math.sin on complex, imaginary part

Percentage Accurate: 54.9% → 99.8%

Time: 14.2s

Alternatives: 14

Speedup: 2.9×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -400 \lor \neg \left(t_0 \leq 0.2\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -400.0) (not (<= t_0 0.2)))
     (* (* (cos re) 0.5) t_0)
     (*
      (cos re)
      (+
       (+
        (* (pow im 7.0) -0.0001984126984126984)
        (* (pow im 5.0) -0.008333333333333333))
       (- (* (pow im 3.0) -0.16666666666666666) im))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -400.0) || !(t_0 <= 0.2)) {
		tmp = (cos(re) * 0.5) * t_0;
	} else {
		tmp = cos(re) * (((pow(im, 7.0) * -0.0001984126984126984) + (pow(im, 5.0) * -0.008333333333333333)) + ((pow(im, 3.0) * -0.16666666666666666) - im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-400.0d0)) .or. (.not. (t_0 <= 0.2d0))) then
        tmp = (cos(re) * 0.5d0) * t_0
    else
        tmp = cos(re) * ((((im ** 7.0d0) * (-0.0001984126984126984d0)) + ((im ** 5.0d0) * (-0.008333333333333333d0))) + (((im ** 3.0d0) * (-0.16666666666666666d0)) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -400.0) || !(t_0 <= 0.2)) {
		tmp = (Math.cos(re) * 0.5) * t_0;
	} else {
		tmp = Math.cos(re) * (((Math.pow(im, 7.0) * -0.0001984126984126984) + (Math.pow(im, 5.0) * -0.008333333333333333)) + ((Math.pow(im, 3.0) * -0.16666666666666666) - im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -400.0) or not (t_0 <= 0.2):
		tmp = (math.cos(re) * 0.5) * t_0
	else:
		tmp = math.cos(re) * (((math.pow(im, 7.0) * -0.0001984126984126984) + (math.pow(im, 5.0) * -0.008333333333333333)) + ((math.pow(im, 3.0) * -0.16666666666666666) - im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -400.0) || !(t_0 <= 0.2))
		tmp = Float64(Float64(cos(re) * 0.5) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64((im ^ 7.0) * -0.0001984126984126984) + Float64((im ^ 5.0) * -0.008333333333333333)) + Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -400.0) || ~((t_0 <= 0.2)))
		tmp = (cos(re) * 0.5) * t_0;
	else
		tmp = cos(re) * ((((im ^ 7.0) * -0.0001984126984126984) + ((im ^ 5.0) * -0.008333333333333333)) + (((im ^ 3.0) * -0.16666666666666666) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -400.0], N[Not[LessEqual[t$95$0, 0.2]], $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, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -400 or 0.20000000000000001 < (-.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 -400 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.20000000000000001

   1. Initial program 9.5%

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

      1. sub0-neg 9.5%

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

   3. Simplified 9.5%

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

   4. Taylor expanded in im around 0 99.7%

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

      1. associate-+r+ 99.7%

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

      2. +-commutative 99.7%

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

      3. mul-1-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-lft-neg-in 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*r* 99.7%

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

      8. distribute-rgt-out 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-*l* 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-*l* 99.7%

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

      13. distribute-lft-out 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -400 \lor \neg \left(e^{-im} - e^{im} \leq 0.2\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \cos re \cdot 0.5\\ \mathbf{if}\;t_0 \leq -400 \lor \neg \left(t_0 \leq 0.02\right):\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* (cos re) 0.5)))
   (if (or (<= t_0 -400.0) (not (<= t_0 0.02)))
     (* t_1 t_0)
     (*
      t_1
      (+
       (* im -2.0)
       (+
        (* (pow im 5.0) -0.016666666666666666)
        (* (pow im 3.0) -0.3333333333333333)))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = cos(re) * 0.5;
	double tmp;
	if ((t_0 <= -400.0) || !(t_0 <= 0.02)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((im * -2.0) + ((pow(im, 5.0) * -0.016666666666666666) + (pow(im, 3.0) * -0.3333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = cos(re) * 0.5d0
    if ((t_0 <= (-400.0d0)) .or. (.not. (t_0 <= 0.02d0))) then
        tmp = t_1 * t_0
    else
        tmp = t_1 * ((im * (-2.0d0)) + (((im ** 5.0d0) * (-0.016666666666666666d0)) + ((im ** 3.0d0) * (-0.3333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.cos(re) * 0.5;
	double tmp;
	if ((t_0 <= -400.0) || !(t_0 <= 0.02)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((im * -2.0) + ((Math.pow(im, 5.0) * -0.016666666666666666) + (Math.pow(im, 3.0) * -0.3333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.cos(re) * 0.5
	tmp = 0
	if (t_0 <= -400.0) or not (t_0 <= 0.02):
		tmp = t_1 * t_0
	else:
		tmp = t_1 * ((im * -2.0) + ((math.pow(im, 5.0) * -0.016666666666666666) + (math.pow(im, 3.0) * -0.3333333333333333)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(cos(re) * 0.5)
	tmp = 0.0
	if ((t_0 <= -400.0) || !(t_0 <= 0.02))
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_1 * Float64(Float64(im * -2.0) + Float64(Float64((im ^ 5.0) * -0.016666666666666666) + Float64((im ^ 3.0) * -0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = cos(re) * 0.5;
	tmp = 0.0;
	if ((t_0 <= -400.0) || ~((t_0 <= 0.02)))
		tmp = t_1 * t_0;
	else
		tmp = t_1 * ((im * -2.0) + (((im ^ 5.0) * -0.016666666666666666) + ((im ^ 3.0) * -0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -400.0], N[Not[LessEqual[t$95$0, 0.02]], $MachinePrecision]], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$1 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -400 or 0.0200000000000000004 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   if -400 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.0200000000000000004

   1. Initial program 8.8%

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

      1. sub0-neg 8.8%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in im around 0 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -400 \lor \neg \left(e^{-im} - e^{im} \leq 0.02\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -400 \lor \neg \left(t_0 \leq 0.001\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -400.0) (not (<= t_0 0.001)))
     (* (* (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 <= -400.0) || !(t_0 <= 0.001)) {
		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 <= (-400.0d0)) .or. (.not. (t_0 <= 0.001d0))) 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 <= -400.0) || !(t_0 <= 0.001)) {
		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 <= -400.0) or not (t_0 <= 0.001):
		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 <= -400.0) || !(t_0 <= 0.001))
		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 <= -400.0) || ~((t_0 <= 0.001)))
		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, -400.0], N[Not[LessEqual[t$95$0, 0.001]], $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)) < -400 or 1e-3 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   if -400 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1e-3

   1. Initial program 8.2%

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

      1. sub0-neg 8.2%

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

   3. Simplified 8.2%

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

   4. Taylor expanded in im around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. distribute-lft-out-- 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -400 \lor \neg \left(e^{-im} - e^{im} \leq 0.001\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re) * ((log(exp(((im ** 3.0d0) * (-0.16666666666666666d0)))) - im) + (((im ** 7.0d0) * (-0.0001984126984126984d0)) + ((im ** 5.0d0) * (-0.008333333333333333d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = cos(re) * ((log(exp(((im ^ 3.0) * -0.16666666666666666))) - im) + (((im ^ 7.0) * -0.0001984126984126984) + ((im ^ 5.0) * -0.008333333333333333)));
end

Wolfram

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

Derivation

1. Initial program 51.2%

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

   1. sub0-neg 51.2%

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

3. Simplified 51.2%

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

4. Taylor expanded in im around 0 95.4%

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

   1. associate-+r+ 95.4%

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

   2. +-commutative 95.4%

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

   3. mul-1-neg 95.4%

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

   4. *-commutative 95.4%

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

   5. distribute-lft-neg-in 95.4%

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

   6. *-commutative 95.4%

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

   7. associate-*r* 95.4%

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

   8. distribute-rgt-out 95.4%

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

   9. *-commutative 95.4%

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

   10. associate-*l* 95.4%

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

   11. *-commutative 95.4%

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

   12. associate-*l* 95.4%

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

   13. distribute-lft-out 95.4%

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

6. Simplified 95.4%

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

   1. log1p-expm1-u 99.1%

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

   2. log1p-udef 98.9%

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

8. Applied egg-rr 98.9%

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

   1. add-exp-log 98.9%

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

   2. log1p-def 98.9%

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

   3. log1p-expm1-u 99.0%

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

10. Applied egg-rr 99.0%

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

11. Final simplification 99.0%

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

Alternative 5: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_2 := t_1 \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{if}\;im \leq -1 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -0.015:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00066:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+172}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_2 (* t_1 (+ (* -0.5 (* re re)) 1.0))))
   (if (<= im -1e+108)
     t_2
     (if (<= im -0.015)
       t_0
       (if (<= im 0.00066)
         (* (cos re) (- im))
         (if (<= im 1e+105) t_0 (if (<= im 5e+172) t_2 t_1)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_2 = t_1 * ((-0.5 * (re * re)) + 1.0);
	double tmp;
	if (im <= -1e+108) {
		tmp = t_2;
	} else if (im <= -0.015) {
		tmp = t_0;
	} else if (im <= 0.00066) {
		tmp = cos(re) * -im;
	} else if (im <= 1e+105) {
		tmp = t_0;
	} else if (im <= 5e+172) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    t_2 = t_1 * (((-0.5d0) * (re * re)) + 1.0d0)
    if (im <= (-1d+108)) then
        tmp = t_2
    else if (im <= (-0.015d0)) then
        tmp = t_0
    else if (im <= 0.00066d0) then
        tmp = cos(re) * -im
    else if (im <= 1d+105) then
        tmp = t_0
    else if (im <= 5d+172) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double t_2 = t_1 * ((-0.5 * (re * re)) + 1.0);
	double tmp;
	if (im <= -1e+108) {
		tmp = t_2;
	} else if (im <= -0.015) {
		tmp = t_0;
	} else if (im <= 0.00066) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1e+105) {
		tmp = t_0;
	} else if (im <= 5e+172) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	t_2 = t_1 * ((-0.5 * (re * re)) + 1.0)
	tmp = 0
	if im <= -1e+108:
		tmp = t_2
	elif im <= -0.015:
		tmp = t_0
	elif im <= 0.00066:
		tmp = math.cos(re) * -im
	elif im <= 1e+105:
		tmp = t_0
	elif im <= 5e+172:
		tmp = t_2
	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(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_2 = Float64(t_1 * Float64(Float64(-0.5 * Float64(re * re)) + 1.0))
	tmp = 0.0
	if (im <= -1e+108)
		tmp = t_2;
	elseif (im <= -0.015)
		tmp = t_0;
	elseif (im <= 0.00066)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1e+105)
		tmp = t_0;
	elseif (im <= 5e+172)
		tmp = t_2;
	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 = ((im ^ 3.0) * -0.16666666666666666) - im;
	t_2 = t_1 * ((-0.5 * (re * re)) + 1.0);
	tmp = 0.0;
	if (im <= -1e+108)
		tmp = t_2;
	elseif (im <= -0.015)
		tmp = t_0;
	elseif (im <= 0.00066)
		tmp = cos(re) * -im;
	elseif (im <= 1e+105)
		tmp = t_0;
	elseif (im <= 5e+172)
		tmp = t_2;
	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[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1e+108], t$95$2, If[LessEqual[im, -0.015], t$95$0, If[LessEqual[im, 0.00066], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1e+105], t$95$0, If[LessEqual[im, 5e+172], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1e108 or 9.9999999999999994e104 < im < 5.0000000000000001e172

   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 0.0%

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

      1. associate--l+ 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-lft1-in 87.3%

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

      4. unpow2 87.3%

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

   9. Simplified 87.3%

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

   if -1e108 < im < -0.014999999999999999 or 6.6e-4 < im < 9.9999999999999994e104

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 81.7%

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

   if -0.014999999999999999 < im < 6.6e-4

   1. Initial program 8.8%

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

      1. sub0-neg 8.8%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in im around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-lft-neg-in 98.9%

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

   6. Simplified 98.9%

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

   if 5.0000000000000001e172 < 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 89.3%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+108}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{elif}\;im \leq -0.015:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.00066:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 10^{+105}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+172}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 6: 95.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.0285:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 5.5:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -1.1e+44)
     t_0
     (if (<= im -0.0285)
       (* 0.5 (- (exp (- im)) (exp im)))
       (if (<= im 5.5)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 7.0) * (cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_0;
	} else if (im <= -0.0285) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else if (im <= 5.5) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im ** 7.0d0) * (cos(re) * (-0.0001984126984126984d0))
    if (im <= (-1.1d+44)) then
        tmp = t_0
    else if (im <= (-0.0285d0)) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else if (im <= 5.5d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 7.0) * (Math.cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_0;
	} else if (im <= -0.0285) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else if (im <= 5.5) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 7.0) * (math.cos(re) * -0.0001984126984126984)
	tmp = 0
	if im <= -1.1e+44:
		tmp = t_0
	elif im <= -0.0285:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	elif im <= 5.5:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 7.0) * Float64(cos(re) * -0.0001984126984126984))
	tmp = 0.0
	if (im <= -1.1e+44)
		tmp = t_0;
	elseif (im <= -0.0285)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	elseif (im <= 5.5)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 7.0) * (cos(re) * -0.0001984126984126984);
	tmp = 0.0;
	if (im <= -1.1e+44)
		tmp = t_0;
	elseif (im <= -0.0285)
		tmp = 0.5 * (exp(-im) - exp(im));
	elseif (im <= 5.5)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 7.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.1e+44], t$95$0, If[LessEqual[im, -0.0285], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.5], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.09999999999999998e44 or 5.5 < im

   1. Initial program 100.0%

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

      1. 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 95.0%

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

      1. associate-+r+ 95.0%

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

      2. +-commutative 95.0%

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

      3. mul-1-neg 95.0%

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

      4. *-commutative 95.0%

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

      5. distribute-lft-neg-in 95.0%

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

      6. *-commutative 95.0%

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

      7. associate-*r* 95.0%

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

      8. distribute-rgt-out 95.0%

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

      9. *-commutative 95.0%

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

      10. associate-*l* 95.0%

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

      11. *-commutative 95.0%

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

      12. associate-*l* 95.0%

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

      13. distribute-lft-out 95.0%

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

   6. Simplified 95.0%

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

   7. Taylor expanded in im around inf 95.0%

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

      1. associate-*r* 95.0%

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

      2. *-commutative 95.0%

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

   9. Simplified 95.0%

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

   if -1.09999999999999998e44 < im < -0.028500000000000001

   1. Initial program 99.1%

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

      1. sub0-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in re around 0 76.5%

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

   if -0.028500000000000001 < im < 5.5

   1. Initial program 8.8%

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

      1. sub0-neg 8.8%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in im around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*l* 99.5%

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

      5. distribute-lft-out-- 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq -0.0285:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 5.5:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 7: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.01:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 4.1:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -1.1e+44)
     t_0
     (if (<= im -0.01)
       (* 0.5 (- (exp (- im)) (exp im)))
       (if (<= im 4.1) (* (cos re) (- im)) t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 7.0) * (cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_0;
	} else if (im <= -0.01) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else if (im <= 4.1) {
		tmp = cos(re) * -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im ** 7.0d0) * (cos(re) * (-0.0001984126984126984d0))
    if (im <= (-1.1d+44)) then
        tmp = t_0
    else if (im <= (-0.01d0)) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else if (im <= 4.1d0) then
        tmp = cos(re) * -im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 7.0) * (Math.cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_0;
	} else if (im <= -0.01) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else if (im <= 4.1) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 7.0) * (math.cos(re) * -0.0001984126984126984)
	tmp = 0
	if im <= -1.1e+44:
		tmp = t_0
	elif im <= -0.01:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	elif im <= 4.1:
		tmp = math.cos(re) * -im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 7.0) * Float64(cos(re) * -0.0001984126984126984))
	tmp = 0.0
	if (im <= -1.1e+44)
		tmp = t_0;
	elseif (im <= -0.01)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	elseif (im <= 4.1)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 7.0) * (cos(re) * -0.0001984126984126984);
	tmp = 0.0;
	if (im <= -1.1e+44)
		tmp = t_0;
	elseif (im <= -0.01)
		tmp = 0.5 * (exp(-im) - exp(im));
	elseif (im <= 4.1)
		tmp = cos(re) * -im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 7.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.1e+44], t$95$0, If[LessEqual[im, -0.01], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.1], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.09999999999999998e44 or 4.0999999999999996 < 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 95.0%

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

      1. associate-+r+ 95.0%

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

      2. +-commutative 95.0%

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

      3. mul-1-neg 95.0%

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

      4. *-commutative 95.0%

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

      5. distribute-lft-neg-in 95.0%

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

      6. *-commutative 95.0%

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

      7. associate-*r* 95.0%

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

      8. distribute-rgt-out 95.0%

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

      9. *-commutative 95.0%

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

      10. associate-*l* 95.0%

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

      11. *-commutative 95.0%

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

      12. associate-*l* 95.0%

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

      13. distribute-lft-out 95.0%

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

   6. Simplified 95.0%

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

   7. Taylor expanded in im around inf 95.0%

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

      1. associate-*r* 95.0%

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

      2. *-commutative 95.0%

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

   9. Simplified 95.0%

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

   if -1.09999999999999998e44 < im < -0.0100000000000000002

   1. Initial program 99.1%

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

      1. sub0-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in re around 0 76.5%

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

   if -0.0100000000000000002 < im < 4.0999999999999996

   1. Initial program 8.8%

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

      1. sub0-neg 8.8%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in im around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-lft-neg-in 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq -0.01:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 4.1:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 8: 81.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot -0.0001984126984126984\\ t_1 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_2 := t_1 \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{if}\;im \leq -5 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -4.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5.8:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+175}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) -0.0001984126984126984))
        (t_1 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_2 (* t_1 (+ (* -0.5 (* re re)) 1.0))))
   (if (<= im -5e+110)
     t_2
     (if (<= im -4.5)
       t_0
       (if (<= im 5.8)
         (* (cos re) (- im))
         (if (<= im 5e+102) t_0 (if (<= im 2e+175) t_2 t_1)))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 7.0) * -0.0001984126984126984;
	double t_1 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_2 = t_1 * ((-0.5 * (re * re)) + 1.0);
	double tmp;
	if (im <= -5e+110) {
		tmp = t_2;
	} else if (im <= -4.5) {
		tmp = t_0;
	} else if (im <= 5.8) {
		tmp = cos(re) * -im;
	} else if (im <= 5e+102) {
		tmp = t_0;
	} else if (im <= 2e+175) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (im ** 7.0d0) * (-0.0001984126984126984d0)
    t_1 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    t_2 = t_1 * (((-0.5d0) * (re * re)) + 1.0d0)
    if (im <= (-5d+110)) then
        tmp = t_2
    else if (im <= (-4.5d0)) then
        tmp = t_0
    else if (im <= 5.8d0) then
        tmp = cos(re) * -im
    else if (im <= 5d+102) then
        tmp = t_0
    else if (im <= 2d+175) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 7.0) * -0.0001984126984126984;
	double t_1 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double t_2 = t_1 * ((-0.5 * (re * re)) + 1.0);
	double tmp;
	if (im <= -5e+110) {
		tmp = t_2;
	} else if (im <= -4.5) {
		tmp = t_0;
	} else if (im <= 5.8) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 5e+102) {
		tmp = t_0;
	} else if (im <= 2e+175) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 7.0) * -0.0001984126984126984
	t_1 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	t_2 = t_1 * ((-0.5 * (re * re)) + 1.0)
	tmp = 0
	if im <= -5e+110:
		tmp = t_2
	elif im <= -4.5:
		tmp = t_0
	elif im <= 5.8:
		tmp = math.cos(re) * -im
	elif im <= 5e+102:
		tmp = t_0
	elif im <= 2e+175:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 7.0) * -0.0001984126984126984)
	t_1 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_2 = Float64(t_1 * Float64(Float64(-0.5 * Float64(re * re)) + 1.0))
	tmp = 0.0
	if (im <= -5e+110)
		tmp = t_2;
	elseif (im <= -4.5)
		tmp = t_0;
	elseif (im <= 5.8)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 5e+102)
		tmp = t_0;
	elseif (im <= 2e+175)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 7.0) * -0.0001984126984126984;
	t_1 = ((im ^ 3.0) * -0.16666666666666666) - im;
	t_2 = t_1 * ((-0.5 * (re * re)) + 1.0);
	tmp = 0.0;
	if (im <= -5e+110)
		tmp = t_2;
	elseif (im <= -4.5)
		tmp = t_0;
	elseif (im <= 5.8)
		tmp = cos(re) * -im;
	elseif (im <= 5e+102)
		tmp = t_0;
	elseif (im <= 2e+175)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5e+110], t$95$2, If[LessEqual[im, -4.5], t$95$0, If[LessEqual[im, 5.8], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 5e+102], t$95$0, If[LessEqual[im, 2e+175], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.99999999999999978e110 or 5e102 < im < 1.9999999999999999e175

   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 0.0%

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

      1. associate--l+ 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-lft1-in 87.3%

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

      4. unpow2 87.3%

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

   9. Simplified 87.3%

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

   if -4.99999999999999978e110 < im < -4.5 or 5.79999999999999982 < im < 5e102

   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 67.6%

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

      1. associate-+r+ 67.6%

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

      2. +-commutative 67.6%

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

      3. mul-1-neg 67.6%

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

      4. *-commutative 67.6%

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

      5. distribute-lft-neg-in 67.6%

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

      6. *-commutative 67.6%

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

      7. associate-*r* 67.6%

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

      8. distribute-rgt-out 67.6%

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

      9. *-commutative 67.6%

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

      10. associate-*l* 67.6%

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

      11. *-commutative 67.6%

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

      12. associate-*l* 67.6%

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

      13. distribute-lft-out 67.6%

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

   6. Simplified 67.6%

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

   7. Taylor expanded in im around inf 67.4%

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

      1. associate-*r* 67.4%

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

      2. *-commutative 67.4%

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

   9. Simplified 67.4%

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

   10. Taylor expanded in re around 0 58.5%

       \[\leadsto {im}^{7} \cdot \color{blue}{-0.0001984126984126984} \]

   if -4.5 < im < 5.79999999999999982

   1. Initial program 10.1%

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

      1. sub0-neg 10.1%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in im around 0 97.9%

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

      1. mul-1-neg 97.9%

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

      2. *-commutative 97.9%

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

      3. distribute-lft-neg-in 97.9%

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

   6. Simplified 97.9%

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

   if 1.9999999999999999e175 < 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 89.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+110}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{elif}\;im \leq -4.5:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{elif}\;im \leq 5.8:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+102}:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 9: 81.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -7.2 \lor \neg \left(im \leq 5.5\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -7.2) (not (<= im 5.5)))
   (* (pow im 7.0) -0.0001984126984126984)
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -7.2) || !(im <= 5.5)) {
		tmp = pow(im, 7.0) * -0.0001984126984126984;
	} 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 <= (-7.2d0)) .or. (.not. (im <= 5.5d0))) then
        tmp = (im ** 7.0d0) * (-0.0001984126984126984d0)
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -7.2) || !(im <= 5.5)) {
		tmp = Math.pow(im, 7.0) * -0.0001984126984126984;
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -7.2) or not (im <= 5.5):
		tmp = math.pow(im, 7.0) * -0.0001984126984126984
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -7.2) || !(im <= 5.5))
		tmp = Float64((im ^ 7.0) * -0.0001984126984126984);
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -7.2) || ~((im <= 5.5)))
		tmp = (im ^ 7.0) * -0.0001984126984126984;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -7.2], N[Not[LessEqual[im, 5.5]], $MachinePrecision]], N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -7.20000000000000018 or 5.5 < im

   1. Initial program 100.0%

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

      1. 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 90.6%

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

      1. associate-+r+ 90.6%

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

      2. +-commutative 90.6%

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

      3. mul-1-neg 90.6%

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

      4. *-commutative 90.6%

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

      5. distribute-lft-neg-in 90.6%

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

      6. *-commutative 90.6%

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

      7. associate-*r* 90.6%

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

      8. distribute-rgt-out 90.6%

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

      9. *-commutative 90.6%

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

      10. associate-*l* 90.6%

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

      11. *-commutative 90.6%

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

      12. associate-*l* 90.6%

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

      13. distribute-lft-out 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in im around inf 90.5%

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

      1. associate-*r* 90.5%

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

      2. *-commutative 90.5%

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

   9. Simplified 90.5%

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

   10. Taylor expanded in re around 0 70.0%

       \[\leadsto {im}^{7} \cdot \color{blue}{-0.0001984126984126984} \]

   if -7.20000000000000018 < im < 5.5

   1. Initial program 10.1%

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

      1. sub0-neg 10.1%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in im around 0 97.9%

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

      1. mul-1-neg 97.9%

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

      2. *-commutative 97.9%

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

      3. distribute-lft-neg-in 97.9%

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

   6. Simplified 97.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.2 \lor \neg \left(im \leq 5.5\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 10: 57.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.75 \cdot 10^{+43}:\\ \;\;\;\;\left(re \cdot im\right) \cdot \left(re \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 660:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.75e+43)
   (- (* (* re im) (* re 0.5)) im)
   (if (<= im 660.0) (* (cos re) (- im)) (+ 13.5 (* (* re re) -6.75)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.75e+43) {
		tmp = ((re * im) * (re * 0.5)) - im;
	} else if (im <= 660.0) {
		tmp = cos(re) * -im;
	} else {
		tmp = 13.5 + ((re * re) * -6.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.75d+43)) then
        tmp = ((re * im) * (re * 0.5d0)) - im
    else if (im <= 660.0d0) then
        tmp = cos(re) * -im
    else
        tmp = 13.5d0 + ((re * re) * (-6.75d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.75e+43) {
		tmp = ((re * im) * (re * 0.5)) - im;
	} else if (im <= 660.0) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = 13.5 + ((re * re) * -6.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.75e+43:
		tmp = ((re * im) * (re * 0.5)) - im
	elif im <= 660.0:
		tmp = math.cos(re) * -im
	else:
		tmp = 13.5 + ((re * re) * -6.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.75e+43)
		tmp = Float64(Float64(Float64(re * im) * Float64(re * 0.5)) - im);
	elseif (im <= 660.0)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = Float64(13.5 + Float64(Float64(re * re) * -6.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.75e+43)
		tmp = ((re * im) * (re * 0.5)) - im;
	elseif (im <= 660.0)
		tmp = cos(re) * -im;
	else
		tmp = 13.5 + ((re * re) * -6.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.75e+43], N[(N[(N[(re * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 660.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(13.5 + N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.7500000000000001e43

   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 5.4%

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

      1. mul-1-neg 5.4%

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

      2. *-commutative 5.4%

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

      3. distribute-lft-neg-in 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in re around 0 27.7%

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

      1. mul-1-neg 27.7%

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

      2. +-commutative 27.7%

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

      3. unsub-neg 27.7%

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

      4. *-commutative 27.7%

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

      5. associate-*l* 27.7%

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

      6. unpow2 27.7%

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

   9. Simplified 27.7%

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

   10. Taylor expanded in re around 0 27.7%

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

       1. *-commutative 27.7%

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

       2. associate-*r* 27.7%

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

       3. unpow2 27.7%

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

       4. associate-*l* 27.7%

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

       5. *-commutative 27.7%

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

       6. associate-*r* 27.7%

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

       7. associate-*l* 27.7%

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

       8. *-commutative 27.7%

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

   12. Simplified 27.7%

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

   if -1.7500000000000001e43 < im < 660

   1. Initial program 14.4%

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

      1. sub0-neg 14.4%

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

   3. Simplified 14.4%

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

   4. Taylor expanded in im around 0 93.5%

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

      1. mul-1-neg 93.5%

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

      2. *-commutative 93.5%

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

      3. distribute-lft-neg-in 93.5%

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

   6. Simplified 93.5%

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

   if 660 < 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%

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

   5. Applied egg-rr 0.6%

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

   6. Taylor expanded in re around 0 16.6%

      \[\leadsto \color{blue}{13.5 + -6.75 \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 16.6%

         \[\leadsto 13.5 + \color{blue}{{re}^{2} \cdot -6.75} \]

      2. unpow2 16.6%

         \[\leadsto 13.5 + \color{blue}{\left(re \cdot re\right)} \cdot -6.75 \]

   8. Simplified 16.6%

      \[\leadsto \color{blue}{13.5 + \left(re \cdot re\right) \cdot -6.75} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.75 \cdot 10^{+43}:\\ \;\;\;\;\left(re \cdot im\right) \cdot \left(re \cdot 0.5\right) - im\\ \mathbf{elif}\;im \leq 660:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 11: 34.0% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8.6 \cdot 10^{+42} \lor \neg \left(im \leq 680\right):\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -8.6e+42) (not (<= im 680.0)))
   (+ 13.5 (* (* re re) -6.75))
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -8.6e+42) || !(im <= 680.0)) {
		tmp = 13.5 + ((re * re) * -6.75);
	} else {
		tmp = -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-8.6d+42)) .or. (.not. (im <= 680.0d0))) then
        tmp = 13.5d0 + ((re * re) * (-6.75d0))
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -8.6e+42) || !(im <= 680.0)) {
		tmp = 13.5 + ((re * re) * -6.75);
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -8.6e+42) or not (im <= 680.0):
		tmp = 13.5 + ((re * re) * -6.75)
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -8.6e+42) || !(im <= 680.0))
		tmp = Float64(13.5 + Float64(Float64(re * re) * -6.75));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -8.6e+42) || ~((im <= 680.0)))
		tmp = 13.5 + ((re * re) * -6.75);
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -8.6e+42], N[Not[LessEqual[im, 680.0]], $MachinePrecision]], N[(13.5 + N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -8.5999999999999996e42 or 680 < 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%

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

   5. Applied egg-rr 1.8%

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

   6. Taylor expanded in re around 0 16.3%

      \[\leadsto \color{blue}{13.5 + -6.75 \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 16.3%

         \[\leadsto 13.5 + \color{blue}{{re}^{2} \cdot -6.75} \]

      2. unpow2 16.3%

         \[\leadsto 13.5 + \color{blue}{\left(re \cdot re\right)} \cdot -6.75 \]

   8. Simplified 16.3%

      \[\leadsto \color{blue}{13.5 + \left(re \cdot re\right) \cdot -6.75} \]

   if -8.5999999999999996e42 < im < 680

   1. Initial program 14.4%

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

      1. sub0-neg 14.4%

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

   3. Simplified 14.4%

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

   4. Taylor expanded in im around 0 93.5%

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

      1. mul-1-neg 93.5%

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

      2. *-commutative 93.5%

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

      3. distribute-lft-neg-in 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in re around 0 54.7%

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

      1. mul-1-neg 54.7%

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

   9. Simplified 54.7%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.6 \cdot 10^{+42} \lor \neg \left(im \leq 680\right):\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 12: 36.4% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+191}:\\ \;\;\;\;\left(re \cdot im\right) \cdot \left(re \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.5e+191)
   (- (* (* re im) (* re 0.5)) im)
   (+ 13.5 (* (* re re) -6.75))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+191) {
		tmp = ((re * im) * (re * 0.5)) - im;
	} else {
		tmp = 13.5 + ((re * re) * -6.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.5d+191) then
        tmp = ((re * im) * (re * 0.5d0)) - im
    else
        tmp = 13.5d0 + ((re * re) * (-6.75d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+191) {
		tmp = ((re * im) * (re * 0.5)) - im;
	} else {
		tmp = 13.5 + ((re * re) * -6.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.5e+191:
		tmp = ((re * im) * (re * 0.5)) - im
	else:
		tmp = 13.5 + ((re * re) * -6.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.5e+191)
		tmp = Float64(Float64(Float64(re * im) * Float64(re * 0.5)) - im);
	else
		tmp = Float64(13.5 + Float64(Float64(re * re) * -6.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.5e+191)
		tmp = ((re * im) * (re * 0.5)) - im;
	else
		tmp = 13.5 + ((re * re) * -6.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.5e+191], N[(N[(N[(re * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(13.5 + N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.4999999999999999e191

   1. Initial program 51.0%

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

      1. sub0-neg 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in im around 0 55.9%

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

      1. mul-1-neg 55.9%

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

      2. *-commutative 55.9%

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

      3. distribute-lft-neg-in 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in re around 0 40.9%

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

      1. mul-1-neg 40.9%

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

      2. +-commutative 40.9%

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

      3. unsub-neg 40.9%

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

      4. *-commutative 40.9%

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

      5. associate-*l* 40.9%

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

      6. unpow2 40.9%

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

   9. Simplified 40.9%

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

   10. Taylor expanded in re around 0 40.9%

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

       1. *-commutative 40.9%

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

       2. associate-*r* 40.9%

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

       3. unpow2 40.9%

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

       4. associate-*l* 40.9%

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

       5. *-commutative 40.9%

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

       6. associate-*r* 40.9%

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

       7. associate-*l* 40.9%

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

       8. *-commutative 40.9%

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

   12. Simplified 40.9%

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

   if 1.4999999999999999e191 < re

   1. Initial program 52.4%

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

      1. sub0-neg 52.4%

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

   3. Simplified 52.4%

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

   5. Applied egg-rr 3.8%

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

   6. Taylor expanded in re around 0 28.0%

      \[\leadsto \color{blue}{13.5 + -6.75 \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 28.0%

         \[\leadsto 13.5 + \color{blue}{{re}^{2} \cdot -6.75} \]

      2. unpow2 28.0%

         \[\leadsto 13.5 + \color{blue}{\left(re \cdot re\right)} \cdot -6.75 \]

   8. Simplified 28.0%

      \[\leadsto \color{blue}{13.5 + \left(re \cdot re\right) \cdot -6.75} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+191}:\\ \;\;\;\;\left(re \cdot im\right) \cdot \left(re \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;13.5 + \left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 13: 29.3% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

   1. sub0-neg 51.2%

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

3. Simplified 51.2%

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

4. Taylor expanded in im around 0 55.7%

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

   1. mul-1-neg 55.7%

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

   2. *-commutative 55.7%

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

   3. distribute-lft-neg-in 55.7%

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

6. Simplified 55.7%

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

7. Taylor expanded in re around 0 33.1%

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

   1. mul-1-neg 33.1%

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

9. Simplified 33.1%

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

10. Final simplification 33.1%

    \[\leadsto -im \]

Alternative 14: 2.8% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 13.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 13.5

Julia

function code(re, im)
	return 13.5
end

MATLAB

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

Wolfram

code[re_, im_] := 13.5

Derivation

1. Initial program 51.2%

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

   1. sub0-neg 51.2%

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

3. Simplified 51.2%

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

5. Applied egg-rr 3.1%

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

6. Taylor expanded in re around 0 3.0%

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

7. Final simplification 3.0%

   \[\leadsto 13.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023182 
(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))))