math.sin on complex, imaginary part

Percentage Accurate: 54.7% → 99.7%

Time: 8.6s

Alternatives: 13

Speedup: 2.9×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+66} \lor \neg \left(t_0 \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-0.0003968253968253968 \cdot {im}^{7} + \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (cos re))))
   (if (or (<= t_0 -5e+66) (not (<= t_0 2e-5)))
     (* t_1 t_0)
     (*
      t_1
      (+
       (* -0.0003968253968253968 (pow im 7.0))
       (+
        (* im -2.0)
        (+
         (* -0.016666666666666666 (pow im 5.0))
         (* -0.3333333333333333 (pow im 3.0)))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = 0.5 * cos(re);
	double tmp;
	if ((t_0 <= -5e+66) || !(t_0 <= 2e-5)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((-0.0003968253968253968 * pow(im, 7.0)) + ((im * -2.0) + ((-0.016666666666666666 * pow(im, 5.0)) + (-0.3333333333333333 * pow(im, 3.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = 0.5d0 * cos(re)
    if ((t_0 <= (-5d+66)) .or. (.not. (t_0 <= 2d-5))) then
        tmp = t_1 * t_0
    else
        tmp = t_1 * (((-0.0003968253968253968d0) * (im ** 7.0d0)) + ((im * (-2.0d0)) + (((-0.016666666666666666d0) * (im ** 5.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = 0.5 * Math.cos(re);
	double tmp;
	if ((t_0 <= -5e+66) || !(t_0 <= 2e-5)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((-0.0003968253968253968 * Math.pow(im, 7.0)) + ((im * -2.0) + ((-0.016666666666666666 * Math.pow(im, 5.0)) + (-0.3333333333333333 * Math.pow(im, 3.0)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = 0.5 * math.cos(re)
	tmp = 0
	if (t_0 <= -5e+66) or not (t_0 <= 2e-5):
		tmp = t_1 * t_0
	else:
		tmp = t_1 * ((-0.0003968253968253968 * math.pow(im, 7.0)) + ((im * -2.0) + ((-0.016666666666666666 * math.pow(im, 5.0)) + (-0.3333333333333333 * math.pow(im, 3.0)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(0.5 * cos(re))
	tmp = 0.0
	if ((t_0 <= -5e+66) || !(t_0 <= 2e-5))
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_1 * Float64(Float64(-0.0003968253968253968 * (im ^ 7.0)) + Float64(Float64(im * -2.0) + Float64(Float64(-0.016666666666666666 * (im ^ 5.0)) + Float64(-0.3333333333333333 * (im ^ 3.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = 0.5 * cos(re);
	tmp = 0.0;
	if ((t_0 <= -5e+66) || ~((t_0 <= 2e-5)))
		tmp = t_1 * t_0;
	else
		tmp = t_1 * ((-0.0003968253968253968 * (im ^ 7.0)) + ((im * -2.0) + ((-0.016666666666666666 * (im ^ 5.0)) + (-0.3333333333333333 * (im ^ 3.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+66], N[Not[LessEqual[t$95$0, 2e-5]], $MachinePrecision]], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$1 * N[(N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(im * -2.0), $MachinePrecision] + N[(N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -4.99999999999999991e66 or 2.00000000000000016e-5 < (-.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 -4.99999999999999991e66 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.00000000000000016e-5

   1. Initial program 7.9%

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

      1. sub0-neg 7.9%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7} + \left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \cdot 10^{+66} \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(-0.0003968253968253968 \cdot {im}^{7} + \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+66} \lor \neg \left(t_0 \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -5e+66) (not (<= t_0 2e-5)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (+
       (- (* (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) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -5e+66) || !(t_0 <= 2e-5)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * (((pow(im, 3.0) * -0.16666666666666666) - im) + ((pow(im, 7.0) * -0.0001984126984126984) + (pow(im, 5.0) * -0.008333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-5d+66)) .or. (.not. (t_0 <= 2d-5))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * ((((im ** 3.0d0) * (-0.16666666666666666d0)) - im) + (((im ** 7.0d0) * (-0.0001984126984126984d0)) + ((im ** 5.0d0) * (-0.008333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -5e+66) || !(t_0 <= 2e-5)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * (((Math.pow(im, 3.0) * -0.16666666666666666) - im) + ((Math.pow(im, 7.0) * -0.0001984126984126984) + (Math.pow(im, 5.0) * -0.008333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -5e+66) or not (t_0 <= 2e-5):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * (((math.pow(im, 3.0) * -0.16666666666666666) - im) + ((math.pow(im, 7.0) * -0.0001984126984126984) + (math.pow(im, 5.0) * -0.008333333333333333)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -5e+66) || !(t_0 <= 2e-5))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) + Float64(Float64((im ^ 7.0) * -0.0001984126984126984) + Float64((im ^ 5.0) * -0.008333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -5e+66) || ~((t_0 <= 2e-5)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((((im ^ 3.0) * -0.16666666666666666) - im) + (((im ^ 7.0) * -0.0001984126984126984) + ((im ^ 5.0) * -0.008333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+66], N[Not[LessEqual[t$95$0, 2e-5]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $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. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -4.99999999999999991e66 or 2.00000000000000016e-5 < (-.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 -4.99999999999999991e66 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.00000000000000016e-5

   1. Initial program 7.9%

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

      1. sub0-neg 7.9%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. associate-+r+ 99.8%

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

      2. +-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. distribute-rgt-out 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

      11. *-commutative 99.8%

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

      12. associate-*l* 99.8%

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

      13. distribute-lft-out 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \cdot 10^{+66} \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.05 \lor \neg \left(t_0 \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.05) (not (<= t_0 2e-5)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.05) || !(t_0 <= 2e-5)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.05d0)) .or. (.not. (t_0 <= 2d-5))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.05) || !(t_0 <= 2e-5)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.05) or not (t_0 <= 2e-5):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.05) || !(t_0 <= 2e-5))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.05) || ~((t_0 <= 2e-5)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.05], N[Not[LessEqual[t$95$0, 2e-5]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.050000000000000003 or 2.00000000000000016e-5 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -0.050000000000000003 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.00000000000000016e-5

   1. Initial program 7.2%

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

      1. sub0-neg 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.05 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4: 96.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -3.3 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.55:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 0.0275:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im)))
        (t_1 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -3.3e+85)
     t_1
     (if (<= im -0.55)
       (* t_0 (+ 0.5 (* re (* re -0.25))))
       (if (<= im 0.0275)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 2.5e+36) (* 0.5 t_0) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = pow(im, 7.0) * (cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -3.3e+85) {
		tmp = t_1;
	} else if (im <= -0.55) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else if (im <= 0.0275) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 2.5e+36) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = (im ** 7.0d0) * (cos(re) * (-0.0001984126984126984d0))
    if (im <= (-3.3d+85)) then
        tmp = t_1
    else if (im <= (-0.55d0)) then
        tmp = t_0 * (0.5d0 + (re * (re * (-0.25d0))))
    else if (im <= 0.0275d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 2.5d+36) then
        tmp = 0.5d0 * t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.pow(im, 7.0) * (Math.cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -3.3e+85) {
		tmp = t_1;
	} else if (im <= -0.55) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else if (im <= 0.0275) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 2.5e+36) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.pow(im, 7.0) * (math.cos(re) * -0.0001984126984126984)
	tmp = 0
	if im <= -3.3e+85:
		tmp = t_1
	elif im <= -0.55:
		tmp = t_0 * (0.5 + (re * (re * -0.25)))
	elif im <= 0.0275:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 2.5e+36:
		tmp = 0.5 * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64((im ^ 7.0) * Float64(cos(re) * -0.0001984126984126984))
	tmp = 0.0
	if (im <= -3.3e+85)
		tmp = t_1;
	elseif (im <= -0.55)
		tmp = Float64(t_0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	elseif (im <= 0.0275)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 2.5e+36)
		tmp = Float64(0.5 * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = (im ^ 7.0) * (cos(re) * -0.0001984126984126984);
	tmp = 0.0;
	if (im <= -3.3e+85)
		tmp = t_1;
	elseif (im <= -0.55)
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	elseif (im <= 0.0275)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 2.5e+36)
		tmp = 0.5 * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 7.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.3e+85], t$95$1, If[LessEqual[im, -0.55], N[(t$95$0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 0.0275], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+36], N[(0.5 * t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.2999999999999999e85 or 2.49999999999999988e36 < 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 99.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7} + \left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\right)} \]

   5. Taylor expanded in im around inf 99.1%

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

      1. *-commutative 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l* 99.1%

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

   7. Simplified 99.1%

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

   if -3.2999999999999999e85 < im < -0.55000000000000004

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 95.7%

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

      4. +-commutative 95.7%

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

      5. *-commutative 95.7%

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

      6. unpow2 95.7%

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

      7. associate-*l* 95.7%

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

   6. Simplified 95.7%

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

   if -0.55000000000000004 < im < 0.0275000000000000001

   1. Initial program 7.2%

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

      1. sub0-neg 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

   if 0.0275000000000000001 < im < 2.49999999999999988e36

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around 0 89.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.3 \cdot 10^{+85}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq -0.55:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 0.0275:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 5: 86.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;im \leq -0.00032:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00112:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im)))))
   (if (<= im -0.00032)
     t_0
     (if (<= im 0.00112)
       (* (cos re) (- im))
       (if (<= im 1.02e+101)
         t_0
         (*
          (- (* (pow im 3.0) -0.16666666666666666) im)
          (+ (* (* re re) -0.5) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double tmp;
	if (im <= -0.00032) {
		tmp = t_0;
	} else if (im <= 0.00112) {
		tmp = cos(re) * -im;
	} else if (im <= 1.02e+101) {
		tmp = t_0;
	} else {
		tmp = ((pow(im, 3.0) * -0.16666666666666666) - im) * (((re * re) * -0.5) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    if (im <= (-0.00032d0)) then
        tmp = t_0
    else if (im <= 0.00112d0) then
        tmp = cos(re) * -im
    else if (im <= 1.02d+101) then
        tmp = t_0
    else
        tmp = (((im ** 3.0d0) * (-0.16666666666666666d0)) - im) * (((re * re) * (-0.5d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (im <= -0.00032) {
		tmp = t_0;
	} else if (im <= 0.00112) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.02e+101) {
		tmp = t_0;
	} else {
		tmp = ((Math.pow(im, 3.0) * -0.16666666666666666) - im) * (((re * re) * -0.5) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	tmp = 0
	if im <= -0.00032:
		tmp = t_0
	elif im <= 0.00112:
		tmp = math.cos(re) * -im
	elif im <= 1.02e+101:
		tmp = t_0
	else:
		tmp = ((math.pow(im, 3.0) * -0.16666666666666666) - im) * (((re * re) * -0.5) + 1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (im <= -0.00032)
		tmp = t_0;
	elseif (im <= 0.00112)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.02e+101)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) * Float64(Float64(Float64(re * re) * -0.5) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	tmp = 0.0;
	if (im <= -0.00032)
		tmp = t_0;
	elseif (im <= 0.00112)
		tmp = cos(re) * -im;
	elseif (im <= 1.02e+101)
		tmp = t_0;
	else
		tmp = (((im ^ 3.0) * -0.16666666666666666) - im) * (((re * re) * -0.5) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -0.00032], t$95$0, If[LessEqual[im, 0.00112], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.02e+101], t$95$0, N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.20000000000000026e-4 or 0.0011199999999999999 < im < 1.02000000000000002e101

   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%

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

   4. Taylor expanded in re around 0 78.2%

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

   if -3.20000000000000026e-4 < im < 0.0011199999999999999

   1. Initial program 7.2%

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

      1. sub0-neg 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in im around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-lft-neg-in 99.3%

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

   6. Simplified 99.3%

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

   if 1.02000000000000002e101 < 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 98.0%

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

      1. mul-1-neg 98.0%

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

      2. unsub-neg 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-*l* 98.0%

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

      5. distribute-lft-out-- 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in re around 0 2.3%

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

      1. sub-neg 2.3%

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

      2. *-commutative 2.3%

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

      3. associate-+r+ 2.3%

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

      4. fma-def 2.3%

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

      5. associate-*r* 2.3%

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

      6. *-commutative 2.3%

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

      7. fma-neg 2.3%

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

      8. *-lft-identity 2.3%

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

      9. distribute-rgt-out 81.4%

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

      10. fma-neg 81.4%

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

      11. *-commutative 81.4%

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

      12. unpow2 81.4%

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

   9. Simplified 81.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00032:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.00112:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \end{array} \]

Alternative 6: 94.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -4.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00089:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -4.2)
     t_0
     (if (<= im 0.00089)
       (* (cos re) (- im))
       (if (<= im 2.5e+36) (* 0.5 (- (exp (- im)) (exp 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 <= -4.2) {
		tmp = t_0;
	} else if (im <= 0.00089) {
		tmp = cos(re) * -im;
	} else if (im <= 2.5e+36) {
		tmp = 0.5 * (exp(-im) - exp(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 <= (-4.2d0)) then
        tmp = t_0
    else if (im <= 0.00089d0) then
        tmp = cos(re) * -im
    else if (im <= 2.5d+36) then
        tmp = 0.5d0 * (exp(-im) - exp(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 <= -4.2) {
		tmp = t_0;
	} else if (im <= 0.00089) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 2.5e+36) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(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 <= -4.2:
		tmp = t_0
	elif im <= 0.00089:
		tmp = math.cos(re) * -im
	elif im <= 2.5e+36:
		tmp = 0.5 * (math.exp(-im) - math.exp(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 <= -4.2)
		tmp = t_0;
	elseif (im <= 0.00089)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 2.5e+36)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(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 <= -4.2)
		tmp = t_0;
	elseif (im <= 0.00089)
		tmp = cos(re) * -im;
	elseif (im <= 2.5e+36)
		tmp = 0.5 * (exp(-im) - exp(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, -4.2], t$95$0, If[LessEqual[im, 0.00089], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 2.5e+36], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.20000000000000018 or 2.49999999999999988e36 < 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 93.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7} + \left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\right)} \]

   5. Taylor expanded in im around inf 93.2%

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

      1. *-commutative 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-*l* 93.2%

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

   7. Simplified 93.2%

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

   if -4.20000000000000018 < im < 8.8999999999999995e-4

   1. Initial program 7.2%

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

      1. sub0-neg 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in im around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-lft-neg-in 99.3%

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

   6. Simplified 99.3%

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

   if 8.8999999999999995e-4 < im < 2.49999999999999988e36

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around 0 89.6%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.2:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq 0.00089:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 7: 94.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -5.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.03:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -5.6)
     t_0
     (if (<= im 0.03)
       (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
       (if (<= im 2.5e+36) (* 0.5 (- (exp (- im)) (exp 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 <= -5.6) {
		tmp = t_0;
	} else if (im <= 0.03) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 2.5e+36) {
		tmp = 0.5 * (exp(-im) - exp(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 <= (-5.6d0)) then
        tmp = t_0
    else if (im <= 0.03d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 2.5d+36) then
        tmp = 0.5d0 * (exp(-im) - exp(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 <= -5.6) {
		tmp = t_0;
	} else if (im <= 0.03) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 2.5e+36) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(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 <= -5.6:
		tmp = t_0
	elif im <= 0.03:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 2.5e+36:
		tmp = 0.5 * (math.exp(-im) - math.exp(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 <= -5.6)
		tmp = t_0;
	elseif (im <= 0.03)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 2.5e+36)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(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 <= -5.6)
		tmp = t_0;
	elseif (im <= 0.03)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 2.5e+36)
		tmp = 0.5 * (exp(-im) - exp(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, -5.6], t$95$0, If[LessEqual[im, 0.03], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+36], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.5999999999999996 or 2.49999999999999988e36 < 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 93.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7} + \left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\right)} \]

   5. Taylor expanded in im around inf 93.2%

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

      1. *-commutative 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-*l* 93.2%

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

   7. Simplified 93.2%

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

   if -5.5999999999999996 < im < 0.029999999999999999

   1. Initial program 7.2%

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

      1. sub0-neg 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

   if 0.029999999999999999 < im < 2.49999999999999988e36

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around 0 89.6%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.6:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq 0.03:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 8: 79.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+23}:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{elif}\;im \leq 0.023:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -9e+23)
   (* (pow im 7.0) -0.0001984126984126984)
   (if (<= im 0.023)
     (* (cos re) (- im))
     (*
      (- (* (pow im 3.0) -0.16666666666666666) im)
      (+ (* (* re re) -0.5) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -9e+23) {
		tmp = pow(im, 7.0) * -0.0001984126984126984;
	} else if (im <= 0.023) {
		tmp = cos(re) * -im;
	} else {
		tmp = ((pow(im, 3.0) * -0.16666666666666666) - im) * (((re * re) * -0.5) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-9d+23)) then
        tmp = (im ** 7.0d0) * (-0.0001984126984126984d0)
    else if (im <= 0.023d0) then
        tmp = cos(re) * -im
    else
        tmp = (((im ** 3.0d0) * (-0.16666666666666666d0)) - im) * (((re * re) * (-0.5d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -9e+23) {
		tmp = Math.pow(im, 7.0) * -0.0001984126984126984;
	} else if (im <= 0.023) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = ((Math.pow(im, 3.0) * -0.16666666666666666) - im) * (((re * re) * -0.5) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -9e+23:
		tmp = math.pow(im, 7.0) * -0.0001984126984126984
	elif im <= 0.023:
		tmp = math.cos(re) * -im
	else:
		tmp = ((math.pow(im, 3.0) * -0.16666666666666666) - im) * (((re * re) * -0.5) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -9e+23)
		tmp = Float64((im ^ 7.0) * -0.0001984126984126984);
	elseif (im <= 0.023)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) * Float64(Float64(Float64(re * re) * -0.5) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -9e+23)
		tmp = (im ^ 7.0) * -0.0001984126984126984;
	elseif (im <= 0.023)
		tmp = cos(re) * -im;
	else
		tmp = (((im ^ 3.0) * -0.16666666666666666) - im) * (((re * re) * -0.5) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -9e+23], N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision], If[LessEqual[im, 0.023], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.99999999999999958e23

   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 97.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7} + \left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\right)} \]

   5. Taylor expanded in im around inf 97.2%

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

      1. *-commutative 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in re around 0 80.3%

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

   if -8.99999999999999958e23 < im < 0.023

   1. Initial program 11.6%

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

      1. sub0-neg 11.6%

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

   3. Simplified 11.6%

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

   4. Taylor expanded in im around 0 94.7%

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

      1. mul-1-neg 94.7%

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

      2. *-commutative 94.7%

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

      3. distribute-lft-neg-in 94.7%

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

   6. Simplified 94.7%

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

   if 0.023 < 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%

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

   4. Taylor expanded in im around 0 68.2%

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

      1. mul-1-neg 68.2%

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

      2. unsub-neg 68.2%

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

      3. *-commutative 68.2%

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

      4. associate-*l* 68.2%

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

      5. distribute-lft-out-- 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in re around 0 8.1%

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

      1. sub-neg 8.1%

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

      2. *-commutative 8.1%

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

      3. associate-+r+ 8.1%

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

      4. fma-def 8.1%

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

      5. associate-*r* 8.1%

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

      6. *-commutative 8.1%

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

      7. fma-neg 8.1%

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

      8. *-lft-identity 8.1%

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

      9. distribute-rgt-out 61.2%

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

      10. fma-neg 61.2%

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

      11. *-commutative 61.2%

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

      12. unpow2 61.2%

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

   9. Simplified 61.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+23}:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{elif}\;im \leq 0.023:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \end{array} \]

Alternative 9: 59.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -480 \lor \neg \left(im \leq 2.65 \cdot 10^{+32}\right):\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -480.0) (not (<= im 2.65e+32)))
   (- (* (* re re) (* im 0.5)) im)
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -480.0) || !(im <= 2.65e+32)) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else {
		tmp = cos(re) * -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-480.0d0)) .or. (.not. (im <= 2.65d+32))) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -480.0) || !(im <= 2.65e+32)) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -480.0) or not (im <= 2.65e+32):
		tmp = ((re * re) * (im * 0.5)) - im
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -480.0) || !(im <= 2.65e+32))
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -480.0) || ~((im <= 2.65e+32)))
		tmp = ((re * re) * (im * 0.5)) - im;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -480.0], N[Not[LessEqual[im, 2.65e+32]], $MachinePrecision]], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -480 or 2.65e32 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.5%

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

      1. mul-1-neg 5.5%

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

      2. *-commutative 5.5%

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

      3. distribute-lft-neg-in 5.5%

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

   6. Simplified 5.5%

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

   7. Taylor expanded in re around 0 23.5%

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

      1. neg-mul-1 23.5%

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

      2. +-commutative 23.5%

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

      3. unsub-neg 23.5%

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

      4. *-commutative 23.5%

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

      5. associate-*l* 23.5%

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

      6. unpow2 23.5%

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

      7. *-commutative 23.5%

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

   9. Simplified 23.5%

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

   if -480 < im < 2.65e32

   1. Initial program 13.6%

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

      1. sub0-neg 13.6%

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

   3. Simplified 13.6%

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

   4. Taylor expanded in im around 0 92.9%

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

      1. mul-1-neg 92.9%

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

      2. *-commutative 92.9%

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

      3. distribute-lft-neg-in 92.9%

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

   6. Simplified 92.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -480 \lor \neg \left(im \leq 2.65 \cdot 10^{+32}\right):\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 10: 81.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.9 \cdot 10^{+20} \lor \neg \left(im \leq 560000000\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.9e+20) (not (<= im 560000000.0)))
   (* (pow im 7.0) -0.0001984126984126984)
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.9e+20) || !(im <= 560000000.0)) {
		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 <= (-1.9d+20)) .or. (.not. (im <= 560000000.0d0))) 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 <= -1.9e+20) || !(im <= 560000000.0)) {
		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 <= -1.9e+20) or not (im <= 560000000.0):
		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 <= -1.9e+20) || !(im <= 560000000.0))
		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 <= -1.9e+20) || ~((im <= 560000000.0)))
		tmp = (im ^ 7.0) * -0.0001984126984126984;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.9e+20], N[Not[LessEqual[im, 560000000.0]], $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 < -1.9e20 or 5.6e8 < 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 91.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7} + \left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\right)} \]

   5. Taylor expanded in im around inf 91.8%

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

      1. *-commutative 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*l* 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in re around 0 68.9%

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

   if -1.9e20 < im < 5.6e8

   1. Initial program 12.9%

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

      1. sub0-neg 12.9%

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

   3. Simplified 12.9%

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

   4. Taylor expanded in im around 0 93.6%

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

      1. mul-1-neg 93.6%

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

      2. *-commutative 93.6%

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

      3. distribute-lft-neg-in 93.6%

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

   6. Simplified 93.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.9 \cdot 10^{+20} \lor \neg \left(im \leq 560000000\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 11: 31.0% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.7 \cdot 10^{+181}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.7e+181) (- im) (* (+ 0.5 (* re (* re -0.25))) -3.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.7e+181) {
		tmp = -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.7d+181) then
        tmp = -im
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.7e+181) {
		tmp = -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.7e+181:
		tmp = -im
	else:
		tmp = (0.5 + (re * (re * -0.25))) * -3.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.7e+181)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.7e+181)
		tmp = -im;
	else
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.7e+181], (-im), N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.70000000000000007e181

   1. Initial program 56.5%

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

      1. sub0-neg 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in im around 0 49.5%

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

      1. mul-1-neg 49.5%

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

      2. *-commutative 49.5%

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

      3. distribute-lft-neg-in 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in re around 0 29.0%

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

      1. neg-mul-1 29.0%

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

   9. Simplified 29.0%

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

   if 2.70000000000000007e181 < re

   1. Initial program 50.0%

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

      1. sub0-neg 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 23.5%

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

      4. +-commutative 23.5%

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

      5. *-commutative 23.5%

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

      6. unpow2 23.5%

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

      7. associate-*l* 23.5%

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

   6. Simplified 23.5%

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

   8. Applied egg-rr 30.3%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.7 \cdot 10^{+181}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \]

Alternative 12: 35.5% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.1 \cdot 10^{+181}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.1e+181)
   (- (* (* re re) (* im 0.5)) im)
   (* (+ 0.5 (* re (* re -0.25))) -3.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.1e+181) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.1d+181) then
        tmp = ((re * re) * (im * 0.5d0)) - im
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.1e+181) {
		tmp = ((re * re) * (im * 0.5)) - im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.1e+181:
		tmp = ((re * re) * (im * 0.5)) - im
	else:
		tmp = (0.5 + (re * (re * -0.25))) * -3.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.1e+181)
		tmp = Float64(Float64(Float64(re * re) * Float64(im * 0.5)) - im);
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.1e+181)
		tmp = ((re * re) * (im * 0.5)) - im;
	else
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.1e+181], N[(N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.09999999999999989e181

   1. Initial program 56.5%

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

      1. sub0-neg 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in im around 0 49.5%

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

      1. mul-1-neg 49.5%

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

      2. *-commutative 49.5%

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

      3. distribute-lft-neg-in 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in re around 0 34.9%

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

      1. neg-mul-1 34.9%

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

      2. +-commutative 34.9%

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

      3. unsub-neg 34.9%

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

      4. *-commutative 34.9%

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

      5. associate-*l* 34.9%

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

      6. unpow2 34.9%

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

      7. *-commutative 34.9%

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

   9. Simplified 34.9%

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

   if 3.09999999999999989e181 < re

   1. Initial program 50.0%

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

      1. sub0-neg 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 23.5%

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

      4. +-commutative 23.5%

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

      5. *-commutative 23.5%

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

      6. unpow2 23.5%

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

      7. associate-*l* 23.5%

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

   6. Simplified 23.5%

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

   8. Applied egg-rr 30.3%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.1 \cdot 10^{+181}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \]

Alternative 13: 29.2% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. sub0-neg 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in im around 0 49.9%

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

   1. mul-1-neg 49.9%

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

   2. *-commutative 49.9%

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

   3. distribute-lft-neg-in 49.9%

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

6. Simplified 49.9%

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

7. Taylor expanded in re around 0 27.8%

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

   1. neg-mul-1 27.8%

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

9. Simplified 27.8%

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

10. Final simplification 27.8%

    \[\leadsto -im \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023240 
(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))))