math.sin on complex, imaginary part

Percentage Accurate: 53.7% → 99.9%

Time: 20.3s

Alternatives: 15

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: 53.7% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.05\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.05)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (+
       (+
        (* (pow im 7.0) -0.0001984126984126984)
        (* (pow im 5.0) -0.008333333333333333))
       (- (* (pow im 3.0) -0.16666666666666666) im))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 0.05)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * (((pow(im, 7.0) * -0.0001984126984126984) + (pow(im, 5.0) * -0.008333333333333333)) + ((pow(im, 3.0) * -0.16666666666666666) - im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-1.0d0)) .or. (.not. (t_0 <= 0.05d0))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * ((((im ** 7.0d0) * (-0.0001984126984126984d0)) + ((im ** 5.0d0) * (-0.008333333333333333d0))) + (((im ** 3.0d0) * (-0.16666666666666666d0)) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 0.05)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * (((Math.pow(im, 7.0) * -0.0001984126984126984) + (Math.pow(im, 5.0) * -0.008333333333333333)) + ((Math.pow(im, 3.0) * -0.16666666666666666) - im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -1.0) or not (t_0 <= 0.05):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * (((math.pow(im, 7.0) * -0.0001984126984126984) + (math.pow(im, 5.0) * -0.008333333333333333)) + ((math.pow(im, 3.0) * -0.16666666666666666) - im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -1.0) || !(t_0 <= 0.05))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64((im ^ 7.0) * -0.0001984126984126984) + Float64((im ^ 5.0) * -0.008333333333333333)) + Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -1.0) || ~((t_0 <= 0.05)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((((im ^ 7.0) * -0.0001984126984126984) + ((im ^ 5.0) * -0.008333333333333333)) + (((im ^ 3.0) * -0.16666666666666666) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1.0], N[Not[LessEqual[t$95$0, 0.05]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -1 or 0.050000000000000003 < (-.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 -1 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.050000000000000003

   1. Initial program 9.7%

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

      1. neg-sub0 9.7%

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

   3. Simplified 9.7%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. associate-+r+ 99.8%

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

      2. +-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. distribute-rgt-out 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

      11. *-commutative 99.8%

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

      12. associate-*l* 99.8%

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

      13. distribute-lft-out 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 0.05\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.05\right):\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (cos re))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.05)))
     (* 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 <= -1.0) || !(t_0 <= 0.05)) {
		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 <= (-1.0d0)) .or. (.not. (t_0 <= 0.05d0))) 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 <= -1.0) || !(t_0 <= 0.05)) {
		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 <= -1.0) or not (t_0 <= 0.05):
		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 <= -1.0) || !(t_0 <= 0.05))
		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 <= -1.0) || ~((t_0 <= 0.05)))
		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, -1.0], N[Not[LessEqual[t$95$0, 0.05]], $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)) < -1 or 0.050000000000000003 < (-.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 -1 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.050000000000000003

   1. Initial program 9.7%

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

      1. neg-sub0 9.7%

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

   3. Simplified 9.7%

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

   4. Taylor expanded in im around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 0.05\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.0005\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.0005)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 0.0005)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-1.0d0)) .or. (.not. (t_0 <= 0.0005d0))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 0.0005)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -1.0) or not (t_0 <= 0.0005):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -1.0) || !(t_0 <= 0.0005))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -1.0) || ~((t_0 <= 0.0005)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1.0], N[Not[LessEqual[t$95$0, 0.0005]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -1 or 5.0000000000000001e-4 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   if -1 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 5.0000000000000001e-4

   1. Initial program 9.0%

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

      1. neg-sub0 9.0%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 0.0005\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4: 94.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{+155} \lor \neg \left(im \leq -0.19\right) \land \left(im \leq 0.66 \lor \neg \left(im \leq 1.65 \cdot 10^{+100}\right)\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.5e+155)
         (and (not (<= im -0.19)) (or (<= im 0.66) (not (<= im 1.65e+100)))))
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* 0.5 (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.5e+155) || (!(im <= -0.19) && ((im <= 0.66) || !(im <= 1.65e+100)))) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-3.5d+155)) .or. (.not. (im <= (-0.19d0))) .and. (im <= 0.66d0) .or. (.not. (im <= 1.65d+100))) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = 0.5d0 * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.5e+155) || (!(im <= -0.19) && ((im <= 0.66) || !(im <= 1.65e+100)))) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.5e+155) or (not (im <= -0.19) and ((im <= 0.66) or not (im <= 1.65e+100))):
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.5e+155) || (!(im <= -0.19) && ((im <= 0.66) || !(im <= 1.65e+100))))
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.5e+155) || (~((im <= -0.19)) && ((im <= 0.66) || ~((im <= 1.65e+100)))))
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = 0.5 * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.5e+155], And[N[Not[LessEqual[im, -0.19]], $MachinePrecision], Or[LessEqual[im, 0.66], N[Not[LessEqual[im, 1.65e+100]], $MachinePrecision]]]], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.49999999999999985e155 or -0.19 < im < 0.660000000000000031 or 1.6500000000000001e100 < 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 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. distribute-lft-out-- 99.0%

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

   6. Simplified 99.0%

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

   if -3.49999999999999985e155 < im < -0.19 or 0.660000000000000031 < im < 1.6500000000000001e100

   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 85.1%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{+155} \lor \neg \left(im \leq -0.19\right) \land \left(im \leq 0.66 \lor \neg \left(im \leq 1.65 \cdot 10^{+100}\right)\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 5: 93.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -6.5 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.6:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 0.66 \lor \neg \left(im \leq 1.65 \cdot 10^{+100}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im)))
        (t_1 (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -6.5e+80)
     t_1
     (if (<= im -0.6)
       (* t_0 (+ 0.5 (* re (* re -0.25))))
       (if (or (<= im 0.66) (not (<= im 1.65e+100))) t_1 (* 0.5 t_0))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -6.5e+80) {
		tmp = t_1;
	} else if (im <= -0.6) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else if ((im <= 0.66) || !(im <= 1.65e+100)) {
		tmp = t_1;
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-6.5d+80)) then
        tmp = t_1
    else if (im <= (-0.6d0)) then
        tmp = t_0 * (0.5d0 + (re * (re * (-0.25d0))))
    else if ((im <= 0.66d0) .or. (.not. (im <= 1.65d+100))) then
        tmp = t_1
    else
        tmp = 0.5d0 * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -6.5e+80) {
		tmp = t_1;
	} else if (im <= -0.6) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else if ((im <= 0.66) || !(im <= 1.65e+100)) {
		tmp = t_1;
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -6.5e+80:
		tmp = t_1
	elif im <= -0.6:
		tmp = t_0 * (0.5 + (re * (re * -0.25)))
	elif (im <= 0.66) or not (im <= 1.65e+100):
		tmp = t_1
	else:
		tmp = 0.5 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -6.5e+80)
		tmp = t_1;
	elseif (im <= -0.6)
		tmp = Float64(t_0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	elseif ((im <= 0.66) || !(im <= 1.65e+100))
		tmp = t_1;
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -6.5e+80)
		tmp = t_1;
	elseif (im <= -0.6)
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	elseif ((im <= 0.66) || ~((im <= 1.65e+100)))
		tmp = t_1;
	else
		tmp = 0.5 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.5e+80], t$95$1, If[LessEqual[im, -0.6], N[(t$95$0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 0.66], N[Not[LessEqual[im, 1.65e+100]], $MachinePrecision]], t$95$1, N[(0.5 * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.4999999999999998e80 or -0.599999999999999978 < im < 0.660000000000000031 or 1.6500000000000001e100 < im

   1. Initial program 48.8%

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

      1. neg-sub0 48.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in im around 0 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*l* 98.6%

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

      5. distribute-lft-out-- 98.6%

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

   6. Simplified 98.6%

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

   if -6.4999999999999998e80 < im < -0.599999999999999978

   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 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. unpow2 100.0%

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

      7. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   if 0.660000000000000031 < im < 1.6500000000000001e100

   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 81.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+80}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -0.6:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 0.66 \lor \neg \left(im \leq 1.65 \cdot 10^{+100}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 6: 87.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;im \leq -0.027:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.66:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im)))))
   (if (<= im -0.027)
     t_0
     (if (<= im 0.66)
       (* (cos re) (- im))
       (if (<= im 1.65e+100)
         t_0
         (*
          (- (* (pow im 3.0) -0.16666666666666666) im)
          (+ (* -0.5 (* re re)) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double tmp;
	if (im <= -0.027) {
		tmp = t_0;
	} else if (im <= 0.66) {
		tmp = cos(re) * -im;
	} else if (im <= 1.65e+100) {
		tmp = t_0;
	} else {
		tmp = ((pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	}
	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 = 0.5d0 * (exp(-im) - exp(im))
    if (im <= (-0.027d0)) then
        tmp = t_0
    else if (im <= 0.66d0) then
        tmp = cos(re) * -im
    else if (im <= 1.65d+100) then
        tmp = t_0
    else
        tmp = (((im ** 3.0d0) * (-0.16666666666666666d0)) - im) * (((-0.5d0) * (re * re)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (im <= -0.027) {
		tmp = t_0;
	} else if (im <= 0.66) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.65e+100) {
		tmp = t_0;
	} else {
		tmp = ((Math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	tmp = 0
	if im <= -0.027:
		tmp = t_0
	elif im <= 0.66:
		tmp = math.cos(re) * -im
	elif im <= 1.65e+100:
		tmp = t_0
	else:
		tmp = ((math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (im <= -0.027)
		tmp = t_0;
	elseif (im <= 0.66)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.65e+100)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	tmp = 0.0;
	if (im <= -0.027)
		tmp = t_0;
	elseif (im <= 0.66)
		tmp = cos(re) * -im;
	elseif (im <= 1.65e+100)
		tmp = t_0;
	else
		tmp = (((im ^ 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -0.027], t$95$0, If[LessEqual[im, 0.66], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.65e+100], t$95$0, N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -0.0269999999999999997 or 0.660000000000000031 < im < 1.6500000000000001e100

   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 80.7%

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

   if -0.0269999999999999997 < im < 0.660000000000000031

   1. Initial program 10.4%

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

      1. neg-sub0 10.4%

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

   3. Simplified 10.4%

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

   4. Taylor expanded in im around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. *-commutative 98.2%

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

      3. distribute-lft-neg-in 98.2%

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

   6. Simplified 98.2%

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

   if 1.6500000000000001e100 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.1%

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

      1. mul-1-neg 98.1%

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

      2. unsub-neg 98.1%

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

      3. *-commutative 98.1%

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

      4. associate-*l* 98.1%

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

      5. distribute-lft-out-- 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in re around 0 2.2%

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

      1. *-commutative 2.2%

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

      2. associate-+r- 2.2%

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

      3. associate-*r* 2.2%

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

      4. *-commutative 2.2%

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

      5. *-lft-identity 2.2%

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

      6. distribute-rgt-out 82.6%

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

      7. *-commutative 82.6%

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

      8. unpow2 82.6%

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

   9. Simplified 82.6%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.027:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.66:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \end{array} \]

Alternative 7: 79.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+31}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -7.5e+38)
   (* 0.5 (+ (* im -2.0) (* (pow im 5.0) -0.016666666666666666)))
   (if (<= im 2.45e+31)
     (* (cos re) (- im))
     (*
      (- (* (pow im 3.0) -0.16666666666666666) im)
      (+ (* -0.5 (* re re)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -7.5e+38) {
		tmp = 0.5 * ((im * -2.0) + (pow(im, 5.0) * -0.016666666666666666));
	} else if (im <= 2.45e+31) {
		tmp = cos(re) * -im;
	} else {
		tmp = ((pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	}
	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+38)) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((im ** 5.0d0) * (-0.016666666666666666d0)))
    else if (im <= 2.45d+31) then
        tmp = cos(re) * -im
    else
        tmp = (((im ** 3.0d0) * (-0.16666666666666666d0)) - im) * (((-0.5d0) * (re * re)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -7.5e+38) {
		tmp = 0.5 * ((im * -2.0) + (Math.pow(im, 5.0) * -0.016666666666666666));
	} else if (im <= 2.45e+31) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = ((Math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -7.5e+38:
		tmp = 0.5 * ((im * -2.0) + (math.pow(im, 5.0) * -0.016666666666666666))
	elif im <= 2.45e+31:
		tmp = math.cos(re) * -im
	else:
		tmp = ((math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -7.5e+38)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((im ^ 5.0) * -0.016666666666666666)));
	elseif (im <= 2.45e+31)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -7.5e+38)
		tmp = 0.5 * ((im * -2.0) + ((im ^ 5.0) * -0.016666666666666666));
	elseif (im <= 2.45e+31)
		tmp = cos(re) * -im;
	else
		tmp = (((im ^ 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -7.5e+38], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.45e+31], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.4999999999999999e38

   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 83.3%

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

   5. Taylor expanded in im around 0 78.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)} \]

   6. Taylor expanded in im around inf 78.4%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.016666666666666666 \cdot {im}^{5}}\right) \]

   if -7.4999999999999999e38 < im < 2.44999999999999998e31

   1. Initial program 18.7%

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

      1. neg-sub0 18.7%

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

   3. Simplified 18.7%

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

   4. Taylor expanded in im around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. *-commutative 89.5%

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

      3. distribute-lft-neg-in 89.5%

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

   6. Simplified 89.5%

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

   if 2.44999999999999998e31 < 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 74.1%

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

      1. mul-1-neg 74.1%

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

      2. unsub-neg 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*l* 74.1%

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

      5. distribute-lft-out-- 74.1%

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

   6. Simplified 74.1%

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

   7. Taylor expanded in re around 0 8.8%

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

      1. *-commutative 8.8%

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

      2. associate-+r- 8.8%

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

      3. associate-*r* 8.8%

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

      4. *-commutative 8.8%

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

      5. *-lft-identity 8.8%

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

      6. distribute-rgt-out 68.4%

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

      7. *-commutative 68.4%

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

      8. unpow2 68.4%

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

   9. Simplified 68.4%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+31}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \end{array} \]

Alternative 8: 73.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+31}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+106} \lor \neg \left(im \leq 5.4 \cdot 10^{+188}\right) \land im \leq 3.3 \cdot 10^{+274}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -1.35e+48)
     t_0
     (if (<= im 2.45e+31)
       (* (cos re) (- im))
       (if (or (<= im 1.6e+106) (and (not (<= im 5.4e+188)) (<= im 3.3e+274)))
         (- (* 0.5 (* re (* im re))) im)
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -1.35e+48) {
		tmp = t_0;
	} else if (im <= 2.45e+31) {
		tmp = cos(re) * -im;
	} else if ((im <= 1.6e+106) || (!(im <= 5.4e+188) && (im <= 3.3e+274))) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-1.35d+48)) then
        tmp = t_0
    else if (im <= 2.45d+31) then
        tmp = cos(re) * -im
    else if ((im <= 1.6d+106) .or. (.not. (im <= 5.4d+188)) .and. (im <= 3.3d+274)) then
        tmp = (0.5d0 * (re * (im * re))) - im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -1.35e+48) {
		tmp = t_0;
	} else if (im <= 2.45e+31) {
		tmp = Math.cos(re) * -im;
	} else if ((im <= 1.6e+106) || (!(im <= 5.4e+188) && (im <= 3.3e+274))) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -1.35e+48:
		tmp = t_0
	elif im <= 2.45e+31:
		tmp = math.cos(re) * -im
	elif (im <= 1.6e+106) or (not (im <= 5.4e+188) and (im <= 3.3e+274)):
		tmp = (0.5 * (re * (im * re))) - im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -1.35e+48)
		tmp = t_0;
	elseif (im <= 2.45e+31)
		tmp = Float64(cos(re) * Float64(-im));
	elseif ((im <= 1.6e+106) || (!(im <= 5.4e+188) && (im <= 3.3e+274)))
		tmp = Float64(Float64(0.5 * Float64(re * Float64(im * re))) - im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -1.35e+48)
		tmp = t_0;
	elseif (im <= 2.45e+31)
		tmp = cos(re) * -im;
	elseif ((im <= 1.6e+106) || (~((im <= 5.4e+188)) && (im <= 3.3e+274)))
		tmp = (0.5 * (re * (im * re))) - im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -1.35e+48], t$95$0, If[LessEqual[im, 2.45e+31], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[Or[LessEqual[im, 1.6e+106], And[N[Not[LessEqual[im, 5.4e+188]], $MachinePrecision], LessEqual[im, 3.3e+274]]], N[(N[(0.5 * N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.35000000000000002e48 or 1.5999999999999999e106 < im < 5.4e188 or 3.30000000000000014e274 < 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 92.4%

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

      1. mul-1-neg 92.4%

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

      2. unsub-neg 92.4%

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

      3. *-commutative 92.4%

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

      4. associate-*l* 92.4%

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

      5. distribute-lft-out-- 92.4%

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

   6. Simplified 92.4%

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

   7. Taylor expanded in re around 0 75.1%

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

   if -1.35000000000000002e48 < im < 2.44999999999999998e31

   1. Initial program 18.7%

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

      1. neg-sub0 18.7%

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

   3. Simplified 18.7%

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

   4. Taylor expanded in im around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. *-commutative 89.5%

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

      3. distribute-lft-neg-in 89.5%

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

   6. Simplified 89.5%

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

   if 2.44999999999999998e31 < im < 1.5999999999999999e106 or 5.4e188 < im < 3.30000000000000014e274

   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 5.0%

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

      1. mul-1-neg 5.0%

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

      2. *-commutative 5.0%

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

      3. distribute-lft-neg-in 5.0%

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

   6. Simplified 5.0%

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

   7. Taylor expanded in re around 0 50.4%

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

      1. neg-mul-1 50.4%

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

      2. +-commutative 50.4%

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

      3. unsub-neg 50.4%

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

      4. *-commutative 50.4%

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

      5. associate-*l* 50.4%

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

      6. unpow2 50.4%

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

   9. Simplified 50.4%

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

   10. Taylor expanded in re around 0 50.4%

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

       1. unpow2 50.4%

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

       2. associate-*l* 50.4%

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

       3. *-commutative 50.4%

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

   12. Simplified 50.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+48}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+31}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+106} \lor \neg \left(im \leq 5.4 \cdot 10^{+188}\right) \land im \leq 3.3 \cdot 10^{+274}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 9: 77.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{if}\;im \leq -1 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 29:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+274}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ (* im -2.0) (* (pow im 5.0) -0.016666666666666666)))))
   (if (<= im -1e+39)
     t_0
     (if (<= im 29.0)
       (* (cos re) (- im))
       (if (<= im 5.4e+188)
         t_0
         (if (<= im 3.5e+274)
           (- (* 0.5 (* re (* im re))) im)
           (- (* (pow im 3.0) -0.16666666666666666) im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((im * -2.0) + (pow(im, 5.0) * -0.016666666666666666));
	double tmp;
	if (im <= -1e+39) {
		tmp = t_0;
	} else if (im <= 29.0) {
		tmp = cos(re) * -im;
	} else if (im <= 5.4e+188) {
		tmp = t_0;
	} else if (im <= 3.5e+274) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((im * (-2.0d0)) + ((im ** 5.0d0) * (-0.016666666666666666d0)))
    if (im <= (-1d+39)) then
        tmp = t_0
    else if (im <= 29.0d0) then
        tmp = cos(re) * -im
    else if (im <= 5.4d+188) then
        tmp = t_0
    else if (im <= 3.5d+274) then
        tmp = (0.5d0 * (re * (im * re))) - im
    else
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((im * -2.0) + (Math.pow(im, 5.0) * -0.016666666666666666));
	double tmp;
	if (im <= -1e+39) {
		tmp = t_0;
	} else if (im <= 29.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 5.4e+188) {
		tmp = t_0;
	} else if (im <= 3.5e+274) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((im * -2.0) + (math.pow(im, 5.0) * -0.016666666666666666))
	tmp = 0
	if im <= -1e+39:
		tmp = t_0
	elif im <= 29.0:
		tmp = math.cos(re) * -im
	elif im <= 5.4e+188:
		tmp = t_0
	elif im <= 3.5e+274:
		tmp = (0.5 * (re * (im * re))) - im
	else:
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((im ^ 5.0) * -0.016666666666666666)))
	tmp = 0.0
	if (im <= -1e+39)
		tmp = t_0;
	elseif (im <= 29.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 5.4e+188)
		tmp = t_0;
	elseif (im <= 3.5e+274)
		tmp = Float64(Float64(0.5 * Float64(re * Float64(im * re))) - im);
	else
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * ((im * -2.0) + ((im ^ 5.0) * -0.016666666666666666));
	tmp = 0.0;
	if (im <= -1e+39)
		tmp = t_0;
	elseif (im <= 29.0)
		tmp = cos(re) * -im;
	elseif (im <= 5.4e+188)
		tmp = t_0;
	elseif (im <= 3.5e+274)
		tmp = (0.5 * (re * (im * re))) - im;
	else
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1e+39], t$95$0, If[LessEqual[im, 29.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 5.4e+188], t$95$0, If[LessEqual[im, 3.5e+274], N[(N[(0.5 * N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -9.9999999999999994e38 or 29 < im < 5.4e188

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 78.6%

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

   5. Taylor expanded in im around 0 63.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)} \]

   6. Taylor expanded in im around inf 63.4%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.016666666666666666 \cdot {im}^{5}}\right) \]

   if -9.9999999999999994e38 < im < 29

   1. Initial program 15.1%

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

      1. neg-sub0 15.1%

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

   3. Simplified 15.1%

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

   4. Taylor expanded in im around 0 93.3%

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

      1. mul-1-neg 93.3%

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

      2. *-commutative 93.3%

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

      3. distribute-lft-neg-in 93.3%

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

   6. Simplified 93.3%

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

   if 5.4e188 < im < 3.4999999999999996e274

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 6.3%

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

      1. mul-1-neg 6.3%

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

      2. *-commutative 6.3%

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

      3. distribute-lft-neg-in 6.3%

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

   6. Simplified 6.3%

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

   7. Taylor expanded in re around 0 61.5%

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

      1. neg-mul-1 61.5%

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

      2. +-commutative 61.5%

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

      3. unsub-neg 61.5%

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

      4. *-commutative 61.5%

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

      5. associate-*l* 61.5%

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

      6. unpow2 61.5%

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

   9. Simplified 61.5%

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

   10. Taylor expanded in re around 0 61.5%

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

       1. unpow2 61.5%

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

       2. associate-*l* 61.5%

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

       3. *-commutative 61.5%

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

   12. Simplified 61.5%

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

   if 3.4999999999999996e274 < 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 0 100.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 29:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+188}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+274}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 10: 79.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.8 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+31}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.8e+48)
   (* 0.5 (+ (* im -2.0) (* (pow im 5.0) -0.016666666666666666)))
   (if (<= im 2.6e+31)
     (* (cos re) (- im))
     (* (* (pow im 3.0) -0.16666666666666666) (+ (* -0.5 (* re re)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.8e+48) {
		tmp = 0.5 * ((im * -2.0) + (pow(im, 5.0) * -0.016666666666666666));
	} else if (im <= 2.6e+31) {
		tmp = cos(re) * -im;
	} else {
		tmp = (pow(im, 3.0) * -0.16666666666666666) * ((-0.5 * (re * re)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.8d+48)) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((im ** 5.0d0) * (-0.016666666666666666d0)))
    else if (im <= 2.6d+31) then
        tmp = cos(re) * -im
    else
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) * (((-0.5d0) * (re * re)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.8e+48) {
		tmp = 0.5 * ((im * -2.0) + (Math.pow(im, 5.0) * -0.016666666666666666));
	} else if (im <= 2.6e+31) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) * ((-0.5 * (re * re)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.8e+48:
		tmp = 0.5 * ((im * -2.0) + (math.pow(im, 5.0) * -0.016666666666666666))
	elif im <= 2.6e+31:
		tmp = math.cos(re) * -im
	else:
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) * ((-0.5 * (re * re)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.8e+48)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((im ^ 5.0) * -0.016666666666666666)));
	elseif (im <= 2.6e+31)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.8e+48)
		tmp = 0.5 * ((im * -2.0) + ((im ^ 5.0) * -0.016666666666666666));
	elseif (im <= 2.6e+31)
		tmp = cos(re) * -im;
	else
		tmp = ((im ^ 3.0) * -0.16666666666666666) * ((-0.5 * (re * re)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.8e+48], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+31], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.79999999999999992e48

   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 83.3%

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

   5. Taylor expanded in im around 0 78.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)} \]

   6. Taylor expanded in im around inf 78.4%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.016666666666666666 \cdot {im}^{5}}\right) \]

   if -1.79999999999999992e48 < im < 2.6e31

   1. Initial program 18.7%

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

      1. neg-sub0 18.7%

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

   3. Simplified 18.7%

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

   4. Taylor expanded in im around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. *-commutative 89.5%

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

      3. distribute-lft-neg-in 89.5%

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

   6. Simplified 89.5%

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

   if 2.6e31 < 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 74.1%

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

      1. mul-1-neg 74.1%

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

      2. unsub-neg 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*l* 74.1%

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

      5. distribute-lft-out-- 74.1%

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

   6. Simplified 74.1%

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

   7. Taylor expanded in re around 0 8.8%

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

      1. *-commutative 8.8%

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

      2. associate-+r- 8.8%

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

      3. associate-*r* 8.8%

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

      4. *-commutative 8.8%

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

      5. *-lft-identity 8.8%

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

      6. distribute-rgt-out 68.4%

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

      7. *-commutative 68.4%

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

      8. unpow2 68.4%

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

   9. Simplified 68.4%

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

   10. Taylor expanded in im around inf 68.4%

       \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right)} \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \]
   11. Step-by-step derivation

       1. *-commutative 68.4%

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

   12. Simplified 68.4%

       \[\leadsto \color{blue}{\left({im}^{3} \cdot -0.16666666666666666\right)} \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.8 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+31}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \end{array} \]

Alternative 11: 62.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot \left(\left(im \cdot -2\right) \cdot -0.25\right)\\ t_1 := \frac{im \cdot im - t_0 \cdot t_0}{\left(-im\right) - t_0}\\ \mathbf{if}\;im \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+31}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+135} \lor \neg \left(im \leq 5 \cdot 10^{+188}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) (* (* im -2.0) -0.25)))
        (t_1 (/ (- (* im im) (* t_0 t_0)) (- (- im) t_0))))
   (if (<= im -1.3e+154)
     t_1
     (if (<= im 3.1e+31)
       (* (cos re) (- im))
       (if (or (<= im 6.8e+135) (not (<= im 5e+188)))
         (- (* 0.5 (* re (* im re))) im)
         t_1)))))

C

double code(double re, double im) {
	double t_0 = (re * re) * ((im * -2.0) * -0.25);
	double t_1 = ((im * im) - (t_0 * t_0)) / (-im - t_0);
	double tmp;
	if (im <= -1.3e+154) {
		tmp = t_1;
	} else if (im <= 3.1e+31) {
		tmp = cos(re) * -im;
	} else if ((im <= 6.8e+135) || !(im <= 5e+188)) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (re * re) * ((im * (-2.0d0)) * (-0.25d0))
    t_1 = ((im * im) - (t_0 * t_0)) / (-im - t_0)
    if (im <= (-1.3d+154)) then
        tmp = t_1
    else if (im <= 3.1d+31) then
        tmp = cos(re) * -im
    else if ((im <= 6.8d+135) .or. (.not. (im <= 5d+188))) then
        tmp = (0.5d0 * (re * (im * re))) - im
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) * ((im * -2.0) * -0.25);
	double t_1 = ((im * im) - (t_0 * t_0)) / (-im - t_0);
	double tmp;
	if (im <= -1.3e+154) {
		tmp = t_1;
	} else if (im <= 3.1e+31) {
		tmp = Math.cos(re) * -im;
	} else if ((im <= 6.8e+135) || !(im <= 5e+188)) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) * ((im * -2.0) * -0.25)
	t_1 = ((im * im) - (t_0 * t_0)) / (-im - t_0)
	tmp = 0
	if im <= -1.3e+154:
		tmp = t_1
	elif im <= 3.1e+31:
		tmp = math.cos(re) * -im
	elif (im <= 6.8e+135) or not (im <= 5e+188):
		tmp = (0.5 * (re * (im * re))) - im
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * Float64(Float64(im * -2.0) * -0.25))
	t_1 = Float64(Float64(Float64(im * im) - Float64(t_0 * t_0)) / Float64(Float64(-im) - t_0))
	tmp = 0.0
	if (im <= -1.3e+154)
		tmp = t_1;
	elseif (im <= 3.1e+31)
		tmp = Float64(cos(re) * Float64(-im));
	elseif ((im <= 6.8e+135) || !(im <= 5e+188))
		tmp = Float64(Float64(0.5 * Float64(re * Float64(im * re))) - im);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) * ((im * -2.0) * -0.25);
	t_1 = ((im * im) - (t_0 * t_0)) / (-im - t_0);
	tmp = 0.0;
	if (im <= -1.3e+154)
		tmp = t_1;
	elseif (im <= 3.1e+31)
		tmp = cos(re) * -im;
	elseif ((im <= 6.8e+135) || ~((im <= 5e+188)))
		tmp = (0.5 * (re * (im * re))) - im;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * N[(N[(im * -2.0), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(im * im), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-im) - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.3e+154], t$95$1, If[LessEqual[im, 3.1e+31], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[Or[LessEqual[im, 6.8e+135], N[Not[LessEqual[im, 5e+188]], $MachinePrecision]], N[(N[(0.5 * N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.29999999999999994e154 or 6.80000000000000019e135 < im < 5.0000000000000001e188

   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 74.4%

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

      4. +-commutative 74.4%

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

      5. *-commutative 74.4%

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

      6. unpow2 74.4%

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

      7. associate-*l* 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in im around 0 20.1%

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

      1. distribute-lft-in 20.1%

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

      2. flip-+ 48.8%

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

   9. Applied egg-rr 48.8%

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

   if -1.29999999999999994e154 < im < 3.1000000000000002e31

   1. Initial program 27.5%

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

      1. neg-sub0 27.5%

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

   3. Simplified 27.5%

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

   4. Taylor expanded in im around 0 80.2%

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

      1. mul-1-neg 80.2%

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

      2. *-commutative 80.2%

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

      3. distribute-lft-neg-in 80.2%

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

   6. Simplified 80.2%

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

   if 3.1000000000000002e31 < im < 6.80000000000000019e135 or 5.0000000000000001e188 < 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 5.1%

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

      1. mul-1-neg 5.1%

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

      2. *-commutative 5.1%

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

      3. distribute-lft-neg-in 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in re around 0 43.1%

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

      1. neg-mul-1 43.1%

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

      2. +-commutative 43.1%

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

      3. unsub-neg 43.1%

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

      4. *-commutative 43.1%

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

      5. associate-*l* 43.1%

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

      6. unpow2 43.1%

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

   9. Simplified 43.1%

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

   10. Taylor expanded in re around 0 43.1%

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

       1. unpow2 43.1%

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

       2. associate-*l* 43.1%

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

       3. *-commutative 43.1%

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

   12. Simplified 43.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{im \cdot im - \left(\left(re \cdot re\right) \cdot \left(\left(im \cdot -2\right) \cdot -0.25\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(\left(im \cdot -2\right) \cdot -0.25\right)\right)}{\left(-im\right) - \left(re \cdot re\right) \cdot \left(\left(im \cdot -2\right) \cdot -0.25\right)}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+31}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+135} \lor \neg \left(im \leq 5 \cdot 10^{+188}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot im - \left(\left(re \cdot re\right) \cdot \left(\left(im \cdot -2\right) \cdot -0.25\right)\right) \cdot \left(\left(re \cdot re\right) \cdot \left(\left(im \cdot -2\right) \cdot -0.25\right)\right)}{\left(-im\right) - \left(re \cdot re\right) \cdot \left(\left(im \cdot -2\right) \cdot -0.25\right)}\\ \end{array} \]

Alternative 12: 33.7% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 720000:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+185}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 720000.0)
   (- im)
   (if (<= re 7e+185)
     (* im (* re (* 0.5 re)))
     (* (+ 0.5 (* re (* re -0.25))) -3.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 720000.0) {
		tmp = -im;
	} else if (re <= 7e+185) {
		tmp = im * (re * (0.5 * re));
	} 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 <= 720000.0d0) then
        tmp = -im
    else if (re <= 7d+185) then
        tmp = im * (re * (0.5d0 * re))
    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 <= 720000.0) {
		tmp = -im;
	} else if (re <= 7e+185) {
		tmp = im * (re * (0.5 * re));
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 720000.0:
		tmp = -im
	elif re <= 7e+185:
		tmp = im * (re * (0.5 * re))
	else:
		tmp = (0.5 + (re * (re * -0.25))) * -3.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 720000.0)
		tmp = Float64(-im);
	elseif (re <= 7e+185)
		tmp = Float64(im * Float64(re * Float64(0.5 * re)));
	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 <= 720000.0)
		tmp = -im;
	elseif (re <= 7e+185)
		tmp = im * (re * (0.5 * re));
	else
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 720000.0], (-im), If[LessEqual[re, 7e+185], N[(im * N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 7.2e5

   1. Initial program 54.2%

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

      1. neg-sub0 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in im around 0 52.5%

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

      1. mul-1-neg 52.5%

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

      2. *-commutative 52.5%

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

      3. distribute-lft-neg-in 52.5%

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

   6. Simplified 52.5%

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

   7. Taylor expanded in re around 0 35.1%

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

      1. neg-mul-1 35.1%

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

   9. Simplified 35.1%

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

   if 7.2e5 < re < 7.00000000000000046e185

   1. Initial program 60.3%

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

      1. neg-sub0 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in re around 0 2.5%

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

      1. *-commutative 2.5%

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

      2. associate-*r* 2.5%

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

      3. distribute-rgt-out 35.0%

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

      4. +-commutative 35.0%

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

      5. *-commutative 35.0%

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

      6. unpow2 35.0%

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

      7. associate-*l* 35.0%

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

   6. Simplified 35.0%

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

   7. Taylor expanded in im around 0 28.9%

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

   8. Taylor expanded in re around inf 28.9%

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

      1. associate-*r* 28.9%

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

      2. *-commutative 28.9%

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

      3. unpow2 28.9%

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

      4. associate-*r* 28.9%

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

   10. Simplified 28.9%

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

   if 7.00000000000000046e185 < re

   1. Initial program 58.7%

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

      1. neg-sub0 58.7%

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

   3. Simplified 58.7%

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

   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 27.1%

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

      4. +-commutative 27.1%

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

      5. *-commutative 27.1%

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

      6. unpow2 27.1%

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

      7. associate-*l* 27.1%

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

   6. Simplified 27.1%

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

   8. Applied egg-rr 35.4%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 720000:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+185}:\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \]

Alternative 13: 37.8% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8500000 \lor \neg \left(im \leq 2.8 \cdot 10^{+31}\right):\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -8500000.0) (not (<= im 2.8e+31)))
   (* im (* re (* 0.5 re)))
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -8500000.0) || !(im <= 2.8e+31)) {
		tmp = im * (re * (0.5 * re));
	} else {
		tmp = -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-8500000.0d0)) .or. (.not. (im <= 2.8d+31))) then
        tmp = im * (re * (0.5d0 * re))
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -8500000.0) || !(im <= 2.8e+31)) {
		tmp = im * (re * (0.5 * re));
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -8500000.0) or not (im <= 2.8e+31):
		tmp = im * (re * (0.5 * re))
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -8500000.0) || !(im <= 2.8e+31))
		tmp = Float64(im * Float64(re * Float64(0.5 * re)));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -8500000.0) || ~((im <= 2.8e+31)))
		tmp = im * (re * (0.5 * re));
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -8500000.0], N[Not[LessEqual[im, 2.8e+31]], $MachinePrecision]], N[(im * N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -8.5e6 or 2.80000000000000017e31 < 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 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 77.7%

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

      4. +-commutative 77.7%

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

      5. *-commutative 77.7%

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

      6. unpow2 77.7%

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

      7. associate-*l* 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in im around 0 30.0%

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

   8. Taylor expanded in re around inf 28.0%

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

      1. associate-*r* 28.0%

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

      2. *-commutative 28.0%

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

      3. unpow2 28.0%

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

      4. associate-*r* 28.0%

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

   10. Simplified 28.0%

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

   if -8.5e6 < im < 2.80000000000000017e31

   1. Initial program 15.7%

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

      1. neg-sub0 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in im around 0 92.6%

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

      1. mul-1-neg 92.6%

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

      2. *-commutative 92.6%

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

      3. distribute-lft-neg-in 92.6%

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

   6. Simplified 92.6%

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

   7. Taylor expanded in re around 0 50.1%

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

      1. neg-mul-1 50.1%

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

   9. Simplified 50.1%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8500000 \lor \neg \left(im \leq 2.8 \cdot 10^{+31}\right):\\ \;\;\;\;im \cdot \left(re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 14: 36.8% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7 \cdot 10^{+185}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7e+185)
   (- (* 0.5 (* re (* im re))) im)
   (* (+ 0.5 (* re (* re -0.25))) -3.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7e+185) {
		tmp = (0.5 * (re * (im * re))) - 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 <= 7d+185) then
        tmp = (0.5d0 * (re * (im * re))) - 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 <= 7e+185) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7e+185:
		tmp = (0.5 * (re * (im * re))) - im
	else:
		tmp = (0.5 + (re * (re * -0.25))) * -3.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7e+185)
		tmp = Float64(Float64(0.5 * Float64(re * Float64(im * re))) - 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 <= 7e+185)
		tmp = (0.5 * (re * (im * re))) - im;
	else
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7e+185], N[(N[(0.5 * N[(re * N[(im * re), $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 < 7.00000000000000046e185

   1. Initial program 55.2%

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

      1. neg-sub0 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in im around 0 51.6%

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

      1. mul-1-neg 51.6%

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

      2. *-commutative 51.6%

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

      3. distribute-lft-neg-in 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in re around 0 39.3%

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

      1. neg-mul-1 39.3%

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

      2. +-commutative 39.3%

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

      3. unsub-neg 39.3%

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

      4. *-commutative 39.3%

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

      5. associate-*l* 39.3%

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

      6. unpow2 39.3%

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

   9. Simplified 39.3%

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

   10. Taylor expanded in re around 0 39.3%

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

       1. unpow2 39.3%

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

       2. associate-*l* 39.3%

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

       3. *-commutative 39.3%

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

   12. Simplified 39.3%

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

   if 7.00000000000000046e185 < re

   1. Initial program 58.7%

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

      1. neg-sub0 58.7%

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

   3. Simplified 58.7%

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

   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 27.1%

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

      4. +-commutative 27.1%

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

      5. *-commutative 27.1%

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

      6. unpow2 27.1%

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

      7. associate-*l* 27.1%

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

   6. Simplified 27.1%

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

   8. Applied egg-rr 35.4%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7 \cdot 10^{+185}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \]

Alternative 15: 30.8% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

   1. neg-sub0 55.6%

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

3. Simplified 55.6%

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

4. Taylor expanded in im around 0 51.4%

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

   1. mul-1-neg 51.4%

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

   2. *-commutative 51.4%

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

   3. distribute-lft-neg-in 51.4%

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

6. Simplified 51.4%

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

7. Taylor expanded in re around 0 28.2%

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

   1. neg-mul-1 28.2%

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

9. Simplified 28.2%

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

10. Final simplification 28.2%

    \[\leadsto -im \]

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 2023274 
(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))))