}^{7} \cdot im\right)\right) + \left(-0.008333333333333333 \cdot \left({re}^{5} \cdot im\right) + -1 \cdot \left(re \cdot im\right)\right) \]

      4. associate-*r* 3.4%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot im + \color{blue}{\left(0.0001984126984126984 \cdot {re}^{7}\right) \cdot im}\right) + \left(-0.008333333333333333 \cdot \left({re}^{5} \cdot im\right) + -1 \cdot \left(re \cdot im\right)\right) \]

      5. distribute-rgt-out 3.4%

         \[\leadsto \color{blue}{im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.0001984126984126984 \cdot {re}^{7}\right)} + \left(-0.008333333333333333 \cdot \left({re}^{5} \cdot im\right) + -1 \cdot \left(re \cdot im\right)\right) \]

      6. associate-*r* 3.4%

         \[\leadsto im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.0001984126984126984 \cdot {re}^{7}\right) + \left(\color{blue}{\left(-0.008333333333333333 \cdot {re}^{5}\right) \cdot im} + -1 \cdot \left(re \cdot im\right)\right) \]

      7. associate-*r* 3.4%

         \[\leadsto im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.0001984126984126984 \cdot {re}^{7}\right) + \left(\left(-0.008333333333333333 \cdot {re}^{5}\right) \cdot im + \color{blue}{\left(-1 \cdot re\right) \cdot im}\right) \]

      8. distribute-rgt-out 3.4%

         \[\leadsto im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.0001984126984126984 \cdot {re}^{7}\right) + \color{blue}{im \cdot \left(-0.008333333333333333 \cdot {re}^{5} + -1 \cdot re\right)} \]

      9. distribute-lft-out 6.7%

         \[\leadsto \color{blue}{im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.0001984126984126984 \cdot {re}^{7}\right) + \left(-0.008333333333333333 \cdot {re}^{5} + -1 \cdot re\right)\right)} \]

      10. mul-1-neg 6.7%

          \[\leadsto im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.0001984126984126984 \cdot {re}^{7}\right) + \left(-0.008333333333333333 \cdot {re}^{5} + \color{blue}{\left(-re\right)}\right)\right) \]

      11. unsub-neg 6.7%

          \[\leadsto im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.0001984126984126984 \cdot {re}^{7}\right) + \color{blue}{\left(-0.008333333333333333 \cdot {re}^{5} - re\right)}\right) \]

   7. Simplified 6.7%

      \[\leadsto \color{blue}{im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.0001984126984126984 \cdot {re}^{7}\right) + \left(-0.008333333333333333 \cdot {re}^{5} - re\right)\right)} \]

   8. Taylor expanded in re around inf 28.2%

      \[\leadsto im \cdot \color{blue}{\left(0.0001984126984126984 \cdot {re}^{7}\right)} \]

   if -1350 < im < 1.22e6

   1. Initial program 31.0%

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

   2. Taylor expanded in im around 0 97.7%

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

      1. mul-1-neg 97.7%

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

      2. *-commutative 97.7%

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

      3. distribute-rgt-neg-in 97.7%

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

   4. Simplified 97.7%

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

   if 1.22e6 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.2%

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

      1. mul-1-neg 4.2%

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

      2. *-commutative 4.2%

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

      3. distribute-rgt-neg-in 4.2%

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

   4. Simplified 4.2%

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

   5. Taylor expanded in re around 0 22.4%

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

      1. mul-1-neg 22.4%

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

      2. distribute-rgt-neg-in 22.4%

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

   7. Simplified 22.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1350:\\ \;\;\;\;im \cdot \left(0.0001984126984126984 \cdot {re}^{7}\right)\\ \mathbf{elif}\;im \leq 1220000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]

Alternative 8: 54.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5200:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5200.0) (* im (- (sin re))) (* im (- re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5200.0) {
		tmp = im * -sin(re);
	} else {
		tmp = im * -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 <= 5200.0d0) then
        tmp = im * -sin(re)
    else
        tmp = im * -re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5200.0) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5200.0:
		tmp = im * -math.sin(re)
	else:
		tmp = im * -re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5200.0)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = Float64(im * Float64(-re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5200.0)
		tmp = im * -sin(re);
	else
		tmp = im * -re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5200.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(im * (-re)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5200

   1. Initial program 52.9%

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

   2. Taylor expanded in im around 0 68.2%

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

      1. mul-1-neg 68.2%

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

      2. *-commutative 68.2%

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

      3. distribute-rgt-neg-in 68.2%

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

   4. Simplified 68.2%

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

   if 5200 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.2%

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

      1. mul-1-neg 4.2%

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

      2. *-commutative 4.2%

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

      3. distribute-rgt-neg-in 4.2%

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

   4. Simplified 4.2%

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

   5. Taylor expanded in re around 0 22.4%

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

      1. mul-1-neg 22.4%

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

      2. distribute-rgt-neg-in 22.4%

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

   7. Simplified 22.4%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5200:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]

Alternative 9: 33.0% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im * -re

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.2%

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

2. Taylor expanded in im around 0 52.9%

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

   1. mul-1-neg 52.9%

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

   2. *-commutative 52.9%

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

   3. distribute-rgt-neg-in 52.9%

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

4. Simplified 52.9%

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

5. Taylor expanded in re around 0 33.9%

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

   1. mul-1-neg 33.9%

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

   2. distribute-rgt-neg-in 33.9%

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

7. Simplified 33.9%

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

8. Final simplification 33.9%

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

Alternative 10: 3.2% accurate, 102.7× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* re 13.5))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re * 13.5

Julia

function code(re, im)
	return Float64(re * 13.5)
end

MATLAB

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

Wolfram

code[re_, im_] := N[(re * 13.5), $MachinePrecision]

Derivation

1. Initial program 64.2%

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

3. Applied egg-rr 3.3%

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

4. Taylor expanded in re around 0 3.0%

   \[\leadsto \color{blue}{13.5 \cdot re} \]
5. Step-by-step derivation

   1. *-commutative 3.0%

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

6. Simplified 3.0%

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

7. Final simplification 3.0%

   \[\leadsto re \cdot 13.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (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 (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-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.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(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 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[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[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

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

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

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))