math.sin on complex, imaginary part

Percentage Accurate: 55.1% → 99.2%

Time: 10.8s

Alternatives: 17

Speedup: 2.8×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% 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: 99.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e-10)))
     (* (* 0.5 (cos re)) t_0)
     (* im (- (cos re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 1e-10)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e-10)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e-10):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 1e-10))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 1e-10)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 1e-10]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 1.00000000000000004e-10 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1.00000000000000004e-10

   1. Initial program 6.6%

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

      1. sub0-neg 6.6%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

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

Alternative 2: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.25 \cdot 10^{+101} \lor \neg \left(im \leq -1.65\right) \land \left(im \leq 0.055 \lor \neg \left(im \leq 5.6 \cdot 10^{+102}\right)\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.25e+101)
         (and (not (<= im -1.65)) (or (<= im 0.055) (not (<= im 5.6e+102)))))
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* (- (exp (- im)) (exp im)) (+ 0.5 (* re (* re -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.25e+101) || (!(im <= -1.65) && ((im <= 0.055) || !(im <= 5.6e+102)))) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (exp(-im) - exp(im)) * (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) :: tmp
    if ((im <= (-1.25d+101)) .or. (.not. (im <= (-1.65d0))) .and. (im <= 0.055d0) .or. (.not. (im <= 5.6d+102))) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = (exp(-im) - exp(im)) * (0.5d0 + (re * (re * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.25e+101) || (!(im <= -1.65) && ((im <= 0.055) || !(im <= 5.6e+102)))) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (Math.exp(-im) - Math.exp(im)) * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.25e+101) or (not (im <= -1.65) and ((im <= 0.055) or not (im <= 5.6e+102))):
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = (math.exp(-im) - math.exp(im)) * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.25e+101) || (!(im <= -1.65) && ((im <= 0.055) || !(im <= 5.6e+102))))
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.25e+101) || (~((im <= -1.65)) && ((im <= 0.055) || ~((im <= 5.6e+102)))))
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = (exp(-im) - exp(im)) * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.25e+101], And[N[Not[LessEqual[im, -1.65]], $MachinePrecision], Or[LessEqual[im, 0.055], N[Not[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[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.24999999999999997e101 or -1.6499999999999999 < im < 0.0550000000000000003 or 5.60000000000000037e102 < im

   1. Initial program 43.9%

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

      1. sub0-neg 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in im around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*l* 99.1%

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

      5. distribute-lft-out-- 99.1%

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

   6. Simplified 99.1%

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

   if -1.24999999999999997e101 < im < -1.6499999999999999 or 0.0550000000000000003 < 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 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 83.7%

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

      4. +-commutative 83.7%

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

      5. *-commutative 83.7%

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

      6. unpow2 83.7%

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

      7. associate-*l* 83.7%

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

   6. Simplified 83.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.25 \cdot 10^{+101} \lor \neg \left(im \leq -1.65\right) \land \left(im \leq 0.055 \lor \neg \left(im \leq 5.6 \cdot 10^{+102}\right)\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 3: 95.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{+102} \lor \neg \left(im \leq -1.65\right) \land \left(im \leq 6600 \lor \neg \left(im \leq 3.3 \cdot 10^{+102}\right)\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5.6e+102)
         (and (not (<= im -1.65)) (or (<= im 6600.0) (not (<= im 3.3e+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 <= -5.6e+102) || (!(im <= -1.65) && ((im <= 6600.0) || !(im <= 3.3e+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 <= (-5.6d+102)) .or. (.not. (im <= (-1.65d0))) .and. (im <= 6600.0d0) .or. (.not. (im <= 3.3d+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 <= -5.6e+102) || (!(im <= -1.65) && ((im <= 6600.0) || !(im <= 3.3e+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 <= -5.6e+102) or (not (im <= -1.65) and ((im <= 6600.0) or not (im <= 3.3e+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 <= -5.6e+102) || (!(im <= -1.65) && ((im <= 6600.0) || !(im <= 3.3e+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 <= -5.6e+102) || (~((im <= -1.65)) && ((im <= 6600.0) || ~((im <= 3.3e+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, -5.6e+102], And[N[Not[LessEqual[im, -1.65]], $MachinePrecision], Or[LessEqual[im, 6600.0], N[Not[LessEqual[im, 3.3e+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 < -5.60000000000000037e102 or -1.6499999999999999 < im < 6600 or 3.29999999999999999e102 < im

   1. Initial program 44.1%

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

      1. sub0-neg 44.1%

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

   3. Simplified 44.1%

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

   4. Taylor expanded in im around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-*l* 98.7%

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

      5. distribute-lft-out-- 98.7%

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

   6. Simplified 98.7%

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

   if -5.60000000000000037e102 < im < -1.6499999999999999 or 6600 < im < 3.29999999999999999e102

   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 71.4%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{+102} \lor \neg \left(im \leq -1.65\right) \land \left(im \leq 6600 \lor \neg \left(im \leq 3.3 \cdot 10^{+102}\right)\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4: 89.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{\frac{im}{\cos re}}\\ \mathbf{elif}\;im \leq -1.65 \lor \neg \left(im \leq 6600\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5.5e+150)
   (/ (- 387420489.0 (* im im)) (/ im (cos re)))
   (if (or (<= im -1.65) (not (<= im 6600.0)))
     (* 0.5 (- (exp (- im)) (exp im)))
     (* im (- (cos re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -5.5e+150) {
		tmp = (387420489.0 - (im * im)) / (im / cos(re));
	} else if ((im <= -1.65) || !(im <= 6600.0)) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-5.5d+150)) then
        tmp = (387420489.0d0 - (im * im)) / (im / cos(re))
    else if ((im <= (-1.65d0)) .or. (.not. (im <= 6600.0d0))) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -5.5e+150) {
		tmp = (387420489.0 - (im * im)) / (im / Math.cos(re));
	} else if ((im <= -1.65) || !(im <= 6600.0)) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -5.5e+150:
		tmp = (387420489.0 - (im * im)) / (im / math.cos(re))
	elif (im <= -1.65) or not (im <= 6600.0):
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -5.5e+150)
		tmp = Float64(Float64(387420489.0 - Float64(im * im)) / Float64(im / cos(re)));
	elseif ((im <= -1.65) || !(im <= 6600.0))
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -5.5e+150)
		tmp = (387420489.0 - (im * im)) / (im / cos(re));
	elseif ((im <= -1.65) || ~((im <= 6600.0)))
		tmp = 0.5 * (exp(-im) - exp(im));
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -5.5e+150], N[(N[(387420489.0 - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(im / N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -1.65], N[Not[LessEqual[im, 6600.0]], $MachinePrecision]], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.50000000000000017e150

   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%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 97.1%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 97.1%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 97.1%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 97.1%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 97.1%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 97.1%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 97.1%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 97.1%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 97.1%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in im around inf 97.1%

       \[\leadsto \frac{387420489 - im \cdot im}{\color{blue}{\frac{im}{\cos re}}} \]

   if -5.50000000000000017e150 < im < -1.6499999999999999 or 6600 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 73.1%

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

   if -1.6499999999999999 < im < 6600

   1. Initial program 8.0%

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

      1. sub0-neg 8.0%

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

   3. Simplified 8.0%

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

   4. Taylor expanded in im around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. distribute-lft-neg-in 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{\frac{im}{\cos re}}\\ \mathbf{elif}\;im \leq -1.65 \lor \neg \left(im \leq 6600\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 5: 83.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ t_1 := \frac{387420489 - im \cdot im}{\frac{im}{\cos re}}\\ \mathbf{if}\;im \leq -7.5 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -580:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 550:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (+ (* -0.5 (* re re)) 1.0)
          (- (* (pow im 3.0) -0.16666666666666666) im)))
        (t_1 (/ (- 387420489.0 (* im im)) (/ im (cos re)))))
   (if (<= im -7.5e+163)
     t_1
     (if (<= im -580.0)
       t_0
       (if (<= im 550.0) (* im (- (cos re))) (if (<= im 2.6e+150) 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 = (387420489.0 - (im * im)) / (im / cos(re));
	double tmp;
	if (im <= -7.5e+163) {
		tmp = t_1;
	} else if (im <= -580.0) {
		tmp = t_0;
	} else if (im <= 550.0) {
		tmp = im * -cos(re);
	} else if (im <= 2.6e+150) {
		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 = (387420489.0d0 - (im * im)) / (im / cos(re))
    if (im <= (-7.5d+163)) then
        tmp = t_1
    else if (im <= (-580.0d0)) then
        tmp = t_0
    else if (im <= 550.0d0) then
        tmp = im * -cos(re)
    else if (im <= 2.6d+150) 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 = (387420489.0 - (im * im)) / (im / Math.cos(re));
	double tmp;
	if (im <= -7.5e+163) {
		tmp = t_1;
	} else if (im <= -580.0) {
		tmp = t_0;
	} else if (im <= 550.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 2.6e+150) {
		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 = (387420489.0 - (im * im)) / (im / math.cos(re))
	tmp = 0
	if im <= -7.5e+163:
		tmp = t_1
	elif im <= -580.0:
		tmp = t_0
	elif im <= 550.0:
		tmp = im * -math.cos(re)
	elif im <= 2.6e+150:
		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(387420489.0 - Float64(im * im)) / Float64(im / cos(re)))
	tmp = 0.0
	if (im <= -7.5e+163)
		tmp = t_1;
	elseif (im <= -580.0)
		tmp = t_0;
	elseif (im <= 550.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 2.6e+150)
		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 = (387420489.0 - (im * im)) / (im / cos(re));
	tmp = 0.0;
	if (im <= -7.5e+163)
		tmp = t_1;
	elseif (im <= -580.0)
		tmp = t_0;
	elseif (im <= 550.0)
		tmp = im * -cos(re);
	elseif (im <= 2.6e+150)
		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[(387420489.0 - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(im / N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.5e+163], t$95$1, If[LessEqual[im, -580.0], t$95$0, If[LessEqual[im, 550.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 2.6e+150], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.50000000000000001e163 or 2.60000000000000006e150 < 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%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 98.4%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 98.4%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 98.4%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 98.4%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 98.4%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 98.4%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 98.4%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 98.4%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 98.4%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in im around inf 98.4%

       \[\leadsto \frac{387420489 - im \cdot im}{\color{blue}{\frac{im}{\cos re}}} \]

   if -7.50000000000000001e163 < im < -580 or 550 < im < 2.60000000000000006e150

   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 39.0%

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

      1. mul-1-neg 39.0%

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

      2. unsub-neg 39.0%

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

      3. *-commutative 39.0%

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

      4. associate-*l* 39.0%

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

      5. distribute-lft-out-- 39.0%

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

   6. Simplified 39.0%

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

   7. Taylor expanded in re around 0 19.7%

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

      1. associate--l+ 19.7%

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

      2. associate-*r* 19.7%

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

      3. distribute-lft1-in 46.2%

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

      4. unpow2 46.2%

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

   9. Simplified 46.2%

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

   if -580 < im < 550

   1. Initial program 7.3%

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

      1. sub0-neg 7.3%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in im around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. distribute-lft-neg-in 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{\frac{im}{\cos re}}\\ \mathbf{elif}\;im \leq -580:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 550:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{\frac{im}{\cos re}}\\ \end{array} \]

Alternative 6: 79.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-0.5 \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ t_1 := \frac{387420489 - im \cdot im}{\frac{im}{\cos re}}\\ \mathbf{if}\;im \leq -2.65 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -88000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* (* -0.5 (* re re)) (- (* (pow im 3.0) -0.16666666666666666) im)))
        (t_1 (/ (- 387420489.0 (* im im)) (/ im (cos re)))))
   (if (<= im -2.65e+154)
     t_1
     (if (<= im -88000000.0)
       t_0
       (if (<= im 700.0) (* im (- (cos re))) (if (<= im 3.3e+145) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (-0.5 * (re * re)) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double t_1 = (387420489.0 - (im * im)) / (im / cos(re));
	double tmp;
	if (im <= -2.65e+154) {
		tmp = t_1;
	} else if (im <= -88000000.0) {
		tmp = t_0;
	} else if (im <= 700.0) {
		tmp = im * -cos(re);
	} else if (im <= 3.3e+145) {
		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)) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    t_1 = (387420489.0d0 - (im * im)) / (im / cos(re))
    if (im <= (-2.65d+154)) then
        tmp = t_1
    else if (im <= (-88000000.0d0)) then
        tmp = t_0
    else if (im <= 700.0d0) then
        tmp = im * -cos(re)
    else if (im <= 3.3d+145) 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)) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double t_1 = (387420489.0 - (im * im)) / (im / Math.cos(re));
	double tmp;
	if (im <= -2.65e+154) {
		tmp = t_1;
	} else if (im <= -88000000.0) {
		tmp = t_0;
	} else if (im <= 700.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 3.3e+145) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (-0.5 * (re * re)) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	t_1 = (387420489.0 - (im * im)) / (im / math.cos(re))
	tmp = 0
	if im <= -2.65e+154:
		tmp = t_1
	elif im <= -88000000.0:
		tmp = t_0
	elif im <= 700.0:
		tmp = im * -math.cos(re)
	elif im <= 3.3e+145:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(-0.5 * Float64(re * re)) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	t_1 = Float64(Float64(387420489.0 - Float64(im * im)) / Float64(im / cos(re)))
	tmp = 0.0
	if (im <= -2.65e+154)
		tmp = t_1;
	elseif (im <= -88000000.0)
		tmp = t_0;
	elseif (im <= 700.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 3.3e+145)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (-0.5 * (re * re)) * (((im ^ 3.0) * -0.16666666666666666) - im);
	t_1 = (387420489.0 - (im * im)) / (im / cos(re));
	tmp = 0.0;
	if (im <= -2.65e+154)
		tmp = t_1;
	elseif (im <= -88000000.0)
		tmp = t_0;
	elseif (im <= 700.0)
		tmp = im * -cos(re);
	elseif (im <= 3.3e+145)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(387420489.0 - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(im / N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.65e+154], t$95$1, If[LessEqual[im, -88000000.0], t$95$0, If[LessEqual[im, 700.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 3.3e+145], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.65000000000000012e154 or 3.30000000000000027e145 < 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%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 96.9%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 96.9%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 96.9%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 96.9%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 96.9%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 96.9%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 96.9%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 96.9%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 96.9%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in im around inf 96.9%

       \[\leadsto \frac{387420489 - im \cdot im}{\color{blue}{\frac{im}{\cos re}}} \]

   if -2.65000000000000012e154 < im < -8.8e7 or 700 < im < 3.30000000000000027e145

   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 36.7%

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

      1. mul-1-neg 36.7%

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

      2. unsub-neg 36.7%

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

      3. *-commutative 36.7%

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

      4. associate-*l* 36.7%

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

      5. distribute-lft-out-- 36.7%

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

   6. Simplified 36.7%

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

   7. Taylor expanded in re around 0 20.9%

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

      1. associate--l+ 20.9%

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

      2. associate-*r* 20.9%

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

      3. distribute-lft1-in 44.4%

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

      4. unpow2 44.4%

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

   9. Simplified 44.4%

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

   10. Taylor expanded in re around inf 35.0%

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

       1. unpow2 35.0%

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

       2. associate-*r* 35.0%

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

       3. *-commutative 35.0%

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

   12. Simplified 35.0%

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

   if -8.8e7 < im < 700

   1. Initial program 8.0%

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

      1. sub0-neg 8.0%

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

   3. Simplified 8.0%

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

   4. Taylor expanded in im around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. distribute-lft-neg-in 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.65 \cdot 10^{+154}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{\frac{im}{\cos re}}\\ \mathbf{elif}\;im \leq -88000000:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right)\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{\frac{im}{\cos re}}\\ \end{array} \]

Alternative 7: 79.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 387420489 - im \cdot im\\ t_1 := -0.5 \cdot \frac{t_0 \cdot \left(re \cdot re\right)}{im + 19683} + \frac{t_0}{im + 19683}\\ t_2 := \frac{t_0}{\frac{im}{\cos re}}\\ \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -650:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 86000000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 387420489.0 (* im im)))
        (t_1
         (+
          (* -0.5 (/ (* t_0 (* re re)) (+ im 19683.0)))
          (/ t_0 (+ im 19683.0))))
        (t_2 (/ t_0 (/ im (cos re)))))
   (if (<= im -1.32e+154)
     t_2
     (if (<= im -650.0)
       t_1
       (if (<= im 86000000000.0)
         (* im (- (cos re)))
         (if (<= im 3.3e+145) t_1 t_2))))))

C

double code(double re, double im) {
	double t_0 = 387420489.0 - (im * im);
	double t_1 = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + (t_0 / (im + 19683.0));
	double t_2 = t_0 / (im / cos(re));
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_2;
	} else if (im <= -650.0) {
		tmp = t_1;
	} else if (im <= 86000000000.0) {
		tmp = im * -cos(re);
	} else if (im <= 3.3e+145) {
		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 = 387420489.0d0 - (im * im)
    t_1 = ((-0.5d0) * ((t_0 * (re * re)) / (im + 19683.0d0))) + (t_0 / (im + 19683.0d0))
    t_2 = t_0 / (im / cos(re))
    if (im <= (-1.32d+154)) then
        tmp = t_2
    else if (im <= (-650.0d0)) then
        tmp = t_1
    else if (im <= 86000000000.0d0) then
        tmp = im * -cos(re)
    else if (im <= 3.3d+145) 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 = 387420489.0 - (im * im);
	double t_1 = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + (t_0 / (im + 19683.0));
	double t_2 = t_0 / (im / Math.cos(re));
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_2;
	} else if (im <= -650.0) {
		tmp = t_1;
	} else if (im <= 86000000000.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 3.3e+145) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 387420489.0 - (im * im)
	t_1 = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + (t_0 / (im + 19683.0))
	t_2 = t_0 / (im / math.cos(re))
	tmp = 0
	if im <= -1.32e+154:
		tmp = t_2
	elif im <= -650.0:
		tmp = t_1
	elif im <= 86000000000.0:
		tmp = im * -math.cos(re)
	elif im <= 3.3e+145:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(387420489.0 - Float64(im * im))
	t_1 = Float64(Float64(-0.5 * Float64(Float64(t_0 * Float64(re * re)) / Float64(im + 19683.0))) + Float64(t_0 / Float64(im + 19683.0)))
	t_2 = Float64(t_0 / Float64(im / cos(re)))
	tmp = 0.0
	if (im <= -1.32e+154)
		tmp = t_2;
	elseif (im <= -650.0)
		tmp = t_1;
	elseif (im <= 86000000000.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 3.3e+145)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 387420489.0 - (im * im);
	t_1 = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + (t_0 / (im + 19683.0));
	t_2 = t_0 / (im / cos(re));
	tmp = 0.0;
	if (im <= -1.32e+154)
		tmp = t_2;
	elseif (im <= -650.0)
		tmp = t_1;
	elseif (im <= 86000000000.0)
		tmp = im * -cos(re);
	elseif (im <= 3.3e+145)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(387420489.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.5 * N[(N[(t$95$0 * N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(im + 19683.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / N[(im + 19683.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(im / N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.32e+154], t$95$2, If[LessEqual[im, -650.0], t$95$1, If[LessEqual[im, 86000000000.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 3.3e+145], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.31999999999999998e154 or 3.30000000000000027e145 < 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%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 97.0%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 97.0%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 97.0%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 97.0%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 97.0%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 97.0%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 97.0%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 97.0%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 97.0%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in im around inf 97.0%

       \[\leadsto \frac{387420489 - im \cdot im}{\color{blue}{\frac{im}{\cos re}}} \]

   if -1.31999999999999998e154 < im < -650 or 8.6e10 < im < 3.30000000000000027e145

   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 36.3%

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

      1. mul-1-neg 36.3%

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

      2. unsub-neg 36.3%

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

      3. *-commutative 36.3%

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

      4. associate-*l* 36.3%

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

      5. distribute-lft-out-- 36.3%

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

   6. Simplified 36.3%

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

   8. Applied egg-rr 3.8%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 3.8%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 3.8%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 3.8%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 3.8%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 3.8%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 3.8%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 3.8%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 3.8%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 3.8%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in re around 0 31.0%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot {re}^{2}}{19683 + im} + 387420489 \cdot \frac{1}{19683 + im}\right) - \frac{{im}^{2}}{19683 + im}} \]
   14. Step-by-step derivation

       1. associate--l+ 31.0%

          \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot {re}^{2}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right)} \]

       2. unpow2 31.0%

          \[\leadsto -0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       3. *-commutative 31.0%

          \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(re \cdot re\right) \cdot \left(387420489 - {im}^{2}\right)}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       4. unpow2 31.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - \color{blue}{im \cdot im}\right)}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       5. +-commutative 31.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       6. associate-*r/ 31.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\color{blue}{\frac{387420489 \cdot 1}{19683 + im}} - \frac{{im}^{2}}{19683 + im}\right) \]

       7. metadata-eval 31.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\frac{\color{blue}{387420489}}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       8. unpow2 31.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\frac{387420489}{19683 + im} - \frac{\color{blue}{im \cdot im}}{19683 + im}\right) \]

       9. div-sub 31.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \color{blue}{\frac{387420489 - im \cdot im}{19683 + im}} \]

       10. +-commutative 31.0%

           \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \frac{387420489 - im \cdot im}{\color{blue}{im + 19683}} \]

   15. Simplified 31.0%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \frac{387420489 - im \cdot im}{im + 19683}} \]

   if -650 < im < 8.6e10

   1. Initial program 8.7%

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

      1. sub0-neg 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in im around 0 97.7%

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

      1. mul-1-neg 97.7%

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

      2. *-commutative 97.7%

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

      3. distribute-lft-neg-in 97.7%

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

   6. Simplified 97.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{\frac{im}{\cos re}}\\ \mathbf{elif}\;im \leq -650:\\ \;\;\;\;-0.5 \cdot \frac{\left(387420489 - im \cdot im\right) \cdot \left(re \cdot re\right)}{im + 19683} + \frac{387420489 - im \cdot im}{im + 19683}\\ \mathbf{elif}\;im \leq 86000000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;-0.5 \cdot \frac{\left(387420489 - im \cdot im\right) \cdot \left(re \cdot re\right)}{im + 19683} + \frac{387420489 - im \cdot im}{im + 19683}\\ \mathbf{else}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{\frac{im}{\cos re}}\\ \end{array} \]

Alternative 8: 76.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_1 := 387420489 - im \cdot im\\ \mathbf{if}\;im \leq -1.3 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 16500000000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;-0.5 \cdot \frac{t_1 \cdot \left(re \cdot re\right)}{im + 19683} + \frac{t_1}{im + 19683}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_1 (- 387420489.0 (* im im))))
   (if (<= im -1.3e+42)
     t_0
     (if (<= im 16500000000000.0)
       (* im (- (cos re)))
       (if (<= im 5.6e+102)
         (+
          (* -0.5 (/ (* t_1 (* re re)) (+ im 19683.0)))
          (/ t_1 (+ im 19683.0)))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = 387420489.0 - (im * im);
	double tmp;
	if (im <= -1.3e+42) {
		tmp = t_0;
	} else if (im <= 16500000000000.0) {
		tmp = im * -cos(re);
	} else if (im <= 5.6e+102) {
		tmp = (-0.5 * ((t_1 * (re * re)) / (im + 19683.0))) + (t_1 / (im + 19683.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    t_1 = 387420489.0d0 - (im * im)
    if (im <= (-1.3d+42)) then
        tmp = t_0
    else if (im <= 16500000000000.0d0) then
        tmp = im * -cos(re)
    else if (im <= 5.6d+102) then
        tmp = ((-0.5d0) * ((t_1 * (re * re)) / (im + 19683.0d0))) + (t_1 / (im + 19683.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = 387420489.0 - (im * im);
	double tmp;
	if (im <= -1.3e+42) {
		tmp = t_0;
	} else if (im <= 16500000000000.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 5.6e+102) {
		tmp = (-0.5 * ((t_1 * (re * re)) / (im + 19683.0))) + (t_1 / (im + 19683.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	t_1 = 387420489.0 - (im * im)
	tmp = 0
	if im <= -1.3e+42:
		tmp = t_0
	elif im <= 16500000000000.0:
		tmp = im * -math.cos(re)
	elif im <= 5.6e+102:
		tmp = (-0.5 * ((t_1 * (re * re)) / (im + 19683.0))) + (t_1 / (im + 19683.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_1 = Float64(387420489.0 - Float64(im * im))
	tmp = 0.0
	if (im <= -1.3e+42)
		tmp = t_0;
	elseif (im <= 16500000000000.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 5.6e+102)
		tmp = Float64(Float64(-0.5 * Float64(Float64(t_1 * Float64(re * re)) / Float64(im + 19683.0))) + Float64(t_1 / Float64(im + 19683.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	t_1 = 387420489.0 - (im * im);
	tmp = 0.0;
	if (im <= -1.3e+42)
		tmp = t_0;
	elseif (im <= 16500000000000.0)
		tmp = im * -cos(re);
	elseif (im <= 5.6e+102)
		tmp = (-0.5 * ((t_1 * (re * re)) / (im + 19683.0))) + (t_1 / (im + 19683.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$1 = N[(387420489.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.3e+42], t$95$0, If[LessEqual[im, 16500000000000.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 5.6e+102], N[(N[(-0.5 * N[(N[(t$95$1 * N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(im + 19683.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / N[(im + 19683.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.29999999999999995e42 or 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 87.4%

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

      1. mul-1-neg 87.4%

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

      2. unsub-neg 87.4%

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

      3. *-commutative 87.4%

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

      4. associate-*l* 87.4%

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

      5. distribute-lft-out-- 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in re around 0 68.4%

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

   if -1.29999999999999995e42 < im < 1.65e13

   1. Initial program 12.7%

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

      1. sub0-neg 12.7%

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

   3. Simplified 12.7%

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

   4. Taylor expanded in im around 0 93.5%

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

      1. mul-1-neg 93.5%

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

      2. *-commutative 93.5%

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

      3. distribute-lft-neg-in 93.5%

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

   6. Simplified 93.5%

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

   if 1.65e13 < 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 im around 0 5.7%

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

      1. mul-1-neg 5.7%

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

      2. unsub-neg 5.7%

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

      3. *-commutative 5.7%

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

      4. associate-*l* 5.7%

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

      5. distribute-lft-out-- 5.7%

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

   6. Simplified 5.7%

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

   8. Applied egg-rr 3.6%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 3.6%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 3.6%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 3.6%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 3.6%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 3.6%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 3.6%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 3.6%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 3.6%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 3.6%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in re around 0 32.4%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot {re}^{2}}{19683 + im} + 387420489 \cdot \frac{1}{19683 + im}\right) - \frac{{im}^{2}}{19683 + im}} \]
   14. Step-by-step derivation

       1. associate--l+ 32.4%

          \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot {re}^{2}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right)} \]

       2. unpow2 32.4%

          \[\leadsto -0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       3. *-commutative 32.4%

          \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(re \cdot re\right) \cdot \left(387420489 - {im}^{2}\right)}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       4. unpow2 32.4%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - \color{blue}{im \cdot im}\right)}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       5. +-commutative 32.4%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       6. associate-*r/ 32.4%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\color{blue}{\frac{387420489 \cdot 1}{19683 + im}} - \frac{{im}^{2}}{19683 + im}\right) \]

       7. metadata-eval 32.4%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\frac{\color{blue}{387420489}}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       8. unpow2 32.4%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\frac{387420489}{19683 + im} - \frac{\color{blue}{im \cdot im}}{19683 + im}\right) \]

       9. div-sub 32.4%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \color{blue}{\frac{387420489 - im \cdot im}{19683 + im}} \]

       10. +-commutative 32.4%

           \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \frac{387420489 - im \cdot im}{\color{blue}{im + 19683}} \]

   15. Simplified 32.4%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \frac{387420489 - im \cdot im}{im + 19683}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+42}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 16500000000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;-0.5 \cdot \frac{\left(387420489 - im \cdot im\right) \cdot \left(re \cdot re\right)}{im + 19683} + \frac{387420489 - im \cdot im}{im + 19683}\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 9: 72.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 387420489 - im \cdot im\\ t_1 := \frac{t_0}{im + 19683}\\ \mathbf{if}\;im \leq -7.5 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -88000000:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 17000000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;-0.5 \cdot \frac{t_0 \cdot \left(re \cdot re\right)}{im + 19683} + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 387420489.0 (* im im))) (t_1 (/ t_0 (+ im 19683.0))))
   (if (<= im -7.5e+163)
     t_1
     (if (<= im -88000000.0)
       (- (* im (* 0.5 (* re re))) im)
       (if (<= im 17000000000.0)
         (* im (- (cos re)))
         (if (<= im 3.3e+145)
           (+ (* -0.5 (/ (* t_0 (* re re)) (+ im 19683.0))) t_1)
           t_1))))))

C

double code(double re, double im) {
	double t_0 = 387420489.0 - (im * im);
	double t_1 = t_0 / (im + 19683.0);
	double tmp;
	if (im <= -7.5e+163) {
		tmp = t_1;
	} else if (im <= -88000000.0) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else if (im <= 17000000000.0) {
		tmp = im * -cos(re);
	} else if (im <= 3.3e+145) {
		tmp = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + t_1;
	} 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 = 387420489.0d0 - (im * im)
    t_1 = t_0 / (im + 19683.0d0)
    if (im <= (-7.5d+163)) then
        tmp = t_1
    else if (im <= (-88000000.0d0)) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else if (im <= 17000000000.0d0) then
        tmp = im * -cos(re)
    else if (im <= 3.3d+145) then
        tmp = ((-0.5d0) * ((t_0 * (re * re)) / (im + 19683.0d0))) + t_1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 387420489.0 - (im * im);
	double t_1 = t_0 / (im + 19683.0);
	double tmp;
	if (im <= -7.5e+163) {
		tmp = t_1;
	} else if (im <= -88000000.0) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else if (im <= 17000000000.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 3.3e+145) {
		tmp = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 387420489.0 - (im * im)
	t_1 = t_0 / (im + 19683.0)
	tmp = 0
	if im <= -7.5e+163:
		tmp = t_1
	elif im <= -88000000.0:
		tmp = (im * (0.5 * (re * re))) - im
	elif im <= 17000000000.0:
		tmp = im * -math.cos(re)
	elif im <= 3.3e+145:
		tmp = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + t_1
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(387420489.0 - Float64(im * im))
	t_1 = Float64(t_0 / Float64(im + 19683.0))
	tmp = 0.0
	if (im <= -7.5e+163)
		tmp = t_1;
	elseif (im <= -88000000.0)
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	elseif (im <= 17000000000.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 3.3e+145)
		tmp = Float64(Float64(-0.5 * Float64(Float64(t_0 * Float64(re * re)) / Float64(im + 19683.0))) + t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 387420489.0 - (im * im);
	t_1 = t_0 / (im + 19683.0);
	tmp = 0.0;
	if (im <= -7.5e+163)
		tmp = t_1;
	elseif (im <= -88000000.0)
		tmp = (im * (0.5 * (re * re))) - im;
	elseif (im <= 17000000000.0)
		tmp = im * -cos(re);
	elseif (im <= 3.3e+145)
		tmp = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + t_1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(387420489.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(im + 19683.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.5e+163], t$95$1, If[LessEqual[im, -88000000.0], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 17000000000.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 3.3e+145], N[(N[(-0.5 * N[(N[(t$95$0 * N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(im + 19683.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -7.50000000000000001e163 or 3.30000000000000027e145 < 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%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 96.8%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 96.8%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 96.8%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 96.8%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 96.8%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 96.8%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 96.8%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 96.8%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 96.8%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in re around 0 85.2%

       \[\leadsto \color{blue}{\frac{387420489 - {im}^{2}}{19683 + im}} \]
   14. Step-by-step derivation

       1. unpow2 85.2%

          \[\leadsto \frac{387420489 - \color{blue}{im \cdot im}}{19683 + im} \]

       2. +-commutative 85.2%

          \[\leadsto \frac{387420489 - im \cdot im}{\color{blue}{im + 19683}} \]

   15. Simplified 85.2%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{im + 19683}} \]

   if -7.50000000000000001e163 < im < -8.8e7

   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 4.0%

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

      1. mul-1-neg 4.0%

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

      2. *-commutative 4.0%

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

      3. distribute-lft-neg-in 4.0%

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

   6. Simplified 4.0%

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

   7. Taylor expanded in re around 0 29.5%

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

      1. neg-mul-1 29.5%

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

      2. +-commutative 29.5%

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

      3. unsub-neg 29.5%

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

      4. *-commutative 29.5%

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

      5. *-commutative 29.5%

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

      6. associate-*l* 29.5%

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

      7. unpow2 29.5%

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

   9. Simplified 29.5%

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

   if -8.8e7 < im < 1.7e10

   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 97.0%

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

      1. mul-1-neg 97.0%

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

      2. *-commutative 97.0%

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

      3. distribute-lft-neg-in 97.0%

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

   6. Simplified 97.0%

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

   if 1.7e10 < im < 3.30000000000000027e145

   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 30.0%

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

      1. mul-1-neg 30.0%

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

      2. unsub-neg 30.0%

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

      3. *-commutative 30.0%

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

      4. associate-*l* 30.0%

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

      5. distribute-lft-out-- 30.0%

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

   6. Simplified 30.0%

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

   8. Applied egg-rr 3.7%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 3.7%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 3.7%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 3.7%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 3.7%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 3.7%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 3.7%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 3.7%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 3.7%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 3.7%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in re around 0 34.0%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot {re}^{2}}{19683 + im} + 387420489 \cdot \frac{1}{19683 + im}\right) - \frac{{im}^{2}}{19683 + im}} \]
   14. Step-by-step derivation

       1. associate--l+ 34.0%

          \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot {re}^{2}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right)} \]

       2. unpow2 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       3. *-commutative 34.0%

          \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(re \cdot re\right) \cdot \left(387420489 - {im}^{2}\right)}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       4. unpow2 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - \color{blue}{im \cdot im}\right)}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       5. +-commutative 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       6. associate-*r/ 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\color{blue}{\frac{387420489 \cdot 1}{19683 + im}} - \frac{{im}^{2}}{19683 + im}\right) \]

       7. metadata-eval 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\frac{\color{blue}{387420489}}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       8. unpow2 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\frac{387420489}{19683 + im} - \frac{\color{blue}{im \cdot im}}{19683 + im}\right) \]

       9. div-sub 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \color{blue}{\frac{387420489 - im \cdot im}{19683 + im}} \]

       10. +-commutative 34.0%

           \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \frac{387420489 - im \cdot im}{\color{blue}{im + 19683}} \]

   15. Simplified 34.0%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \frac{387420489 - im \cdot im}{im + 19683}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{im + 19683}\\ \mathbf{elif}\;im \leq -88000000:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 17000000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;-0.5 \cdot \frac{\left(387420489 - im \cdot im\right) \cdot \left(re \cdot re\right)}{im + 19683} + \frac{387420489 - im \cdot im}{im + 19683}\\ \mathbf{else}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{im + 19683}\\ \end{array} \]

Alternative 10: 50.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 387420489 - im \cdot im\\ t_1 := \frac{t_0}{im + 19683}\\ \mathbf{if}\;im \leq -7.5 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -88000000:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 17000000000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;-0.5 \cdot \frac{t_0 \cdot \left(re \cdot re\right)}{im + 19683} + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 387420489.0 (* im im))) (t_1 (/ t_0 (+ im 19683.0))))
   (if (<= im -7.5e+163)
     t_1
     (if (<= im -88000000.0)
       (- (* im (* 0.5 (* re re))) im)
       (if (<= im 17000000000.0)
         (- im)
         (if (<= im 3.3e+145)
           (+ (* -0.5 (/ (* t_0 (* re re)) (+ im 19683.0))) t_1)
           t_1))))))

C

double code(double re, double im) {
	double t_0 = 387420489.0 - (im * im);
	double t_1 = t_0 / (im + 19683.0);
	double tmp;
	if (im <= -7.5e+163) {
		tmp = t_1;
	} else if (im <= -88000000.0) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else if (im <= 17000000000.0) {
		tmp = -im;
	} else if (im <= 3.3e+145) {
		tmp = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + t_1;
	} 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 = 387420489.0d0 - (im * im)
    t_1 = t_0 / (im + 19683.0d0)
    if (im <= (-7.5d+163)) then
        tmp = t_1
    else if (im <= (-88000000.0d0)) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else if (im <= 17000000000.0d0) then
        tmp = -im
    else if (im <= 3.3d+145) then
        tmp = ((-0.5d0) * ((t_0 * (re * re)) / (im + 19683.0d0))) + t_1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 387420489.0 - (im * im);
	double t_1 = t_0 / (im + 19683.0);
	double tmp;
	if (im <= -7.5e+163) {
		tmp = t_1;
	} else if (im <= -88000000.0) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else if (im <= 17000000000.0) {
		tmp = -im;
	} else if (im <= 3.3e+145) {
		tmp = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 387420489.0 - (im * im)
	t_1 = t_0 / (im + 19683.0)
	tmp = 0
	if im <= -7.5e+163:
		tmp = t_1
	elif im <= -88000000.0:
		tmp = (im * (0.5 * (re * re))) - im
	elif im <= 17000000000.0:
		tmp = -im
	elif im <= 3.3e+145:
		tmp = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + t_1
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(387420489.0 - Float64(im * im))
	t_1 = Float64(t_0 / Float64(im + 19683.0))
	tmp = 0.0
	if (im <= -7.5e+163)
		tmp = t_1;
	elseif (im <= -88000000.0)
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	elseif (im <= 17000000000.0)
		tmp = Float64(-im);
	elseif (im <= 3.3e+145)
		tmp = Float64(Float64(-0.5 * Float64(Float64(t_0 * Float64(re * re)) / Float64(im + 19683.0))) + t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 387420489.0 - (im * im);
	t_1 = t_0 / (im + 19683.0);
	tmp = 0.0;
	if (im <= -7.5e+163)
		tmp = t_1;
	elseif (im <= -88000000.0)
		tmp = (im * (0.5 * (re * re))) - im;
	elseif (im <= 17000000000.0)
		tmp = -im;
	elseif (im <= 3.3e+145)
		tmp = (-0.5 * ((t_0 * (re * re)) / (im + 19683.0))) + t_1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(387420489.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(im + 19683.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.5e+163], t$95$1, If[LessEqual[im, -88000000.0], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 17000000000.0], (-im), If[LessEqual[im, 3.3e+145], N[(N[(-0.5 * N[(N[(t$95$0 * N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(im + 19683.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -7.50000000000000001e163 or 3.30000000000000027e145 < 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%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 96.8%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 96.8%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 96.8%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 96.8%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 96.8%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 96.8%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 96.8%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 96.8%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 96.8%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in re around 0 85.2%

       \[\leadsto \color{blue}{\frac{387420489 - {im}^{2}}{19683 + im}} \]
   14. Step-by-step derivation

       1. unpow2 85.2%

          \[\leadsto \frac{387420489 - \color{blue}{im \cdot im}}{19683 + im} \]

       2. +-commutative 85.2%

          \[\leadsto \frac{387420489 - im \cdot im}{\color{blue}{im + 19683}} \]

   15. Simplified 85.2%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{im + 19683}} \]

   if -7.50000000000000001e163 < im < -8.8e7

   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 4.0%

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

      1. mul-1-neg 4.0%

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

      2. *-commutative 4.0%

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

      3. distribute-lft-neg-in 4.0%

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

   6. Simplified 4.0%

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

   7. Taylor expanded in re around 0 29.5%

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

      1. neg-mul-1 29.5%

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

      2. +-commutative 29.5%

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

      3. unsub-neg 29.5%

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

      4. *-commutative 29.5%

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

      5. *-commutative 29.5%

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

      6. associate-*l* 29.5%

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

      7. unpow2 29.5%

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

   9. Simplified 29.5%

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

   if -8.8e7 < im < 1.7e10

   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 97.0%

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

      1. mul-1-neg 97.0%

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

      2. *-commutative 97.0%

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

      3. distribute-lft-neg-in 97.0%

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

   6. Simplified 97.0%

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

   7. Taylor expanded in re around 0 51.7%

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

      1. neg-mul-1 51.7%

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

   9. Simplified 51.7%

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

   if 1.7e10 < im < 3.30000000000000027e145

   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 30.0%

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

      1. mul-1-neg 30.0%

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

      2. unsub-neg 30.0%

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

      3. *-commutative 30.0%

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

      4. associate-*l* 30.0%

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

      5. distribute-lft-out-- 30.0%

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

   6. Simplified 30.0%

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

   8. Applied egg-rr 3.7%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 3.7%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 3.7%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 3.7%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 3.7%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 3.7%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 3.7%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 3.7%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 3.7%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 3.7%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in re around 0 34.0%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot {re}^{2}}{19683 + im} + 387420489 \cdot \frac{1}{19683 + im}\right) - \frac{{im}^{2}}{19683 + im}} \]
   14. Step-by-step derivation

       1. associate--l+ 34.0%

          \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot {re}^{2}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right)} \]

       2. unpow2 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(387420489 - {im}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       3. *-commutative 34.0%

          \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(re \cdot re\right) \cdot \left(387420489 - {im}^{2}\right)}}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       4. unpow2 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - \color{blue}{im \cdot im}\right)}{19683 + im} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       5. +-commutative 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} + \left(387420489 \cdot \frac{1}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       6. associate-*r/ 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\color{blue}{\frac{387420489 \cdot 1}{19683 + im}} - \frac{{im}^{2}}{19683 + im}\right) \]

       7. metadata-eval 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\frac{\color{blue}{387420489}}{19683 + im} - \frac{{im}^{2}}{19683 + im}\right) \]

       8. unpow2 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \left(\frac{387420489}{19683 + im} - \frac{\color{blue}{im \cdot im}}{19683 + im}\right) \]

       9. div-sub 34.0%

          \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \color{blue}{\frac{387420489 - im \cdot im}{19683 + im}} \]

       10. +-commutative 34.0%

           \[\leadsto -0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \frac{387420489 - im \cdot im}{\color{blue}{im + 19683}} \]

   15. Simplified 34.0%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(387420489 - im \cdot im\right)}{im + 19683} + \frac{387420489 - im \cdot im}{im + 19683}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{im + 19683}\\ \mathbf{elif}\;im \leq -88000000:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 17000000000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;-0.5 \cdot \frac{\left(387420489 - im \cdot im\right) \cdot \left(re \cdot re\right)}{im + 19683} + \frac{387420489 - im \cdot im}{im + 19683}\\ \mathbf{else}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{im + 19683}\\ \end{array} \]

Alternative 11: 50.2% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{387420489 - im \cdot im}{im + 19683}\\ t_1 := im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{if}\;im \leq -7.5 \cdot 10^{+163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2150000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 8 \cdot 10^{-51}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ (- 387420489.0 (* im im)) (+ im 19683.0)))
        (t_1 (- (* im (* 0.5 (* re re))) im)))
   (if (<= im -7.5e+163)
     t_0
     (if (<= im -2150000000.0)
       t_1
       (if (<= im 8e-51) (- im) (if (<= im 3.3e+145) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = (387420489.0 - (im * im)) / (im + 19683.0);
	double t_1 = (im * (0.5 * (re * re))) - im;
	double tmp;
	if (im <= -7.5e+163) {
		tmp = t_0;
	} else if (im <= -2150000000.0) {
		tmp = t_1;
	} else if (im <= 8e-51) {
		tmp = -im;
	} else if (im <= 3.3e+145) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (387420489.0d0 - (im * im)) / (im + 19683.0d0)
    t_1 = (im * (0.5d0 * (re * re))) - im
    if (im <= (-7.5d+163)) then
        tmp = t_0
    else if (im <= (-2150000000.0d0)) then
        tmp = t_1
    else if (im <= 8d-51) then
        tmp = -im
    else if (im <= 3.3d+145) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (387420489.0 - (im * im)) / (im + 19683.0);
	double t_1 = (im * (0.5 * (re * re))) - im;
	double tmp;
	if (im <= -7.5e+163) {
		tmp = t_0;
	} else if (im <= -2150000000.0) {
		tmp = t_1;
	} else if (im <= 8e-51) {
		tmp = -im;
	} else if (im <= 3.3e+145) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (387420489.0 - (im * im)) / (im + 19683.0)
	t_1 = (im * (0.5 * (re * re))) - im
	tmp = 0
	if im <= -7.5e+163:
		tmp = t_0
	elif im <= -2150000000.0:
		tmp = t_1
	elif im <= 8e-51:
		tmp = -im
	elif im <= 3.3e+145:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(387420489.0 - Float64(im * im)) / Float64(im + 19683.0))
	t_1 = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im)
	tmp = 0.0
	if (im <= -7.5e+163)
		tmp = t_0;
	elseif (im <= -2150000000.0)
		tmp = t_1;
	elseif (im <= 8e-51)
		tmp = Float64(-im);
	elseif (im <= 3.3e+145)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (387420489.0 - (im * im)) / (im + 19683.0);
	t_1 = (im * (0.5 * (re * re))) - im;
	tmp = 0.0;
	if (im <= -7.5e+163)
		tmp = t_0;
	elseif (im <= -2150000000.0)
		tmp = t_1;
	elseif (im <= 8e-51)
		tmp = -im;
	elseif (im <= 3.3e+145)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(387420489.0 - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(im + 19683.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -7.5e+163], t$95$0, If[LessEqual[im, -2150000000.0], t$95$1, If[LessEqual[im, 8e-51], (-im), If[LessEqual[im, 3.3e+145], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.50000000000000001e163 or 3.30000000000000027e145 < 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%

      \[\leadsto \cos re \cdot \left(\color{blue}{19683} - im\right) \]
   9. Step-by-step derivation

      1. flip-- 96.8%

         \[\leadsto \cos re \cdot \color{blue}{\frac{19683 \cdot 19683 - im \cdot im}{19683 + im}} \]

      2. associate-*r/ 96.8%

         \[\leadsto \color{blue}{\frac{\cos re \cdot \left(19683 \cdot 19683 - im \cdot im\right)}{19683 + im}} \]

      3. metadata-eval 96.8%

         \[\leadsto \frac{\cos re \cdot \left(\color{blue}{387420489} - im \cdot im\right)}{19683 + im} \]

      4. +-commutative 96.8%

         \[\leadsto \frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{\color{blue}{im + 19683}} \]

   10. Applied egg-rr 96.8%

       \[\leadsto \color{blue}{\frac{\cos re \cdot \left(387420489 - im \cdot im\right)}{im + 19683}} \]
   11. Step-by-step derivation

       1. *-commutative 96.8%

          \[\leadsto \frac{\color{blue}{\left(387420489 - im \cdot im\right) \cdot \cos re}}{im + 19683} \]

       2. associate-/l* 96.8%

          \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{im + 19683}{\cos re}}} \]

       3. +-commutative 96.8%

          \[\leadsto \frac{387420489 - im \cdot im}{\frac{\color{blue}{19683 + im}}{\cos re}} \]

   12. Simplified 96.8%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{\frac{19683 + im}{\cos re}}} \]

   13. Taylor expanded in re around 0 85.2%

       \[\leadsto \color{blue}{\frac{387420489 - {im}^{2}}{19683 + im}} \]
   14. Step-by-step derivation

       1. unpow2 85.2%

          \[\leadsto \frac{387420489 - \color{blue}{im \cdot im}}{19683 + im} \]

       2. +-commutative 85.2%

          \[\leadsto \frac{387420489 - im \cdot im}{\color{blue}{im + 19683}} \]

   15. Simplified 85.2%

       \[\leadsto \color{blue}{\frac{387420489 - im \cdot im}{im + 19683}} \]

   if -7.50000000000000001e163 < im < -2.15e9 or 8.0000000000000001e-51 < im < 3.30000000000000027e145

   1. Initial program 92.7%

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

      1. sub0-neg 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in im around 0 11.8%

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

      1. mul-1-neg 11.8%

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

      2. *-commutative 11.8%

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

      3. distribute-lft-neg-in 11.8%

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

   6. Simplified 11.8%

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

   7. Taylor expanded in re around 0 32.8%

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

      1. neg-mul-1 32.8%

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

      2. +-commutative 32.8%

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

      3. unsub-neg 32.8%

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

      4. *-commutative 32.8%

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

      5. *-commutative 32.8%

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

      6. associate-*l* 32.8%

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

      7. unpow2 32.8%

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

   9. Simplified 32.8%

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

   if -2.15e9 < im < 8.0000000000000001e-51

   1. Initial program 7.8%

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

      1. sub0-neg 7.8%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in im around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. *-commutative 98.3%

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

      3. distribute-lft-neg-in 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in re around 0 51.7%

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

      1. neg-mul-1 51.7%

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

   9. Simplified 51.7%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{im + 19683}\\ \mathbf{elif}\;im \leq -2150000000:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 8 \cdot 10^{-51}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\frac{387420489 - im \cdot im}{im + 19683}\\ \end{array} \]

Alternative 12: 34.5% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot -6.75\\ \mathbf{if}\;im \leq -88000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-10}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+113}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) -6.75)))
   (if (<= im -88000000.0)
     t_0
     (if (<= im 2e-10)
       (- im)
       (if (<= im 5.5e+113) (* (+ 0.5 (* re (* re -0.25))) -3.0) t_0)))))

C

double code(double re, double im) {
	double t_0 = (re * re) * -6.75;
	double tmp;
	if (im <= -88000000.0) {
		tmp = t_0;
	} else if (im <= 2e-10) {
		tmp = -im;
	} else if (im <= 5.5e+113) {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * re) * (-6.75d0)
    if (im <= (-88000000.0d0)) then
        tmp = t_0
    else if (im <= 2d-10) then
        tmp = -im
    else if (im <= 5.5d+113) then
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * (-3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) * -6.75;
	double tmp;
	if (im <= -88000000.0) {
		tmp = t_0;
	} else if (im <= 2e-10) {
		tmp = -im;
	} else if (im <= 5.5e+113) {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) * -6.75
	tmp = 0
	if im <= -88000000.0:
		tmp = t_0
	elif im <= 2e-10:
		tmp = -im
	elif im <= 5.5e+113:
		tmp = (0.5 + (re * (re * -0.25))) * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * -6.75)
	tmp = 0.0
	if (im <= -88000000.0)
		tmp = t_0;
	elseif (im <= 2e-10)
		tmp = Float64(-im);
	elseif (im <= 5.5e+113)
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) * -6.75;
	tmp = 0.0;
	if (im <= -88000000.0)
		tmp = t_0;
	elseif (im <= 2e-10)
		tmp = -im;
	elseif (im <= 5.5e+113)
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]}, If[LessEqual[im, -88000000.0], t$95$0, If[LessEqual[im, 2e-10], (-im), If[LessEqual[im, 5.5e+113], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.8e7 or 5.5000000000000001e113 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 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%

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

   8. Applied egg-rr 18.1%

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

   9. Taylor expanded in re around inf 17.7%

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

       1. unpow2 17.7%

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

   11. Simplified 17.7%

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

   if -8.8e7 < im < 2.00000000000000007e-10

   1. Initial program 7.7%

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

      1. sub0-neg 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in im around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. distribute-lft-neg-in 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in re around 0 52.8%

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

      1. neg-mul-1 52.8%

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

   9. Simplified 52.8%

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

   if 2.00000000000000007e-10 < im < 5.5000000000000001e113

   1. Initial program 98.4%

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

      1. sub0-neg 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in re around 0 0.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   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 76.1%

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

      4. +-commutative 76.1%

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

      5. *-commutative 76.1%

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

      6. unpow2 76.1%

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

      7. associate-*l* 76.1%

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

   6. Simplified 76.1%

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

   8. Applied egg-rr 26.4%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -88000000:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-10}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+113}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 13: 34.7% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + re \cdot \left(re \cdot -0.25\right)\\ \mathbf{if}\;im \leq -420000000:\\ \;\;\;\;t_0 \cdot 27\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-10}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+113}:\\ \;\;\;\;t_0 \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* re (* re -0.25)))))
   (if (<= im -420000000.0)
     (* t_0 27.0)
     (if (<= im 2e-10)
       (- im)
       (if (<= im 3e+113) (* t_0 -3.0) (* (* re re) -6.75))))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (re * (re * -0.25));
	double tmp;
	if (im <= -420000000.0) {
		tmp = t_0 * 27.0;
	} else if (im <= 2e-10) {
		tmp = -im;
	} else if (im <= 3e+113) {
		tmp = t_0 * -3.0;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (re * (re * (-0.25d0)))
    if (im <= (-420000000.0d0)) then
        tmp = t_0 * 27.0d0
    else if (im <= 2d-10) then
        tmp = -im
    else if (im <= 3d+113) then
        tmp = t_0 * (-3.0d0)
    else
        tmp = (re * re) * (-6.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (re * (re * -0.25));
	double tmp;
	if (im <= -420000000.0) {
		tmp = t_0 * 27.0;
	} else if (im <= 2e-10) {
		tmp = -im;
	} else if (im <= 3e+113) {
		tmp = t_0 * -3.0;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (re * (re * -0.25))
	tmp = 0
	if im <= -420000000.0:
		tmp = t_0 * 27.0
	elif im <= 2e-10:
		tmp = -im
	elif im <= 3e+113:
		tmp = t_0 * -3.0
	else:
		tmp = (re * re) * -6.75
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(re * Float64(re * -0.25)))
	tmp = 0.0
	if (im <= -420000000.0)
		tmp = Float64(t_0 * 27.0);
	elseif (im <= 2e-10)
		tmp = Float64(-im);
	elseif (im <= 3e+113)
		tmp = Float64(t_0 * -3.0);
	else
		tmp = Float64(Float64(re * re) * -6.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (re * (re * -0.25));
	tmp = 0.0;
	if (im <= -420000000.0)
		tmp = t_0 * 27.0;
	elseif (im <= 2e-10)
		tmp = -im;
	elseif (im <= 3e+113)
		tmp = t_0 * -3.0;
	else
		tmp = (re * re) * -6.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -420000000.0], N[(t$95$0 * 27.0), $MachinePrecision], If[LessEqual[im, 2e-10], (-im), If[LessEqual[im, 3e+113], N[(t$95$0 * -3.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.2e8

   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 75.8%

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

      4. +-commutative 75.8%

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

      5. *-commutative 75.8%

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

      6. unpow2 75.8%

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

      7. associate-*l* 75.8%

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

   6. Simplified 75.8%

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

   8. Applied egg-rr 16.6%

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

   if -4.2e8 < im < 2.00000000000000007e-10

   1. Initial program 7.7%

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

      1. sub0-neg 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in im around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. distribute-lft-neg-in 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in re around 0 52.8%

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

      1. neg-mul-1 52.8%

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

   9. Simplified 52.8%

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

   if 2.00000000000000007e-10 < im < 3e113

   1. Initial program 98.4%

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

      1. sub0-neg 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in re around 0 0.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   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 76.1%

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

      4. +-commutative 76.1%

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

      5. *-commutative 76.1%

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

      6. unpow2 76.1%

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

      7. associate-*l* 76.1%

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

   6. Simplified 76.1%

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

   8. Applied egg-rr 26.4%

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

   if 3e113 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 63.9%

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

      4. +-commutative 63.9%

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

      5. *-commutative 63.9%

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

      6. unpow2 63.9%

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

      7. associate-*l* 63.9%

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

   6. Simplified 63.9%

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

   8. Applied egg-rr 20.6%

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

   9. Taylor expanded in re around inf 21.2%

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

       1. unpow2 21.2%

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

   11. Simplified 21.2%

       \[\leadsto \color{blue}{-6.75 \cdot \left(re \cdot re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -420000000:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-10}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 14: 38.3% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -88000000 \lor \neg \left(im \leq 8 \cdot 10^{-51}\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -88000000.0) (not (<= im 8e-51)))
   (- (* im (* 0.5 (* re re))) im)
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -88000000.0) || !(im <= 8e-51)) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-88000000.0d0)) .or. (.not. (im <= 8d-51))) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -88000000.0) || !(im <= 8e-51)) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -88000000.0) or not (im <= 8e-51):
		tmp = (im * (0.5 * (re * re))) - im
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -88000000.0) || !(im <= 8e-51))
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -88000000.0) || ~((im <= 8e-51)))
		tmp = (im * (0.5 * (re * re))) - im;
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -88000000.0], N[Not[LessEqual[im, 8e-51]], $MachinePrecision]], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -8.8e7 or 8.0000000000000001e-51 < im

   1. Initial program 96.0%

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

      1. sub0-neg 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in im around 0 9.6%

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

      1. mul-1-neg 9.6%

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

      2. *-commutative 9.6%

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

      3. distribute-lft-neg-in 9.6%

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

   6. Simplified 9.6%

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

   7. Taylor expanded in re around 0 24.3%

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

      1. neg-mul-1 24.3%

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

      2. +-commutative 24.3%

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

      3. unsub-neg 24.3%

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

      4. *-commutative 24.3%

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

      5. *-commutative 24.3%

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

      6. associate-*l* 24.3%

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

      7. unpow2 24.3%

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

   9. Simplified 24.3%

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

   if -8.8e7 < im < 8.0000000000000001e-51

   1. Initial program 7.8%

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

      1. sub0-neg 7.8%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in im around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. *-commutative 98.3%

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

      3. distribute-lft-neg-in 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in re around 0 51.7%

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

      1. neg-mul-1 51.7%

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

   9. Simplified 51.7%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -88000000 \lor \neg \left(im \leq 8 \cdot 10^{-51}\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 15: 31.9% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.05 \cdot 10^{+155}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.05e+155) (- im) (* (* re re) -6.75)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.05e+155) {
		tmp = -im;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.05d+155) then
        tmp = -im
    else
        tmp = (re * re) * (-6.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.05e+155) {
		tmp = -im;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.05e+155:
		tmp = -im
	else:
		tmp = (re * re) * -6.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.05e+155)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(re * re) * -6.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.05e+155)
		tmp = -im;
	else
		tmp = (re * re) * -6.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.05e+155], (-im), N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.0499999999999999e155

   1. Initial program 53.8%

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

      1. sub0-neg 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in im around 0 52.3%

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

      1. mul-1-neg 52.3%

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

      2. *-commutative 52.3%

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

      3. distribute-lft-neg-in 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in re around 0 31.8%

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

      1. neg-mul-1 31.8%

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

   9. Simplified 31.8%

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

   if 2.0499999999999999e155 < re

   1. Initial program 49.8%

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

      1. sub0-neg 49.8%

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

   3. Simplified 49.8%

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

   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 26.5%

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

      4. +-commutative 26.5%

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

      5. *-commutative 26.5%

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

      6. unpow2 26.5%

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

      7. associate-*l* 26.5%

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

   6. Simplified 26.5%

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

   8. Applied egg-rr 27.2%

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

   9. Taylor expanded in re around inf 27.2%

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

       1. unpow2 27.2%

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

   11. Simplified 27.2%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.05 \cdot 10^{+155}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 16: 29.4% 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 53.3%

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

   1. sub0-neg 53.3%

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

3. Simplified 53.3%

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

4. Taylor expanded in im around 0 52.6%

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

   1. mul-1-neg 52.6%

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

   2. *-commutative 52.6%

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

   3. distribute-lft-neg-in 52.6%

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

6. Simplified 52.6%

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

7. Taylor expanded in re around 0 28.7%

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

   1. neg-mul-1 28.7%

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

9. Simplified 28.7%

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

10. Final simplification 28.7%

    \[\leadsto -im \]

Alternative 17: 2.8% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 13.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 13.5

Julia

function code(re, im)
	return 13.5
end

MATLAB

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

Wolfram

code[re_, im_] := 13.5

Derivation

1. Initial program 53.3%

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

   1. sub0-neg 53.3%

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

3. Simplified 53.3%

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

4. Taylor expanded in re around 0 2.2%

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

   1. *-commutative 2.2%

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

   2. associate-*r* 2.2%

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

   3. distribute-rgt-out 38.5%

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

   4. +-commutative 38.5%

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

   5. *-commutative 38.5%

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

   6. unpow2 38.5%

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

   7. associate-*l* 38.5%

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

6. Simplified 38.5%

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

8. Applied egg-rr 8.8%

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

9. Taylor expanded in re around 0 2.9%

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

10. Final simplification 2.9%

    \[\leadsto 13.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023217 
(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))))