math.sin on complex, imaginary part

Percentage Accurate: 53.8% → 99.3%

Time: 12.7s

Alternatives: 21

Speedup: 2.8×

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

Specification

Math

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

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: 53.8% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right) - im \cdot \cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-14)))
     (* (* 0.5 (cos re)) t_0)
     (- (* (pow im 3.0) (* (cos re) -0.16666666666666666)) (* im (cos re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e-14)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = (pow(im, 3.0) * (cos(re) * -0.16666666666666666)) - (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 <= 5e-14)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = (Math.pow(im, 3.0) * (Math.cos(re) * -0.16666666666666666)) - (im * Math.cos(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e-14):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = (math.pow(im, 3.0) * (math.cos(re) * -0.16666666666666666)) - (im * math.cos(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e-14))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(Float64((im ^ 3.0) * Float64(cos(re) * -0.16666666666666666)) - Float64(im * cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e-14)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = ((im ^ 3.0) * (cos(re) * -0.16666666666666666)) - (im * cos(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-14]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 5.0000000000000002e-14 < (-.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)) < 5.0000000000000002e-14

   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 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right) - im \cdot \cos re\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-14)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e-14)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	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 <= 5e-14)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e-14):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e-14))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e-14)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-14]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 5.0000000000000002e-14 < (-.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)) < 5.0000000000000002e-14

   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 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. distribute-lft-out-- 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.7 \cdot 10^{+101} \lor \neg \left(im \leq -0.07\right) \land \left(im \leq 0.039 \lor \neg \left(im \leq 5.5 \cdot 10^{+102}\right)\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.7e+101)
         (and (not (<= im -0.07)) (or (<= im 0.039) (not (<= im 5.5e+102)))))
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* 0.5 (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.7e+101) || (!(im <= -0.07) && ((im <= 0.039) || !(im <= 5.5e+102)))) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-2.7d+101)) .or. (.not. (im <= (-0.07d0))) .and. (im <= 0.039d0) .or. (.not. (im <= 5.5d+102))) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = 0.5d0 * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.7e+101) || (!(im <= -0.07) && ((im <= 0.039) || !(im <= 5.5e+102)))) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.7e+101) or (not (im <= -0.07) and ((im <= 0.039) or not (im <= 5.5e+102))):
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.7e+101) || (!(im <= -0.07) && ((im <= 0.039) || !(im <= 5.5e+102))))
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.7e+101) || (~((im <= -0.07)) && ((im <= 0.039) || ~((im <= 5.5e+102)))))
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = 0.5 * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.7e+101], And[N[Not[LessEqual[im, -0.07]], $MachinePrecision], Or[LessEqual[im, 0.039], N[Not[LessEqual[im, 5.5e+102]], $MachinePrecision]]]], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.70000000000000006e101 or -0.070000000000000007 < im < 0.0389999999999999999 or 5.49999999999999981e102 < im

   1. Initial program 46.3%

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

      1. sub0-neg 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in im around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*l* 99.5%

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

      5. distribute-lft-out-- 99.5%

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

   6. Simplified 99.5%

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

   if -2.70000000000000006e101 < im < -0.070000000000000007 or 0.0389999999999999999 < im < 5.49999999999999981e102

   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 81.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.7 \cdot 10^{+101} \lor \neg \left(im \leq -0.07\right) \land \left(im \leq 0.039 \lor \neg \left(im \leq 5.5 \cdot 10^{+102}\right)\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4: 87.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;im \leq -0.00035:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00355:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot -2 + {im}^{3} \cdot -0.3333333333333333\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im)))))
   (if (<= im -0.00035)
     t_0
     (if (<= im 0.00355)
       (* im (- (cos re)))
       (if (<= im 1e+139)
         t_0
         (*
          (+ (* im -2.0) (* (pow im 3.0) -0.3333333333333333))
          (+ 0.5 (* re (* re -0.25)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double tmp;
	if (im <= -0.00035) {
		tmp = t_0;
	} else if (im <= 0.00355) {
		tmp = im * -cos(re);
	} else if (im <= 1e+139) {
		tmp = t_0;
	} else {
		tmp = ((im * -2.0) + (pow(im, 3.0) * -0.3333333333333333)) * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    if (im <= (-0.00035d0)) then
        tmp = t_0
    else if (im <= 0.00355d0) then
        tmp = im * -cos(re)
    else if (im <= 1d+139) then
        tmp = t_0
    else
        tmp = ((im * (-2.0d0)) + ((im ** 3.0d0) * (-0.3333333333333333d0))) * (0.5d0 + (re * (re * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (im <= -0.00035) {
		tmp = t_0;
	} else if (im <= 0.00355) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1e+139) {
		tmp = t_0;
	} else {
		tmp = ((im * -2.0) + (Math.pow(im, 3.0) * -0.3333333333333333)) * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	tmp = 0
	if im <= -0.00035:
		tmp = t_0
	elif im <= 0.00355:
		tmp = im * -math.cos(re)
	elif im <= 1e+139:
		tmp = t_0
	else:
		tmp = ((im * -2.0) + (math.pow(im, 3.0) * -0.3333333333333333)) * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (im <= -0.00035)
		tmp = t_0;
	elseif (im <= 0.00355)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1e+139)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(im * -2.0) + Float64((im ^ 3.0) * -0.3333333333333333)) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	tmp = 0.0;
	if (im <= -0.00035)
		tmp = t_0;
	elseif (im <= 0.00355)
		tmp = im * -cos(re);
	elseif (im <= 1e+139)
		tmp = t_0;
	else
		tmp = ((im * -2.0) + ((im ^ 3.0) * -0.3333333333333333)) * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -0.00035], t$95$0, If[LessEqual[im, 0.00355], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1e+139], t$95$0, N[(N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.49999999999999996e-4 or 0.0035500000000000002 < im < 1.00000000000000003e139

   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 84.0%

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

   if -3.49999999999999996e-4 < im < 0.0035500000000000002

   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 99.7%

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

      1. mul-1-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. distribute-lft-neg-in 99.7%

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

   6. Simplified 99.7%

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

   if 1.00000000000000003e139 < 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 87.2%

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

      4. +-commutative 87.2%

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

      5. *-commutative 87.2%

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

      6. unpow2 87.2%

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

      7. associate-*l* 87.2%

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

   6. Simplified 87.2%

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

   7. Taylor expanded in im around 0 87.2%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00035:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.00355:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 10^{+139}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot -2 + {im}^{3} \cdot -0.3333333333333333\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 5: 75.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.3333333333333333\\ \mathbf{if}\;im \leq -1 \cdot 10^{+166}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -21500000:\\ \;\;\;\;t_0 \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+138}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right) + {re}^{4} \cdot -0.041666666666666664\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot -2 + t_0\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.3333333333333333)))
   (if (<= im -1e+166)
     (* (pow im 3.0) -0.16666666666666666)
     (if (<= im -21500000.0)
       (* t_0 (+ 0.5 (* -0.25 (* re re))))
       (if (<= im 4.2e-7)
         (* im (- (cos re)))
         (if (<= im 7.5e+138)
           (-
            (* im (+ (* re (* 0.5 re)) (* (pow re 4.0) -0.041666666666666664)))
            im)
           (* (+ (* im -2.0) t_0) (+ 0.5 (* re (* re -0.25))))))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.3333333333333333;
	double tmp;
	if (im <= -1e+166) {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	} else if (im <= -21500000.0) {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	} else if (im <= 4.2e-7) {
		tmp = im * -cos(re);
	} else if (im <= 7.5e+138) {
		tmp = (im * ((re * (0.5 * re)) + (pow(re, 4.0) * -0.041666666666666664))) - im;
	} else {
		tmp = ((im * -2.0) + t_0) * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.3333333333333333d0)
    if (im <= (-1d+166)) then
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    else if (im <= (-21500000.0d0)) then
        tmp = t_0 * (0.5d0 + ((-0.25d0) * (re * re)))
    else if (im <= 4.2d-7) then
        tmp = im * -cos(re)
    else if (im <= 7.5d+138) then
        tmp = (im * ((re * (0.5d0 * re)) + ((re ** 4.0d0) * (-0.041666666666666664d0)))) - im
    else
        tmp = ((im * (-2.0d0)) + t_0) * (0.5d0 + (re * (re * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.3333333333333333;
	double tmp;
	if (im <= -1e+166) {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	} else if (im <= -21500000.0) {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	} else if (im <= 4.2e-7) {
		tmp = im * -Math.cos(re);
	} else if (im <= 7.5e+138) {
		tmp = (im * ((re * (0.5 * re)) + (Math.pow(re, 4.0) * -0.041666666666666664))) - im;
	} else {
		tmp = ((im * -2.0) + t_0) * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.3333333333333333
	tmp = 0
	if im <= -1e+166:
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	elif im <= -21500000.0:
		tmp = t_0 * (0.5 + (-0.25 * (re * re)))
	elif im <= 4.2e-7:
		tmp = im * -math.cos(re)
	elif im <= 7.5e+138:
		tmp = (im * ((re * (0.5 * re)) + (math.pow(re, 4.0) * -0.041666666666666664))) - im
	else:
		tmp = ((im * -2.0) + t_0) * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.3333333333333333)
	tmp = 0.0
	if (im <= -1e+166)
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	elseif (im <= -21500000.0)
		tmp = Float64(t_0 * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	elseif (im <= 4.2e-7)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 7.5e+138)
		tmp = Float64(Float64(im * Float64(Float64(re * Float64(0.5 * re)) + Float64((re ^ 4.0) * -0.041666666666666664))) - im);
	else
		tmp = Float64(Float64(Float64(im * -2.0) + t_0) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.3333333333333333;
	tmp = 0.0;
	if (im <= -1e+166)
		tmp = (im ^ 3.0) * -0.16666666666666666;
	elseif (im <= -21500000.0)
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	elseif (im <= 4.2e-7)
		tmp = im * -cos(re);
	elseif (im <= 7.5e+138)
		tmp = (im * ((re * (0.5 * re)) + ((re ^ 4.0) * -0.041666666666666664))) - im;
	else
		tmp = ((im * -2.0) + t_0) * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]}, If[LessEqual[im, -1e+166], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[im, -21500000.0], N[(t$95$0 * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e-7], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 7.5e+138], N[(N[(im * N[(N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] + N[(N[Power[re, 4.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(N[(N[(im * -2.0), $MachinePrecision] + t$95$0), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -9.9999999999999994e165

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 73.1%

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

      4. +-commutative 73.1%

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

      5. *-commutative 73.1%

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

      6. unpow2 73.1%

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

      7. associate-*l* 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in im around 0 73.1%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around inf 73.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 73.1%

         \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      2. unpow2 73.1%

         \[\leadsto \left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   10. Simplified 73.1%

       \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around 0 92.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   12. Step-by-step derivation

       1. *-commutative 92.3%

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

   13. Simplified 92.3%

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

   if -9.9999999999999994e165 < im < -2.15e7

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 77.3%

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

      4. +-commutative 77.3%

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

      5. *-commutative 77.3%

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

      6. unpow2 77.3%

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

      7. associate-*l* 77.3%

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

   6. Simplified 77.3%

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

   7. Taylor expanded in im around 0 40.9%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around inf 40.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 40.9%

         \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      2. unpow2 40.9%

         \[\leadsto \left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   10. Simplified 40.9%

       \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   if -2.15e7 < im < 4.2e-7

   1. Initial program 8.3%

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

      1. sub0-neg 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in im around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. *-commutative 98.2%

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

      3. distribute-lft-neg-in 98.2%

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

   6. Simplified 98.2%

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

   if 4.2e-7 < im < 7.4999999999999999e138

   1. Initial program 99.0%

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

      1. sub0-neg 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in im around 0 6.6%

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

      1. mul-1-neg 6.6%

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

      2. *-commutative 6.6%

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

      3. distribute-lft-neg-in 6.6%

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

   6. Simplified 6.6%

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

   7. Taylor expanded in re around 0 21.9%

      \[\leadsto \color{blue}{-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 21.9%

         \[\leadsto -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(\color{blue}{\left(-im\right)} + 0.5 \cdot \left({re}^{2} \cdot im\right)\right) \]

      2. associate-+r+ 21.9%

         \[\leadsto \color{blue}{\left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-im\right)\right) + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]

      3. +-commutative 21.9%

         \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) + \left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-im\right)\right)} \]

      4. unsub-neg 21.9%

         \[\leadsto 0.5 \cdot \left({re}^{2} \cdot im\right) + \color{blue}{\left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) - im\right)} \]

      5. associate-+r- 21.9%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left({re}^{2} \cdot im\right) + -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right)\right) - im} \]

      6. associate-*r* 21.9%

         \[\leadsto \left(\color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot im} + -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right)\right) - im \]

      7. associate-*r* 21.9%

         \[\leadsto \left(\left(0.5 \cdot {re}^{2}\right) \cdot im + \color{blue}{\left(-0.041666666666666664 \cdot {re}^{4}\right) \cdot im}\right) - im \]

      8. distribute-rgt-out 28.8%

         \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot {re}^{2} + -0.041666666666666664 \cdot {re}^{4}\right)} - im \]

      9. *-commutative 28.8%

         \[\leadsto im \cdot \left(\color{blue}{{re}^{2} \cdot 0.5} + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      10. unpow2 28.8%

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot 0.5 + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      11. associate-*l* 28.8%

          \[\leadsto im \cdot \left(\color{blue}{re \cdot \left(re \cdot 0.5\right)} + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      12. *-commutative 28.8%

          \[\leadsto im \cdot \left(re \cdot \left(re \cdot 0.5\right) + \color{blue}{{re}^{4} \cdot -0.041666666666666664}\right) - im \]

   9. Simplified 28.8%

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(re \cdot 0.5\right) + {re}^{4} \cdot -0.041666666666666664\right) - im} \]

   if 7.4999999999999999e138 < 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 87.2%

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

      4. +-commutative 87.2%

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

      5. *-commutative 87.2%

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

      6. unpow2 87.2%

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

      7. associate-*l* 87.2%

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

   6. Simplified 87.2%

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

   7. Taylor expanded in im around 0 87.2%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+166}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -21500000:\\ \;\;\;\;\left({im}^{3} \cdot -0.3333333333333333\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+138}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right) + {re}^{4} \cdot -0.041666666666666664\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot -2 + {im}^{3} \cdot -0.3333333333333333\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 6: 75.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({im}^{3} \cdot -0.3333333333333333\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;im \leq -4 \cdot 10^{+163}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -54000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+138}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right) + {re}^{4} \cdot -0.041666666666666664\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* (* (pow im 3.0) -0.3333333333333333) (+ 0.5 (* -0.25 (* re re))))))
   (if (<= im -4e+163)
     (* (pow im 3.0) -0.16666666666666666)
     (if (<= im -54000000.0)
       t_0
       (if (<= im 4.2e-7)
         (* im (- (cos re)))
         (if (<= im 7.5e+138)
           (-
            (* im (+ (* re (* 0.5 re)) (* (pow re 4.0) -0.041666666666666664)))
            im)
           t_0))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.3333333333333333) * (0.5 + (-0.25 * (re * re)));
	double tmp;
	if (im <= -4e+163) {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	} else if (im <= -54000000.0) {
		tmp = t_0;
	} else if (im <= 4.2e-7) {
		tmp = im * -cos(re);
	} else if (im <= 7.5e+138) {
		tmp = (im * ((re * (0.5 * re)) + (pow(re, 4.0) * -0.041666666666666664))) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.3333333333333333d0)) * (0.5d0 + ((-0.25d0) * (re * re)))
    if (im <= (-4d+163)) then
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    else if (im <= (-54000000.0d0)) then
        tmp = t_0
    else if (im <= 4.2d-7) then
        tmp = im * -cos(re)
    else if (im <= 7.5d+138) then
        tmp = (im * ((re * (0.5d0 * re)) + ((re ** 4.0d0) * (-0.041666666666666664d0)))) - im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.3333333333333333) * (0.5 + (-0.25 * (re * re)));
	double tmp;
	if (im <= -4e+163) {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	} else if (im <= -54000000.0) {
		tmp = t_0;
	} else if (im <= 4.2e-7) {
		tmp = im * -Math.cos(re);
	} else if (im <= 7.5e+138) {
		tmp = (im * ((re * (0.5 * re)) + (Math.pow(re, 4.0) * -0.041666666666666664))) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.3333333333333333) * (0.5 + (-0.25 * (re * re)))
	tmp = 0
	if im <= -4e+163:
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	elif im <= -54000000.0:
		tmp = t_0
	elif im <= 4.2e-7:
		tmp = im * -math.cos(re)
	elif im <= 7.5e+138:
		tmp = (im * ((re * (0.5 * re)) + (math.pow(re, 4.0) * -0.041666666666666664))) - im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.3333333333333333) * Float64(0.5 + Float64(-0.25 * Float64(re * re))))
	tmp = 0.0
	if (im <= -4e+163)
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	elseif (im <= -54000000.0)
		tmp = t_0;
	elseif (im <= 4.2e-7)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 7.5e+138)
		tmp = Float64(Float64(im * Float64(Float64(re * Float64(0.5 * re)) + Float64((re ^ 4.0) * -0.041666666666666664))) - im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.3333333333333333) * (0.5 + (-0.25 * (re * re)));
	tmp = 0.0;
	if (im <= -4e+163)
		tmp = (im ^ 3.0) * -0.16666666666666666;
	elseif (im <= -54000000.0)
		tmp = t_0;
	elseif (im <= 4.2e-7)
		tmp = im * -cos(re);
	elseif (im <= 7.5e+138)
		tmp = (im * ((re * (0.5 * re)) + ((re ^ 4.0) * -0.041666666666666664))) - im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4e+163], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[im, -54000000.0], t$95$0, If[LessEqual[im, 4.2e-7], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 7.5e+138], N[(N[(im * N[(N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] + N[(N[Power[re, 4.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.9999999999999998e163

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 73.1%

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

      4. +-commutative 73.1%

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

      5. *-commutative 73.1%

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

      6. unpow2 73.1%

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

      7. associate-*l* 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in im around 0 73.1%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around inf 73.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 73.1%

         \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      2. unpow2 73.1%

         \[\leadsto \left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   10. Simplified 73.1%

       \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around 0 92.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   12. Step-by-step derivation

       1. *-commutative 92.3%

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

   13. Simplified 92.3%

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

   if -3.9999999999999998e163 < im < -5.4e7 or 7.4999999999999999e138 < 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 81.9%

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

      4. +-commutative 81.9%

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

      5. *-commutative 81.9%

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

      6. unpow2 81.9%

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

      7. associate-*l* 81.9%

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

   6. Simplified 81.9%

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

   7. Taylor expanded in im around 0 62.6%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around inf 62.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 62.6%

         \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      2. unpow2 62.6%

         \[\leadsto \left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   10. Simplified 62.6%

       \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   if -5.4e7 < im < 4.2e-7

   1. Initial program 8.3%

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

      1. sub0-neg 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in im around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. *-commutative 98.2%

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

      3. distribute-lft-neg-in 98.2%

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

   6. Simplified 98.2%

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

   if 4.2e-7 < im < 7.4999999999999999e138

   1. Initial program 99.0%

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

      1. sub0-neg 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in im around 0 6.6%

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

      1. mul-1-neg 6.6%

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

      2. *-commutative 6.6%

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

      3. distribute-lft-neg-in 6.6%

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

   6. Simplified 6.6%

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

   7. Taylor expanded in re around 0 21.9%

      \[\leadsto \color{blue}{-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 21.9%

         \[\leadsto -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(\color{blue}{\left(-im\right)} + 0.5 \cdot \left({re}^{2} \cdot im\right)\right) \]

      2. associate-+r+ 21.9%

         \[\leadsto \color{blue}{\left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-im\right)\right) + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]

      3. +-commutative 21.9%

         \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) + \left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-im\right)\right)} \]

      4. unsub-neg 21.9%

         \[\leadsto 0.5 \cdot \left({re}^{2} \cdot im\right) + \color{blue}{\left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) - im\right)} \]

      5. associate-+r- 21.9%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left({re}^{2} \cdot im\right) + -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right)\right) - im} \]

      6. associate-*r* 21.9%

         \[\leadsto \left(\color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot im} + -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right)\right) - im \]

      7. associate-*r* 21.9%

         \[\leadsto \left(\left(0.5 \cdot {re}^{2}\right) \cdot im + \color{blue}{\left(-0.041666666666666664 \cdot {re}^{4}\right) \cdot im}\right) - im \]

      8. distribute-rgt-out 28.8%

         \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot {re}^{2} + -0.041666666666666664 \cdot {re}^{4}\right)} - im \]

      9. *-commutative 28.8%

         \[\leadsto im \cdot \left(\color{blue}{{re}^{2} \cdot 0.5} + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      10. unpow2 28.8%

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot 0.5 + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      11. associate-*l* 28.8%

          \[\leadsto im \cdot \left(\color{blue}{re \cdot \left(re \cdot 0.5\right)} + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      12. *-commutative 28.8%

          \[\leadsto im \cdot \left(re \cdot \left(re \cdot 0.5\right) + \color{blue}{{re}^{4} \cdot -0.041666666666666664}\right) - im \]

   9. Simplified 28.8%

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(re \cdot 0.5\right) + {re}^{4} \cdot -0.041666666666666664\right) - im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{+163}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -54000000:\\ \;\;\;\;\left({im}^{3} \cdot -0.3333333333333333\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+138}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right) + {re}^{4} \cdot -0.041666666666666664\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.3333333333333333\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 7: 76.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := \left({im}^{3} \cdot -0.3333333333333333\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;im \leq -3.8 \cdot 10^{+167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -21500000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 10^{+139}:\\ \;\;\;\;t_0 - im\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1
         (* (* (pow im 3.0) -0.3333333333333333) (+ 0.5 (* -0.25 (* re re))))))
   (if (<= im -3.8e+167)
     t_0
     (if (<= im -21500000.0)
       t_1
       (if (<= im 4.2e-7)
         (* im (- (cos re)))
         (if (<= im 1e+139) (- t_0 im) t_1))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (pow(im, 3.0) * -0.3333333333333333) * (0.5 + (-0.25 * (re * re)));
	double tmp;
	if (im <= -3.8e+167) {
		tmp = t_0;
	} else if (im <= -21500000.0) {
		tmp = t_1;
	} else if (im <= 4.2e-7) {
		tmp = im * -cos(re);
	} else if (im <= 1e+139) {
		tmp = t_0 - im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = ((im ** 3.0d0) * (-0.3333333333333333d0)) * (0.5d0 + ((-0.25d0) * (re * re)))
    if (im <= (-3.8d+167)) then
        tmp = t_0
    else if (im <= (-21500000.0d0)) then
        tmp = t_1
    else if (im <= 4.2d-7) then
        tmp = im * -cos(re)
    else if (im <= 1d+139) then
        tmp = t_0 - im
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (Math.pow(im, 3.0) * -0.3333333333333333) * (0.5 + (-0.25 * (re * re)));
	double tmp;
	if (im <= -3.8e+167) {
		tmp = t_0;
	} else if (im <= -21500000.0) {
		tmp = t_1;
	} else if (im <= 4.2e-7) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1e+139) {
		tmp = t_0 - im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = (math.pow(im, 3.0) * -0.3333333333333333) * (0.5 + (-0.25 * (re * re)))
	tmp = 0
	if im <= -3.8e+167:
		tmp = t_0
	elif im <= -21500000.0:
		tmp = t_1
	elif im <= 4.2e-7:
		tmp = im * -math.cos(re)
	elif im <= 1e+139:
		tmp = t_0 - im
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(Float64((im ^ 3.0) * -0.3333333333333333) * Float64(0.5 + Float64(-0.25 * Float64(re * re))))
	tmp = 0.0
	if (im <= -3.8e+167)
		tmp = t_0;
	elseif (im <= -21500000.0)
		tmp = t_1;
	elseif (im <= 4.2e-7)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1e+139)
		tmp = Float64(t_0 - im);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = ((im ^ 3.0) * -0.3333333333333333) * (0.5 + (-0.25 * (re * re)));
	tmp = 0.0;
	if (im <= -3.8e+167)
		tmp = t_0;
	elseif (im <= -21500000.0)
		tmp = t_1;
	elseif (im <= 4.2e-7)
		tmp = im * -cos(re);
	elseif (im <= 1e+139)
		tmp = t_0 - im;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.8e+167], t$95$0, If[LessEqual[im, -21500000.0], t$95$1, If[LessEqual[im, 4.2e-7], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1e+139], N[(t$95$0 - im), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.79999999999999994e167

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 73.1%

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

      4. +-commutative 73.1%

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

      5. *-commutative 73.1%

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

      6. unpow2 73.1%

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

      7. associate-*l* 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in im around 0 73.1%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around inf 73.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 73.1%

         \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      2. unpow2 73.1%

         \[\leadsto \left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   10. Simplified 73.1%

       \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around 0 92.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   12. Step-by-step derivation

       1. *-commutative 92.3%

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

   13. Simplified 92.3%

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

   if -3.79999999999999994e167 < im < -2.15e7 or 1.00000000000000003e139 < 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 81.9%

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

      4. +-commutative 81.9%

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

      5. *-commutative 81.9%

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

      6. unpow2 81.9%

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

      7. associate-*l* 81.9%

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

   6. Simplified 81.9%

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

   7. Taylor expanded in im around 0 62.6%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around inf 62.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 62.6%

         \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      2. unpow2 62.6%

         \[\leadsto \left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   10. Simplified 62.6%

       \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   if -2.15e7 < im < 4.2e-7

   1. Initial program 8.3%

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

      1. sub0-neg 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in im around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. *-commutative 98.2%

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

      3. distribute-lft-neg-in 98.2%

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

   6. Simplified 98.2%

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

   if 4.2e-7 < im < 1.00000000000000003e139

   1. Initial program 99.0%

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

      1. sub0-neg 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in im around 0 28.0%

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

      1. mul-1-neg 28.0%

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

      2. unsub-neg 28.0%

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

      3. *-commutative 28.0%

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

      4. *-commutative 28.0%

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

      5. *-commutative 28.0%

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

      6. associate-*l* 28.0%

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

   6. Simplified 28.0%

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

   7. Taylor expanded in re around 0 27.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{+167}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -21500000:\\ \;\;\;\;\left({im}^{3} \cdot -0.3333333333333333\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 10^{+139}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.3333333333333333\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 8: 38.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ t_1 := \frac{im \cdot im - 4 \cdot \left(t_0 \cdot t_0\right)}{\left(-im\right) - -2 \cdot t_0}\\ \mathbf{if}\;im \leq -3 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.52 \cdot 10^{+55}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+220}:\\ \;\;\;\;{im}^{3} \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* -0.25 (* re re))))
        (t_1 (/ (- (* im im) (* 4.0 (* t_0 t_0))) (- (- im) (* -2.0 t_0)))))
   (if (<= im -3e+25)
     t_1
     (if (<= im 1.52e+55)
       (- (* re (* re (* im 0.5))) im)
       (if (<= im 9.5e+139)
         (* (+ 0.5 (* re (* re -0.25))) 27.0)
         (if (<= im 8e+220) (* (pow im 3.0) 0.0625) t_1))))))

C

double code(double re, double im) {
	double t_0 = im * (-0.25 * (re * re));
	double t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0));
	double tmp;
	if (im <= -3e+25) {
		tmp = t_1;
	} else if (im <= 1.52e+55) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else if (im <= 9.5e+139) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else if (im <= 8e+220) {
		tmp = pow(im, 3.0) * 0.0625;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * ((-0.25d0) * (re * re))
    t_1 = ((im * im) - (4.0d0 * (t_0 * t_0))) / (-im - ((-2.0d0) * t_0))
    if (im <= (-3d+25)) then
        tmp = t_1
    else if (im <= 1.52d+55) then
        tmp = (re * (re * (im * 0.5d0))) - im
    else if (im <= 9.5d+139) then
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    else if (im <= 8d+220) then
        tmp = (im ** 3.0d0) * 0.0625d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (-0.25 * (re * re));
	double t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0));
	double tmp;
	if (im <= -3e+25) {
		tmp = t_1;
	} else if (im <= 1.52e+55) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else if (im <= 9.5e+139) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else if (im <= 8e+220) {
		tmp = Math.pow(im, 3.0) * 0.0625;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (-0.25 * (re * re))
	t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0))
	tmp = 0
	if im <= -3e+25:
		tmp = t_1
	elif im <= 1.52e+55:
		tmp = (re * (re * (im * 0.5))) - im
	elif im <= 9.5e+139:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	elif im <= 8e+220:
		tmp = math.pow(im, 3.0) * 0.0625
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(-0.25 * Float64(re * re)))
	t_1 = Float64(Float64(Float64(im * im) - Float64(4.0 * Float64(t_0 * t_0))) / Float64(Float64(-im) - Float64(-2.0 * t_0)))
	tmp = 0.0
	if (im <= -3e+25)
		tmp = t_1;
	elseif (im <= 1.52e+55)
		tmp = Float64(Float64(re * Float64(re * Float64(im * 0.5))) - im);
	elseif (im <= 9.5e+139)
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	elseif (im <= 8e+220)
		tmp = Float64((im ^ 3.0) * 0.0625);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (-0.25 * (re * re));
	t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0));
	tmp = 0.0;
	if (im <= -3e+25)
		tmp = t_1;
	elseif (im <= 1.52e+55)
		tmp = (re * (re * (im * 0.5))) - im;
	elseif (im <= 9.5e+139)
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	elseif (im <= 8e+220)
		tmp = (im ^ 3.0) * 0.0625;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(im * im), $MachinePrecision] - N[(4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-im) - N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3e+25], t$95$1, If[LessEqual[im, 1.52e+55], N[(N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 9.5e+139], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision], If[LessEqual[im, 8e+220], N[(N[Power[im, 3.0], $MachinePrecision] * 0.0625), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.00000000000000006e25 or 8e220 < 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 82.3%

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

      4. +-commutative 82.3%

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

      5. *-commutative 82.3%

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

      6. unpow2 82.3%

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

      7. associate-*l* 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in im around 0 18.0%

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

      1. distribute-lft-in 18.0%

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

      2. flip-+ 39.1%

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

   9. Applied egg-rr 39.1%

      \[\leadsto \color{blue}{\frac{\left(im \cdot -1\right) \cdot \left(im \cdot -1\right) - \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right) \cdot \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. swap-sqr 39.1%

          \[\leadsto \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(-1 \cdot -1\right)} - \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right) \cdot \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       2. metadata-eval 39.1%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot \color{blue}{1} - \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right) \cdot \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       3. swap-sqr 39.1%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - \color{blue}{\left(-2 \cdot -2\right) \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       4. metadata-eval 39.1%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - \color{blue}{4} \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       5. *-commutative 39.1%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\color{blue}{-1 \cdot im} - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       6. mul-1-neg 39.1%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\color{blue}{\left(-im\right)} - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

   11. Simplified 39.1%

       \[\leadsto \color{blue}{\frac{\left(im \cdot im\right) \cdot 1 - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\left(-im\right) - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}} \]

   if -3.00000000000000006e25 < im < 1.5200000000000001e55

   1. Initial program 21.4%

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

      1. sub0-neg 21.4%

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

   3. Simplified 21.4%

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

   4. Taylor expanded in im around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. *-commutative 85.1%

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

      3. distribute-lft-neg-in 85.1%

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

   6. Simplified 85.1%

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

   7. Taylor expanded in re around 0 52.0%

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

      1. neg-mul-1 52.0%

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

      2. +-commutative 52.0%

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

      3. unsub-neg 52.0%

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

      4. *-commutative 52.0%

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

      5. associate-*l* 52.0%

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

      6. metadata-eval 52.0%

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

      7. distribute-rgt-neg-in 52.0%

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

      8. *-commutative 52.0%

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

      9. unpow2 52.0%

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

      10. associate-*l* 52.1%

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

      11. *-commutative 52.1%

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

      12. distribute-rgt-neg-in 52.1%

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

      13. metadata-eval 52.1%

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

   9. Simplified 52.1%

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

   if 1.5200000000000001e55 < im < 9.5000000000000002e139

   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 41.2%

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

      4. +-commutative 41.2%

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

      5. *-commutative 41.2%

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

      6. unpow2 41.2%

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

      7. associate-*l* 41.2%

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

   6. Simplified 41.2%

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

   8. Applied egg-rr 37.1%

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

   if 9.5000000000000002e139 < im < 8e220

   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%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 50.0%

      \[\leadsto {im}^{3} \cdot \color{blue}{0.0625} - im \cdot \cos re \]

   9. Taylor expanded in im around inf 50.0%

      \[\leadsto \color{blue}{0.0625 \cdot {im}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\frac{im \cdot im - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\left(-im\right) - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}\\ \mathbf{elif}\;im \leq 1.52 \cdot 10^{+55}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+220}:\\ \;\;\;\;{im}^{3} \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot im - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\left(-im\right) - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}\\ \end{array} \]

Alternative 9: 62.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ t_1 := \frac{im \cdot im - 4 \cdot \left(t_0 \cdot t_0\right)}{\left(-im\right) - -2 \cdot t_0}\\ \mathbf{if}\;im \leq -4.8 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{+51}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 7.8 \cdot 10^{+138}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+221}:\\ \;\;\;\;{im}^{3} \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* -0.25 (* re re))))
        (t_1 (/ (- (* im im) (* 4.0 (* t_0 t_0))) (- (- im) (* -2.0 t_0)))))
   (if (<= im -4.8e+23)
     t_1
     (if (<= im 2.35e+51)
       (* im (- (cos re)))
       (if (<= im 7.8e+138)
         (* (+ 0.5 (* re (* re -0.25))) 27.0)
         (if (<= im 1.35e+221) (* (pow im 3.0) 0.0625) t_1))))))

C

double code(double re, double im) {
	double t_0 = im * (-0.25 * (re * re));
	double t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0));
	double tmp;
	if (im <= -4.8e+23) {
		tmp = t_1;
	} else if (im <= 2.35e+51) {
		tmp = im * -cos(re);
	} else if (im <= 7.8e+138) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else if (im <= 1.35e+221) {
		tmp = pow(im, 3.0) * 0.0625;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * ((-0.25d0) * (re * re))
    t_1 = ((im * im) - (4.0d0 * (t_0 * t_0))) / (-im - ((-2.0d0) * t_0))
    if (im <= (-4.8d+23)) then
        tmp = t_1
    else if (im <= 2.35d+51) then
        tmp = im * -cos(re)
    else if (im <= 7.8d+138) then
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    else if (im <= 1.35d+221) then
        tmp = (im ** 3.0d0) * 0.0625d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (-0.25 * (re * re));
	double t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0));
	double tmp;
	if (im <= -4.8e+23) {
		tmp = t_1;
	} else if (im <= 2.35e+51) {
		tmp = im * -Math.cos(re);
	} else if (im <= 7.8e+138) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else if (im <= 1.35e+221) {
		tmp = Math.pow(im, 3.0) * 0.0625;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (-0.25 * (re * re))
	t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0))
	tmp = 0
	if im <= -4.8e+23:
		tmp = t_1
	elif im <= 2.35e+51:
		tmp = im * -math.cos(re)
	elif im <= 7.8e+138:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	elif im <= 1.35e+221:
		tmp = math.pow(im, 3.0) * 0.0625
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(-0.25 * Float64(re * re)))
	t_1 = Float64(Float64(Float64(im * im) - Float64(4.0 * Float64(t_0 * t_0))) / Float64(Float64(-im) - Float64(-2.0 * t_0)))
	tmp = 0.0
	if (im <= -4.8e+23)
		tmp = t_1;
	elseif (im <= 2.35e+51)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 7.8e+138)
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	elseif (im <= 1.35e+221)
		tmp = Float64((im ^ 3.0) * 0.0625);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (-0.25 * (re * re));
	t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0));
	tmp = 0.0;
	if (im <= -4.8e+23)
		tmp = t_1;
	elseif (im <= 2.35e+51)
		tmp = im * -cos(re);
	elseif (im <= 7.8e+138)
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	elseif (im <= 1.35e+221)
		tmp = (im ^ 3.0) * 0.0625;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(im * im), $MachinePrecision] - N[(4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-im) - N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.8e+23], t$95$1, If[LessEqual[im, 2.35e+51], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 7.8e+138], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision], If[LessEqual[im, 1.35e+221], N[(N[Power[im, 3.0], $MachinePrecision] * 0.0625), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.8e23 or 1.35e221 < 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 82.5%

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

      4. +-commutative 82.5%

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

      5. *-commutative 82.5%

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

      6. unpow2 82.5%

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

      7. associate-*l* 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in im around 0 17.8%

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

      1. distribute-lft-in 17.8%

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

      2. flip-+ 38.6%

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

   9. Applied egg-rr 38.6%

      \[\leadsto \color{blue}{\frac{\left(im \cdot -1\right) \cdot \left(im \cdot -1\right) - \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right) \cdot \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. swap-sqr 38.6%

          \[\leadsto \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(-1 \cdot -1\right)} - \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right) \cdot \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       2. metadata-eval 38.6%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot \color{blue}{1} - \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right) \cdot \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       3. swap-sqr 38.6%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - \color{blue}{\left(-2 \cdot -2\right) \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       4. metadata-eval 38.6%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - \color{blue}{4} \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       5. *-commutative 38.6%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\color{blue}{-1 \cdot im} - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       6. mul-1-neg 38.6%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\color{blue}{\left(-im\right)} - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

   11. Simplified 38.6%

       \[\leadsto \color{blue}{\frac{\left(im \cdot im\right) \cdot 1 - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\left(-im\right) - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}} \]

   if -4.8e23 < im < 2.3500000000000001e51

   1. Initial program 20.8%

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

      1. sub0-neg 20.8%

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

   3. Simplified 20.8%

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

   4. Taylor expanded in im around 0 85.7%

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

      1. mul-1-neg 85.7%

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

      2. *-commutative 85.7%

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

      3. distribute-lft-neg-in 85.7%

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

   6. Simplified 85.7%

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

   if 2.3500000000000001e51 < im < 7.7999999999999996e138

   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 41.2%

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

      4. +-commutative 41.2%

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

      5. *-commutative 41.2%

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

      6. unpow2 41.2%

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

      7. associate-*l* 41.2%

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

   6. Simplified 41.2%

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

   8. Applied egg-rr 37.1%

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

   if 7.7999999999999996e138 < im < 1.35e221

   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%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 50.0%

      \[\leadsto {im}^{3} \cdot \color{blue}{0.0625} - im \cdot \cos re \]

   9. Taylor expanded in im around inf 50.0%

      \[\leadsto \color{blue}{0.0625 \cdot {im}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{im \cdot im - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\left(-im\right) - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{+51}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 7.8 \cdot 10^{+138}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+221}:\\ \;\;\;\;{im}^{3} \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot im - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\left(-im\right) - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}\\ \end{array} \]

Alternative 10: 75.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -2.4 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.18 \cdot 10^{+44}:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - re \cdot \left(re \cdot 0.75\right)}\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -2.4e+89)
     t_0
     (if (<= im -1.18e+44)
       (/
        (- 2.25 (* (* (* re re) (* re re)) 0.5625))
        (- -1.5 (* re (* re 0.75))))
       (if (<= im 4.2e-7) (* im (- (cos re))) (- t_0 im))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -2.4e+89) {
		tmp = t_0;
	} else if (im <= -1.18e+44) {
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - (re * (re * 0.75)));
	} else if (im <= 4.2e-7) {
		tmp = im * -cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    if (im <= (-2.4d+89)) then
        tmp = t_0
    else if (im <= (-1.18d+44)) then
        tmp = (2.25d0 - (((re * re) * (re * re)) * 0.5625d0)) / ((-1.5d0) - (re * (re * 0.75d0)))
    else if (im <= 4.2d-7) then
        tmp = im * -cos(re)
    else
        tmp = t_0 - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -2.4e+89) {
		tmp = t_0;
	} else if (im <= -1.18e+44) {
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - (re * (re * 0.75)));
	} else if (im <= 4.2e-7) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -2.4e+89:
		tmp = t_0
	elif im <= -1.18e+44:
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - (re * (re * 0.75)))
	elif im <= 4.2e-7:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0 - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -2.4e+89)
		tmp = t_0;
	elseif (im <= -1.18e+44)
		tmp = Float64(Float64(2.25 - Float64(Float64(Float64(re * re) * Float64(re * re)) * 0.5625)) / Float64(-1.5 - Float64(re * Float64(re * 0.75))));
	elseif (im <= 4.2e-7)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -2.4e+89)
		tmp = t_0;
	elseif (im <= -1.18e+44)
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - (re * (re * 0.75)));
	elseif (im <= 4.2e-7)
		tmp = im * -cos(re);
	else
		tmp = t_0 - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -2.4e+89], t$95$0, If[LessEqual[im, -1.18e+44], N[(N[(2.25 - N[(N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.5625), $MachinePrecision]), $MachinePrecision] / N[(-1.5 - N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e-7], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(t$95$0 - im), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.40000000000000004e89

   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 79.1%

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

      4. +-commutative 79.1%

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

      5. *-commutative 79.1%

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

      6. unpow2 79.1%

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

      7. associate-*l* 79.1%

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

   6. Simplified 79.1%

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

   7. Taylor expanded in im around 0 72.8%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around inf 72.8%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 72.8%

         \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      2. unpow2 72.8%

         \[\leadsto \left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   10. Simplified 72.8%

       \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around 0 79.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   12. Step-by-step derivation

       1. *-commutative 79.7%

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

   13. Simplified 79.7%

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

   if -2.40000000000000004e89 < im < -1.17999999999999997e44

   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 61.5%

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

      4. +-commutative 61.5%

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

      5. *-commutative 61.5%

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

      6. unpow2 61.5%

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

      7. associate-*l* 61.5%

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

   6. Simplified 61.5%

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

   8. Applied egg-rr 3.2%

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

      1. distribute-lft-in 3.2%

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

      2. flip-+ 38.8%

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

      3. metadata-eval 38.8%

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

      4. metadata-eval 38.8%

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

      5. metadata-eval 38.8%

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

      6. *-commutative 38.8%

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

      7. associate-*r* 38.8%

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

      8. associate-*l* 38.8%

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

      9. metadata-eval 38.8%

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

      10. *-commutative 38.8%

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

      11. associate-*r* 38.8%

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

      12. associate-*l* 38.8%

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

      13. metadata-eval 38.8%

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

      14. metadata-eval 38.8%

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

      15. *-commutative 38.8%

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

      16. associate-*r* 38.8%

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

      17. associate-*l* 38.8%

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

      18. metadata-eval 38.8%

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

   10. Applied egg-rr 38.8%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}} \]
   11. Step-by-step derivation

       1. swap-sqr 38.8%

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

       2. metadata-eval 38.8%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{0.5625}}{-1.5 - \left(re \cdot re\right) \cdot 0.75} \]

       3. associate-*l* 38.8%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - \color{blue}{re \cdot \left(re \cdot 0.75\right)}} \]

   12. Simplified 38.8%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - re \cdot \left(re \cdot 0.75\right)}} \]

   if -1.17999999999999997e44 < im < 4.2e-7

   1. Initial program 18.0%

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

      1. sub0-neg 18.0%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in im around 0 88.1%

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

      1. mul-1-neg 88.1%

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

      2. *-commutative 88.1%

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

      3. distribute-lft-neg-in 88.1%

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

   6. Simplified 88.1%

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

   if 4.2e-7 < im

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 69.3%

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

      1. mul-1-neg 69.3%

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

      2. unsub-neg 69.3%

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

      3. *-commutative 69.3%

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

      4. *-commutative 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in re around 0 44.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.4 \cdot 10^{+89}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -1.18 \cdot 10^{+44}:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - re \cdot \left(re \cdot 0.75\right)}\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 11: 76.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -3.5 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -720:\\ \;\;\;\;\left|im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\right|\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -3.5e+91)
     t_0
     (if (<= im -720.0)
       (fabs (* im (* 0.5 (* re re))))
       (if (<= im 4.2e-7) (* im (- (cos re))) (- t_0 im))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -3.5e+91) {
		tmp = t_0;
	} else if (im <= -720.0) {
		tmp = fabs((im * (0.5 * (re * re))));
	} else if (im <= 4.2e-7) {
		tmp = im * -cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    if (im <= (-3.5d+91)) then
        tmp = t_0
    else if (im <= (-720.0d0)) then
        tmp = abs((im * (0.5d0 * (re * re))))
    else if (im <= 4.2d-7) then
        tmp = im * -cos(re)
    else
        tmp = t_0 - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -3.5e+91) {
		tmp = t_0;
	} else if (im <= -720.0) {
		tmp = Math.abs((im * (0.5 * (re * re))));
	} else if (im <= 4.2e-7) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -3.5e+91:
		tmp = t_0
	elif im <= -720.0:
		tmp = math.fabs((im * (0.5 * (re * re))))
	elif im <= 4.2e-7:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0 - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -3.5e+91)
		tmp = t_0;
	elseif (im <= -720.0)
		tmp = abs(Float64(im * Float64(0.5 * Float64(re * re))));
	elseif (im <= 4.2e-7)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -3.5e+91)
		tmp = t_0;
	elseif (im <= -720.0)
		tmp = abs((im * (0.5 * (re * re))));
	elseif (im <= 4.2e-7)
		tmp = im * -cos(re);
	else
		tmp = t_0 - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -3.5e+91], t$95$0, If[LessEqual[im, -720.0], N[Abs[N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 4.2e-7], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(t$95$0 - im), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.50000000000000001e91

   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 79.1%

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

      4. +-commutative 79.1%

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

      5. *-commutative 79.1%

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

      6. unpow2 79.1%

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

      7. associate-*l* 79.1%

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

   6. Simplified 79.1%

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

   7. Taylor expanded in im around 0 72.8%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around inf 72.8%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 72.8%

         \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      2. unpow2 72.8%

         \[\leadsto \left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   10. Simplified 72.8%

       \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around 0 79.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   12. Step-by-step derivation

       1. *-commutative 79.7%

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

   13. Simplified 79.7%

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

   if -3.50000000000000001e91 < im < -720

   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 65.5%

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

      4. +-commutative 65.5%

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

      5. *-commutative 65.5%

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

      6. unpow2 65.5%

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

      7. associate-*l* 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in im around 0 12.7%

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

   8. Taylor expanded in re around inf 11.8%

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

      1. unpow2 11.8%

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

      2. associate-*r* 11.8%

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

      3. metadata-eval 11.8%

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

      4. associate-*r* 11.8%

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

      5. *-commutative 11.8%

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

      6. associate-*r* 11.8%

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

      7. *-commutative 11.8%

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

      8. associate-*r* 11.8%

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

      9. *-commutative 11.8%

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

      10. associate-*l* 11.8%

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

      11. metadata-eval 11.8%

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

   10. Simplified 11.8%

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

       1. associate-*r* 11.8%

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

       2. *-commutative 11.8%

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

       3. associate-*r* 11.8%

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

       4. *-commutative 11.8%

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

       5. *-commutative 11.8%

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

       6. add-sqr-sqrt 0.3%

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

       7. sqrt-unprod 32.0%

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

       8. associate-*r* 32.0%

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

       9. associate-*r* 32.0%

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

       10. swap-sqr 31.9%

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

       11. pow2 31.9%

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

       12. pow2 31.9%

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

       13. pow-prod-up 31.9%

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

       14. metadata-eval 31.9%

           \[\leadsto \sqrt{{re}^{\color{blue}{4}} \cdot \left(\left(im \cdot 0.5\right) \cdot \left(im \cdot 0.5\right)\right)} \]

       15. swap-sqr 31.9%

           \[\leadsto \sqrt{{re}^{4} \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot 0.5\right)\right)}} \]

       16. metadata-eval 31.9%

           \[\leadsto \sqrt{{re}^{4} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{0.25}\right)} \]

   12. Applied egg-rr 31.9%

       \[\leadsto \color{blue}{\sqrt{{re}^{4} \cdot \left(\left(im \cdot im\right) \cdot 0.25\right)}} \]
   13. Step-by-step derivation

       1. metadata-eval 31.9%

          \[\leadsto \sqrt{{re}^{4} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(0.5 \cdot 0.5\right)}\right)} \]

       2. swap-sqr 31.9%

          \[\leadsto \sqrt{{re}^{4} \cdot \color{blue}{\left(\left(im \cdot 0.5\right) \cdot \left(im \cdot 0.5\right)\right)}} \]

       3. metadata-eval 31.9%

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

       4. pow-plus 31.9%

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

       5. unpow3 31.9%

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

       6. associate-*r* 31.9%

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

       7. associate-*l* 32.0%

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

       8. swap-sqr 32.0%

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

       9. associate-*l* 32.0%

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

       10. *-commutative 32.0%

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

       11. *-commutative 32.0%

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

       12. associate-*l* 32.0%

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

       13. *-commutative 32.0%

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

       14. *-commutative 32.0%

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

       15. swap-sqr 32.0%

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

       16. rem-sqrt-square 25.6%

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

       17. associate-*r* 25.6%

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

       18. *-commutative 25.6%

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

       19. associate-*r* 25.6%

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

   14. Simplified 25.6%

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

   if -720 < im < 4.2e-7

   1. Initial program 6.7%

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

      1. sub0-neg 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in im around 0 99.8%

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

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

   if 4.2e-7 < im

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 69.3%

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

      1. mul-1-neg 69.3%

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

      2. unsub-neg 69.3%

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

      3. *-commutative 69.3%

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

      4. *-commutative 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in re around 0 44.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{+91}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -720:\\ \;\;\;\;\left|im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\right|\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 12: 75.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -1.25 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.18 \cdot 10^{+44}:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - re \cdot \left(re \cdot 0.75\right)}\\ \mathbf{elif}\;im \leq 36000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -1.25e+91)
     t_0
     (if (<= im -1.18e+44)
       (/
        (- 2.25 (* (* (* re re) (* re re)) 0.5625))
        (- -1.5 (* re (* re 0.75))))
       (if (<= im 36000000.0) (* im (- (cos re))) t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -1.25e+91) {
		tmp = t_0;
	} else if (im <= -1.18e+44) {
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - (re * (re * 0.75)));
	} else if (im <= 36000000.0) {
		tmp = im * -cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    if (im <= (-1.25d+91)) then
        tmp = t_0
    else if (im <= (-1.18d+44)) then
        tmp = (2.25d0 - (((re * re) * (re * re)) * 0.5625d0)) / ((-1.5d0) - (re * (re * 0.75d0)))
    else if (im <= 36000000.0d0) then
        tmp = im * -cos(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -1.25e+91) {
		tmp = t_0;
	} else if (im <= -1.18e+44) {
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - (re * (re * 0.75)));
	} else if (im <= 36000000.0) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -1.25e+91:
		tmp = t_0
	elif im <= -1.18e+44:
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - (re * (re * 0.75)))
	elif im <= 36000000.0:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -1.25e+91)
		tmp = t_0;
	elseif (im <= -1.18e+44)
		tmp = Float64(Float64(2.25 - Float64(Float64(Float64(re * re) * Float64(re * re)) * 0.5625)) / Float64(-1.5 - Float64(re * Float64(re * 0.75))));
	elseif (im <= 36000000.0)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -1.25e+91)
		tmp = t_0;
	elseif (im <= -1.18e+44)
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - (re * (re * 0.75)));
	elseif (im <= 36000000.0)
		tmp = im * -cos(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -1.25e+91], t$95$0, If[LessEqual[im, -1.18e+44], N[(N[(2.25 - N[(N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.5625), $MachinePrecision]), $MachinePrecision] / N[(-1.5 - N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 36000000.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.2500000000000001e91 or 3.6e7 < 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 74.5%

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

      4. +-commutative 74.5%

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

      5. *-commutative 74.5%

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

      6. unpow2 74.5%

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

      7. associate-*l* 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in im around 0 60.9%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around inf 60.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left({im}^{3} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 60.9%

         \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      2. unpow2 60.9%

         \[\leadsto \left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   10. Simplified 60.9%

       \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   11. Taylor expanded in re around 0 57.6%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   12. Step-by-step derivation

       1. *-commutative 57.6%

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

   13. Simplified 57.6%

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

   if -1.2500000000000001e91 < im < -1.17999999999999997e44

   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 61.5%

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

      4. +-commutative 61.5%

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

      5. *-commutative 61.5%

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

      6. unpow2 61.5%

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

      7. associate-*l* 61.5%

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

   6. Simplified 61.5%

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

   8. Applied egg-rr 3.2%

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

      1. distribute-lft-in 3.2%

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

      2. flip-+ 38.8%

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

      3. metadata-eval 38.8%

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

      4. metadata-eval 38.8%

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

      5. metadata-eval 38.8%

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

      6. *-commutative 38.8%

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

      7. associate-*r* 38.8%

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

      8. associate-*l* 38.8%

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

      9. metadata-eval 38.8%

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

      10. *-commutative 38.8%

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

      11. associate-*r* 38.8%

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

      12. associate-*l* 38.8%

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

      13. metadata-eval 38.8%

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

      14. metadata-eval 38.8%

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

      15. *-commutative 38.8%

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

      16. associate-*r* 38.8%

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

      17. associate-*l* 38.8%

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

      18. metadata-eval 38.8%

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

   10. Applied egg-rr 38.8%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}} \]
   11. Step-by-step derivation

       1. swap-sqr 38.8%

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

       2. metadata-eval 38.8%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{0.5625}}{-1.5 - \left(re \cdot re\right) \cdot 0.75} \]

       3. associate-*l* 38.8%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - \color{blue}{re \cdot \left(re \cdot 0.75\right)}} \]

   12. Simplified 38.8%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - re \cdot \left(re \cdot 0.75\right)}} \]

   if -1.17999999999999997e44 < im < 3.6e7

   1. Initial program 18.4%

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

      1. sub0-neg 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in im around 0 88.1%

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

      1. mul-1-neg 88.1%

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

      2. *-commutative 88.1%

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

      3. distribute-lft-neg-in 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.25 \cdot 10^{+91}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -1.18 \cdot 10^{+44}:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - re \cdot \left(re \cdot 0.75\right)}\\ \mathbf{elif}\;im \leq 36000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 13: 38.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ t_1 := \frac{im \cdot im - 4 \cdot \left(t_0 \cdot t_0\right)}{\left(-im\right) - -2 \cdot t_0}\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{+53}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+138}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+220}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* -0.25 (* re re))))
        (t_1 (/ (- (* im im) (* 4.0 (* t_0 t_0))) (- (- im) (* -2.0 t_0)))))
   (if (<= im -5.6e+25)
     t_1
     (if (<= im 4.7e+53)
       (- (* re (* re (* im 0.5))) im)
       (if (<= im 9.5e+138)
         (* (+ 0.5 (* re (* re -0.25))) 27.0)
         (if (<= im 8.2e+220) (- (* im (* re (* 0.5 re))) im) t_1))))))

C

double code(double re, double im) {
	double t_0 = im * (-0.25 * (re * re));
	double t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0));
	double tmp;
	if (im <= -5.6e+25) {
		tmp = t_1;
	} else if (im <= 4.7e+53) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else if (im <= 9.5e+138) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else if (im <= 8.2e+220) {
		tmp = (im * (re * (0.5 * re))) - im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * ((-0.25d0) * (re * re))
    t_1 = ((im * im) - (4.0d0 * (t_0 * t_0))) / (-im - ((-2.0d0) * t_0))
    if (im <= (-5.6d+25)) then
        tmp = t_1
    else if (im <= 4.7d+53) then
        tmp = (re * (re * (im * 0.5d0))) - im
    else if (im <= 9.5d+138) then
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    else if (im <= 8.2d+220) then
        tmp = (im * (re * (0.5d0 * re))) - im
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (-0.25 * (re * re));
	double t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0));
	double tmp;
	if (im <= -5.6e+25) {
		tmp = t_1;
	} else if (im <= 4.7e+53) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else if (im <= 9.5e+138) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else if (im <= 8.2e+220) {
		tmp = (im * (re * (0.5 * re))) - im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (-0.25 * (re * re))
	t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0))
	tmp = 0
	if im <= -5.6e+25:
		tmp = t_1
	elif im <= 4.7e+53:
		tmp = (re * (re * (im * 0.5))) - im
	elif im <= 9.5e+138:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	elif im <= 8.2e+220:
		tmp = (im * (re * (0.5 * re))) - im
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(-0.25 * Float64(re * re)))
	t_1 = Float64(Float64(Float64(im * im) - Float64(4.0 * Float64(t_0 * t_0))) / Float64(Float64(-im) - Float64(-2.0 * t_0)))
	tmp = 0.0
	if (im <= -5.6e+25)
		tmp = t_1;
	elseif (im <= 4.7e+53)
		tmp = Float64(Float64(re * Float64(re * Float64(im * 0.5))) - im);
	elseif (im <= 9.5e+138)
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	elseif (im <= 8.2e+220)
		tmp = Float64(Float64(im * Float64(re * Float64(0.5 * re))) - im);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (-0.25 * (re * re));
	t_1 = ((im * im) - (4.0 * (t_0 * t_0))) / (-im - (-2.0 * t_0));
	tmp = 0.0;
	if (im <= -5.6e+25)
		tmp = t_1;
	elseif (im <= 4.7e+53)
		tmp = (re * (re * (im * 0.5))) - im;
	elseif (im <= 9.5e+138)
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	elseif (im <= 8.2e+220)
		tmp = (im * (re * (0.5 * re))) - im;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(im * im), $MachinePrecision] - N[(4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-im) - N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.6e+25], t$95$1, If[LessEqual[im, 4.7e+53], N[(N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 9.5e+138], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision], If[LessEqual[im, 8.2e+220], N[(N[(im * N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -5.6000000000000003e25 or 8.19999999999999962e220 < 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 82.3%

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

      4. +-commutative 82.3%

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

      5. *-commutative 82.3%

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

      6. unpow2 82.3%

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

      7. associate-*l* 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in im around 0 18.0%

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

      1. distribute-lft-in 18.0%

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

      2. flip-+ 39.1%

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

   9. Applied egg-rr 39.1%

      \[\leadsto \color{blue}{\frac{\left(im \cdot -1\right) \cdot \left(im \cdot -1\right) - \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right) \cdot \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. swap-sqr 39.1%

          \[\leadsto \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(-1 \cdot -1\right)} - \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right) \cdot \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       2. metadata-eval 39.1%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot \color{blue}{1} - \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right) \cdot \left(-2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       3. swap-sqr 39.1%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - \color{blue}{\left(-2 \cdot -2\right) \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       4. metadata-eval 39.1%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - \color{blue}{4} \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{im \cdot -1 - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       5. *-commutative 39.1%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\color{blue}{-1 \cdot im} - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

       6. mul-1-neg 39.1%

          \[\leadsto \frac{\left(im \cdot im\right) \cdot 1 - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\color{blue}{\left(-im\right)} - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

   11. Simplified 39.1%

       \[\leadsto \color{blue}{\frac{\left(im \cdot im\right) \cdot 1 - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\left(-im\right) - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}} \]

   if -5.6000000000000003e25 < im < 4.69999999999999976e53

   1. Initial program 21.4%

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

      1. sub0-neg 21.4%

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

   3. Simplified 21.4%

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

   4. Taylor expanded in im around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. *-commutative 85.1%

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

      3. distribute-lft-neg-in 85.1%

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

   6. Simplified 85.1%

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

   7. Taylor expanded in re around 0 52.0%

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

      1. neg-mul-1 52.0%

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

      2. +-commutative 52.0%

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

      3. unsub-neg 52.0%

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

      4. *-commutative 52.0%

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

      5. associate-*l* 52.0%

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

      6. metadata-eval 52.0%

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

      7. distribute-rgt-neg-in 52.0%

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

      8. *-commutative 52.0%

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

      9. unpow2 52.0%

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

      10. associate-*l* 52.1%

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

      11. *-commutative 52.1%

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

      12. distribute-rgt-neg-in 52.1%

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

      13. metadata-eval 52.1%

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

   9. Simplified 52.1%

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

   if 4.69999999999999976e53 < im < 9.49999999999999998e138

   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 41.2%

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

      4. +-commutative 41.2%

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

      5. *-commutative 41.2%

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

      6. unpow2 41.2%

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

      7. associate-*l* 41.2%

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

   6. Simplified 41.2%

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

   8. Applied egg-rr 37.1%

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

   if 9.49999999999999998e138 < im < 8.19999999999999962e220

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 77.3%

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

      4. +-commutative 77.3%

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

      5. *-commutative 77.3%

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

      6. unpow2 77.3%

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

      7. associate-*l* 77.3%

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

   6. Simplified 77.3%

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

   7. Taylor expanded in im around 0 77.3%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around 0 38.9%

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

      1. distribute-lft-in 38.9%

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

      2. *-commutative 38.9%

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

      3. unpow2 38.9%

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

      4. distribute-lft-in 38.9%

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

      5. associate-*r* 38.9%

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

      6. metadata-eval 38.9%

         \[\leadsto \color{blue}{-1} \cdot im + -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \]

      7. *-commutative 38.9%

         \[\leadsto -1 \cdot im + -2 \cdot \color{blue}{\left(\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot im\right)} \]

      8. associate-*l* 38.9%

         \[\leadsto -1 \cdot im + -2 \cdot \color{blue}{\left(-0.25 \cdot \left(\left(re \cdot re\right) \cdot im\right)\right)} \]

      9. associate-*r* 38.9%

         \[\leadsto -1 \cdot im + \color{blue}{\left(-2 \cdot -0.25\right) \cdot \left(\left(re \cdot re\right) \cdot im\right)} \]

      10. metadata-eval 38.9%

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

      11. unpow2 38.9%

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

      12. +-commutative 38.9%

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

      13. mul-1-neg 38.9%

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

      14. unsub-neg 38.9%

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

   10. Simplified 38.9%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{im \cdot im - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\left(-im\right) - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{+53}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+138}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+220}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot im - 4 \cdot \left(\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)\right)}{\left(-im\right) - -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right)}\\ \end{array} \]

Alternative 14: 35.9% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.75\right)\\ \mathbf{if}\;re \leq 6.6 \cdot 10^{+81}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right) - im\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+154}:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.75))))
   (if (<= re 6.6e+81)
     (- (* im (* re (* 0.5 re))) im)
     (if (<= re 1.55e+154)
       (/ (- 2.25 (* (* (* re re) (* re re)) 0.5625)) (- -1.5 t_0))
       t_0))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.75);
	double tmp;
	if (re <= 6.6e+81) {
		tmp = (im * (re * (0.5 * re))) - im;
	} else if (re <= 1.55e+154) {
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - t_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 * 0.75d0)
    if (re <= 6.6d+81) then
        tmp = (im * (re * (0.5d0 * re))) - im
    else if (re <= 1.55d+154) then
        tmp = (2.25d0 - (((re * re) * (re * re)) * 0.5625d0)) / ((-1.5d0) - t_0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.75);
	double tmp;
	if (re <= 6.6e+81) {
		tmp = (im * (re * (0.5 * re))) - im;
	} else if (re <= 1.55e+154) {
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - t_0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.75)
	tmp = 0
	if re <= 6.6e+81:
		tmp = (im * (re * (0.5 * re))) - im
	elif re <= 1.55e+154:
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - t_0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.75))
	tmp = 0.0
	if (re <= 6.6e+81)
		tmp = Float64(Float64(im * Float64(re * Float64(0.5 * re))) - im);
	elseif (re <= 1.55e+154)
		tmp = Float64(Float64(2.25 - Float64(Float64(Float64(re * re) * Float64(re * re)) * 0.5625)) / Float64(-1.5 - t_0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.75);
	tmp = 0.0;
	if (re <= 6.6e+81)
		tmp = (im * (re * (0.5 * re))) - im;
	elseif (re <= 1.55e+154)
		tmp = (2.25 - (((re * re) * (re * re)) * 0.5625)) / (-1.5 - t_0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 6.6e+81], N[(N[(im * N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[re, 1.55e+154], N[(N[(2.25 - N[(N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.5625), $MachinePrecision]), $MachinePrecision] / N[(-1.5 - t$95$0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if re < 6.6e81

   1. Initial program 57.7%

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

      1. sub0-neg 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in re around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-*r* 2.7%

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

      3. distribute-rgt-out 45.7%

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

      4. +-commutative 45.7%

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

      5. *-commutative 45.7%

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

      6. unpow2 45.7%

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

      7. associate-*l* 45.7%

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

   6. Simplified 45.7%

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

   7. Taylor expanded in im around 0 61.2%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around 0 39.9%

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

      1. distribute-lft-in 39.9%

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

      2. *-commutative 39.9%

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

      3. unpow2 39.9%

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

      4. distribute-lft-in 39.9%

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

      5. associate-*r* 39.9%

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

      6. metadata-eval 39.9%

         \[\leadsto \color{blue}{-1} \cdot im + -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \]

      7. *-commutative 39.9%

         \[\leadsto -1 \cdot im + -2 \cdot \color{blue}{\left(\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot im\right)} \]

      8. associate-*l* 39.9%

         \[\leadsto -1 \cdot im + -2 \cdot \color{blue}{\left(-0.25 \cdot \left(\left(re \cdot re\right) \cdot im\right)\right)} \]

      9. associate-*r* 39.9%

         \[\leadsto -1 \cdot im + \color{blue}{\left(-2 \cdot -0.25\right) \cdot \left(\left(re \cdot re\right) \cdot im\right)} \]

      10. metadata-eval 39.9%

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

      11. unpow2 39.9%

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

      12. +-commutative 39.9%

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

      13. mul-1-neg 39.9%

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

      14. unsub-neg 39.9%

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

   10. Simplified 39.9%

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

   if 6.6e81 < re < 1.5500000000000001e154

   1. Initial program 49.3%

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

      1. sub0-neg 49.3%

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

   3. Simplified 49.3%

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

   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 14.0%

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

      4. +-commutative 14.0%

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

      5. *-commutative 14.0%

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

      6. unpow2 14.0%

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

      7. associate-*l* 14.0%

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

   6. Simplified 14.0%

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

   8. Applied egg-rr 4.1%

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

      1. distribute-lft-in 4.1%

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

      2. flip-+ 36.3%

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

      3. metadata-eval 36.3%

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

      4. metadata-eval 36.3%

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

      5. metadata-eval 36.3%

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

      6. *-commutative 36.3%

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

      7. associate-*r* 36.3%

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

      8. associate-*l* 36.3%

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

      9. metadata-eval 36.3%

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

      10. *-commutative 36.3%

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

      11. associate-*r* 36.3%

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

      12. associate-*l* 36.3%

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

      13. metadata-eval 36.3%

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

      14. metadata-eval 36.3%

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

      15. *-commutative 36.3%

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

      16. associate-*r* 36.3%

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

      17. associate-*l* 36.3%

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

      18. metadata-eval 36.3%

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

   10. Applied egg-rr 36.3%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}} \]
   11. Step-by-step derivation

       1. swap-sqr 36.3%

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

       2. metadata-eval 36.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{0.5625}}{-1.5 - \left(re \cdot re\right) \cdot 0.75} \]

       3. associate-*l* 36.3%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - \color{blue}{re \cdot \left(re \cdot 0.75\right)}} \]

   12. Simplified 36.3%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - re \cdot \left(re \cdot 0.75\right)}} \]

   if 1.5500000000000001e154 < re

   1. Initial program 62.5%

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

      1. sub0-neg 62.5%

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

   3. Simplified 62.5%

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

   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 28.2%

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

      4. +-commutative 28.2%

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

      5. *-commutative 28.2%

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

      6. unpow2 28.2%

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

      7. associate-*l* 28.2%

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

   6. Simplified 28.2%

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

   8. Applied egg-rr 28.8%

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

   9. Taylor expanded in re around inf 28.8%

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

       1. unpow2 28.8%

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

       2. *-commutative 28.8%

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

       3. associate-*l* 28.8%

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

   11. Simplified 28.8%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.6 \cdot 10^{+81}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right) - im\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+154}:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.5625}{-1.5 - re \cdot \left(re \cdot 0.75\right)}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \]

Alternative 15: 36.6% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{if}\;im \leq -126000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{+52}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+139}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* re (* 0.5 re)))))
   (if (<= im -126000000.0)
     t_0
     (if (<= im 9.8e+52)
       (- im)
       (if (<= im 4.3e+139) (* (+ 0.5 (* re (* re -0.25))) 27.0) t_0)))))

C

double code(double re, double im) {
	double t_0 = im * (re * (0.5 * re));
	double tmp;
	if (im <= -126000000.0) {
		tmp = t_0;
	} else if (im <= 9.8e+52) {
		tmp = -im;
	} else if (im <= 4.3e+139) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} 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 = im * (re * (0.5d0 * re))
    if (im <= (-126000000.0d0)) then
        tmp = t_0
    else if (im <= 9.8d+52) then
        tmp = -im
    else if (im <= 4.3d+139) then
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (re * (0.5 * re));
	double tmp;
	if (im <= -126000000.0) {
		tmp = t_0;
	} else if (im <= 9.8e+52) {
		tmp = -im;
	} else if (im <= 4.3e+139) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (re * (0.5 * re))
	tmp = 0
	if im <= -126000000.0:
		tmp = t_0
	elif im <= 9.8e+52:
		tmp = -im
	elif im <= 4.3e+139:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(re * Float64(0.5 * re)))
	tmp = 0.0
	if (im <= -126000000.0)
		tmp = t_0;
	elseif (im <= 9.8e+52)
		tmp = Float64(-im);
	elseif (im <= 4.3e+139)
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (re * (0.5 * re));
	tmp = 0.0;
	if (im <= -126000000.0)
		tmp = t_0;
	elseif (im <= 9.8e+52)
		tmp = -im;
	elseif (im <= 4.3e+139)
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -126000000.0], t$95$0, If[LessEqual[im, 9.8e+52], (-im), If[LessEqual[im, 4.3e+139], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.26e8 or 4.2999999999999998e139 < 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 79.8%

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

      4. +-commutative 79.8%

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

      5. *-commutative 79.8%

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

      6. unpow2 79.8%

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

      7. associate-*l* 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in im around 0 23.7%

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

   8. Taylor expanded in re around inf 21.3%

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

      1. unpow2 21.3%

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

      2. associate-*r* 21.3%

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

      3. metadata-eval 21.3%

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

      4. associate-*r* 21.3%

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

      5. *-commutative 21.3%

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

      6. associate-*r* 21.3%

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

      7. *-commutative 21.3%

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

      8. associate-*r* 21.3%

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

      9. *-commutative 21.3%

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

      10. associate-*l* 21.3%

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

      11. metadata-eval 21.3%

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

   10. Simplified 21.3%

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

   if -1.26e8 < im < 9.79999999999999993e52

   1. Initial program 16.6%

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

      1. sub0-neg 16.6%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in im around 0 90.1%

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

      1. mul-1-neg 90.1%

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

      2. *-commutative 90.1%

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

      3. distribute-lft-neg-in 90.1%

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

   6. Simplified 90.1%

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

   7. Taylor expanded in re around 0 55.0%

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

      1. neg-mul-1 55.0%

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

   9. Simplified 55.0%

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

   if 9.79999999999999993e52 < im < 4.2999999999999998e139

   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 41.2%

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

      4. +-commutative 41.2%

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

      5. *-commutative 41.2%

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

      6. unpow2 41.2%

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

      7. associate-*l* 41.2%

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

   6. Simplified 41.2%

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

   8. Applied egg-rr 37.1%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -126000000:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{+52}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+139}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right)\\ \end{array} \]

Alternative 16: 35.8% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+238}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right) - im\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+279}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.4e+238)
   (- (* im (* re (* 0.5 re))) im)
   (if (<= re 1.1e+279)
     (* re (* re 0.75))
     (* (+ 0.5 (* re (* re -0.25))) 27.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.4e+238) {
		tmp = (im * (re * (0.5 * re))) - im;
	} else if (re <= 1.1e+279) {
		tmp = re * (re * 0.75);
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.4d+238) then
        tmp = (im * (re * (0.5d0 * re))) - im
    else if (re <= 1.1d+279) then
        tmp = re * (re * 0.75d0)
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.4e+238) {
		tmp = (im * (re * (0.5 * re))) - im;
	} else if (re <= 1.1e+279) {
		tmp = re * (re * 0.75);
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.4e+238:
		tmp = (im * (re * (0.5 * re))) - im
	elif re <= 1.1e+279:
		tmp = re * (re * 0.75)
	else:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.4e+238)
		tmp = Float64(Float64(im * Float64(re * Float64(0.5 * re))) - im);
	elseif (re <= 1.1e+279)
		tmp = Float64(re * Float64(re * 0.75));
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.4e+238)
		tmp = (im * (re * (0.5 * re))) - im;
	elseif (re <= 1.1e+279)
		tmp = re * (re * 0.75);
	else
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.4e+238], N[(N[(im * N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[re, 1.1e+279], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.39999999999999995e238

   1. Initial program 56.9%

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

      1. sub0-neg 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in re around 0 2.5%

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

      1. *-commutative 2.5%

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

      2. associate-*r* 2.5%

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

      3. distribute-rgt-out 42.8%

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

      4. +-commutative 42.8%

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

      5. *-commutative 42.8%

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

      6. unpow2 42.8%

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

      7. associate-*l* 42.8%

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

   6. Simplified 42.8%

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

   7. Taylor expanded in im around 0 56.2%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around 0 37.7%

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

      1. distribute-lft-in 37.7%

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

      2. *-commutative 37.7%

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

      3. unpow2 37.7%

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

      4. distribute-lft-in 37.7%

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

      5. associate-*r* 37.7%

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

      6. metadata-eval 37.7%

         \[\leadsto \color{blue}{-1} \cdot im + -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \]

      7. *-commutative 37.7%

         \[\leadsto -1 \cdot im + -2 \cdot \color{blue}{\left(\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot im\right)} \]

      8. associate-*l* 37.7%

         \[\leadsto -1 \cdot im + -2 \cdot \color{blue}{\left(-0.25 \cdot \left(\left(re \cdot re\right) \cdot im\right)\right)} \]

      9. associate-*r* 37.7%

         \[\leadsto -1 \cdot im + \color{blue}{\left(-2 \cdot -0.25\right) \cdot \left(\left(re \cdot re\right) \cdot im\right)} \]

      10. metadata-eval 37.7%

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

      11. unpow2 37.7%

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

      12. +-commutative 37.7%

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

      13. mul-1-neg 37.7%

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

      14. unsub-neg 37.7%

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

   10. Simplified 37.7%

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

   if 1.39999999999999995e238 < re < 1.1e279

   1. Initial program 90.3%

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

      1. sub0-neg 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 28.7%

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

      4. +-commutative 28.7%

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

      5. *-commutative 28.7%

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

      6. unpow2 28.7%

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

      7. associate-*l* 28.7%

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

   6. Simplified 28.7%

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

   8. Applied egg-rr 57.6%

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

   9. Taylor expanded in re around inf 57.6%

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

       1. unpow2 57.6%

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

       2. *-commutative 57.6%

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

       3. associate-*l* 57.6%

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

   11. Simplified 57.6%

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

   if 1.1e279 < re

   1. Initial program 37.6%

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

      1. sub0-neg 37.6%

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

   3. Simplified 37.6%

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

   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 0.0%

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

      4. +-commutative 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. associate-*l* 0.0%

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

   6. Simplified 0.0%

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

   8. Applied egg-rr 34.1%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+238}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right) - im\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+279}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \end{array} \]

Alternative 17: 35.7% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{+54}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;im \leq 1.86 \cdot 10^{+139}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right) - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9e+54)
   (- (* re (* re (* im 0.5))) im)
   (if (<= im 1.86e+139)
     (* (+ 0.5 (* re (* re -0.25))) 27.0)
     (- (* im (* re (* 0.5 re))) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9e+54) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else if (im <= 1.86e+139) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else {
		tmp = (im * (re * (0.5 * re))) - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9d+54) then
        tmp = (re * (re * (im * 0.5d0))) - im
    else if (im <= 1.86d+139) then
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    else
        tmp = (im * (re * (0.5d0 * re))) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9e+54) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else if (im <= 1.86e+139) {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	} else {
		tmp = (im * (re * (0.5 * re))) - im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9e+54:
		tmp = (re * (re * (im * 0.5))) - im
	elif im <= 1.86e+139:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	else:
		tmp = (im * (re * (0.5 * re))) - im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9e+54)
		tmp = Float64(Float64(re * Float64(re * Float64(im * 0.5))) - im);
	elseif (im <= 1.86e+139)
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	else
		tmp = Float64(Float64(im * Float64(re * Float64(0.5 * re))) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9e+54)
		tmp = (re * (re * (im * 0.5))) - im;
	elseif (im <= 1.86e+139)
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	else
		tmp = (im * (re * (0.5 * re))) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9e+54], N[(N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 1.86e+139], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision], N[(N[(im * N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8.99999999999999968e54

   1. Initial program 45.8%

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

      1. sub0-neg 45.8%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in im around 0 60.3%

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

      1. mul-1-neg 60.3%

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

      2. *-commutative 60.3%

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

      3. distribute-lft-neg-in 60.3%

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

   6. Simplified 60.3%

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

   7. Taylor expanded in re around 0 39.5%

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

      1. neg-mul-1 39.5%

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

      2. +-commutative 39.5%

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

      3. unsub-neg 39.5%

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

      4. *-commutative 39.5%

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

      5. associate-*l* 39.5%

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

      6. metadata-eval 39.5%

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

      7. distribute-rgt-neg-in 39.5%

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

      8. *-commutative 39.5%

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

      9. unpow2 39.5%

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

      10. associate-*l* 39.6%

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

      11. *-commutative 39.6%

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

      12. distribute-rgt-neg-in 39.6%

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

      13. metadata-eval 39.6%

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

   9. Simplified 39.6%

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

   if 8.99999999999999968e54 < im < 1.8600000000000001e139

   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 41.2%

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

      4. +-commutative 41.2%

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

      5. *-commutative 41.2%

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

      6. unpow2 41.2%

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

      7. associate-*l* 41.2%

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

   6. Simplified 41.2%

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

   8. Applied egg-rr 37.1%

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

   if 1.8600000000000001e139 < 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 87.2%

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

      4. +-commutative 87.2%

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

      5. *-commutative 87.2%

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

      6. unpow2 87.2%

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

      7. associate-*l* 87.2%

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

   6. Simplified 87.2%

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

   7. Taylor expanded in im around 0 87.2%

      \[\leadsto \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   8. Taylor expanded in im around 0 39.5%

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

      1. distribute-lft-in 39.5%

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

      2. *-commutative 39.5%

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

      3. unpow2 39.5%

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

      4. distribute-lft-in 39.5%

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

      5. associate-*r* 39.5%

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

      6. metadata-eval 39.5%

         \[\leadsto \color{blue}{-1} \cdot im + -2 \cdot \left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\right) \]

      7. *-commutative 39.5%

         \[\leadsto -1 \cdot im + -2 \cdot \color{blue}{\left(\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot im\right)} \]

      8. associate-*l* 39.5%

         \[\leadsto -1 \cdot im + -2 \cdot \color{blue}{\left(-0.25 \cdot \left(\left(re \cdot re\right) \cdot im\right)\right)} \]

      9. associate-*r* 39.5%

         \[\leadsto -1 \cdot im + \color{blue}{\left(-2 \cdot -0.25\right) \cdot \left(\left(re \cdot re\right) \cdot im\right)} \]

      10. metadata-eval 39.5%

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

      11. unpow2 39.5%

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

      12. +-commutative 39.5%

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

      13. mul-1-neg 39.5%

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

      14. unsub-neg 39.5%

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

   10. Simplified 39.5%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{+54}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;im \leq 1.86 \cdot 10^{+139}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right) - im\\ \end{array} \]

Alternative 18: 36.8% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -54000000 \lor \neg \left(im \leq 4.6 \cdot 10^{+41}\right):\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -54000000.0) (not (<= im 4.6e+41)))
   (* im (* re (* 0.5 re)))
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -54000000.0) || !(im <= 4.6e+41)) {
		tmp = im * (re * (0.5 * re));
	} 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 <= (-54000000.0d0)) .or. (.not. (im <= 4.6d+41))) then
        tmp = im * (re * (0.5d0 * re))
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -54000000.0) || !(im <= 4.6e+41)) {
		tmp = im * (re * (0.5 * re));
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -54000000.0) or not (im <= 4.6e+41):
		tmp = im * (re * (0.5 * re))
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -54000000.0) || !(im <= 4.6e+41))
		tmp = Float64(im * Float64(re * Float64(0.5 * re)));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -54000000.0) || ~((im <= 4.6e+41)))
		tmp = im * (re * (0.5 * re));
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -54000000.0], N[Not[LessEqual[im, 4.6e+41]], $MachinePrecision]], N[(im * N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -5.4e7 or 4.5999999999999997e41 < 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 75.0%

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

      4. +-commutative 75.0%

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

      5. *-commutative 75.0%

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

      6. unpow2 75.0%

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

      7. associate-*l* 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in im around 0 21.2%

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

   8. Taylor expanded in re around inf 19.0%

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

      1. unpow2 19.0%

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

      2. associate-*r* 19.0%

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

      3. metadata-eval 19.0%

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

      4. associate-*r* 19.0%

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

      5. *-commutative 19.0%

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

      6. associate-*r* 19.0%

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

      7. *-commutative 19.0%

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

      8. associate-*r* 19.0%

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

      9. *-commutative 19.0%

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

      10. associate-*l* 19.0%

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

      11. metadata-eval 19.0%

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

   10. Simplified 19.0%

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

   if -5.4e7 < im < 4.5999999999999997e41

   1. Initial program 15.2%

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

      1. sub0-neg 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in im around 0 91.5%

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

      1. mul-1-neg 91.5%

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

      2. *-commutative 91.5%

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

      3. distribute-lft-neg-in 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in re around 0 55.9%

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

      1. neg-mul-1 55.9%

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

   9. Simplified 55.9%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -54000000 \lor \neg \left(im \leq 4.6 \cdot 10^{+41}\right):\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 19: 31.7% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.8 \cdot 10^{+165}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.8e+165) (- im) (* re (* re 0.75))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.8e+165) {
		tmp = -im;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 4.8d+165) then
        tmp = -im
    else
        tmp = re * (re * 0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.8e+165) {
		tmp = -im;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.8e+165:
		tmp = -im
	else:
		tmp = re * (re * 0.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.8e+165)
		tmp = Float64(-im);
	else
		tmp = Float64(re * Float64(re * 0.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.8e+165)
		tmp = -im;
	else
		tmp = re * (re * 0.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.8e+165], (-im), N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.80000000000000001e165

   1. Initial program 56.8%

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

      1. sub0-neg 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in im around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. *-commutative 48.7%

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

      3. distribute-lft-neg-in 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in re around 0 32.2%

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

      1. neg-mul-1 32.2%

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

   9. Simplified 32.2%

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

   if 4.80000000000000001e165 < re

   1. Initial program 66.0%

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

      1. sub0-neg 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in re around 0 0.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   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 32.0%

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

      4. +-commutative 32.0%

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

      5. *-commutative 32.0%

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

      6. unpow2 32.0%

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

      7. associate-*l* 32.0%

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

   6. Simplified 32.0%

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

   8. Applied egg-rr 32.6%

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

   9. Taylor expanded in re around inf 32.6%

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

       1. unpow2 32.6%

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

       2. *-commutative 32.6%

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

       3. associate-*l* 32.6%

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

   11. Simplified 32.6%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.8 \cdot 10^{+165}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \]

Alternative 20: 29.8% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 57.6%

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

   1. sub0-neg 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in im around 0 48.3%

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

   1. mul-1-neg 48.3%

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

   2. *-commutative 48.3%

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

   3. distribute-lft-neg-in 48.3%

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

6. Simplified 48.3%

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

7. Taylor expanded in re around 0 29.9%

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

   1. neg-mul-1 29.9%

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

9. Simplified 29.9%

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

10. Final simplification 29.9%

    \[\leadsto -im \]

Alternative 21: 2.9% accurate, 309.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

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

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 57.6%

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

   1. sub0-neg 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in re around 0 2.4%

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

   1. *-commutative 2.4%

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

   2. associate-*r* 2.4%

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

   3. distribute-rgt-out 41.9%

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

   4. +-commutative 41.9%

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

   5. *-commutative 41.9%

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

   6. unpow2 41.9%

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

   7. associate-*l* 41.9%

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

6. Simplified 41.9%

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

8. Applied egg-rr 9.4%

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

9. Taylor expanded in re around 0 2.6%

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

10. Final simplification 2.6%

    \[\leadsto -1.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023257 
(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))))