math.sin on complex, imaginary part

Percentage Accurate: 53.6% → 99.6%

Time: 17.5s

Alternatives: 18

Speedup: 2.9×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.6% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -5 \lor \neg \left(t_0 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -5.0) (not (<= t_0 2e-12)))
     (* (* 0.5 (cos re)) t_0)
     (+
      (* -0.008333333333333333 (* (cos re) (pow im 5.0)))
      (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 2e-12)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = (-0.008333333333333333 * (cos(re) * pow(im, 5.0))) + (cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-5.0d0)) .or. (.not. (t_0 <= 2d-12))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = ((-0.008333333333333333d0) * (cos(re) * (im ** 5.0d0))) + (cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - 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 <= -5.0) || !(t_0 <= 2e-12)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = (-0.008333333333333333 * (Math.cos(re) * Math.pow(im, 5.0))) + (Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -5.0) or not (t_0 <= 2e-12):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = (-0.008333333333333333 * (math.cos(re) * math.pow(im, 5.0))) + (math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -5.0) || !(t_0 <= 2e-12))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(Float64(-0.008333333333333333 * Float64(cos(re) * (im ^ 5.0))) + Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -5.0) || ~((t_0 <= 2e-12)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = (-0.008333333333333333 * (cos(re) * (im ^ 5.0))) + (cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - 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, -5.0], N[Not[LessEqual[t$95$0, 2e-12]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(-0.008333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -5 or 1.99999999999999996e-12 < (-.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 -5 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1.99999999999999996e-12

   1. Initial program 9.3%

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

      1. neg-sub0 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. *-commutative 99.9%

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

      7. mul-1-neg 99.9%

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

      8. unsub-neg 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

      11. distribute-lft-out-- 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in re around inf 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;t_0 \leq -5 \lor \neg \left(t_0 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (cos re))))
   (if (or (<= t_0 -5.0) (not (<= t_0 2e-12)))
     (* t_1 t_0)
     (*
      t_1
      (+
       (* im -2.0)
       (+
        (* (pow im 5.0) -0.016666666666666666)
        (* (pow im 3.0) -0.3333333333333333)))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = 0.5 * cos(re);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 2e-12)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((im * -2.0) + ((pow(im, 5.0) * -0.016666666666666666) + (pow(im, 3.0) * -0.3333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = 0.5d0 * cos(re)
    if ((t_0 <= (-5.0d0)) .or. (.not. (t_0 <= 2d-12))) then
        tmp = t_1 * t_0
    else
        tmp = t_1 * ((im * (-2.0d0)) + (((im ** 5.0d0) * (-0.016666666666666666d0)) + ((im ** 3.0d0) * (-0.3333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = 0.5 * Math.cos(re);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 2e-12)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((im * -2.0) + ((Math.pow(im, 5.0) * -0.016666666666666666) + (Math.pow(im, 3.0) * -0.3333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = 0.5 * math.cos(re)
	tmp = 0
	if (t_0 <= -5.0) or not (t_0 <= 2e-12):
		tmp = t_1 * t_0
	else:
		tmp = t_1 * ((im * -2.0) + ((math.pow(im, 5.0) * -0.016666666666666666) + (math.pow(im, 3.0) * -0.3333333333333333)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(0.5 * cos(re))
	tmp = 0.0
	if ((t_0 <= -5.0) || !(t_0 <= 2e-12))
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_1 * Float64(Float64(im * -2.0) + Float64(Float64((im ^ 5.0) * -0.016666666666666666) + Float64((im ^ 3.0) * -0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = 0.5 * cos(re);
	tmp = 0.0;
	if ((t_0 <= -5.0) || ~((t_0 <= 2e-12)))
		tmp = t_1 * t_0;
	else
		tmp = t_1 * ((im * -2.0) + (((im ^ 5.0) * -0.016666666666666666) + ((im ^ 3.0) * -0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5.0], N[Not[LessEqual[t$95$0, 2e-12]], $MachinePrecision]], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$1 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -5 or 1.99999999999999996e-12 < (-.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 -5 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1.99999999999999996e-12

   1. Initial program 9.3%

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

      1. neg-sub0 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.002 \lor \neg \left(t_0 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.002) (not (<= t_0 2e-12)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.002) || !(t_0 <= 2e-12)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.002d0)) .or. (.not. (t_0 <= 2d-12))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.002) || !(t_0 <= 2e-12)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.002) or not (t_0 <= 2e-12):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.002) || !(t_0 <= 2e-12))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.002) || ~((t_0 <= 2e-12)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.002], N[Not[LessEqual[t$95$0, 2e-12]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -2e-3 or 1.99999999999999996e-12 < (-.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 -2e-3 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1.99999999999999996e-12

   1. Initial program 8.7%

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

      1. neg-sub0 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. distribute-lft-out-- 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.002 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 4: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+94} \lor \neg \left(im \leq -130 \lor \neg \left(im \leq 2\right) \land im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -7.5e+94)
         (not (or (<= im -130.0) (and (not (<= im 2.0)) (<= im 5.6e+102)))))
   (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
   (* (- (exp (- im)) (exp im)) (+ 0.5 (* re (* re -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -7.5e+94) || !((im <= -130.0) || (!(im <= 2.0) && (im <= 5.6e+102)))) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else {
		tmp = (exp(-im) - exp(im)) * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-7.5d+94)) .or. (.not. (im <= (-130.0d0)) .or. (.not. (im <= 2.0d0)) .and. (im <= 5.6d+102))) then
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else
        tmp = (exp(-im) - exp(im)) * (0.5d0 + (re * (re * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -7.5e+94) || !((im <= -130.0) || (!(im <= 2.0) && (im <= 5.6e+102)))) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else {
		tmp = (Math.exp(-im) - Math.exp(im)) * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -7.5e+94) or not ((im <= -130.0) or (not (im <= 2.0) and (im <= 5.6e+102))):
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	else:
		tmp = (math.exp(-im) - math.exp(im)) * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -7.5e+94) || !((im <= -130.0) || (!(im <= 2.0) && (im <= 5.6e+102))))
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	else
		tmp = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -7.5e+94) || ~(((im <= -130.0) || (~((im <= 2.0)) && (im <= 5.6e+102)))))
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	else
		tmp = (exp(-im) - exp(im)) * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -7.5e+94], N[Not[Or[LessEqual[im, -130.0], And[N[Not[LessEqual[im, 2.0]], $MachinePrecision], LessEqual[im, 5.6e+102]]]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -7.49999999999999978e94 or -130 < im < 2 or 5.60000000000000037e102 < im

   1. Initial program 46.3%

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

      1. neg-sub0 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in im around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*l* 98.3%

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

      5. distribute-lft-out-- 98.3%

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

   6. Simplified 98.3%

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

   if -7.49999999999999978e94 < im < -130 or 2 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

      1. 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 3.0%

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

      1. *-commutative 3.0%

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

      2. associate-*r* 3.0%

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

      3. distribute-rgt-out 84.8%

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

      4. +-commutative 84.8%

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

      5. *-commutative 84.8%

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

      6. unpow2 84.8%

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

      7. associate-*l* 84.8%

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

   6. Simplified 84.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+94} \lor \neg \left(im \leq -130 \lor \neg \left(im \leq 2\right) \land im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 5: 93.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\\ t_1 := \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{if}\;im \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.072:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 6.5 \lor \neg \left(im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \frac{t_0 \cdot t_0 - \left(im \cdot -2\right) \cdot \left(im \cdot -2\right)}{t_0 - im \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* -0.3333333333333333 (* im im))))
        (t_1 (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))))
   (if (<= im -5.5e+102)
     t_1
     (if (<= im -0.072)
       (* 0.5 (- (exp (- im)) (exp im)))
       (if (or (<= im 6.5) (not (<= im 5.6e+102)))
         t_1
         (*
          (fma (* re re) -0.25 0.5)
          (/
           (- (* t_0 t_0) (* (* im -2.0) (* im -2.0)))
           (- t_0 (* im -2.0)))))))))

C

double code(double re, double im) {
	double t_0 = im * (-0.3333333333333333 * (im * im));
	double t_1 = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	double tmp;
	if (im <= -5.5e+102) {
		tmp = t_1;
	} else if (im <= -0.072) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else if ((im <= 6.5) || !(im <= 5.6e+102)) {
		tmp = t_1;
	} else {
		tmp = fma((re * re), -0.25, 0.5) * (((t_0 * t_0) - ((im * -2.0) * (im * -2.0))) / (t_0 - (im * -2.0)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * Float64(-0.3333333333333333 * Float64(im * im)))
	t_1 = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im))
	tmp = 0.0
	if (im <= -5.5e+102)
		tmp = t_1;
	elseif (im <= -0.072)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	elseif ((im <= 6.5) || !(im <= 5.6e+102))
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(Float64(Float64(t_0 * t_0) - Float64(Float64(im * -2.0) * Float64(im * -2.0))) / Float64(t_0 - Float64(im * -2.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.5e+102], t$95$1, If[LessEqual[im, -0.072], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 6.5], N[Not[LessEqual[im, 5.6e+102]], $MachinePrecision]], t$95$1, N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[(im * -2.0), $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.49999999999999981e102 or -0.0719999999999999946 < im < 6.5 or 5.60000000000000037e102 < im

   1. Initial program 45.6%

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

      1. neg-sub0 45.6%

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

   3. Simplified 45.6%

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

   4. Taylor expanded in im around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*l* 99.5%

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

      5. distribute-lft-out-- 99.5%

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

   6. Simplified 99.5%

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

   if -5.49999999999999981e102 < im < -0.0719999999999999946

   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 69.0%

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

   if 6.5 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

      1. 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 7.1%

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

      1. *-commutative 7.1%

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

      2. associate-*r* 7.1%

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

      3. distribute-rgt-out 85.7%

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

      4. +-commutative 85.7%

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

      5. *-commutative 85.7%

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

      6. unpow2 85.7%

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

      7. associate-*l* 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in im around 0 32.9%

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

      1. associate-*r* 32.9%

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

      2. associate-*r* 32.9%

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

      3. distribute-rgt-out 32.9%

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

      4. +-commutative 32.9%

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

      5. *-commutative 32.9%

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

      6. fma-def 32.9%

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

      7. unpow2 32.9%

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

      8. unpow3 32.9%

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

      9. associate-*r* 32.9%

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

      10. distribute-rgt-out 32.9%

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

   9. Simplified 32.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-in 32.9%

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

       2. flip-+ 72.9%

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \color{blue}{\frac{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right) - \left(-2 \cdot im\right) \cdot \left(-2 \cdot im\right)}{\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im - -2 \cdot im}} \]

   11. Applied egg-rr 72.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq -0.072:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 6.5 \lor \neg \left(im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \frac{\left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right) - \left(im \cdot -2\right) \cdot \left(im \cdot -2\right)}{im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) - im \cdot -2}\\ \end{array} \]

Alternative 6: 85.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3 \cdot 10^{+234}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{elif}\;im \leq -2.1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \left(-2 + -0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq -0.00105 \lor \neg \left(im \leq 2\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3e+234)
   (* -0.16666666666666666 (pow im 3.0))
   (if (<= im -2.1e+70)
     (*
      (fma (* re re) -0.25 0.5)
      (* im (+ -2.0 (* -0.3333333333333333 (* im im)))))
     (if (or (<= im -0.00105) (not (<= im 2.0)))
       (* 0.5 (- (exp (- im)) (exp im)))
       (* im (- (cos re)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3e+234) {
		tmp = -0.16666666666666666 * pow(im, 3.0);
	} else if (im <= -2.1e+70) {
		tmp = fma((re * re), -0.25, 0.5) * (im * (-2.0 + (-0.3333333333333333 * (im * im))));
	} else if ((im <= -0.00105) || !(im <= 2.0)) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3e+234)
		tmp = Float64(-0.16666666666666666 * (im ^ 3.0));
	elseif (im <= -2.1e+70)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im * Float64(-2.0 + Float64(-0.3333333333333333 * Float64(im * im)))));
	elseif ((im <= -0.00105) || !(im <= 2.0))
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3e+234], N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -2.1e+70], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * N[(-2.0 + N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -0.00105], N[Not[LessEqual[im, 2.0]], $MachinePrecision]], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.9999999999999999e234

   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%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 90.0%

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

   8. Taylor expanded in im around inf 90.0%

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

      1. *-commutative 90.0%

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

   10. Simplified 90.0%

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

   if -2.9999999999999999e234 < im < -2.10000000000000008e70

   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%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 79.4%

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

      4. +-commutative 79.4%

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

      5. *-commutative 79.4%

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

      6. unpow2 79.4%

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

      7. associate-*l* 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in im around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. associate-*r* 76.6%

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

      3. distribute-rgt-out 76.6%

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

      4. +-commutative 76.6%

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

      5. *-commutative 76.6%

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

      6. fma-def 76.6%

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

      7. unpow2 76.6%

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

      8. unpow3 76.6%

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

      9. associate-*r* 76.6%

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

      10. distribute-rgt-out 76.6%

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

   9. Simplified 76.6%

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

   if -2.10000000000000008e70 < im < -0.00104999999999999994 or 2 < 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 72.7%

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

   if -0.00104999999999999994 < im < 2

   1. Initial program 10.0%

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

      1. neg-sub0 10.0%

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

   3. Simplified 10.0%

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

   4. Taylor expanded in im around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. *-commutative 98.5%

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

      3. distribute-lft-neg-in 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3 \cdot 10^{+234}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{elif}\;im \leq -2.1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \left(-2 + -0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq -0.00105 \lor \neg \left(im \leq 2\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 7: 80.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ t_1 := -0.16666666666666666 \cdot {im}^{3}\\ t_2 := -0.3333333333333333 \cdot \left(im \cdot im\right)\\ t_3 := im \cdot t_2\\ \mathbf{if}\;im \leq -7.8 \cdot 10^{+233}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -49000000:\\ \;\;\;\;t_0 \cdot \left(im \cdot \left(-2 + t_2\right)\right)\\ \mathbf{elif}\;im \leq 6.5:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;t_0 \cdot \frac{t_3 \cdot t_3 - \left(im \cdot -2\right) \cdot \left(im \cdot -2\right)}{t_3 - im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;t_1 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* re re) -0.25 0.5))
        (t_1 (* -0.16666666666666666 (pow im 3.0)))
        (t_2 (* -0.3333333333333333 (* im im)))
        (t_3 (* im t_2)))
   (if (<= im -7.8e+233)
     t_1
     (if (<= im -49000000.0)
       (* t_0 (* im (+ -2.0 t_2)))
       (if (<= im 6.5)
         (* im (- (cos re)))
         (if (<= im 8.2e+102)
           (*
            t_0
            (/
             (- (* t_3 t_3) (* (* im -2.0) (* im -2.0)))
             (- t_3 (* im -2.0))))
           (- t_1 im)))))))

C

double code(double re, double im) {
	double t_0 = fma((re * re), -0.25, 0.5);
	double t_1 = -0.16666666666666666 * pow(im, 3.0);
	double t_2 = -0.3333333333333333 * (im * im);
	double t_3 = im * t_2;
	double tmp;
	if (im <= -7.8e+233) {
		tmp = t_1;
	} else if (im <= -49000000.0) {
		tmp = t_0 * (im * (-2.0 + t_2));
	} else if (im <= 6.5) {
		tmp = im * -cos(re);
	} else if (im <= 8.2e+102) {
		tmp = t_0 * (((t_3 * t_3) - ((im * -2.0) * (im * -2.0))) / (t_3 - (im * -2.0)));
	} else {
		tmp = t_1 - im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(re * re), -0.25, 0.5)
	t_1 = Float64(-0.16666666666666666 * (im ^ 3.0))
	t_2 = Float64(-0.3333333333333333 * Float64(im * im))
	t_3 = Float64(im * t_2)
	tmp = 0.0
	if (im <= -7.8e+233)
		tmp = t_1;
	elseif (im <= -49000000.0)
		tmp = Float64(t_0 * Float64(im * Float64(-2.0 + t_2)));
	elseif (im <= 6.5)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 8.2e+102)
		tmp = Float64(t_0 * Float64(Float64(Float64(t_3 * t_3) - Float64(Float64(im * -2.0) * Float64(im * -2.0))) / Float64(t_3 - Float64(im * -2.0))));
	else
		tmp = Float64(t_1 - im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(im * t$95$2), $MachinePrecision]}, If[LessEqual[im, -7.8e+233], t$95$1, If[LessEqual[im, -49000000.0], N[(t$95$0 * N[(im * N[(-2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.5], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 8.2e+102], N[(t$95$0 * N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(im * -2.0), $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 - N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - im), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -7.7999999999999998e233

   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%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 90.0%

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

   8. Taylor expanded in im around inf 90.0%

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

      1. *-commutative 90.0%

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

   10. Simplified 90.0%

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

   if -7.7999999999999998e233 < im < -4.9e7

   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%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 80.0%

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

      4. +-commutative 80.0%

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

      5. *-commutative 80.0%

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

      6. unpow2 80.0%

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

      7. associate-*l* 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in im around 0 63.0%

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

      1. associate-*r* 63.0%

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

      2. associate-*r* 63.0%

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

      3. distribute-rgt-out 63.0%

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

      4. +-commutative 63.0%

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

      5. *-commutative 63.0%

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

      6. fma-def 63.0%

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

      7. unpow2 63.0%

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

      8. unpow3 63.0%

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

      9. associate-*r* 63.0%

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

      10. distribute-rgt-out 63.0%

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

   9. Simplified 63.0%

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

   if -4.9e7 < im < 6.5

   1. Initial program 13.3%

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

      1. neg-sub0 13.3%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in im around 0 95.1%

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

      1. mul-1-neg 95.1%

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

      2. *-commutative 95.1%

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

      3. distribute-lft-neg-in 95.1%

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

   6. Simplified 95.1%

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

   if 6.5 < im < 8.1999999999999999e102

   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 7.1%

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

      1. *-commutative 7.1%

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

      2. associate-*r* 7.1%

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

      3. distribute-rgt-out 85.7%

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

      4. +-commutative 85.7%

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

      5. *-commutative 85.7%

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

      6. unpow2 85.7%

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

      7. associate-*l* 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in im around 0 32.9%

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

      1. associate-*r* 32.9%

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

      2. associate-*r* 32.9%

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

      3. distribute-rgt-out 32.9%

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

      4. +-commutative 32.9%

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

      5. *-commutative 32.9%

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

      6. fma-def 32.9%

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

      7. unpow2 32.9%

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

      8. unpow3 32.9%

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

      9. associate-*r* 32.9%

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

      10. distribute-rgt-out 32.9%

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

   9. Simplified 32.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) + -2\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-in 32.9%

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

       2. flip-+ 72.9%

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \color{blue}{\frac{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right) - \left(-2 \cdot im\right) \cdot \left(-2 \cdot im\right)}{\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im - -2 \cdot im}} \]

   11. Applied egg-rr 72.9%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \color{blue}{\frac{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right) - \left(-2 \cdot im\right) \cdot \left(-2 \cdot im\right)}{\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im - -2 \cdot im}} \]

   if 8.1999999999999999e102 < 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%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 71.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.8 \cdot 10^{+233}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{elif}\;im \leq -49000000:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \left(-2 + -0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.5:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \frac{\left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right) - \left(im \cdot -2\right) \cdot \left(im \cdot -2\right)}{im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) - im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3} - im\\ \end{array} \]

Alternative 8: 77.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{+233}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{elif}\;im \leq -110000000 \lor \neg \left(im \leq 6.5\right):\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \left(-2 + -0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.6e+233)
   (* -0.16666666666666666 (pow im 3.0))
   (if (or (<= im -110000000.0) (not (<= im 6.5)))
     (*
      (fma (* re re) -0.25 0.5)
      (* im (+ -2.0 (* -0.3333333333333333 (* im im)))))
     (* im (- (cos re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.6e+233) {
		tmp = -0.16666666666666666 * pow(im, 3.0);
	} else if ((im <= -110000000.0) || !(im <= 6.5)) {
		tmp = fma((re * re), -0.25, 0.5) * (im * (-2.0 + (-0.3333333333333333 * (im * im))));
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.6e+233)
		tmp = Float64(-0.16666666666666666 * (im ^ 3.0));
	elseif ((im <= -110000000.0) || !(im <= 6.5))
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im * Float64(-2.0 + Float64(-0.3333333333333333 * Float64(im * im)))));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.6e+233], N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -110000000.0], N[Not[LessEqual[im, 6.5]], $MachinePrecision]], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * N[(-2.0 + N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.60000000000000009e233

   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%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 90.0%

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

   8. Taylor expanded in im around inf 90.0%

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

      1. *-commutative 90.0%

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

   10. Simplified 90.0%

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

   if -1.60000000000000009e233 < im < -1.1e8 or 6.5 < 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 1.0%

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

      1. *-commutative 1.0%

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

      2. associate-*r* 1.0%

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

      3. distribute-rgt-out 75.5%

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

      4. +-commutative 75.5%

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

      5. *-commutative 75.5%

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

      6. unpow2 75.5%

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

      7. associate-*l* 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in im around 0 60.2%

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

      1. associate-*r* 60.2%

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

      2. associate-*r* 60.2%

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

      3. distribute-rgt-out 60.2%

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

      4. +-commutative 60.2%

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

      5. *-commutative 60.2%

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

      6. fma-def 60.2%

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

      7. unpow2 60.2%

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

      8. unpow3 60.2%

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

      9. associate-*r* 60.2%

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

      10. distribute-rgt-out 60.2%

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

   9. Simplified 60.2%

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

   if -1.1e8 < im < 6.5

   1. Initial program 13.3%

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

      1. neg-sub0 13.3%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in im around 0 95.1%

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

      1. mul-1-neg 95.1%

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

      2. *-commutative 95.1%

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

      3. distribute-lft-neg-in 95.1%

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

   6. Simplified 95.1%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{+233}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{elif}\;im \leq -110000000 \lor \neg \left(im \leq 6.5\right):\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \left(-2 + -0.3333333333333333 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 9: 75.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot {im}^{3}\\ \mathbf{if}\;im \leq -2.8 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;-0.25 \cdot \left(\left(im \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+50}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (pow im 3.0))))
   (if (<= im -2.8e+122)
     t_0
     (if (<= im -8.5e+16)
       (* -0.25 (* (* im (* re re)) (fma -0.3333333333333333 (* im im) -2.0)))
       (if (<= im 3.3e+50) (* im (- (cos re))) (- t_0 im))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * pow(im, 3.0);
	double tmp;
	if (im <= -2.8e+122) {
		tmp = t_0;
	} else if (im <= -8.5e+16) {
		tmp = -0.25 * ((im * (re * re)) * fma(-0.3333333333333333, (im * im), -2.0));
	} else if (im <= 3.3e+50) {
		tmp = im * -cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * (im ^ 3.0))
	tmp = 0.0
	if (im <= -2.8e+122)
		tmp = t_0;
	elseif (im <= -8.5e+16)
		tmp = Float64(-0.25 * Float64(Float64(im * Float64(re * re)) * fma(-0.3333333333333333, Float64(im * im), -2.0)));
	elseif (im <= 3.3e+50)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.8e+122], t$95$0, If[LessEqual[im, -8.5e+16], N[(-0.25 * N[(N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.3e+50], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(t$95$0 - im), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.8e122

   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%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 77.8%

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

   8. Taylor expanded in im around inf 77.8%

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

      1. *-commutative 77.8%

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

   10. Simplified 77.8%

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

   if -2.8e122 < im < -8.5e16

   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%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 77.8%

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

      4. +-commutative 77.8%

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

      5. *-commutative 77.8%

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

      6. unpow2 77.8%

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

      7. associate-*l* 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in im around 0 46.0%

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

      1. associate-*r* 46.0%

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

      2. associate-*r* 46.0%

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

      3. distribute-rgt-out 46.0%

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

      4. +-commutative 46.0%

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

      5. *-commutative 46.0%

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

      6. fma-def 46.0%

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

      7. unpow2 46.0%

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

      8. unpow3 46.0%

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

      9. associate-*r* 46.0%

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

      10. distribute-rgt-out 46.0%

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

   9. Simplified 46.0%

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

   10. Taylor expanded in re around inf 39.4%

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

       1. *-commutative 39.4%

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

       2. *-commutative 39.4%

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

       3. unpow2 39.4%

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

       4. fma-neg 39.4%

          \[\leadsto -0.25 \cdot \left(\left(im \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right)}\right) \]

       5. unpow2 39.4%

          \[\leadsto -0.25 \cdot \left(\left(im \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(-0.3333333333333333, \color{blue}{im \cdot im}, -2\right)\right) \]

       6. metadata-eval 39.4%

          \[\leadsto -0.25 \cdot \left(\left(im \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, \color{blue}{-2}\right)\right) \]

   12. Simplified 39.4%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\left(im \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right)} \]

   if -8.5e16 < im < 3.3e50

   1. Initial program 18.0%

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

      1. neg-sub0 18.0%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in im around 0 90.1%

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

      1. mul-1-neg 90.1%

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

      2. *-commutative 90.1%

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

      3. distribute-lft-neg-in 90.1%

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

   6. Simplified 90.1%

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

   if 3.3e50 < 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 84.2%

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

      1. mul-1-neg 84.2%

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

      2. unsub-neg 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-*l* 84.2%

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

      5. distribute-lft-out-- 84.2%

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

   6. Simplified 84.2%

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

   7. Taylor expanded in re around 0 60.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+122}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{elif}\;im \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;-0.25 \cdot \left(\left(im \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+50}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3} - im\\ \end{array} \]

Alternative 10: 75.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot {im}^{3}\\ \mathbf{if}\;im \leq -9.6 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (pow im 3.0))))
   (if (<= im -9.6e+86)
     t_0
     (if (<= im 2.25e+52) (* im (- (cos re))) (- t_0 im)))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * pow(im, 3.0);
	double tmp;
	if (im <= -9.6e+86) {
		tmp = t_0;
	} else if (im <= 2.25e+52) {
		tmp = im * -cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.16666666666666666d0) * (im ** 3.0d0)
    if (im <= (-9.6d+86)) then
        tmp = t_0
    else if (im <= 2.25d+52) then
        tmp = im * -cos(re)
    else
        tmp = t_0 - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.16666666666666666 * Math.pow(im, 3.0);
	double tmp;
	if (im <= -9.6e+86) {
		tmp = t_0;
	} else if (im <= 2.25e+52) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * math.pow(im, 3.0)
	tmp = 0
	if im <= -9.6e+86:
		tmp = t_0
	elif im <= 2.25e+52:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0 - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * (im ^ 3.0))
	tmp = 0.0
	if (im <= -9.6e+86)
		tmp = t_0;
	elseif (im <= 2.25e+52)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (im ^ 3.0);
	tmp = 0.0;
	if (im <= -9.6e+86)
		tmp = t_0;
	elseif (im <= 2.25e+52)
		tmp = im * -cos(re);
	else
		tmp = t_0 - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9.6e+86], t$95$0, If[LessEqual[im, 2.25e+52], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(t$95$0 - im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.6000000000000001e86

   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 96.4%

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

      1. mul-1-neg 96.4%

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

      2. unsub-neg 96.4%

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

      3. *-commutative 96.4%

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

      4. associate-*l* 96.4%

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

      5. distribute-lft-out-- 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in re around 0 72.4%

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

   8. Taylor expanded in im around inf 72.4%

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

      1. *-commutative 72.4%

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

   10. Simplified 72.4%

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

   if -9.6000000000000001e86 < im < 2.25e52

   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 83.0%

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

      1. mul-1-neg 83.0%

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

      2. *-commutative 83.0%

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

      3. distribute-lft-neg-in 83.0%

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

   6. Simplified 83.0%

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

   if 2.25e52 < 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 84.2%

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

      1. mul-1-neg 84.2%

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

      2. unsub-neg 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-*l* 84.2%

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

      5. distribute-lft-out-- 84.2%

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

   6. Simplified 84.2%

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

   7. Taylor expanded in re around 0 60.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.6 \cdot 10^{+86}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3} - im\\ \end{array} \]

Alternative 11: 60.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{+16} \lor \neg \left(im \leq 6.5\right):\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -6.8e+16) (not (<= im 6.5)))
   (- (* re (* re (* im 0.5))) im)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -6.8e+16) || !(im <= 6.5)) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-6.8d+16)) .or. (.not. (im <= 6.5d0))) then
        tmp = (re * (re * (im * 0.5d0))) - im
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -6.8e+16) || !(im <= 6.5)) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -6.8e+16) or not (im <= 6.5):
		tmp = (re * (re * (im * 0.5))) - im
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -6.8e+16) || !(im <= 6.5))
		tmp = Float64(Float64(re * Float64(re * Float64(im * 0.5))) - im);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -6.8e+16) || ~((im <= 6.5)))
		tmp = (re * (re * (im * 0.5))) - im;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -6.8e+16], N[Not[LessEqual[im, 6.5]], $MachinePrecision]], N[(N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -6.8e16 or 6.5 < 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 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. distribute-lft-neg-in 5.8%

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

   6. Simplified 5.8%

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

   7. Taylor expanded in re around 0 26.3%

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

      1. neg-mul-1 26.3%

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

      2. +-commutative 26.3%

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

      3. unsub-neg 26.3%

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

      4. *-commutative 26.3%

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

      5. associate-*l* 26.3%

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

      6. unpow2 26.3%

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

      7. *-commutative 26.3%

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

      8. associate-*l* 26.3%

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

      9. *-commutative 26.3%

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

   9. Simplified 26.3%

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

   if -6.8e16 < im < 6.5

   1. Initial program 14.5%

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

      1. neg-sub0 14.5%

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

   3. Simplified 14.5%

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

   4. Taylor expanded in im around 0 93.8%

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

      1. mul-1-neg 93.8%

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

      2. *-commutative 93.8%

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

      3. distribute-lft-neg-in 93.8%

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

   6. Simplified 93.8%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{+16} \lor \neg \left(im \leq 6.5\right):\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 12: 75.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+86} \lor \neg \left(im \leq 8.4 \cdot 10^{+43}\right):\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -6.5e+86) (not (<= im 8.4e+43)))
   (* -0.16666666666666666 (pow im 3.0))
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -6.5e+86) || !(im <= 8.4e+43)) {
		tmp = -0.16666666666666666 * pow(im, 3.0);
	} 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 <= (-6.5d+86)) .or. (.not. (im <= 8.4d+43))) then
        tmp = (-0.16666666666666666d0) * (im ** 3.0d0)
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -6.5e+86) || !(im <= 8.4e+43)) {
		tmp = -0.16666666666666666 * Math.pow(im, 3.0);
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -6.5e+86) or not (im <= 8.4e+43):
		tmp = -0.16666666666666666 * math.pow(im, 3.0)
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -6.5e+86) || !(im <= 8.4e+43))
		tmp = Float64(-0.16666666666666666 * (im ^ 3.0));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -6.5e+86) || ~((im <= 8.4e+43)))
		tmp = -0.16666666666666666 * (im ^ 3.0);
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -6.5e+86], N[Not[LessEqual[im, 8.4e+43]], $MachinePrecision]], N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -6.49999999999999996e86 or 8.40000000000000007e43 < 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 90.4%

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

      1. mul-1-neg 90.4%

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

      2. unsub-neg 90.4%

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

      3. *-commutative 90.4%

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

      4. associate-*l* 90.4%

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

      5. distribute-lft-out-- 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in re around 0 66.7%

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

   8. Taylor expanded in im around inf 66.7%

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

      1. *-commutative 66.7%

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

   10. Simplified 66.7%

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

   if -6.49999999999999996e86 < im < 8.40000000000000007e43

   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 83.0%

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

      1. mul-1-neg 83.0%

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

      2. *-commutative 83.0%

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

      3. distribute-lft-neg-in 83.0%

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

   6. Simplified 83.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+86} \lor \neg \left(im \leq 8.4 \cdot 10^{+43}\right):\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 13: 36.2% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot -0.25\right)\\ \mathbf{if}\;re \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{+153}:\\ \;\;\;\;\frac{182.25 - 729 \cdot \left(t_0 \cdot t_0\right)}{13.5 - \left(re \cdot -0.25\right) \cdot \left(re \cdot 27\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + t_0\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re -0.25))))
   (if (<= re 1.05e+80)
     (- (* re (* re (* im 0.5))) im)
     (if (<= re 3.1e+153)
       (/
        (- 182.25 (* 729.0 (* t_0 t_0)))
        (- 13.5 (* (* re -0.25) (* re 27.0))))
       (* (+ 0.5 t_0) -3.0)))))

C

double code(double re, double im) {
	double t_0 = re * (re * -0.25);
	double tmp;
	if (re <= 1.05e+80) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else if (re <= 3.1e+153) {
		tmp = (182.25 - (729.0 * (t_0 * t_0))) / (13.5 - ((re * -0.25) * (re * 27.0)));
	} else {
		tmp = (0.5 + t_0) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (re * (-0.25d0))
    if (re <= 1.05d+80) then
        tmp = (re * (re * (im * 0.5d0))) - im
    else if (re <= 3.1d+153) then
        tmp = (182.25d0 - (729.0d0 * (t_0 * t_0))) / (13.5d0 - ((re * (-0.25d0)) * (re * 27.0d0)))
    else
        tmp = (0.5d0 + t_0) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * -0.25);
	double tmp;
	if (re <= 1.05e+80) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else if (re <= 3.1e+153) {
		tmp = (182.25 - (729.0 * (t_0 * t_0))) / (13.5 - ((re * -0.25) * (re * 27.0)));
	} else {
		tmp = (0.5 + t_0) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * -0.25)
	tmp = 0
	if re <= 1.05e+80:
		tmp = (re * (re * (im * 0.5))) - im
	elif re <= 3.1e+153:
		tmp = (182.25 - (729.0 * (t_0 * t_0))) / (13.5 - ((re * -0.25) * (re * 27.0)))
	else:
		tmp = (0.5 + t_0) * -3.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * -0.25))
	tmp = 0.0
	if (re <= 1.05e+80)
		tmp = Float64(Float64(re * Float64(re * Float64(im * 0.5))) - im);
	elseif (re <= 3.1e+153)
		tmp = Float64(Float64(182.25 - Float64(729.0 * Float64(t_0 * t_0))) / Float64(13.5 - Float64(Float64(re * -0.25) * Float64(re * 27.0))));
	else
		tmp = Float64(Float64(0.5 + t_0) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * -0.25);
	tmp = 0.0;
	if (re <= 1.05e+80)
		tmp = (re * (re * (im * 0.5))) - im;
	elseif (re <= 3.1e+153)
		tmp = (182.25 - (729.0 * (t_0 * t_0))) / (13.5 - ((re * -0.25) * (re * 27.0)));
	else
		tmp = (0.5 + t_0) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 1.05e+80], N[(N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[re, 3.1e+153], N[(N[(182.25 - N[(729.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(13.5 - N[(N[(re * -0.25), $MachinePrecision] * N[(re * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + t$95$0), $MachinePrecision] * -3.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.05000000000000001e80

   1. Initial program 54.3%

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

      1. neg-sub0 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in im around 0 52.7%

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

      1. mul-1-neg 52.7%

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

      2. *-commutative 52.7%

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

      3. distribute-lft-neg-in 52.7%

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

   6. Simplified 52.7%

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

   7. Taylor expanded in re around 0 40.8%

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

      1. neg-mul-1 40.8%

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

      2. +-commutative 40.8%

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

      3. unsub-neg 40.8%

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

      4. *-commutative 40.8%

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

      5. associate-*l* 40.8%

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

      6. unpow2 40.8%

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

      7. *-commutative 40.8%

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

      8. associate-*l* 40.8%

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

      9. *-commutative 40.8%

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

   9. Simplified 40.8%

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

   if 1.05000000000000001e80 < re < 3.1e153

   1. Initial program 39.0%

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

      1. neg-sub0 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in re around 0 3.9%

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

      1. *-commutative 3.9%

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

      2. associate-*r* 3.9%

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

      3. distribute-rgt-out 17.2%

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

      4. +-commutative 17.2%

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

      5. *-commutative 17.2%

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

      6. unpow2 17.2%

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

      7. associate-*l* 17.2%

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

   6. Simplified 17.2%

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

   8. Applied egg-rr 2.8%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in 2.8%

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

      2. flip-+ 21.4%

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

      3. metadata-eval 21.4%

         \[\leadsto \frac{\color{blue}{13.5} \cdot \left(27 \cdot 0.5\right) - \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{27 \cdot 0.5 - 27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      4. metadata-eval 21.4%

         \[\leadsto \frac{13.5 \cdot \color{blue}{13.5} - \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{27 \cdot 0.5 - 27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      5. metadata-eval 21.4%

         \[\leadsto \frac{\color{blue}{182.25} - \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{27 \cdot 0.5 - 27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      6. metadata-eval 21.4%

         \[\leadsto \frac{182.25 - \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{\color{blue}{13.5} - 27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

   10. Applied egg-rr 21.4%

       \[\leadsto \color{blue}{\frac{182.25 - \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{13.5 - 27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 21.4%

          \[\leadsto \frac{182.25 - \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\right)} \cdot \left(27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{13.5 - 27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

       2. *-commutative 21.4%

          \[\leadsto \frac{182.25 - \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\right) \cdot \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\right)}}{13.5 - 27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

       3. *-commutative 21.4%

          \[\leadsto \frac{182.25 - \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\right) \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\right)}{13.5 - \color{blue}{\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot 27}} \]

       4. swap-sqr 21.4%

          \[\leadsto \frac{182.25 - \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(27 \cdot 27\right)}}{13.5 - \left(re \cdot \left(re \cdot -0.25\right)\right) \cdot 27} \]

       5. *-commutative 21.4%

          \[\leadsto \frac{182.25 - \color{blue}{\left(27 \cdot 27\right) \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}}{13.5 - \left(re \cdot \left(re \cdot -0.25\right)\right) \cdot 27} \]

       6. metadata-eval 21.4%

          \[\leadsto \frac{182.25 - \color{blue}{729} \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{13.5 - \left(re \cdot \left(re \cdot -0.25\right)\right) \cdot 27} \]

       7. *-commutative 21.4%

          \[\leadsto \frac{182.25 - 729 \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{13.5 - \color{blue}{27 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)}} \]

       8. associate-*r* 21.4%

          \[\leadsto \frac{182.25 - 729 \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{13.5 - \color{blue}{\left(27 \cdot re\right) \cdot \left(re \cdot -0.25\right)}} \]

   12. Simplified 21.4%

       \[\leadsto \color{blue}{\frac{182.25 - 729 \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{13.5 - \left(27 \cdot re\right) \cdot \left(re \cdot -0.25\right)}} \]

   if 3.1e153 < re

   1. Initial program 53.1%

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

      1. neg-sub0 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 27.9%

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

      4. +-commutative 27.9%

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

      5. *-commutative 27.9%

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

      6. unpow2 27.9%

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

      7. associate-*l* 27.9%

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

   6. Simplified 27.9%

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

   8. Applied egg-rr 31.8%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{+153}:\\ \;\;\;\;\frac{182.25 - 729 \cdot \left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{13.5 - \left(re \cdot -0.25\right) \cdot \left(re \cdot 27\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \]

Alternative 14: 32.0% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.5 \cdot 10^{+162}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.5e+162) (- im) (* (+ 0.5 (* re (* re -0.25))) -3.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.5e+162) {
		tmp = -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.5d+162) then
        tmp = -im
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.5e+162) {
		tmp = -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.5e+162:
		tmp = -im
	else:
		tmp = (0.5 + (re * (re * -0.25))) * -3.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.5e+162)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.5e+162)
		tmp = -im;
	else
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.5e+162], (-im), N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.49999999999999966e162

   1. Initial program 53.4%

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

      1. neg-sub0 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in im around 0 53.7%

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

      1. mul-1-neg 53.7%

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

      2. *-commutative 53.7%

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

      3. distribute-lft-neg-in 53.7%

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

   6. Simplified 53.7%

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

   7. Taylor expanded in re around 0 33.3%

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

      1. neg-mul-1 33.3%

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

   9. Simplified 33.3%

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

   if 5.49999999999999966e162 < re

   1. Initial program 51.9%

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

      1. neg-sub0 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 25.9%

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

      4. +-commutative 25.9%

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

      5. *-commutative 25.9%

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

      6. unpow2 25.9%

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

      7. associate-*l* 25.9%

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

   6. Simplified 25.9%

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

   8. Applied egg-rr 36.3%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.5 \cdot 10^{+162}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \]

Alternative 15: 35.8% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.9 \cdot 10^{+158}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.9e+158)
   (- (* re (* re (* im 0.5))) im)
   (* (+ 0.5 (* re (* re -0.25))) -3.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.9e+158) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.9d+158) then
        tmp = (re * (re * (im * 0.5d0))) - im
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.9e+158) {
		tmp = (re * (re * (im * 0.5))) - im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.9e+158:
		tmp = (re * (re * (im * 0.5))) - im
	else:
		tmp = (0.5 + (re * (re * -0.25))) * -3.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.9e+158)
		tmp = Float64(Float64(re * Float64(re * Float64(im * 0.5))) - im);
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.9e+158)
		tmp = (re * (re * (im * 0.5))) - im;
	else
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.9e+158], N[(N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.9e158

   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 53.5%

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

      1. mul-1-neg 53.5%

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

      2. *-commutative 53.5%

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

      3. distribute-lft-neg-in 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in re around 0 39.0%

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

      1. neg-mul-1 39.0%

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

      2. +-commutative 39.0%

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

      3. unsub-neg 39.0%

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

      4. *-commutative 39.0%

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

      5. associate-*l* 39.0%

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

      6. unpow2 39.0%

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

      7. *-commutative 39.0%

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

      8. associate-*l* 39.0%

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

      9. *-commutative 39.0%

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

   9. Simplified 39.0%

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

   if 3.9e158 < re

   1. Initial program 51.7%

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

      1. neg-sub0 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 24.4%

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

      4. +-commutative 24.4%

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

      5. *-commutative 24.4%

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

      6. unpow2 24.4%

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

      7. associate-*l* 24.4%

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

   6. Simplified 24.4%

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

   8. Applied egg-rr 34.2%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.9 \cdot 10^{+158}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \]

Alternative 16: 32.4% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot -6.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.8e+152) (- im) (* re (* re -6.75))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.8e+152) {
		tmp = -im;
	} else {
		tmp = re * (re * -6.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.8d+152) then
        tmp = -im
    else
        tmp = re * (re * (-6.75d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.8e+152) {
		tmp = -im;
	} else {
		tmp = re * (re * -6.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.8e+152:
		tmp = -im
	else:
		tmp = re * (re * -6.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.8e+152)
		tmp = Float64(-im);
	else
		tmp = Float64(re * Float64(re * -6.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.8e+152)
		tmp = -im;
	else
		tmp = re * (re * -6.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.8e+152], (-im), N[(re * N[(re * -6.75), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.8000000000000002e152

   1. Initial program 53.3%

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

      1. neg-sub0 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in im around 0 53.8%

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

      1. mul-1-neg 53.8%

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

      2. *-commutative 53.8%

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

      3. distribute-lft-neg-in 53.8%

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

   6. Simplified 53.8%

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

   7. Taylor expanded in re around 0 33.8%

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

      1. neg-mul-1 33.8%

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

   9. Simplified 33.8%

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

   if 2.8000000000000002e152 < re

   1. Initial program 53.1%

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

      1. neg-sub0 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 27.9%

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

      4. +-commutative 27.9%

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

      5. *-commutative 27.9%

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

      6. unpow2 27.9%

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

      7. associate-*l* 27.9%

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

   6. Simplified 27.9%

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

   8. Applied egg-rr 12.1%

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

   9. Taylor expanded in re around inf 12.1%

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

       1. *-commutative 12.1%

          \[\leadsto \color{blue}{{re}^{2} \cdot -6.75} \]

       2. unpow2 12.1%

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

   11. Simplified 12.1%

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

   12. Taylor expanded in re around 0 12.1%

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

       1. *-commutative 12.1%

          \[\leadsto \color{blue}{{re}^{2} \cdot -6.75} \]

       2. unpow2 12.1%

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

       3. associate-*r* 12.1%

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

   14. Simplified 12.1%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot -6.75\right)\\ \end{array} \]

Alternative 17: 30.2% 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 53.2%

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

   1. neg-sub0 53.2%

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

3. Simplified 53.2%

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

4. Taylor expanded in im around 0 53.9%

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

   1. mul-1-neg 53.9%

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

   2. *-commutative 53.9%

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

   3. distribute-lft-neg-in 53.9%

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

6. Simplified 53.9%

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

7. Taylor expanded in re around 0 30.2%

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

   1. neg-mul-1 30.2%

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

9. Simplified 30.2%

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

10. Final simplification 30.2%

    \[\leadsto -im \]

Alternative 18: 2.9% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 13.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 13.5

Julia

function code(re, im)
	return 13.5
end

MATLAB

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

Wolfram

code[re_, im_] := 13.5

Derivation

1. Initial program 53.2%

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

   1. neg-sub0 53.2%

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

3. Simplified 53.2%

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

4. Taylor expanded in re around 0 3.4%

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

   1. *-commutative 3.4%

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

   2. associate-*r* 3.4%

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

   3. distribute-rgt-out 37.8%

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

   4. +-commutative 37.8%

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

   5. *-commutative 37.8%

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

   6. unpow2 37.8%

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

   7. associate-*l* 37.8%

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

6. Simplified 37.8%

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

8. Applied egg-rr 6.8%

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

9. Taylor expanded in re around 0 2.9%

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

10. Final simplification 2.9%

    \[\leadsto 13.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023279 
(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))))