math.sin on complex, imaginary part

Percentage Accurate: 52.8% → 99.8%

Time: 8.9s

Alternatives: 16

Speedup: 2.9×

• 47.8% 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: 52.8% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.2 \lor \neg \left(t_0 \leq 10^{-7}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.2) (not (<= t_0 1e-7)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 1e-7)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.2d0)) .or. (.not. (t_0 <= 1d-7))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 1e-7)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.2) or not (t_0 <= 1e-7):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.2) || !(t_0 <= 1e-7))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.2) || ~((t_0 <= 1e-7)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 1e-7]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.20000000000000001 or 9.9999999999999995e-8 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 9.9999999999999995e-8

   1. Initial program 8.0%

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

      1. sub0-neg 8.0%

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

   3. Simplified 8.0%

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

   4. Taylor expanded in im around 0 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2 \lor \neg \left(e^{-im} - e^{im} \leq 10^{-7}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 95.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -6.8 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0225:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;im \leq 0.043 \lor \neg \left(im \leq 5.8 \cdot 10^{+102}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im)))
        (t_1 (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -6.8e+95)
     t_1
     (if (<= im -0.0225)
       (* 0.5 t_0)
       (if (or (<= im 0.043) (not (<= im 5.8e+102)))
         t_1
         (* t_0 (+ 0.5 (* re (* re -0.25)))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -6.8e+95) {
		tmp = t_1;
	} else if (im <= -0.0225) {
		tmp = 0.5 * t_0;
	} else if ((im <= 0.043) || !(im <= 5.8e+102)) {
		tmp = t_1;
	} else {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-6.8d+95)) then
        tmp = t_1
    else if (im <= (-0.0225d0)) then
        tmp = 0.5d0 * t_0
    else if ((im <= 0.043d0) .or. (.not. (im <= 5.8d+102))) then
        tmp = t_1
    else
        tmp = t_0 * (0.5d0 + (re * (re * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -6.8e+95) {
		tmp = t_1;
	} else if (im <= -0.0225) {
		tmp = 0.5 * t_0;
	} else if ((im <= 0.043) || !(im <= 5.8e+102)) {
		tmp = t_1;
	} else {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -6.8e+95:
		tmp = t_1
	elif im <= -0.0225:
		tmp = 0.5 * t_0
	elif (im <= 0.043) or not (im <= 5.8e+102):
		tmp = t_1
	else:
		tmp = t_0 * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -6.8e+95)
		tmp = t_1;
	elseif (im <= -0.0225)
		tmp = Float64(0.5 * t_0);
	elseif ((im <= 0.043) || !(im <= 5.8e+102))
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -6.8e+95)
		tmp = t_1;
	elseif (im <= -0.0225)
		tmp = 0.5 * t_0;
	elseif ((im <= 0.043) || ~((im <= 5.8e+102)))
		tmp = t_1;
	else
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.8e+95], t$95$1, If[LessEqual[im, -0.0225], N[(0.5 * t$95$0), $MachinePrecision], If[Or[LessEqual[im, 0.043], N[Not[LessEqual[im, 5.8e+102]], $MachinePrecision]], t$95$1, N[(t$95$0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.80000000000000043e95 or -0.022499999999999999 < im < 0.042999999999999997 or 5.8000000000000005e102 < im

   1. Initial program 46.4%

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

      1. sub0-neg 46.4%

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

   3. Simplified 46.4%

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

   4. Taylor expanded in im around 0 99.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   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 -6.80000000000000043e95 < im < -0.022499999999999999

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 83.9%

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

   if 0.042999999999999997 < im < 5.8000000000000005e102

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 7.5%

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

      1. *-commutative 7.5%

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

      2. associate-*r* 7.5%

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

      3. distribute-rgt-out 92.1%

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

      4. +-commutative 92.1%

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

      5. *-commutative 92.1%

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

      6. unpow2 92.1%

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

      7. associate-*l* 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{+95}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -0.0225:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.043 \lor \neg \left(im \leq 5.8 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 3: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{+95} \lor \neg \left(im \leq -0.04 \lor \neg \left(im \leq 0.016\right) \land im \leq 5.8 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -6.8e+95)
         (not (or (<= im -0.04) (and (not (<= im 0.016)) (<= im 5.8e+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 <= -6.8e+95) || !((im <= -0.04) || (!(im <= 0.016) && (im <= 5.8e+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 <= (-6.8d+95)) .or. (.not. (im <= (-0.04d0)) .or. (.not. (im <= 0.016d0)) .and. (im <= 5.8d+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 <= -6.8e+95) || !((im <= -0.04) || (!(im <= 0.016) && (im <= 5.8e+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 <= -6.8e+95) or not ((im <= -0.04) or (not (im <= 0.016) and (im <= 5.8e+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 <= -6.8e+95) || !((im <= -0.04) || (!(im <= 0.016) && (im <= 5.8e+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 <= -6.8e+95) || ~(((im <= -0.04) || (~((im <= 0.016)) && (im <= 5.8e+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, -6.8e+95], N[Not[Or[LessEqual[im, -0.04], And[N[Not[LessEqual[im, 0.016]], $MachinePrecision], LessEqual[im, 5.8e+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 < -6.80000000000000043e95 or -0.0400000000000000008 < im < 0.016 or 5.8000000000000005e102 < im

   1. Initial program 46.4%

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

      1. sub0-neg 46.4%

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

   3. Simplified 46.4%

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

   4. Taylor expanded in im around 0 99.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   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 -6.80000000000000043e95 < im < -0.0400000000000000008 or 0.016 < im < 5.8000000000000005e102

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 76.2%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{+95} \lor \neg \left(im \leq -0.04 \lor \neg \left(im \leq 0.016\right) \land im \leq 5.8 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4: 89.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;im \leq -0.0004:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im)))))
   (if (<= im -0.0004)
     t_0
     (if (<= im 4.7e-5)
       (* im (- (cos re)))
       (if (<= im 1.8e+150)
         t_0
         (/
          (* (cos re) (- 5.960464477539063e-8 (* im im)))
          (+ im 0.000244140625)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double tmp;
	if (im <= -0.0004) {
		tmp = t_0;
	} else if (im <= 4.7e-5) {
		tmp = im * -cos(re);
	} else if (im <= 1.8e+150) {
		tmp = t_0;
	} else {
		tmp = (cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	}
	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 <= (-0.0004d0)) then
        tmp = t_0
    else if (im <= 4.7d-5) then
        tmp = im * -cos(re)
    else if (im <= 1.8d+150) then
        tmp = t_0
    else
        tmp = (cos(re) * (5.960464477539063d-8 - (im * im))) / (im + 0.000244140625d0)
    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 <= -0.0004) {
		tmp = t_0;
	} else if (im <= 4.7e-5) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.8e+150) {
		tmp = t_0;
	} else {
		tmp = (Math.cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	tmp = 0
	if im <= -0.0004:
		tmp = t_0
	elif im <= 4.7e-5:
		tmp = im * -math.cos(re)
	elif im <= 1.8e+150:
		tmp = t_0
	else:
		tmp = (math.cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (im <= -0.0004)
		tmp = t_0;
	elseif (im <= 4.7e-5)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.8e+150)
		tmp = t_0;
	else
		tmp = Float64(Float64(cos(re) * Float64(5.960464477539063e-8 - Float64(im * im))) / Float64(im + 0.000244140625));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	tmp = 0.0;
	if (im <= -0.0004)
		tmp = t_0;
	elseif (im <= 4.7e-5)
		tmp = im * -cos(re);
	elseif (im <= 1.8e+150)
		tmp = t_0;
	else
		tmp = (cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	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, -0.0004], t$95$0, If[LessEqual[im, 4.7e-5], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.8e+150], t$95$0, N[(N[(N[Cos[re], $MachinePrecision] * N[(5.960464477539063e-8 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + 0.000244140625), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.00000000000000019e-4 or 4.69999999999999972e-5 < im < 1.79999999999999993e150

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 77.8%

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

   if -4.00000000000000019e-4 < im < 4.69999999999999972e-5

   1. Initial program 7.4%

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

      1. sub0-neg 7.4%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. distribute-lft-neg-in 99.9%

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

   6. Simplified 99.9%

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

   if 1.79999999999999993e150 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.5%

      \[\leadsto \cos re \cdot \left(\color{blue}{0.000244140625} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.5%

         \[\leadsto \color{blue}{\left(0.000244140625 - im\right) \cdot \cos re} \]

      2. flip-- 96.8%

         \[\leadsto \color{blue}{\frac{0.000244140625 \cdot 0.000244140625 - im \cdot im}{0.000244140625 + im}} \cdot \cos re \]

      3. associate-*l/ 96.8%

         \[\leadsto \color{blue}{\frac{\left(0.000244140625 \cdot 0.000244140625 - im \cdot im\right) \cdot \cos re}{0.000244140625 + im}} \]

      4. metadata-eval 96.8%

         \[\leadsto \frac{\left(\color{blue}{5.960464477539063 \cdot 10^{-8}} - im \cdot im\right) \cdot \cos re}{0.000244140625 + im} \]

      5. +-commutative 96.8%

         \[\leadsto \frac{\left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + 0.000244140625}} \]

   10. Applied egg-rr 96.8%

       \[\leadsto \color{blue}{\frac{\left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right) \cdot \cos re}{im + 0.000244140625}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.0004:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \end{array} \]

Alternative 5: 83.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{{im}^{6} \cdot 0.027777777777777776}\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -4e+49)
   (sqrt (* (pow im 6.0) 0.027777777777777776))
   (if (<= im 4.7e-5)
     (* im (- (cos re)))
     (if (<= im 1.35e+154)
       (*
        (+ (* -0.5 (* re re)) 1.0)
        (- (* (pow im 3.0) -0.16666666666666666) im))
       (/
        (* (cos re) (- 5.960464477539063e-8 (* im im)))
        (+ im 0.000244140625))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -4e+49) {
		tmp = sqrt((pow(im, 6.0) * 0.027777777777777776));
	} else if (im <= 4.7e-5) {
		tmp = im * -cos(re);
	} else if (im <= 1.35e+154) {
		tmp = ((-0.5 * (re * re)) + 1.0) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-4d+49)) then
        tmp = sqrt(((im ** 6.0d0) * 0.027777777777777776d0))
    else if (im <= 4.7d-5) then
        tmp = im * -cos(re)
    else if (im <= 1.35d+154) then
        tmp = (((-0.5d0) * (re * re)) + 1.0d0) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = (cos(re) * (5.960464477539063d-8 - (im * im))) / (im + 0.000244140625d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -4e+49) {
		tmp = Math.sqrt((Math.pow(im, 6.0) * 0.027777777777777776));
	} else if (im <= 4.7e-5) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = ((-0.5 * (re * re)) + 1.0) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (Math.cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -4e+49:
		tmp = math.sqrt((math.pow(im, 6.0) * 0.027777777777777776))
	elif im <= 4.7e-5:
		tmp = im * -math.cos(re)
	elif im <= 1.35e+154:
		tmp = ((-0.5 * (re * re)) + 1.0) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = (math.cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -4e+49)
		tmp = sqrt(Float64((im ^ 6.0) * 0.027777777777777776));
	elseif (im <= 4.7e-5)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(Float64(-0.5 * Float64(re * re)) + 1.0) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(Float64(cos(re) * Float64(5.960464477539063e-8 - Float64(im * im))) / Float64(im + 0.000244140625));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -4e+49)
		tmp = sqrt(((im ^ 6.0) * 0.027777777777777776));
	elseif (im <= 4.7e-5)
		tmp = im * -cos(re);
	elseif (im <= 1.35e+154)
		tmp = ((-0.5 * (re * re)) + 1.0) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = (cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -4e+49], N[Sqrt[N[(N[Power[im, 6.0], $MachinePrecision] * 0.027777777777777776), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 4.7e-5], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[(5.960464477539063e-8 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + 0.000244140625), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.99999999999999979e49

   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 80.9%

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

      1. mul-1-neg 80.9%

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

      2. unsub-neg 80.9%

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

      3. *-commutative 80.9%

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

      4. associate-*l* 80.9%

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

      5. distribute-lft-out-- 80.9%

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

   6. Simplified 80.9%

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

   7. Taylor expanded in re around 0 64.0%

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

   8. Taylor expanded in im around inf 64.0%

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

      1. add-sqr-sqrt 64.0%

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

      2. sqrt-unprod 79.6%

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

      3. *-commutative 79.6%

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

      4. *-commutative 79.6%

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

      5. swap-sqr 79.6%

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

      6. pow-prod-up 79.6%

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

      7. metadata-eval 79.6%

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

      8. metadata-eval 79.6%

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

   10. Applied egg-rr 79.6%

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

   if -3.99999999999999979e49 < im < 4.69999999999999972e-5

   1. Initial program 17.2%

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

      1. sub0-neg 17.2%

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

   3. Simplified 17.2%

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

   4. Taylor expanded in im around 0 89.8%

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

      1. mul-1-neg 89.8%

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

      2. *-commutative 89.8%

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

      3. distribute-lft-neg-in 89.8%

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

   6. Simplified 89.8%

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

   if 4.69999999999999972e-5 < im < 1.35000000000000003e154

   1. Initial program 99.3%

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

      1. sub0-neg 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in im around 0 62.5%

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

      1. mul-1-neg 62.5%

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

      2. unsub-neg 62.5%

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

      3. *-commutative 62.5%

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

      4. associate-*l* 62.5%

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

      5. distribute-lft-out-- 62.5%

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

   6. Simplified 62.5%

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

   7. Taylor expanded in re around 0 18.0%

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

      1. associate--l+ 18.0%

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

      2. associate-*r* 18.0%

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

      3. distribute-lft1-in 58.7%

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

      4. unpow2 58.7%

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

   9. Simplified 58.7%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.6%

      \[\leadsto \cos re \cdot \left(\color{blue}{0.000244140625} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.6%

         \[\leadsto \color{blue}{\left(0.000244140625 - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{0.000244140625 \cdot 0.000244140625 - im \cdot im}{0.000244140625 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.000244140625 \cdot 0.000244140625 - im \cdot im\right) \cdot \cos re}{0.000244140625 + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{5.960464477539063 \cdot 10^{-8}} - im \cdot im\right) \cdot \cos re}{0.000244140625 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + 0.000244140625}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right) \cdot \cos re}{im + 0.000244140625}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{{im}^{6} \cdot 0.027777777777777776}\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \end{array} \]

Alternative 6: 85.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ t_1 := \frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.03:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (+ (* -0.5 (* re re)) 1.0)
          (- (* (pow im 3.0) -0.16666666666666666) im)))
        (t_1
         (/
          (* (cos re) (- 5.960464477539063e-8 (* im im)))
          (+ im 0.000244140625))))
   (if (<= im -4.5e+163)
     t_1
     (if (<= im -0.03)
       t_0
       (if (<= im 4.7e-5)
         (* im (- (cos re)))
         (if (<= im 1.35e+154) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = ((-0.5 * (re * re)) + 1.0) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double t_1 = (cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	double tmp;
	if (im <= -4.5e+163) {
		tmp = t_1;
	} else if (im <= -0.03) {
		tmp = t_0;
	} else if (im <= 4.7e-5) {
		tmp = im * -cos(re);
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((-0.5d0) * (re * re)) + 1.0d0) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    t_1 = (cos(re) * (5.960464477539063d-8 - (im * im))) / (im + 0.000244140625d0)
    if (im <= (-4.5d+163)) then
        tmp = t_1
    else if (im <= (-0.03d0)) then
        tmp = t_0
    else if (im <= 4.7d-5) then
        tmp = im * -cos(re)
    else if (im <= 1.35d+154) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = ((-0.5 * (re * re)) + 1.0) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double t_1 = (Math.cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	double tmp;
	if (im <= -4.5e+163) {
		tmp = t_1;
	} else if (im <= -0.03) {
		tmp = t_0;
	} else if (im <= 4.7e-5) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((-0.5 * (re * re)) + 1.0) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	t_1 = (math.cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625)
	tmp = 0
	if im <= -4.5e+163:
		tmp = t_1
	elif im <= -0.03:
		tmp = t_0
	elif im <= 4.7e-5:
		tmp = im * -math.cos(re)
	elif im <= 1.35e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(-0.5 * Float64(re * re)) + 1.0) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	t_1 = Float64(Float64(cos(re) * Float64(5.960464477539063e-8 - Float64(im * im))) / Float64(im + 0.000244140625))
	tmp = 0.0
	if (im <= -4.5e+163)
		tmp = t_1;
	elseif (im <= -0.03)
		tmp = t_0;
	elseif (im <= 4.7e-5)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((-0.5 * (re * re)) + 1.0) * (((im ^ 3.0) * -0.16666666666666666) - im);
	t_1 = (cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	tmp = 0.0;
	if (im <= -4.5e+163)
		tmp = t_1;
	elseif (im <= -0.03)
		tmp = t_0;
	elseif (im <= 4.7e-5)
		tmp = im * -cos(re);
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(5.960464477539063e-8 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + 0.000244140625), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.5e+163], t$95$1, If[LessEqual[im, -0.03], t$95$0, If[LessEqual[im, 4.7e-5], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.49999999999999988e163 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.5%

      \[\leadsto \cos re \cdot \left(\color{blue}{0.000244140625} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.5%

         \[\leadsto \color{blue}{\left(0.000244140625 - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{0.000244140625 \cdot 0.000244140625 - im \cdot im}{0.000244140625 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.000244140625 \cdot 0.000244140625 - im \cdot im\right) \cdot \cos re}{0.000244140625 + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{5.960464477539063 \cdot 10^{-8}} - im \cdot im\right) \cdot \cos re}{0.000244140625 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + 0.000244140625}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right) \cdot \cos re}{im + 0.000244140625}} \]

   if -4.49999999999999988e163 < im < -0.029999999999999999 or 4.69999999999999972e-5 < im < 1.35000000000000003e154

   1. Initial program 99.7%

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

      1. sub0-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in im around 0 53.5%

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

      1. mul-1-neg 53.5%

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

      2. unsub-neg 53.5%

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

      3. *-commutative 53.5%

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

      4. associate-*l* 53.5%

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

      5. distribute-lft-out-- 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in re around 0 15.1%

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

      1. associate--l+ 15.1%

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

      2. associate-*r* 15.1%

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

      3. distribute-lft1-in 48.5%

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

      4. unpow2 48.5%

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

   9. Simplified 48.5%

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

   if -0.029999999999999999 < im < 4.69999999999999972e-5

   1. Initial program 7.4%

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

      1. sub0-neg 7.4%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. distribute-lft-neg-in 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \mathbf{elif}\;im \leq -0.03:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \end{array} \]

Alternative 7: 84.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ t_1 := \frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 520:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* (+ (* -0.5 (* re re)) 1.0) (* (pow im 3.0) -0.16666666666666666)))
        (t_1
         (/
          (* (cos re) (- 5.960464477539063e-8 (* im im)))
          (+ im 0.000244140625))))
   (if (<= im -4.5e+163)
     t_1
     (if (<= im -3.2e+18)
       t_0
       (if (<= im 520.0)
         (* im (- (cos re)))
         (if (<= im 1.35e+154) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = ((-0.5 * (re * re)) + 1.0) * (pow(im, 3.0) * -0.16666666666666666);
	double t_1 = (cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	double tmp;
	if (im <= -4.5e+163) {
		tmp = t_1;
	} else if (im <= -3.2e+18) {
		tmp = t_0;
	} else if (im <= 520.0) {
		tmp = im * -cos(re);
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((-0.5d0) * (re * re)) + 1.0d0) * ((im ** 3.0d0) * (-0.16666666666666666d0))
    t_1 = (cos(re) * (5.960464477539063d-8 - (im * im))) / (im + 0.000244140625d0)
    if (im <= (-4.5d+163)) then
        tmp = t_1
    else if (im <= (-3.2d+18)) then
        tmp = t_0
    else if (im <= 520.0d0) then
        tmp = im * -cos(re)
    else if (im <= 1.35d+154) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = ((-0.5 * (re * re)) + 1.0) * (Math.pow(im, 3.0) * -0.16666666666666666);
	double t_1 = (Math.cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	double tmp;
	if (im <= -4.5e+163) {
		tmp = t_1;
	} else if (im <= -3.2e+18) {
		tmp = t_0;
	} else if (im <= 520.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((-0.5 * (re * re)) + 1.0) * (math.pow(im, 3.0) * -0.16666666666666666)
	t_1 = (math.cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625)
	tmp = 0
	if im <= -4.5e+163:
		tmp = t_1
	elif im <= -3.2e+18:
		tmp = t_0
	elif im <= 520.0:
		tmp = im * -math.cos(re)
	elif im <= 1.35e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(-0.5 * Float64(re * re)) + 1.0) * Float64((im ^ 3.0) * -0.16666666666666666))
	t_1 = Float64(Float64(cos(re) * Float64(5.960464477539063e-8 - Float64(im * im))) / Float64(im + 0.000244140625))
	tmp = 0.0
	if (im <= -4.5e+163)
		tmp = t_1;
	elseif (im <= -3.2e+18)
		tmp = t_0;
	elseif (im <= 520.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((-0.5 * (re * re)) + 1.0) * ((im ^ 3.0) * -0.16666666666666666);
	t_1 = (cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	tmp = 0.0;
	if (im <= -4.5e+163)
		tmp = t_1;
	elseif (im <= -3.2e+18)
		tmp = t_0;
	elseif (im <= 520.0)
		tmp = im * -cos(re);
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(5.960464477539063e-8 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + 0.000244140625), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.5e+163], t$95$1, If[LessEqual[im, -3.2e+18], t$95$0, If[LessEqual[im, 520.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.49999999999999988e163 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.5%

      \[\leadsto \cos re \cdot \left(\color{blue}{0.000244140625} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.5%

         \[\leadsto \color{blue}{\left(0.000244140625 - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{0.000244140625 \cdot 0.000244140625 - im \cdot im}{0.000244140625 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.000244140625 \cdot 0.000244140625 - im \cdot im\right) \cdot \cos re}{0.000244140625 + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{5.960464477539063 \cdot 10^{-8}} - im \cdot im\right) \cdot \cos re}{0.000244140625 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + 0.000244140625}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right) \cdot \cos re}{im + 0.000244140625}} \]

   if -4.49999999999999988e163 < im < -3.2e18 or 520 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 55.1%

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

      1. mul-1-neg 55.1%

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

      2. unsub-neg 55.1%

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

      3. *-commutative 55.1%

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

      4. associate-*l* 55.1%

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

      5. distribute-lft-out-- 55.1%

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

   6. Simplified 55.1%

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

   7. Taylor expanded in re around 0 13.6%

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

      1. associate--l+ 13.6%

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

      2. associate-*r* 13.6%

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

      3. distribute-lft1-in 49.7%

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

      4. unpow2 49.7%

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

   9. Simplified 49.7%

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

   10. Taylor expanded in im around inf 49.7%

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

   if -3.2e18 < im < 520

   1. Initial program 11.4%

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

      1. sub0-neg 11.4%

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

   3. Simplified 11.4%

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

   4. Taylor expanded in im around 0 96.4%

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

      1. mul-1-neg 96.4%

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

      2. *-commutative 96.4%

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

      3. distribute-lft-neg-in 96.4%

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

   6. Simplified 96.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \mathbf{elif}\;im \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq 520:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \end{array} \]

Alternative 8: 78.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -0.000225:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -0.000225)
     t_0
     (if (<= im 4.7e-5)
       (* im (- (cos re)))
       (if (<= im 1.8e+150)
         t_0
         (/
          (* (cos re) (- 5.960464477539063e-8 (* im im)))
          (+ im 0.000244140625)))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -0.000225) {
		tmp = t_0;
	} else if (im <= 4.7e-5) {
		tmp = im * -cos(re);
	} else if (im <= 1.8e+150) {
		tmp = t_0;
	} else {
		tmp = (cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-0.000225d0)) then
        tmp = t_0
    else if (im <= 4.7d-5) then
        tmp = im * -cos(re)
    else if (im <= 1.8d+150) then
        tmp = t_0
    else
        tmp = (cos(re) * (5.960464477539063d-8 - (im * im))) / (im + 0.000244140625d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -0.000225) {
		tmp = t_0;
	} else if (im <= 4.7e-5) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.8e+150) {
		tmp = t_0;
	} else {
		tmp = (Math.cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -0.000225:
		tmp = t_0
	elif im <= 4.7e-5:
		tmp = im * -math.cos(re)
	elif im <= 1.8e+150:
		tmp = t_0
	else:
		tmp = (math.cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -0.000225)
		tmp = t_0;
	elseif (im <= 4.7e-5)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.8e+150)
		tmp = t_0;
	else
		tmp = Float64(Float64(cos(re) * Float64(5.960464477539063e-8 - Float64(im * im))) / Float64(im + 0.000244140625));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -0.000225)
		tmp = t_0;
	elseif (im <= 4.7e-5)
		tmp = im * -cos(re);
	elseif (im <= 1.8e+150)
		tmp = t_0;
	else
		tmp = (cos(re) * (5.960464477539063e-8 - (im * im))) / (im + 0.000244140625);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -0.000225], t$95$0, If[LessEqual[im, 4.7e-5], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.8e+150], t$95$0, N[(N[(N[Cos[re], $MachinePrecision] * N[(5.960464477539063e-8 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + 0.000244140625), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.2499999999999999e-4 or 4.69999999999999972e-5 < im < 1.79999999999999993e150

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in im around 0 63.7%

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

      1. mul-1-neg 63.7%

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

      2. unsub-neg 63.7%

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

      3. *-commutative 63.7%

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

      4. associate-*l* 63.7%

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

      5. distribute-lft-out-- 63.7%

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

   6. Simplified 63.7%

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

   7. Taylor expanded in re around 0 51.3%

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

   if -2.2499999999999999e-4 < im < 4.69999999999999972e-5

   1. Initial program 7.4%

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

      1. sub0-neg 7.4%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. distribute-lft-neg-in 99.9%

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

   6. Simplified 99.9%

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

   if 1.79999999999999993e150 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 6.5%

      \[\leadsto \cos re \cdot \left(\color{blue}{0.000244140625} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 6.5%

         \[\leadsto \color{blue}{\left(0.000244140625 - im\right) \cdot \cos re} \]

      2. flip-- 96.8%

         \[\leadsto \color{blue}{\frac{0.000244140625 \cdot 0.000244140625 - im \cdot im}{0.000244140625 + im}} \cdot \cos re \]

      3. associate-*l/ 96.8%

         \[\leadsto \color{blue}{\frac{\left(0.000244140625 \cdot 0.000244140625 - im \cdot im\right) \cdot \cos re}{0.000244140625 + im}} \]

      4. metadata-eval 96.8%

         \[\leadsto \frac{\left(\color{blue}{5.960464477539063 \cdot 10^{-8}} - im \cdot im\right) \cdot \cos re}{0.000244140625 + im} \]

      5. +-commutative 96.8%

         \[\leadsto \frac{\left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right) \cdot \cos re}{\color{blue}{im + 0.000244140625}} \]

   10. Applied egg-rr 96.8%

       \[\leadsto \color{blue}{\frac{\left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right) \cdot \cos re}{im + 0.000244140625}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.000225:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(5.960464477539063 \cdot 10^{-8} - im \cdot im\right)}{im + 0.000244140625}\\ \end{array} \]

Alternative 9: 40.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(0.5 \cdot re\right) - im\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02) (- (* (* im re) (* 0.5 re)) im) (- im)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = ((im * re) * (0.5 * re)) - im;
	} else {
		tmp = -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= (-0.02d0)) then
        tmp = ((im * re) * (0.5d0 * re)) - im
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= -0.02) {
		tmp = ((im * re) * (0.5 * re)) - im;
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -0.02:
		tmp = ((im * re) * (0.5 * re)) - im
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(Float64(Float64(im * re) * Float64(0.5 * re)) - im);
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= -0.02)
		tmp = ((im * re) * (0.5 * re)) - im;
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(N[(im * re), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 62.9%

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

      1. sub0-neg 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in im around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. *-commutative 43.1%

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

      3. distribute-lft-neg-in 43.1%

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

   6. Simplified 43.1%

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

   7. Taylor expanded in re around 0 46.8%

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

      1. neg-mul-1 46.8%

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

      2. +-commutative 46.8%

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

      3. unsub-neg 46.8%

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

      4. *-commutative 46.8%

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

      5. associate-*l* 46.8%

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

      6. unpow2 46.8%

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

   9. Simplified 46.8%

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

   10. Taylor expanded in re around 0 46.8%

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

       1. unpow2 46.8%

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

       2. associate-*r* 46.8%

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

       3. *-commutative 46.8%

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

       4. *-commutative 46.8%

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

       5. associate-*l* 46.8%

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

       6. associate-*r* 46.8%

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

   12. Simplified 46.8%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 52.1%

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

      1. sub0-neg 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in im around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. unsub-neg 86.8%

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

      3. *-commutative 86.8%

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

      4. associate-*l* 86.8%

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

      5. distribute-lft-out-- 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in re around 0 73.3%

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

   8. Taylor expanded in im around 0 40.9%

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

      1. neg-mul-1 40.9%

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

   10. Simplified 40.9%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(0.5 \cdot re\right) - im\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 10: 75.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.00165 \lor \neg \left(im \leq 2.3 \cdot 10^{-5}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.00165) (not (<= im 2.3e-5)))
   (- (* (pow im 3.0) -0.16666666666666666) im)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.00165) || !(im <= 2.3e-5)) {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-0.00165d0)) .or. (.not. (im <= 2.3d-5))) then
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.00165) || !(im <= 2.3e-5)) {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.00165) or not (im <= 2.3e-5):
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.00165) || !(im <= 2.3e-5))
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.00165) || ~((im <= 2.3e-5)))
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.00165], N[Not[LessEqual[im, 2.3e-5]], $MachinePrecision]], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -0.00165 or 2.3e-5 < im

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in im around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. unsub-neg 72.1%

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

      3. *-commutative 72.1%

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

      4. associate-*l* 72.1%

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

      5. distribute-lft-out-- 72.1%

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

   6. Simplified 72.1%

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

   7. Taylor expanded in re around 0 55.6%

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

   if -0.00165 < im < 2.3e-5

   1. Initial program 7.4%

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

      1. sub0-neg 7.4%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. distribute-lft-neg-in 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00165 \lor \neg \left(im \leq 2.3 \cdot 10^{-5}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 11: 52.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+101} \lor \neg \left(im \leq 2.8 \cdot 10^{+70}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(0.5 \cdot re\right) - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -9e+101) (not (<= im 2.8e+70)))
   (* (pow im 3.0) -0.16666666666666666)
   (- (* (* im re) (* 0.5 re)) im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -9e+101) || !(im <= 2.8e+70)) {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	} 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+101)) .or. (.not. (im <= 2.8d+70))) then
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    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+101) || !(im <= 2.8e+70)) {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = ((im * re) * (0.5 * re)) - im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -9e+101) or not (im <= 2.8e+70):
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	else:
		tmp = ((im * re) * (0.5 * re)) - im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -9e+101) || !(im <= 2.8e+70))
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	else
		tmp = Float64(Float64(Float64(im * re) * Float64(0.5 * re)) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -9e+101) || ~((im <= 2.8e+70)))
		tmp = (im ^ 3.0) * -0.16666666666666666;
	else
		tmp = ((im * re) * (0.5 * re)) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -9e+101], N[Not[LessEqual[im, 2.8e+70]], $MachinePrecision]], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(N[(N[(im * re), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -9.0000000000000004e101 or 2.7999999999999999e70 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.0%

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

      1. mul-1-neg 98.0%

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

      2. unsub-neg 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-*l* 98.0%

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

      5. distribute-lft-out-- 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in re around 0 75.1%

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

   8. Taylor expanded in im around inf 75.1%

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

   if -9.0000000000000004e101 < im < 2.7999999999999999e70

   1. Initial program 28.7%

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

      1. sub0-neg 28.7%

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

   3. Simplified 28.7%

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

   4. Taylor expanded in im around 0 78.2%

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

      1. mul-1-neg 78.2%

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

      2. *-commutative 78.2%

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

      3. distribute-lft-neg-in 78.2%

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

   6. Simplified 78.2%

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

   7. Taylor expanded in re around 0 49.2%

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

      1. neg-mul-1 49.2%

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

      2. +-commutative 49.2%

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

      3. unsub-neg 49.2%

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

      4. *-commutative 49.2%

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

      5. associate-*l* 49.2%

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

      6. unpow2 49.2%

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

   9. Simplified 49.2%

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

   10. Taylor expanded in re around 0 49.2%

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

       1. unpow2 49.2%

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

       2. associate-*r* 49.2%

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

       3. *-commutative 49.2%

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

       4. *-commutative 49.2%

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

       5. associate-*l* 49.2%

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

       6. associate-*r* 49.2%

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

   12. Simplified 49.2%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+101} \lor \neg \left(im \leq 2.8 \cdot 10^{+70}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(0.5 \cdot re\right) - im\\ \end{array} \]

Alternative 12: 75.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{+50} \lor \neg \left(im \leq 1.7 \cdot 10^{+62}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.2e+50) (not (<= im 1.7e+62)))
   (* (pow im 3.0) -0.16666666666666666)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.2e+50) || !(im <= 1.7e+62)) {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.2d+50)) .or. (.not. (im <= 1.7d+62))) then
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.2e+50) || !(im <= 1.7e+62)) {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.2e+50) or not (im <= 1.7e+62):
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.2e+50) || !(im <= 1.7e+62))
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.2e+50) || ~((im <= 1.7e+62)))
		tmp = (im ^ 3.0) * -0.16666666666666666;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.2e+50], N[Not[LessEqual[im, 1.7e+62]], $MachinePrecision]], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.2000000000000001e50 or 1.70000000000000007e62 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 88.2%

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

      1. mul-1-neg 88.2%

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

      2. unsub-neg 88.2%

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

      3. *-commutative 88.2%

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

      4. associate-*l* 88.2%

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

      5. distribute-lft-out-- 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in re around 0 67.6%

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

   8. Taylor expanded in im around inf 67.6%

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

   if -1.2000000000000001e50 < im < 1.70000000000000007e62

   1. Initial program 23.6%

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

      1. sub0-neg 23.6%

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

   3. Simplified 23.6%

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

   4. Taylor expanded in im around 0 83.6%

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

      1. mul-1-neg 83.6%

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

      2. *-commutative 83.6%

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

      3. distribute-lft-neg-in 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{+50} \lor \neg \left(im \leq 1.7 \cdot 10^{+62}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 13: 32.5% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+201}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.2e+153)
   (- im)
   (if (<= re 2.3e+201)
     (* (* re re) -6.75)
     (* (+ 0.5 (* re (* re -0.25))) -3.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.2e+153) {
		tmp = -im;
	} else if (re <= 2.3e+201) {
		tmp = (re * re) * -6.75;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.2d+153) then
        tmp = -im
    else if (re <= 2.3d+201) then
        tmp = (re * re) * (-6.75d0)
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.2e+153) {
		tmp = -im;
	} else if (re <= 2.3e+201) {
		tmp = (re * re) * -6.75;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.2e+153:
		tmp = -im
	elif re <= 2.3e+201:
		tmp = (re * re) * -6.75
	else:
		tmp = (0.5 + (re * (re * -0.25))) * -3.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.2e+153)
		tmp = Float64(-im);
	elseif (re <= 2.3e+201)
		tmp = Float64(Float64(re * re) * -6.75);
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.2e+153)
		tmp = -im;
	elseif (re <= 2.3e+201)
		tmp = (re * re) * -6.75;
	else
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.2e+153], (-im), If[LessEqual[re, 2.3e+201], N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 5.1999999999999998e153

   1. Initial program 52.3%

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

      1. sub0-neg 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in im around 0 87.1%

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

      1. mul-1-neg 87.1%

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

      2. unsub-neg 87.1%

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

      3. *-commutative 87.1%

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

      4. associate-*l* 87.1%

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

      5. distribute-lft-out-- 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in re around 0 64.1%

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

   8. Taylor expanded in im around 0 37.0%

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

      1. neg-mul-1 37.0%

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

   10. Simplified 37.0%

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

   if 5.1999999999999998e153 < re < 2.3000000000000001e201

   1. Initial program 88.2%

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

      1. sub0-neg 88.2%

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

   3. Simplified 88.2%

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

   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 33.4%

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

      4. +-commutative 33.4%

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

      5. *-commutative 33.4%

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

      6. unpow2 33.4%

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

      7. associate-*l* 33.4%

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

   6. Simplified 33.4%

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

   8. Applied egg-rr 83.8%

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

   9. Taylor expanded in re around inf 83.8%

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

       1. unpow2 83.8%

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

   11. Simplified 83.8%

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

   if 2.3000000000000001e201 < re

   1. Initial program 63.7%

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

      1. sub0-neg 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in re around 0 0.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   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 33.4%

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

      4. +-commutative 33.4%

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

      5. *-commutative 33.4%

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

      6. unpow2 33.4%

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

      7. associate-*l* 33.4%

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

   6. Simplified 33.4%

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

   8. Applied egg-rr 45.2%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+201}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \]

Alternative 14: 32.5% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.8e+153) (- im) (* (* re re) -6.75)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+153) {
		tmp = -im;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.8d+153) then
        tmp = -im
    else
        tmp = (re * re) * (-6.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+153) {
		tmp = -im;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.8e+153:
		tmp = -im
	else:
		tmp = (re * re) * -6.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.8e+153)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(re * re) * -6.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.8e+153)
		tmp = -im;
	else
		tmp = (re * re) * -6.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.8e+153], (-im), N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.80000000000000004e153

   1. Initial program 52.3%

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

      1. sub0-neg 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in im around 0 87.1%

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

      1. mul-1-neg 87.1%

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

      2. unsub-neg 87.1%

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

      3. *-commutative 87.1%

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

      4. associate-*l* 87.1%

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

      5. distribute-lft-out-- 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in re around 0 64.1%

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

   8. Taylor expanded in im around 0 37.0%

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

      1. neg-mul-1 37.0%

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

   10. Simplified 37.0%

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

   if 5.80000000000000004e153 < re

   1. Initial program 68.2%

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

      1. sub0-neg 68.2%

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

   3. Simplified 68.2%

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

   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 33.4%

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

      4. +-commutative 33.4%

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

      5. *-commutative 33.4%

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

      6. unpow2 33.4%

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

      7. associate-*l* 33.4%

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

   6. Simplified 33.4%

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

   8. Applied egg-rr 27.9%

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

   9. Taylor expanded in re around inf 27.9%

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

       1. unpow2 27.9%

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

   11. Simplified 27.9%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 15: 30.7% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 54.3%

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

   1. sub0-neg 54.3%

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

3. Simplified 54.3%

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

4. Taylor expanded in im around 0 85.8%

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

   1. mul-1-neg 85.8%

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

   2. unsub-neg 85.8%

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

   3. *-commutative 85.8%

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

   4. associate-*l* 85.8%

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

   5. distribute-lft-out-- 85.8%

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

6. Simplified 85.8%

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

7. Taylor expanded in re around 0 58.8%

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

8. Taylor expanded in im around 0 33.0%

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

   1. neg-mul-1 33.0%

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

10. Simplified 33.0%

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

11. Final simplification 33.0%

    \[\leadsto -im \]

Alternative 16: 2.9% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 13.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 13.5

Julia

function code(re, im)
	return 13.5
end

MATLAB

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

Wolfram

code[re_, im_] := 13.5

Derivation

1. Initial program 54.3%

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

   1. sub0-neg 54.3%

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

3. Simplified 54.3%

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

4. Taylor expanded in re around 0 3.7%

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

   1. *-commutative 3.7%

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

   2. associate-*r* 3.7%

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

   3. distribute-rgt-out 39.3%

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

   4. +-commutative 39.3%

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

   5. *-commutative 39.3%

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

   6. unpow2 39.3%

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

   7. associate-*l* 39.3%

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

6. Simplified 39.3%

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

8. Applied egg-rr 8.2%

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

9. Taylor expanded in re around 0 3.1%

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

10. Final simplification 3.1%

    \[\leadsto 13.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023189 
(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))))