math.sin on complex, imaginary part

Percentage Accurate: 55.2% → 99.3%

Time: 16.3s

Alternatives: 20

Speedup: 2.8×

• 50.6% 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: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+15} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1e+15) (not (<= t_0 0.0)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (+
       (+
        (* (pow im 7.0) -0.0001984126984126984)
        (* (pow im 5.0) -0.008333333333333333))
       (- (* (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 <= -1e+15) || !(t_0 <= 0.0)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * (((pow(im, 7.0) * -0.0001984126984126984) + (pow(im, 5.0) * -0.008333333333333333)) + ((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 <= (-1d+15)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * ((((im ** 7.0d0) * (-0.0001984126984126984d0)) + ((im ** 5.0d0) * (-0.008333333333333333d0))) + (((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 <= -1e+15) || !(t_0 <= 0.0)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * (((Math.pow(im, 7.0) * -0.0001984126984126984) + (Math.pow(im, 5.0) * -0.008333333333333333)) + ((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 <= -1e+15) or not (t_0 <= 0.0):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * (((math.pow(im, 7.0) * -0.0001984126984126984) + (math.pow(im, 5.0) * -0.008333333333333333)) + ((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 <= -1e+15) || !(t_0 <= 0.0))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64((im ^ 7.0) * -0.0001984126984126984) + Float64((im ^ 5.0) * -0.008333333333333333)) + 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 <= -1e+15) || ~((t_0 <= 0.0)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((((im ^ 7.0) * -0.0001984126984126984) + ((im ^ 5.0) * -0.008333333333333333)) + (((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, -1e+15], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -1e15 or 0.0 < (-.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 -1e15 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.0

   1. Initial program 7.6%

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

      1. neg-sub0 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      3. associate-*r* 99.8%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      4. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-1 \cdot im\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      5. neg-mul-1 99.8%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-im\right)} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      7. +-commutative 99.8%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\left(-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]

      8. associate-*r* 99.8%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) \]

      9. associate-*r* 99.8%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re}\right) \]

      10. distribute-rgt-out 99.8%

          \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\cos re \cdot \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)} \]

      11. distribute-lft-out 99.8%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)\right)} \]

      12. +-commutative 99.8%

          \[\leadsto \cos re \cdot \color{blue}{\left(\left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)\right)} \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\mathsf{fma}\left({im}^{5}, -0.008333333333333333, {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]
   7. Step-by-step derivation

      1. fma-udef 99.8%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right) \]

      2. +-commutative 99.8%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right) \]

   8. Applied egg-rr 99.8%

      \[\leadsto \cos re \cdot \left(\color{blue}{\left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+15} \lor \neg \left(e^{-im} - e^{im} \leq 0\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+15} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) + {im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1e+15) (not (<= t_0 0.0)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (+
       (- (* (* im im) (* im -0.16666666666666666)) im)
       (* (pow im 5.0) -0.008333333333333333))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -1e+15) || !(t_0 <= 0.0)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((((im * im) * (im * -0.16666666666666666)) - im) + (pow(im, 5.0) * -0.008333333333333333));
	}
	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 <= (-1d+15)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * ((((im * im) * (im * (-0.16666666666666666d0))) - im) + ((im ** 5.0d0) * (-0.008333333333333333d0)))
    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 <= -1e+15) || !(t_0 <= 0.0)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((((im * im) * (im * -0.16666666666666666)) - im) + (Math.pow(im, 5.0) * -0.008333333333333333));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -1e+15) or not (t_0 <= 0.0):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((((im * im) * (im * -0.16666666666666666)) - im) + (math.pow(im, 5.0) * -0.008333333333333333))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -1e+15) || !(t_0 <= 0.0))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64(Float64(im * im) * Float64(im * -0.16666666666666666)) - im) + Float64((im ^ 5.0) * -0.008333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -1e+15) || ~((t_0 <= 0.0)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((((im * im) * (im * -0.16666666666666666)) - im) + ((im ^ 5.0) * -0.008333333333333333));
	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, -1e+15], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -1e15 or 0.0 < (-.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 -1e15 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.0

   1. Initial program 7.6%

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

      1. neg-sub0 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      3. associate-*r* 99.8%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      4. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-1 \cdot im\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      5. neg-mul-1 99.8%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-im\right)} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      7. +-commutative 99.8%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\left(-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]

      8. associate-*r* 99.8%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) \]

      9. associate-*r* 99.8%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re}\right) \]

      10. distribute-rgt-out 99.8%

          \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\cos re \cdot \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)} \]

      11. distribute-lft-out 99.8%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)\right)} \]

      12. +-commutative 99.8%

          \[\leadsto \cos re \cdot \color{blue}{\left(\left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)\right)} \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\mathsf{fma}\left({im}^{5}, -0.008333333333333333, {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   8. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)} \]

      2. associate-*r* 99.8%

         \[\leadsto \left(\color{blue}{\left(-1 \cdot im\right) \cdot \cos re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      3. neg-mul-1 99.8%

         \[\leadsto \left(\color{blue}{\left(-im\right)} \cdot \cos re + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      4. associate-*r* 99.8%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re}\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      5. *-commutative 99.8%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + \color{blue}{\left({im}^{3} \cdot -0.16666666666666666\right)} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(-im\right) + {im}^{3} \cdot -0.16666666666666666\right)} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      7. +-commutative 99.8%

         \[\leadsto \cos re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 + \left(-im\right)\right)} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      8. sub-neg 99.8%

         \[\leadsto \cos re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right)} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      9. associate-*r* 99.8%

         \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re} \]

      10. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) + \color{blue}{\left({im}^{5} \cdot -0.008333333333333333\right)} \cdot \cos re \]

      11. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) + \color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]

      12. distribute-lft-out 99.8%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + {im}^{5} \cdot -0.008333333333333333\right)} \]

      13. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{-0.16666666666666666 \cdot {im}^{3}} - im\right) + {im}^{5} \cdot -0.008333333333333333\right) \]

      14. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} - im\right) + \color{blue}{-0.008333333333333333 \cdot {im}^{5}}\right) \]

   9. Simplified 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right)} \]
   10. Step-by-step derivation

       1. +-rgt-identity 99.8%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} + 0\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       2. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left(\left(\left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} + 0\right) - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       3. unpow3 99.8%

          \[\leadsto \cos re \cdot \left(\left(\left(\color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} \cdot -0.16666666666666666 + 0\right) - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       4. associate-*l* 99.8%

          \[\leadsto \cos re \cdot \left(\left(\left(\color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right)} + 0\right) - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       5. fma-def 99.8%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, 0\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       6. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left(\left(\mathsf{fma}\left(im \cdot im, \color{blue}{-0.16666666666666666 \cdot im}, 0\right) - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

   11. Applied egg-rr 99.8%

       \[\leadsto \cos re \cdot \left(\left(\color{blue}{\mathsf{fma}\left(im \cdot im, -0.16666666666666666 \cdot im, 0\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]
   12. Step-by-step derivation

       1. fma-udef 99.8%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right) + 0\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       2. +-rgt-identity 99.8%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

   13. Simplified 99.8%

       \[\leadsto \cos re \cdot \left(\left(\color{blue}{\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+15} \lor \neg \left(e^{-im} - e^{im} \leq 0\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) + {im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \]

Alternative 3: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.122:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.33:\\ \;\;\;\;\cos re \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) + {im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re)))))
        (t_1 (* -0.0001984126984126984 (* (cos re) (pow im 7.0)))))
   (if (<= im -1.1e+44)
     t_1
     (if (<= im -0.122)
       t_0
       (if (<= im 0.33)
         (*
          (cos re)
          (+
           (- (* (* im im) (* im -0.16666666666666666)) im)
           (* (pow im 5.0) -0.008333333333333333)))
         (if (<= im 1.1e+44) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) - exp(im)) * (0.5 + (-0.25 * (re * re)));
	double t_1 = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -0.122) {
		tmp = t_0;
	} else if (im <= 0.33) {
		tmp = cos(re) * ((((im * im) * (im * -0.16666666666666666)) - im) + (pow(im, 5.0) * -0.008333333333333333));
	} else if (im <= 1.1e+44) {
		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 = (exp(-im) - exp(im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    t_1 = (-0.0001984126984126984d0) * (cos(re) * (im ** 7.0d0))
    if (im <= (-1.1d+44)) then
        tmp = t_1
    else if (im <= (-0.122d0)) then
        tmp = t_0
    else if (im <= 0.33d0) then
        tmp = cos(re) * ((((im * im) * (im * (-0.16666666666666666d0))) - im) + ((im ** 5.0d0) * (-0.008333333333333333d0)))
    else if (im <= 1.1d+44) 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.exp(-im) - Math.exp(im)) * (0.5 + (-0.25 * (re * re)));
	double t_1 = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -0.122) {
		tmp = t_0;
	} else if (im <= 0.33) {
		tmp = Math.cos(re) * ((((im * im) * (im * -0.16666666666666666)) - im) + (Math.pow(im, 5.0) * -0.008333333333333333));
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) - math.exp(im)) * (0.5 + (-0.25 * (re * re)))
	t_1 = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.1e+44:
		tmp = t_1
	elif im <= -0.122:
		tmp = t_0
	elif im <= 0.33:
		tmp = math.cos(re) * ((((im * im) * (im * -0.16666666666666666)) - im) + (math.pow(im, 5.0) * -0.008333333333333333))
	elif im <= 1.1e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))))
	t_1 = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -0.122)
		tmp = t_0;
	elseif (im <= 0.33)
		tmp = Float64(cos(re) * Float64(Float64(Float64(Float64(im * im) * Float64(im * -0.16666666666666666)) - im) + Float64((im ^ 5.0) * -0.008333333333333333)));
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) - exp(im)) * (0.5 + (-0.25 * (re * re)));
	t_1 = -0.0001984126984126984 * (cos(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -0.122)
		tmp = t_0;
	elseif (im <= 0.33)
		tmp = cos(re) * ((((im * im) * (im * -0.16666666666666666)) - im) + ((im ^ 5.0) * -0.008333333333333333));
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.1e+44], t$95$1, If[LessEqual[im, -0.122], t$95$0, If[LessEqual[im, 0.33], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.09999999999999998e44 or 1.09999999999999998e44 < 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%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-1 \cdot im\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      5. neg-mul-1 100.0%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-im\right)} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      6. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      7. +-commutative 100.0%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\left(-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) \]

      9. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re}\right) \]

      10. distribute-rgt-out 100.0%

          \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\cos re \cdot \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)} \]

      11. distribute-lft-out 100.0%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)\right)} \]

      12. +-commutative 100.0%

          \[\leadsto \cos re \cdot \color{blue}{\left(\left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\mathsf{fma}\left({im}^{5}, -0.008333333333333333, {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]

   if -1.09999999999999998e44 < im < -0.122 or 0.330000000000000016 < im < 1.09999999999999998e44

   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 5.0%

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

      1. +-commutative 5.0%

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

      2. associate-*r* 5.0%

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

      3. distribute-rgt-out 85.0%

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

      4. unpow2 85.0%

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

   6. Simplified 85.0%

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

   if -0.122 < im < 0.330000000000000016

   1. Initial program 7.6%

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

      1. neg-sub0 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      3. associate-*r* 99.8%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      4. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-1 \cdot im\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      5. neg-mul-1 99.8%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-im\right)} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      7. +-commutative 99.8%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\left(-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]

      8. associate-*r* 99.8%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) \]

      9. associate-*r* 99.8%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re}\right) \]

      10. distribute-rgt-out 99.8%

          \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\cos re \cdot \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)} \]

      11. distribute-lft-out 99.8%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)\right)} \]

      12. +-commutative 99.8%

          \[\leadsto \cos re \cdot \color{blue}{\left(\left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)\right)} \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\mathsf{fma}\left({im}^{5}, -0.008333333333333333, {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   8. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)} \]

      2. associate-*r* 99.8%

         \[\leadsto \left(\color{blue}{\left(-1 \cdot im\right) \cdot \cos re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      3. neg-mul-1 99.8%

         \[\leadsto \left(\color{blue}{\left(-im\right)} \cdot \cos re + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      4. associate-*r* 99.8%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re}\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      5. *-commutative 99.8%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + \color{blue}{\left({im}^{3} \cdot -0.16666666666666666\right)} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(-im\right) + {im}^{3} \cdot -0.16666666666666666\right)} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      7. +-commutative 99.8%

         \[\leadsto \cos re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 + \left(-im\right)\right)} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      8. sub-neg 99.8%

         \[\leadsto \cos re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right)} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) \]

      9. associate-*r* 99.8%

         \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re} \]

      10. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) + \color{blue}{\left({im}^{5} \cdot -0.008333333333333333\right)} \cdot \cos re \]

      11. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) + \color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]

      12. distribute-lft-out 99.8%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + {im}^{5} \cdot -0.008333333333333333\right)} \]

      13. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{-0.16666666666666666 \cdot {im}^{3}} - im\right) + {im}^{5} \cdot -0.008333333333333333\right) \]

      14. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} - im\right) + \color{blue}{-0.008333333333333333 \cdot {im}^{5}}\right) \]

   9. Simplified 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right)} \]
   10. Step-by-step derivation

       1. +-rgt-identity 99.8%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} + 0\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       2. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left(\left(\left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} + 0\right) - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       3. unpow3 99.8%

          \[\leadsto \cos re \cdot \left(\left(\left(\color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} \cdot -0.16666666666666666 + 0\right) - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       4. associate-*l* 99.8%

          \[\leadsto \cos re \cdot \left(\left(\left(\color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right)} + 0\right) - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       5. fma-def 99.8%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, 0\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       6. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left(\left(\mathsf{fma}\left(im \cdot im, \color{blue}{-0.16666666666666666 \cdot im}, 0\right) - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

   11. Applied egg-rr 99.8%

       \[\leadsto \cos re \cdot \left(\left(\color{blue}{\mathsf{fma}\left(im \cdot im, -0.16666666666666666 \cdot im, 0\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]
   12. Step-by-step derivation

       1. fma-udef 99.8%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right) + 0\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

       2. +-rgt-identity 99.8%

          \[\leadsto \cos re \cdot \left(\left(\color{blue}{\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]

   13. Simplified 99.8%

       \[\leadsto \cos re \cdot \left(\left(\color{blue}{\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)} - im\right) + -0.008333333333333333 \cdot {im}^{5}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.122:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 0.33:\\ \;\;\;\;\cos re \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) + {im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.025:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.065:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re)))))
        (t_1 (* -0.0001984126984126984 (* (cos re) (pow im 7.0)))))
   (if (<= im -1.1e+44)
     t_1
     (if (<= im -0.025)
       t_0
       (if (<= im 0.065)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 1.1e+44) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) - exp(im)) * (0.5 + (-0.25 * (re * re)));
	double t_1 = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -0.025) {
		tmp = t_0;
	} else if (im <= 0.065) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		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 = (exp(-im) - exp(im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    t_1 = (-0.0001984126984126984d0) * (cos(re) * (im ** 7.0d0))
    if (im <= (-1.1d+44)) then
        tmp = t_1
    else if (im <= (-0.025d0)) then
        tmp = t_0
    else if (im <= 0.065d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.1d+44) 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.exp(-im) - Math.exp(im)) * (0.5 + (-0.25 * (re * re)));
	double t_1 = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -0.025) {
		tmp = t_0;
	} else if (im <= 0.065) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) - math.exp(im)) * (0.5 + (-0.25 * (re * re)))
	t_1 = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.1e+44:
		tmp = t_1
	elif im <= -0.025:
		tmp = t_0
	elif im <= 0.065:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))))
	t_1 = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -0.025)
		tmp = t_0;
	elseif (im <= 0.065)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) - exp(im)) * (0.5 + (-0.25 * (re * re)));
	t_1 = -0.0001984126984126984 * (cos(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -0.025)
		tmp = t_0;
	elseif (im <= 0.065)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.1e+44], t$95$1, If[LessEqual[im, -0.025], t$95$0, If[LessEqual[im, 0.065], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.09999999999999998e44 or 1.09999999999999998e44 < 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%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-1 \cdot im\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      5. neg-mul-1 100.0%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-im\right)} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      6. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      7. +-commutative 100.0%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\left(-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) \]

      9. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re}\right) \]

      10. distribute-rgt-out 100.0%

          \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\cos re \cdot \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)} \]

      11. distribute-lft-out 100.0%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)\right)} \]

      12. +-commutative 100.0%

          \[\leadsto \cos re \cdot \color{blue}{\left(\left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\mathsf{fma}\left({im}^{5}, -0.008333333333333333, {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]

   if -1.09999999999999998e44 < im < -0.025000000000000001 or 0.065000000000000002 < im < 1.09999999999999998e44

   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 5.0%

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

      1. +-commutative 5.0%

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

      2. associate-*r* 5.0%

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

      3. distribute-rgt-out 85.0%

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

      4. unpow2 85.0%

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

   6. Simplified 85.0%

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

   if -0.025000000000000001 < im < 0.065000000000000002

   1. Initial program 7.6%

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

      1. neg-sub0 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. associate-*r* 99.6%

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

      5. distribute-rgt-out-- 99.6%

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

      6. *-commutative 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.025:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 0.065:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -3.8 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0008:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.011:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* -0.0001984126984126984 (* (cos re) (pow im 7.0)))))
   (if (<= im -3.8e+43)
     t_1
     (if (<= im -0.0008)
       t_0
       (if (<= im 0.011) (* im (- (cos re))) (if (<= im 1.1e+44) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	double tmp;
	if (im <= -3.8e+43) {
		tmp = t_1;
	} else if (im <= -0.0008) {
		tmp = t_0;
	} else if (im <= 0.011) {
		tmp = im * -cos(re);
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (-0.0001984126984126984d0) * (cos(re) * (im ** 7.0d0))
    if (im <= (-3.8d+43)) then
        tmp = t_1
    else if (im <= (-0.0008d0)) then
        tmp = t_0
    else if (im <= 0.011d0) then
        tmp = im * -cos(re)
    else if (im <= 1.1d+44) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -3.8e+43) {
		tmp = t_1;
	} else if (im <= -0.0008) {
		tmp = t_0;
	} else if (im <= 0.011) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -3.8e+43:
		tmp = t_1
	elif im <= -0.0008:
		tmp = t_0
	elif im <= 0.011:
		tmp = im * -math.cos(re)
	elif im <= 1.1e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -3.8e+43)
		tmp = t_1;
	elseif (im <= -0.0008)
		tmp = t_0;
	elseif (im <= 0.011)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = -0.0001984126984126984 * (cos(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -3.8e+43)
		tmp = t_1;
	elseif (im <= -0.0008)
		tmp = t_0;
	elseif (im <= 0.011)
		tmp = im * -cos(re);
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.8e+43], t$95$1, If[LessEqual[im, -0.0008], t$95$0, If[LessEqual[im, 0.011], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.1e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.80000000000000008e43 or 1.09999999999999998e44 < 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 99.2%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.2%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

      2. +-commutative 99.2%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      3. associate-*r* 99.2%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      4. associate-*r* 99.2%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-1 \cdot im\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      5. neg-mul-1 99.2%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-im\right)} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      6. distribute-rgt-out 99.2%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      7. +-commutative 99.2%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\left(-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]

      8. associate-*r* 99.2%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) \]

      9. associate-*r* 99.2%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re}\right) \]

      10. distribute-rgt-out 99.2%

          \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\cos re \cdot \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)} \]

      11. distribute-lft-out 99.2%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)\right)} \]

      12. +-commutative 99.2%

          \[\leadsto \cos re \cdot \color{blue}{\left(\left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)\right)} \]

   6. Simplified 99.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(\mathsf{fma}\left({im}^{5}, -0.008333333333333333, {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around inf 99.2%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]

   if -3.80000000000000008e43 < im < -8.00000000000000038e-4 or 0.010999999999999999 < im < 1.09999999999999998e44

   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 78.9%

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

   if -8.00000000000000038e-4 < im < 0.010999999999999999

   1. Initial program 7.6%

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

      1. neg-sub0 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in im around 0 99.4%

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

      1. associate-*r* 99.4%

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

      2. neg-mul-1 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{+43}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.0008:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.011:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -3.8 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.125:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.082:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* -0.0001984126984126984 (* (cos re) (pow im 7.0)))))
   (if (<= im -3.8e+43)
     t_1
     (if (<= im -0.125)
       t_0
       (if (<= im 0.082)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 1.1e+44) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	double tmp;
	if (im <= -3.8e+43) {
		tmp = t_1;
	} else if (im <= -0.125) {
		tmp = t_0;
	} else if (im <= 0.082) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (-0.0001984126984126984d0) * (cos(re) * (im ** 7.0d0))
    if (im <= (-3.8d+43)) then
        tmp = t_1
    else if (im <= (-0.125d0)) then
        tmp = t_0
    else if (im <= 0.082d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.1d+44) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -3.8e+43) {
		tmp = t_1;
	} else if (im <= -0.125) {
		tmp = t_0;
	} else if (im <= 0.082) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -3.8e+43:
		tmp = t_1
	elif im <= -0.125:
		tmp = t_0
	elif im <= 0.082:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -3.8e+43)
		tmp = t_1;
	elseif (im <= -0.125)
		tmp = t_0;
	elseif (im <= 0.082)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = -0.0001984126984126984 * (cos(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -3.8e+43)
		tmp = t_1;
	elseif (im <= -0.125)
		tmp = t_0;
	elseif (im <= 0.082)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.8e+43], t$95$1, If[LessEqual[im, -0.125], t$95$0, If[LessEqual[im, 0.082], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.80000000000000008e43 or 1.09999999999999998e44 < 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 99.2%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.2%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

      2. +-commutative 99.2%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      3. associate-*r* 99.2%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      4. associate-*r* 99.2%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-1 \cdot im\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      5. neg-mul-1 99.2%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-im\right)} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      6. distribute-rgt-out 99.2%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      7. +-commutative 99.2%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\left(-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]

      8. associate-*r* 99.2%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) \]

      9. associate-*r* 99.2%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re}\right) \]

      10. distribute-rgt-out 99.2%

          \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\cos re \cdot \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)} \]

      11. distribute-lft-out 99.2%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)\right)} \]

      12. +-commutative 99.2%

          \[\leadsto \cos re \cdot \color{blue}{\left(\left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)\right)} \]

   6. Simplified 99.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(\mathsf{fma}\left({im}^{5}, -0.008333333333333333, {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around inf 99.2%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]

   if -3.80000000000000008e43 < im < -0.125 or 0.0820000000000000034 < im < 1.09999999999999998e44

   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 78.9%

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

   if -0.125 < im < 0.0820000000000000034

   1. Initial program 7.6%

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

      1. neg-sub0 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. associate-*r* 99.6%

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

      5. distribute-rgt-out-- 99.6%

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

      6. *-commutative 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{+43}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.125:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.082:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 7: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.1 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.1) (not (<= im 4.2)))
   (* -0.0001984126984126984 (* (cos re) (pow im 7.0)))
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.1) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-4.1d0)) .or. (.not. (im <= 4.2d0))) then
        tmp = (-0.0001984126984126984d0) * (cos(re) * (im ** 7.0d0))
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -4.1) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.1) or not (im <= 4.2):
		tmp = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.1) || !(im <= 4.2))
		tmp = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.1) || ~((im <= 4.2)))
		tmp = -0.0001984126984126984 * (cos(re) * (im ^ 7.0));
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.1], N[Not[LessEqual[im, 4.2]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.0999999999999996 or 4.20000000000000018 < 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 85.3%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 85.3%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

      2. +-commutative 85.3%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      3. associate-*r* 85.3%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      4. associate-*r* 85.3%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-1 \cdot im\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      5. neg-mul-1 85.3%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-im\right)} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      6. distribute-rgt-out 85.3%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      7. +-commutative 85.3%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\left(-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]

      8. associate-*r* 85.3%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) \]

      9. associate-*r* 85.3%

         \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re}\right) \]

      10. distribute-rgt-out 85.3%

          \[\leadsto \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \color{blue}{\cos re \cdot \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)} \]

      11. distribute-lft-out 85.3%

          \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right) + \left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right)\right)} \]

      12. +-commutative 85.3%

          \[\leadsto \cos re \cdot \color{blue}{\left(\left(-0.0001984126984126984 \cdot {im}^{7} + -0.008333333333333333 \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot {im}^{3} + \left(-im\right)\right)\right)} \]

   6. Simplified 85.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(\mathsf{fma}\left({im}^{5}, -0.008333333333333333, {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around inf 85.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]

   if -4.0999999999999996 < im < 4.20000000000000018

   1. Initial program 7.6%

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

      1. neg-sub0 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in im around 0 99.4%

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

      1. associate-*r* 99.4%

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

      2. neg-mul-1 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.1 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 8: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\cos re\\ t_1 := \left(im \cdot im\right) \cdot \frac{t_0}{im}\\ t_2 := \langle \left( \langle \left( im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -780:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 35:\\ \;\;\;\;im \cdot t_0\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (cos re)))
        (t_1 (* (* im im) (/ t_0 im)))
        (t_2
         (cast
          (!
           :precision
           binary32
           (cast (! :precision binary64 (- (* im (* 0.5 (* re re))) im)))))))
   (if (<= im -2.45e+179)
     t_1
     (if (<= im -780.0)
       t_2
       (if (<= im 35.0) (* im t_0) (if (<= im 1.35e+154) t_2 t_1))))))

C

double code(double re, double im) {
	double t_0 = -cos(re);
	double t_1 = (im * im) * (t_0 / im);
	double tmp_2 = (im * (0.5 * (re * re))) - im;
	double tmp_1 = (float) tmp_2;
	double t_2 = (double) tmp_1;
	double tmp_3;
	if (im <= -2.45e+179) {
		tmp_3 = t_1;
	} else if (im <= -780.0) {
		tmp_3 = t_2;
	} else if (im <= 35.0) {
		tmp_3 = im * t_0;
	} else if (im <= 1.35e+154) {
		tmp_3 = t_2;
	} else {
		tmp_3 = t_1;
	}
	return tmp_3;
}

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
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = -cos(re)
    t_1 = (im * im) * (t_0 / im)
    tmp_2 = (im * (0.5d0 * (re * re))) - im
    tmp_1 = real(tmp_2, 4)
    t_2 = real(tmp_1, 8)
    if (im <= (-2.45d+179)) then
        tmp_3 = t_1
    else if (im <= (-780.0d0)) then
        tmp_3 = t_2
    else if (im <= 35.0d0) then
        tmp_3 = im * t_0
    else if (im <= 1.35d+154) then
        tmp_3 = t_2
    else
        tmp_3 = t_1
    end if
    code = tmp_3
end function

Julia

function code(re, im)
	t_0 = Float64(-cos(re))
	t_1 = Float64(Float64(im * im) * Float64(t_0 / im))
	tmp_2 = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im)
	tmp_1 = Float32(tmp_2)
	t_2 = Float64(tmp_1)
	tmp_3 = 0.0
	if (im <= -2.45e+179)
		tmp_3 = t_1;
	elseif (im <= -780.0)
		tmp_3 = t_2;
	elseif (im <= 35.0)
		tmp_3 = Float64(im * t_0);
	elseif (im <= 1.35e+154)
		tmp_3 = t_2;
	else
		tmp_3 = t_1;
	end
	return tmp_3
end

MATLAB

function tmp_5 = code(re, im)
	t_0 = -cos(re);
	t_1 = (im * im) * (t_0 / im);
	tmp_3 = (im * (0.5 * (re * re))) - im;
	tmp_2 = single(tmp_3);
	t_2 = double(tmp_2);
	tmp_4 = 0.0;
	if (im <= -2.45e+179)
		tmp_4 = t_1;
	elseif (im <= -780.0)
		tmp_4 = t_2;
	elseif (im <= 35.0)
		tmp_4 = im * t_0;
	elseif (im <= 1.35e+154)
		tmp_4 = t_2;
	else
		tmp_4 = t_1;
	end
	tmp_5 = tmp_4;
end

Derivation

1. Split input into 3 regimes

2. if im < -2.4499999999999999e179 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. 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 7.3%

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

      1. associate-*r* 7.3%

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

      2. neg-mul-1 7.3%

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

   6. Simplified 7.3%

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

      1. *-commutative 7.3%

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

      2. neg-sub0 7.3%

         \[\leadsto \cos re \cdot \color{blue}{\left(0 - im\right)} \]

      3. flip-- 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \cos re \cdot \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

      5. neg-sub0 100.0%

         \[\leadsto \cos re \cdot \frac{\color{blue}{-im \cdot im}}{0 + im} \]

      6. +-lft-identity 100.0%

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

      7. associate-*r/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*l* 100.0%

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

      10. neg-mul-1 100.0%

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

      11. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

   10. Simplified 100.0%

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

   if -2.4499999999999999e179 < im < -780 or 35 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. 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 4.0%

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

      1. associate-*r* 4.0%

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

      2. neg-mul-1 4.0%

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

   6. Simplified 4.0%

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

   7. Taylor expanded in re around 0 19.4%

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

      1. neg-mul-1 19.4%

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

      2. +-commutative 19.4%

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

      3. unsub-neg 19.4%

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

      4. *-commutative 19.4%

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

      5. unpow2 19.4%

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

      6. associate-*l* 19.4%

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

   9. Simplified 19.4%

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

       1. rewrite-binary64/binary32-simplify 64.0%

          \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right) - im} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   11. Applied rewrite-once 64.0%

       \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right) - im} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   if -780 < im < 35

   1. Initial program 8.3%

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

      1. neg-sub0 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in im around 0 98.6%

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

      1. associate-*r* 98.6%

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

      2. neg-mul-1 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-\cos re}{im}\\ \mathbf{elif}\;im \leq -780:\\ \;\;\;\;\langle \left( \langle \left( im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \mathbf{elif}\;im \leq 35:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\langle \left( \langle \left( im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-\cos re}{im}\\ \end{array} \]

Alternative 9: 78.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot -2 + {im}^{3} \cdot -0.3333333333333333\\ t_1 := -\cos re\\ \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{t_1}{im}\\ \mathbf{elif}\;im \leq -6 \cdot 10^{+36}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot t_0\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+67}:\\ \;\;\;\;im \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* im -2.0) (* (pow im 3.0) -0.3333333333333333)))
        (t_1 (- (cos re))))
   (if (<= im -2.45e+179)
     (* (* im im) (/ t_1 im))
     (if (<= im -6e+36)
       (* (+ 0.5 (* -0.25 (* re re))) t_0)
       (if (<= im 6.5e+67) (* im t_1) (* 0.5 t_0))))))

C

double code(double re, double im) {
	double t_0 = (im * -2.0) + (pow(im, 3.0) * -0.3333333333333333);
	double t_1 = -cos(re);
	double tmp;
	if (im <= -2.45e+179) {
		tmp = (im * im) * (t_1 / im);
	} else if (im <= -6e+36) {
		tmp = (0.5 + (-0.25 * (re * re))) * t_0;
	} else if (im <= 6.5e+67) {
		tmp = im * 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 = (im * (-2.0d0)) + ((im ** 3.0d0) * (-0.3333333333333333d0))
    t_1 = -cos(re)
    if (im <= (-2.45d+179)) then
        tmp = (im * im) * (t_1 / im)
    else if (im <= (-6d+36)) then
        tmp = (0.5d0 + ((-0.25d0) * (re * re))) * t_0
    else if (im <= 6.5d+67) then
        tmp = im * 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 = (im * -2.0) + (Math.pow(im, 3.0) * -0.3333333333333333);
	double t_1 = -Math.cos(re);
	double tmp;
	if (im <= -2.45e+179) {
		tmp = (im * im) * (t_1 / im);
	} else if (im <= -6e+36) {
		tmp = (0.5 + (-0.25 * (re * re))) * t_0;
	} else if (im <= 6.5e+67) {
		tmp = im * t_1;
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * -2.0) + (math.pow(im, 3.0) * -0.3333333333333333)
	t_1 = -math.cos(re)
	tmp = 0
	if im <= -2.45e+179:
		tmp = (im * im) * (t_1 / im)
	elif im <= -6e+36:
		tmp = (0.5 + (-0.25 * (re * re))) * t_0
	elif im <= 6.5e+67:
		tmp = im * t_1
	else:
		tmp = 0.5 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * -2.0) + Float64((im ^ 3.0) * -0.3333333333333333))
	t_1 = Float64(-cos(re))
	tmp = 0.0
	if (im <= -2.45e+179)
		tmp = Float64(Float64(im * im) * Float64(t_1 / im));
	elseif (im <= -6e+36)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * Float64(re * re))) * t_0);
	elseif (im <= 6.5e+67)
		tmp = Float64(im * t_1);
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * -2.0) + ((im ^ 3.0) * -0.3333333333333333);
	t_1 = -cos(re);
	tmp = 0.0;
	if (im <= -2.45e+179)
		tmp = (im * im) * (t_1 / im);
	elseif (im <= -6e+36)
		tmp = (0.5 + (-0.25 * (re * re))) * t_0;
	elseif (im <= 6.5e+67)
		tmp = im * t_1;
	else
		tmp = 0.5 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Cos[re], $MachinePrecision])}, If[LessEqual[im, -2.45e+179], N[(N[(im * im), $MachinePrecision] * N[(t$95$1 / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -6e+36], N[(N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[im, 6.5e+67], N[(im * t$95$1), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.4499999999999999e179

   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 7.0%

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

      1. associate-*r* 7.0%

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

      2. neg-mul-1 7.0%

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

   6. Simplified 7.0%

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

      1. *-commutative 7.0%

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

      2. neg-sub0 7.0%

         \[\leadsto \cos re \cdot \color{blue}{\left(0 - im\right)} \]

      3. flip-- 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. metadata-eval 100.0%

         \[\leadsto \cos re \cdot \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

      5. neg-sub0 100.0%

         \[\leadsto \cos re \cdot \frac{\color{blue}{-im \cdot im}}{0 + im} \]

      6. +-lft-identity 100.0%

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

      7. associate-*r/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*l* 100.0%

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

      10. neg-mul-1 100.0%

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

      11. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

   10. Simplified 100.0%

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

   if -2.4499999999999999e179 < im < -6e36

   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 80.0%

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

      4. unpow2 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in im around 0 68.7%

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

   if -6e36 < im < 6.4999999999999995e67

   1. Initial program 20.6%

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

      1. neg-sub0 20.6%

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

   3. Simplified 20.6%

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

   4. Taylor expanded in im around 0 85.9%

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

      1. associate-*r* 85.9%

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

      2. neg-mul-1 85.9%

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

   6. Simplified 85.9%

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

   if 6.4999999999999995e67 < 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 re around 0 92.3%

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

   5. Taylor expanded in im around 0 83.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-\cos re}{im}\\ \mathbf{elif}\;im \leq -6 \cdot 10^{+36}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot -2 + {im}^{3} \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+67}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{3} \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 10: 77.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\cos re\\ \mathbf{if}\;im \leq -3.9 \cdot 10^{+141}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{t_0}{im}\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{+44} \lor \neg \left(im \leq 2.3 \cdot 10^{+67}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{3} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (cos re))))
   (if (<= im -3.9e+141)
     (* (* im im) (/ t_0 im))
     (if (or (<= im -6.4e+44) (not (<= im 2.3e+67)))
       (* 0.5 (+ (* im -2.0) (* (pow im 3.0) -0.3333333333333333)))
       (* im t_0)))))

C

double code(double re, double im) {
	double t_0 = -cos(re);
	double tmp;
	if (im <= -3.9e+141) {
		tmp = (im * im) * (t_0 / im);
	} else if ((im <= -6.4e+44) || !(im <= 2.3e+67)) {
		tmp = 0.5 * ((im * -2.0) + (pow(im, 3.0) * -0.3333333333333333));
	} else {
		tmp = im * 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 = -cos(re)
    if (im <= (-3.9d+141)) then
        tmp = (im * im) * (t_0 / im)
    else if ((im <= (-6.4d+44)) .or. (.not. (im <= 2.3d+67))) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((im ** 3.0d0) * (-0.3333333333333333d0)))
    else
        tmp = im * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -Math.cos(re);
	double tmp;
	if (im <= -3.9e+141) {
		tmp = (im * im) * (t_0 / im);
	} else if ((im <= -6.4e+44) || !(im <= 2.3e+67)) {
		tmp = 0.5 * ((im * -2.0) + (Math.pow(im, 3.0) * -0.3333333333333333));
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -math.cos(re)
	tmp = 0
	if im <= -3.9e+141:
		tmp = (im * im) * (t_0 / im)
	elif (im <= -6.4e+44) or not (im <= 2.3e+67):
		tmp = 0.5 * ((im * -2.0) + (math.pow(im, 3.0) * -0.3333333333333333))
	else:
		tmp = im * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-cos(re))
	tmp = 0.0
	if (im <= -3.9e+141)
		tmp = Float64(Float64(im * im) * Float64(t_0 / im));
	elseif ((im <= -6.4e+44) || !(im <= 2.3e+67))
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((im ^ 3.0) * -0.3333333333333333)));
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -cos(re);
	tmp = 0.0;
	if (im <= -3.9e+141)
		tmp = (im * im) * (t_0 / im);
	elseif ((im <= -6.4e+44) || ~((im <= 2.3e+67)))
		tmp = 0.5 * ((im * -2.0) + ((im ^ 3.0) * -0.3333333333333333));
	else
		tmp = im * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = (-N[Cos[re], $MachinePrecision])}, If[LessEqual[im, -3.9e+141], N[(N[(im * im), $MachinePrecision] * N[(t$95$0 / im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -6.4e+44], N[Not[LessEqual[im, 2.3e+67]], $MachinePrecision]], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.89999999999999991e141

   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 6.6%

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

      1. associate-*r* 6.6%

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

      2. neg-mul-1 6.6%

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

   6. Simplified 6.6%

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

      1. *-commutative 6.6%

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

      2. neg-sub0 6.6%

         \[\leadsto \cos re \cdot \color{blue}{\left(0 - im\right)} \]

      3. flip-- 97.4%

         \[\leadsto \cos re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. metadata-eval 97.4%

         \[\leadsto \cos re \cdot \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

      5. neg-sub0 97.4%

         \[\leadsto \cos re \cdot \frac{\color{blue}{-im \cdot im}}{0 + im} \]

      6. +-lft-identity 97.4%

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

      7. associate-*r/ 97.4%

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

      8. neg-mul-1 97.4%

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

      9. associate-*l* 97.4%

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

      10. neg-mul-1 97.4%

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

      11. *-commutative 97.4%

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

   8. Applied egg-rr 97.4%

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

      1. associate-/l* 97.4%

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

      2. associate-/r/ 97.4%

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

      3. distribute-rgt-neg-out 97.4%

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

   10. Simplified 97.4%

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

   if -3.89999999999999991e141 < im < -6.40000000000000009e44 or 2.2999999999999999e67 < 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 re around 0 89.9%

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

   5. Taylor expanded in im around 0 74.9%

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

   if -6.40000000000000009e44 < im < 2.2999999999999999e67

   1. Initial program 21.1%

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

      1. neg-sub0 21.1%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in im around 0 85.3%

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

      1. associate-*r* 85.3%

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

      2. neg-mul-1 85.3%

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

   6. Simplified 85.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.9 \cdot 10^{+141}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-\cos re}{im}\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{+44} \lor \neg \left(im \leq 2.3 \cdot 10^{+67}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{3} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 11: 74.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\cos re\\ \mathbf{if}\;im \leq -2 \cdot 10^{+141} \lor \neg \left(im \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{t_0}{im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (cos re))))
   (if (or (<= im -2e+141) (not (<= im 1.1e-6)))
     (* (* im im) (/ t_0 im))
     (* im t_0))))

C

double code(double re, double im) {
	double t_0 = -cos(re);
	double tmp;
	if ((im <= -2e+141) || !(im <= 1.1e-6)) {
		tmp = (im * im) * (t_0 / im);
	} else {
		tmp = im * 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 = -cos(re)
    if ((im <= (-2d+141)) .or. (.not. (im <= 1.1d-6))) then
        tmp = (im * im) * (t_0 / im)
    else
        tmp = im * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -Math.cos(re);
	double tmp;
	if ((im <= -2e+141) || !(im <= 1.1e-6)) {
		tmp = (im * im) * (t_0 / im);
	} else {
		tmp = im * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -math.cos(re)
	tmp = 0
	if (im <= -2e+141) or not (im <= 1.1e-6):
		tmp = (im * im) * (t_0 / im)
	else:
		tmp = im * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-cos(re))
	tmp = 0.0
	if ((im <= -2e+141) || !(im <= 1.1e-6))
		tmp = Float64(Float64(im * im) * Float64(t_0 / im));
	else
		tmp = Float64(im * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -cos(re);
	tmp = 0.0;
	if ((im <= -2e+141) || ~((im <= 1.1e-6)))
		tmp = (im * im) * (t_0 / im);
	else
		tmp = im * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = (-N[Cos[re], $MachinePrecision])}, If[Or[LessEqual[im, -2e+141], N[Not[LessEqual[im, 1.1e-6]], $MachinePrecision]], N[(N[(im * im), $MachinePrecision] * N[(t$95$0 / im), $MachinePrecision]), $MachinePrecision], N[(im * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < -2.00000000000000003e141 or 1.1000000000000001e-6 < im

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in im around 0 6.7%

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

      1. associate-*r* 6.7%

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

      2. neg-mul-1 6.7%

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

   6. Simplified 6.7%

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

      1. *-commutative 6.7%

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

      2. neg-sub0 6.7%

         \[\leadsto \cos re \cdot \color{blue}{\left(0 - im\right)} \]

      3. flip-- 75.2%

         \[\leadsto \cos re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. metadata-eval 75.2%

         \[\leadsto \cos re \cdot \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

      5. neg-sub0 75.2%

         \[\leadsto \cos re \cdot \frac{\color{blue}{-im \cdot im}}{0 + im} \]

      6. +-lft-identity 75.2%

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

      7. associate-*r/ 75.2%

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

      8. neg-mul-1 75.2%

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

      9. associate-*l* 75.2%

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

      10. neg-mul-1 75.2%

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

      11. *-commutative 75.2%

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

   8. Applied egg-rr 75.2%

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

      1. associate-/l* 75.2%

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

      2. associate-/r/ 75.2%

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

      3. distribute-rgt-neg-out 75.2%

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

   10. Simplified 75.2%

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

   if -2.00000000000000003e141 < im < 1.1000000000000001e-6

   1. Initial program 24.7%

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

      1. neg-sub0 24.7%

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

   3. Simplified 24.7%

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

   4. Taylor expanded in im around 0 81.4%

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

      1. associate-*r* 81.4%

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

      2. neg-mul-1 81.4%

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

   6. Simplified 81.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{+141} \lor \neg \left(im \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-\cos re}{im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 12: 74.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\cos re\\ \mathbf{if}\;im \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{t_0}{im}\\ \mathbf{elif}\;im \leq 3 \cdot 10^{-10}:\\ \;\;\;\;im \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re}{\frac{im}{im \cdot \left(-im\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (cos re))))
   (if (<= im -2e+141)
     (* (* im im) (/ t_0 im))
     (if (<= im 3e-10) (* im t_0) (/ (cos re) (/ im (* im (- im))))))))

C

double code(double re, double im) {
	double t_0 = -cos(re);
	double tmp;
	if (im <= -2e+141) {
		tmp = (im * im) * (t_0 / im);
	} else if (im <= 3e-10) {
		tmp = im * t_0;
	} else {
		tmp = cos(re) / (im / (im * -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 = -cos(re)
    if (im <= (-2d+141)) then
        tmp = (im * im) * (t_0 / im)
    else if (im <= 3d-10) then
        tmp = im * t_0
    else
        tmp = cos(re) / (im / (im * -im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -Math.cos(re);
	double tmp;
	if (im <= -2e+141) {
		tmp = (im * im) * (t_0 / im);
	} else if (im <= 3e-10) {
		tmp = im * t_0;
	} else {
		tmp = Math.cos(re) / (im / (im * -im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -math.cos(re)
	tmp = 0
	if im <= -2e+141:
		tmp = (im * im) * (t_0 / im)
	elif im <= 3e-10:
		tmp = im * t_0
	else:
		tmp = math.cos(re) / (im / (im * -im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-cos(re))
	tmp = 0.0
	if (im <= -2e+141)
		tmp = Float64(Float64(im * im) * Float64(t_0 / im));
	elseif (im <= 3e-10)
		tmp = Float64(im * t_0);
	else
		tmp = Float64(cos(re) / Float64(im / Float64(im * Float64(-im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -cos(re);
	tmp = 0.0;
	if (im <= -2e+141)
		tmp = (im * im) * (t_0 / im);
	elseif (im <= 3e-10)
		tmp = im * t_0;
	else
		tmp = cos(re) / (im / (im * -im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = (-N[Cos[re], $MachinePrecision])}, If[LessEqual[im, -2e+141], N[(N[(im * im), $MachinePrecision] * N[(t$95$0 / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3e-10], N[(im * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] / N[(im / N[(im * (-im)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.00000000000000003e141

   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 6.6%

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

      1. associate-*r* 6.6%

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

      2. neg-mul-1 6.6%

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

   6. Simplified 6.6%

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

      1. *-commutative 6.6%

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

      2. neg-sub0 6.6%

         \[\leadsto \cos re \cdot \color{blue}{\left(0 - im\right)} \]

      3. flip-- 97.4%

         \[\leadsto \cos re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. metadata-eval 97.4%

         \[\leadsto \cos re \cdot \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

      5. neg-sub0 97.4%

         \[\leadsto \cos re \cdot \frac{\color{blue}{-im \cdot im}}{0 + im} \]

      6. +-lft-identity 97.4%

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

      7. associate-*r/ 97.4%

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

      8. neg-mul-1 97.4%

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

      9. associate-*l* 97.4%

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

      10. neg-mul-1 97.4%

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

      11. *-commutative 97.4%

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

   8. Applied egg-rr 97.4%

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

      1. associate-/l* 97.4%

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

      2. associate-/r/ 97.4%

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

      3. distribute-rgt-neg-out 97.4%

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

   10. Simplified 97.4%

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

   if -2.00000000000000003e141 < im < 3e-10

   1. Initial program 24.2%

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

      1. neg-sub0 24.2%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in im around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. neg-mul-1 81.2%

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

   6. Simplified 81.2%

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

   if 3e-10 < im

   1. Initial program 98.8%

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

      1. neg-sub0 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in im around 0 9.7%

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

      1. associate-*r* 9.7%

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

      2. neg-mul-1 9.7%

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

   6. Simplified 9.7%

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

      1. *-commutative 9.7%

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

      2. neg-sub0 9.7%

         \[\leadsto \cos re \cdot \color{blue}{\left(0 - im\right)} \]

      3. flip-- 63.2%

         \[\leadsto \cos re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. metadata-eval 63.2%

         \[\leadsto \cos re \cdot \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

      5. neg-sub0 63.2%

         \[\leadsto \cos re \cdot \frac{\color{blue}{-im \cdot im}}{0 + im} \]

      6. +-lft-identity 63.2%

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

      7. associate-*r/ 63.2%

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

      8. neg-mul-1 63.2%

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

      9. associate-*l* 63.2%

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

      10. neg-mul-1 63.2%

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

      11. *-commutative 63.2%

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

   8. Applied egg-rr 63.2%

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

      1. associate-/l* 63.2%

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

   10. Simplified 63.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-\cos re}{im}\\ \mathbf{elif}\;im \leq 3 \cdot 10^{-10}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re}{\frac{im}{im \cdot \left(-im\right)}}\\ \end{array} \]

Alternative 13: 68.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \frac{-1}{im}\\ \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{+35}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right) + -1\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+132}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (/ -1.0 im))))
   (if (<= im -2.45e+179)
     t_0
     (if (<= im -5.5e+35)
       (* im (+ (* re (* 0.5 re)) -1.0))
       (if (<= im 2.1e+132) (* im (- (cos re))) t_0)))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (-1.0 / im);
	double tmp;
	if (im <= -2.45e+179) {
		tmp = t_0;
	} else if (im <= -5.5e+35) {
		tmp = im * ((re * (0.5 * re)) + -1.0);
	} else if (im <= 2.1e+132) {
		tmp = im * -cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * ((-1.0d0) / im)
    if (im <= (-2.45d+179)) then
        tmp = t_0
    else if (im <= (-5.5d+35)) then
        tmp = im * ((re * (0.5d0 * re)) + (-1.0d0))
    else if (im <= 2.1d+132) then
        tmp = im * -cos(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (-1.0 / im);
	double tmp;
	if (im <= -2.45e+179) {
		tmp = t_0;
	} else if (im <= -5.5e+35) {
		tmp = im * ((re * (0.5 * re)) + -1.0);
	} else if (im <= 2.1e+132) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (-1.0 / im)
	tmp = 0
	if im <= -2.45e+179:
		tmp = t_0
	elif im <= -5.5e+35:
		tmp = im * ((re * (0.5 * re)) + -1.0)
	elif im <= 2.1e+132:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(-1.0 / im))
	tmp = 0.0
	if (im <= -2.45e+179)
		tmp = t_0;
	elseif (im <= -5.5e+35)
		tmp = Float64(im * Float64(Float64(re * Float64(0.5 * re)) + -1.0));
	elseif (im <= 2.1e+132)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (-1.0 / im);
	tmp = 0.0;
	if (im <= -2.45e+179)
		tmp = t_0;
	elseif (im <= -5.5e+35)
		tmp = im * ((re * (0.5 * re)) + -1.0);
	elseif (im <= 2.1e+132)
		tmp = im * -cos(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(-1.0 / im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.45e+179], t$95$0, If[LessEqual[im, -5.5e+35], N[(im * N[(N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.1e+132], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.4499999999999999e179 or 2.09999999999999993e132 < 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 7.2%

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

      1. associate-*r* 7.2%

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

      2. neg-mul-1 7.2%

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

   6. Simplified 7.2%

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

      1. *-commutative 7.2%

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

      2. neg-sub0 7.2%

         \[\leadsto \cos re \cdot \color{blue}{\left(0 - im\right)} \]

      3. flip-- 94.6%

         \[\leadsto \cos re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. metadata-eval 94.6%

         \[\leadsto \cos re \cdot \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

      5. neg-sub0 94.6%

         \[\leadsto \cos re \cdot \frac{\color{blue}{-im \cdot im}}{0 + im} \]

      6. +-lft-identity 94.6%

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

      7. associate-*r/ 94.6%

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

      8. neg-mul-1 94.6%

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

      9. associate-*l* 94.6%

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

      10. neg-mul-1 94.6%

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

      11. *-commutative 94.6%

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

   8. Applied egg-rr 94.6%

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

      1. associate-/l* 94.6%

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

      2. associate-/r/ 94.6%

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

      3. distribute-rgt-neg-out 94.6%

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

   10. Simplified 94.6%

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

   11. Taylor expanded in re around 0 80.5%

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

   if -2.4499999999999999e179 < im < -5.50000000000000001e35

   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%

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

      1. associate-*r* 4.2%

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

      2. neg-mul-1 4.2%

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

   6. Simplified 4.2%

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

   7. Taylor expanded in re around 0 30.3%

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

      1. neg-mul-1 30.3%

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

      2. +-commutative 30.3%

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

      3. unsub-neg 30.3%

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

      4. *-commutative 30.3%

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

      5. unpow2 30.3%

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

      6. associate-*l* 30.3%

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

   9. Simplified 30.3%

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

       1. sub-neg 30.3%

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

       2. *-commutative 30.3%

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

       3. neg-mul-1 30.3%

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

       4. distribute-rgt-out 30.3%

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

       5. associate-*l* 30.3%

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

   11. Applied egg-rr 30.3%

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

   if -5.50000000000000001e35 < im < 2.09999999999999993e132

   1. Initial program 26.1%

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

      1. neg-sub0 26.1%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in im around 0 80.2%

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

      1. associate-*r* 80.2%

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

      2. neg-mul-1 80.2%

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

   6. Simplified 80.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-1}{im}\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{+35}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right) + -1\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+132}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-1}{im}\\ \end{array} \]

Alternative 14: 49.4% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \frac{-1}{im}\\ t_1 := im \cdot \left(re \cdot \left(0.5 \cdot re\right) + -1\right)\\ \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 8 \cdot 10^{-32}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (/ -1.0 im)))
        (t_1 (* im (+ (* re (* 0.5 re)) -1.0))))
   (if (<= im -2.45e+179)
     t_0
     (if (<= im -3e-21)
       t_1
       (if (<= im 8e-32) (- im) (if (<= im 1.95e+130) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (-1.0 / im);
	double t_1 = im * ((re * (0.5 * re)) + -1.0);
	double tmp;
	if (im <= -2.45e+179) {
		tmp = t_0;
	} else if (im <= -3e-21) {
		tmp = t_1;
	} else if (im <= 8e-32) {
		tmp = -im;
	} else if (im <= 1.95e+130) {
		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 * im) * ((-1.0d0) / im)
    t_1 = im * ((re * (0.5d0 * re)) + (-1.0d0))
    if (im <= (-2.45d+179)) then
        tmp = t_0
    else if (im <= (-3d-21)) then
        tmp = t_1
    else if (im <= 8d-32) then
        tmp = -im
    else if (im <= 1.95d+130) 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 = (im * im) * (-1.0 / im);
	double t_1 = im * ((re * (0.5 * re)) + -1.0);
	double tmp;
	if (im <= -2.45e+179) {
		tmp = t_0;
	} else if (im <= -3e-21) {
		tmp = t_1;
	} else if (im <= 8e-32) {
		tmp = -im;
	} else if (im <= 1.95e+130) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (-1.0 / im)
	t_1 = im * ((re * (0.5 * re)) + -1.0)
	tmp = 0
	if im <= -2.45e+179:
		tmp = t_0
	elif im <= -3e-21:
		tmp = t_1
	elif im <= 8e-32:
		tmp = -im
	elif im <= 1.95e+130:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(-1.0 / im))
	t_1 = Float64(im * Float64(Float64(re * Float64(0.5 * re)) + -1.0))
	tmp = 0.0
	if (im <= -2.45e+179)
		tmp = t_0;
	elseif (im <= -3e-21)
		tmp = t_1;
	elseif (im <= 8e-32)
		tmp = Float64(-im);
	elseif (im <= 1.95e+130)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (-1.0 / im);
	t_1 = im * ((re * (0.5 * re)) + -1.0);
	tmp = 0.0;
	if (im <= -2.45e+179)
		tmp = t_0;
	elseif (im <= -3e-21)
		tmp = t_1;
	elseif (im <= 8e-32)
		tmp = -im;
	elseif (im <= 1.95e+130)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(-1.0 / im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.45e+179], t$95$0, If[LessEqual[im, -3e-21], t$95$1, If[LessEqual[im, 8e-32], (-im), If[LessEqual[im, 1.95e+130], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.4499999999999999e179 or 1.9500000000000001e130 < 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 7.2%

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

      1. associate-*r* 7.2%

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

      2. neg-mul-1 7.2%

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

   6. Simplified 7.2%

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

      1. *-commutative 7.2%

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

      2. neg-sub0 7.2%

         \[\leadsto \cos re \cdot \color{blue}{\left(0 - im\right)} \]

      3. flip-- 94.6%

         \[\leadsto \cos re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. metadata-eval 94.6%

         \[\leadsto \cos re \cdot \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

      5. neg-sub0 94.6%

         \[\leadsto \cos re \cdot \frac{\color{blue}{-im \cdot im}}{0 + im} \]

      6. +-lft-identity 94.6%

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

      7. associate-*r/ 94.6%

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

      8. neg-mul-1 94.6%

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

      9. associate-*l* 94.6%

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

      10. neg-mul-1 94.6%

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

      11. *-commutative 94.6%

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

   8. Applied egg-rr 94.6%

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

      1. associate-/l* 94.6%

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

      2. associate-/r/ 94.6%

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

      3. distribute-rgt-neg-out 94.6%

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

   10. Simplified 94.6%

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

   11. Taylor expanded in re around 0 80.5%

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

   if -2.4499999999999999e179 < im < -2.99999999999999991e-21 or 8.00000000000000045e-32 < im < 1.9500000000000001e130

   1. Initial program 92.5%

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

      1. neg-sub0 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in im around 0 14.9%

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

      1. associate-*r* 14.9%

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

      2. neg-mul-1 14.9%

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

   6. Simplified 14.9%

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

   7. Taylor expanded in re around 0 22.7%

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

      1. neg-mul-1 22.7%

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

      2. +-commutative 22.7%

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

      3. unsub-neg 22.7%

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

      4. *-commutative 22.7%

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

      5. unpow2 22.7%

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

      6. associate-*l* 22.7%

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

   9. Simplified 22.7%

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

       1. sub-neg 22.7%

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

       2. *-commutative 22.7%

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

       3. neg-mul-1 22.7%

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

       4. distribute-rgt-out 22.7%

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

       5. associate-*l* 22.7%

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

   11. Applied egg-rr 22.7%

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

   if -2.99999999999999991e-21 < im < 8.00000000000000045e-32

   1. Initial program 5.5%

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

      1. neg-sub0 5.5%

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

   3. Simplified 5.5%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in re around 0 54.2%

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

      1. neg-mul-1 54.2%

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

   9. Simplified 54.2%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-1}{im}\\ \mathbf{elif}\;im \leq -3 \cdot 10^{-21}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right) + -1\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{-32}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+130}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-1}{im}\\ \end{array} \]

Alternative 15: 49.3% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \frac{-1}{im}\\ \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3 \cdot 10^{-21}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right) + -1\right)\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{-32}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+126}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (/ -1.0 im))))
   (if (<= im -2.45e+179)
     t_0
     (if (<= im -3e-21)
       (* im (+ (* re (* 0.5 re)) -1.0))
       (if (<= im 8.2e-32)
         (- im)
         (if (<= im 7e+126) (- (* re (* 0.5 (* im re))) im) t_0))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (-1.0 / im);
	double tmp;
	if (im <= -2.45e+179) {
		tmp = t_0;
	} else if (im <= -3e-21) {
		tmp = im * ((re * (0.5 * re)) + -1.0);
	} else if (im <= 8.2e-32) {
		tmp = -im;
	} else if (im <= 7e+126) {
		tmp = (re * (0.5 * (im * re))) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * ((-1.0d0) / im)
    if (im <= (-2.45d+179)) then
        tmp = t_0
    else if (im <= (-3d-21)) then
        tmp = im * ((re * (0.5d0 * re)) + (-1.0d0))
    else if (im <= 8.2d-32) then
        tmp = -im
    else if (im <= 7d+126) then
        tmp = (re * (0.5d0 * (im * re))) - im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (-1.0 / im);
	double tmp;
	if (im <= -2.45e+179) {
		tmp = t_0;
	} else if (im <= -3e-21) {
		tmp = im * ((re * (0.5 * re)) + -1.0);
	} else if (im <= 8.2e-32) {
		tmp = -im;
	} else if (im <= 7e+126) {
		tmp = (re * (0.5 * (im * re))) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (-1.0 / im)
	tmp = 0
	if im <= -2.45e+179:
		tmp = t_0
	elif im <= -3e-21:
		tmp = im * ((re * (0.5 * re)) + -1.0)
	elif im <= 8.2e-32:
		tmp = -im
	elif im <= 7e+126:
		tmp = (re * (0.5 * (im * re))) - im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(-1.0 / im))
	tmp = 0.0
	if (im <= -2.45e+179)
		tmp = t_0;
	elseif (im <= -3e-21)
		tmp = Float64(im * Float64(Float64(re * Float64(0.5 * re)) + -1.0));
	elseif (im <= 8.2e-32)
		tmp = Float64(-im);
	elseif (im <= 7e+126)
		tmp = Float64(Float64(re * Float64(0.5 * Float64(im * re))) - im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (-1.0 / im);
	tmp = 0.0;
	if (im <= -2.45e+179)
		tmp = t_0;
	elseif (im <= -3e-21)
		tmp = im * ((re * (0.5 * re)) + -1.0);
	elseif (im <= 8.2e-32)
		tmp = -im;
	elseif (im <= 7e+126)
		tmp = (re * (0.5 * (im * re))) - im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(-1.0 / im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.45e+179], t$95$0, If[LessEqual[im, -3e-21], N[(im * N[(N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.2e-32], (-im), If[LessEqual[im, 7e+126], N[(N[(re * N[(0.5 * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.4499999999999999e179 or 7.0000000000000005e126 < 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 7.2%

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

      1. associate-*r* 7.2%

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

      2. neg-mul-1 7.2%

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

   6. Simplified 7.2%

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

      1. *-commutative 7.2%

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

      2. neg-sub0 7.2%

         \[\leadsto \cos re \cdot \color{blue}{\left(0 - im\right)} \]

      3. flip-- 94.6%

         \[\leadsto \cos re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. metadata-eval 94.6%

         \[\leadsto \cos re \cdot \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

      5. neg-sub0 94.6%

         \[\leadsto \cos re \cdot \frac{\color{blue}{-im \cdot im}}{0 + im} \]

      6. +-lft-identity 94.6%

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

      7. associate-*r/ 94.6%

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

      8. neg-mul-1 94.6%

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

      9. associate-*l* 94.6%

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

      10. neg-mul-1 94.6%

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

      11. *-commutative 94.6%

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

   8. Applied egg-rr 94.6%

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

      1. associate-/l* 94.6%

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

      2. associate-/r/ 94.6%

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

      3. distribute-rgt-neg-out 94.6%

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

   10. Simplified 94.6%

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

   11. Taylor expanded in re around 0 80.5%

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

   if -2.4499999999999999e179 < im < -2.99999999999999991e-21

   1. Initial program 95.0%

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

      1. neg-sub0 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in im around 0 8.8%

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

      1. associate-*r* 8.8%

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

      2. neg-mul-1 8.8%

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

   6. Simplified 8.8%

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

   7. Taylor expanded in re around 0 20.6%

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

      1. neg-mul-1 20.6%

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

      2. +-commutative 20.6%

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

      3. unsub-neg 20.6%

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

      4. *-commutative 20.6%

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

      5. unpow2 20.6%

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

      6. associate-*l* 20.6%

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

   9. Simplified 20.6%

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

       1. sub-neg 20.6%

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

       2. *-commutative 20.6%

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

       3. neg-mul-1 20.6%

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

       4. distribute-rgt-out 20.6%

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

       5. associate-*l* 20.6%

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

   11. Applied egg-rr 20.6%

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

   if -2.99999999999999991e-21 < im < 8.1999999999999995e-32

   1. Initial program 5.5%

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

      1. neg-sub0 5.5%

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

   3. Simplified 5.5%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in re around 0 54.2%

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

      1. neg-mul-1 54.2%

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

   9. Simplified 54.2%

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

   if 8.1999999999999995e-32 < im < 7.0000000000000005e126

   1. Initial program 88.7%

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

      1. neg-sub0 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in im around 0 24.0%

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

      1. associate-*r* 24.0%

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

      2. neg-mul-1 24.0%

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

   6. Simplified 24.0%

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

   7. Taylor expanded in re around 0 26.0%

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

      1. neg-mul-1 26.0%

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

      2. +-commutative 26.0%

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

      3. unsub-neg 26.0%

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

      4. *-commutative 26.0%

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

      5. unpow2 26.0%

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

      6. associate-*l* 26.0%

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

   9. Simplified 26.0%

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

   10. Taylor expanded in im around 0 26.0%

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

       1. *-commutative 26.0%

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

       2. unpow2 26.0%

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

       3. associate-*r* 26.0%

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

       4. *-commutative 26.0%

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

       5. associate-*r* 26.0%

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

       6. *-commutative 26.0%

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

       7. *-commutative 26.0%

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

   12. Simplified 26.0%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-1}{im}\\ \mathbf{elif}\;im \leq -3 \cdot 10^{-21}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right) + -1\right)\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{-32}:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+126}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-1}{im}\\ \end{array} \]

Alternative 16: 47.4% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \frac{-1}{im}\\ \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6.2 \cdot 10^{+35}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-8}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (/ -1.0 im))))
   (if (<= im -2.45e+179)
     t_0
     (if (<= im -6.2e+35)
       (* (* re re) (* im 0.5))
       (if (<= im 2e-8) (- im) t_0)))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (-1.0 / im);
	double tmp;
	if (im <= -2.45e+179) {
		tmp = t_0;
	} else if (im <= -6.2e+35) {
		tmp = (re * re) * (im * 0.5);
	} else if (im <= 2e-8) {
		tmp = -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * ((-1.0d0) / im)
    if (im <= (-2.45d+179)) then
        tmp = t_0
    else if (im <= (-6.2d+35)) then
        tmp = (re * re) * (im * 0.5d0)
    else if (im <= 2d-8) then
        tmp = -im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (-1.0 / im);
	double tmp;
	if (im <= -2.45e+179) {
		tmp = t_0;
	} else if (im <= -6.2e+35) {
		tmp = (re * re) * (im * 0.5);
	} else if (im <= 2e-8) {
		tmp = -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (-1.0 / im)
	tmp = 0
	if im <= -2.45e+179:
		tmp = t_0
	elif im <= -6.2e+35:
		tmp = (re * re) * (im * 0.5)
	elif im <= 2e-8:
		tmp = -im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(-1.0 / im))
	tmp = 0.0
	if (im <= -2.45e+179)
		tmp = t_0;
	elseif (im <= -6.2e+35)
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	elseif (im <= 2e-8)
		tmp = Float64(-im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (-1.0 / im);
	tmp = 0.0;
	if (im <= -2.45e+179)
		tmp = t_0;
	elseif (im <= -6.2e+35)
		tmp = (re * re) * (im * 0.5);
	elseif (im <= 2e-8)
		tmp = -im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(-1.0 / im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.45e+179], t$95$0, If[LessEqual[im, -6.2e+35], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2e-8], (-im), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.4499999999999999e179 or 2e-8 < im

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in im around 0 6.9%

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

      1. associate-*r* 6.9%

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

      2. neg-mul-1 6.9%

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

   6. Simplified 6.9%

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

      1. *-commutative 6.9%

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

      2. neg-sub0 6.9%

         \[\leadsto \cos re \cdot \color{blue}{\left(0 - im\right)} \]

      3. flip-- 74.4%

         \[\leadsto \cos re \cdot \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

      4. metadata-eval 74.4%

         \[\leadsto \cos re \cdot \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

      5. neg-sub0 74.4%

         \[\leadsto \cos re \cdot \frac{\color{blue}{-im \cdot im}}{0 + im} \]

      6. +-lft-identity 74.4%

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

      7. associate-*r/ 74.4%

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

      8. neg-mul-1 74.4%

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

      9. associate-*l* 74.4%

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

      10. neg-mul-1 74.4%

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

      11. *-commutative 74.4%

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

   8. Applied egg-rr 74.4%

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

      1. associate-/l* 74.4%

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

      2. associate-/r/ 74.4%

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

      3. distribute-rgt-neg-out 74.4%

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

   10. Simplified 74.4%

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

   11. Taylor expanded in re around 0 62.9%

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

   if -2.4499999999999999e179 < im < -6.19999999999999973e35

   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%

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

      1. associate-*r* 4.2%

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

      2. neg-mul-1 4.2%

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

   6. Simplified 4.2%

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

   7. Taylor expanded in re around 0 30.3%

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

      1. neg-mul-1 30.3%

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

      2. +-commutative 30.3%

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

      3. unsub-neg 30.3%

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

      4. *-commutative 30.3%

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

      5. unpow2 30.3%

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

      6. associate-*l* 30.3%

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

   9. Simplified 30.3%

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

       1. sub-neg 30.3%

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

       2. +-commutative 30.3%

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

       3. flip-+ 13.5%

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

       4. sqr-neg 13.5%

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

       5. pow2 13.5%

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

       6. associate-*l* 13.5%

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

       7. associate-*r* 13.5%

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

       8. *-commutative 13.5%

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

       9. *-commutative 13.5%

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

       10. associate-*l* 13.5%

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

       11. associate-*r* 13.5%

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

       12. *-commutative 13.5%

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

       13. *-commutative 13.5%

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

   11. Applied egg-rr 13.5%

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

   12. Taylor expanded in re around inf 29.0%

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

       1. unpow2 29.0%

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

       2. associate-*r* 29.0%

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

       3. *-commutative 29.0%

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

       4. *-commutative 29.0%

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

   14. Simplified 29.0%

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

   if -6.19999999999999973e35 < im < 2e-8

   1. Initial program 14.9%

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

      1. neg-sub0 14.9%

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

   3. Simplified 14.9%

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

   4. Taylor expanded in im around 0 91.5%

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

      1. associate-*r* 91.5%

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

      2. neg-mul-1 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in re around 0 49.2%

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

      1. neg-mul-1 49.2%

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

   9. Simplified 49.2%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.45 \cdot 10^{+179}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-1}{im}\\ \mathbf{elif}\;im \leq -6.2 \cdot 10^{+35}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-8}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-1}{im}\\ \end{array} \]

Alternative 17: 32.1% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.8 \cdot 10^{+117}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+231}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.8e+117)
   (- im)
   (if (<= re 8.5e+231) (* (* re re) (* im 0.5)) (* re (* re 0.75)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.8e+117) {
		tmp = -im;
	} else if (re <= 8.5e+231) {
		tmp = (re * re) * (im * 0.5);
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 6.8d+117) then
        tmp = -im
    else if (re <= 8.5d+231) then
        tmp = (re * re) * (im * 0.5d0)
    else
        tmp = re * (re * 0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.8e+117) {
		tmp = -im;
	} else if (re <= 8.5e+231) {
		tmp = (re * re) * (im * 0.5);
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.8e+117:
		tmp = -im
	elif re <= 8.5e+231:
		tmp = (re * re) * (im * 0.5)
	else:
		tmp = re * (re * 0.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.8e+117)
		tmp = Float64(-im);
	elseif (re <= 8.5e+231)
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	else
		tmp = Float64(re * Float64(re * 0.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.8e+117)
		tmp = -im;
	elseif (re <= 8.5e+231)
		tmp = (re * re) * (im * 0.5);
	else
		tmp = re * (re * 0.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.8e+117], (-im), If[LessEqual[re, 8.5e+231], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 6.8000000000000002e117

   1. Initial program 53.9%

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

      1. neg-sub0 53.9%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in im around 0 52.6%

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

      1. associate-*r* 52.6%

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

      2. neg-mul-1 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in re around 0 31.9%

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

      1. neg-mul-1 31.9%

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

   9. Simplified 31.9%

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

   if 6.8000000000000002e117 < re < 8.4999999999999994e231

   1. Initial program 46.9%

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

      1. neg-sub0 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in im around 0 58.1%

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

      1. associate-*r* 58.1%

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

      2. neg-mul-1 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in re around 0 32.6%

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

      1. neg-mul-1 32.6%

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

      2. +-commutative 32.6%

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

      3. unsub-neg 32.6%

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

      4. *-commutative 32.6%

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

      5. unpow2 32.6%

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

      6. associate-*l* 32.6%

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

   9. Simplified 32.6%

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

       1. sub-neg 32.6%

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

       2. +-commutative 32.6%

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

       3. flip-+ 0.7%

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

       4. sqr-neg 0.7%

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

       5. pow2 0.7%

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

       6. associate-*l* 0.7%

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

       7. associate-*r* 1.2%

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

       8. *-commutative 1.2%

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

       9. *-commutative 1.2%

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

       10. associate-*l* 1.2%

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

       11. associate-*r* 1.0%

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

       12. *-commutative 1.0%

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

       13. *-commutative 1.0%

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

   11. Applied egg-rr 1.0%

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

   12. Taylor expanded in re around inf 32.6%

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

       1. unpow2 32.6%

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

       2. associate-*r* 32.6%

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

       3. *-commutative 32.6%

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

       4. *-commutative 32.6%

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

   14. Simplified 32.6%

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

   if 8.4999999999999994e231 < re

   1. Initial program 58.9%

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

      1. neg-sub0 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in re around 0 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 18.8%

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

      4. unpow2 18.8%

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

   6. Simplified 18.8%

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

   8. Applied egg-rr 32.0%

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

   9. Taylor expanded in re around inf 32.0%

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

       1. *-commutative 32.0%

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

       2. unpow2 32.0%

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

       3. associate-*l* 32.0%

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

   11. Simplified 32.0%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.8 \cdot 10^{+117}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+231}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \]

Alternative 18: 31.4% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.2 \cdot 10^{+171}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.2e+171) (- im) (* re (* re 0.75))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.2e+171) {
		tmp = -im;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 4.2d+171) then
        tmp = -im
    else
        tmp = re * (re * 0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.2e+171) {
		tmp = -im;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.2e+171:
		tmp = -im
	else:
		tmp = re * (re * 0.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.2e+171)
		tmp = Float64(-im);
	else
		tmp = Float64(re * Float64(re * 0.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.2e+171)
		tmp = -im;
	else
		tmp = re * (re * 0.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.2e+171], (-im), N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.2000000000000003e171

   1. Initial program 53.5%

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

      1. neg-sub0 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in im around 0 53.0%

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

      1. associate-*r* 53.0%

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

      2. neg-mul-1 53.0%

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

   6. Simplified 53.0%

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

   7. Taylor expanded in re around 0 31.2%

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

      1. neg-mul-1 31.2%

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

   9. Simplified 31.2%

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

   if 4.2000000000000003e171 < re

   1. Initial program 56.9%

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

      1. neg-sub0 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in re around 0 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 20.8%

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

      4. unpow2 20.8%

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

   6. Simplified 20.8%

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

   8. Applied egg-rr 25.8%

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

   9. Taylor expanded in re around inf 25.8%

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

       1. *-commutative 25.8%

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

       2. unpow2 25.8%

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

       3. associate-*l* 25.8%

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

   11. Simplified 25.8%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.2 \cdot 10^{+171}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \]

Alternative 19: 29.3% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

   1. neg-sub0 53.8%

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

3. Simplified 53.8%

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

4. Taylor expanded in im around 0 52.5%

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

   1. associate-*r* 52.5%

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

   2. neg-mul-1 52.5%

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

6. Simplified 52.5%

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

7. Taylor expanded in re around 0 28.9%

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

   1. neg-mul-1 28.9%

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

9. Simplified 28.9%

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

10. Final simplification 28.9%

    \[\leadsto -im \]

Alternative 20: 2.8% accurate, 309.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

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

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 53.8%

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

   1. neg-sub0 53.8%

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

3. Simplified 53.8%

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

4. Taylor expanded in re around 0 3.0%

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

   1. +-commutative 3.0%

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

   2. associate-*r* 3.0%

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

   3. distribute-rgt-out 42.1%

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

   4. unpow2 42.1%

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

6. Simplified 42.1%

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

8. Applied egg-rr 7.8%

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

9. Taylor expanded in re around 0 2.9%

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

10. Final simplification 2.9%

    \[\leadsto -1.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(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))))