math.sin on complex, imaginary part

Percentage Accurate: 53.9% → 99.8%

Time: 12.8s

Alternatives: 18

Speedup: 2.9×

• 49.0% 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: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -10 \lor \neg \left(t_0 \leq 0.0002\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -10.0) (not (<= t_0 0.0002)))
     (* (* (cos re) 0.5) t_0)
     (*
      (cos re)
      (+
       (+
        (* (pow im 5.0) -0.008333333333333333)
        (* (pow im 7.0) -0.0001984126984126984))
       (- (* (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 <= -10.0) || !(t_0 <= 0.0002)) {
		tmp = (cos(re) * 0.5) * t_0;
	} else {
		tmp = cos(re) * (((pow(im, 5.0) * -0.008333333333333333) + (pow(im, 7.0) * -0.0001984126984126984)) + ((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 <= (-10.0d0)) .or. (.not. (t_0 <= 0.0002d0))) then
        tmp = (cos(re) * 0.5d0) * t_0
    else
        tmp = cos(re) * ((((im ** 5.0d0) * (-0.008333333333333333d0)) + ((im ** 7.0d0) * (-0.0001984126984126984d0))) + (((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 <= -10.0) || !(t_0 <= 0.0002)) {
		tmp = (Math.cos(re) * 0.5) * t_0;
	} else {
		tmp = Math.cos(re) * (((Math.pow(im, 5.0) * -0.008333333333333333) + (Math.pow(im, 7.0) * -0.0001984126984126984)) + ((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 <= -10.0) or not (t_0 <= 0.0002):
		tmp = (math.cos(re) * 0.5) * t_0
	else:
		tmp = math.cos(re) * (((math.pow(im, 5.0) * -0.008333333333333333) + (math.pow(im, 7.0) * -0.0001984126984126984)) + ((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 <= -10.0) || !(t_0 <= 0.0002))
		tmp = Float64(Float64(cos(re) * 0.5) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64((im ^ 5.0) * -0.008333333333333333) + Float64((im ^ 7.0) * -0.0001984126984126984)) + 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 <= -10.0) || ~((t_0 <= 0.0002)))
		tmp = (cos(re) * 0.5) * t_0;
	else
		tmp = cos(re) * ((((im ^ 5.0) * -0.008333333333333333) + ((im ^ 7.0) * -0.0001984126984126984)) + (((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, -10.0], N[Not[LessEqual[t$95$0, 0.0002]], $MachinePrecision]], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $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)) < -10 or 2.0000000000000001e-4 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -10 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.0000000000000001e-4

   1. Initial program 8.7%

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

      1. sub0-neg 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in im around 0 99.8%

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

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l* 99.8%

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

      7. distribute-lft-out 99.8%

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

      8. mul-1-neg 99.8%

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

      9. unsub-neg 99.8%

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

      10. *-commutative 99.8%

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

      11. associate-*l* 99.8%

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

      12. distribute-lft-out-- 99.8%

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

      13. distribute-lft-out 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -10 \lor \neg \left(e^{-im} - e^{im} \leq 0.0002\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.05 \lor \neg \left(t_0 \leq 0.0002\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.05) (not (<= t_0 0.0002)))
     (* (* (cos re) 0.5) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.05) || !(t_0 <= 0.0002)) {
		tmp = (cos(re) * 0.5) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.05d0)) .or. (.not. (t_0 <= 0.0002d0))) then
        tmp = (cos(re) * 0.5d0) * t_0
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.05) || !(t_0 <= 0.0002)) {
		tmp = (Math.cos(re) * 0.5) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.05) or not (t_0 <= 0.0002):
		tmp = (math.cos(re) * 0.5) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.05) || !(t_0 <= 0.0002))
		tmp = Float64(Float64(cos(re) * 0.5) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.05) || ~((t_0 <= 0.0002)))
		tmp = (cos(re) * 0.5) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.05], N[Not[LessEqual[t$95$0, 0.0002]], $MachinePrecision]], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.050000000000000003 or 2.0000000000000001e-4 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -0.050000000000000003 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.0000000000000001e-4

   1. Initial program 8.0%

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

      1. sub0-neg 8.0%

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

   3. Simplified 8.0%

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

   4. Taylor expanded in im around 0 99.8%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.05 \lor \neg \left(e^{-im} - e^{im} \leq 0.0002\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left(\log \left(1 + \mathsf{expm1}\left({im}^{3} \cdot -0.16666666666666666\right)\right) - im\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (cos re)
  (+
   (+
    (* (pow im 5.0) -0.008333333333333333)
    (* (pow im 7.0) -0.0001984126984126984))
   (- (log (+ 1.0 (expm1 (* (pow im 3.0) -0.16666666666666666)))) im))))

C

double code(double re, double im) {
	return cos(re) * (((pow(im, 5.0) * -0.008333333333333333) + (pow(im, 7.0) * -0.0001984126984126984)) + (log((1.0 + expm1((pow(im, 3.0) * -0.16666666666666666)))) - im));
}

Java

public static double code(double re, double im) {
	return Math.cos(re) * (((Math.pow(im, 5.0) * -0.008333333333333333) + (Math.pow(im, 7.0) * -0.0001984126984126984)) + (Math.log((1.0 + Math.expm1((Math.pow(im, 3.0) * -0.16666666666666666)))) - im));
}

Python

def code(re, im):
	return math.cos(re) * (((math.pow(im, 5.0) * -0.008333333333333333) + (math.pow(im, 7.0) * -0.0001984126984126984)) + (math.log((1.0 + math.expm1((math.pow(im, 3.0) * -0.16666666666666666)))) - im))

Julia

function code(re, im)
	return Float64(cos(re) * Float64(Float64(Float64((im ^ 5.0) * -0.008333333333333333) + Float64((im ^ 7.0) * -0.0001984126984126984)) + Float64(log(Float64(1.0 + expm1(Float64((im ^ 3.0) * -0.16666666666666666)))) - im)))
end

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(1.0 + N[(Exp[N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.3%

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

   1. sub0-neg 49.3%

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

3. Simplified 49.3%

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

4. Taylor expanded in im around 0 90.8%

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

   1. associate-+r+ 90.8%

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

   2. +-commutative 90.8%

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

   3. *-commutative 90.8%

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

   4. associate-*l* 90.8%

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

   5. *-commutative 90.8%

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

   6. associate-*l* 90.8%

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

   7. distribute-lft-out 90.8%

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

   8. mul-1-neg 90.8%

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

   9. unsub-neg 90.8%

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

   10. *-commutative 90.8%

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

   11. associate-*l* 90.8%

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

   12. distribute-lft-out-- 90.8%

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

   13. distribute-lft-out 90.8%

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

6. Simplified 90.8%

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

   1. log1p-expm1-u 99.3%

      \[\leadsto \cos re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({im}^{3} \cdot -0.16666666666666666\right)\right)} - im\right)\right) \]

   2. log1p-udef 99.0%

      \[\leadsto \cos re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left(\color{blue}{\log \left(1 + \mathsf{expm1}\left({im}^{3} \cdot -0.16666666666666666\right)\right)} - im\right)\right) \]

8. Applied egg-rr 99.0%

   \[\leadsto \cos re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left(\color{blue}{\log \left(1 + \mathsf{expm1}\left({im}^{3} \cdot -0.16666666666666666\right)\right)} - im\right)\right) \]

9. Final simplification 99.0%

   \[\leadsto \cos re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \left(\log \left(1 + \mathsf{expm1}\left({im}^{3} \cdot -0.16666666666666666\right)\right) - im\right)\right) \]

Alternative 4: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -4.1 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.054:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 0.04:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im)))
        (t_1 (* -0.0001984126984126984 (* (cos re) (pow im 7.0)))))
   (if (<= im -4.1e+46)
     t_1
     (if (<= im -0.054)
       (* t_0 (+ 0.5 (* re (* re -0.25))))
       (if (<= im 0.04)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 1.1e+44) (* 0.5 t_0) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	double tmp;
	if (im <= -4.1e+46) {
		tmp = t_1;
	} else if (im <= -0.054) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else if (im <= 0.04) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = (-0.0001984126984126984d0) * (cos(re) * (im ** 7.0d0))
    if (im <= (-4.1d+46)) then
        tmp = t_1
    else if (im <= (-0.054d0)) then
        tmp = t_0 * (0.5d0 + (re * (re * (-0.25d0))))
    else if (im <= 0.04d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.1d+44) then
        tmp = 0.5d0 * t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -4.1e+46) {
		tmp = t_1;
	} else if (im <= -0.054) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else if (im <= 0.04) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -4.1e+46:
		tmp = t_1
	elif im <= -0.054:
		tmp = t_0 * (0.5 + (re * (re * -0.25)))
	elif im <= 0.04:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		tmp = 0.5 * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -4.1e+46)
		tmp = t_1;
	elseif (im <= -0.054)
		tmp = Float64(t_0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	elseif (im <= 0.04)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		tmp = Float64(0.5 * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = -0.0001984126984126984 * (cos(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -4.1e+46)
		tmp = t_1;
	elseif (im <= -0.054)
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	elseif (im <= 0.04)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.1e+44)
		tmp = 0.5 * t_0;
	else
		tmp = t_1;
	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[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.1e+46], t$95$1, If[LessEqual[im, -0.054], N[(t$95$0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 0.04], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], N[(0.5 * t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.1e46 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. mul-1-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. distribute-lft-out-- 100.0%

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

      13. distribute-lft-out 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -4.1e46 < im < -0.0539999999999999994

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in 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%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 86.7%

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

      5. *-commutative 86.7%

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

      6. unpow2 86.7%

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

      7. associate-*l* 86.7%

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

   6. Simplified 86.7%

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

   if -0.0539999999999999994 < im < 0.0400000000000000008

   1. Initial program 8.7%

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

      1. sub0-neg 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. distribute-lft-out-- 99.6%

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

   6. Simplified 99.6%

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

   if 0.0400000000000000008 < im < 1.09999999999999998e44

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 74.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.1 \cdot 10^{+46}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.054:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 0.04:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 5: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.033:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* -0.0001984126984126984 (* (cos re) (pow im 7.0)))))
   (if (<= im -1.1e+44)
     t_1
     (if (<= im -2e-5)
       t_0
       (if (<= im 0.033) (* (cos re) (- im)) (if (<= im 1.1e+44) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -2e-5) {
		tmp = t_0;
	} else if (im <= 0.033) {
		tmp = cos(re) * -im;
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (-0.0001984126984126984d0) * (cos(re) * (im ** 7.0d0))
    if (im <= (-1.1d+44)) then
        tmp = t_1
    else if (im <= (-2d-5)) then
        tmp = t_0
    else if (im <= 0.033d0) then
        tmp = cos(re) * -im
    else if (im <= 1.1d+44) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -2e-5) {
		tmp = t_0;
	} else if (im <= 0.033) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.1e+44:
		tmp = t_1
	elif im <= -2e-5:
		tmp = t_0
	elif im <= 0.033:
		tmp = math.cos(re) * -im
	elif im <= 1.1e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -2e-5)
		tmp = t_0;
	elseif (im <= 0.033)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = -0.0001984126984126984 * (cos(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -2e-5)
		tmp = t_0;
	elseif (im <= 0.033)
		tmp = cos(re) * -im;
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.1e+44], t$95$1, If[LessEqual[im, -2e-5], t$95$0, If[LessEqual[im, 0.033], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.1e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. mul-1-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. distribute-lft-out-- 100.0%

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

      13. distribute-lft-out 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -1.09999999999999998e44 < im < -2.00000000000000016e-5 or 0.033000000000000002 < im < 1.09999999999999998e44

   1. Initial program 98.6%

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

      1. sub0-neg 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in re around 0 77.2%

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

   if -2.00000000000000016e-5 < im < 0.033000000000000002

   1. Initial program 7.6%

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

      1. sub0-neg 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in im around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. *-commutative 99.2%

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

      3. distribute-lft-neg-in 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.033:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.11:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.053:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* -0.0001984126984126984 (* (cos re) (pow im 7.0)))))
   (if (<= im -1.1e+44)
     t_1
     (if (<= im -0.11)
       t_0
       (if (<= im 0.053)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 2e+44) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -0.11) {
		tmp = t_0;
	} else if (im <= 0.053) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 2e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (-0.0001984126984126984d0) * (cos(re) * (im ** 7.0d0))
    if (im <= (-1.1d+44)) then
        tmp = t_1
    else if (im <= (-0.11d0)) then
        tmp = t_0
    else if (im <= 0.053d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 2d+44) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -0.11) {
		tmp = t_0;
	} else if (im <= 0.053) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 2e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.1e+44:
		tmp = t_1
	elif im <= -0.11:
		tmp = t_0
	elif im <= 0.053:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 2e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -0.11)
		tmp = t_0;
	elseif (im <= 0.053)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 2e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = -0.0001984126984126984 * (cos(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -0.11)
		tmp = t_0;
	elseif (im <= 0.053)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 2e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.1e+44], t$95$1, If[LessEqual[im, -0.11], t$95$0, If[LessEqual[im, 0.053], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.09999999999999998e44 or 2.0000000000000002e44 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. mul-1-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. distribute-lft-out-- 100.0%

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

      13. distribute-lft-out 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -1.09999999999999998e44 < im < -0.110000000000000001 or 0.0529999999999999985 < im < 2.0000000000000002e44

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 77.0%

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

   if -0.110000000000000001 < im < 0.0529999999999999985

   1. Initial program 8.7%

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

      1. sub0-neg 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. distribute-lft-out-- 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.11:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.053:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 7: 92.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if im < -4.20000000000000018 or 4.20000000000000018 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 79.9%

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

      1. associate-+r+ 79.9%

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

      2. +-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-*l* 79.9%

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

      5. *-commutative 79.9%

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

      6. associate-*l* 79.9%

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

      7. distribute-lft-out 79.9%

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

      8. mul-1-neg 79.9%

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

      9. unsub-neg 79.9%

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

      10. *-commutative 79.9%

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

      11. associate-*l* 79.9%

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

      12. distribute-lft-out-- 79.9%

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

      13. distribute-lft-out 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in im around inf 79.9%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 79.9%

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

   9. Simplified 79.9%

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

   if -4.20000000000000018 < im < 4.20000000000000018

   1. Initial program 9.3%

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

      1. sub0-neg 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in im around 0 98.2%

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

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

4. Final simplification 90.1%

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

Alternative 8: 80.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\\ t_1 := 0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{if}\;im \leq -3.8 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3.15 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (- (* (pow im 3.0) -0.16666666666666666) im)
          (+ 1.0 (* -0.5 (* re re)))))
        (t_1 (* 0.5 (+ (* im -2.0) (* (pow im 5.0) -0.016666666666666666)))))
   (if (<= im -3.8e+72)
     t_1
     (if (<= im -3.15e-6)
       t_0
       (if (<= im 1.3e+15)
         (* (cos re) (- im))
         (if (<= im 1.2e+112) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = ((pow(im, 3.0) * -0.16666666666666666) - im) * (1.0 + (-0.5 * (re * re)));
	double t_1 = 0.5 * ((im * -2.0) + (pow(im, 5.0) * -0.016666666666666666));
	double tmp;
	if (im <= -3.8e+72) {
		tmp = t_1;
	} else if (im <= -3.15e-6) {
		tmp = t_0;
	} else if (im <= 1.3e+15) {
		tmp = cos(re) * -im;
	} else if (im <= 1.2e+112) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((im ** 3.0d0) * (-0.16666666666666666d0)) - im) * (1.0d0 + ((-0.5d0) * (re * re)))
    t_1 = 0.5d0 * ((im * (-2.0d0)) + ((im ** 5.0d0) * (-0.016666666666666666d0)))
    if (im <= (-3.8d+72)) then
        tmp = t_1
    else if (im <= (-3.15d-6)) then
        tmp = t_0
    else if (im <= 1.3d+15) then
        tmp = cos(re) * -im
    else if (im <= 1.2d+112) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = ((Math.pow(im, 3.0) * -0.16666666666666666) - im) * (1.0 + (-0.5 * (re * re)));
	double t_1 = 0.5 * ((im * -2.0) + (Math.pow(im, 5.0) * -0.016666666666666666));
	double tmp;
	if (im <= -3.8e+72) {
		tmp = t_1;
	} else if (im <= -3.15e-6) {
		tmp = t_0;
	} else if (im <= 1.3e+15) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.2e+112) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((math.pow(im, 3.0) * -0.16666666666666666) - im) * (1.0 + (-0.5 * (re * re)))
	t_1 = 0.5 * ((im * -2.0) + (math.pow(im, 5.0) * -0.016666666666666666))
	tmp = 0
	if im <= -3.8e+72:
		tmp = t_1
	elif im <= -3.15e-6:
		tmp = t_0
	elif im <= 1.3e+15:
		tmp = math.cos(re) * -im
	elif im <= 1.2e+112:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) * Float64(1.0 + Float64(-0.5 * Float64(re * re))))
	t_1 = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((im ^ 5.0) * -0.016666666666666666)))
	tmp = 0.0
	if (im <= -3.8e+72)
		tmp = t_1;
	elseif (im <= -3.15e-6)
		tmp = t_0;
	elseif (im <= 1.3e+15)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.2e+112)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (((im ^ 3.0) * -0.16666666666666666) - im) * (1.0 + (-0.5 * (re * re)));
	t_1 = 0.5 * ((im * -2.0) + ((im ^ 5.0) * -0.016666666666666666));
	tmp = 0.0;
	if (im <= -3.8e+72)
		tmp = t_1;
	elseif (im <= -3.15e-6)
		tmp = t_0;
	elseif (im <= 1.3e+15)
		tmp = cos(re) * -im;
	elseif (im <= 1.2e+112)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.8e+72], t$95$1, If[LessEqual[im, -3.15e-6], t$95$0, If[LessEqual[im, 1.3e+15], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.2e+112], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.80000000000000006e72 or 1.3e15 < im < 1.2e112

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 95.9%

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

   5. Taylor expanded in im around 0 82.5%

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

   6. Taylor expanded in im around inf 82.5%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.016666666666666666 \cdot {im}^{5}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 82.5%

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

   8. Simplified 82.5%

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

   if -3.80000000000000006e72 < im < -3.14999999999999991e-6 or 1.2e112 < im

   1. Initial program 99.0%

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

      1. sub0-neg 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in im around 0 61.4%

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

      1. mul-1-neg 61.4%

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

      2. unsub-neg 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-*l* 61.4%

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

      5. distribute-lft-out-- 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in re around 0 18.7%

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

      1. *-commutative 18.7%

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

      2. associate-+r- 18.7%

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

      3. associate-*r* 18.7%

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

      4. *-commutative 18.7%

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

      5. *-lft-identity 18.7%

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

      6. distribute-rgt-out 62.2%

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

      7. *-commutative 62.2%

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

      8. unpow2 62.2%

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

   9. Simplified 62.2%

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

   if -3.14999999999999991e-6 < im < 1.3e15

   1. Initial program 11.0%

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

      1. sub0-neg 11.0%

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

   3. Simplified 11.0%

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

   4. Taylor expanded in im around 0 95.5%

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

      1. mul-1-neg 95.5%

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

      2. *-commutative 95.5%

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

      3. distribute-lft-neg-in 95.5%

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

   6. Simplified 95.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{elif}\;im \leq -3.15 \cdot 10^{-6}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 9: 78.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ t_1 := 0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{if}\;im \leq -3.6 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.12 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.9 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+196}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (* im (* re re))) im))
        (t_1 (* 0.5 (+ (* im -2.0) (* (pow im 5.0) -0.016666666666666666)))))
   (if (<= im -3.6e+72)
     t_1
     (if (<= im -1.12e+15)
       t_0
       (if (<= im 1.5e+15)
         (* (cos re) (- im))
         (if (<= im 5.9e+175)
           t_1
           (if (<= im 1.75e+196)
             t_0
             (- (* (pow im 3.0) -0.16666666666666666) im))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * (im * (re * re))) - im;
	double t_1 = 0.5 * ((im * -2.0) + (pow(im, 5.0) * -0.016666666666666666));
	double tmp;
	if (im <= -3.6e+72) {
		tmp = t_1;
	} else if (im <= -1.12e+15) {
		tmp = t_0;
	} else if (im <= 1.5e+15) {
		tmp = cos(re) * -im;
	} else if (im <= 5.9e+175) {
		tmp = t_1;
	} else if (im <= 1.75e+196) {
		tmp = t_0;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (0.5d0 * (im * (re * re))) - im
    t_1 = 0.5d0 * ((im * (-2.0d0)) + ((im ** 5.0d0) * (-0.016666666666666666d0)))
    if (im <= (-3.6d+72)) then
        tmp = t_1
    else if (im <= (-1.12d+15)) then
        tmp = t_0
    else if (im <= 1.5d+15) then
        tmp = cos(re) * -im
    else if (im <= 5.9d+175) then
        tmp = t_1
    else if (im <= 1.75d+196) then
        tmp = t_0
    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 * (re * re))) - im;
	double t_1 = 0.5 * ((im * -2.0) + (Math.pow(im, 5.0) * -0.016666666666666666));
	double tmp;
	if (im <= -3.6e+72) {
		tmp = t_1;
	} else if (im <= -1.12e+15) {
		tmp = t_0;
	} else if (im <= 1.5e+15) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 5.9e+175) {
		tmp = t_1;
	} else if (im <= 1.75e+196) {
		tmp = t_0;
	} else {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * (im * (re * re))) - im
	t_1 = 0.5 * ((im * -2.0) + (math.pow(im, 5.0) * -0.016666666666666666))
	tmp = 0
	if im <= -3.6e+72:
		tmp = t_1
	elif im <= -1.12e+15:
		tmp = t_0
	elif im <= 1.5e+15:
		tmp = math.cos(re) * -im
	elif im <= 5.9e+175:
		tmp = t_1
	elif im <= 1.75e+196:
		tmp = t_0
	else:
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * Float64(im * Float64(re * re))) - im)
	t_1 = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((im ^ 5.0) * -0.016666666666666666)))
	tmp = 0.0
	if (im <= -3.6e+72)
		tmp = t_1;
	elseif (im <= -1.12e+15)
		tmp = t_0;
	elseif (im <= 1.5e+15)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 5.9e+175)
		tmp = t_1;
	elseif (im <= 1.75e+196)
		tmp = t_0;
	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 * (re * re))) - im;
	t_1 = 0.5 * ((im * -2.0) + ((im ^ 5.0) * -0.016666666666666666));
	tmp = 0.0;
	if (im <= -3.6e+72)
		tmp = t_1;
	elseif (im <= -1.12e+15)
		tmp = t_0;
	elseif (im <= 1.5e+15)
		tmp = cos(re) * -im;
	elseif (im <= 5.9e+175)
		tmp = t_1;
	elseif (im <= 1.75e+196)
		tmp = t_0;
	else
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.6e+72], t$95$1, If[LessEqual[im, -1.12e+15], t$95$0, If[LessEqual[im, 1.5e+15], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 5.9e+175], t$95$1, If[LessEqual[im, 1.75e+196], t$95$0, N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.60000000000000035e72 or 1.5e15 < im < 5.9000000000000003e175

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 89.1%

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

   5. Taylor expanded in im around 0 78.8%

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

   6. Taylor expanded in im around inf 78.8%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.016666666666666666 \cdot {im}^{5}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 78.8%

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

   8. Simplified 78.8%

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

   if -3.60000000000000035e72 < im < -1.12e15 or 5.9000000000000003e175 < im < 1.7499999999999999e196

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.8%

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

      1. mul-1-neg 3.8%

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

      2. *-commutative 3.8%

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

      3. distribute-lft-neg-in 3.8%

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

   6. Simplified 3.8%

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

   7. Taylor expanded in re around 0 44.3%

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

      1. neg-mul-1 44.3%

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

      2. +-commutative 44.3%

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

      3. unsub-neg 44.3%

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

      4. *-commutative 44.3%

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

      5. unpow2 44.3%

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

   9. Simplified 44.3%

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

   if -1.12e15 < im < 1.5e15

   1. Initial program 14.7%

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

      1. sub0-neg 14.7%

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

   3. Simplified 14.7%

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

   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%

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

   if 1.7499999999999999e196 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 78.6%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.6 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{elif}\;im \leq -1.12 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.9 \cdot 10^{+175}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+196}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 10: 74.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_1 := 0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{if}\;im \leq -9.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8.6 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 0.058:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.9 \cdot 10^{+175} \lor \neg \left(im \leq 1.75 \cdot 10^{+196}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_1 (- (* 0.5 (* im (* re re))) im)))
   (if (<= im -9.2e+77)
     t_0
     (if (<= im -8.6e+16)
       t_1
       (if (<= im 0.058)
         (* (cos re) (- im))
         (if (or (<= im 5.9e+175) (not (<= im 1.75e+196))) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = (0.5 * (im * (re * re))) - im;
	double tmp;
	if (im <= -9.2e+77) {
		tmp = t_0;
	} else if (im <= -8.6e+16) {
		tmp = t_1;
	} else if (im <= 0.058) {
		tmp = cos(re) * -im;
	} else if ((im <= 5.9e+175) || !(im <= 1.75e+196)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    t_1 = (0.5d0 * (im * (re * re))) - im
    if (im <= (-9.2d+77)) then
        tmp = t_0
    else if (im <= (-8.6d+16)) then
        tmp = t_1
    else if (im <= 0.058d0) then
        tmp = cos(re) * -im
    else if ((im <= 5.9d+175) .or. (.not. (im <= 1.75d+196))) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = (0.5 * (im * (re * re))) - im;
	double tmp;
	if (im <= -9.2e+77) {
		tmp = t_0;
	} else if (im <= -8.6e+16) {
		tmp = t_1;
	} else if (im <= 0.058) {
		tmp = Math.cos(re) * -im;
	} else if ((im <= 5.9e+175) || !(im <= 1.75e+196)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	t_1 = (0.5 * (im * (re * re))) - im
	tmp = 0
	if im <= -9.2e+77:
		tmp = t_0
	elif im <= -8.6e+16:
		tmp = t_1
	elif im <= 0.058:
		tmp = math.cos(re) * -im
	elif (im <= 5.9e+175) or not (im <= 1.75e+196):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_1 = Float64(Float64(0.5 * Float64(im * Float64(re * re))) - im)
	tmp = 0.0
	if (im <= -9.2e+77)
		tmp = t_0;
	elseif (im <= -8.6e+16)
		tmp = t_1;
	elseif (im <= 0.058)
		tmp = Float64(cos(re) * Float64(-im));
	elseif ((im <= 5.9e+175) || !(im <= 1.75e+196))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	t_1 = (0.5 * (im * (re * re))) - im;
	tmp = 0.0;
	if (im <= -9.2e+77)
		tmp = t_0;
	elseif (im <= -8.6e+16)
		tmp = t_1;
	elseif (im <= 0.058)
		tmp = cos(re) * -im;
	elseif ((im <= 5.9e+175) || ~((im <= 1.75e+196)))
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -9.2e+77], t$95$0, If[LessEqual[im, -8.6e+16], t$95$1, If[LessEqual[im, 0.058], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[Or[LessEqual[im, 5.9e+175], N[Not[LessEqual[im, 1.75e+196]], $MachinePrecision]], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.19999999999999979e77 or 0.0580000000000000029 < im < 5.9000000000000003e175 or 1.7499999999999999e196 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 76.6%

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

      1. mul-1-neg 76.6%

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

      2. unsub-neg 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-*l* 76.6%

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

      5. distribute-lft-out-- 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in re around 0 64.7%

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

   if -9.19999999999999979e77 < im < -8.6e16 or 5.9000000000000003e175 < im < 1.7499999999999999e196

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.8%

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

      1. mul-1-neg 3.8%

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

      2. *-commutative 3.8%

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

      3. distribute-lft-neg-in 3.8%

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

   6. Simplified 3.8%

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

   7. Taylor expanded in re around 0 45.9%

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

      1. neg-mul-1 45.9%

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

      2. +-commutative 45.9%

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

      3. unsub-neg 45.9%

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

      4. *-commutative 45.9%

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

      5. unpow2 45.9%

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

   9. Simplified 45.9%

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

   if -8.6e16 < im < 0.0580000000000000029

   1. Initial program 11.8%

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

      1. sub0-neg 11.8%

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

   3. Simplified 11.8%

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

   4. Taylor expanded in im around 0 95.5%

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

      1. mul-1-neg 95.5%

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

      2. *-commutative 95.5%

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

      3. distribute-lft-neg-in 95.5%

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

   6. Simplified 95.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.2 \cdot 10^{+77}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -8.6 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 0.058:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.9 \cdot 10^{+175} \lor \neg \left(im \leq 1.75 \cdot 10^{+196}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \]

Alternative 11: 60.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot 0.75\\ t_1 := 0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{if}\;im \leq -2.25 \cdot 10^{+123}:\\ \;\;\;\;im \cdot \left({re}^{4} \cdot -0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -8500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1700000000:\\ \;\;\;\;\frac{2.25 - t_0 \cdot t_0}{-1.5 - t_0}\\ \mathbf{elif}\;im \leq 250:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) 0.75)) (t_1 (- (* 0.5 (* im (* re re))) im)))
   (if (<= im -2.25e+123)
     (* im (* (pow re 4.0) -0.041666666666666664))
     (if (<= im -8500000000.0)
       t_1
       (if (<= im -1700000000.0)
         (/ (- 2.25 (* t_0 t_0)) (- -1.5 t_0))
         (if (<= im 250.0) (* (cos re) (- im)) t_1))))))

C

double code(double re, double im) {
	double t_0 = (re * re) * 0.75;
	double t_1 = (0.5 * (im * (re * re))) - im;
	double tmp;
	if (im <= -2.25e+123) {
		tmp = im * (pow(re, 4.0) * -0.041666666666666664);
	} else if (im <= -8500000000.0) {
		tmp = t_1;
	} else if (im <= -1700000000.0) {
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	} else if (im <= 250.0) {
		tmp = cos(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) * 0.75d0
    t_1 = (0.5d0 * (im * (re * re))) - im
    if (im <= (-2.25d+123)) then
        tmp = im * ((re ** 4.0d0) * (-0.041666666666666664d0))
    else if (im <= (-8500000000.0d0)) then
        tmp = t_1
    else if (im <= (-1700000000.0d0)) then
        tmp = (2.25d0 - (t_0 * t_0)) / ((-1.5d0) - t_0)
    else if (im <= 250.0d0) then
        tmp = cos(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) * 0.75;
	double t_1 = (0.5 * (im * (re * re))) - im;
	double tmp;
	if (im <= -2.25e+123) {
		tmp = im * (Math.pow(re, 4.0) * -0.041666666666666664);
	} else if (im <= -8500000000.0) {
		tmp = t_1;
	} else if (im <= -1700000000.0) {
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	} else if (im <= 250.0) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) * 0.75
	t_1 = (0.5 * (im * (re * re))) - im
	tmp = 0
	if im <= -2.25e+123:
		tmp = im * (math.pow(re, 4.0) * -0.041666666666666664)
	elif im <= -8500000000.0:
		tmp = t_1
	elif im <= -1700000000.0:
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0)
	elif im <= 250.0:
		tmp = math.cos(re) * -im
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * 0.75)
	t_1 = Float64(Float64(0.5 * Float64(im * Float64(re * re))) - im)
	tmp = 0.0
	if (im <= -2.25e+123)
		tmp = Float64(im * Float64((re ^ 4.0) * -0.041666666666666664));
	elseif (im <= -8500000000.0)
		tmp = t_1;
	elseif (im <= -1700000000.0)
		tmp = Float64(Float64(2.25 - Float64(t_0 * t_0)) / Float64(-1.5 - t_0));
	elseif (im <= 250.0)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) * 0.75;
	t_1 = (0.5 * (im * (re * re))) - im;
	tmp = 0.0;
	if (im <= -2.25e+123)
		tmp = im * ((re ^ 4.0) * -0.041666666666666664);
	elseif (im <= -8500000000.0)
		tmp = t_1;
	elseif (im <= -1700000000.0)
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	elseif (im <= 250.0)
		tmp = cos(re) * -im;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -2.25e+123], N[(im * N[(N[Power[re, 4.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -8500000000.0], t$95$1, If[LessEqual[im, -1700000000.0], N[(N[(2.25 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.5 - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 250.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.24999999999999991e123

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 6.1%

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

      1. mul-1-neg 6.1%

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

      2. *-commutative 6.1%

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

      3. distribute-lft-neg-in 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in re around 0 6.4%

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

      1. neg-mul-1 6.4%

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

      2. associate-+r+ 6.4%

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

      3. +-commutative 6.4%

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

      4. associate-*r* 6.4%

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

      5. *-commutative 6.4%

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

      6. associate-*r* 6.4%

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

      7. neg-mul-1 6.4%

         \[\leadsto im \cdot \left(0.5 \cdot {re}^{2}\right) + \left(\left(-0.041666666666666664 \cdot {re}^{4}\right) \cdot im + \color{blue}{-1 \cdot im}\right) \]

      8. distribute-rgt-out 6.4%

         \[\leadsto im \cdot \left(0.5 \cdot {re}^{2}\right) + \color{blue}{im \cdot \left(-0.041666666666666664 \cdot {re}^{4} + -1\right)} \]

      9. distribute-lft-out 29.8%

         \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot {re}^{2} + \left(-0.041666666666666664 \cdot {re}^{4} + -1\right)\right)} \]

      10. *-commutative 29.8%

          \[\leadsto im \cdot \left(\color{blue}{{re}^{2} \cdot 0.5} + \left(-0.041666666666666664 \cdot {re}^{4} + -1\right)\right) \]

      11. unpow2 29.8%

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot 0.5 + \left(-0.041666666666666664 \cdot {re}^{4} + -1\right)\right) \]

      12. *-commutative 29.8%

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot 0.5 + \left(\color{blue}{{re}^{4} \cdot -0.041666666666666664} + -1\right)\right) \]

   9. Simplified 29.8%

      \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5 + \left({re}^{4} \cdot -0.041666666666666664 + -1\right)\right)} \]

   10. Taylor expanded in re around inf 41.1%

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

       1. *-commutative 41.1%

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

   12. Simplified 41.1%

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

   if -2.24999999999999991e123 < im < -8.5e9 or 250 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.5%

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

      1. mul-1-neg 4.5%

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

      2. *-commutative 4.5%

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

      3. distribute-lft-neg-in 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in re around 0 25.0%

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

      1. neg-mul-1 25.0%

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

      2. +-commutative 25.0%

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

      3. unsub-neg 25.0%

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

      4. *-commutative 25.0%

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

      5. unpow2 25.0%

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

   9. Simplified 25.0%

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

   if -8.5e9 < im < -1.7e9

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in 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%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. associate-*l* 0.0%

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

   6. Simplified 0.0%

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

   8. Applied egg-rr 6.9%

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

      1. distribute-lft-in 6.9%

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

      2. flip-+ 100.0%

         \[\leadsto \color{blue}{\frac{\left(-3 \cdot 0.5\right) \cdot \left(-3 \cdot 0.5\right) - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)}} \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{-1.5} \cdot \left(-3 \cdot 0.5\right) - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{-1.5} - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{2.25} - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      6. *-commutative 100.0%

         \[\leadsto \frac{2.25 - \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)} \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      7. associate-*r* 100.0%

         \[\leadsto \frac{2.25 - \left(\color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      8. associate-*l* 100.0%

         \[\leadsto \frac{2.25 - \color{blue}{\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)\right)} \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \color{blue}{0.75}\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      10. *-commutative 100.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      11. associate-*r* 100.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      12. associate-*l* 100.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)\right)}}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{0.75}\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{\color{blue}{-1.5} - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      15. *-commutative 100.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3}} \]

      16. associate-*r* 100.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3} \]

      17. associate-*l* 100.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)}} \]

      18. metadata-eval 100.0%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot \color{blue}{0.75}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}} \]

   if -1.7e9 < im < 250

   1. Initial program 9.9%

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

      1. sub0-neg 9.9%

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

   3. Simplified 9.9%

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

   4. Taylor expanded in im around 0 97.6%

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

      1. mul-1-neg 97.6%

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

      2. *-commutative 97.6%

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

      3. distribute-lft-neg-in 97.6%

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

   6. Simplified 97.6%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.25 \cdot 10^{+123}:\\ \;\;\;\;im \cdot \left({re}^{4} \cdot -0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -8500000000:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq -1700000000:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}\\ \mathbf{elif}\;im \leq 250:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \]

Alternative 12: 60.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.36 \cdot 10^{+17} \lor \neg \left(im \leq 250\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.36e+17) (not (<= im 250.0)))
   (- (* 0.5 (* im (* re re))) im)
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.36e+17) || !(im <= 250.0)) {
		tmp = (0.5 * (im * (re * re))) - im;
	} else {
		tmp = cos(re) * -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.36d+17)) .or. (.not. (im <= 250.0d0))) then
        tmp = (0.5d0 * (im * (re * re))) - im
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.36e+17) || !(im <= 250.0)) {
		tmp = (0.5 * (im * (re * re))) - im;
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.36e+17) or not (im <= 250.0):
		tmp = (0.5 * (im * (re * re))) - im
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.36e+17) || !(im <= 250.0))
		tmp = Float64(Float64(0.5 * Float64(im * Float64(re * re))) - im);
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.36e+17) || ~((im <= 250.0)))
		tmp = (0.5 * (im * (re * re))) - im;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.36e+17], N[Not[LessEqual[im, 250.0]], $MachinePrecision]], N[(N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.36e17 or 250 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 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 21.3%

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

      1. neg-mul-1 21.3%

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

      2. +-commutative 21.3%

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

      3. unsub-neg 21.3%

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

      4. *-commutative 21.3%

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

      5. unpow2 21.3%

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

   9. Simplified 21.3%

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

   if -1.36e17 < im < 250

   1. Initial program 12.4%

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

      1. sub0-neg 12.4%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in im around 0 95.0%

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

      1. mul-1-neg 95.0%

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

      2. *-commutative 95.0%

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

      3. distribute-lft-neg-in 95.0%

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

   6. Simplified 95.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.36 \cdot 10^{+17} \lor \neg \left(im \leq 250\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 13: 33.0% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot 0.75\\ \mathbf{if}\;re \leq 9.6 \cdot 10^{+76}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{2.25 - t_0 \cdot t_0}{-1.5 - t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) 0.75)))
   (if (<= re 9.6e+76)
     (- im)
     (if (<= re 8.5e+153)
       (/ (- 2.25 (* t_0 t_0)) (- -1.5 t_0))
       (* (+ 0.5 (* re (* re -0.25))) 27.0)))))

C

double code(double re, double im) {
	double t_0 = (re * re) * 0.75;
	double tmp;
	if (re <= 9.6e+76) {
		tmp = -im;
	} else if (re <= 8.5e+153) {
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * re) * 0.75d0
    if (re <= 9.6d+76) then
        tmp = -im
    else if (re <= 8.5d+153) then
        tmp = (2.25d0 - (t_0 * t_0)) / ((-1.5d0) - t_0)
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) * 0.75;
	double tmp;
	if (re <= 9.6e+76) {
		tmp = -im;
	} else if (re <= 8.5e+153) {
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) * 0.75
	tmp = 0
	if re <= 9.6e+76:
		tmp = -im
	elif re <= 8.5e+153:
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0)
	else:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * 0.75)
	tmp = 0.0
	if (re <= 9.6e+76)
		tmp = Float64(-im);
	elseif (re <= 8.5e+153)
		tmp = Float64(Float64(2.25 - Float64(t_0 * t_0)) / Float64(-1.5 - t_0));
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) * 0.75;
	tmp = 0.0;
	if (re <= 9.6e+76)
		tmp = -im;
	elseif (re <= 8.5e+153)
		tmp = (2.25 - (t_0 * t_0)) / (-1.5 - t_0);
	else
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]}, If[LessEqual[re, 9.6e+76], (-im), If[LessEqual[re, 8.5e+153], N[(N[(2.25 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.5 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 9.5999999999999999e76

   1. Initial program 49.5%

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

      1. sub0-neg 49.5%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in im around 0 56.9%

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

      1. mul-1-neg 56.9%

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

      2. *-commutative 56.9%

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

      3. distribute-lft-neg-in 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in re around 0 42.6%

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

      1. neg-mul-1 42.6%

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

   9. Simplified 42.6%

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

   if 9.5999999999999999e76 < re < 8.49999999999999935e153

   1. Initial program 63.4%

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

      1. sub0-neg 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in re around 0 2.2%

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

      1. +-commutative 2.2%

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

      2. *-commutative 2.2%

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

      3. associate-*r* 2.2%

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

      4. distribute-rgt-out 32.9%

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

      5. *-commutative 32.9%

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

      6. unpow2 32.9%

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

      7. associate-*l* 32.9%

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

   6. Simplified 32.9%

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

   8. Applied egg-rr 2.5%

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

      1. distribute-lft-in 2.5%

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

      2. flip-+ 31.4%

         \[\leadsto \color{blue}{\frac{\left(-3 \cdot 0.5\right) \cdot \left(-3 \cdot 0.5\right) - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)}} \]

      3. metadata-eval 31.4%

         \[\leadsto \frac{\color{blue}{-1.5} \cdot \left(-3 \cdot 0.5\right) - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      4. metadata-eval 31.4%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{-1.5} - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      5. metadata-eval 31.4%

         \[\leadsto \frac{\color{blue}{2.25} - \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      6. *-commutative 31.4%

         \[\leadsto \frac{2.25 - \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)} \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      7. associate-*r* 31.4%

         \[\leadsto \frac{2.25 - \left(\color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      8. associate-*l* 31.4%

         \[\leadsto \frac{2.25 - \color{blue}{\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)\right)} \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      9. metadata-eval 31.4%

         \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot \color{blue}{0.75}\right) \cdot \left(-3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      10. *-commutative 31.4%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \color{blue}{\left(\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\right)}}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      11. associate-*r* 31.4%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      12. associate-*l* 31.4%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)\right)}}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      13. metadata-eval 31.4%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{0.75}\right)}{-3 \cdot 0.5 - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      14. metadata-eval 31.4%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{\color{blue}{-1.5} - -3 \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)} \]

      15. *-commutative 31.4%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(re \cdot \left(re \cdot -0.25\right)\right) \cdot -3}} \]

      16. associate-*r* 31.4%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)} \cdot -3} \]

      17. associate-*l* 31.4%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot -3\right)}} \]

      18. metadata-eval 31.4%

          \[\leadsto \frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot \color{blue}{0.75}} \]

   10. Applied egg-rr 31.4%

       \[\leadsto \color{blue}{\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}} \]

   if 8.49999999999999935e153 < re

   1. Initial program 42.6%

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

      1. sub0-neg 42.6%

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

   3. Simplified 42.6%

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

   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%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 19.4%

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

      5. *-commutative 19.4%

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

      6. unpow2 19.4%

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

      7. associate-*l* 19.4%

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

   6. Simplified 19.4%

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

   8. Applied egg-rr 23.7%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9.6 \cdot 10^{+76}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{2.25 - \left(\left(re \cdot re\right) \cdot 0.75\right) \cdot \left(\left(re \cdot re\right) \cdot 0.75\right)}{-1.5 - \left(re \cdot re\right) \cdot 0.75}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \end{array} \]

Alternative 14: 31.9% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.85 \cdot 10^{+158}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.85e+158) (- im) (* (+ 0.5 (* re (* re -0.25))) 27.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.85e+158) {
		tmp = -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.85d+158) then
        tmp = -im
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.85e+158) {
		tmp = -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.85e+158:
		tmp = -im
	else:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.85e+158)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.85e+158)
		tmp = -im;
	else
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.85e+158], (-im), N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.8500000000000001e158

   1. Initial program 50.1%

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

      1. sub0-neg 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in im around 0 56.4%

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

      1. mul-1-neg 56.4%

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

      2. *-commutative 56.4%

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

      3. distribute-lft-neg-in 56.4%

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

   6. Simplified 56.4%

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

   7. Taylor expanded in re around 0 40.3%

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

      1. neg-mul-1 40.3%

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

   9. Simplified 40.3%

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

   if 2.8500000000000001e158 < re

   1. Initial program 43.8%

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

      1. sub0-neg 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in re around 0 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%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 20.0%

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

      5. *-commutative 20.0%

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

      6. unpow2 20.0%

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

      7. associate-*l* 20.0%

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

   6. Simplified 20.0%

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

   8. Applied egg-rr 24.4%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.85 \cdot 10^{+158}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \end{array} \]

Alternative 15: 35.7% accurate, 34.3× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (- (* 0.5 (* im (* re re))) im))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return (0.5 * (im * (re * re))) - im;
}

Python

def code(re, im):
	return (0.5 * (im * (re * re))) - im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.3%

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

   1. sub0-neg 49.3%

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

3. Simplified 49.3%

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

4. Taylor expanded in im around 0 57.0%

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

   1. mul-1-neg 57.0%

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

   2. *-commutative 57.0%

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

   3. distribute-lft-neg-in 57.0%

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

6. Simplified 57.0%

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

7. Taylor expanded in re around 0 41.6%

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

   1. neg-mul-1 41.6%

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

   2. +-commutative 41.6%

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

   3. unsub-neg 41.6%

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

   4. *-commutative 41.6%

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

   5. unpow2 41.6%

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

9. Simplified 41.6%

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

10. Final simplification 41.6%

    \[\leadsto 0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im \]

Alternative 16: 31.0% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.75 \cdot 10^{+190}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.75e+190) (- im) (* re (* re 0.75))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.75e+190) {
		tmp = -im;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.75d+190) then
        tmp = -im
    else
        tmp = re * (re * 0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.75e+190) {
		tmp = -im;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.75e+190:
		tmp = -im
	else:
		tmp = re * (re * 0.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.75e+190)
		tmp = Float64(-im);
	else
		tmp = Float64(re * Float64(re * 0.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.75e+190)
		tmp = -im;
	else
		tmp = re * (re * 0.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.75e+190], (-im), N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.7499999999999999e190

   1. Initial program 49.7%

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

      1. sub0-neg 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in im around 0 56.7%

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

      1. mul-1-neg 56.7%

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

      2. *-commutative 56.7%

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

      3. distribute-lft-neg-in 56.7%

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

   6. Simplified 56.7%

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

   7. Taylor expanded in re around 0 40.0%

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

      1. neg-mul-1 40.0%

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

   9. Simplified 40.0%

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

   if 1.7499999999999999e190 < re

   1. Initial program 46.7%

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

      1. sub0-neg 46.7%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 21.5%

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

      5. *-commutative 21.5%

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

      6. unpow2 21.5%

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

      7. associate-*l* 21.5%

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

   6. Simplified 21.5%

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

   8. Applied egg-rr 18.8%

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

   9. Taylor expanded in re around inf 18.8%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 18.8%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 18.8%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

       3. associate-*l* 18.8%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   11. Simplified 18.8%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.75 \cdot 10^{+190}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \]

Alternative 17: 29.4% 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 49.3%

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

   1. sub0-neg 49.3%

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

3. Simplified 49.3%

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

4. Taylor expanded in im around 0 57.0%

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

   1. mul-1-neg 57.0%

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

   2. *-commutative 57.0%

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

   3. distribute-lft-neg-in 57.0%

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

6. Simplified 57.0%

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

7. Taylor expanded in re around 0 36.7%

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

   1. neg-mul-1 36.7%

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

9. Simplified 36.7%

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

10. Final simplification 36.7%

    \[\leadsto -im \]

Alternative 18: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.5)

C

double code(double re, double im) {
	return -1.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.5d0
end function

Java

public static double code(double re, double im) {
	return -1.5;
}

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

function tmp = code(re, im)
	tmp = -1.5;
end

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 49.3%

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

   1. sub0-neg 49.3%

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

3. Simplified 49.3%

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

4. Taylor expanded in re around 0 4.1%

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

   1. +-commutative 4.1%

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

   2. *-commutative 4.1%

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

   3. associate-*r* 4.1%

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

   4. distribute-rgt-out 37.3%

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

   5. *-commutative 37.3%

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

   6. unpow2 37.3%

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

   7. associate-*l* 37.3%

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

6. Simplified 37.3%

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

8. Applied egg-rr 6.0%

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

9. Taylor expanded in re around 0 2.8%

   \[\leadsto \color{blue}{-1.5} \]

10. Final simplification 2.8%

    \[\leadsto -1.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023199 
(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))))