math.sin on complex, imaginary part

Percentage Accurate: 54.0% → 99.8%

Time: 13.3s

Alternatives: 21

Speedup: 2.7×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.2 \lor \neg \left(t_0 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.2) (not (<= t_0 2e-6)))
     (* (* 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 <= 2e-6)) {
		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 <= 2d-6))) 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 <= 2e-6)) {
		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 <= 2e-6):
		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 <= 2e-6))
		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 <= 2e-6)))
		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, 2e-6]], $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 1.99999999999999991e-6 < (-.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. neg-sub0 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)) < 1.99999999999999991e-6

   1. Initial program 7.9%

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

      1. neg-sub0 7.9%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out-- 99.9%

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

      6. *-commutative 99.9%

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

   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 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos re \cdot \frac{68719476736 - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im \cdot im + 262144}}{im + -512}\\ t_1 := \frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ t_2 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.00195:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 235000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (/
          (*
           (cos re)
           (/
            (- 68719476736.0 (* (* im im) (* im im)))
            (+ (* im im) 262144.0)))
          (+ im -512.0)))
        (t_1 (/ (* (cos re) (- (* im im))) (+ im -512.0)))
        (t_2 (* 0.5 (- (exp (- im)) (exp im)))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -1.15e+77)
       t_0
       (if (<= im -0.00195)
         t_2
         (if (<= im 235000000000.0)
           (- (* im (cos re)))
           (if (<= im 1.15e+77) t_2 (if (<= im 1.4e+154) t_0 t_1))))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0);
	double t_1 = (cos(re) * -(im * im)) / (im + -512.0);
	double t_2 = 0.5 * (exp(-im) - exp(im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -1.15e+77) {
		tmp = t_0;
	} else if (im <= -0.00195) {
		tmp = t_2;
	} else if (im <= 235000000000.0) {
		tmp = -(im * cos(re));
	} else if (im <= 1.15e+77) {
		tmp = t_2;
	} else if (im <= 1.4e+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) :: t_2
    real(8) :: tmp
    t_0 = (cos(re) * ((68719476736.0d0 - ((im * im) * (im * im))) / ((im * im) + 262144.0d0))) / (im + (-512.0d0))
    t_1 = (cos(re) * -(im * im)) / (im + (-512.0d0))
    t_2 = 0.5d0 * (exp(-im) - exp(im))
    if (im <= (-1.35d+154)) then
        tmp = t_1
    else if (im <= (-1.15d+77)) then
        tmp = t_0
    else if (im <= (-0.00195d0)) then
        tmp = t_2
    else if (im <= 235000000000.0d0) then
        tmp = -(im * cos(re))
    else if (im <= 1.15d+77) then
        tmp = t_2
    else if (im <= 1.4d+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 = (Math.cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0);
	double t_1 = (Math.cos(re) * -(im * im)) / (im + -512.0);
	double t_2 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -1.15e+77) {
		tmp = t_0;
	} else if (im <= -0.00195) {
		tmp = t_2;
	} else if (im <= 235000000000.0) {
		tmp = -(im * Math.cos(re));
	} else if (im <= 1.15e+77) {
		tmp = t_2;
	} else if (im <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0)
	t_1 = (math.cos(re) * -(im * im)) / (im + -512.0)
	t_2 = 0.5 * (math.exp(-im) - math.exp(im))
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -1.15e+77:
		tmp = t_0
	elif im <= -0.00195:
		tmp = t_2
	elif im <= 235000000000.0:
		tmp = -(im * math.cos(re))
	elif im <= 1.15e+77:
		tmp = t_2
	elif im <= 1.4e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * Float64(Float64(68719476736.0 - Float64(Float64(im * im) * Float64(im * im))) / Float64(Float64(im * im) + 262144.0))) / Float64(im + -512.0))
	t_1 = Float64(Float64(cos(re) * Float64(-Float64(im * im))) / Float64(im + -512.0))
	t_2 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -1.15e+77)
		tmp = t_0;
	elseif (im <= -0.00195)
		tmp = t_2;
	elseif (im <= 235000000000.0)
		tmp = Float64(-Float64(im * cos(re)));
	elseif (im <= 1.15e+77)
		tmp = t_2;
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0);
	t_1 = (cos(re) * -(im * im)) / (im + -512.0);
	t_2 = 0.5 * (exp(-im) - exp(im));
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -1.15e+77)
		tmp = t_0;
	elseif (im <= -0.00195)
		tmp = t_2;
	elseif (im <= 235000000000.0)
		tmp = -(im * cos(re));
	elseif (im <= 1.15e+77)
		tmp = t_2;
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(68719476736.0 - N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(im * im), $MachinePrecision] + 262144.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * (-N[(im * im), $MachinePrecision])), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -1.15e+77], t$95$0, If[LessEqual[im, -0.00195], t$95$2, If[LessEqual[im, 235000000000.0], (-N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), If[LessEqual[im, 1.15e+77], t$95$2, If[LessEqual[im, 1.4e+154], t$95$0, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.35000000000000003e154 or 1.4e154 < 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. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.6%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.6%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in im around inf 100.0%

       \[\leadsto \frac{\color{blue}{\left(-1 \cdot {im}^{2}\right)} \cdot \cos re}{im + -512} \]
   12. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto \frac{\left(-1 \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \cos re}{im + -512} \]

       2. neg-mul-1 100.0%

          \[\leadsto \frac{\color{blue}{\left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   if -1.35000000000000003e154 < im < -1.14999999999999997e77 or 1.14999999999999997e77 < im < 1.4e154

   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. neg-sub0 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 59.6%

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

      1. +-commutative 59.6%

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

      2. mul-1-neg 59.6%

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

      3. unsub-neg 59.6%

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

      4. associate-*r* 59.6%

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

      5. distribute-rgt-out-- 59.6%

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

      6. *-commutative 59.6%

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

   6. Simplified 59.6%

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

   8. Applied egg-rr 4.2%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 4.2%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 4.2%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 4.2%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 4.2%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 4.2%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 4.2%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]
   11. Step-by-step derivation

       1. sub-neg 4.2%

          \[\leadsto \frac{\color{blue}{\left(262144 + \left(-im \cdot im\right)\right)} \cdot \cos re}{im + -512} \]

       2. flip-+ 100.0%

          \[\leadsto \frac{\color{blue}{\frac{262144 \cdot 262144 - \left(-im \cdot im\right) \cdot \left(-im \cdot im\right)}{262144 - \left(-im \cdot im\right)}} \cdot \cos re}{im + -512} \]

       3. metadata-eval 100.0%

          \[\leadsto \frac{\frac{\color{blue}{68719476736} - \left(-im \cdot im\right) \cdot \left(-im \cdot im\right)}{262144 - \left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   12. Applied egg-rr 100.0%

       \[\leadsto \frac{\color{blue}{\frac{68719476736 - \left(-im \cdot im\right) \cdot \left(-im \cdot im\right)}{262144 - \left(-im \cdot im\right)}} \cdot \cos re}{im + -512} \]

   if -1.14999999999999997e77 < im < -0.0019499999999999999 or 2.35e11 < im < 1.14999999999999997e77

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 73.7%

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

   if -0.0019499999999999999 < im < 2.35e11

   1. Initial program 9.3%

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

      1. neg-sub0 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in im around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \mathbf{elif}\;im \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{\cos re \cdot \frac{68719476736 - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im \cdot im + 262144}}{im + -512}\\ \mathbf{elif}\;im \leq -0.00195:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 235000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \frac{68719476736 - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im \cdot im + 262144}}{im + -512}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \end{array} \]

Alternative 3: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+113} \lor \neg \left(im \leq -0.07 \lor \neg \left(im \leq 235000000000\right) \land im \leq 2.25 \cdot 10^{+101}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1e+113)
         (not
          (or (<= im -0.07)
              (and (not (<= im 235000000000.0)) (<= im 2.25e+101)))))
   (* (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 <= -1e+113) || !((im <= -0.07) || (!(im <= 235000000000.0) && (im <= 2.25e+101)))) {
		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 <= (-1d+113)) .or. (.not. (im <= (-0.07d0)) .or. (.not. (im <= 235000000000.0d0)) .and. (im <= 2.25d+101))) 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 <= -1e+113) || !((im <= -0.07) || (!(im <= 235000000000.0) && (im <= 2.25e+101)))) {
		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 <= -1e+113) or not ((im <= -0.07) or (not (im <= 235000000000.0) and (im <= 2.25e+101))):
		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 <= -1e+113) || !((im <= -0.07) || (!(im <= 235000000000.0) && (im <= 2.25e+101))))
		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 <= -1e+113) || ~(((im <= -0.07) || (~((im <= 235000000000.0)) && (im <= 2.25e+101)))))
		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, -1e+113], N[Not[Or[LessEqual[im, -0.07], And[N[Not[LessEqual[im, 235000000000.0]], $MachinePrecision], LessEqual[im, 2.25e+101]]]], $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 < -1e113 or -0.070000000000000007 < im < 2.35e11 or 2.2500000000000001e101 < im

   1. Initial program 42.7%

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

      1. neg-sub0 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in im around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. associate-*r* 98.7%

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

      5. distribute-rgt-out-- 98.7%

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

      6. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   if -1e113 < im < -0.070000000000000007 or 2.35e11 < im < 2.2500000000000001e101

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 74.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+113} \lor \neg \left(im \leq -0.07 \lor \neg \left(im \leq 235000000000\right) \land im \leq 2.25 \cdot 10^{+101}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -5.7 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1000:\\ \;\;\;\;t_0 \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 235000000000 \lor \neg \left(im \leq 2.25 \cdot 10^{+101}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \end{array} \]

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 -5.7e+102)
     t_1
     (if (<= im -1000.0)
       (* t_0 (+ 0.5 (* -0.25 (* re re))))
       (if (or (<= im 235000000000.0) (not (<= im 2.25e+101)))
         t_1
         (* 0.5 t_0))))))

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 <= -5.7e+102) {
		tmp = t_1;
	} else if (im <= -1000.0) {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	} else if ((im <= 235000000000.0) || !(im <= 2.25e+101)) {
		tmp = t_1;
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-5.7d+102)) then
        tmp = t_1
    else if (im <= (-1000.0d0)) then
        tmp = t_0 * (0.5d0 + ((-0.25d0) * (re * re)))
    else if ((im <= 235000000000.0d0) .or. (.not. (im <= 2.25d+101))) then
        tmp = t_1
    else
        tmp = 0.5d0 * t_0
    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 <= -5.7e+102) {
		tmp = t_1;
	} else if (im <= -1000.0) {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	} else if ((im <= 235000000000.0) || !(im <= 2.25e+101)) {
		tmp = t_1;
	} else {
		tmp = 0.5 * t_0;
	}
	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 <= -5.7e+102:
		tmp = t_1
	elif im <= -1000.0:
		tmp = t_0 * (0.5 + (-0.25 * (re * re)))
	elif (im <= 235000000000.0) or not (im <= 2.25e+101):
		tmp = t_1
	else:
		tmp = 0.5 * t_0
	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 <= -5.7e+102)
		tmp = t_1;
	elseif (im <= -1000.0)
		tmp = Float64(t_0 * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	elseif ((im <= 235000000000.0) || !(im <= 2.25e+101))
		tmp = t_1;
	else
		tmp = Float64(0.5 * t_0);
	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 <= -5.7e+102)
		tmp = t_1;
	elseif (im <= -1000.0)
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	elseif ((im <= 235000000000.0) || ~((im <= 2.25e+101)))
		tmp = t_1;
	else
		tmp = 0.5 * t_0;
	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, -5.7e+102], t$95$1, If[LessEqual[im, -1000.0], N[(t$95$0 * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 235000000000.0], N[Not[LessEqual[im, 2.25e+101]], $MachinePrecision]], t$95$1, N[(0.5 * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.6999999999999999e102 or -1e3 < im < 2.35e11 or 2.2500000000000001e101 < im

   1. Initial program 43.5%

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

      1. neg-sub0 43.5%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in im around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. mul-1-neg 98.3%

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

      3. unsub-neg 98.3%

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

      4. associate-*r* 98.3%

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

      5. distribute-rgt-out-- 98.3%

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

      6. *-commutative 98.3%

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

   6. Simplified 98.3%

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

   if -5.6999999999999999e102 < im < -1e3

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 87.0%

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

      4. unpow2 87.0%

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

   6. Simplified 87.0%

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

   if 2.35e11 < im < 2.2500000000000001e101

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 79.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.7 \cdot 10^{+102}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -1000:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 235000000000 \lor \neg \left(im \leq 2.25 \cdot 10^{+101}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 5: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos re \cdot \frac{68719476736 - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im \cdot im + 262144}}{im + -512}\\ t_1 := \frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1000:\\ \;\;\;\;\left(re \cdot \left(re \cdot -0.5\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 750000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \left(re \cdot -6.75\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (/
          (*
           (cos re)
           (/
            (- 68719476736.0 (* (* im im) (* im im)))
            (+ (* im im) 262144.0)))
          (+ im -512.0)))
        (t_1 (/ (* (cos re) (- (* im im))) (+ im -512.0))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -8e+76)
       t_0
       (if (<= im -1000.0)
         (*
          (+ (* re (* re -0.5)) 1.0)
          (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 750000000000.0)
           (- (* im (cos re)))
           (if (<= im 1.15e+77)
             (log1p (expm1 (* re (* re -6.75))))
             (if (<= im 1.4e+154) t_0 t_1))))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0);
	double t_1 = (cos(re) * -(im * im)) / (im + -512.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -8e+76) {
		tmp = t_0;
	} else if (im <= -1000.0) {
		tmp = ((re * (re * -0.5)) + 1.0) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 750000000000.0) {
		tmp = -(im * cos(re));
	} else if (im <= 1.15e+77) {
		tmp = log1p(expm1((re * (re * -6.75))));
	} else if (im <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (Math.cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0);
	double t_1 = (Math.cos(re) * -(im * im)) / (im + -512.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -8e+76) {
		tmp = t_0;
	} else if (im <= -1000.0) {
		tmp = ((re * (re * -0.5)) + 1.0) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 750000000000.0) {
		tmp = -(im * Math.cos(re));
	} else if (im <= 1.15e+77) {
		tmp = Math.log1p(Math.expm1((re * (re * -6.75))));
	} else if (im <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0)
	t_1 = (math.cos(re) * -(im * im)) / (im + -512.0)
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -8e+76:
		tmp = t_0
	elif im <= -1000.0:
		tmp = ((re * (re * -0.5)) + 1.0) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 750000000000.0:
		tmp = -(im * math.cos(re))
	elif im <= 1.15e+77:
		tmp = math.log1p(math.expm1((re * (re * -6.75))))
	elif im <= 1.4e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * Float64(Float64(68719476736.0 - Float64(Float64(im * im) * Float64(im * im))) / Float64(Float64(im * im) + 262144.0))) / Float64(im + -512.0))
	t_1 = Float64(Float64(cos(re) * Float64(-Float64(im * im))) / Float64(im + -512.0))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -8e+76)
		tmp = t_0;
	elseif (im <= -1000.0)
		tmp = Float64(Float64(Float64(re * Float64(re * -0.5)) + 1.0) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 750000000000.0)
		tmp = Float64(-Float64(im * cos(re)));
	elseif (im <= 1.15e+77)
		tmp = log1p(expm1(Float64(re * Float64(re * -6.75))));
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(68719476736.0 - N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(im * im), $MachinePrecision] + 262144.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * (-N[(im * im), $MachinePrecision])), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -8e+76], t$95$0, If[LessEqual[im, -1000.0], N[(N[(N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 750000000000.0], (-N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), If[LessEqual[im, 1.15e+77], N[Log[1 + N[(Exp[N[(re * N[(re * -6.75), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.4e+154], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -1.35000000000000003e154 or 1.4e154 < 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. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.6%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.6%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in im around inf 100.0%

       \[\leadsto \frac{\color{blue}{\left(-1 \cdot {im}^{2}\right)} \cdot \cos re}{im + -512} \]
   12. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto \frac{\left(-1 \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \cos re}{im + -512} \]

       2. neg-mul-1 100.0%

          \[\leadsto \frac{\color{blue}{\left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   if -1.35000000000000003e154 < im < -8.0000000000000004e76 or 1.14999999999999997e77 < im < 1.4e154

   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. neg-sub0 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 58.1%

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

      1. +-commutative 58.1%

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

      2. mul-1-neg 58.1%

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

      3. unsub-neg 58.1%

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

      4. associate-*r* 58.1%

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

      5. distribute-rgt-out-- 58.1%

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

      6. *-commutative 58.1%

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

   6. Simplified 58.1%

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

   8. Applied egg-rr 4.2%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 4.2%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 4.2%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 4.2%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 4.2%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 4.2%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 4.2%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]
   11. Step-by-step derivation

       1. sub-neg 4.2%

          \[\leadsto \frac{\color{blue}{\left(262144 + \left(-im \cdot im\right)\right)} \cdot \cos re}{im + -512} \]

       2. flip-+ 97.4%

          \[\leadsto \frac{\color{blue}{\frac{262144 \cdot 262144 - \left(-im \cdot im\right) \cdot \left(-im \cdot im\right)}{262144 - \left(-im \cdot im\right)}} \cdot \cos re}{im + -512} \]

       3. metadata-eval 97.4%

          \[\leadsto \frac{\frac{\color{blue}{68719476736} - \left(-im \cdot im\right) \cdot \left(-im \cdot im\right)}{262144 - \left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   12. Applied egg-rr 97.4%

       \[\leadsto \frac{\color{blue}{\frac{68719476736 - \left(-im \cdot im\right) \cdot \left(-im \cdot im\right)}{262144 - \left(-im \cdot im\right)}} \cdot \cos re}{im + -512} \]

   if -8.0000000000000004e76 < im < -1e3

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.2%

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

      1. +-commutative 4.2%

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

      2. mul-1-neg 4.2%

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

      3. unsub-neg 4.2%

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

      4. associate-*r* 4.2%

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

      5. distribute-rgt-out-- 4.2%

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

      6. *-commutative 4.2%

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

   6. Simplified 4.2%

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

   7. Taylor expanded in re around 0 38.4%

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

      1. associate--l+ 38.4%

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

      2. associate-*r* 38.4%

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

      3. distribute-lft1-in 38.4%

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

      4. unpow2 38.4%

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

      5. associate-*r* 38.4%

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

   9. Simplified 38.4%

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

   if -1e3 < im < 7.5e11

   1. Initial program 10.7%

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

      1. neg-sub0 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in im around 0 97.0%

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

      1. associate-*r* 97.0%

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

      2. neg-mul-1 97.0%

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

   6. Simplified 97.0%

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

   if 7.5e11 < im < 1.14999999999999997e77

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 57.1%

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

      4. unpow2 57.1%

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

   6. Simplified 57.1%

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

   8. Applied egg-rr 20.4%

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

   9. Taylor expanded in re around inf 20.9%

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

       1. unpow2 20.9%

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

       2. *-commutative 20.9%

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

       3. associate-*l* 20.9%

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

   11. Simplified 20.9%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot -6.75\right)} \]
   12. Step-by-step derivation

       1. log1p-expm1-u 43.6%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \left(re \cdot -6.75\right)\right)\right)} \]

   13. Applied egg-rr 43.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \left(re \cdot -6.75\right)\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\cos re \cdot \frac{68719476736 - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im \cdot im + 262144}}{im + -512}\\ \mathbf{elif}\;im \leq -1000:\\ \;\;\;\;\left(re \cdot \left(re \cdot -0.5\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 750000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \left(re \cdot -6.75\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \frac{68719476736 - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im \cdot im + 262144}}{im + -512}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \end{array} \]

Alternative 6: 89.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos re \cdot \frac{68719476736 - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im \cdot im + 262144}}{im + -512}\\ t_1 := \frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1000:\\ \;\;\;\;\left(re \cdot \left(re \cdot -0.5\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 750000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (/
          (*
           (cos re)
           (/
            (- 68719476736.0 (* (* im im) (* im im)))
            (+ (* im im) 262144.0)))
          (+ im -512.0)))
        (t_1 (/ (* (cos re) (- (* im im))) (+ im -512.0))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -8e+76)
       t_0
       (if (<= im -1000.0)
         (*
          (+ (* re (* re -0.5)) 1.0)
          (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 750000000000.0)
           (- (* im (cos re)))
           (if (<= im 1.15e+77)
             (-
              (* (* re re) (* im (+ 0.5 (* (* re re) -0.041666666666666664))))
              im)
             (if (<= im 1.4e+154) t_0 t_1))))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0);
	double t_1 = (cos(re) * -(im * im)) / (im + -512.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -8e+76) {
		tmp = t_0;
	} else if (im <= -1000.0) {
		tmp = ((re * (re * -0.5)) + 1.0) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 750000000000.0) {
		tmp = -(im * cos(re));
	} else if (im <= 1.15e+77) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 1.4e+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 = (cos(re) * ((68719476736.0d0 - ((im * im) * (im * im))) / ((im * im) + 262144.0d0))) / (im + (-512.0d0))
    t_1 = (cos(re) * -(im * im)) / (im + (-512.0d0))
    if (im <= (-1.35d+154)) then
        tmp = t_1
    else if (im <= (-8d+76)) then
        tmp = t_0
    else if (im <= (-1000.0d0)) then
        tmp = ((re * (re * (-0.5d0))) + 1.0d0) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 750000000000.0d0) then
        tmp = -(im * cos(re))
    else if (im <= 1.15d+77) then
        tmp = ((re * re) * (im * (0.5d0 + ((re * re) * (-0.041666666666666664d0))))) - im
    else if (im <= 1.4d+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 = (Math.cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0);
	double t_1 = (Math.cos(re) * -(im * im)) / (im + -512.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -8e+76) {
		tmp = t_0;
	} else if (im <= -1000.0) {
		tmp = ((re * (re * -0.5)) + 1.0) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 750000000000.0) {
		tmp = -(im * Math.cos(re));
	} else if (im <= 1.15e+77) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0)
	t_1 = (math.cos(re) * -(im * im)) / (im + -512.0)
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -8e+76:
		tmp = t_0
	elif im <= -1000.0:
		tmp = ((re * (re * -0.5)) + 1.0) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 750000000000.0:
		tmp = -(im * math.cos(re))
	elif im <= 1.15e+77:
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im
	elif im <= 1.4e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * Float64(Float64(68719476736.0 - Float64(Float64(im * im) * Float64(im * im))) / Float64(Float64(im * im) + 262144.0))) / Float64(im + -512.0))
	t_1 = Float64(Float64(cos(re) * Float64(-Float64(im * im))) / Float64(im + -512.0))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -8e+76)
		tmp = t_0;
	elseif (im <= -1000.0)
		tmp = Float64(Float64(Float64(re * Float64(re * -0.5)) + 1.0) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 750000000000.0)
		tmp = Float64(-Float64(im * cos(re)));
	elseif (im <= 1.15e+77)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * Float64(0.5 + Float64(Float64(re * re) * -0.041666666666666664)))) - im);
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (cos(re) * ((68719476736.0 - ((im * im) * (im * im))) / ((im * im) + 262144.0))) / (im + -512.0);
	t_1 = (cos(re) * -(im * im)) / (im + -512.0);
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -8e+76)
		tmp = t_0;
	elseif (im <= -1000.0)
		tmp = ((re * (re * -0.5)) + 1.0) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 750000000000.0)
		tmp = -(im * cos(re));
	elseif (im <= 1.15e+77)
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(68719476736.0 - N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(im * im), $MachinePrecision] + 262144.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * (-N[(im * im), $MachinePrecision])), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -8e+76], t$95$0, If[LessEqual[im, -1000.0], N[(N[(N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 750000000000.0], (-N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), If[LessEqual[im, 1.15e+77], N[(N[(N[(re * re), $MachinePrecision] * N[(im * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 1.4e+154], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -1.35000000000000003e154 or 1.4e154 < 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. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.6%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.6%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in im around inf 100.0%

       \[\leadsto \frac{\color{blue}{\left(-1 \cdot {im}^{2}\right)} \cdot \cos re}{im + -512} \]
   12. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto \frac{\left(-1 \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \cos re}{im + -512} \]

       2. neg-mul-1 100.0%

          \[\leadsto \frac{\color{blue}{\left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   if -1.35000000000000003e154 < im < -8.0000000000000004e76 or 1.14999999999999997e77 < im < 1.4e154

   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. neg-sub0 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 58.1%

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

      1. +-commutative 58.1%

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

      2. mul-1-neg 58.1%

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

      3. unsub-neg 58.1%

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

      4. associate-*r* 58.1%

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

      5. distribute-rgt-out-- 58.1%

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

      6. *-commutative 58.1%

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

   6. Simplified 58.1%

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

   8. Applied egg-rr 4.2%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 4.2%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 4.2%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 4.2%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 4.2%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 4.2%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 4.2%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]
   11. Step-by-step derivation

       1. sub-neg 4.2%

          \[\leadsto \frac{\color{blue}{\left(262144 + \left(-im \cdot im\right)\right)} \cdot \cos re}{im + -512} \]

       2. flip-+ 97.4%

          \[\leadsto \frac{\color{blue}{\frac{262144 \cdot 262144 - \left(-im \cdot im\right) \cdot \left(-im \cdot im\right)}{262144 - \left(-im \cdot im\right)}} \cdot \cos re}{im + -512} \]

       3. metadata-eval 97.4%

          \[\leadsto \frac{\frac{\color{blue}{68719476736} - \left(-im \cdot im\right) \cdot \left(-im \cdot im\right)}{262144 - \left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   12. Applied egg-rr 97.4%

       \[\leadsto \frac{\color{blue}{\frac{68719476736 - \left(-im \cdot im\right) \cdot \left(-im \cdot im\right)}{262144 - \left(-im \cdot im\right)}} \cdot \cos re}{im + -512} \]

   if -8.0000000000000004e76 < im < -1e3

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.2%

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

      1. +-commutative 4.2%

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

      2. mul-1-neg 4.2%

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

      3. unsub-neg 4.2%

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

      4. associate-*r* 4.2%

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

      5. distribute-rgt-out-- 4.2%

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

      6. *-commutative 4.2%

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

   6. Simplified 4.2%

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

   7. Taylor expanded in re around 0 38.4%

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

      1. associate--l+ 38.4%

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

      2. associate-*r* 38.4%

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

      3. distribute-lft1-in 38.4%

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

      4. unpow2 38.4%

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

      5. associate-*r* 38.4%

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

   9. Simplified 38.4%

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

   if -1e3 < im < 7.5e11

   1. Initial program 10.7%

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

      1. neg-sub0 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in im around 0 97.0%

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

      1. associate-*r* 97.0%

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

      2. neg-mul-1 97.0%

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

   6. Simplified 97.0%

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

   if 7.5e11 < im < 1.14999999999999997e77

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

      1. associate-*r* 3.5%

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

      2. neg-mul-1 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 12.0%

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

      1. neg-mul-1 12.0%

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

      2. +-commutative 12.0%

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

      3. unsub-neg 12.0%

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

      4. associate-*r* 12.0%

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

      5. metadata-eval 12.0%

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

      6. pow-sqr 12.0%

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

      7. associate-*r* 12.0%

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

      8. associate-*r* 12.0%

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

      9. distribute-rgt-out 31.0%

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

      10. unpow2 31.0%

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

      11. *-commutative 31.0%

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

      12. associate-*l* 31.0%

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

      13. *-commutative 31.0%

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

      14. distribute-lft-out 31.0%

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

      15. unpow2 31.0%

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

   9. Simplified 31.0%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot \left(-0.041666666666666664 \cdot \left(re \cdot re\right) + 0.5\right)\right) - im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\cos re \cdot \frac{68719476736 - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im \cdot im + 262144}}{im + -512}\\ \mathbf{elif}\;im \leq -1000:\\ \;\;\;\;\left(re \cdot \left(re \cdot -0.5\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 750000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \frac{68719476736 - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im \cdot im + 262144}}{im + -512}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \end{array} \]

Alternative 7: 84.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot \left(re \cdot -0.5\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ t_1 := \frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 750000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+92}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (+ (* re (* re -0.5)) 1.0)
          (- (* (pow im 3.0) -0.16666666666666666) im)))
        (t_1 (/ (* (cos re) (- (* im im))) (+ im -512.0))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -1000.0)
       t_0
       (if (<= im 750000000000.0)
         (- (* im (cos re)))
         (if (<= im 3.1e+92)
           (-
            (* (* re re) (* im (+ 0.5 (* (* re re) -0.041666666666666664))))
            im)
           (if (<= im 1.4e+154) t_0 t_1)))))))

C

double code(double re, double im) {
	double t_0 = ((re * (re * -0.5)) + 1.0) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double t_1 = (cos(re) * -(im * im)) / (im + -512.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -1000.0) {
		tmp = t_0;
	} else if (im <= 750000000000.0) {
		tmp = -(im * cos(re));
	} else if (im <= 3.1e+92) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 1.4e+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 = ((re * (re * (-0.5d0))) + 1.0d0) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    t_1 = (cos(re) * -(im * im)) / (im + (-512.0d0))
    if (im <= (-1.35d+154)) then
        tmp = t_1
    else if (im <= (-1000.0d0)) then
        tmp = t_0
    else if (im <= 750000000000.0d0) then
        tmp = -(im * cos(re))
    else if (im <= 3.1d+92) then
        tmp = ((re * re) * (im * (0.5d0 + ((re * re) * (-0.041666666666666664d0))))) - im
    else if (im <= 1.4d+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 = ((re * (re * -0.5)) + 1.0) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double t_1 = (Math.cos(re) * -(im * im)) / (im + -512.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -1000.0) {
		tmp = t_0;
	} else if (im <= 750000000000.0) {
		tmp = -(im * Math.cos(re));
	} else if (im <= 3.1e+92) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((re * (re * -0.5)) + 1.0) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	t_1 = (math.cos(re) * -(im * im)) / (im + -512.0)
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -1000.0:
		tmp = t_0
	elif im <= 750000000000.0:
		tmp = -(im * math.cos(re))
	elif im <= 3.1e+92:
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im
	elif im <= 1.4e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(re * Float64(re * -0.5)) + 1.0) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	t_1 = Float64(Float64(cos(re) * Float64(-Float64(im * im))) / Float64(im + -512.0))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -1000.0)
		tmp = t_0;
	elseif (im <= 750000000000.0)
		tmp = Float64(-Float64(im * cos(re)));
	elseif (im <= 3.1e+92)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * Float64(0.5 + Float64(Float64(re * re) * -0.041666666666666664)))) - im);
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((re * (re * -0.5)) + 1.0) * (((im ^ 3.0) * -0.16666666666666666) - im);
	t_1 = (cos(re) * -(im * im)) / (im + -512.0);
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -1000.0)
		tmp = t_0;
	elseif (im <= 750000000000.0)
		tmp = -(im * cos(re));
	elseif (im <= 3.1e+92)
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(re * N[(re * -0.5), $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[(im * im), $MachinePrecision])), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -1000.0], t$95$0, If[LessEqual[im, 750000000000.0], (-N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), If[LessEqual[im, 3.1e+92], N[(N[(N[(re * re), $MachinePrecision] * N[(im * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 1.4e+154], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.35000000000000003e154 or 1.4e154 < 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. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.6%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.6%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in im around inf 100.0%

       \[\leadsto \frac{\color{blue}{\left(-1 \cdot {im}^{2}\right)} \cdot \cos re}{im + -512} \]
   12. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto \frac{\left(-1 \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \cos re}{im + -512} \]

       2. neg-mul-1 100.0%

          \[\leadsto \frac{\color{blue}{\left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   if -1.35000000000000003e154 < im < -1e3 or 3.1000000000000002e92 < im < 1.4e154

   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. neg-sub0 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 45.6%

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

      1. +-commutative 45.6%

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

      2. mul-1-neg 45.6%

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

      3. unsub-neg 45.6%

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

      4. associate-*r* 45.6%

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

      5. distribute-rgt-out-- 45.6%

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

      6. *-commutative 45.6%

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

   6. Simplified 45.6%

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

   7. Taylor expanded in re around 0 22.8%

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

      1. associate--l+ 22.8%

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

      2. associate-*r* 22.8%

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

      3. distribute-lft1-in 58.3%

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

      4. unpow2 58.3%

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

      5. associate-*r* 58.3%

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

   9. Simplified 58.3%

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

   if -1e3 < im < 7.5e11

   1. Initial program 10.7%

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

      1. neg-sub0 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in im around 0 97.0%

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

      1. associate-*r* 97.0%

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

      2. neg-mul-1 97.0%

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

   6. Simplified 97.0%

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

   if 7.5e11 < im < 3.1000000000000002e92

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

      1. associate-*r* 3.5%

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

      2. neg-mul-1 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 14.8%

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

      1. neg-mul-1 14.8%

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

      2. +-commutative 14.8%

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

      3. unsub-neg 14.8%

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

      4. associate-*r* 14.8%

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

      5. metadata-eval 14.8%

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

      6. pow-sqr 14.8%

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

      7. associate-*r* 14.8%

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

      8. associate-*r* 14.8%

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

      9. distribute-rgt-out 35.6%

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

      10. unpow2 35.6%

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

      11. *-commutative 35.6%

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

      12. associate-*l* 35.6%

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

      13. *-commutative 35.6%

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

      14. distribute-lft-out 35.6%

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

      15. unpow2 35.6%

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

   9. Simplified 35.6%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \mathbf{elif}\;im \leq -1000:\\ \;\;\;\;\left(re \cdot \left(re \cdot -0.5\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 750000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+92}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\left(re \cdot \left(re \cdot -0.5\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \end{array} \]

Alternative 8: 79.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := \frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{if}\;im \leq -6.4 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -150000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 750000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+95}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1
         (/
          (+
           262144.0
           (- (* -0.5 (* (* re re) (- 262144.0 (* im im)))) (* im im)))
          (+ im -512.0))))
   (if (<= im -6.4e+100)
     t_0
     (if (<= im -150000000000.0)
       t_1
       (if (<= im 750000000000.0)
         (- (* im (cos re)))
         (if (<= im 1.35e+95)
           (-
            (* (* re re) (* im (+ 0.5 (* (* re re) -0.041666666666666664))))
            im)
           (if (<= im 5.4e+133)
             t_1
             (if (<= im 1.4e+154)
               t_0
               (/ (* (cos re) (- (* im im))) (+ im -512.0))))))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	double tmp;
	if (im <= -6.4e+100) {
		tmp = t_0;
	} else if (im <= -150000000000.0) {
		tmp = t_1;
	} else if (im <= 750000000000.0) {
		tmp = -(im * cos(re));
	} else if (im <= 1.35e+95) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 5.4e+133) {
		tmp = t_1;
	} else if (im <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = (cos(re) * -(im * im)) / (im + -512.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = (262144.0d0 + (((-0.5d0) * ((re * re) * (262144.0d0 - (im * im)))) - (im * im))) / (im + (-512.0d0))
    if (im <= (-6.4d+100)) then
        tmp = t_0
    else if (im <= (-150000000000.0d0)) then
        tmp = t_1
    else if (im <= 750000000000.0d0) then
        tmp = -(im * cos(re))
    else if (im <= 1.35d+95) then
        tmp = ((re * re) * (im * (0.5d0 + ((re * re) * (-0.041666666666666664d0))))) - im
    else if (im <= 5.4d+133) then
        tmp = t_1
    else if (im <= 1.4d+154) then
        tmp = t_0
    else
        tmp = (cos(re) * -(im * im)) / (im + (-512.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	double tmp;
	if (im <= -6.4e+100) {
		tmp = t_0;
	} else if (im <= -150000000000.0) {
		tmp = t_1;
	} else if (im <= 750000000000.0) {
		tmp = -(im * Math.cos(re));
	} else if (im <= 1.35e+95) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 5.4e+133) {
		tmp = t_1;
	} else if (im <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = (Math.cos(re) * -(im * im)) / (im + -512.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0)
	tmp = 0
	if im <= -6.4e+100:
		tmp = t_0
	elif im <= -150000000000.0:
		tmp = t_1
	elif im <= 750000000000.0:
		tmp = -(im * math.cos(re))
	elif im <= 1.35e+95:
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im
	elif im <= 5.4e+133:
		tmp = t_1
	elif im <= 1.4e+154:
		tmp = t_0
	else:
		tmp = (math.cos(re) * -(im * im)) / (im + -512.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(Float64(262144.0 + Float64(Float64(-0.5 * Float64(Float64(re * re) * Float64(262144.0 - Float64(im * im)))) - Float64(im * im))) / Float64(im + -512.0))
	tmp = 0.0
	if (im <= -6.4e+100)
		tmp = t_0;
	elseif (im <= -150000000000.0)
		tmp = t_1;
	elseif (im <= 750000000000.0)
		tmp = Float64(-Float64(im * cos(re)));
	elseif (im <= 1.35e+95)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * Float64(0.5 + Float64(Float64(re * re) * -0.041666666666666664)))) - im);
	elseif (im <= 5.4e+133)
		tmp = t_1;
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(cos(re) * Float64(-Float64(im * im))) / Float64(im + -512.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	tmp = 0.0;
	if (im <= -6.4e+100)
		tmp = t_0;
	elseif (im <= -150000000000.0)
		tmp = t_1;
	elseif (im <= 750000000000.0)
		tmp = -(im * cos(re));
	elseif (im <= 1.35e+95)
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	elseif (im <= 5.4e+133)
		tmp = t_1;
	elseif (im <= 1.4e+154)
		tmp = t_0;
	else
		tmp = (cos(re) * -(im * im)) / (im + -512.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[(262144.0 + N[(N[(-0.5 * N[(N[(re * re), $MachinePrecision] * N[(262144.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.4e+100], t$95$0, If[LessEqual[im, -150000000000.0], t$95$1, If[LessEqual[im, 750000000000.0], (-N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), If[LessEqual[im, 1.35e+95], N[(N[(N[(re * re), $MachinePrecision] * N[(im * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 5.4e+133], t$95$1, If[LessEqual[im, 1.4e+154], t$95$0, N[(N[(N[Cos[re], $MachinePrecision] * (-N[(im * im), $MachinePrecision])), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -6.3999999999999998e100 or 5.4000000000000004e133 < im < 1.4e154

   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. neg-sub0 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%

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

      1. +-commutative 98.0%

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

      2. mul-1-neg 98.0%

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

      3. unsub-neg 98.0%

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

      4. associate-*r* 98.0%

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

      5. distribute-rgt-out-- 98.0%

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

      6. *-commutative 98.0%

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

   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 86.6%

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

   8. Taylor expanded in im around inf 86.6%

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

   if -6.3999999999999998e100 < im < -1.5e11 or 1.35e95 < im < 5.4000000000000004e133

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 30.0%

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

      1. +-commutative 30.0%

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

      2. mul-1-neg 30.0%

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

      3. unsub-neg 30.0%

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

      4. associate-*r* 30.0%

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

      5. distribute-rgt-out-- 30.0%

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

      6. *-commutative 30.0%

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

   6. Simplified 30.0%

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

   8. Applied egg-rr 3.8%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 3.8%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 3.8%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 3.8%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 3.8%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 3.8%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in re around 0 51.5%

       \[\leadsto \frac{\color{blue}{\left(262144 + -0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right)\right) - {im}^{2}}}{im + -512} \]
   12. Step-by-step derivation

       1. unpow2 51.5%

          \[\leadsto \frac{\left(262144 + -0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right)\right) - \color{blue}{im \cdot im}}{im + -512} \]

       2. associate--l+ 51.5%

          \[\leadsto \frac{\color{blue}{262144 + \left(-0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right) - im \cdot im\right)}}{im + -512} \]

       3. unpow2 51.5%

          \[\leadsto \frac{262144 + \left(-0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(262144 - {im}^{2}\right)\right) - im \cdot im\right)}{im + -512} \]

       4. unpow2 51.5%

          \[\leadsto \frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - \color{blue}{im \cdot im}\right)\right) - im \cdot im\right)}{im + -512} \]

   13. Simplified 51.5%

       \[\leadsto \frac{\color{blue}{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}}{im + -512} \]

   if -1.5e11 < im < 7.5e11

   1. Initial program 12.1%

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

      1. neg-sub0 12.1%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in im around 0 95.6%

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

      1. associate-*r* 95.6%

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

      2. neg-mul-1 95.6%

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

   6. Simplified 95.6%

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

   if 7.5e11 < im < 1.35e95

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

      1. associate-*r* 3.5%

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

      2. neg-mul-1 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 14.4%

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

      1. neg-mul-1 14.4%

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

      2. +-commutative 14.4%

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

      3. unsub-neg 14.4%

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

      4. associate-*r* 14.4%

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

      5. metadata-eval 14.4%

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

      6. pow-sqr 14.4%

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

      7. associate-*r* 14.4%

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

      8. associate-*r* 14.4%

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

      9. distribute-rgt-out 34.4%

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

      10. unpow2 34.4%

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

      11. *-commutative 34.4%

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

      12. associate-*l* 34.4%

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

      13. *-commutative 34.4%

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

      14. distribute-lft-out 34.4%

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

      15. unpow2 34.4%

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

   9. Simplified 34.4%

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

   if 1.4e154 < 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. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.4%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.4%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in im around inf 100.0%

       \[\leadsto \frac{\color{blue}{\left(-1 \cdot {im}^{2}\right)} \cdot \cos re}{im + -512} \]
   12. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto \frac{\left(-1 \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \cos re}{im + -512} \]

       2. neg-mul-1 100.0%

          \[\leadsto \frac{\color{blue}{\left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\left(-im \cdot im\right)} \cdot \cos re}{im + -512} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.4 \cdot 10^{+100}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -150000000000:\\ \;\;\;\;\frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{elif}\;im \leq 750000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+95}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;\frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(-im \cdot im\right)}{im + -512}\\ \end{array} \]

Alternative 9: 76.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := \frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{if}\;im \leq -6.8 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -150000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1500000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+93}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+134}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1
         (/
          (+
           262144.0
           (- (* -0.5 (* (* re re) (- 262144.0 (* im im)))) (* im im)))
          (+ im -512.0))))
   (if (<= im -6.8e+100)
     t_0
     (if (<= im -150000000000.0)
       t_1
       (if (<= im 1500000000000.0)
         (- (* im (cos re)))
         (if (<= im 1.5e+93)
           (-
            (* (* re re) (* im (+ 0.5 (* (* re re) -0.041666666666666664))))
            im)
           (if (<= im 3e+134) t_1 (- t_0 im))))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	double tmp;
	if (im <= -6.8e+100) {
		tmp = t_0;
	} else if (im <= -150000000000.0) {
		tmp = t_1;
	} else if (im <= 1500000000000.0) {
		tmp = -(im * cos(re));
	} else if (im <= 1.5e+93) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 3e+134) {
		tmp = t_1;
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = (262144.0d0 + (((-0.5d0) * ((re * re) * (262144.0d0 - (im * im)))) - (im * im))) / (im + (-512.0d0))
    if (im <= (-6.8d+100)) then
        tmp = t_0
    else if (im <= (-150000000000.0d0)) then
        tmp = t_1
    else if (im <= 1500000000000.0d0) then
        tmp = -(im * cos(re))
    else if (im <= 1.5d+93) then
        tmp = ((re * re) * (im * (0.5d0 + ((re * re) * (-0.041666666666666664d0))))) - im
    else if (im <= 3d+134) then
        tmp = t_1
    else
        tmp = t_0 - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	double tmp;
	if (im <= -6.8e+100) {
		tmp = t_0;
	} else if (im <= -150000000000.0) {
		tmp = t_1;
	} else if (im <= 1500000000000.0) {
		tmp = -(im * Math.cos(re));
	} else if (im <= 1.5e+93) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 3e+134) {
		tmp = t_1;
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0)
	tmp = 0
	if im <= -6.8e+100:
		tmp = t_0
	elif im <= -150000000000.0:
		tmp = t_1
	elif im <= 1500000000000.0:
		tmp = -(im * math.cos(re))
	elif im <= 1.5e+93:
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im
	elif im <= 3e+134:
		tmp = t_1
	else:
		tmp = t_0 - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(Float64(262144.0 + Float64(Float64(-0.5 * Float64(Float64(re * re) * Float64(262144.0 - Float64(im * im)))) - Float64(im * im))) / Float64(im + -512.0))
	tmp = 0.0
	if (im <= -6.8e+100)
		tmp = t_0;
	elseif (im <= -150000000000.0)
		tmp = t_1;
	elseif (im <= 1500000000000.0)
		tmp = Float64(-Float64(im * cos(re)));
	elseif (im <= 1.5e+93)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * Float64(0.5 + Float64(Float64(re * re) * -0.041666666666666664)))) - im);
	elseif (im <= 3e+134)
		tmp = t_1;
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	tmp = 0.0;
	if (im <= -6.8e+100)
		tmp = t_0;
	elseif (im <= -150000000000.0)
		tmp = t_1;
	elseif (im <= 1500000000000.0)
		tmp = -(im * cos(re));
	elseif (im <= 1.5e+93)
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	elseif (im <= 3e+134)
		tmp = t_1;
	else
		tmp = t_0 - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[(262144.0 + N[(N[(-0.5 * N[(N[(re * re), $MachinePrecision] * N[(262144.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.8e+100], t$95$0, If[LessEqual[im, -150000000000.0], t$95$1, If[LessEqual[im, 1500000000000.0], (-N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), If[LessEqual[im, 1.5e+93], N[(N[(N[(re * re), $MachinePrecision] * N[(im * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 3e+134], t$95$1, N[(t$95$0 - im), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -6.79999999999999988e100

   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. neg-sub0 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 97.7%

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

      1. +-commutative 97.7%

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

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

      4. associate-*r* 97.7%

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

      5. distribute-rgt-out-- 97.7%

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

      6. *-commutative 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in re around 0 84.9%

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

   8. Taylor expanded in im around inf 84.9%

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

   if -6.79999999999999988e100 < im < -1.5e11 or 1.49999999999999989e93 < im < 2.99999999999999997e134

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 30.0%

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

      1. +-commutative 30.0%

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

      2. mul-1-neg 30.0%

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

      3. unsub-neg 30.0%

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

      4. associate-*r* 30.0%

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

      5. distribute-rgt-out-- 30.0%

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

      6. *-commutative 30.0%

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

   6. Simplified 30.0%

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

   8. Applied egg-rr 3.8%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 3.8%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 3.8%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 3.8%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 3.8%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 3.8%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in re around 0 51.5%

       \[\leadsto \frac{\color{blue}{\left(262144 + -0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right)\right) - {im}^{2}}}{im + -512} \]
   12. Step-by-step derivation

       1. unpow2 51.5%

          \[\leadsto \frac{\left(262144 + -0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right)\right) - \color{blue}{im \cdot im}}{im + -512} \]

       2. associate--l+ 51.5%

          \[\leadsto \frac{\color{blue}{262144 + \left(-0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right) - im \cdot im\right)}}{im + -512} \]

       3. unpow2 51.5%

          \[\leadsto \frac{262144 + \left(-0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(262144 - {im}^{2}\right)\right) - im \cdot im\right)}{im + -512} \]

       4. unpow2 51.5%

          \[\leadsto \frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - \color{blue}{im \cdot im}\right)\right) - im \cdot im\right)}{im + -512} \]

   13. Simplified 51.5%

       \[\leadsto \frac{\color{blue}{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}}{im + -512} \]

   if -1.5e11 < im < 1.5e12

   1. Initial program 12.1%

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

      1. neg-sub0 12.1%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in im around 0 95.6%

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

      1. associate-*r* 95.6%

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

      2. neg-mul-1 95.6%

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

   6. Simplified 95.6%

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

   if 1.5e12 < im < 1.49999999999999989e93

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

      1. associate-*r* 3.5%

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

      2. neg-mul-1 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 14.4%

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

      1. neg-mul-1 14.4%

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

      2. +-commutative 14.4%

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

      3. unsub-neg 14.4%

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

      4. associate-*r* 14.4%

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

      5. metadata-eval 14.4%

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

      6. pow-sqr 14.4%

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

      7. associate-*r* 14.4%

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

      8. associate-*r* 14.4%

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

      9. distribute-rgt-out 34.4%

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

      10. unpow2 34.4%

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

      11. *-commutative 34.4%

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

      12. associate-*l* 34.4%

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

      13. *-commutative 34.4%

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

      14. distribute-lft-out 34.4%

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

      15. unpow2 34.4%

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

   9. Simplified 34.4%

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

   if 2.99999999999999997e134 < 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. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 82.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{+100}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -150000000000:\\ \;\;\;\;\frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{elif}\;im \leq 1500000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+93}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+134}:\\ \;\;\;\;\frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 10: 54.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := \frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{if}\;im \leq -7 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -150000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 235000000000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+93}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1
         (/
          (+
           262144.0
           (- (* -0.5 (* (* re re) (- 262144.0 (* im im)))) (* im im)))
          (+ im -512.0))))
   (if (<= im -7e+100)
     t_0
     (if (<= im -150000000000.0)
       t_1
       (if (<= im 235000000000.0)
         (- im)
         (if (<= im 3.9e+93)
           (-
            (* (* re re) (* im (+ 0.5 (* (* re re) -0.041666666666666664))))
            im)
           (if (<= im 5.4e+133) t_1 t_0)))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	double tmp;
	if (im <= -7e+100) {
		tmp = t_0;
	} else if (im <= -150000000000.0) {
		tmp = t_1;
	} else if (im <= 235000000000.0) {
		tmp = -im;
	} else if (im <= 3.9e+93) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 5.4e+133) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = (262144.0d0 + (((-0.5d0) * ((re * re) * (262144.0d0 - (im * im)))) - (im * im))) / (im + (-512.0d0))
    if (im <= (-7d+100)) then
        tmp = t_0
    else if (im <= (-150000000000.0d0)) then
        tmp = t_1
    else if (im <= 235000000000.0d0) then
        tmp = -im
    else if (im <= 3.9d+93) then
        tmp = ((re * re) * (im * (0.5d0 + ((re * re) * (-0.041666666666666664d0))))) - im
    else if (im <= 5.4d+133) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	double tmp;
	if (im <= -7e+100) {
		tmp = t_0;
	} else if (im <= -150000000000.0) {
		tmp = t_1;
	} else if (im <= 235000000000.0) {
		tmp = -im;
	} else if (im <= 3.9e+93) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 5.4e+133) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0)
	tmp = 0
	if im <= -7e+100:
		tmp = t_0
	elif im <= -150000000000.0:
		tmp = t_1
	elif im <= 235000000000.0:
		tmp = -im
	elif im <= 3.9e+93:
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im
	elif im <= 5.4e+133:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(Float64(262144.0 + Float64(Float64(-0.5 * Float64(Float64(re * re) * Float64(262144.0 - Float64(im * im)))) - Float64(im * im))) / Float64(im + -512.0))
	tmp = 0.0
	if (im <= -7e+100)
		tmp = t_0;
	elseif (im <= -150000000000.0)
		tmp = t_1;
	elseif (im <= 235000000000.0)
		tmp = Float64(-im);
	elseif (im <= 3.9e+93)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * Float64(0.5 + Float64(Float64(re * re) * -0.041666666666666664)))) - im);
	elseif (im <= 5.4e+133)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	tmp = 0.0;
	if (im <= -7e+100)
		tmp = t_0;
	elseif (im <= -150000000000.0)
		tmp = t_1;
	elseif (im <= 235000000000.0)
		tmp = -im;
	elseif (im <= 3.9e+93)
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	elseif (im <= 5.4e+133)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[(262144.0 + N[(N[(-0.5 * N[(N[(re * re), $MachinePrecision] * N[(262144.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7e+100], t$95$0, If[LessEqual[im, -150000000000.0], t$95$1, If[LessEqual[im, 235000000000.0], (-im), If[LessEqual[im, 3.9e+93], N[(N[(N[(re * re), $MachinePrecision] * N[(im * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 5.4e+133], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -6.99999999999999953e100 or 5.4000000000000004e133 < 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. neg-sub0 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 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. associate-*r* 98.7%

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

      5. distribute-rgt-out-- 98.7%

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

      6. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in re around 0 84.0%

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

   8. Taylor expanded in im around inf 84.0%

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

   if -6.99999999999999953e100 < im < -1.5e11 or 3.9000000000000002e93 < im < 5.4000000000000004e133

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 30.0%

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

      1. +-commutative 30.0%

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

      2. mul-1-neg 30.0%

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

      3. unsub-neg 30.0%

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

      4. associate-*r* 30.0%

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

      5. distribute-rgt-out-- 30.0%

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

      6. *-commutative 30.0%

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

   6. Simplified 30.0%

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

   8. Applied egg-rr 3.8%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 3.8%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 3.8%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 3.8%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 3.8%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 3.8%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in re around 0 51.5%

       \[\leadsto \frac{\color{blue}{\left(262144 + -0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right)\right) - {im}^{2}}}{im + -512} \]
   12. Step-by-step derivation

       1. unpow2 51.5%

          \[\leadsto \frac{\left(262144 + -0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right)\right) - \color{blue}{im \cdot im}}{im + -512} \]

       2. associate--l+ 51.5%

          \[\leadsto \frac{\color{blue}{262144 + \left(-0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right) - im \cdot im\right)}}{im + -512} \]

       3. unpow2 51.5%

          \[\leadsto \frac{262144 + \left(-0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(262144 - {im}^{2}\right)\right) - im \cdot im\right)}{im + -512} \]

       4. unpow2 51.5%

          \[\leadsto \frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - \color{blue}{im \cdot im}\right)\right) - im \cdot im\right)}{im + -512} \]

   13. Simplified 51.5%

       \[\leadsto \frac{\color{blue}{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}}{im + -512} \]

   if -1.5e11 < im < 2.35e11

   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. neg-sub0 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 96.5%

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

      1. +-commutative 96.5%

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

      2. mul-1-neg 96.5%

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

      3. unsub-neg 96.5%

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

      4. associate-*r* 96.5%

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

      5. distribute-rgt-out-- 96.5%

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

      6. *-commutative 96.5%

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

   6. Simplified 96.5%

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

   7. Taylor expanded in re around 0 57.0%

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

   8. Taylor expanded in im around 0 57.0%

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

      1. neg-mul-1 57.0%

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

   10. Simplified 57.0%

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

   if 2.35e11 < im < 3.9000000000000002e93

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

      1. associate-*r* 3.5%

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

      2. neg-mul-1 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 13.9%

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

      1. neg-mul-1 13.9%

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

      2. +-commutative 13.9%

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

      3. unsub-neg 13.9%

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

      4. associate-*r* 13.9%

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

      5. metadata-eval 13.9%

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

      6. pow-sqr 13.9%

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

      7. associate-*r* 13.9%

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

      8. associate-*r* 13.9%

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

      9. distribute-rgt-out 33.2%

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

      10. unpow2 33.2%

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

      11. *-commutative 33.2%

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

      12. associate-*l* 33.2%

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

      13. *-commutative 33.2%

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

      14. distribute-lft-out 33.2%

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

      15. unpow2 33.2%

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

   9. Simplified 33.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7 \cdot 10^{+100}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -150000000000:\\ \;\;\;\;\frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{elif}\;im \leq 235000000000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+93}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;\frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 11: 76.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := \frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{if}\;im \leq -7 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -150000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 750000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 10^{+97}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1
         (/
          (+
           262144.0
           (- (* -0.5 (* (* re re) (- 262144.0 (* im im)))) (* im im)))
          (+ im -512.0))))
   (if (<= im -7e+100)
     t_0
     (if (<= im -150000000000.0)
       t_1
       (if (<= im 750000000000.0)
         (- (* im (cos re)))
         (if (<= im 1e+97)
           (-
            (* (* re re) (* im (+ 0.5 (* (* re re) -0.041666666666666664))))
            im)
           (if (<= im 5.4e+133) t_1 t_0)))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	double tmp;
	if (im <= -7e+100) {
		tmp = t_0;
	} else if (im <= -150000000000.0) {
		tmp = t_1;
	} else if (im <= 750000000000.0) {
		tmp = -(im * cos(re));
	} else if (im <= 1e+97) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 5.4e+133) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = (262144.0d0 + (((-0.5d0) * ((re * re) * (262144.0d0 - (im * im)))) - (im * im))) / (im + (-512.0d0))
    if (im <= (-7d+100)) then
        tmp = t_0
    else if (im <= (-150000000000.0d0)) then
        tmp = t_1
    else if (im <= 750000000000.0d0) then
        tmp = -(im * cos(re))
    else if (im <= 1d+97) then
        tmp = ((re * re) * (im * (0.5d0 + ((re * re) * (-0.041666666666666664d0))))) - im
    else if (im <= 5.4d+133) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	double tmp;
	if (im <= -7e+100) {
		tmp = t_0;
	} else if (im <= -150000000000.0) {
		tmp = t_1;
	} else if (im <= 750000000000.0) {
		tmp = -(im * Math.cos(re));
	} else if (im <= 1e+97) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 5.4e+133) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0)
	tmp = 0
	if im <= -7e+100:
		tmp = t_0
	elif im <= -150000000000.0:
		tmp = t_1
	elif im <= 750000000000.0:
		tmp = -(im * math.cos(re))
	elif im <= 1e+97:
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im
	elif im <= 5.4e+133:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(Float64(262144.0 + Float64(Float64(-0.5 * Float64(Float64(re * re) * Float64(262144.0 - Float64(im * im)))) - Float64(im * im))) / Float64(im + -512.0))
	tmp = 0.0
	if (im <= -7e+100)
		tmp = t_0;
	elseif (im <= -150000000000.0)
		tmp = t_1;
	elseif (im <= 750000000000.0)
		tmp = Float64(-Float64(im * cos(re)));
	elseif (im <= 1e+97)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * Float64(0.5 + Float64(Float64(re * re) * -0.041666666666666664)))) - im);
	elseif (im <= 5.4e+133)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = (262144.0 + ((-0.5 * ((re * re) * (262144.0 - (im * im)))) - (im * im))) / (im + -512.0);
	tmp = 0.0;
	if (im <= -7e+100)
		tmp = t_0;
	elseif (im <= -150000000000.0)
		tmp = t_1;
	elseif (im <= 750000000000.0)
		tmp = -(im * cos(re));
	elseif (im <= 1e+97)
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	elseif (im <= 5.4e+133)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[(262144.0 + N[(N[(-0.5 * N[(N[(re * re), $MachinePrecision] * N[(262144.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7e+100], t$95$0, If[LessEqual[im, -150000000000.0], t$95$1, If[LessEqual[im, 750000000000.0], (-N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), If[LessEqual[im, 1e+97], N[(N[(N[(re * re), $MachinePrecision] * N[(im * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 5.4e+133], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -6.99999999999999953e100 or 5.4000000000000004e133 < 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. neg-sub0 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 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. associate-*r* 98.7%

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

      5. distribute-rgt-out-- 98.7%

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

      6. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in re around 0 84.0%

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

   8. Taylor expanded in im around inf 84.0%

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

   if -6.99999999999999953e100 < im < -1.5e11 or 1.0000000000000001e97 < im < 5.4000000000000004e133

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 30.0%

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

      1. +-commutative 30.0%

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

      2. mul-1-neg 30.0%

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

      3. unsub-neg 30.0%

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

      4. associate-*r* 30.0%

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

      5. distribute-rgt-out-- 30.0%

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

      6. *-commutative 30.0%

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

   6. Simplified 30.0%

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

   8. Applied egg-rr 3.8%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 3.8%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 3.8%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 3.8%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 3.8%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 3.8%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in re around 0 51.5%

       \[\leadsto \frac{\color{blue}{\left(262144 + -0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right)\right) - {im}^{2}}}{im + -512} \]
   12. Step-by-step derivation

       1. unpow2 51.5%

          \[\leadsto \frac{\left(262144 + -0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right)\right) - \color{blue}{im \cdot im}}{im + -512} \]

       2. associate--l+ 51.5%

          \[\leadsto \frac{\color{blue}{262144 + \left(-0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right) - im \cdot im\right)}}{im + -512} \]

       3. unpow2 51.5%

          \[\leadsto \frac{262144 + \left(-0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(262144 - {im}^{2}\right)\right) - im \cdot im\right)}{im + -512} \]

       4. unpow2 51.5%

          \[\leadsto \frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - \color{blue}{im \cdot im}\right)\right) - im \cdot im\right)}{im + -512} \]

   13. Simplified 51.5%

       \[\leadsto \frac{\color{blue}{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}}{im + -512} \]

   if -1.5e11 < im < 7.5e11

   1. Initial program 12.1%

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

      1. neg-sub0 12.1%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in im around 0 95.6%

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

      1. associate-*r* 95.6%

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

      2. neg-mul-1 95.6%

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

   6. Simplified 95.6%

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

   if 7.5e11 < im < 1.0000000000000001e97

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

      1. associate-*r* 3.5%

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

      2. neg-mul-1 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 14.4%

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

      1. neg-mul-1 14.4%

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

      2. +-commutative 14.4%

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

      3. unsub-neg 14.4%

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

      4. associate-*r* 14.4%

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

      5. metadata-eval 14.4%

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

      6. pow-sqr 14.4%

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

      7. associate-*r* 14.4%

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

      8. associate-*r* 14.4%

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

      9. distribute-rgt-out 34.4%

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

      10. unpow2 34.4%

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

      11. *-commutative 34.4%

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

      12. associate-*l* 34.4%

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

      13. *-commutative 34.4%

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

      14. distribute-lft-out 34.4%

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

      15. unpow2 34.4%

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

   9. Simplified 34.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7 \cdot 10^{+100}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -150000000000:\\ \;\;\;\;\frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{elif}\;im \leq 750000000000:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{elif}\;im \leq 10^{+97}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;\frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 12: 51.2% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 262144 - im \cdot im\\ t_1 := \frac{t_0}{im + -512}\\ t_2 := \frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot t_0\right) - im \cdot im\right)}{im + -512}\\ \mathbf{if}\;im \leq -7.2 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -150000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 235000000000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+96}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 262144.0 (* im im)))
        (t_1 (/ t_0 (+ im -512.0)))
        (t_2
         (/
          (+ 262144.0 (- (* -0.5 (* (* re re) t_0)) (* im im)))
          (+ im -512.0))))
   (if (<= im -7.2e+141)
     t_1
     (if (<= im -150000000000.0)
       t_2
       (if (<= im 235000000000.0)
         (- im)
         (if (<= im 2.5e+96)
           (-
            (* (* re re) (* im (+ 0.5 (* (* re re) -0.041666666666666664))))
            im)
           (if (<= im 1.55e+135) t_2 t_1)))))))

C

double code(double re, double im) {
	double t_0 = 262144.0 - (im * im);
	double t_1 = t_0 / (im + -512.0);
	double t_2 = (262144.0 + ((-0.5 * ((re * re) * t_0)) - (im * im))) / (im + -512.0);
	double tmp;
	if (im <= -7.2e+141) {
		tmp = t_1;
	} else if (im <= -150000000000.0) {
		tmp = t_2;
	} else if (im <= 235000000000.0) {
		tmp = -im;
	} else if (im <= 2.5e+96) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 1.55e+135) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 262144.0d0 - (im * im)
    t_1 = t_0 / (im + (-512.0d0))
    t_2 = (262144.0d0 + (((-0.5d0) * ((re * re) * t_0)) - (im * im))) / (im + (-512.0d0))
    if (im <= (-7.2d+141)) then
        tmp = t_1
    else if (im <= (-150000000000.0d0)) then
        tmp = t_2
    else if (im <= 235000000000.0d0) then
        tmp = -im
    else if (im <= 2.5d+96) then
        tmp = ((re * re) * (im * (0.5d0 + ((re * re) * (-0.041666666666666664d0))))) - im
    else if (im <= 1.55d+135) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 262144.0 - (im * im);
	double t_1 = t_0 / (im + -512.0);
	double t_2 = (262144.0 + ((-0.5 * ((re * re) * t_0)) - (im * im))) / (im + -512.0);
	double tmp;
	if (im <= -7.2e+141) {
		tmp = t_1;
	} else if (im <= -150000000000.0) {
		tmp = t_2;
	} else if (im <= 235000000000.0) {
		tmp = -im;
	} else if (im <= 2.5e+96) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 1.55e+135) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 262144.0 - (im * im)
	t_1 = t_0 / (im + -512.0)
	t_2 = (262144.0 + ((-0.5 * ((re * re) * t_0)) - (im * im))) / (im + -512.0)
	tmp = 0
	if im <= -7.2e+141:
		tmp = t_1
	elif im <= -150000000000.0:
		tmp = t_2
	elif im <= 235000000000.0:
		tmp = -im
	elif im <= 2.5e+96:
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im
	elif im <= 1.55e+135:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(262144.0 - Float64(im * im))
	t_1 = Float64(t_0 / Float64(im + -512.0))
	t_2 = Float64(Float64(262144.0 + Float64(Float64(-0.5 * Float64(Float64(re * re) * t_0)) - Float64(im * im))) / Float64(im + -512.0))
	tmp = 0.0
	if (im <= -7.2e+141)
		tmp = t_1;
	elseif (im <= -150000000000.0)
		tmp = t_2;
	elseif (im <= 235000000000.0)
		tmp = Float64(-im);
	elseif (im <= 2.5e+96)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * Float64(0.5 + Float64(Float64(re * re) * -0.041666666666666664)))) - im);
	elseif (im <= 1.55e+135)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 262144.0 - (im * im);
	t_1 = t_0 / (im + -512.0);
	t_2 = (262144.0 + ((-0.5 * ((re * re) * t_0)) - (im * im))) / (im + -512.0);
	tmp = 0.0;
	if (im <= -7.2e+141)
		tmp = t_1;
	elseif (im <= -150000000000.0)
		tmp = t_2;
	elseif (im <= 235000000000.0)
		tmp = -im;
	elseif (im <= 2.5e+96)
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	elseif (im <= 1.55e+135)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(262144.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(262144.0 + N[(N[(-0.5 * N[(N[(re * re), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.2e+141], t$95$1, If[LessEqual[im, -150000000000.0], t$95$2, If[LessEqual[im, 235000000000.0], (-im), If[LessEqual[im, 2.5e+96], N[(N[(N[(re * re), $MachinePrecision] * N[(im * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 1.55e+135], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -7.2000000000000003e141 or 1.55000000000000011e135 < 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. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.2%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.2%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 87.9%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 87.9%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 87.9%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 87.9%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 87.9%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in re around 0 73.6%

       \[\leadsto \color{blue}{\frac{262144 - {im}^{2}}{im - 512}} \]
   12. Step-by-step derivation

       1. unpow2 73.6%

          \[\leadsto \frac{262144 - \color{blue}{im \cdot im}}{im - 512} \]

       2. sub-neg 73.6%

          \[\leadsto \frac{262144 - im \cdot im}{\color{blue}{im + \left(-512\right)}} \]

       3. metadata-eval 73.6%

          \[\leadsto \frac{262144 - im \cdot im}{im + \color{blue}{-512}} \]

   13. Simplified 73.6%

       \[\leadsto \color{blue}{\frac{262144 - im \cdot im}{im + -512}} \]

   if -7.2000000000000003e141 < im < -1.5e11 or 2.5000000000000002e96 < im < 1.55000000000000011e135

   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. neg-sub0 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 37.1%

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

      1. +-commutative 37.1%

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

      2. mul-1-neg 37.1%

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

      3. unsub-neg 37.1%

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

      4. associate-*r* 37.1%

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

      5. distribute-rgt-out-- 37.1%

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

      6. *-commutative 37.1%

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

   6. Simplified 37.1%

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

   8. Applied egg-rr 3.9%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 3.9%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 3.9%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 3.9%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 3.9%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 3.9%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 3.9%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in re around 0 47.5%

       \[\leadsto \frac{\color{blue}{\left(262144 + -0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right)\right) - {im}^{2}}}{im + -512} \]
   12. Step-by-step derivation

       1. unpow2 47.5%

          \[\leadsto \frac{\left(262144 + -0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right)\right) - \color{blue}{im \cdot im}}{im + -512} \]

       2. associate--l+ 47.5%

          \[\leadsto \frac{\color{blue}{262144 + \left(-0.5 \cdot \left({re}^{2} \cdot \left(262144 - {im}^{2}\right)\right) - im \cdot im\right)}}{im + -512} \]

       3. unpow2 47.5%

          \[\leadsto \frac{262144 + \left(-0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(262144 - {im}^{2}\right)\right) - im \cdot im\right)}{im + -512} \]

       4. unpow2 47.5%

          \[\leadsto \frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - \color{blue}{im \cdot im}\right)\right) - im \cdot im\right)}{im + -512} \]

   13. Simplified 47.5%

       \[\leadsto \frac{\color{blue}{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}}{im + -512} \]

   if -1.5e11 < im < 2.35e11

   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. neg-sub0 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 96.5%

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

      1. +-commutative 96.5%

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

      2. mul-1-neg 96.5%

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

      3. unsub-neg 96.5%

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

      4. associate-*r* 96.5%

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

      5. distribute-rgt-out-- 96.5%

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

      6. *-commutative 96.5%

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

   6. Simplified 96.5%

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

   7. Taylor expanded in re around 0 57.0%

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

   8. Taylor expanded in im around 0 57.0%

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

      1. neg-mul-1 57.0%

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

   10. Simplified 57.0%

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

   if 2.35e11 < im < 2.5000000000000002e96

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

      1. associate-*r* 3.5%

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

      2. neg-mul-1 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 13.9%

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

      1. neg-mul-1 13.9%

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

      2. +-commutative 13.9%

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

      3. unsub-neg 13.9%

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

      4. associate-*r* 13.9%

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

      5. metadata-eval 13.9%

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

      6. pow-sqr 13.9%

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

      7. associate-*r* 13.9%

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

      8. associate-*r* 13.9%

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

      9. distribute-rgt-out 33.2%

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

      10. unpow2 33.2%

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

      11. *-commutative 33.2%

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

      12. associate-*l* 33.2%

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

      13. *-commutative 33.2%

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

      14. distribute-lft-out 33.2%

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

      15. unpow2 33.2%

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

   9. Simplified 33.2%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{262144 - im \cdot im}{im + -512}\\ \mathbf{elif}\;im \leq -150000000000:\\ \;\;\;\;\frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{elif}\;im \leq 235000000000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+96}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;\frac{262144 + \left(-0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(262144 - im \cdot im\right)\right) - im \cdot im\right)}{im + -512}\\ \mathbf{else}:\\ \;\;\;\;\frac{262144 - im \cdot im}{im + -512}\\ \end{array} \]

Alternative 13: 50.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{262144 - im \cdot im}{im + -512}\\ t_1 := \left(im \cdot 0.5\right) \cdot \left(re \cdot re + -2\right)\\ \mathbf{if}\;im \leq -7.2 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 235000000000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+93}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ (- 262144.0 (* im im)) (+ im -512.0)))
        (t_1 (* (* im 0.5) (+ (* re re) -2.0))))
   (if (<= im -7.2e+141)
     t_0
     (if (<= im -5.5e-14)
       t_1
       (if (<= im 235000000000.0)
         (- im)
         (if (<= im 2.9e+93)
           (-
            (* (* re re) (* im (+ 0.5 (* (* re re) -0.041666666666666664))))
            im)
           (if (<= im 1.55e+135) t_1 t_0)))))))

C

double code(double re, double im) {
	double t_0 = (262144.0 - (im * im)) / (im + -512.0);
	double t_1 = (im * 0.5) * ((re * re) + -2.0);
	double tmp;
	if (im <= -7.2e+141) {
		tmp = t_0;
	} else if (im <= -5.5e-14) {
		tmp = t_1;
	} else if (im <= 235000000000.0) {
		tmp = -im;
	} else if (im <= 2.9e+93) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 1.55e+135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (262144.0d0 - (im * im)) / (im + (-512.0d0))
    t_1 = (im * 0.5d0) * ((re * re) + (-2.0d0))
    if (im <= (-7.2d+141)) then
        tmp = t_0
    else if (im <= (-5.5d-14)) then
        tmp = t_1
    else if (im <= 235000000000.0d0) then
        tmp = -im
    else if (im <= 2.9d+93) then
        tmp = ((re * re) * (im * (0.5d0 + ((re * re) * (-0.041666666666666664d0))))) - im
    else if (im <= 1.55d+135) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (262144.0 - (im * im)) / (im + -512.0);
	double t_1 = (im * 0.5) * ((re * re) + -2.0);
	double tmp;
	if (im <= -7.2e+141) {
		tmp = t_0;
	} else if (im <= -5.5e-14) {
		tmp = t_1;
	} else if (im <= 235000000000.0) {
		tmp = -im;
	} else if (im <= 2.9e+93) {
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	} else if (im <= 1.55e+135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (262144.0 - (im * im)) / (im + -512.0)
	t_1 = (im * 0.5) * ((re * re) + -2.0)
	tmp = 0
	if im <= -7.2e+141:
		tmp = t_0
	elif im <= -5.5e-14:
		tmp = t_1
	elif im <= 235000000000.0:
		tmp = -im
	elif im <= 2.9e+93:
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im
	elif im <= 1.55e+135:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(262144.0 - Float64(im * im)) / Float64(im + -512.0))
	t_1 = Float64(Float64(im * 0.5) * Float64(Float64(re * re) + -2.0))
	tmp = 0.0
	if (im <= -7.2e+141)
		tmp = t_0;
	elseif (im <= -5.5e-14)
		tmp = t_1;
	elseif (im <= 235000000000.0)
		tmp = Float64(-im);
	elseif (im <= 2.9e+93)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * Float64(0.5 + Float64(Float64(re * re) * -0.041666666666666664)))) - im);
	elseif (im <= 1.55e+135)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (262144.0 - (im * im)) / (im + -512.0);
	t_1 = (im * 0.5) * ((re * re) + -2.0);
	tmp = 0.0;
	if (im <= -7.2e+141)
		tmp = t_0;
	elseif (im <= -5.5e-14)
		tmp = t_1;
	elseif (im <= 235000000000.0)
		tmp = -im;
	elseif (im <= 2.9e+93)
		tmp = ((re * re) * (im * (0.5 + ((re * re) * -0.041666666666666664)))) - im;
	elseif (im <= 1.55e+135)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(262144.0 - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * 0.5), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.2e+141], t$95$0, If[LessEqual[im, -5.5e-14], t$95$1, If[LessEqual[im, 235000000000.0], (-im), If[LessEqual[im, 2.9e+93], N[(N[(N[(re * re), $MachinePrecision] * N[(im * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 1.55e+135], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -7.2000000000000003e141 or 1.55000000000000011e135 < 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. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.2%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.2%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 87.9%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 87.9%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 87.9%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 87.9%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 87.9%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in re around 0 73.6%

       \[\leadsto \color{blue}{\frac{262144 - {im}^{2}}{im - 512}} \]
   12. Step-by-step derivation

       1. unpow2 73.6%

          \[\leadsto \frac{262144 - \color{blue}{im \cdot im}}{im - 512} \]

       2. sub-neg 73.6%

          \[\leadsto \frac{262144 - im \cdot im}{\color{blue}{im + \left(-512\right)}} \]

       3. metadata-eval 73.6%

          \[\leadsto \frac{262144 - im \cdot im}{im + \color{blue}{-512}} \]

   13. Simplified 73.6%

       \[\leadsto \color{blue}{\frac{262144 - im \cdot im}{im + -512}} \]

   if -7.2000000000000003e141 < im < -5.49999999999999991e-14 or 2.8999999999999998e93 < im < 1.55000000000000011e135

   1. Initial program 95.7%

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

      1. neg-sub0 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in im around 0 12.2%

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

      1. associate-*r* 12.2%

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

      2. neg-mul-1 12.2%

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

   6. Simplified 12.2%

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

   7. Taylor expanded in re around 0 38.7%

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

      1. neg-mul-1 38.7%

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

      2. +-commutative 38.7%

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

      3. associate-*r* 38.7%

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

      4. neg-mul-1 38.7%

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

      5. metadata-eval 38.7%

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

      6. associate-*r* 38.7%

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

      7. *-commutative 38.7%

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

      8. distribute-lft-out 38.7%

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

      9. *-commutative 38.7%

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

      10. unpow2 38.7%

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

   9. Simplified 38.7%

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

   if -5.49999999999999991e-14 < im < 2.35e11

   1. Initial program 7.9%

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

      1. neg-sub0 7.9%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in im around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. associate-*r* 98.7%

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

      5. distribute-rgt-out-- 98.7%

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

      6. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in re around 0 58.5%

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

   8. Taylor expanded in im around 0 58.5%

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

      1. neg-mul-1 58.5%

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

   10. Simplified 58.5%

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

   if 2.35e11 < im < 2.8999999999999998e93

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

      1. associate-*r* 3.5%

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

      2. neg-mul-1 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 13.9%

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

      1. neg-mul-1 13.9%

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

      2. +-commutative 13.9%

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

      3. unsub-neg 13.9%

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

      4. associate-*r* 13.9%

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

      5. metadata-eval 13.9%

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

      6. pow-sqr 13.9%

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

      7. associate-*r* 13.9%

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

      8. associate-*r* 13.9%

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

      9. distribute-rgt-out 33.2%

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

      10. unpow2 33.2%

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

      11. *-commutative 33.2%

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

      12. associate-*l* 33.2%

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

      13. *-commutative 33.2%

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

      14. distribute-lft-out 33.2%

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

      15. unpow2 33.2%

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

   9. Simplified 33.2%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{262144 - im \cdot im}{im + -512}\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{-14}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \left(re \cdot re + -2\right)\\ \mathbf{elif}\;im \leq 235000000000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+93}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \left(re \cdot re + -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{262144 - im \cdot im}{im + -512}\\ \end{array} \]

Alternative 14: 50.4% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{262144 - im \cdot im}{im + -512}\\ \mathbf{if}\;im \leq -7.2 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -9 \cdot 10^{-14}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \left(re \cdot re + -2\right)\\ \mathbf{elif}\;im \leq 0.175:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;\left(im + 512\right) \cdot \left(-1 + re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ (- 262144.0 (* im im)) (+ im -512.0))))
   (if (<= im -7.2e+141)
     t_0
     (if (<= im -9e-14)
       (* (* im 0.5) (+ (* re re) -2.0))
       (if (<= im 0.175)
         (- im)
         (if (<= im 1.55e+135)
           (* (+ im 512.0) (+ -1.0 (* re (* 0.5 re))))
           t_0))))))

C

double code(double re, double im) {
	double t_0 = (262144.0 - (im * im)) / (im + -512.0);
	double tmp;
	if (im <= -7.2e+141) {
		tmp = t_0;
	} else if (im <= -9e-14) {
		tmp = (im * 0.5) * ((re * re) + -2.0);
	} else if (im <= 0.175) {
		tmp = -im;
	} else if (im <= 1.55e+135) {
		tmp = (im + 512.0) * (-1.0 + (re * (0.5 * re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (262144.0d0 - (im * im)) / (im + (-512.0d0))
    if (im <= (-7.2d+141)) then
        tmp = t_0
    else if (im <= (-9d-14)) then
        tmp = (im * 0.5d0) * ((re * re) + (-2.0d0))
    else if (im <= 0.175d0) then
        tmp = -im
    else if (im <= 1.55d+135) then
        tmp = (im + 512.0d0) * ((-1.0d0) + (re * (0.5d0 * re)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (262144.0 - (im * im)) / (im + -512.0);
	double tmp;
	if (im <= -7.2e+141) {
		tmp = t_0;
	} else if (im <= -9e-14) {
		tmp = (im * 0.5) * ((re * re) + -2.0);
	} else if (im <= 0.175) {
		tmp = -im;
	} else if (im <= 1.55e+135) {
		tmp = (im + 512.0) * (-1.0 + (re * (0.5 * re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (262144.0 - (im * im)) / (im + -512.0)
	tmp = 0
	if im <= -7.2e+141:
		tmp = t_0
	elif im <= -9e-14:
		tmp = (im * 0.5) * ((re * re) + -2.0)
	elif im <= 0.175:
		tmp = -im
	elif im <= 1.55e+135:
		tmp = (im + 512.0) * (-1.0 + (re * (0.5 * re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(262144.0 - Float64(im * im)) / Float64(im + -512.0))
	tmp = 0.0
	if (im <= -7.2e+141)
		tmp = t_0;
	elseif (im <= -9e-14)
		tmp = Float64(Float64(im * 0.5) * Float64(Float64(re * re) + -2.0));
	elseif (im <= 0.175)
		tmp = Float64(-im);
	elseif (im <= 1.55e+135)
		tmp = Float64(Float64(im + 512.0) * Float64(-1.0 + Float64(re * Float64(0.5 * re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (262144.0 - (im * im)) / (im + -512.0);
	tmp = 0.0;
	if (im <= -7.2e+141)
		tmp = t_0;
	elseif (im <= -9e-14)
		tmp = (im * 0.5) * ((re * re) + -2.0);
	elseif (im <= 0.175)
		tmp = -im;
	elseif (im <= 1.55e+135)
		tmp = (im + 512.0) * (-1.0 + (re * (0.5 * re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(262144.0 - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.2e+141], t$95$0, If[LessEqual[im, -9e-14], N[(N[(im * 0.5), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 0.175], (-im), If[LessEqual[im, 1.55e+135], N[(N[(im + 512.0), $MachinePrecision] * N[(-1.0 + N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -7.2000000000000003e141 or 1.55000000000000011e135 < 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. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.2%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.2%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 87.9%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 87.9%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 87.9%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 87.9%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 87.9%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in re around 0 73.6%

       \[\leadsto \color{blue}{\frac{262144 - {im}^{2}}{im - 512}} \]
   12. Step-by-step derivation

       1. unpow2 73.6%

          \[\leadsto \frac{262144 - \color{blue}{im \cdot im}}{im - 512} \]

       2. sub-neg 73.6%

          \[\leadsto \frac{262144 - im \cdot im}{\color{blue}{im + \left(-512\right)}} \]

       3. metadata-eval 73.6%

          \[\leadsto \frac{262144 - im \cdot im}{im + \color{blue}{-512}} \]

   13. Simplified 73.6%

       \[\leadsto \color{blue}{\frac{262144 - im \cdot im}{im + -512}} \]

   if -7.2000000000000003e141 < im < -8.9999999999999995e-14

   1. Initial program 94.0%

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

      1. neg-sub0 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in im around 0 15.2%

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

      1. associate-*r* 15.2%

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

      2. neg-mul-1 15.2%

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

   6. Simplified 15.2%

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

   7. Taylor expanded in re around 0 33.3%

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

      1. neg-mul-1 33.3%

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

      2. +-commutative 33.3%

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

      3. associate-*r* 33.3%

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

      4. neg-mul-1 33.3%

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

      5. metadata-eval 33.3%

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

      6. associate-*r* 33.3%

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

      7. *-commutative 33.3%

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

      8. distribute-lft-out 33.3%

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

      9. *-commutative 33.3%

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

      10. unpow2 33.3%

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

   9. Simplified 33.3%

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

   if -8.9999999999999995e-14 < im < 0.17499999999999999

   1. Initial program 6.4%

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

      1. neg-sub0 6.4%

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

   3. Simplified 6.4%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out-- 99.9%

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

      6. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in re around 0 59.5%

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

   8. Taylor expanded in im around 0 59.5%

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

      1. neg-mul-1 59.5%

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

   10. Simplified 59.5%

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

   if 0.17499999999999999 < im < 1.55000000000000011e135

   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. neg-sub0 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 25.4%

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

      1. +-commutative 25.4%

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

      2. mul-1-neg 25.4%

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

      3. unsub-neg 25.4%

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

      4. associate-*r* 25.4%

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

      5. distribute-rgt-out-- 25.4%

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

      6. *-commutative 25.4%

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

   6. Simplified 25.4%

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

   8. Applied egg-rr 3.9%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]

   9. Taylor expanded in re around 0 20.0%

      \[\leadsto \color{blue}{-1 \cdot \left(512 + im\right) + 0.5 \cdot \left({re}^{2} \cdot \left(512 + im\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 20.0%

          \[\leadsto -1 \cdot \left(512 + im\right) + 0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(512 + im\right)\right) \]

       2. associate-*r* 20.0%

          \[\leadsto -1 \cdot \left(512 + im\right) + \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(512 + im\right)} \]

       3. distribute-rgt-out 20.0%

          \[\leadsto \color{blue}{\left(512 + im\right) \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)} \]

       4. +-commutative 20.0%

          \[\leadsto \color{blue}{\left(im + 512\right)} \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right) \]

       5. metadata-eval 20.0%

          \[\leadsto \left(im + 512\right) \cdot \left(-1 + \color{blue}{\left(--0.5\right)} \cdot \left(re \cdot re\right)\right) \]

       6. distribute-lft-neg-in 20.0%

          \[\leadsto \left(im + 512\right) \cdot \left(-1 + \color{blue}{\left(--0.5 \cdot \left(re \cdot re\right)\right)}\right) \]

       7. associate-*r* 20.0%

          \[\leadsto \left(im + 512\right) \cdot \left(-1 + \left(-\color{blue}{\left(-0.5 \cdot re\right) \cdot re}\right)\right) \]

       8. *-commutative 20.0%

          \[\leadsto \left(im + 512\right) \cdot \left(-1 + \left(-\color{blue}{re \cdot \left(-0.5 \cdot re\right)}\right)\right) \]

       9. distribute-rgt-neg-in 20.0%

          \[\leadsto \left(im + 512\right) \cdot \left(-1 + \color{blue}{re \cdot \left(--0.5 \cdot re\right)}\right) \]

       10. distribute-lft-neg-in 20.0%

           \[\leadsto \left(im + 512\right) \cdot \left(-1 + re \cdot \color{blue}{\left(\left(--0.5\right) \cdot re\right)}\right) \]

       11. metadata-eval 20.0%

           \[\leadsto \left(im + 512\right) \cdot \left(-1 + re \cdot \left(\color{blue}{0.5} \cdot re\right)\right) \]

   11. Simplified 20.0%

       \[\leadsto \color{blue}{\left(im + 512\right) \cdot \left(-1 + re \cdot \left(0.5 \cdot re\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{262144 - im \cdot im}{im + -512}\\ \mathbf{elif}\;im \leq -9 \cdot 10^{-14}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \left(re \cdot re + -2\right)\\ \mathbf{elif}\;im \leq 0.175:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;\left(im + 512\right) \cdot \left(-1 + re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{262144 - im \cdot im}{im + -512}\\ \end{array} \]

Alternative 15: 50.4% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{262144 - im \cdot im}{im + -512}\\ t_1 := \left(im \cdot 0.5\right) \cdot \left(re \cdot re + -2\right)\\ \mathbf{if}\;im \leq -7.2 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.02 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 0.175:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ (- 262144.0 (* im im)) (+ im -512.0)))
        (t_1 (* (* im 0.5) (+ (* re re) -2.0))))
   (if (<= im -7.2e+141)
     t_0
     (if (<= im -1.02e-13)
       t_1
       (if (<= im 0.175) (- im) (if (<= im 1.55e+135) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = (262144.0 - (im * im)) / (im + -512.0);
	double t_1 = (im * 0.5) * ((re * re) + -2.0);
	double tmp;
	if (im <= -7.2e+141) {
		tmp = t_0;
	} else if (im <= -1.02e-13) {
		tmp = t_1;
	} else if (im <= 0.175) {
		tmp = -im;
	} else if (im <= 1.55e+135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (262144.0d0 - (im * im)) / (im + (-512.0d0))
    t_1 = (im * 0.5d0) * ((re * re) + (-2.0d0))
    if (im <= (-7.2d+141)) then
        tmp = t_0
    else if (im <= (-1.02d-13)) then
        tmp = t_1
    else if (im <= 0.175d0) then
        tmp = -im
    else if (im <= 1.55d+135) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (262144.0 - (im * im)) / (im + -512.0);
	double t_1 = (im * 0.5) * ((re * re) + -2.0);
	double tmp;
	if (im <= -7.2e+141) {
		tmp = t_0;
	} else if (im <= -1.02e-13) {
		tmp = t_1;
	} else if (im <= 0.175) {
		tmp = -im;
	} else if (im <= 1.55e+135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (262144.0 - (im * im)) / (im + -512.0)
	t_1 = (im * 0.5) * ((re * re) + -2.0)
	tmp = 0
	if im <= -7.2e+141:
		tmp = t_0
	elif im <= -1.02e-13:
		tmp = t_1
	elif im <= 0.175:
		tmp = -im
	elif im <= 1.55e+135:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(262144.0 - Float64(im * im)) / Float64(im + -512.0))
	t_1 = Float64(Float64(im * 0.5) * Float64(Float64(re * re) + -2.0))
	tmp = 0.0
	if (im <= -7.2e+141)
		tmp = t_0;
	elseif (im <= -1.02e-13)
		tmp = t_1;
	elseif (im <= 0.175)
		tmp = Float64(-im);
	elseif (im <= 1.55e+135)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (262144.0 - (im * im)) / (im + -512.0);
	t_1 = (im * 0.5) * ((re * re) + -2.0);
	tmp = 0.0;
	if (im <= -7.2e+141)
		tmp = t_0;
	elseif (im <= -1.02e-13)
		tmp = t_1;
	elseif (im <= 0.175)
		tmp = -im;
	elseif (im <= 1.55e+135)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(262144.0 - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(im + -512.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * 0.5), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.2e+141], t$95$0, If[LessEqual[im, -1.02e-13], t$95$1, If[LessEqual[im, 0.175], (-im), If[LessEqual[im, 1.55e+135], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.2000000000000003e141 or 1.55000000000000011e135 < 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. neg-sub0 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%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.2%

      \[\leadsto \cos re \cdot \left(\color{blue}{-512} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.2%

         \[\leadsto \color{blue}{\left(-512 - im\right) \cdot \cos re} \]

      2. flip-- 87.9%

         \[\leadsto \color{blue}{\frac{-512 \cdot -512 - im \cdot im}{-512 + im}} \cdot \cos re \]

      3. associate-*l/ 87.9%

         \[\leadsto \color{blue}{\frac{\left(-512 \cdot -512 - im \cdot im\right) \cdot \cos re}{-512 + im}} \]

      4. metadata-eval 87.9%

         \[\leadsto \frac{\left(\color{blue}{262144} - im \cdot im\right) \cdot \cos re}{-512 + im} \]

      5. +-commutative 87.9%

         \[\leadsto \frac{\left(262144 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -512}} \]

   10. Applied egg-rr 87.9%

       \[\leadsto \color{blue}{\frac{\left(262144 - im \cdot im\right) \cdot \cos re}{im + -512}} \]

   11. Taylor expanded in re around 0 73.6%

       \[\leadsto \color{blue}{\frac{262144 - {im}^{2}}{im - 512}} \]
   12. Step-by-step derivation

       1. unpow2 73.6%

          \[\leadsto \frac{262144 - \color{blue}{im \cdot im}}{im - 512} \]

       2. sub-neg 73.6%

          \[\leadsto \frac{262144 - im \cdot im}{\color{blue}{im + \left(-512\right)}} \]

       3. metadata-eval 73.6%

          \[\leadsto \frac{262144 - im \cdot im}{im + \color{blue}{-512}} \]

   13. Simplified 73.6%

       \[\leadsto \color{blue}{\frac{262144 - im \cdot im}{im + -512}} \]

   if -7.2000000000000003e141 < im < -1.0199999999999999e-13 or 0.17499999999999999 < im < 1.55000000000000011e135

   1. Initial program 97.3%

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

      1. neg-sub0 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in im around 0 9.2%

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

      1. associate-*r* 9.2%

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

      2. neg-mul-1 9.2%

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

   6. Simplified 9.2%

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

   7. Taylor expanded in re around 0 25.9%

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

      1. neg-mul-1 25.9%

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

      2. +-commutative 25.9%

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

      3. associate-*r* 25.9%

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

      4. neg-mul-1 25.9%

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

      5. metadata-eval 25.9%

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

      6. associate-*r* 25.9%

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

      7. *-commutative 25.9%

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

      8. distribute-lft-out 25.9%

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

      9. *-commutative 25.9%

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

      10. unpow2 25.9%

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

   9. Simplified 25.9%

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

   if -1.0199999999999999e-13 < im < 0.17499999999999999

   1. Initial program 6.4%

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

      1. neg-sub0 6.4%

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

   3. Simplified 6.4%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out-- 99.9%

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

      6. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in re around 0 59.5%

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

   8. Taylor expanded in im around 0 59.5%

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

      1. neg-mul-1 59.5%

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

   10. Simplified 59.5%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{262144 - im \cdot im}{im + -512}\\ \mathbf{elif}\;im \leq -1.02 \cdot 10^{-13}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \left(re \cdot re + -2\right)\\ \mathbf{elif}\;im \leq 0.175:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \left(re \cdot re + -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{262144 - im \cdot im}{im + -512}\\ \end{array} \]

Alternative 16: 36.1% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.5 \cdot 10^{+157}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \left(re \cdot re + -2\right)\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{+233}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.5e+157)
   (* (* im 0.5) (+ (* re re) -2.0))
   (if (<= re 1.75e+233)
     (* (* re re) -6.75)
     (* (+ 0.5 (* -0.25 (* re re))) -3.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.5e+157) {
		tmp = (im * 0.5) * ((re * re) + -2.0);
	} else if (re <= 1.75e+233) {
		tmp = (re * re) * -6.75;
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * -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 <= 6.5d+157) then
        tmp = (im * 0.5d0) * ((re * re) + (-2.0d0))
    else if (re <= 1.75d+233) then
        tmp = (re * re) * (-6.75d0)
    else
        tmp = (0.5d0 + ((-0.25d0) * (re * re))) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.5e+157) {
		tmp = (im * 0.5) * ((re * re) + -2.0);
	} else if (re <= 1.75e+233) {
		tmp = (re * re) * -6.75;
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.5e+157:
		tmp = (im * 0.5) * ((re * re) + -2.0)
	elif re <= 1.75e+233:
		tmp = (re * re) * -6.75
	else:
		tmp = (0.5 + (-0.25 * (re * re))) * -3.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.5e+157)
		tmp = Float64(Float64(im * 0.5) * Float64(Float64(re * re) + -2.0));
	elseif (re <= 1.75e+233)
		tmp = Float64(Float64(re * re) * -6.75);
	else
		tmp = Float64(Float64(0.5 + Float64(-0.25 * Float64(re * re))) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.5e+157)
		tmp = (im * 0.5) * ((re * re) + -2.0);
	elseif (re <= 1.75e+233)
		tmp = (re * re) * -6.75;
	else
		tmp = (0.5 + (-0.25 * (re * re))) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.5e+157], N[(N[(im * 0.5), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.75e+233], N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision], N[(N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 6.5e157

   1. Initial program 55.9%

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

      1. neg-sub0 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in im around 0 50.7%

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

      1. associate-*r* 50.7%

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

      2. neg-mul-1 50.7%

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

   6. Simplified 50.7%

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

   7. Taylor expanded in re around 0 39.1%

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

      1. neg-mul-1 39.1%

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

      2. +-commutative 39.1%

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

      3. associate-*r* 39.1%

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

      4. neg-mul-1 39.1%

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

      5. metadata-eval 39.1%

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

      6. associate-*r* 39.1%

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

      7. *-commutative 39.1%

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

      8. distribute-lft-out 39.1%

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

      9. *-commutative 39.1%

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

      10. unpow2 39.1%

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

   9. Simplified 39.1%

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

   if 6.5e157 < re < 1.7499999999999999e233

   1. Initial program 39.0%

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

      1. neg-sub0 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 21.4%

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

      4. unpow2 21.4%

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

   6. Simplified 21.4%

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

   8. Applied egg-rr 29.8%

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

   9. Taylor expanded in re around inf 29.8%

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

       1. unpow2 29.8%

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

       2. *-commutative 29.8%

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

       3. associate-*l* 29.8%

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

   11. Simplified 29.8%

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

   12. Taylor expanded in re around 0 29.8%

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

       1. unpow2 29.8%

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

   14. Simplified 29.8%

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

   if 1.7499999999999999e233 < re

   1. Initial program 56.5%

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

      1. neg-sub0 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in re around 0 0.2%

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

      1. +-commutative 0.2%

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

      2. associate-*r* 0.2%

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

      3. distribute-rgt-out 26.9%

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

      4. unpow2 26.9%

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

   6. Simplified 26.9%

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

   8. Applied egg-rr 27.6%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.5 \cdot 10^{+157}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \left(re \cdot re + -2\right)\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{+233}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot -3\\ \end{array} \]

Alternative 17: 36.1% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.6 \cdot 10^{+171}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+231}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.6e+171)
   (- (* 0.5 (* re (* im re))) im)
   (if (<= re 2.3e+231)
     (* (* re re) -6.75)
     (* (+ 0.5 (* -0.25 (* re re))) -3.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.6e+171) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else if (re <= 2.3e+231) {
		tmp = (re * re) * -6.75;
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * -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 <= 3.6d+171) then
        tmp = (0.5d0 * (re * (im * re))) - im
    else if (re <= 2.3d+231) then
        tmp = (re * re) * (-6.75d0)
    else
        tmp = (0.5d0 + ((-0.25d0) * (re * re))) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.6e+171) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else if (re <= 2.3e+231) {
		tmp = (re * re) * -6.75;
	} else {
		tmp = (0.5 + (-0.25 * (re * re))) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.6e+171:
		tmp = (0.5 * (re * (im * re))) - im
	elif re <= 2.3e+231:
		tmp = (re * re) * -6.75
	else:
		tmp = (0.5 + (-0.25 * (re * re))) * -3.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.6e+171)
		tmp = Float64(Float64(0.5 * Float64(re * Float64(im * re))) - im);
	elseif (re <= 2.3e+231)
		tmp = Float64(Float64(re * re) * -6.75);
	else
		tmp = Float64(Float64(0.5 + Float64(-0.25 * Float64(re * re))) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.6e+171)
		tmp = (0.5 * (re * (im * re))) - im;
	elseif (re <= 2.3e+231)
		tmp = (re * re) * -6.75;
	else
		tmp = (0.5 + (-0.25 * (re * re))) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.6e+171], N[(N[(0.5 * N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[re, 2.3e+231], N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision], N[(N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 3.60000000000000018e171

   1. Initial program 56.1%

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

      1. neg-sub0 56.1%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in im around 0 50.6%

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

      1. associate-*r* 50.6%

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

      2. neg-mul-1 50.6%

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

   6. Simplified 50.6%

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

   7. Taylor expanded in re around 0 39.4%

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

      1. neg-mul-1 39.4%

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

      2. +-commutative 39.4%

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

      3. unsub-neg 39.4%

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

      4. *-commutative 39.4%

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

      5. unpow2 39.4%

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

      6. associate-*l* 39.4%

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

   9. Simplified 39.4%

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

   if 3.60000000000000018e171 < re < 2.29999999999999999e231

   1. Initial program 34.3%

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

      1. neg-sub0 34.3%

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

   3. Simplified 34.3%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 15.4%

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

      4. unpow2 15.4%

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

   6. Simplified 15.4%

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

   8. Applied egg-rr 24.4%

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

   9. Taylor expanded in re around inf 24.4%

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

       1. unpow2 24.4%

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

       2. *-commutative 24.4%

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

       3. associate-*l* 24.4%

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

   11. Simplified 24.4%

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

   12. Taylor expanded in re around 0 24.4%

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

       1. unpow2 24.4%

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

   14. Simplified 24.4%

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

   if 2.29999999999999999e231 < re

   1. Initial program 56.5%

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

      1. neg-sub0 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in re around 0 0.2%

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

      1. +-commutative 0.2%

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

      2. associate-*r* 0.2%

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

      3. distribute-rgt-out 26.9%

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

      4. unpow2 26.9%

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

   6. Simplified 26.9%

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

   8. Applied egg-rr 27.6%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.6 \cdot 10^{+171}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+231}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot -3\\ \end{array} \]

Alternative 18: 31.5% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.05e+162) (- im) (* (* re re) -6.75)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.05e+162) {
		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 <= 1.05d+162) 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 <= 1.05e+162) {
		tmp = -im;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.05e+162:
		tmp = -im
	else:
		tmp = (re * re) * -6.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.05e+162)
		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 <= 1.05e+162)
		tmp = -im;
	else
		tmp = (re * re) * -6.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.05e+162], (-im), N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.05e162

   1. Initial program 55.9%

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

      1. neg-sub0 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in im around 0 79.2%

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

      1. +-commutative 79.2%

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

      2. mul-1-neg 79.2%

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

      3. unsub-neg 79.2%

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

      4. associate-*r* 79.2%

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

      5. distribute-rgt-out-- 79.2%

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

      6. *-commutative 79.2%

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

   6. Simplified 79.2%

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

   7. Taylor expanded in re around 0 56.9%

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

   8. Taylor expanded in im around 0 34.1%

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

      1. neg-mul-1 34.1%

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

   10. Simplified 34.1%

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

   if 1.05e162 < re

   1. Initial program 48.0%

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

      1. neg-sub0 48.0%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 24.2%

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

      4. unpow2 24.2%

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

   6. Simplified 24.2%

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

   8. Applied egg-rr 25.1%

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

   9. Taylor expanded in re around inf 25.1%

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

       1. unpow2 25.1%

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

       2. *-commutative 25.1%

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

       3. associate-*l* 25.1%

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

   11. Simplified 25.1%

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

   12. Taylor expanded in re around 0 25.1%

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

       1. unpow2 25.1%

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

   14. Simplified 25.1%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 19: 31.5% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8.6 \cdot 10^{+161}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8.6e+161) (* 0.5 (* im -2.0)) (* (* re re) -6.75)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8.6e+161) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 8.6d+161) then
        tmp = 0.5d0 * (im * (-2.0d0))
    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 <= 8.6e+161) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8.6e+161:
		tmp = 0.5 * (im * -2.0)
	else:
		tmp = (re * re) * -6.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8.6e+161)
		tmp = Float64(0.5 * Float64(im * -2.0));
	else
		tmp = Float64(Float64(re * re) * -6.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 8.6e+161)
		tmp = 0.5 * (im * -2.0);
	else
		tmp = (re * re) * -6.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8.6e+161], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.6e161

   1. Initial program 55.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 55.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 55.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 44.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

   5. Taylor expanded in im around 0 34.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im\right)} \]

   if 8.6e161 < re

   1. Initial program 48.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 48.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 48.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 0.1%

         \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right) + -0.25 \cdot \left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      2. associate-*r* 0.1%

         \[\leadsto 0.5 \cdot \left(e^{-im} - e^{im}\right) + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      3. distribute-rgt-out 24.2%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. unpow2 24.2%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   6. Simplified 24.2%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   8. Applied egg-rr 25.1%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]

   9. Taylor expanded in re around inf 25.1%

      \[\leadsto \color{blue}{-6.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 25.1%

          \[\leadsto -6.75 \cdot \color{blue}{\left(re \cdot re\right)} \]

       2. *-commutative 25.1%

          \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot -6.75} \]

       3. associate-*l* 25.1%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot -6.75\right)} \]

   11. Simplified 25.1%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot -6.75\right)} \]

   12. Taylor expanded in re around 0 25.1%

       \[\leadsto \color{blue}{-6.75 \cdot {re}^{2}} \]
   13. Step-by-step derivation

       1. unpow2 25.1%

          \[\leadsto -6.75 \cdot \color{blue}{\left(re \cdot re\right)} \]

   14. Simplified 25.1%

       \[\leadsto \color{blue}{-6.75 \cdot \left(re \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8.6 \cdot 10^{+161}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 20: 29.8% accurate, 154.5× speedup

Math

\[\begin{array}{l} \\ -im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (- im))

C

double code(double re, double im) {
	return -im;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -im
end function

Java

public static double code(double re, double im) {
	return -im;
}

Python

def code(re, im):
	return -im

Julia

function code(re, im)
	return Float64(-im)
end

MATLAB

function tmp = code(re, im)
	tmp = -im;
end

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 55.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. neg-sub0 55.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

3. Simplified 55.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

4. Taylor expanded in im around 0 79.5%

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
5. Step-by-step derivation

   1. +-commutative 79.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

   2. mul-1-neg 79.5%

      \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

   3. unsub-neg 79.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

   4. associate-*r* 79.5%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

   5. distribute-rgt-out-- 79.5%

      \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

   6. *-commutative 79.5%

      \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

6. Simplified 79.5%

   \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

7. Taylor expanded in re around 0 52.8%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

8. Taylor expanded in im around 0 31.1%

   \[\leadsto \color{blue}{-1 \cdot im} \]
9. Step-by-step derivation

   1. neg-mul-1 31.1%

      \[\leadsto \color{blue}{-im} \]

10. Simplified 31.1%

    \[\leadsto \color{blue}{-im} \]

11. Final simplification 31.1%

    \[\leadsto -im \]

Alternative 21: 2.8% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 13.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 13.5)

C

double code(double re, double im) {
	return 13.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 13.5d0
end function

Java

public static double code(double re, double im) {
	return 13.5;
}

Python

def code(re, im):
	return 13.5

Julia

function code(re, im)
	return 13.5
end

MATLAB

function tmp = code(re, im)
	tmp = 13.5;
end

Wolfram

code[re_, im_] := 13.5

Derivation

1. Initial program 55.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. neg-sub0 55.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

3. Simplified 55.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

4. Taylor expanded in re around 0 2.4%

   \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
5. Step-by-step derivation

   1. +-commutative 2.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right) + -0.25 \cdot \left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} \]

   2. associate-*r* 2.4%

      \[\leadsto 0.5 \cdot \left(e^{-im} - e^{im}\right) + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   3. distribute-rgt-out 39.5%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

   4. unpow2 39.5%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

6. Simplified 39.5%

   \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

8. Applied egg-rr 8.4%

   \[\leadsto \color{blue}{27} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]

9. Taylor expanded in re around 0 2.8%

   \[\leadsto \color{blue}{13.5} \]

10. Final simplification 2.8%

    \[\leadsto 13.5 \]

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 2023271 
(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))))