math.sin on complex, imaginary part

Percentage Accurate: 54.1% → 99.8%

Time: 9.3s

Alternatives: 15

Speedup: 2.9×

• 49.4% 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: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.2 \lor \neg \left(t_0 \leq 0.0005\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.2) (not (<= t_0 0.0005)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 0.0005)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.2d0)) .or. (.not. (t_0 <= 0.0005d0))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 0.0005)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.2) or not (t_0 <= 0.0005):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.2) || !(t_0 <= 0.0005))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.2) || ~((t_0 <= 0.0005)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 0.0005]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.20000000000000001 or 5.0000000000000001e-4 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 5.0000000000000001e-4

   1. Initial program 9.4%

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

      1. sub0-neg 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. distribute-lft-out-- 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2 \lor \neg \left(e^{-im} - e^{im} \leq 0.0005\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 95.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -3.5 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.265:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;im \leq 0.102 \lor \neg \left(im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im)))
        (t_1 (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -3.5e+128)
     t_1
     (if (<= im -0.265)
       (* 0.5 t_0)
       (if (or (<= im 0.102) (not (<= im 5.6e+102)))
         t_1
         (* t_0 (+ 0.5 (* re (* re -0.25)))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -3.5e+128) {
		tmp = t_1;
	} else if (im <= -0.265) {
		tmp = 0.5 * t_0;
	} else if ((im <= 0.102) || !(im <= 5.6e+102)) {
		tmp = t_1;
	} else {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-3.5d+128)) then
        tmp = t_1
    else if (im <= (-0.265d0)) then
        tmp = 0.5d0 * t_0
    else if ((im <= 0.102d0) .or. (.not. (im <= 5.6d+102))) then
        tmp = t_1
    else
        tmp = t_0 * (0.5d0 + (re * (re * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -3.5e+128) {
		tmp = t_1;
	} else if (im <= -0.265) {
		tmp = 0.5 * t_0;
	} else if ((im <= 0.102) || !(im <= 5.6e+102)) {
		tmp = t_1;
	} else {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -3.5e+128:
		tmp = t_1
	elif im <= -0.265:
		tmp = 0.5 * t_0
	elif (im <= 0.102) or not (im <= 5.6e+102):
		tmp = t_1
	else:
		tmp = t_0 * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -3.5e+128)
		tmp = t_1;
	elseif (im <= -0.265)
		tmp = Float64(0.5 * t_0);
	elseif ((im <= 0.102) || !(im <= 5.6e+102))
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -3.5e+128)
		tmp = t_1;
	elseif (im <= -0.265)
		tmp = 0.5 * t_0;
	elseif ((im <= 0.102) || ~((im <= 5.6e+102)))
		tmp = t_1;
	else
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.5e+128], t$95$1, If[LessEqual[im, -0.265], N[(0.5 * t$95$0), $MachinePrecision], If[Or[LessEqual[im, 0.102], N[Not[LessEqual[im, 5.6e+102]], $MachinePrecision]], t$95$1, N[(t$95$0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.49999999999999969e128 or -0.26500000000000001 < im < 0.101999999999999993 or 5.60000000000000037e102 < im

   1. Initial program 43.5%

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

      1. sub0-neg 43.5%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

   if -3.49999999999999969e128 < im < -0.26500000000000001

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 82.1%

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

   if 0.101999999999999993 < im < 5.60000000000000037e102

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 9.4%

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

      1. *-commutative 9.4%

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

      2. associate-*r* 9.4%

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

      3. distribute-rgt-out 71.4%

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

      4. +-commutative 71.4%

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

      5. *-commutative 71.4%

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

      6. unpow2 71.4%

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

      7. associate-*l* 71.4%

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

   6. Simplified 71.4%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{+128}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -0.265:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.102 \lor \neg \left(im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 3: 95.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{+128} \lor \neg \left(im \leq -0.092 \lor \neg \left(im \leq 0.039\right) \land im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.5e+128)
         (not (or (<= im -0.092) (and (not (<= im 0.039)) (<= im 5.6e+102)))))
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* 0.5 (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.5e+128) || !((im <= -0.092) || (!(im <= 0.039) && (im <= 5.6e+102)))) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-3.5d+128)) .or. (.not. (im <= (-0.092d0)) .or. (.not. (im <= 0.039d0)) .and. (im <= 5.6d+102))) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = 0.5d0 * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.5e+128) || !((im <= -0.092) || (!(im <= 0.039) && (im <= 5.6e+102)))) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.5e+128) or not ((im <= -0.092) or (not (im <= 0.039) and (im <= 5.6e+102))):
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.5e+128) || !((im <= -0.092) || (!(im <= 0.039) && (im <= 5.6e+102))))
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.5e+128) || ~(((im <= -0.092) || (~((im <= 0.039)) && (im <= 5.6e+102)))))
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = 0.5 * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.5e+128], N[Not[Or[LessEqual[im, -0.092], And[N[Not[LessEqual[im, 0.039]], $MachinePrecision], LessEqual[im, 5.6e+102]]]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.49999999999999969e128 or -0.091999999999999998 < im < 0.0389999999999999999 or 5.60000000000000037e102 < im

   1. Initial program 43.5%

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

      1. sub0-neg 43.5%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

   if -3.49999999999999969e128 < im < -0.091999999999999998 or 0.0389999999999999999 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{+128} \lor \neg \left(im \leq -0.092 \lor \neg \left(im \leq 0.039\right) \land im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4: 92.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{if}\;im \leq -6.8 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.00192:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0018:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0))))
   (if (<= im -6.8e+143)
     t_1
     (if (<= im -0.00192)
       t_0
       (if (<= im 0.0018)
         (* im (- (cos re)))
         (if (<= im 1.3e+152) 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) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -6.8e+143) {
		tmp = t_1;
	} else if (im <= -0.00192) {
		tmp = t_0;
	} else if (im <= 0.0018) {
		tmp = im * -cos(re);
	} else if (im <= 1.3e+152) {
		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) * (9.0d0 - (im * im))) / (im + (-3.0d0))
    if (im <= (-6.8d+143)) then
        tmp = t_1
    else if (im <= (-0.00192d0)) then
        tmp = t_0
    else if (im <= 0.0018d0) then
        tmp = im * -cos(re)
    else if (im <= 1.3d+152) 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) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -6.8e+143) {
		tmp = t_1;
	} else if (im <= -0.00192) {
		tmp = t_0;
	} else if (im <= 0.0018) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.3e+152) {
		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) * (9.0 - (im * im))) / (im + -3.0)
	tmp = 0
	if im <= -6.8e+143:
		tmp = t_1
	elif im <= -0.00192:
		tmp = t_0
	elif im <= 0.0018:
		tmp = im * -math.cos(re)
	elif im <= 1.3e+152:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	tmp = 0.0
	if (im <= -6.8e+143)
		tmp = t_1;
	elseif (im <= -0.00192)
		tmp = t_0;
	elseif (im <= 0.0018)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.3e+152)
		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) * (9.0 - (im * im))) / (im + -3.0);
	tmp = 0.0;
	if (im <= -6.8e+143)
		tmp = t_1;
	elseif (im <= -0.00192)
		tmp = t_0;
	elseif (im <= 0.0018)
		tmp = im * -cos(re);
	elseif (im <= 1.3e+152)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.8e+143], t$95$1, If[LessEqual[im, -0.00192], t$95$0, If[LessEqual[im, 0.0018], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.3e+152], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.79999999999999964e143 or 1.3e152 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.9%

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

      1. *-commutative 6.9%

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

      2. flip-- 95.6%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 95.6%

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

      4. metadata-eval 95.6%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 95.6%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 95.6%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -6.79999999999999964e143 < im < -0.00192000000000000005 or 0.0018 < im < 1.3e152

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 75.8%

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

   if -0.00192000000000000005 < im < 0.0018

   1. Initial program 9.4%

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

      1. sub0-neg 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. *-commutative 99.2%

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

      3. distribute-lft-neg-in 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{+143}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -0.00192:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.0018:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 5: 81.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_1 := im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ t_2 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{if}\;im \leq -6.8 \cdot 10^{+143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -5.6 \cdot 10^{+24}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_1 (- (* im (* 0.5 (* re re))) im))
        (t_2 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0))))
   (if (<= im -6.8e+143)
     t_2
     (if (<= im -5.6e+102)
       t_0
       (if (<= im -5.6e+24)
         (* re (* re 0.75))
         (if (<= im 3.2e+30)
           (* im (- (cos re)))
           (if (<= im 2.8e+64)
             t_1
             (if (<= im 7.5e+144) t_0 (if (<= im 1.35e+154) t_1 t_2)))))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = (im * (0.5 * (re * re))) - im;
	double t_2 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -6.8e+143) {
		tmp = t_2;
	} else if (im <= -5.6e+102) {
		tmp = t_0;
	} else if (im <= -5.6e+24) {
		tmp = re * (re * 0.75);
	} else if (im <= 3.2e+30) {
		tmp = im * -cos(re);
	} else if (im <= 2.8e+64) {
		tmp = t_1;
	} else if (im <= 7.5e+144) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    t_1 = (im * (0.5d0 * (re * re))) - im
    t_2 = (cos(re) * (9.0d0 - (im * im))) / (im + (-3.0d0))
    if (im <= (-6.8d+143)) then
        tmp = t_2
    else if (im <= (-5.6d+102)) then
        tmp = t_0
    else if (im <= (-5.6d+24)) then
        tmp = re * (re * 0.75d0)
    else if (im <= 3.2d+30) then
        tmp = im * -cos(re)
    else if (im <= 2.8d+64) then
        tmp = t_1
    else if (im <= 7.5d+144) then
        tmp = t_0
    else if (im <= 1.35d+154) then
        tmp = t_1
    else
        tmp = t_2
    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 t_1 = (im * (0.5 * (re * re))) - im;
	double t_2 = (Math.cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -6.8e+143) {
		tmp = t_2;
	} else if (im <= -5.6e+102) {
		tmp = t_0;
	} else if (im <= -5.6e+24) {
		tmp = re * (re * 0.75);
	} else if (im <= 3.2e+30) {
		tmp = im * -Math.cos(re);
	} else if (im <= 2.8e+64) {
		tmp = t_1;
	} else if (im <= 7.5e+144) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	t_1 = (im * (0.5 * (re * re))) - im
	t_2 = (math.cos(re) * (9.0 - (im * im))) / (im + -3.0)
	tmp = 0
	if im <= -6.8e+143:
		tmp = t_2
	elif im <= -5.6e+102:
		tmp = t_0
	elif im <= -5.6e+24:
		tmp = re * (re * 0.75)
	elif im <= 3.2e+30:
		tmp = im * -math.cos(re)
	elif im <= 2.8e+64:
		tmp = t_1
	elif im <= 7.5e+144:
		tmp = t_0
	elif im <= 1.35e+154:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_1 = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im)
	t_2 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	tmp = 0.0
	if (im <= -6.8e+143)
		tmp = t_2;
	elseif (im <= -5.6e+102)
		tmp = t_0;
	elseif (im <= -5.6e+24)
		tmp = Float64(re * Float64(re * 0.75));
	elseif (im <= 3.2e+30)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 2.8e+64)
		tmp = t_1;
	elseif (im <= 7.5e+144)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	t_1 = (im * (0.5 * (re * re))) - im;
	t_2 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	tmp = 0.0;
	if (im <= -6.8e+143)
		tmp = t_2;
	elseif (im <= -5.6e+102)
		tmp = t_0;
	elseif (im <= -5.6e+24)
		tmp = re * (re * 0.75);
	elseif (im <= 3.2e+30)
		tmp = im * -cos(re);
	elseif (im <= 2.8e+64)
		tmp = t_1;
	elseif (im <= 7.5e+144)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_2;
	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]}, Block[{t$95$1 = N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.8e+143], t$95$2, If[LessEqual[im, -5.6e+102], t$95$0, If[LessEqual[im, -5.6e+24], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.2e+30], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 2.8e+64], t$95$1, If[LessEqual[im, 7.5e+144], t$95$0, If[LessEqual[im, 1.35e+154], t$95$1, t$95$2]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -6.79999999999999964e143 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.0%

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

      1. *-commutative 7.0%

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

      2. flip-- 98.5%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 98.5%

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

      4. metadata-eval 98.5%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 98.5%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 98.5%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -6.79999999999999964e143 < im < -5.60000000000000037e102 or 2.80000000000000024e64 < im < 7.5000000000000006e144

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 76.6%

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

      1. mul-1-neg 76.6%

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

      2. unsub-neg 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-*l* 76.6%

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

      5. distribute-lft-out-- 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in re around 0 68.0%

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

   if -5.60000000000000037e102 < im < -5.6000000000000003e24

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 64.7%

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

      4. +-commutative 64.7%

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

      5. *-commutative 64.7%

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

      6. unpow2 64.7%

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

      7. associate-*l* 64.7%

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

   6. Simplified 64.7%

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

   8. Applied egg-rr 29.9%

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

   9. Taylor expanded in re around inf 30.3%

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

       1. *-commutative 30.3%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 30.3%

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

       3. associate-*l* 30.3%

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

   11. Simplified 30.3%

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

   if -5.6000000000000003e24 < im < 3.19999999999999973e30

   1. Initial program 15.9%

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

      1. sub0-neg 15.9%

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

   3. Simplified 15.9%

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

   4. Taylor expanded in im around 0 92.6%

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

      1. mul-1-neg 92.6%

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

      2. *-commutative 92.6%

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

      3. distribute-lft-neg-in 92.6%

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

   6. Simplified 92.6%

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

   if 3.19999999999999973e30 < im < 2.80000000000000024e64 or 7.5000000000000006e144 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.8%

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

      1. mul-1-neg 3.8%

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

      2. *-commutative 3.8%

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

      3. distribute-lft-neg-in 3.8%

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

   6. Simplified 3.8%

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

   7. Taylor expanded in re around 0 69.8%

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

      1. neg-mul-1 69.8%

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

      2. +-commutative 69.8%

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

      3. unsub-neg 69.8%

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

      4. *-commutative 69.8%

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

      5. *-commutative 69.8%

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

      6. associate-*l* 69.8%

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

      7. unpow2 69.8%

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

   9. Simplified 69.8%

      \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right) - im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{+143}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -5.6 \cdot 10^{+24}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+64}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+144}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 6: 82.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_1 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ t_2 := \left(-0.5 \cdot \left(re \cdot re\right)\right) \cdot t_0\\ \mathbf{if}\;im \leq -6.8 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3.05 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_1 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0)))
        (t_2 (* (* -0.5 (* re re)) t_0)))
   (if (<= im -6.8e+143)
     t_1
     (if (<= im -3.05e+92)
       t_0
       (if (<= im -3e+20)
         t_2
         (if (<= im 5e+30)
           (* im (- (cos re)))
           (if (<= im 1.8e+96)
             t_2
             (if (<= im 7.5e+144)
               t_0
               (if (<= im 1.35e+154)
                 (- (* im (* 0.5 (* re re))) im)
                 t_1)))))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double t_2 = (-0.5 * (re * re)) * t_0;
	double tmp;
	if (im <= -6.8e+143) {
		tmp = t_1;
	} else if (im <= -3.05e+92) {
		tmp = t_0;
	} else if (im <= -3e+20) {
		tmp = t_2;
	} else if (im <= 5e+30) {
		tmp = im * -cos(re);
	} else if (im <= 1.8e+96) {
		tmp = t_2;
	} else if (im <= 7.5e+144) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    t_1 = (cos(re) * (9.0d0 - (im * im))) / (im + (-3.0d0))
    t_2 = ((-0.5d0) * (re * re)) * t_0
    if (im <= (-6.8d+143)) then
        tmp = t_1
    else if (im <= (-3.05d+92)) then
        tmp = t_0
    else if (im <= (-3d+20)) then
        tmp = t_2
    else if (im <= 5d+30) then
        tmp = im * -cos(re)
    else if (im <= 1.8d+96) then
        tmp = t_2
    else if (im <= 7.5d+144) then
        tmp = t_0
    else if (im <= 1.35d+154) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = (Math.cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double t_2 = (-0.5 * (re * re)) * t_0;
	double tmp;
	if (im <= -6.8e+143) {
		tmp = t_1;
	} else if (im <= -3.05e+92) {
		tmp = t_0;
	} else if (im <= -3e+20) {
		tmp = t_2;
	} else if (im <= 5e+30) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.8e+96) {
		tmp = t_2;
	} else if (im <= 7.5e+144) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	t_1 = (math.cos(re) * (9.0 - (im * im))) / (im + -3.0)
	t_2 = (-0.5 * (re * re)) * t_0
	tmp = 0
	if im <= -6.8e+143:
		tmp = t_1
	elif im <= -3.05e+92:
		tmp = t_0
	elif im <= -3e+20:
		tmp = t_2
	elif im <= 5e+30:
		tmp = im * -math.cos(re)
	elif im <= 1.8e+96:
		tmp = t_2
	elif im <= 7.5e+144:
		tmp = t_0
	elif im <= 1.35e+154:
		tmp = (im * (0.5 * (re * re))) - im
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_1 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	t_2 = Float64(Float64(-0.5 * Float64(re * re)) * t_0)
	tmp = 0.0
	if (im <= -6.8e+143)
		tmp = t_1;
	elseif (im <= -3.05e+92)
		tmp = t_0;
	elseif (im <= -3e+20)
		tmp = t_2;
	elseif (im <= 5e+30)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.8e+96)
		tmp = t_2;
	elseif (im <= 7.5e+144)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	t_2 = (-0.5 * (re * re)) * t_0;
	tmp = 0.0;
	if (im <= -6.8e+143)
		tmp = t_1;
	elseif (im <= -3.05e+92)
		tmp = t_0;
	elseif (im <= -3e+20)
		tmp = t_2;
	elseif (im <= 5e+30)
		tmp = im * -cos(re);
	elseif (im <= 1.8e+96)
		tmp = t_2;
	elseif (im <= 7.5e+144)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = (im * (0.5 * (re * re))) - im;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[im, -6.8e+143], t$95$1, If[LessEqual[im, -3.05e+92], t$95$0, If[LessEqual[im, -3e+20], t$95$2, If[LessEqual[im, 5e+30], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.8e+96], t$95$2, If[LessEqual[im, 7.5e+144], t$95$0, If[LessEqual[im, 1.35e+154], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$1]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -6.79999999999999964e143 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.0%

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

      1. *-commutative 7.0%

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

      2. flip-- 98.5%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 98.5%

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

      4. metadata-eval 98.5%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 98.5%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 98.5%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -6.79999999999999964e143 < im < -3.04999999999999987e92 or 1.80000000000000007e96 < im < 7.5000000000000006e144

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 83.7%

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

      1. mul-1-neg 83.7%

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

      2. unsub-neg 83.7%

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

      3. *-commutative 83.7%

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

      4. associate-*l* 83.7%

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

      5. distribute-lft-out-- 83.7%

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

   6. Simplified 83.7%

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

   7. Taylor expanded in re around 0 74.6%

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

   if -3.04999999999999987e92 < im < -3e20 or 4.9999999999999998e30 < im < 1.80000000000000007e96

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.1%

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

      1. mul-1-neg 5.1%

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

      2. unsub-neg 5.1%

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

      3. *-commutative 5.1%

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

      4. associate-*l* 5.1%

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

      5. distribute-lft-out-- 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in re around 0 43.3%

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

      1. associate--l+ 43.3%

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

      2. associate-*r* 43.3%

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

      3. distribute-lft1-in 43.3%

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

      4. unpow2 43.3%

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

   9. Simplified 43.3%

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

   10. Taylor expanded in re around inf 41.8%

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

       1. fma-neg 41.8%

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

       2. associate-*r* 41.8%

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

       3. unpow2 41.8%

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

       4. fma-neg 41.8%

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

   12. Simplified 41.8%

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

   if -3e20 < im < 4.9999999999999998e30

   1. Initial program 15.3%

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

      1. sub0-neg 15.3%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in im around 0 93.2%

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

      1. mul-1-neg 93.2%

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

      2. *-commutative 93.2%

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

      3. distribute-lft-neg-in 93.2%

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

   6. Simplified 93.2%

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

   if 7.5000000000000006e144 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.6%

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

      1. mul-1-neg 4.6%

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

      2. *-commutative 4.6%

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

      3. distribute-lft-neg-in 4.6%

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

   6. Simplified 4.6%

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

   7. Taylor expanded in re around 0 76.1%

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

      1. neg-mul-1 76.1%

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

      2. +-commutative 76.1%

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

      3. unsub-neg 76.1%

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

      4. *-commutative 76.1%

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

      5. *-commutative 76.1%

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

      6. associate-*l* 76.1%

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

      7. unpow2 76.1%

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

   9. Simplified 76.1%

      \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right) - im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{+143}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -3.05 \cdot 10^{+92}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -3 \cdot 10^{+20}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+96}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+144}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 7: 83.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ t_1 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{if}\;im \leq -8 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -22:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (+ (* -0.5 (* re re)) 1.0)
          (- (* (pow im 3.0) -0.16666666666666666) im)))
        (t_1 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0))))
   (if (<= im -8e+179)
     t_1
     (if (<= im -22.0)
       t_0
       (if (<= im 1.7e-7)
         (* im (- (cos re)))
         (if (<= im 1.35e+154) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = ((-0.5 * (re * re)) + 1.0) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -8e+179) {
		tmp = t_1;
	} else if (im <= -22.0) {
		tmp = t_0;
	} else if (im <= 1.7e-7) {
		tmp = im * -cos(re);
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((-0.5d0) * (re * re)) + 1.0d0) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    t_1 = (cos(re) * (9.0d0 - (im * im))) / (im + (-3.0d0))
    if (im <= (-8d+179)) then
        tmp = t_1
    else if (im <= (-22.0d0)) then
        tmp = t_0
    else if (im <= 1.7d-7) then
        tmp = im * -cos(re)
    else if (im <= 1.35d+154) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = ((-0.5 * (re * re)) + 1.0) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double t_1 = (Math.cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -8e+179) {
		tmp = t_1;
	} else if (im <= -22.0) {
		tmp = t_0;
	} else if (im <= 1.7e-7) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((-0.5 * (re * re)) + 1.0) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	t_1 = (math.cos(re) * (9.0 - (im * im))) / (im + -3.0)
	tmp = 0
	if im <= -8e+179:
		tmp = t_1
	elif im <= -22.0:
		tmp = t_0
	elif im <= 1.7e-7:
		tmp = im * -math.cos(re)
	elif im <= 1.35e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(-0.5 * Float64(re * re)) + 1.0) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	t_1 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	tmp = 0.0
	if (im <= -8e+179)
		tmp = t_1;
	elseif (im <= -22.0)
		tmp = t_0;
	elseif (im <= 1.7e-7)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((-0.5 * (re * re)) + 1.0) * (((im ^ 3.0) * -0.16666666666666666) - im);
	t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	tmp = 0.0;
	if (im <= -8e+179)
		tmp = t_1;
	elseif (im <= -22.0)
		tmp = t_0;
	elseif (im <= 1.7e-7)
		tmp = im * -cos(re);
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -8e+179], t$95$1, If[LessEqual[im, -22.0], t$95$0, If[LessEqual[im, 1.7e-7], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.99999999999999984e179 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.1%

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

      1. *-commutative 7.1%

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

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -7.99999999999999984e179 < im < -22 or 1.69999999999999987e-7 < im < 1.35000000000000003e154

   1. Initial program 99.1%

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

      1. sub0-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in im around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. unsub-neg 43.1%

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

      3. *-commutative 43.1%

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

      4. associate-*l* 43.1%

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

      5. distribute-lft-out-- 43.1%

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

   6. Simplified 43.1%

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

   7. Taylor expanded in re around 0 22.4%

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

      1. associate--l+ 22.4%

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

      2. associate-*r* 22.4%

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

      3. distribute-lft1-in 49.9%

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

      4. unpow2 49.9%

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

   9. Simplified 49.9%

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

   if -22 < im < 1.69999999999999987e-7

   1. Initial program 8.4%

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

      1. sub0-neg 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in im around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-lft-neg-in 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{+179}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -22:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 8: 76.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -1.55 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -5.8 \cdot 10^{+26}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+64}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -1.55e+102)
     t_0
     (if (<= im -5.8e+26)
       (* re (* re 0.75))
       (if (<= im 3.3e+30)
         (* im (- (cos re)))
         (if (<= im 2.8e+64) (- (* im (* 0.5 (* re re))) im) t_0))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -1.55e+102) {
		tmp = t_0;
	} else if (im <= -5.8e+26) {
		tmp = re * (re * 0.75);
	} else if (im <= 3.3e+30) {
		tmp = im * -cos(re);
	} else if (im <= 2.8e+64) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-1.55d+102)) then
        tmp = t_0
    else if (im <= (-5.8d+26)) then
        tmp = re * (re * 0.75d0)
    else if (im <= 3.3d+30) then
        tmp = im * -cos(re)
    else if (im <= 2.8d+64) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -1.55e+102) {
		tmp = t_0;
	} else if (im <= -5.8e+26) {
		tmp = re * (re * 0.75);
	} else if (im <= 3.3e+30) {
		tmp = im * -Math.cos(re);
	} else if (im <= 2.8e+64) {
		tmp = (im * (0.5 * (re * re))) - 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 <= -1.55e+102:
		tmp = t_0
	elif im <= -5.8e+26:
		tmp = re * (re * 0.75)
	elif im <= 3.3e+30:
		tmp = im * -math.cos(re)
	elif im <= 2.8e+64:
		tmp = (im * (0.5 * (re * re))) - 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 <= -1.55e+102)
		tmp = t_0;
	elseif (im <= -5.8e+26)
		tmp = Float64(re * Float64(re * 0.75));
	elseif (im <= 3.3e+30)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 2.8e+64)
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - 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 <= -1.55e+102)
		tmp = t_0;
	elseif (im <= -5.8e+26)
		tmp = re * (re * 0.75);
	elseif (im <= 3.3e+30)
		tmp = im * -cos(re);
	elseif (im <= 2.8e+64)
		tmp = (im * (0.5 * (re * re))) - 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, -1.55e+102], t$95$0, If[LessEqual[im, -5.8e+26], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.3e+30], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 2.8e+64], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.54999999999999993e102 or 2.80000000000000024e64 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 93.8%

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

      1. mul-1-neg 93.8%

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

      2. unsub-neg 93.8%

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

      3. *-commutative 93.8%

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

      4. associate-*l* 93.8%

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

      5. distribute-lft-out-- 93.8%

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

   6. Simplified 93.8%

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

   7. Taylor expanded in re around 0 72.9%

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

   if -1.54999999999999993e102 < im < -5.8e26

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 64.7%

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

      4. +-commutative 64.7%

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

      5. *-commutative 64.7%

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

      6. unpow2 64.7%

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

      7. associate-*l* 64.7%

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

   6. Simplified 64.7%

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

   8. Applied egg-rr 29.9%

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

   9. Taylor expanded in re around inf 30.3%

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

       1. *-commutative 30.3%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 30.3%

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

       3. associate-*l* 30.3%

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

   11. Simplified 30.3%

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

   if -5.8e26 < im < 3.30000000000000026e30

   1. Initial program 15.9%

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

      1. sub0-neg 15.9%

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

   3. Simplified 15.9%

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

   4. Taylor expanded in im around 0 92.6%

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

      1. mul-1-neg 92.6%

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

      2. *-commutative 92.6%

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

      3. distribute-lft-neg-in 92.6%

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

   6. Simplified 92.6%

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

   if 3.30000000000000026e30 < im < 2.80000000000000024e64

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. *-commutative 3.5%

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

      3. distribute-lft-neg-in 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 67.0%

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

      1. neg-mul-1 67.0%

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

      2. +-commutative 67.0%

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

      3. unsub-neg 67.0%

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

      4. *-commutative 67.0%

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

      5. *-commutative 67.0%

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

      6. associate-*l* 67.0%

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

      7. unpow2 67.0%

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

   9. Simplified 67.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.55 \cdot 10^{+102}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -5.8 \cdot 10^{+26}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+64}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 9: 58.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{+24}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5.6e+24)
   (* re (* re 0.75))
   (if (<= im 3.7e+30) (* im (- (cos re))) (- (* im (* 0.5 (* re re))) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -5.6e+24) {
		tmp = re * (re * 0.75);
	} else if (im <= 3.7e+30) {
		tmp = im * -cos(re);
	} else {
		tmp = (im * (0.5 * (re * re))) - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-5.6d+24)) then
        tmp = re * (re * 0.75d0)
    else if (im <= 3.7d+30) then
        tmp = im * -cos(re)
    else
        tmp = (im * (0.5d0 * (re * re))) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -5.6e+24) {
		tmp = re * (re * 0.75);
	} else if (im <= 3.7e+30) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = (im * (0.5 * (re * re))) - im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -5.6e+24:
		tmp = re * (re * 0.75)
	elif im <= 3.7e+30:
		tmp = im * -math.cos(re)
	else:
		tmp = (im * (0.5 * (re * re))) - im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -5.6e+24)
		tmp = Float64(re * Float64(re * 0.75));
	elseif (im <= 3.7e+30)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -5.6e+24)
		tmp = re * (re * 0.75);
	elseif (im <= 3.7e+30)
		tmp = im * -cos(re);
	else
		tmp = (im * (0.5 * (re * re))) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -5.6e+24], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.7e+30], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.6000000000000003e24

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 68.9%

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

      4. +-commutative 68.9%

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

      5. *-commutative 68.9%

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

      6. unpow2 68.9%

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

      7. associate-*l* 68.9%

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

   6. Simplified 68.9%

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

   8. Applied egg-rr 16.0%

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

   9. Taylor expanded in re around inf 16.6%

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

       1. *-commutative 16.6%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 16.6%

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

       3. associate-*l* 16.6%

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

   11. Simplified 16.6%

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

   if -5.6000000000000003e24 < im < 3.70000000000000016e30

   1. Initial program 15.9%

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

      1. sub0-neg 15.9%

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

   3. Simplified 15.9%

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

   4. Taylor expanded in im around 0 92.6%

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

      1. mul-1-neg 92.6%

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

      2. *-commutative 92.6%

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

      3. distribute-lft-neg-in 92.6%

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

   6. Simplified 92.6%

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

   if 3.70000000000000016e30 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.7%

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

      1. mul-1-neg 5.7%

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

      2. *-commutative 5.7%

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

      3. distribute-lft-neg-in 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 30.0%

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

      1. neg-mul-1 30.0%

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

      2. +-commutative 30.0%

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

      3. unsub-neg 30.0%

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

      4. *-commutative 30.0%

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

      5. *-commutative 30.0%

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

      6. associate-*l* 30.0%

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

      7. unpow2 30.0%

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

   9. Simplified 30.0%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{+24}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \]

Alternative 10: 36.4% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot 0.75\\ \mathbf{if}\;re \leq 3.3 \cdot 10^{+77}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2.25 - t_0 \cdot t_0}{-1.5 - t_0}\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+229}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) 0.75)))
   (if (<= re 3.3e+77)
     (- (* im (* 0.5 (* re re))) im)
     (if (<= re 1.35e+154)
       (/ (- 2.25 (* t_0 t_0)) (- -1.5 t_0))
       (if (<= re 1.05e+229)
         (* (+ 0.5 (* re (* re -0.25))) 27.0)
         (* re (* re 0.75)))))))

C

double code(double re, double im) {
	double t_0 = (re * re) * 0.75;
	double tmp;
	if (re <= 3.3e+77) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else if (re <= 1.35e+154) {
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	} else if (re <= 1.05e+229) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * re) * 0.75d0
    if (re <= 3.3d+77) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else if (re <= 1.35d+154) then
        tmp = (2.25d0 - (t_0 * t_0)) / ((-1.5d0) - t_0)
    else if (re <= 1.05d+229) then
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    else
        tmp = re * (re * 0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) * 0.75;
	double tmp;
	if (re <= 3.3e+77) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else if (re <= 1.35e+154) {
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	} else if (re <= 1.05e+229) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) * 0.75
	tmp = 0
	if re <= 3.3e+77:
		tmp = (im * (0.5 * (re * re))) - im
	elif re <= 1.35e+154:
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0)
	elif re <= 1.05e+229:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	else:
		tmp = re * (re * 0.75)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * 0.75)
	tmp = 0.0
	if (re <= 3.3e+77)
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	elseif (re <= 1.35e+154)
		tmp = Float64(Float64(2.25 - Float64(t_0 * t_0)) / Float64(-1.5 - t_0));
	elseif (re <= 1.05e+229)
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	else
		tmp = Float64(re * Float64(re * 0.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) * 0.75;
	tmp = 0.0;
	if (re <= 3.3e+77)
		tmp = (im * (0.5 * (re * re))) - im;
	elseif (re <= 1.35e+154)
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	elseif (re <= 1.05e+229)
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	else
		tmp = re * (re * 0.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]}, If[LessEqual[re, 3.3e+77], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[re, 1.35e+154], N[(N[(2.25 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.5 - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e+229], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < 3.2999999999999998e77

   1. Initial program 57.2%

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

      1. sub0-neg 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in im around 0 50.2%

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

      1. mul-1-neg 50.2%

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

      2. *-commutative 50.2%

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

      3. distribute-lft-neg-in 50.2%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in re around 0 35.7%

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

      1. neg-mul-1 35.7%

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

      2. +-commutative 35.7%

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

      3. unsub-neg 35.7%

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

      4. *-commutative 35.7%

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

      5. *-commutative 35.7%

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

      6. associate-*l* 35.7%

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

      7. unpow2 35.7%

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

   9. Simplified 35.7%

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

   if 3.2999999999999998e77 < re < 1.35000000000000003e154

   1. Initial program 36.9%

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

      1. sub0-neg 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in re around 0 3.6%

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

      1. *-commutative 3.6%

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

      2. associate-*r* 3.6%

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

      3. distribute-rgt-out 3.6%

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

      4. +-commutative 3.6%

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

      5. *-commutative 3.6%

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

      6. unpow2 3.6%

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

      7. associate-*l* 3.6%

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

   6. Simplified 3.6%

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

   8. Applied egg-rr 2.9%

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

      1. distribute-lft-in 2.9%

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

      2. flip-+ 25.9%

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

      3. metadata-eval 25.9%

         \[\leadsto \frac{\color{blue}{-1.5} \cdot \left(-3 \cdot 0.5\right) - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      4. metadata-eval 25.9%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{-1.5} - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      5. metadata-eval 25.9%

         \[\leadsto \frac{\color{blue}{2.25} - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      6. *-commutative 25.9%

         \[\leadsto \frac{2.25 - \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)} \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      7. *-commutative 25.9%

         \[\leadsto \frac{2.25 - \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right) \cdot \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      8. associate-*r* 25.9%

         \[\leadsto \frac{2.25 - \left(\color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3\right) \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      9. associate-*l* 25.9%

         \[\leadsto \frac{2.25 - \color{blue}{\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)\right)} \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      10. metadata-eval 25.9%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \color{blue}{0.75}\right) \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      11. associate-*r* 25.9%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      12. associate-*l* 25.9%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)\right)}}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      13. metadata-eval 25.9%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{0.75}\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      14. metadata-eval 25.9%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{\color{blue}{-1.5} - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      15. *-commutative 25.9%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3}} \]

      16. associate-*r* 25.9%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3} \]

      17. associate-*l* 25.9%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)}} \]

      18. metadata-eval 25.9%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot \color{blue}{0.75}} \]

   10. Applied egg-rr 25.9%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}} \]

   if 1.35000000000000003e154 < re < 1.04999999999999994e229

   1. Initial program 43.4%

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

      1. sub0-neg 43.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 24.0%

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

      4. +-commutative 24.0%

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

      5. *-commutative 24.0%

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

      6. unpow2 24.0%

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

      7. associate-*l* 24.0%

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

   6. Simplified 24.0%

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

   8. Applied egg-rr 25.2%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   if 1.04999999999999994e229 < re

   1. Initial program 49.7%

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

      1. sub0-neg 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in re around 0 0.2%

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

      1. *-commutative 0.2%

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

      2. associate-*r* 0.2%

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

      3. distribute-rgt-out 21.2%

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

      4. +-commutative 21.2%

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

      5. *-commutative 21.2%

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

      6. unpow2 21.2%

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

      7. associate-*l* 21.2%

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

   6. Simplified 21.2%

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

   8. Applied egg-rr 22.0%

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

   9. Taylor expanded in re around inf 22.0%

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

       1. *-commutative 22.0%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 22.0%

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

       3. associate-*l* 22.0%

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

   11. Simplified 22.0%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+77}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+229}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \]

Alternative 11: 31.9% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.1 \cdot 10^{+155}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 7.6 \cdot 10^{+228}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.1e+155)
   (- im)
   (if (<= re 7.6e+228)
     (* (+ 0.5 (* re (* re -0.25))) 27.0)
     (* re (* re 0.75)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.1e+155) {
		tmp = -im;
	} else if (re <= 7.6e+228) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.1d+155) then
        tmp = -im
    else if (re <= 7.6d+228) then
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    else
        tmp = re * (re * 0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.1e+155) {
		tmp = -im;
	} else if (re <= 7.6e+228) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.1e+155:
		tmp = -im
	elif re <= 7.6e+228:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	else:
		tmp = re * (re * 0.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.1e+155)
		tmp = Float64(-im);
	elseif (re <= 7.6e+228)
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	else
		tmp = Float64(re * Float64(re * 0.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.1e+155)
		tmp = -im;
	elseif (re <= 7.6e+228)
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	else
		tmp = re * (re * 0.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.1e+155], (-im), If[LessEqual[re, 7.6e+228], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.1000000000000001e155

   1. Initial program 56.0%

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

      1. sub0-neg 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in im around 0 51.2%

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

      1. mul-1-neg 51.2%

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

      2. *-commutative 51.2%

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

      3. distribute-lft-neg-in 51.2%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in re around 0 29.3%

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

      1. neg-mul-1 29.3%

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

   9. Simplified 29.3%

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

   if 1.1000000000000001e155 < re < 7.6000000000000004e228

   1. Initial program 43.4%

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

      1. sub0-neg 43.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 24.0%

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

      4. +-commutative 24.0%

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

      5. *-commutative 24.0%

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

      6. unpow2 24.0%

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

      7. associate-*l* 24.0%

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

   6. Simplified 24.0%

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

   8. Applied egg-rr 25.2%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   if 7.6000000000000004e228 < re

   1. Initial program 49.7%

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

      1. sub0-neg 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in re around 0 0.2%

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

      1. *-commutative 0.2%

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

      2. associate-*r* 0.2%

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

      3. distribute-rgt-out 21.2%

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

      4. +-commutative 21.2%

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

      5. *-commutative 21.2%

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

      6. unpow2 21.2%

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

      7. associate-*l* 21.2%

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

   6. Simplified 21.2%

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

   8. Applied egg-rr 22.0%

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

   9. Taylor expanded in re around inf 22.0%

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

       1. *-commutative 22.0%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 22.0%

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

       3. associate-*l* 22.0%

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

   11. Simplified 22.0%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.1 \cdot 10^{+155}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 7.6 \cdot 10^{+228}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \]

Alternative 12: 35.9% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{+24}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+30}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5.6e+24)
   (* re (* re 0.75))
   (if (<= im 3.2e+30) (- im) (- (* im (* 0.5 (* re re))) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -5.6e+24) {
		tmp = re * (re * 0.75);
	} else if (im <= 3.2e+30) {
		tmp = -im;
	} else {
		tmp = (im * (0.5 * (re * re))) - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-5.6d+24)) then
        tmp = re * (re * 0.75d0)
    else if (im <= 3.2d+30) then
        tmp = -im
    else
        tmp = (im * (0.5d0 * (re * re))) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -5.6e+24) {
		tmp = re * (re * 0.75);
	} else if (im <= 3.2e+30) {
		tmp = -im;
	} else {
		tmp = (im * (0.5 * (re * re))) - im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -5.6e+24:
		tmp = re * (re * 0.75)
	elif im <= 3.2e+30:
		tmp = -im
	else:
		tmp = (im * (0.5 * (re * re))) - im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -5.6e+24)
		tmp = Float64(re * Float64(re * 0.75));
	elseif (im <= 3.2e+30)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -5.6e+24)
		tmp = re * (re * 0.75);
	elseif (im <= 3.2e+30)
		tmp = -im;
	else
		tmp = (im * (0.5 * (re * re))) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -5.6e+24], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.2e+30], (-im), N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.6000000000000003e24

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 68.9%

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

      4. +-commutative 68.9%

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

      5. *-commutative 68.9%

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

      6. unpow2 68.9%

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

      7. associate-*l* 68.9%

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

   6. Simplified 68.9%

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

   8. Applied egg-rr 16.0%

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

   9. Taylor expanded in re around inf 16.6%

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

       1. *-commutative 16.6%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 16.6%

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

       3. associate-*l* 16.6%

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

   11. Simplified 16.6%

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

   if -5.6000000000000003e24 < im < 3.19999999999999973e30

   1. Initial program 15.9%

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

      1. sub0-neg 15.9%

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

   3. Simplified 15.9%

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

   4. Taylor expanded in im around 0 92.6%

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

      1. mul-1-neg 92.6%

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

      2. *-commutative 92.6%

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

      3. distribute-lft-neg-in 92.6%

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

   6. Simplified 92.6%

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

   7. Taylor expanded in re around 0 44.2%

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

      1. neg-mul-1 44.2%

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

   9. Simplified 44.2%

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

   if 3.19999999999999973e30 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.7%

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

      1. mul-1-neg 5.7%

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

      2. *-commutative 5.7%

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

      3. distribute-lft-neg-in 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 30.0%

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

      1. neg-mul-1 30.0%

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

      2. +-commutative 30.0%

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

      3. unsub-neg 30.0%

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

      4. *-commutative 30.0%

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

      5. *-commutative 30.0%

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

      6. associate-*l* 30.0%

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

      7. unpow2 30.0%

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

   9. Simplified 30.0%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{+24}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+30}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \]

Alternative 13: 31.4% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{+166}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2e+166) (- im) (* re (* re 0.75))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2e+166) {
		tmp = -im;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2d+166) then
        tmp = -im
    else
        tmp = re * (re * 0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2e+166) {
		tmp = -im;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2e+166:
		tmp = -im
	else:
		tmp = re * (re * 0.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2e+166)
		tmp = Float64(-im);
	else
		tmp = Float64(re * Float64(re * 0.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2e+166)
		tmp = -im;
	else
		tmp = re * (re * 0.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2e+166], (-im), N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.99999999999999988e166

   1. Initial program 56.3%

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

      1. sub0-neg 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in im around 0 50.8%

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

      1. mul-1-neg 50.8%

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

      2. *-commutative 50.8%

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

      3. distribute-lft-neg-in 50.8%

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

   6. Simplified 50.8%

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

   7. Taylor expanded in re around 0 28.6%

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

      1. neg-mul-1 28.6%

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

   9. Simplified 28.6%

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

   if 1.99999999999999988e166 < re

   1. Initial program 42.0%

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

      1. sub0-neg 42.0%

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

   3. Simplified 42.0%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 16.8%

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

      4. +-commutative 16.8%

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

      5. *-commutative 16.8%

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

      6. unpow2 16.8%

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

      7. associate-*l* 16.8%

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

   6. Simplified 16.8%

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

   8. Applied egg-rr 23.2%

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

   9. Taylor expanded in re around inf 23.2%

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

       1. *-commutative 23.2%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 23.2%

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

       3. associate-*l* 23.2%

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

   11. Simplified 23.2%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{+166}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \]

Alternative 14: 29.6% 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 54.3%

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

   1. sub0-neg 54.3%

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

3. Simplified 54.3%

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

4. Taylor expanded in im around 0 52.8%

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

   1. mul-1-neg 52.8%

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

   2. *-commutative 52.8%

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

   3. distribute-lft-neg-in 52.8%

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

6. Simplified 52.8%

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

7. Taylor expanded in re around 0 25.9%

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

   1. neg-mul-1 25.9%

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

9. Simplified 25.9%

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

10. Final simplification 25.9%

    \[\leadsto -im \]

Alternative 15: 2.9% accurate, 309.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

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

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 54.3%

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

   1. sub0-neg 54.3%

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

3. Simplified 54.3%

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

4. Taylor expanded in re around 0 4.2%

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

   1. *-commutative 4.2%

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

   2. associate-*r* 4.2%

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

   3. distribute-rgt-out 39.7%

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

   4. +-commutative 39.7%

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

   5. *-commutative 39.7%

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

   6. unpow2 39.7%

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

   7. associate-*l* 39.7%

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

6. Simplified 39.7%

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

8. Applied egg-rr 9.0%

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

9. Taylor expanded in re around 0 2.8%

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

10. Final simplification 2.8%

    \[\leadsto -1.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023230 
(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))))